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[ [ "Topological Magnon Bands in Ferromagnetic Star Lattice" ], [ "Abstract The experimental observation of topological magnon bands and thermal Hall effect in a kagom\\'e lattice ferromagnet Cu(1-3, bdc) has inspired the search for topological magnon effects in various insulating ferromagnets that lack an inversion center allowing a Dzyaloshinskii-Moriya (DM) spin-orbit interaction.", "The star lattice (also known as the decorated honeycomb lattice) ferromagnets is an ideal candidate for this purpose because it is a variant of the kagom\\'e lattice with additional links that connect the up-pointing and down-pointing triangles.", "This gives rise to twice the unit cell of the kagom\\'e lattice, hence a more interesting topological magnon effects.", "In particular, the triangular bridges on the star lattice can be coupled either ferromagnetically or antiferromagnetically which is not possible on the kagome lattice ferromagnets.", "Here, we study DM-induced topological magnon bands, chiral edge modes, and thermal magnon Hall effect on the star lattice ferromagnet in different parameter regimes.", "The star lattice can also be visualized as the parent material from which topological magnon bands can be realized for the kagom\\'e and honeycomb lattices in some limiting cases." ], [ " Introduction", "Topological magnon matter is the magnonic analog of topological fermionic matter.", "In contrast, magnons are charge-neutral bosonic excitations of ordered quantum magnets.", "Thus, the transport properties of magnons are believed to be the new direction for dissipationless transports in insulating ferromagnets applicable to modern technology such as magnon spintronics and magnon thermal devices [1], [3], [2].", "In insulating collinear quantum magnets the DM interaction [4], [5] arising from spin-orbit coupling (SOC) [5] is the key ingredient that leads to thermal magnon Hall effect [6], [7], [8], [9] and topological magnon bands [11], [12], [10], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].", "The DM interaction is present in magnetic systems that lack inversion symmetry between magnetic ions.", "The kagomé lattice is built with this structure, because the midpoint of the bonds connecting two nearest-neighbour magnetic ions is not a center of inversion.", "Therefore, a DM interaction is intrinsic to the kagomé lattice.", "The thermal magnon Hall effect induced by the DM interaction was first observed experimentally in a number of three-dimensional (3D) ferromagnetic pyrochlores — Lu$_2$ V$_2$ O$_7$ , Ho$_2$ V$_2$ O$_7$ , and In$_2$ Mn$_2$ O$_7$ [7], [8].", "Subsequently, the thermal magnon Hall effect was realized in a kagomé ferromagnet Cu(1-3, bdc) [9] followed by the first experimental realization of topological magnon bands [10] in the same material.", "On the other hand, no topologcal magnon bands have been observed in pyrochlore ferromagnets, but recent studies have proposed Weyl magnons in 3D pyrochlore ferromagnets [23], [24].", "In the same way a pyrochlore lattice can be visualized as an alternating kagomé and triangular layers, the star lattice (also known as the decorated honeycomb lattice) can be visualized as an interpolating lattice between the honeycomb lattice and the kagomé lattice.", "The number of sites per unit cell in star lattice is six, three times as in honeycomb and twice as in kagome.", "Most importantly, the star lattice plays a prominent role in different branches of physics.", "The Kitaev model on the star lattice has an exact solution as a chiral spin liquid [25].", "The star lattice also plays an important role in ultracold atoms [26].", "In particular, quantum magnets [27], [25], [28], [29], [26] and topological insulators [30], [31], [32] on the star-lattice show distinctive remarkable features different from the kagome and honeycomb lattices.", "Motivated by these interesting features on the star lattice, the study of topological magnon bands on the star lattice is necessary and unavoidable.", "In this work, we study the topological magnon bands and the thermal magnon Hall response in insulating quantum ferromagnet on the star lattice.", "We show that lack of an inversion center allows a DM interaction between the midpoint of the triangular bonds.", "The DM interaction induces a fictitious magnetic flux in the unit cell and leads to nontrivial topological magnon bands.", "This directly leads to the existence of nonzero Berry curvatures and Chern numbers accompanied by topologically protected gapless edge modes.", "Indeed, it is feasible to synthesize magnetic materials with the star structure and directly confirm the present theoretical results.", "Figure: Color online.", "Schematics of star lattice.", "(a) Coupled ferromagnets within and between triangles of the star lattice.", "(b) Coupled antiferromagnets between the triangles.", "(c).", "The Brillouin zone of the star lattice.Figure: The possible configurations of the DM-induced flux φ ij \\phi _{ij} (see Appendix ) in the presence of a magnetic field, where θ\\theta is the field-induced canting angle.", "Bold arrows indicate the sign of the fictitious magnetic flux φ ij \\phi _{ij} and the small arrows show the magnetic-field-induced spin canting in the xx-zz plane.", "(a)(a) Uniform flux.", "(b)(b) Staggered flux." ], [ "Ferromagnetic Hamiltonian", "Quantum magnets on the star lattice are known to possess magnetic long-range orders [27], [29].", "However, the effects of the intrinsic DM perturbation have not been considered on the star lattice.", "Here, we study a ferromagnetic model which is a magnetically ordered state on the star lattice.", "The Hamiltonian is given by $\\mathcal {H}&=\\mathcal {H}_0+\\mathcal {H}_{Z}+ \\mathcal {H}_{\\text{pert}},$ where $\\mathcal {H}_0=-J\\sum _{\\langle ij\\rangle }{\\bf S}_{i}\\cdot {\\bf S}_{j}+J^\\prime \\sum _{\\langle ij\\rangle }{\\bf S}_{i}\\cdot {\\bf S}_{j},$ and $J,J^\\prime $ are exchange couplings within sites on the triangles “$\\Delta $ ” and between triangles “$\\Delta \\leftrightarrow \\nabla $ ” as shown in Fig.", "REF .", "The Zeeman magnetic field is given by $\\mathcal {H}_{Z}=- {H}\\cdot \\sum _{i} {\\bf S}_i,$ where ${H}=h{\\hat{z}}$ with $h=g\\mu _B H$ is the strength of the out-of-plane magnetic field.", "The last term $\\mathcal {H}_{\\text{pert}}$ represents all the perturbative interactions to the Heisenberg exchange.", "The DM interaction is usually the dominant perturbative anisotropy.", "It is allowed on the star-lattice due to lack of inversion center between magnetic ions according to the Moriya rule [5].", "The magneto-crystalline anisotropy is second order in perturbation which can be neglected.", "Therefore, we will consider only the DM perturbation term.", "Hence, $\\mathcal {H}_{\\text{pert}}= \\sum _{ \\langle ij\\rangle }D_{ij} \\cdot {\\bf S}_{i}\\times {\\bf S}_{j}.$ There are two ferromagnetic ordered states on the star-lattice.", "The first one corresponds to $J>0$ and $J^\\prime <0$ , i.e.", "fully polarized ferromagnets on sublattice $\\mathcal {A}$ (down pointing triangles indicated with blue sites) and $\\mathcal {B}$ (up pointing triangles indicated with red sites) as shown in Fig.", "REF .", "The second one corresponds to $J>0$ and $J^\\prime >0$ , i.e., antiferromagnetic interaction between triangles and ferromagnetic interaction on each triangle.", "In the latter case the spins on sublattice $\\mathcal {A}$ are oriented in the opposite direction to those on sublattice $\\mathcal {B}$ , and they also cant along the magnetic field direction as shown in Fig.", "REF .", "As we will show in the subsequent sections, the latter case recovers the former case at the saturation field $h_s=2J^\\prime S$ ." ], [ "Ground state energy", "At zero field the spins are aligned along the star plane chosen as the $x$ -$y$ plane with the quantization axis chosen along the $x$ -direction.", "The ground state is a collinear ferromagnet unaffected by the DM interaction.", "However, if the sublattices $\\mathcal {A}$ and $ \\mathcal {B}$ are coupled antiferromagnetically, that is $J>0$ and $J^\\prime >0$ , then a small magnetic field induces canting along the direction of the field and the ground state is no longer the collinear ferromagnets.", "The collinear ferromagnet is only recovered at the saturation field $h_s$ .", "In the large-$S$ limit, the spin operators can be approximated as classical vectors, written as ${S}_{\\tau }= S{n}_\\tau $ , where ${n}_\\tau =\\left(\\sin \\theta \\cos \\vartheta _\\tau , \\sin \\theta \\sin \\vartheta _\\tau ,\\cos \\theta \\right)$ is a unit vector and $\\tau $ denotes the down ($\\mathcal {A}$ ) and up ($\\mathcal {B}$ ) triangles, with $\\vartheta _{\\mathcal {A}}=0$ and $\\vartheta _{\\mathcal {B}}=\\pi $ and $\\theta $ is the magnetic-field-induced canting angle.", "As the system is ordered ferromagnetically on each triangle, the DM interaction does not contribute to the classical energy given by $e_0=-JS^2+\\frac{J^\\prime S^2}{2}\\cos 2\\theta - hS\\cos \\theta ,$ where $e_0=E/6N$ is energy per site and $N$ is the number of sites per unit cell.", "Minimizing the classical energy yields the canting angle $\\cos \\theta = h/h_s$ ." ], [ "Magnetic excitations", "In the low temperature regime the magnetic excitations above the classical ground state are magnons and higher order magnon-magnon interactions are negligible.", "Thus, the linearized Holstein-Primakoff (HP) spin-boson representation [33] is valid.", "Due to the magnetic-field-induced spin canting, we have to rotate the coordinate axes such that the $z$ -axis coincides with the local direction of the classical polarization.", "This involves rotating the spins from laboratory frame to local frame by the spin oriented angles $\\vartheta _\\tau $ about the $z$ -axis.", "This rotation is followed by another rotation about the $y$ -axis with the canting angle $\\theta $ , and the resulting transformation is given by $&S_{i\\tau }^x=\\pm S_{i\\tau }^{\\prime x}\\cos \\theta \\pm S_{i\\tau }^{\\prime z}\\sin \\theta ,\\nonumber \\\\&S_{i\\tau }^y=\\pm S_{i\\tau }^{\\prime y},\\\\&\\nonumber S_{i\\tau }^z=- S_{i\\tau }^{\\prime x}\\sin \\theta + S_{i\\tau }^{\\prime z}\\cos \\theta ,$ where $+(-)$ applies to the spins residing on the sublattice $\\mathcal {A}$ ($\\mathcal {B}$ ) triangles respectively.", "It is easily checked that this rotation does not affect the collinear ferromagnetic ordering on each triangle, that is the $J$ term.", "Usually, in collinear ferromagnets the perpendicular-to-field DM component does to contribute to the free magnon theory because the magnetic moments are polarized along the magnetic field direction [7], [8], [9].", "Therefore, only the parallel-to-field DM component has a significant contribution to the free magnon dispersion [11], [12], [10], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [6], [7], [8], [9].", "In the present model, however, the situation is different.", "Due to antiferromagnetic coupling between triangles an applied magnetic field induces two spin components — one parallel to the field and the other perpendicular to the field.", "As we pointed out above, the DM interaction has a finite contribution to the free magnon model only when the magnetic moments are along its direction.", "Therefore the parallel-to-field DM component (${\\bf D}\\parallel {\\bf H}$ ) and the perpendicular-to-field DM component (${\\bf D}\\perp {H}$ ) will have a finite contribution to the free magnon theory due to spin canting.", "The former is rescaled as $D_{\\parallel ,\\theta }\\rightarrow D\\cos \\theta $ and the latter is rescaled as $D_{\\perp ,\\theta }\\rightarrow D\\sin \\theta $ by the magnetic field.", "Implementing the spin transformation (REF ) the terms that contribute to the free magnon model are given by $&\\mathcal {H}_{0}=-J\\sum _{\\langle i, j\\rangle \\tau }{\\bf S}_{i\\tau }^\\prime \\cdot {\\bf S}_{j\\tau }^\\prime \\nonumber \\\\&-J^\\prime \\sum _{\\langle i,j\\rangle \\tau \\tau ^\\prime }[\\cos 2\\theta (S_{i\\tau }^{\\prime x}S_{j\\tau ^\\prime }^{\\prime x}-S_{i\\tau }^{\\prime z}S_{j\\tau ^\\prime }^{\\prime z})+S_{i\\tau }^{\\prime y}S_{j\\tau ^\\prime }^{\\prime y}],\\\\& ~\\mathcal {H}_Z= -h_{\\theta }\\sum _{i\\tau } S_{i\\tau }^{\\prime z},\\\\&\\mathcal {H}_{\\text{pert},\\parallel }= D_{\\parallel ,\\theta }\\sum _{\\langle i, j\\rangle \\tau }\\chi _{ij\\tau }^{\\prime z},~\\mathcal {H}_{\\text{pert},\\perp }= \\pm D_{\\perp ,\\theta }\\sum _{\\langle i, j\\rangle \\tau }\\chi _{ij\\tau }^{\\prime z},$ where $h_{\\theta }=h\\cos \\theta $ and $\\chi _{ij\\tau }^{\\prime z}$ is the $z$ -component of the vector spin chirality $[{\\bf S}_i^\\prime \\times {\\bf S}_j^\\prime ]_z$ .", "We note that $H_{\\text{pert},\\parallel }$ and $H_{\\text{pert},\\perp }$ DM components are oriented perpendicular to the bond due to the magnetic field.", "From these expressions it is evident that at zero magnetic field ($\\theta =\\pi /2$ ) the spins are along the $x$ -$y$ plane of the star lattice, therefore only the DM component parallel to the spin axis has a finite contribution to the free magnon model.", "The same is true at the saturation field ($\\theta =0$ ) when the spins are fully aligned along the $z$ -direction.", "In Appendix we have shown the tight binding magnon model.", "In the following section we will consider the two field-induced spin components separately.", "Figure: Color online.", "Magnon bands in the fully polarized ferromagnet at the saturation field h=h s h=h_s.", "Top panel: D=0 D=0.", "Bottom panel: D/J=0.15D/J=0.15.", "The circled points are gapped.Figure: Color online.", "Magnon bands in the canted antiferromagnet for h<h s =0.75Jh<h_s=0.75J.", "Top panel: Perpendicular-to-field spin components.", "Bottom panel: Parallel-to-field spin components.", "The DM value is D/J=0.15D/J=0.15.", "The circled points are gapped." ], [ "Topological magnon bands", "We note that the first topological magnon bands have been measured in the kagomé ferromagnet Cu(1-3, bdc) [10], thus paving the way to search for topological magnon bands in other systems.", "In the following we study the topological magnon bands in the star lattice ferromagnet.", "We consider spin-$1/2$ and take $J$ as the unit of energy while varying $J^\\prime /J$ .", "We also take the DM value of the kagomé ferromagnet Cu(1-3, bdc) $D/J=0.15$ [9], [10].", "In Fig.", "REF we have shown the magnon bands at $D=0$ and $D/J=0.15$ with varying $J^\\prime /J$ .", "In the former (top panel) the lower bands for $J^\\prime /J<1.5$ in Fig.", "REF (a) looks like Dirac magnon on the honeycomb lattice [35].", "On the other hand, for $J^\\prime /J>1.5$ the bands in Fig.", "REF (c) resemble two copies of magnon bands on the kagomé lattice ferromagnet with a flat band and two dispersive Dirac magnon bands on each copy.", "The gap closes at $J^\\prime /J=1.5$ as shown in Fig.", "REF (b).", "For nonzero DM interaction (lower panel) the the magnon bands are separated by a finite energy gap proportional to the DM interaction in all the parameter regimes.", "Notice that the collinear ferromagnet at $h=h_s$ has a quadratic dispersion (Goldstone mode) at the ${\\bf \\Gamma }$ point due spontaneous breaking of U(1) symmetry about the $z$ -axis.", "In contrast, for $h<h_s$ the spins are canted due to antiferromagnetic interaction between the triangles.", "This leads to two spin components — perpendicular-to-field spin components and parallel-to-field spin components.", "In Fig.", "REF we have shown the magnon bands in the canted antiferromagnetic phase.", "Indeed, we recover a linear Goldstone mode at the ${\\bf \\Gamma }$ point which signifies an antiferromagnetic spin order.", "In this case the model is no longer an analog of topological fermion insulator on the star lattice [30], [31], [32].", "Furthermore, the perpendicular-to-field spin components (top panel of Fig.", "REF ) show gapless magnon bands even in the presence of DM interaction.", "A similar gapless magnon bands was reported in the kagomé ferromagnet Cu(1-3, bdc) [10] when the spins are polarized along the kagomé plane by an in-plane magnetic field.", "In contrast, the parallel-to-field spin components (bottom panel of Fig.", "REF ) show a finite gap separating the magnon bands due to the presence of DM interaction along the spin polarization." ], [ "Chiral edge modes", "The topological aspects of gap Dirac points can be studied by the defining the Berry curvature and the Chern number.", "In the present model we started from an antiferromagnetic coupled ferromagnets and then recovered a collinear ferromagnetic model.", "Therefore the Hamiltonian has an off-diagonal terms and the diagonalization requires the generalized Bogoliubov transformation and the eigenfunctions define the Berry curvatures and Chern numbers (see Appendix ).", "The Chern number vanishes for all bands at zero DM interaction (top panel of Fig.", "REF ).", "We also find that the Chern number vanishes for the magnon bands of perpendicular-to-field spin components (top panel of Fig.", "REF ).", "This signifies a trivial magnon insulator.", "However, we find nonzero Chern numbers for the other magnon bands (bottom panel of Figs.", "REF and REF ) which defines a topological magnon insulator.", "A nonzero Chern number is associated with topological chiral gapless magnon edge modes which appear at the DM-induced gaps as shown in Figs.", "REF and REF .", "The edge modes are solved for a strip geometry on the star-lattice with open boundary conditions along the $y$ direction and infinite along the $x$ direction.", "Figure: Color online.", "The corresponding chiral magnon edge modes for Fig.", ".Figure: Color online.", "The corresponding chiral magnon edge modes for Fig.", ".Figure: Color online.", "Low temperature dependence of the thermal Hall conductivity for several values of J ' /JJ^\\prime /J and D/J=0.15D/J=0.15." ], [ "Thermal magnon Hall effect", "Inelastic neutron scattering experiment has measured the thermal Hall conductivity in the kagomé and pyrochlore ferromagnets [7], [8], [9].", "Theoretically, the thermal Hall effect in insulating ferromagnets is understood as a consequence of the topological magnon bands induced by the DM interaction [6].", "A temperature gradient $-\\nabla T$ induces a transverse heat current $J^Q$ and the DM-induced Berry curvature acts as an effective magnetic field that deflects the propagation of magnon in the system giving rise to a thermal Hall effect similar to Hall effect in electronic systems.", "From linear response theory, one obtains $\\mathcal {J}_{\\alpha }^Q=-\\sum _{\\beta }\\kappa _{\\alpha \\beta }\\nabla _{\\beta } T$ , where $\\kappa _{\\alpha \\beta }$ is the thermal conductivity and the transverse component $\\kappa _{xy}$ is associated with thermal Hall conductivity given explicitly as [11], [15] $\\kappa _{xy}=-\\frac{k_B^2 T}{\\hbar V}\\sum _{{k}}\\sum _{\\alpha =1}^N \\left(c_2[ g\\left(\\omega _{{k}\\alpha }\\right)]-\\frac{\\pi ^2}{3}\\right)\\Omega _{xy;\\alpha }(k),$ where $V$ is the volume of the system, $k_B$ is the Boltzmann constant, $T$ is the temperature, $g(\\omega _{\\alpha {k}})=[e^{{\\omega _{\\alpha {k}}}/k_BT}-1]^{-1}$ is the Bose function, and $c_2(x)$ is defined as $c_2(x)=(1+x)\\left(\\ln \\frac{1+x}{x}\\right)^2-(\\ln x)^2-2\\text{Li}_2(-x),$ with $\\text{Li}_2(x)$ being the dilogarithm.", "The Berry curvature $\\Omega _{xy;\\alpha }(k)$ is defined in Appendix .", "The thermal Hall conductivity is finite only in the fully polarized collinear ferromagnets at $h=h_s$ and the parallel-to-field spin components in the canted antiferromagnet for $h<h_s$ .", "As shown in Fig.", "REF the thermal Hall conductivity shows a sharp peak and a sign change in the canted phase and vanishes at zero temperature as there are no thermal excitations.", "This is consistent with the trend observed in previous experiments on the kagomé and pyrochlore ferromagnets [7], [8], [9]." ], [ "Conclusion", "The star lattice has attracted considerable attention in recent years as an exact solution of Kitaev model [25].", "Chiral spin liquids, topological fermion insulators and quantum anomalous Hall effect have been proposed [27], [25], [28], [29], [26], [30], [31], [32].", "However, there is no experimental realizations at the moment.", "In this work, we have contributed to the list of interesting proposals on the star lattice.", "We have shown that insulating quantum ferromagnets on the star lattice are candidates for topological magnon insulators and thermal magnon Hall transports.", "We showed that the intrinsic DM interaction which is allowed on the star lattice gives rise to magnetic excitations that exhibit nontrivial magnon bands with non-vanishing Berry curvatures and Chern numbers.", "We believe that the synthesis of magnetic materials with a star structure is feasible.", "In fact, experiment has previously realized polymeric iron (III) acetate as a star lattice antiferromagnet with both spin frustration and magnetic long-range order [27].", "A strong applied magnetic field is sufficient to induce a ferromagnetic ordered phase in this material and the topological magnon bands can be realized.", "Unfortunately, inelastic neutron scattering is a bulk sensitive method and the chiral magnon edge modes have not been measured in any topological magnon insulator [10].", "It is possible that edge sensitive methods such as light [36] or electronic [37] scattering method can see the chiral magnon edge modes in topological magnon insulators." ], [ " Acknowledgments", "Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation." ], [ "Free magnon theory", "The corresponding free magnon model is achieved by mapping the spin operators to boson operators [33]: $S_{i,\\tau }^{\\prime x}=\\sqrt{S/2}(b_{i,\\tau }^\\dagger +b_{i,\\tau })$ , $S_{i,\\tau }^{\\prime y}=i\\sqrt{S/2}(b_{i,\\tau }^\\dagger -b_{i,\\tau })$ , and $S_{i,\\tau }^{\\prime z}=S-b_{i,\\tau }^\\dagger b_{i,\\tau }$ .", "The resulting free magnon model is given by $\\mathcal {H}&=v_0\\sum _{j,\\tau }n_{j\\tau } - v_t\\sum _{\\langle ij\\rangle ,\\tau }(e^{-i\\sigma \\phi _{ij}} b^\\dagger _{i\\tau } b_{j\\tau }+h.c.", ")\\nonumber \\\\& -v^\\prime \\sum _{\\langle i j\\rangle \\tau \\tau ^\\prime }[( b_{i\\tau }^\\dagger b_{j\\tau ^\\prime }+ h.c.)\\cos ^2\\theta -( b_{i\\tau }^\\dagger b_{j\\tau ^\\prime }^\\dagger + h.c.)\\sin ^2\\theta ],$ where $n_{j\\tau }=b_{j\\tau }^\\dagger b_{j\\tau }$ , $v_0=2v_s-v_s^\\prime \\cos 2\\theta + h\\cos \\theta =2v_s +v_s^\\prime $ , $v_s(v_s^\\prime )= JS(J^\\prime S)$ , $v_t=\\sqrt{v_s^{ 2} +v_{D_{z(x),\\theta }}^2}=JS/\\cos (\\phi _{ij}),~v_{D_{\\parallel (\\perp ),\\theta }}=D_{\\parallel (\\perp ),\\theta }S$ ; $\\phi _{ij}=\\pm \\phi =\\arctan (D_{\\parallel (\\perp ),\\theta }/J)$ is a magnetic flux generated by the DM interaction within the triangular plaquettes, similar to Haldane model [34].", "For ${\\bf D}\\parallel {\\bf H}$ , $\\sigma =1$ and for ${\\bf D}\\perp {\\bf H}$ , $\\sigma =\\pm 1$ for sublattice $\\mathcal {A}$ and $\\mathcal {B}$ respectively.", "The configurations of $\\phi _{ij}$ for both cases are depicted in Fig.", "REF .", "The total flux in the dodecagon consisting of twelve sites is $-2\\phi $ and 0 respectively.", "Indeed, $\\phi _{ij}$ vanishes along the $J^\\prime $ link as it contains no triangular plaquettes.", "The momentum space Hamiltonian can be written as $ \\mathcal {H}=\\frac{1}{2}\\sum _{k}\\psi ^\\dagger _{k}\\cdot \\mathcal {H}(k)\\cdot \\psi _{k},$ with $\\psi ^\\dagger _{k}= (b_{\\mu ,{k}}^{\\dagger },\\,b_{\\mu ^\\prime ,{k}}^{\\dagger }, b_{\\mu ,-{k}},\\,b_{\\mu ^\\prime ,-{k}})$ , where $\\mu =1,2,3$ and $\\mu ^\\prime =4,5,6$ .", "The Bogoliubov Hamiltonian $\\mathcal {H}(k)$ is a $2N\\times 2N$ matrix given by $\\mathcal {H}(k)=\\begin{pmatrix}{A}({k}, \\phi ) & {B}({k})\\\\{B}^(-{k})&{A}^(-{k},\\phi )\\end{pmatrix},$ where ${A}(k)=\\begin{pmatrix}{a}(\\sigma \\phi ) & {b}_1({k})\\\\{b}_1(-{k})&{a}(\\sigma \\phi )\\end{pmatrix},~{B}(k)=\\begin{pmatrix}{\\bf 0} & {b}_2({k})\\\\{b}_2(-{k})& {\\bf 0}\\end{pmatrix},$ $&{a}(\\sigma \\phi )=\\begin{pmatrix}v_0&-v_t e^{-i\\sigma \\phi }&-v_t e^{i\\sigma \\phi }\\\\-v_t e^{i\\sigma \\phi }&v_0&-v_t e^{-i\\sigma \\phi }\\\\-v_t e^{-i\\sigma \\phi }&-v_t e^{i\\sigma \\phi }&v_0\\end{pmatrix},\\\\&{b}_1({k})=-v_s^\\prime \\cos ^2\\theta \\begin{pmatrix}e^{ik_2}&0&0\\\\0& e^{ik_1}&0\\\\0&0&1\\end{pmatrix},\\\\&{b}_2({k})=v_s^\\prime \\sin ^2\\theta \\begin{pmatrix}e^{ik_2}&0&0\\\\0& e^{ik_1}&0\\\\0&0&1\\\\\\end{pmatrix},$ where $k_{1}={k}\\cdot {a}_{1}$ and $k_{2}={k}\\cdot {a}_{2}$ .", "The lattice basis vectors are chosen as $a_1=2\\hat{x}$ and $a_2= \\hat{x} + \\sqrt{3}\\hat{y}$ .", "At the saturation field $h=h_s (\\theta =0)$ the spins are fully aligned along the $z$ -axis and ${b}_2({k})=0$ .", "We therefore recover collinear ferromagnet along the $z$ -direction.", "The Hamiltonian is diagonalized below." ], [ "Berry curvature and Chern number", "To diagonalize the Hamiltonian we make a linear transformation $\\psi _{k}= \\mathcal {P}_{k}Q_{k}$ , where $\\mathcal {P}_{k}$ is a $2N\\times 2N$ paraunitary matrix and $Q^\\dagger _{k}= (\\mathcal {Q}_{k}^\\dagger ,\\,\\mathcal {Q}_{-{k}})$ with $ \\mathcal {Q}_{k}^\\dagger =(\\gamma _{{k}\\mu }^{\\dagger },\\,\\gamma _{{k}\\mu ^\\prime }^{\\dagger })$ being the quasiparticle operators.", "The matrix $\\mathcal {P}_{k}$ satisfies the relations, $&\\mathcal {P}_{k}^\\dagger \\mathcal {H}({k}) \\mathcal {P}_{k}=\\mathcal {E}_{k},\\\\ &\\mathcal {P}_{k}^\\dagger \\tau _3 \\mathcal {P}_{k}= \\tau _3,$ where $\\mathcal {E}_{k}= \\text{diag}(\\omega _{{k}\\alpha },~\\omega _{-{k}\\alpha })$ , $ \\tau _3=\\text{diag}(\\mathbf {I}_{N\\times N}, -\\mathbf {I}_{N\\times N} )$ , and $\\omega _{{k}\\alpha }$ are the energy eigenvalues and $\\alpha $ labels the bands.", "From Eq.", "we get $\\mathcal {P}_{k}^\\dagger = \\tau _3 \\mathcal {P}_{k}^{-1} \\tau _3$ , and Eq.", "REF is equivalent to saying that we need to diagonalize the Hamiltonian $\\mathcal {H}^\\prime ({k})= \\tau _3\\mathcal {H}({k}),$ whose eigenvalues are given by $ \\tau _3\\mathcal {E}_{k}$ and the columns of $\\mathcal {P}_{k}$ are the corresponding eigenvectors.", "The eigenvalues of this Hamiltonian cannot be obtained analytically except at zero field.", "The paraunitary operator $\\mathcal {P}_{k}$ defines a Berry curvature given by $\\Omega _{ij;\\alpha }({k})=-2\\text{Im}[ \\tau _3\\mathcal {(}\\partial _{k_i}\\mathcal {P}_{{k}\\alpha }^\\dagger ) \\tau _3(\\partial _{k_j}\\mathcal {P}_{{k}\\alpha })]_{\\alpha \\alpha },$ with $i,j=\\lbrace x,y\\rbrace $ and $\\mathcal {P}_{{k}\\alpha }$ are the columns of $\\mathcal {P}_{{k}}$ .", "In this form, the Berry curvature simply extracts the diagonal components which are the most important.", "From Eq.", "REF the Berry curvature can be written alternatively as $\\Omega _{ij;\\alpha }(k)=-\\sum _{\\alpha ^\\prime \\ne \\alpha }\\frac{2\\text{Im}[ \\mathinner {\\langle {\\mathcal {P}_{{k}\\alpha }\|v_i\|\\mathcal {P}_{{k}\\alpha ^\\prime }}\\rangle }\\mathinner {\\langle {\\mathcal {P}_{{k}\\alpha ^\\prime }\|v_j\|\\mathcal {P}_{{k}\\alpha }}\\rangle }]}{\\left(\\omega _{{k}\\alpha }-\\omega _{{k}\\alpha ^\\prime }\\right)^2},$ where $v=\\partial \\mathcal {H}^\\prime ({k})/\\partial k$ defines the velocity operators.", "The Berry curvature is related to DM interaction $\\Omega (k)\\propto \\phi $ and the Chern number is defined as, $\\mathcal {C}_\\alpha = \\frac{1}{2\\pi }\\int _{{BZ}} dk_xdk_y~ \\Omega _{xy;\\alpha }(k).$" ] ]	1606.04904
[ [ "Presupernova neutrino events relating to the final evolution of massive\n stars" ], [ "Abstract When a supernova explosion occurs in neighbors around hundreds pc, current and future neutrino detectors are expected to observe neutrinos from the presupernova star before the explosion.", "We show a possibility for obtaining the evidence for burning processes in the central region of presupernova stars though the observations of neutrino signals by current and future neutrino detectors such as KamLAND, JUNO, and Hyper-Kamiokande.", "We also investigate supernova alarms using neutrinos from presupernova stars in neighbors.", "If a supernova explodes at ~ 200 pc, future 20 kton size liquid scintillation detectors are expected to observe hundreds neutrino events.", "We also propose a possibility of the detection of neutrino events by Gd-loaded Hyper-Kamiokande using delayed $\\gamma$-ray signals.", "These detectors could observe detailed time variation of neutrino events.", "The neutrino emission rate increases by the core contraction in the final evolution stage.", "However, the O and Si shell burnings suppress the neutrino emission for a moment.", "The observed decrease in the neutrino event rate before hours to the explosion is possibly evidence for the shell burnings.", "The observations of detailed time evolution of presupernova neutrino events could reveal properties of burning processes in the central region of presupernova stars." ], [ "Introduction", "Neutrinos emitted from core-collapse (CC) supernovae (SNe) give the information of the central interior of the collapsing core in an evolved massive star.", "Kamiokande and Irvine-Michigan-Brookhaven (IMB) experiment observed twelve and eight neutrino events in the explosion SN 1987A [1], [2].", "These observations confirmed basic characteristics of current SN models and neutron star formation.", "If a SN explodes at the Galactic center, thousands of neutrino events will be detected by Super-Kamiokande (e.g., [3]) and hundreds events will be by KamLAND and other neutrino detectors.", "There are also evolved massive stars such as red supergiants and Wolf–Rayet (WR) stars in the distance of hundreds pc.", "One famous example is Betelgeuse, a red supergiant at $197 \\pm 45$ pc and its initial mass is considered to be $\\sim $ 17 $M_\\odot $ [4].", "If these massive stars become SNe, millions neutrinos will be detected by Super-Kamiokande.", "On the other hand, neutrinos are also emitted from SN progenitors before the SN explosion.", "Indeed, neutrino emission is the most efficient cooling process after the He burning.", "The neutrino luminosity during the Si burning becomes the order of $10^{47}$ erg s$^{-1}$ .", "Recently, properties of neutrino spectra emitted by various neutrino emission processes in evolved massive stars have been investigated: pair neutrino process [5], [6], plasma neutrinos [7], and weak interactions of nuclei such as electron captures in nuclear statistical equilibrium [8], [9], [10].", "The time evolution of the neutrino emission rate has been investigated [9].", "Neutrinos from presupernova (preSN) stars are less energetic and less luminous than SN neutrinos.", "Nevertheless, since liquid scintillation neutrino detectors such as KamLAND have low threshold energy, they are expected to detect preSN neutrinos if a SN explodes in neighbors.", "Depending on the stellar models and the distance, tens $\\sim 100$ neutrino events from preSN stars of nearby SNe have been expected to be detected by KamLAND [7], [11], [12].", "The detectability of the neutrinos from the progenitor of an electron capture SN is also investigated [11].", "A SN alarm using the detection of neutrinos from a preSN star could be possible by KamLAND within a few to tens of hours before a SN explosion [12].", "Table: Properties of the 12, 15, and 20 M ⊙ M_\\odot models.L f L_{\\rm f} is the luminosity at the last step.We still have no ways to observe directly deep interior of evolved stars.", "The advanced stellar evolution of massive stars has been understood through findings of theoretical studies on stellar models (e.g., [13], [14], [15], [16], [17], [18]).", "At the final stage, the Si core burning, the Si shell burning as well as the O shell burning proceeds complicatedly in the central region.", "The Si core burning proceeds for a few days to one week to form an iron core.", "The iron core grows up with the Si shell burning and collapses for hours.", "The structure change by burning processes and the collapse affects properties of neutrinos emitted from a preSN star.", "If the evolution of detailed neutrino emissivity or neutrino spectra is observed by current or future neutrino detectors, these observational data will give constraints to the core structure and burning processes of preSN stars.", "KamLAND is a one kton size liquid scintillation neutrino detector [12].", "Twenty kton size liquid scintillation detectors such as JUNO [19] and RENO-50 [20] are planned.", "A hundreds kton size water Cherenkov detector Hyper-Kamiokande is also planned [21].", "Thus, it is important to investigate the relations between detailed neutrino properties from preSN stars and the final evolution of the stars.", "The purposes of this study are to estimate preSN neutrino events observed by current and future neutrino detectors such as KamLAND, JUNO, and Hyper-Kamiokande and to find the relations between detailed time variation of the neutrino events observed by these detectors and burning processes occurred in the central region of preSN stars.", "We also estimate the SN alarms using the observations of preSN neutrinos.", "We investigate the time evolution of the spectra of neutrinos emitted through pair-neutrino process from the Si burning of 12, 15, and 20 $M_\\odot $ stars taking into account the stellar structure.", "Then, we estimate preSN neutrino events observed by current and future neutrino detectors.", "We discuss the relation between the evolution of preSN neutrinos and burning processes after the Si core burning.", "We will organize this article as follows.", "We present the stellar evolution model and properties of neutrinos produced by pair neutrino process in Sec.", "II.", "In Sec.", "III, we show properties of neutrino spectra and the corresponding internal structure during the evolution from the Si burning in the 15 $M_\\odot $ model.", "We also show the stellar mass dependence of the neutrino properties.", "In Sec.", "IV, we estimate the neutrino events by current and future neutrino detectors assuming the distance to a preSN star of 200 pc.", "We discuss burning processes suggested by the evolution of preSN neutrino events.", "The SN alarm using preSN neutrinos by KamLAND and JUNO is also discussed.", "In Sec.", "V, we discuss preSN neutrino events from Betelgeuse considering the uncertainty of the distance.", "We also discuss the relation between the preSN neutrino events and parameters characterizing SN explosion.", "We give conclusions in Sec.", "VI.", "We calculate the evolution of massive stars with the initial mass of 12, 15, and 20 $M_\\odot $ and the solar metallicity from the zero-age main sequence to the onset of the core-collapse when the central temperature reaches $\\sim 10^{9.8}$ K. The main input physics of the stellar evolution is written in [22].", "We use the nuclear reaction network of 300 species of nuclei [23].", "The stellar mass reduces by the mass loss process during the evolution up to the C burning.", "The mass loss rate for the main-sequence stage and red supergiant stage is adopted from [24], [25].", "In the stellar evolution models, the neutrino energy loss rate by pair, photo, plasma, Bremsstrahlung, and recombination neutrino processes are included.", "These rates are calculated using the approximated formula in [26].", "We also calculate the neutrino energy loss by weak interactions of nuclei in calculating the abundance evolution using tables in [27], [28], [29].", "Figure: The evolution of the central density and temperature in the 12 (blue line), 15 (red line), and20 (green line) M ⊙ M_\\odot models after the C burning.Triangles and rectangles indicate the ignition and termination of the Si core burning, respectively.Mass fraction distribution and neutrino properties at the stages assigned by circleswith symbols (a)–(e) in the 15 M ⊙ M_\\odot model are explained in Sec.", "III A 2.Table: Stellar properties and average neutrino energy at five different stages of the 15 M ⊙ M_\\odot model.The final mass, the masses of He, CO, Si, and Fe cores are listed in Table REF .", "We set the outer boundaries of the He, CO, and Si cores as the outermost mass coordinates where the mass fractions of H, He, and O are smaller than 0.1, respectively.", "For Fe-core mass, we set the boundary as the outermost mass coordinate where the mass fraction of Fe-peak elements, denoted as “Fe,\" the elements with atomic number $Z \\ge 22$ , is larger than 0.5.", "In these models, the masses of the He, CO, and Si cores monotonically increase with the initial mass.", "We also list the periods between the ignition and termination of the Si core burning $t_{\\rm Si-b}$ and from the termination of the Si core burning until the last step of the calculations $t_{\\rm col}$ , and the luminosity at the last step.", "The period of the Si-burning is one day to one week.", "It decreases with increasing the initial stellar mass.", "We show the evolution of the central temperature and density after the C burning to the core-collapse in Fig.", "REF .", "We also list some stellar properties at five different stages (a)–(e) of the 15 $M_\\odot $ model in Table REF .", "During most of the period of the Si core burning, assigned by triangle and rectangle in Fig.", "REF , the central temperature slightly raises and the central density decreases.", "Stage (a) is located during the expansion by the Si core burning.", "When the Si core burning weakens, the density turns to raise again.", "Stage (b) is located just after the termination of the Si core burning.", "Then, the central temperature decreases once at around $\\log T_{\\rm C} \\sim 9.6$ close to stage (c) in Fig.", "REF .", "At this time, the O shell burning starts at $M_{\\rm r} \\sim 1.4 M_\\odot $ .", "There are also dips around $\\log T_{\\rm C} \\sim 9.65$ in the 12 and 15 $M_\\odot $ models [see stage (d) for the 15 $M_\\odot $ model].", "The main burning process changes to the Si shell burning in $M_{r} \\gtrsim 1 M_\\odot $ .", "Stage (e) is located at the last step of the calculation." ], [ "Neutrino emission by pair neutrino process", "Pair neutrino process is the neutrino emission process through the pair annihilation of electrons and positrons.", "This process is a dominant neutrino emission process during most advanced stages of massive stars (e.g., [26]).", "Here, we evaluate the spectra of neutrinos and antineutrinos emitted through pair neutrino process.", "The neutrino emission rate has been evaluated in [30] (see also [31]).", "We use the description of the emission rate of a given flavor of neutrinos per unit volume as $r(\\varepsilon _{\\nu }, \\varepsilon _{\\bar{\\nu }}) && = \\\\&& \\frac{c}{16 (2\\pi \\hbar )^{12}} \\int f_{\\rm e^{-}} f_{\\rm e^{+}}(2\\pi )^4 \\delta ^4(p_{\\rm e^{-}}+p_{\\rm e^{+}}-p_{\\nu }-p_{\\bar{\\nu }}) \\nonumber \\\\&& \\times \\frac{\|M\|^2}{\\varepsilon _{\\rm e^{-}} \\varepsilon _{\\rm e^{+}} \\varepsilon _{\\nu } \\varepsilon _{\\bar{\\nu }} }d^3 p_{\\rm e^{-}} d^3 p_{\\rm e^{+}} d\\Omega _{\\nu } d\\Omega _{\\bar{\\nu }} , \\nonumber $ where $\\varepsilon _\\nu $ and $\\varepsilon _{\\bar{\\nu }}$ are the energies of neutrinos and antineutrinos of a given flavor, $\\hbar $ is the reduced Planck constant, $c$ is the light speed, $f_i$ , $p_i$ , $\\varepsilon _i$ , and $\\Omega _i$ are the Fermi-Dirac distribution, four-dimensional momentum, energy, and solid angle of particle $i$ .", "The matrix element $\|M\|^2$ is written as $\|M\|^2 &=& 16 G_{\\rm F}^2 (\\hbar c)^2 \\lbrace (C_{\\rm A} - C_{\\rm V})^2 (p_{\\rm e^{-}} \\cdot p_{\\nu }) (p_{\\rm e^{+}} \\cdot p_{\\bar{\\nu }}) \\\\&& + (C_{\\rm A} + C_{\\rm V})^2 (p_{\\rm e^{+}} \\cdot p_{\\nu }) (p_{\\rm e^{-}} \\cdot p_{\\bar{\\nu }}) \\nonumber \\\\&& + m_e^2 c^4 (C_{\\rm A}^2 + C_{\\rm V}^2) (p_{\\nu } \\cdot p_{\\bar{\\nu }}) \\rbrace , \\nonumber $ where $G_{\\rm F}$ is the Fermi-coupling constant, $C_{\\rm V}$ and $C_{\\rm A}$ are the vector and axial-vector coupling constants, respectively, $m_e$ is the electron mass.", "The Fermi-Dirac distribution function depends on the temperature $T$ and the multiple of the density $\\rho $ and the electron mole fraction $Y_e$ .", "The value of $C_{\\rm V}$ is set to be $1/2+2\\sin ^2\\theta _{\\rm W}$ and $1/2-2\\sin ^2\\theta _{\\rm W}$ for $\\nu _e\\bar{\\nu }_e$ pair and $\\nu _{\\mu }\\bar{\\nu }_{\\mu }$ and $\\nu _{\\tau }\\bar{\\nu }_{\\tau }$ pairs, respectively, where $\\theta _{\\rm W}$ is the Weinberg angle and we set $\\sin ^2\\theta _{\\rm W} = 0.23126$ [32].", "The value of $C_{\\rm A}$ is set to be 1/2.", "The $\\nu _e\\bar{\\nu }_e$ pair is produced through neutral-current and charged-current processes.", "The pairs of $\\nu _{\\mu }\\bar{\\nu }_{\\mu }$ and $\\nu _{\\tau }\\bar{\\nu }_{\\tau }$ are produced through neutral-current process.", "Figure: Contours of the energy loss rate in the unit mass dϵ ν /dtd\\epsilon _{\\nu }/dtby ν e ν ¯ e \\nu _{e}\\bar{\\nu }_{e} pairs (red lines) andν x ν ¯ x \\nu _{x}\\bar{\\nu }_{x} pairs where xx=μ\\mu or τ\\tau (blue lines)on the logρ\\log \\rho – logT\\log T plane.The Y e Y_e value is assumed to be 0.5.The numbers attached to lines indicate log(dϵ ν /dt)\\log (d\\epsilon _{\\nu }/dt).The neutrino spectra at the points assigned by small green circles on this plane are calculatedin this study.Figure: Contours of the average energy of ν e \\nu _{e} (red lines) and ν ¯ e \\bar{\\nu }_{e} (blue lines)emitted by pair neutrino process in units of MeV on the logρ\\log \\rho – logT\\log T plane.The Y e Y_e value is assumed to be 0.5.Black numbers indicate the energy value of both of ν e \\nu _e and ν ¯ e \\bar{\\nu }_e.Red and blue numbers indicate the values of the corresponding contours of ν e \\nu _e and ν ¯ e \\bar{\\nu }_e,respectively.In this study, we calculate the neutrino spectra at 1581 points on the plain of $\\log \\rho $ and $\\log T$ assigned by small green circles in Figs.", "REF and REF and assuming $Y_e = 0.5$ .", "Integration of the phase space has been performed using Monte-Carlo method.", "This region covers the ranges of the density and temperature in the stellar evolution models where the temperature is larger than $1 \\times 10^9$ K. We evaluate the neutrino spectra at each time and each mass coordinate by interpolating the spectra of four neighboring points on the $\\log \\rho Y_{e}$ and $\\log T$ plain.", "Figure REF shows contours of the energy loss rate $d\\epsilon _{\\nu }/dt$ (erg s$^{-1}$ g$^{-1}$ ) by pair neutrino process.", "The energy loss rate increases with temperature and decreases with increase in density.", "When the density is high, the electron degeneracy becomes large and positron number decreases.", "Thus, the rate of pair neutrino process and the energy loss rate decreases.", "Figure REF shows contours of the average energies of $\\nu _e$ and $\\bar{\\nu }_e$ .", "At a given density, the average energies of $\\nu _e$ and $\\bar{\\nu }_e$ increase with temperature.", "The average energy of $\\nu _e$ is larger than that of $\\bar{\\nu }_e$ .", "When the electron degeneracy is small, the difference of the average $\\nu _e$ energy and $\\bar{\\nu }_e$ energy is small.", "The difference becomes larger for larger electron degeneracy.", "The higher $\\nu _e$ energy is due to the fact that the forward emission of $\\nu _e$ against electrons is favored in the pair neutrino process [33].", "In the temperature range of the Si core burning ($9.5 \\lesssim \\log T_{\\rm C} \\lesssim 9.6$ ) the average $\\bar{\\nu }_e$ temperature is less than the threshold energy 1.8 MeV of $p (\\bar{\\nu }_e, e^+)n$ reaction." ], [ "15 $M_\\odot $ model", "The evolution of the central core during the final stage is qualitatively in common among 12–20 $M_\\odot $ stars.", "We present neutrino spectra and the structure in five different stages listed in Table REF in the 15 $M_\\odot $ model.", "Figure: Time evolution of neutrino luminosity until the onset of the core-collapse(logT C \\log T_{\\rm C} = 9.8) of the 15 M ⊙ M_\\odot model.The dotted line indicates the total luminosity.Red, blue, pink, green, and orange lines are the contributions of pair neutrinos, photo neutrinos,neutrino Bremsstrahlung, plasma neutrinos, and weak interactions of nuclei.Triangle and rectangle correspond to the ignition and termination of the Si core burning,respectively.First, we show the contributions of the neutrino emission processes adopted in the 15 $M_\\odot $ stellar evolution models to the energy loss by neutrinos.", "Figure REF shows the time variation of the neutrino luminosity by the above neutrino emission processes from the central Ne burning to the collapse.", "Pair neutrino process dominates the neutrino luminosity for most of the advanced stellar evolution.", "For last several minutes, the luminosity of weak interaction reactions of nuclei exceeds that of pair neutrino process.", "Note that pair neutrino process produces all flavors of neutrinos.", "However, weak interactions of nuclei mainly produce $\\nu _e$ because electron captures rather than $\\beta ^-$ -decays occur in the collapsing iron core.", "Since current neutrino detectors such as KamLAND and Super-Kamiokande mainly detect $\\bar{\\nu }_e$ events through $p(\\bar{\\nu }_e,e^+)n$ reaction, the main sources of the neutrino events from preSN stars will be pair neutrinos.", "Investigating properties of neutrinos produced through weak interactions of nuclei is beyond the scope of this study.", "Figure: Time evolution of the emission rates for ν ¯ e \\bar{\\nu }_e and ν ¯ μ,τ \\bar{\\nu }_{\\mu ,\\tau }in the 15 M ⊙ M_\\odot model.Top panel shows the neutrino emission rates.Bottom panel shows the rate dN ν ¯ α σ ν /dtdN_{\\bar{\\nu }_{\\alpha }} \\sigma _{\\nu }/dt defined in Equation ().Points (a)–(d) indicate different evolution stages explained in Sec.", "III A 2.Next, we show the time evolution of the pair neutrino emission rate.", "The top panel of Fig.", "REF shows the time evolution of the emission rate for $\\bar{\\nu }_e$ and $\\bar{\\nu }_{\\mu ,\\tau }$ produced through pair neutrino process.", "Note that the neutrino emission rate of $\\bar{\\nu }_{\\mu ,\\tau }$ is the sum of the rates of $\\bar{\\nu }_{\\mu }$ and $\\bar{\\nu }_{\\tau }$ .", "The neutrino emission rates increase with time for most of the time because the star gradually contracts and the temperature in the central region rises.", "On the other hand, the rates decrease temporally when the main burning process changes.", "The Si core burning ignites before point (a).", "The O shell burning starts between points (b) and (c).", "The Si shell burning around $M_r \\sim 1 M_\\odot $ starts at a time just before point (d).", "At the ignitions of the Si core burning, the O shell burning, and the Si shell burning, the $\\bar{\\nu }_e$ emission rate decreases by factors of 1.2, 1.6, and 1.1, respectively.", "The neutrino event rate is determined by the multiple of the neutrino emission rate and the neutrino cross section.", "Most current and future neutrino detectors detect $\\bar{\\nu }_e$ from preSN stars through the $p(\\bar{\\nu }_e, e^+)n$ reaction.", "Therefore, it is useful to evaluate the quantity considering the weight of the neutrino cross section to the neutrino emission rate.", "We consider the rate $dN_{\\bar{\\nu }_{\\alpha }} \\sigma _{\\nu }/dt$ of $\\bar{\\nu }_\\alpha $ defined by $\\frac{dN_{\\bar{\\nu }_{\\alpha }}\\sigma _\\nu }{dt} &=&\\int _0^{M} \\frac{d^2N_{\\bar{\\nu }_{\\alpha }}\\sigma _\\nu (t,M_r)}{dt dM_r} dM_r \\\\&=&\\int _0^{M} \\left\\lbrace \\int _0^\\infty \\frac{d^3N_{\\bar{\\nu }_{\\alpha }}(t, M_r, \\varepsilon _\\nu )}{dt dM_r d\\varepsilon _\\nu } \\sigma _{p+\\bar{\\nu }_e}(\\varepsilon _\\nu ) d\\varepsilon _{\\nu } \\right\\rbrace dM_r, \\nonumber $ where $d^3N_{\\bar{\\nu }_{\\alpha }}(t, M_r, \\varepsilon _\\nu )/d\\varepsilon _\\nu dM_r dt$ is the emission rate of $\\bar{\\nu }_{\\alpha }$ with the neutrino energy $\\varepsilon _\\nu $ , at the mass coordinate $M_r$ , and at a time $t$ , $\\sigma _{p + \\bar{\\nu }_e}(\\varepsilon _\\nu )$ is the cross section of $p(\\bar{\\nu }_e,e^+)n$ as a function of the neutrino energy, and $M$ is the mass of the star.", "We call $dN_{\\bar{\\nu }_{\\alpha }} \\sigma _{\\nu }/dt$ “the detected $\\bar{\\nu }_{\\alpha }$ emission rate\" and call $d^2N_{\\bar{\\nu }_{\\alpha }} \\sigma _{\\nu }/dM_r dt$ the detected $\\bar{\\nu }_{\\alpha }$ emission rate at the mass coordinate $M_r$ .", "The cross section of $p(\\bar{\\nu }_e, e^+)n$ is adopted from [34].", "The threshold energy of this reaction is 1.8 MeV.", "The bottom panel of Fig.", "REF shows the time evolution of the detected neutrino emission rate.", "This rate also increases toward the collapse for most of time.", "This rate steeply rises at the ignition of the Si core burning [see point (a)], whereas the emission rate drops.", "Before the Si core burning, the neutrinos are mainly emitted from off-centered region where the O shell burning proceeds.", "When the Si core burning ignites, the main neutrino emission region changes to the center.", "Since the temperature in the main neutrino emission region becomes high, the average neutrino energy also becomes high.", "Combining to larger cross section for higher energy neutrinos, the detected emission rate rises steeply by a factor of 1.9 at the Si core ignition.", "On the other hand, the drop of the rate $dN_{\\bar{\\nu }_{\\alpha }} \\sigma _{\\nu }/dt$ between points (b) and (c) is more prominent than that of the neutrino emission rate.", "The rate drops by a factor of 3.2.", "During the stellar evolution in this period, the region of the main neutrino emission changes from the stellar center to the outer region $M_r \\sim 1.4 M_\\odot $ where the O shell burning occurs.", "Since the temperature of the O shell burning region is lower than the central temperature, the neutrinos from the outer region are more difficult to be detected than the neutrinos from the center.", "More details will be explained in Sec.", "III A 2.", "Figure: Same as Fig.", "but for the stage after the central Si burning (20.1 hours before the last step;point (b) in Figs.", "and )." ], [ "Neutrino energy spectra", "We present neutrino properties at five different stages (a)–(e) in Table REF during the final evolution stage of the 15 $M_\\odot $ model.", "The central temperature and density, and the neutrino emission rate in these stages are shown in Figs.", "REF and REF .", "Here, we will show the mass fraction distribution, the fraction of the $\\bar{\\nu }_e$ emission rate in the region of $M_r \\le 2 M_\\odot $ , and the neutrino spectra.", "We define the fractions of the $\\bar{\\nu }_e$ emission rate $\\psi _{\\rm N}(M_{\\rm in}, M_{\\rm out})$ and the detected $\\bar{\\nu }_e$ rate $\\psi _{\\rm D}(M_{\\rm in}, M_{\\rm out})$ in the mass range between $M_{\\rm in}$ and $M_{\\rm out}$ as $\\psi _{\\rm N}(M_{\\rm in}, M_{\\rm out}) =\\frac{\\int _{M_{\\rm in}}^{M_{\\rm out}} \\frac{d^2 N_{\\bar{\\nu }_e}(t, M_r^{\\prime })}{dt dM_r^{\\prime }} dM_r^{\\prime }}{\\int _{0 M_\\odot }^{2 M_\\odot } \\frac{d^2 N_{\\bar{\\nu }_e}(t, M_r^{\\prime })}{dt dM_r^{\\prime }} dM_r^{\\prime }}$ and $\\psi _{\\rm D}(M_{\\rm in}, M_{\\rm out}) =\\frac{\\int _{M_{\\rm in}}^{M_{\\rm out}} \\frac{d^2 N_{\\bar{\\nu }_e}\\sigma _\\nu (t, M_r^{\\prime })}{dt dM_r^{\\prime }} dM_r^{\\prime }}{\\int _{0 M_\\odot }^{2 M_\\odot } \\frac{d^2 N_{\\bar{\\nu }_e}\\sigma _\\nu (t, M_r^{\\prime })}{dt dM_r^{\\prime }} dM_r^{\\prime }} ,$ where $d^2 N_{\\bar{\\nu }_e}(t, M_r)/dt dM_r$ is the $\\bar{\\nu }_e$ emission rate at the mass coordinate $M_r$ , and $d^2 N_{\\bar{\\nu }_e}\\sigma _\\nu (t, M_r)/dt dM_r$ is the detected $\\bar{\\nu }_e$ rate defined at Eq.", "(3).", "We also define the $\\bar{\\nu }_e$ emission fraction $\\phi _{\\rm N}(M_r)$ and the detected $\\bar{\\nu }_e$ fraction $\\phi _{\\rm D}(M_r)$ in the interval of 0.1 $M_\\odot $ below the mass coordinate $M_r$ as $\\phi _{({\\rm N, D})}(M_r) = \\psi _{({\\rm N, D})}(M_r - 0.1 M_\\odot , M_r).$ They are indicators of the location where electron antineutrinos are mainly emitted from.", "In the following figures on the fractions of the $\\bar{\\nu }_e$ emission rate and the detected $\\bar{\\nu }_e$ rate, we will show the distributions of $\\phi _{\\rm N}(M_r)$ and $\\phi _{\\rm D}(M_r)$ , respectively, taking discrete values with the interval of 0.1 $M_\\odot $ for $M_r$ , i.e., $M_r = 0.1$ , 0.2, ..., 2.0 $M_\\odot $ .", "Stage (a) is at 3.61 days before the last step.", "This star is in the convective Si core burning.", "The left panel of Fig.", "REF shows the mass fraction distribution in $M_r \\le 2 M_\\odot $ .", "The convection region extends to 0.57 $M_\\odot $ .", "The center panel shows the $\\bar{\\nu }_e$ emission fraction and the detected $\\bar{\\nu }_e$ fraction.", "There is a peak at the center for the both fractions.", "Electron antineutrinos of 54 % are emitted from the convective Si/Fe core.", "The rest is emitted from the surrounding Si and O-rich layers.", "On the other hand, almost all electron antineutrinos detected through $p(\\bar{\\nu }_e,e^+)n$ are emitted from the convective Si/Fe core.", "About half are emitted from the central region, $\\phi _{\\rm D}(0.1M_\\odot ) = 0.52$ .", "The right panel shows the neutrino energy spectra.", "The neutrino emission rate is the order of $10^{50}$ s$^{-1}$ at this time.", "The average energy is 1.1–1.3 MeV, that is smaller than the threshold energy of $p(\\bar{\\nu }_e,e^+)n$ .", "Stage (b) is at 20.1 hours before the last step.", "The Si core burning has ceased 1.2 hours before this time and the central region of the star is contracting.", "The left panel of Fig.", "REF indicates that the Fe core of 1.02 $M_\\odot $ has been formed.", "The intermediate elements denoting “Si\" have been exhausted through the Si core burning.", "The surrounding layer enriched in “Si\" extends to 1.37 $M_\\odot $ .", "The O mass fraction increases outwards there.", "In the center panel, most of electron antineutrinos are emitted from the Fe core and the detected ones are from the central region of the core.", "We obtain $\\psi _{\\rm N}(0M_\\odot , 1.0M_\\odot ) = 0.89$ and $\\psi _{\\rm D}(0M_\\odot , 0.5M_\\odot ) = 0.94$ .", "The right panel indicates that the neutrino emission rate increases by a factor of four from the previous stage.", "The average energy slightly increases from the Si core burning but is still smaller than the threshold energy of $p(\\bar{\\nu }_e,e^+)n$ .", "Figure: Same as Fig.", "but for the stage at 23.8 minutes before the last step[point (d) in Figs.", "and ]In the left panel, black line indicates He.Stage (c) is at 9.78 hours before the last step.", "Before this time Oxygen ignited at $M_r \\sim 1.36 M_\\odot $ and the convective Si/O layer extended outwards.", "The O shell burning expands the star and the central temperature and density decrease [see the range between (b) and (c) in Fig.", "REF ].", "The neutrino emission rate also decreases at that time (see Fig.", "REF ).", "We see in the left panel of Fig.", "REF that the mass fractions of Si and O are almost constant between 1.36 and 1.66 $M_\\odot $ .", "The convective O shell burning occurs in this region.", "The temperature in this region rises and the high temperature raises the neutrino emissivity.", "The center panel indicates that the $\\bar{\\nu }_e$ emission is not concentrated to the center and it is broadly distributed.", "We also see a peak in the O shell burning region for $\\phi _{\\rm N}(M_r)$ .", "We see the decrease in $\\psi _{\\rm N}(0M_\\odot , 1.0M_\\odot )$ to 0.52 and the increase in $\\psi _{\\rm N}(1.3M_\\odot , 1.6M_\\odot )$ to 0.41.", "However, the distribution of the detected $\\bar{\\nu }_e$ fraction has a peak at the center and the contribution of the convective Si/O layer is still small.", "Most of the detected $\\bar{\\nu }_e$ are from the Fe core, i.e., $\\psi _{\\rm D}(0M_\\odot , 1.0M_\\odot ) = 0.93$ and $\\psi _{\\rm D}(1.3M_\\odot , 1.6M_\\odot ) = 0.06$ .", "This is because the electron antineutrinos emitted from the central region have higher energy than the ones from the Si/O layer.", "The right panel shows the neutrino spectra at this time.", "Comparing with the spectra in the previous stage, we do not see clear difference in the maximum value for each neutrino flavor.", "On the other hand, the spectra become less energetic from the previous stage.", "This is due to the contribution of the neutrinos from the Si/O layer.", "Thus, when the O shell burning starts, the neutrino energy spectra become less energetic and the detectability of the neutrino events will decrease.", "Stage (d) is at 23.8 minutes before the last step.", "After stage (c), the O shell burning ceases, the star contracts again, and the Si shell burning ignites at $M_r \\sim 1 M_\\odot $ .", "The Fe core and the surrounding Si layer grow up through the shell burnings.", "The left panel of Fig.", "REF shows that the Fe core mass and the outer boundary of the Si layer become 1.37 and 1.65 $M_\\odot $ , respectively.", "We also see the O/Si-rich and O/Ne-rich layers outside the Si layer.", "In the center panel, we see two peaks at $M_r \\sim 1$ and 1.4 $M_\\odot $ for the $\\bar{\\nu }_e$ emission.", "At this stage, the central temperature is $\\log T_{\\rm C} = 9.66$ and there are two peaks at $M_r = 1.02$ and 1.36 $M_\\odot $ in the temperature distribution.", "At $M_r = 1.02 M_\\odot $ the temperature is $\\log T = 9.64$ and the density is $\\log \\rho = 6.79$ .", "The high temperature and low density in this region produce the large neutrino emission rate.", "The $\\bar{\\nu }_e$ emission rate in this region is much larger than that at the center (see Fig.", "REF ).", "Different from the stage (c), we also see a peak for the detected $\\bar{\\nu }_e$ fraction in this region.", "This corresponds to $\\phi _{\\rm D}(1.0M_\\odot , 1.2M_\\odot ) = 0.56$ .", "Some detected electron antineutrinos are also produced inside the Si shell; $\\phi _{\\rm D}(0M_\\odot , 1.0M_\\odot ) = 0.33$ .", "The right panel shows that the neutrino emission rate increases to the order of $10^{51}$ s$^{-1}$ .", "Although the neutrino emission rate decreases when the Si shell burning starts, the star has turned to the contraction at this stage.", "The neutrino spectra shift to higher energy from the previous stage.", "This is also due to the contraction after the O and Si shell burnings.", "Figure: Same as Fig.", "but for the last step(point (e) in Figs.", "and ).In the left panel, black and grey lines indicate He and H.Stage (e) is the last step of the stellar evolution calculation.", "The central region including the Fe core and Si and O/Si shells is contracting and the central temperature exceeds $\\log T_{\\rm C} = 9.81$ .", "The left panel of Fig.", "REF shows the mass fraction distribution at this time.", "The Fe core grows to 1.50 $M_\\odot $ through the Si shell burning.", "The distributions of the $\\bar{\\nu }_e$ emission fraction and the detected $\\bar{\\nu }_e$ fraction, seen in the center panel, are similar to stage (d); there are two peaks at $M_r \\sim 1.0$ and 1.4 $M_\\odot $ for the $\\bar{\\nu }_e$ emission and a peak at $M_r \\sim 1.0 M_\\odot $ for the detected $\\bar{\\nu }_e$ .", "The two peaks of the temperature distribution seen in the previous stage still remain and the peak temperatures rise.", "The detected rate of $\\bar{\\nu }_e$ from the Si shell burning region is slightly larger than the rate of $\\bar{\\nu }_e$ from inside the region.", "We obtain $\\phi _{\\rm D}(0M_\\odot , 1.0M_\\odot ) = 0.41$ and $\\phi _{\\rm D}(1.0M_\\odot , 1.2M_\\odot ) = 0.45$ at this stage.", "The right panel shows that the emission rates of $\\nu _e$ and $\\bar{\\nu }_e$ increase to about $10^{52}$ s$^{-1}$ and the spectra shift to high energy direction further.", "The central high density region contributes the production of high energy neutrinos.", "The average energy of the neutrino spectra exceeds the threshold energy of $p(\\bar{\\nu }_e,e^+)n$ reaction (see Table REF ).", "Massive stars form an Fe core in their final stage of the evolution and the evolution time scale depends on the initial mass.", "The period of the Si core burning is 7.4 days and 16 hours for the 12 and 20 $M_\\odot $ models, respectively (see Table REF ).", "Thus, quantitative properties of neutrinos emitted in the final stage also depend on the stellar mass.", "Here, we show the dependence of the neutrino emission rate and average energy on the stellar mass.", "Figure: Time evolution of the ν ¯ e \\bar{\\nu }_e emission rate (top panel) and the detected ν ¯ e \\bar{\\nu }_eemission rate (bottom panel)of the 12 (blue line), 15 (red line), and 20 (green line) M ⊙ M_\\odot models.Blue and green arrows indicate the time when the neutrino emission rate decreases in the 12 and20 M ⊙ _\\odot models, respectively.See text for details.Figure REF shows the time evolution of the $\\bar{\\nu }_e$ emission rate and the detected $\\bar{\\nu }_e$ emission rate of the 12, 15, and 20 $M_\\odot $ model.", "Since the stellar mass dependence of the emission rate for other flavors is similar to that of $\\bar{\\nu }_e$ , we discuss only the dependence of $\\bar{\\nu }_e$ .", "Both the $\\bar{\\nu }_e$ emission rate and the detected $\\bar{\\nu }_e$ emission rate increase with time for most of the evolution period among the three models.", "In the 12 $M_\\odot $ model, the $\\bar{\\nu }_e$ emission rate is smaller than the 15 $M_\\odot $ model for a given time.", "The $\\bar{\\nu }_e$ emission rate decreases by a factor of 1.5 (see the left blue arrow in the top panel) and the detected $\\bar{\\nu }_e$ rate increases by a factor of 2.5 (see the left blue arrow in the bottom panel) around 8.2 days before the last step.", "At this time the Si core burning starts.", "Although this change is also seen in the 15 $M_\\odot $ model, the corresponding time is different.", "This is due to the longer period of the Si core burning in the 12 $M_\\odot $ model.", "We see the decrease in both rates in 10 hours before the last step, assigned by the center blue arrow in each panel.", "The decreases in the $\\bar{\\nu }_e$ emission rate and the detected emission rate are factors of 1.6 and 3.4, respectively.", "This is evidence for the ignition of the O shell burning.", "In 50 minutes before the last step, we also see slight decreases in both rates, assigned by the right blue arrow.", "The $\\bar{\\nu }_e$ emission rate and the detected emission rate decrease by factors of 1.1 and 1.3, respectively.", "This corresponds to the ignition of the Si shell burning.", "The trend of these decreases is similar to the 15 $M_\\odot $ model.", "In the 20 $M_\\odot $ model, the $\\bar{\\nu }_e$ emission rate is generally larger than the 15 $M_\\odot $ model and the neutrino emission rate has some properties different from the other stellar models.", "We see the decrease in the emission rate is seen by a factor of 1.2 at 0.9 days before the last step (the left green arrow in the top panel).", "However, the steep increase in the detected $\\bar{\\nu }_e$ emission rate is not seen at that time (the left green arrow in the bottom panel).", "The $\\bar{\\nu }_e$ emission rate slightly decreases at 6.5 and 2.4 hours before the last step (see the center and right arrows in each panel).", "The corresponding decreases of the detected $\\bar{\\nu }_e$ emission rate are small.", "This seems to be due to differences of burning processes in the advanced evolution.", "In the 20 $M_\\odot $ model, the convective shell of the O shell burning before the Si core burning extended to 1.5 $M_\\odot $ and the oxygen in this region has been exhausted.", "The Si shell grows up to 1.59 $M_\\odot $ when the Si core burning ended.", "Then, the O shell burning occurs in the region of $M_r \\gtrsim 1.6 M_\\odot $ .", "However, this burning scarcely prevents the core contraction and, thus, the neutrino emission rate scarcely decreases.", "We do not see the decrease in the central temperature at this time.", "We note that, in the 12 and 15 $M_\\odot $ models, the O and Si-enriched region remains outside $M_r \\sim 1.36 M_\\odot $ even after the Si core burning and, then, the O shell burning in this region is stronger.", "The decrease at 2.4 hours before the last step corresponds to the Si shell burning.", "The Si shell burning occurs at $M_r \\sim 0.7 M_\\odot $ that is deeper than the 12 and 15 $M_\\odot $ models.", "We consider that stronger Si shell burning prevents decreasing the neutrino emissivity.", "We show the time evolution of the average $\\bar{\\nu }_e$ energy $\\langle \\epsilon _{\\nu } \\rangle $ of these three models in Fig.", "REF .", "Before the Si core burning, all models indicate the average energy with less than $\\sim $ 1 MeV.", "When the Si core burning ignites, the neutrino emission from the center increases and the average energy becomes large for the 12 and 15 $M_\\odot $ models.", "The average energy of the 20 $M_\\odot $ model also increases more steeply than before, although the steepness is less than the less massive models.", "The average energy during the Si core burning is lower for more massive star models.", "As shown in Fig.", "REF , the evolution track of the more massive one passes through a lower density and higher temperature path.", "The average $\\bar{\\nu }_e$ energy is higher in higher density for a given temperature.", "Thus, we consider that lower density structure of more massive star provides lower average $\\bar{\\nu }_e$ energy during the Si burning.", "The average energy becomes small during the O shell burning.", "The main region of the neutrino emission is the O burning shell, where the temperature and density is lower than the center.", "During the Si shell burning to collapse, we do not see clear dependence on the stellar mass.", "This is partly due to the difference of the contraction time scale in these models.", "For a given central temperature after the ignition of the Si shell burning, the average energy is smaller for more a massive model.", "Figure: Time evolution of the average ν ¯ e \\bar{\\nu }_e energyof the 12 (blue line), 15 (red line), and 20 (green line) M ⊙ M_\\odot models.We showed that the time evolution of the neutrino emission rate and the detected neutrino emission rate from the Si core burning to the core collapse is influenced by burning processes in the central region of massive stars.", "The ignition of the Si core burning raises the average $\\bar{\\nu }_e$ energy and the neutrino detectability.", "The onset of O and Si shell burnings decreases the neutrino emission rate as well as the neutrino detectability for a moment.", "The stellar mass dependence of the above burning processes brings about the dependence of the neutrino properties from preSN stars." ], [ "Evaluation of neutrino events by neutrino observatories", "We investigate the neutrino event rate and the total events by current and future neutrino detectors.", "Current and most future neutrino detectors mainly detect $\\bar{\\nu }_e$ signals through $p(\\bar{\\nu }_e, e^{+})n$ reaction.", "So, we evaluate the neutrino events through this reaction.", "In this study, we assume that the distance of a preSN star is 200 pc.", "This distance corresponds to the distance to Betelgeuse.", "When we investigate the neutrino events of preSN stars, we need to consider the flavor change by the Mikheyev-Smirnov-Wolfenstein (MSW) effect during the passage of the stellar interior.", "The flavor change mainly occurs at resonance layers.", "In preSN stars, the flavors change at the high (H) and low (L) resonances [35], [36].", "The adiabaticity and, thus, the transition probability depend on the mixing angles.", "Recent neutrino experiments confirmed that the mixing angle $\\sin ^2\\theta _{13} \\sim 0.02$ [37], [38], [39], [40] and the large $\\sin ^2\\theta _{13}$ value indicates adiabatic flavor change at the both resonances.", "The density of the H resonance is written as $\\rho _{\\rm H-res} \\sim 3.0 \\times 10^{4} \\left( \\frac{\\rm 1 MeV}{\\varepsilon _\\nu } \\right) \\, {\\rm g \\, cm^{-3}}.$ Here, we set squared mass difference between mass eigenstates 1 and 3 as $\|\\Delta m_{31}^2\| c^4 = 2.43 \\times 10^{-3}$ eV$^2$ , the mixing angle $\\theta _{13}$ as $\\sin ^2 \\theta _{13} = 0.024$ [32], and the electron fraction $Y_e$ as 0.5.", "Since the density inside the O/Ne layer is larger than the density of the H resonance with $\\varepsilon _\\nu > 1$ MeV, it is reasonable to assume that the transition probability does not depend on the neutrino energy.", "Thus, we evaluate the transition probability of $\\bar{\\nu }_e \\rightarrow \\bar{\\nu }_e$ in normal and inverted mass hierarchies as $P_{(\\bar{\\nu }_{e} \\rightarrow \\bar{\\nu }_{e})} = \\left\\lbrace \\begin{array}{ll}\\cos ^2\\theta _{12} \\cos ^2\\theta _{13} = 0.675 & ({\\rm normal}) \\\\\\sin ^2\\theta _{13} = 0.024 & ({\\rm inverted}),\\end{array}\\right.$ where $\\sin ^2\\theta _{12} = 0.308$ [32].", "The sum of the transition probabilities from $\\bar{\\nu }_\\mu $ and $\\bar{\\nu }_\\tau $ is $\\sum _{\\alpha =\\mu ,\\tau } P_{(\\bar{\\nu }_\\alpha \\rightarrow \\bar{\\nu }_e)} =1 - P_{(\\bar{\\nu }_e \\rightarrow \\bar{\\nu }_e)}.$" ], [ "KamLAND", "The Kamioka Liquid-scintillator Antineutrino Detector (KamLAND) is a one kton size neutrino detector located in the Kamioka Mine, Japan.", "This detector has a detectability of low energy $\\gamma $ -rays using its liquid-scintillator.", "KamLAND detects $\\bar{\\nu }_e$ events through $p(\\bar{\\nu }_e, e^+)n$ from a preSN star.", "First, the produced positron is pair-annihilated to produce $\\gamma $ -rays.", "Then, the neutron produced through the neutrino reaction is captured by a proton through $n(p, \\gamma )d$ and 2.2 MeV $\\gamma $ -rays are emitted.", "KamLAND identifies a $\\bar{\\nu }_e$ event using both the prompt $\\gamma $ -rays produced by the pair-annihilation and the delayed 2.2 MeV $\\gamma $ -rays produced by the neutron capture.", "The low threshold energy enables to detect $\\bar{\\nu }_e$ events from preSN stars with the distance of Betelgeuse (e.g., [11], [12]).", "We calculate the spectrum of the $\\bar{\\nu }_e$ events detected by KamLAND in accordance with [12]: $\\frac{d^2 N(t,\\varepsilon _{p})}{dt d\\varepsilon _{p}} &=&\\epsilon _{\\rm live} \\epsilon _{\\rm s}(\\varepsilon _{p}) \\frac{N_{\\rm T}}{4 \\pi d^2} \\\\&\\times & \\sum _{\\alpha } \\int \\frac{d^2N_{\\bar{\\nu }_\\alpha }(t,\\varepsilon _{\\nu })}{dt d\\varepsilon _{\\nu }}P_{(\\bar{\\nu }_{\\alpha } \\rightarrow \\bar{\\nu }_e)}\\sigma _{p + \\bar{\\nu }_e}(\\varepsilon _{\\nu }) \\nonumber \\\\&\\times & \\left( \\frac{d\\varepsilon _{\\nu }}{d\\varepsilon ^{\\prime }_{p}} \\right)R(\\varepsilon _{p},\\varepsilon ^{\\prime }_{p})d\\varepsilon ^{\\prime }_{p} , \\nonumber $ where $\\epsilon _{\\rm live}$ is mean livetime-to-runtime ratio, $\\epsilon _{\\rm s}(\\varepsilon _p)$ is the total detection efficiency, $N_{\\rm T} = 5.98 \\times 10^{31}$ is the fiducial proton number of KamLAND, $d$ is the distance to a preSN star, that is assumed to be 200 pc, $\\varepsilon ^{\\prime }_p$ is the expected energy of the prompt event with the relation of $\\varepsilon ^{\\prime }_p = \\varepsilon _\\nu - 0.78$ MeV, $R(\\varepsilon _p, \\varepsilon ^{\\prime }_{p})$ is the detector response assumed to be the Gaussian distribution of the energy resolution of $6.4 \\% / \\sqrt{\\varepsilon ^{\\prime }_p \\, {\\rm (MeV)}}$ .", "The neutrino event rate in the energy range $\\varepsilon _{pL} \\le \\varepsilon _p \\le \\varepsilon _{pU}$ is obtained using $\\frac{dN(t; \\varepsilon _{pL}:\\varepsilon _{pU})}{dt} =\\int _{\\varepsilon _{pL}}^{\\varepsilon _{pU}}\\frac{d^2 N(t,\\varepsilon _{p})}{dt d\\varepsilon _{p}} d\\varepsilon _{p},$ where $\\varepsilon _{pL}$ and $\\varepsilon _{pU}$ are the lower and upper limits of the event energy.", "Table: The p(ν ¯ e ,e + )np(\\bar{\\nu }_e,e^+)n events integrated for seven days before SN explosionsby current and future neutrino observatories.We discuss the time evolution of the expected neutrino events by KamLAND.", "Here, we consider the neutrino events integrated from the last step of the evolution calculation to a time before the stellar collapse: $N(t_r; \\varepsilon _{pL}:\\varepsilon _{pU}) =\\int _{t_f-t_r}^{t_f} \\frac{dN(t^{\\prime }; \\varepsilon _{pL}:\\varepsilon _{pU})}{dt^{\\prime }} dt^{\\prime },$ where $t_f$ is the time at the last step and $t_r$ is equal to $t_f - t$ , i.e., the period from a time to the last-step time.", "Figure REF shows the integrated neutrino events by KamLAND with the energy range of $0 \\le \\varepsilon _p \\le \\infty $ .", "It is convenient to use this figure when we evaluate the neutrino events during a given period to the core collapse.", "We assume that the effective livetime $\\epsilon _{\\rm live}$ is 0.903 and use the average value 0.64 for the detection efficiency $\\epsilon _{\\rm s}$ [12], which we call “Average efficiency\".", "The $p(\\bar{\\nu }_e,e^+)n$ events integrated for seven days before SN explosions observed by current and future neutrino observatories are listed in Table REF .", "We expect the neutrino events of 7–14 in the normal mass hierarchy and 4–7 in the inverted mass hierarchy for one week before a SN explosion if the SN explode at the distance of $\\sim 200$ pc.", "The smaller event number in the inverted mass hierarchy is due to the fact that almost all electron antineutrinos have been converted from $\\mu $ or $\\tau $ antineutrinos through the MSW effect and that the emission rate of the $\\mu $ and $\\tau $ antineutrinos is smaller than that of electron antineutrinos (see Sec.", "II and Sec.", "III).", "For more massive stars, the number of the total neutrino events is larger and the period of observable neutrino events is shorter.", "Thus, combined with these two features, we could constrain the stellar mass of a SN from the observations of preSN neutrinos.", "We should note that only several events will be observed by KamLAND at most in the inverted mass hierarchy.", "In this case, it will be quite difficult to constrain the stellar mass of the preSN star because of statistical and systematic errors of the neutrino events.", "This difficulty also should be considered in the observations by other neutrino detectors.", "Uncertainties of the neutrino events by the stellar evolution models will be discussed in Sec.", "V. We show the expected spectrum of neutrino events detected by KamLAND in Fig.", "REF .", "Although the average energy is slightly larger for more massive models, the stellar mass dependence of the spectrum feature is small.", "The difference is mainly the total event number.", "Owing to this reason, higher energy events may be observed for a more massive model.", "Figure: Expected spectrum of neutrino events detected by KamLAND.Blue, red, and green lines indicate the 12, 15, and 20 M ⊙ M_\\odot models.Solid and dashed lines correspond to the normal and inverted mass hierarchies, respectively.At present, KamLAND contains an inner balloon for neutrinoless double beta-decay experiment (KamLAND-Zen) [41].", "Asakura et al.", "[12] adopted the energy dependent efficiency $\\varepsilon _{\\rm s}(\\epsilon _p)$ due to the Likelihood selection.", "The main effect of the efficiency loss is the inner balloon cut [12].", "The observed neutrino events depend on the detection efficiency determined by the current installed system.", "So, we also consider the efficiency without the inner balloon cut (No balloon).", "In this case, the efficiency is about 0.9 [12].", "This efficiency gives more neutrino events than the average efficiency.", "The detection efficiency of KamLAND can increase depending on the future experiment system.", "An alarm of a SN explosion using the observation of preSN neutrinos has been discussed [12].", "They expected that the $3\\sigma $ detection of preSN neutrinos 2–90 hours before the SN explosion is possible with counting the neutrino events for 48 hours.", "The main background for preSN neutrinos is reactor neutrinos and the background rate is $B_{\\rm low} = 0.071$ events day$^{-1}$ in the low-reactor phase and 0.355 events day$^{-1}$ in the high-reactor phase.", "Here, we also estimate an alarm of a nearby SN explosion using the detection of three preSN neutrino events by KamLAND for 48 hours.", "Three events for 48 hours correspond to the detection significance of $3.7 \\sigma $ and $2.1 \\sigma $ for low-reactor phase and high-reactor phase, respectively.", "So, three events for 48 hours by KamLAND can be a good indicator of a SN alarm.", "When we evaluate the time, we consider the energy range of the neutrino signal as $0.9 \\le \\varepsilon _p \\le 3.5$ MeV in accordance with [12].", "We also consider a SN alarm in the case of No balloon.", "In this case, larger efficiency also raises the background events.", "We expect $B_{\\rm low} = 0.103$ and $B_{\\rm high} = 0.516$ using the detection efficiency averaged in the detected energy of 0.93.", "We obtain that three events for 48 hours in low reactor phase correspond to $3.2 \\sigma $ detection significance and that six events in high reactor phase correspond to $3.3 \\sigma $ .", "We consider that the observations of these events for 48 hours give a SN alarm.", "Table: Expected SN alarm time by KamLAND.Norm: normal mass hierarchy, Inv: inverted mass hierarchy.Table REF shows the expected SN alarm time, i.e., the time when three neutrino events are expected in 48 hours.", "When we consider averaged efficiency case, the expected time for the SN alarm is 3.5–18.1 hours and less than 3.6 hours in the normal and inverted mass hierarchies, respectively.", "In the normal mass hierarchy, the alarm may be provided just after the termination of the Si core burning.", "The 15 $M_\\odot $ model gives the longest time prior to the explosion.", "This is due to moderate Si-burning period and moderate neutrino emission.", "In the 12 $M_\\odot $ model, the period of the Si burning is longer but the neutrino emission rate is low, so neutrino event number within a given period increases less steeply than the 15 $M_\\odot $ model.", "In the inverted mass hierarchy, it may be difficult to send a SN neutrino alarm because of less neutrino events.", "In the 20 $M_\\odot $ model, the short period prior to the explosion is mainly due to the short Si-burning period.", "When we consider no balloon case, three and six events correspond to low and high reactor phases, respectively.", "In low reactor phase, the expected SN alarm time is earlier than the average efficiency case owing to the larger detection efficiency.", "The difference is larger for the SN alarm from a less massive star.", "Even in high reactor phase, the SN alarm several hours before the explosion is possible for the SN of more massive than a $\\sim 15 M_\\odot $ star and in the normal mass hierarchy.", "The SN alarm using preSN neutrinos will extend the possibility of the SuperNova Early Warning System (SNEWS) [42]." ], [ "SNO+ and Borexino", "SNO+ is a neutrino experiment in SNOWLAB, Sudbury, Canada (recent review: [43]).", "This experiment is planned to detect neutrinos using 780 tons of liquid scintillator.", "The main target is a search for the neutrinoless double-beta decays of $^{130}$ Te, and other broad topics of neutrino experiments will be performed.", "The detection of SN neutrinos is one topic of the experiments.", "SN neutrinos are mainly observed through $p(\\bar{\\nu }_e,e^+)n$ as a prompt signal and $n(p,\\gamma )d$ as a delayed signal.", "The experimental facility of SNO+ such as the vessel volume and the use of liquid scintillator is similar to KamLAND.", "Table REF shows the expected preSN neutrino events for seven days before the SN explosion in SNO+.", "Here, we assume 780 ton fiducial mass and the detection efficiency of $\\varepsilon _{\\rm live} = \\varepsilon _{\\rm S}(\\varepsilon _p) = 1$ for simplicity.", "These event numbers are scaled proportionally to the fiducial volume of KamLAND with the same assumptions.", "So, the preSN neutrino events will be smaller than the cases of KamLAND with no balloon.", "We expect that SNO+ can also give a SN alarm using preSN neutrinos.", "In this case, the background is mainly determined by reactor neutrinos.", "The number of reactor $\\bar{\\nu }_e$ events in SNO+ is expected to be around 90 events per year [43], corresponding to 0.25 events per day.", "In this background, the detection significance of three $\\bar{\\nu }_e$ events for 48 hours is $2.5\\sigma $ .", "Thus, if the detectability of preSN neutrinos by SNO+ is similar to KamLAND, SNO+ will also observe the preSN neutrino events expected in Fig.", "REF and will give a SN alarm.", "Observing preSN neutrinos by the two neutrino experiments will increase the reliability of the SN alarm system.", "Borexino is a liquid scintillation neutrino detector in the Laboratori Nazionali del Gran Sasso, Italy.", "This experiment observes low energy solar neutrinos [44] as well as geo-neutrinos [45].", "The geo-neutrino experiment observes electron antineutrinos through $p(\\bar{\\nu }_e,e^+)n$ similar to KamLAND, so preSN neutrinos will be observed through this reaction.", "This experiment uses 278 tons liquid scintillator and the target proton number is $\\sim 1.7 \\times 10^{31}$ [46].", "This proton number corresponds to about one third of KamLAND, so the number of preSN neutrino events is also expected to be one third of KamLAND.", "Thus, 2–7 preSN neutrino events are expected to be observed in Borexino if the detection efficiency is one (see Table REF ).", "Figure: Expected neutrino events per one hour detected by JUNO in 30 hours prior to a SN explosion.Left panel shows the 15 M ⊙ M_\\odot model and right panel shows the 20 M ⊙ M_\\odot model.Red and blue lines indicate the cases of the normal and inverted mass hierarchies.Dashed and dash-dotted green lines indicate the background events per hourin high and low reactor phases." ], [ "JUNO, RENO-50, and LENA", "There are planned and proposed neutrino experiments using large size liquid scintillation detectors.", "JUNO in China [19] and RENO-50 in Korea [20] have plans for constructing 20 and 18 kton size detectors, respectively.", "LENA in Europe has a plan of the construction of a 50 kton size detector [47].", "These detectors will enable to raise the neutrino detectability owing to the large size comparably to the Super-Kamiokande and the low energy threshold similar to KamLAND.", "If these neutrino experiments operate, the observed events of preSN neutrinos will increase drastically.", "Here, we will evaluate preSN neutrino events by these detectors based on the current proposals of JUNO [19].", "JUNO observes electron antineutrinos through $p(\\bar{\\nu }_e,e^+)n$ as a prompt reaction and $n(p,\\gamma )d$ as a delayed reaction similar to KamLAND.", "The fiducial mass is 20 kton and the corresponding target proton number is $1.5 \\times 10^{33}$ .", "The detection efficiency from the fiducial mass to reduce the background is 0.79 (see Table 2-1 in [19]).", "In this case, the event number is expected to be 34 times as large as in KamLAND with average detection efficiency.", "When reactor neutrino experiments are conducted in JUNO, neutrinos from reactors become background against preSN neutrinos.", "The background during reactor neutrino experiments is estimated to be 60 events per day [19].", "When the reactors are turned off, the background is estimated to be 3.8 events per day [19].", "So, we consider that the background neutrino events are 63.8 and 3.8 for high- and low-reactor phases.", "These events correspond to 2.66 and 0.16 events per hour.", "The number of neutrino events with the significance more than $3\\sigma $ is nine and three.", "The event number of preSN neutrinos by JUNO is estimated from the result of KamLAND (see Fig.", "REF ) and the ratio of the target proton numbers.", "We show the event number by JUNO for seven days before the SN explosion in Table REF .", "The event number by RENO-50 and LENA is also calculated using the ratio of the fiducial mass and is listed in Table REF .", "We expect hundreds of preSN neutrino events will be observed by these neutrino observatories.", "We show the expected neutrino events per one hour detected by JUNO in 30 hours prior to the SN explosions of the 15 and 20 $M_\\odot $ models in Fig.", "REF .", "We see a minimum of the neutrino events around ten hours prior to the explosion in the 15 $M_\\odot $ model.", "The minimum value is less than five and the maximum event number prior to the minimum is about ten.", "This minimum corresponds to the decrease in the high energy neutrino emission due to the ignition of the O shell burning after the termination of the Si core burning.", "In the case of the 12 $M_\\odot $ model, there is a peak with eight (four) events in the normal (inverted) mass hierarchy at about sixteen hours before the explosion.", "The event number per hour becomes a minimum around ten hours before the explosion.", "Since the peak height is smaller than the 15 $M_\\odot $ model, it is more difficult to observe it.", "Figure: Expected neutrino events per ten minutes detected by JUNO in 120 minutes prior to a SN explosionof the 15 M ⊙ M_\\odot model.Red and blue lines indicate the cases of the normal and inverted mass hierarchies.We also see a minimum of the neutrino events around seven hours prior to the explosion in the 20 $M_\\odot $ model.", "This change is less prominent than the 15 $M_\\odot $ model but still we would recognize the change of burning processes in the central region.", "The 20 $M_\\odot $ model indicates weak neutrino emission from the O shell.", "Thus, we could observe the evolution of burning processes in the central region of collapsing stars through the preSN neutrino events and could constrain shell burnings after the Si core burning.", "We present the time evolution of the neutrino events just before the SN explosion.", "Figure REF shows the expected neutrino events for ten minutes detected by JUNO for two hours prior to the explosion in the 15 $M_\\odot $ model.", "We see a minimum of the event around 50 minutes before the explosion but the difference of the event number at the maximum around 70 minutes is small.", "This corresponds to the ignition of the Si shell burning at $M_r \\sim 1 M_\\odot $ .", "This signal also would be observed by Hyper-Kamiokande if low threshold energy is set.", "The multiple observations of preSN neutrinos will raise the reliability of observed neutrino signals.", "We note that the neutrino events per ten minutes monotonically increase for two hours in the 20 $M_\\odot $ model.", "Since the convection of the Si shell burning of the 20 $M_\\odot $ model is not so strong, the efficiency of the convection would be constrained from the preSN neutrino signals.", "Figure: Same as Fig.", "but for the detection by Hyper-Kamiokande with the neutrino thresholdenergy of 4.79 MeV.Dashed and dash-dotted green lines indicate the neutrino events with the neutrino threshold energy of5.29 and 6.29 MeV, respectively.A SN alarm using preSN neutrinos is also possible for JUNO.", "We consider the time for the SN alarm using the neutrino events per hour of the three models.", "The energy range for the alarm is set to be $0.9 \\le \\varepsilon _p \\le 3.5$ MeV similar to KamLAND.", "The energy resolution is assumed to be the same as KamLAND.", "Table REF lists the expected SN alarm time given by JUNO.", "In the low reactor phase, the SN alarm will be sent earlier than KamLAND.", "Except for the case of the inverted mass hierarchy of the 12 $M_\\odot $ model, the alarm will be sent before the star starts the O shell burning.", "Table: Expected SN alarm time by JUNO (hours prior to the explosion).We note that one of the main targets in JUNO and RENO-50 experiments is reactor neutrino experiment.", "The neutrino background will be high during the reactor neutrino experiment.", "Since preSN neutrinos are hidden in the high background, the identification and sending a SN alarm will be delayed.", "In high-reactor phase, it may be difficult to observe a minimum of the neutrino events after the Si core burning by JUNO (see Fig.", "REF ).", "The SN alarm by JUNO is delayed to KamLAND in the low reactor phase in some cases.", "Although the power of nuclear plants for RENO-50 is smaller than JUNO, the SN alarm time is still similar to KamLAND or worse.", "So, we consider that monitoring preSN neutrinos and sending an alarm by KamLAND and SNO+ are also important.", "In order to send a SN alarm in an early time before the SN explosion, monitoring by many neutrino detectors with a low energy threshold is desirable." ], [ "Super-Kamiokande and Hyper-Kamiokande", "Super-Kamiokande is a water Cherenkov detector located in the Kamioka Mine, Japan.", "In the fourth phase of solar neutrino experiment, the fiducial volume is 22.5 kton for most of the energy range and the threshold energy of the recoil electrons reduces to 3.5 MeV [48].", "Supernova neutrinos will be detected mainly by $p(\\bar{\\nu }_e,e^+)n$ reaction.", "In this case, the observable threshold for neutrinos is 4.79 MeV.", "Hyper-Kamiokande is proposed as a next generation water Cherenkov detector.", "The proposed fiducial volume has been recently changed to 380 kton.", "The threshold energy is expected to be lower than the previous plan [21].", "Here, we estimate the neutrino events of Super-Kamiokande and Hyper-Kamiokande.", "We assume for Hyper-Kamiokande that the fiducial volume is 380 kton and the neutrino threshold energy is 4.79 MeV.", "The neutrino events of Super-Kamiokande can be estimated by scaling with a factor 22.5/380 = 0.06.", "Figure: Same as Fig.", "for the detection by Hyper-Kamiokandewith the neutrino energy threshold of 4.79 MeV.We estimate the integrated $\\bar{\\nu }_e$ events by Hyper-Kamiokande in Fig.", "REF .", "The neutrino events for seven days before the explosion by Super-Kamiokande and Hyper-Kamiokande are listed in Table REF .", "Most of the preSN neutrino events will be observed in one day before the explosion.", "Although the threshold energy is higher than liquid-scintillation observatories, the large fiducial volume makes it possible to observe hundreds of neutrino events.", "In Super-Kamiokande, several to tens of neutrino events will be observed.", "Figure: Same as Fig.", "but for the detection using delayed signals by Gd-loaded Hyper-Kamiokande.Figure: Expected neutrino events per hour detected using delayed signals by Super-Kamiokande with Gd(top panel) and neutrino events per 10 minutes by Hyper-Kamiokande with Gd(bottom panel) in 24 hours prior to a SN explosion of the 15 M ⊙ M_\\odot model.Red and blue bins indicate the cases of the normal and inverted mass hierarchies.The green horizontal line is the event number for the unit interval of the significancemore than 3σ3\\sigma (see text for details).The expected neutrino events strongly depend on the threshold energy.", "We also show the integrated $\\bar{\\nu }_e$ events for the 20 $M_\\odot $ model in $E_{\\nu ,{\\rm th}}$ = 5.29 and 6.29 MeV in Fig.", "REF .", "The neutrino event number is smaller by factors of about two and ten.", "Thus, it is very important for the observation of preSN neutrino events to achieve low energy threshold.", "Owing to the large fiducial volume, Hyper-Kamiokande also would observe the change of burning processes during the final stage of massive star evolution.", "Figure REF shows the expected $\\bar{\\nu }_e$ events per ten minutes of the 15 $M_\\odot $ model by Hyper-Kamiokande.", "We see the decrease in the neutrino events before about one hour to the explosion and the observed neutrino events are similar to JUNO (see Fig.", "REF ).", "Observations of different types of neutrino detectors will increase the reliability of the neutrino events caused by processes during the massive star evolution." ], [ "Gd-loaded Super-Kamiokande and Hyper-Kamiokande", "The Gadzooks!", "project is an establishment of the identification of $p(\\bar{\\nu }_e,e^+)n$ event by neutron tagging by Gd [49].", "When small amount of Gd is contained in water Cherenkov detector, neutrons produced through the inverse beta-decay are captured by Gd and $\\gamma $ -rays with $\\sim 8$ MeV are released.", "The prompt signal by $e^-e^+$ -annihilation tagged by the $\\gamma $ -rays by the neutron capture is identified as this event.", "In preSN neutrinos, since the average $\\bar{\\nu }_e$ energy is below the observation threshold, most of the prompt signal will not be detected.", "However, the delayed signals are detectable.", "If many delayed signals are detected, these signals will be recognized as preSN neutrinos even after the corresponding SN explosion.", "We estimate preSN neutrino events detected by the delayed signal in Gd-loaded Super-Kamiokande and Hyper-Kamiokande.", "We assume the detection efficiency of 0.5 for Gd-loaded Super-Kamiokande and Hyper-Kamiokande.", "This efficiency roughly corresponds to the detection of lower energy region in the energy spectrum of the delayed $\\gamma $ -ray signals, since the $\\gamma $ -rays have the energy spectrum peaked at $\\sim 5$ MeV [50].", "The background events are considered to be the events below 5 MeV in the third phase solar-neutrino experiment [51].", "The average background events are 0.21 and 3.61 per hour for Super-Kamiokande and Hyper-Kamiokande, respectively.", "The event number of more than $3\\sigma $ significance is three and eleven per hour.", "We note that the assumption on the detection efficiency and the background is rough.", "The threshold for the neutrino energy is determined by $p(\\bar{\\nu }_e,e^+)n$ to be 1.8 MeV.", "Figure REF shows the integrated neutrino events by Hyper-Kamiokande with Gd.", "The neutrino events for seven days before the explosion in Super-Kamiokande and Hyper-Kamiokande are also listed in Table REF .", "Large fiducial volume and low energy threshold give many events.", "We assume here that the distance to the SN is 200 pc.", "Since the neutrino event number declines at inverse square of the distance, more than ten events are expected in the distance of $\\sim 3$ kpc for Hyper-Kamiokande.", "Thus, the detectable region of preSN neutrino events increases by Hyper-Kamiokande with Gd.", "We also investigate detailed time evolution of the preSN neutrino events.", "Figure REF shows the neutrino events of the 15 $M_\\odot $ model.", "We investigate neutrino events per one hour for Super-Kamiokande and per ten minutes for Hyper-Kamiokande.", "We see a peak around 17 hours before the explosion for Super-Kamiokande.", "The peak is expected to be detectable in the normal mass hierarchy.", "The neutrino events could be identified even in the inverted mass hierarchy.", "Combined with the observations by other neutrino detectors such as JUNO, we will confirm the time evolution of the neutrino events.", "In Hyper-Kamiokande, we could see a peak at 17 hours before the explosion more clearly and another peak around one hour before the explosion.", "These peaks appear just before the ignitions of the O shell burning and the Si shell burning.", "They are above $3\\sigma $ background events even in the inverted mass hierarchy.", "Thus, the time evolution of the neutrino events could constrain burning processes during the final evolution of massive stars.", "The observation using delayed neutrino signals by Hyper-Kamiokande with Gd is a powerful tool to observe preSN neutrinos.", "With this method, thousands of preSN neutrino events are expected.", "Although we do not show the SN alarm by Hyper-Kamiokande with Gd because of rough estimation of the background, we expect that the SN alarm earlier than JUNO and KamLAND is possible.", "After the investigation of the signal to noise ratio in Hyper-Kamiokande, we will discuss the possibility of long period preSN neutrino observations." ], [ "DUNE", "Deep Underground Neutrino Experiment (DUNE) is a neutrino experiment proposed in the United States [52].", "The main characteristic of this experiment is observing electron neutrinos using a massive liquid argon time-projection chamber.", "Four detectors with the fiducial mass of 10 kton will be constructed until around 2028, so the total fiducial mass is planned to be 40 kton.", "DUNE detector observes charged-current (CC) and neutral-current reactions of $^{40}$ Ar and electron scatterings.", "A charged current reaction $^{40}$ Ar($\\nu _e,e^-)^{40}$ K$^$ has the threshold of 1.5 MeV and its cross section is the largest among the reactions.", "We evaluate the expected electron neutrino events through the CC reaction of $^{40}$ Ar by DUNE.", "We assume that the fiducial mass of the detector is 40 kton and the threshold of the neutrino energy is 5 MeV.", "The cross section of $^{40}$ Ar($\\nu _e,e^-)^{40}$ K$^$ is adopted from [53] as numerical data in SNOwGLoBES http://www.phy.duke.edu/ schol/snowglobes/ (see also a software package GLoBES [55]).", "The transition probability of $\\nu _e \\rightarrow \\nu _e$ is evaluated as $P_{(\\nu _{e} \\rightarrow \\nu _{e})} = \\left\\lbrace \\begin{array}{ll}\\sin ^2\\theta _{13} = 0.024 & ({\\rm normal}) \\\\\\sin ^2\\theta _{12} \\cos ^2\\theta _{13} = 0.301 & ({\\rm inverted}).\\end{array}\\right.$ The expected neutrino events are 0.11–0.32 (0.16–0.48) for one day prior to the explosion in the normal (inverted) mass hierarchy.", "The event number from seven days to the previous day of the explosion is much smaller.", "Thus, we consider that it is quite difficult to observe the pair neutrino events by DUNE.", "However, we should note that electron neutrino events through electron captures of nuclei are not taken into account in this study.", "The neutrino luminosity by weak interactions by nuclei becomes larger than that by the pair neutrinos five minutes prior to the explosion in the 15 $M_\\odot $ model (see Fig.", "REF ).", "If the average neutrino energy is higher, the observed neutrino events will be much larger.", "We will investigate the neutrino events by weak interactions of nuclei in future study." ], [ "Neutrino events from Betelgeuse", "Betelgeuse is one of our neighboring red supergiants.", "The bolometric luminosity of this star is $\\log L/L_\\odot = 5.10 \\pm 0.22$ [4] and the effective temperature is $3641 \\pm 53$ K [56].", "The evolution of nonrotating and rotating massive stars like Betelgeuse has recently been discussed [57].", "The distance to Betelgeuse has been deduced as $197 \\pm 45$ pc combined with their very large array (VLA) radio positions, the published VLA positions, and the Hipparcos Intermediate Astrometric data [4].", "The distance has a large uncertainty and the uncertainty relates to uncertainties on the stellar luminosity and, thus, the stellar mass.", "In the cases of the shortest (152 pc) and longest (242 pc) distance, the stellar luminosity $\\log L/L_\\odot $ is estimated as 4.87 and 5.28, respectively.", "Compared with the final luminosity of stellar evolution models, we can estimate the initial mass range of the stellar evolution model (see Table REF ).", "In our models, the luminosity of 5.10 corresponds to about a 17 $M_\\odot $ star.", "The minimum and maximum luminosities correspond to about 13 and 20 $M_\\odot $ stars, respectively.", "If the distance to Betelgeuse is 200 pc, the initial stellar mass is expected to be 17 $M_\\odot $ in our models.", "The total neutrino events will be larger and the duration of the neutrino emission will be shorter than the case of the 15 $M_\\odot $ model.", "A SN alarm using preSN neutrinos could be sent within 24 hours prior to the explosion by KamLAND, SNO+, JUNO, and RENO-50.", "The evolution of shell burnings after the Si core burning could be observed by JUNO and RENO-50 if the mass hierarchy is normal.", "If the distance to Betelgeuse is $\\sim 150$ pc, the neutrino flux is 1.8 times as large as the case of 200 pc.", "On the other hand, the initial stellar mass is expected to be $\\sim 13 M_\\odot $ .", "From the integrated neutrino events of the three models, we consider that the total neutrino events could be larger than the case of a $\\sim 20 M_\\odot $ star at 200 pc.", "The estimation of the SN alarm time is difficult because the period of the Si core burning is longer for less massive stars.", "When the neutrino flux is 1.8 times larger, the alarm time for the 12 and 15 $M_\\odot $ models by KamLAND in the low-reactor phase is 21.7 (2.6) and 27.6 (17.1) hours before the explosion in the normal (inverted) mass hierarchy.", "We expect from the estimated alarm time that the SN alarm may be sent from KamLAND and SNO+ before the O shell burning starts even in the 13 $M_\\odot $ star in the case of the normal mass hierarchy.", "The closer distance also raises the possibility of the observations of the time evolution after the Si core burning.", "The evolution of shell burnings after the Si core burning could be observed more clearly by JUNO and RENO-50.", "Combined analysis of JUNO, RENO-50, and Hyper-Kamiokande could reveal evidence for the Si shell burning within one hour before the explosion.", "If Hyper-Kamiokande with Gd detects more preSN neutrino events, these events give constraints of burning processes from the Si core burning.", "If the distance to Betelgeuse is $\\sim 250$ pc, the neutrino flux is 0.64 times as large as the case of 200 pc, although the initial stellar mass is expected to be $\\sim 20 M_\\odot $ .", "In this case, the expected neutrino events by KamLAND are 7 and 3 in the normal and inverted mass hierarchies, respectively.", "So, the preSN neutrino events will be observed by KamLAND and SNO+ even in the distance to Betelgeuse of $\\sim 250$ pc.", "Larger events could be observed by JUNO, RENO-50, and Hyper-Kamiokande with a low energy threshold.", "We see from the right panel of Fig.", "REF that about twelve events per hour could be observed at $\\sim 8$ hours and, then, the event rate per hour could continue for three hours in the normal mass hierarchy.", "So, even in this distance, we could obtain the information in the central region of the preSN star from the neutrino events.", "We should note that the delayed alarm time by KamLAND may make it difficult to observe burning processes by JUNO and RENO-50.", "The SN alarm by KamLAND in the low-reactor phase will be sent at 5.9 (0.4) hours before the explosion in the normal (inverted) mass hierarchy.", "In this case, the O shell burning has started and it is difficult to observe it.", "If JUNO is in the low-reactor phase, it will send a SN alarm at 12.5 (10.0) hours before the explosion in the normal (inverted) mass hierarchy.", "This time is prior to a minimum of the neutrino events per hour in Fig.", "REF .", "Thus, the evidence for the burning processes after the Si core burning may be still possible to be observed by JUNO and RENO-50.", "Although stellar evolution models explain many stellar phenomena, there are still uncertainties in their parameters.", "The uncertainties will affect the predictions on the neutrino emission in the final stage of the evolution.", "The 15 $M_\\odot $ model in [11] used larger overshoot effect and different convection treatment.", "The CO-core mass in their model is larger than our model and, thus, the period from the Si core burning to the core collapse is shorter.", "They do not see clear decrease in the neutrino events by the O and Si shell burnings.", "The 15 $M_\\odot $ model in [12], originally in [16], seems to indicate the Si burning period similar to our model.", "The evidence for the Si shell burning is shown in around 1–2 hours before the explosion in the neutrino luminosity (see Fig.", "2 in [12]).", "A small dip of the neutrino luminosity around four hours before the explosion could be the evidence for the O shell burning.", "Even in a fixed stellar mass, the CO core mass has an uncertainty due to the material mixing by convection and stellar rotation.", "This uncertainty affects the time scale of the advanced evolution.", "The efficiency of the convection also affects the radial distribution of the neutrino emission through burning processes.", "The observation of preSN neutrinos will be the first direct observation of the central region of a collapsing massive star.", "If detailed time evolution of the neutrino events from an evolved massive star is observed, evolution signatures such as the period of the Si core burning and Si and/or O shell burnings could be revealed.", "We considered the MSW effect of neutrino oscillations in SN progenitors.", "However, we do not consider the Earth effect of neutrino oscillations.", "The time scale of preSN neutrinos is the order of day, which is much longer than SN neutrinos.", "The neutrino path in the Earth changes in one day period.", "We would like to take into account this effect in future study.", "Neutrino forward scattering in SN neutrinos changes the neutrino flavors (collective oscillations) and the final neutrino spectra (e.g., [58]).", "In preSN stage, since the neutrino flux is still much smaller than in the SN neutrinos, the effect of neutrino self-interaction is expected to be negligible for neutrino oscillations.", "It is an interesting problem when neutrino scattering becomes important for neutrino oscillations during the SN explosion.", "Although there is a large uncertainty in the distance to Betelgeuse, we expect in this study that neutrino events will be observed for several to tens hours before Betelgeuse explodes as a SN.", "The time of the SN alarm strongly depends on the distance, neutrino mass hierarchy, and the background of each neutrino detector.", "If the mass hierarchy is normal and the background of KamLAND and JUNO is low, the burning processes just before the SN explosion could be revealed through the preSN neutrino observations.", "Hyper-Kamiokande with Gd could observe thousands preSN neutrino events using delayed signals from the Gd+$n$ reaction despite the uncertainty of the distance.", "The neutrino events also become a constraint of the burning processes.", "The possibility of nearby SN explosions is expected to be one event per tens-thousand years; there are about ten evolved stars such as red supergiants and WR stars around a few hundred pc (e.g., [12]) and the lifetime of the He burning is 10$^5$ –10$^6$ years.", "This possibility is certainly small, but we hope that we observe the interior of the evolved massive star directly when a SN explodes in neighbors." ], [ "Relation between the neutrino emission from preSN stars and stellar structure", "We investigated the dependence of the neutrino emission of preSN stars on the initial stellar mass among the three models.", "The neutrino events and the period from the Si core burning to the collapse phase correlate with the initial mass and final mass.", "However, the final mass does not correlate with the initial mass for more massive stars because of larger mass loss effect.", "Further, even in the three models the Fe core mass does not correlate with the initial stellar mass.", "On the other hand, structural characteristics of preSN stars for the likelihood of explosion have been investigated.", "The compactness parameter [59] and the mass derivative at the location of the dimensionless entropy per nucleon of $s=4$ [60] are parameters indicating a likelihood of SN explosion.", "Here, we discuss whether these parameters become indicators of the neutrino emission from preSN stars.", "Figure: The time variation of the compactness parameters ξ 1.5 \\xi _{1.5} (solid lines) andξ 2.5 \\xi _{2.5} (dashed lines).Blue, red, and green lines correspond to the 12, 15, and 20 M ⊙ M_\\odot models, respectively.The compactness parameter $\\xi _M$ is defined by $\\xi _M \\equiv \\frac{M/M_\\odot }{R(M)/1000 {\\rm km}}$ at the core bounce, where $M$ is the specified mass coordinate and $R(M)$ is the corresponding radius.", "However, we are now interested in the neutrino emission during the Si core burning and later phase.", "So, we investigate the time variation of this parameter.", "We consider two cases of the compactness parameter: $M$ = 1.5 and 2.5 $M_\\odot $ .", "Figure REF shows the time variation of the compact parameters $\\xi _{1.5}$ and $\\xi _{2.5}$ .", "The value of $\\xi _{1.5}$ increases during core contraction and decreases during core and shell burnings among the three models.", "Although the final values of the parameter have a correlation to the neutrino emission of preSN stars, they do not reflect the structure outside the final Fe core.", "On the other hand, the time variation of $\\xi _{2.5}$ is much smaller and the values correlate with the neutrino emission: larger $\\xi _{2.5}$ indicates high neutrino emissivity and short time scale of neutrino emission.", "It was pointed out that $\\xi _{2.5}$ might have a correlation with the period from the core Si depletion to the core-collapse [61].", "So, $\\xi _{2.5}$ could be an indicator characterizing the neutrino emission from preSN stars.", "The relation between the normalized mass inside a dimensionless entropy per nucleon of $s=4$ , $M_4 \\equiv M(s=4)/M_\\odot $ , and the mass derivative at this location, $\\mu _{4} \\equiv (dM/M_\\odot )/(dr/ 1000 \\, {\\rm km}) \\vert _{s=4}$ , was considered in [60].", "The parameter $\\mu _4$ is linked to the accretion rate of matter [62].", "We investigated the time evolution of $M_4$ and $\\mu _4$ in the three models.", "In the 15 $M_\\odot $ model, $M_4$ changes from the location of the outer boundary of the O/Ne layer to the inner boundary of the O/Ne layer by the Ne shell burning during the Si shell burning.", "The change of $M_4$ during the evolution is also seen in the 20 $M_\\odot $ model.", "The $M_4$ value of the models before the Si core burning increases with the order of the 12, 20, and 15 $M_\\odot $ model but it increases with the 20, 15, and 12 $M_\\odot $ model at the last step.", "The mass derivative $\\mu _4$ shows a correlation with the stellar mass but the corresponding values change with the $M_4$ value.", "So, $M_4$ and $\\mu _4$ indicate characteristics at the collapsing phase rather than the Si burning.", "Thus, we consider that the compact parameter $\\xi _{2.5}$ indicates structure characteristics from the Si core burning to the collapsing phase and that it could characterize the neutrino emission from preSN stars.", "PreSN stars with larger $\\xi _{2.5}$ will emit more neutrinos with shorter time scale of the Si burning." ], [ "Conclusions", "We investigated the neutrino emission of preSN stars with the initial mass of 12, 15, and 20 $M_\\odot $ from the Si core burning until the core collapse.", "We showed for the first time detailed time variation of the neutrino emission relating the stellar evolution during the final stage.", "The neutrino emission rate and the neutrino average energy increase during the Si core burning and the collapsing stage.", "However, they decrease during the O and Si shell burnings.", "These nuclear burnings affect properties of the neutrino emission.", "In these three models, larger stellar mass model indicates stronger neutrino emission and shorter period from the Si burning to the core collapse.", "Thus, the observations of preSN neutrinos could constrain the structure and burning processes in the central region of a preSN star.", "Then, we estimated the neutrino events that will be observed by current and future neutrino detectors.", "After a massive star at $\\sim 200$ pc forms an Fe core, the neutrino emission becomes so strong that a few to tens electron antineutrinos will be observed by KamLAND.", "Future larger size scintillation detectors such as JUNO and RENO-50 could observe about 30 times as large as neutrino events than KamLAND.", "Hyper-Kamiokande could also observe hundreds neutrino events if low observable threshold for preSN neutrinos is achieved.", "We proposed for the first time a possibility of the observations of preSN neutrinos by Gd-loaded Hyper-Kamiokande using delayed $\\gamma $ -ray signals from the neutron capture of Gd.", "Although there are still many uncertainties in the threshold energy, detection efficiency, and the background events, Hyper-Kamiokande has a potential of observing thousands preSN neutrino events owing to the large fiducial mass and the energy range of the delayed $\\gamma $ -ray signals.", "If more than several neutrino events are observed in a unit time and the event number exceeds the background event number, detailed variation of the neutrino events could be observed.", "The decrease in the neutrino event rate will indicate the ignition of the O or Si shell burning.", "These neutrino observations will be the first direct observation of the central region of a fully evolved star.", "SN alarm using preSN neutrino events is possible by KamLAND, SNO+, JUNO, and RENO-50 in one day before the explosion.", "The alarm time will strongly depend on background events, especially on reactor neutrino experiments.", "Monitoring transient neutrino events by multiple neutrino observatories raises the reliability of the SN alarm.", "Betelgeuse is a red supergiant with the initial mass of 13–20 $M_\\odot $ .", "The distance is evaluated as $197 \\pm 45$ pc.", "Although the uncertainty in the distance is large, neutrino events could be observed by KamLAND, SNO+, JUNO, and RENO-50 for at most tens hours before the SN explosion.", "The time variation of the neutrino events per hour might be observed by JUNO and RENO-50 if the neutrino mass hierarchy is normal and the background by reactors is low.", "Hyper-Kamiokande with Gd could observe thousands neutrino events using the detection of delayed signals.", "This observation could reveal burning processes in the central region of Betelgeuse.", "We thank Chinami Kato and Shoichi Yamada for valuable discussions.", "We are grateful to Masayuki Nakahata, Yusuke Koshio, and Hajime Yano for valuable comments about neutrino experiments.", "This work has been partly supported by Grant-in-Aid for Scientific Research on Innovative Areas (26104007) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Grants-in-Aid for Scientific Research (24244028, 26400271) from Japan Society for the Promotion of Science." ] ]	1606.04915
[ [ "Minimization of Akaike's Information Criterion in Linear Regression\n Analysis via Mixed Integer Nonlinear Program" ], [ "Abstract Akaike's information criterion (AIC) is a measure of the quality of a statistical model for a given set of data.", "We can determine the best statistical model for a particular data set by the minimization of the AIC.", "Since we need to evaluate exponentially many candidates of the model by the minimization of the AIC, the minimization is unreasonable.", "Instead, stepwise methods, which are local search algorithms, are commonly used to find a better statistical model though it may not be the best.", "We propose a branch and bound search algorithm for a mixed integer nonlinear programming formulation of the AIC minimization by Miyashiro and Takano (2015).", "More concretely, we propose methods to find lower and upper bounds, and branching rules for this minimization.", "We then combine them with SCIP, which is a mathematical optimization software and a branch-and-bound framework.", "We show that the proposed method can provide the best statistical model based on AIC for small-sized or medium-sized benchmark data sets in UCI Machine Learning Repository.", "Furthermore, we show that this method finds good quality solutions for large-sized benchmark data sets." ], [ "Introduction", "Selecting the best statistical model from a number of candidate statistical models for a given set of data is one of the most important problems solved in statistical applications, e.g.", "regression analysis.", "This is called variable selection.", "The purposes of variable selection are to provide the simplest statistical model for a given data set and to improve the prediction performance while keeping the goodness-of-fit for a given data set.", "See [8] for more details on variable selection.", "In variable selection based on an information criterion, all the candidates are evaluated by the information criterion and select a statistical model by using those evaluations.", "Akaike's information criterion (AIC) is one of the information criteria and proposed in [3].", "An AIC value is computed for each candidate, and the model whose AIC value is the smallest is selected as the best statistical model.", "Since we often need to handle too many candidates of statistical models in practical applications, the global minimization based on AIC is not practical.", "Instead of the global minimization, stepwise methods, which are local search algorithms, are commonly used to find a statistical model which has as small AIC as possible, but it may not be the smallest.", "The contribution of our study is to propose a branch and bound search algorithm for a mixed integer nonlinear programming (MINLP) formulation of the minimization of AIC in linear regression by Miyashiro and Takano [12].", "Miyashiro and Takano [12] propose a mixed integer second-order cone programming (MISOCP) formulation from the MINLP formulation and solve the resulting problems by CPLEX [9], while we propose procedures to find lower and upper bounds of the MINLP problems and define branching rules for efficient computation.", "In addition, we provide an implementation to solve it efficiently via SCIP.", "SCIP is a mathematical optimization software and a branch-and-bound framework.", "SCIP has high flexibility of user plugin and control on various parameters in the branch-and-bound framework for efficient computation.", "We also propose an efficient computation for a set of data which has linear dependency.", "By applying our proposed method to benchmark data sets in [16], we can obtain the best statistical models for some of them.", "Our implementation is available at [18].", "We introduce some related work.", "Miyashiro and Takano [12] propose a MISOCP formulation for variable selection based on some information criteria in linear regression.", "Bertsimas and Shioda [6] and Bertsimas, King and Mazumder [5] provide a mixed integer quadratic programming (MIQP) formulation for linear regression with a cardinality constraint.", "Their formulation is available to our problems by fixing the number of explanatory variables.", "We compare our proposed method with MIQP and MISOCP formulations, and observe that our proposed method outperforms MIQP and MISOCP formulations.", "The organization of this manuscript is as follows: We give a brief introduction of linear regression based on AIC in Section 2.", "We introduce the MINLP formulation of the AIC minimization and ways to find lower and upper bounds used in the branch-and-bound framework in Section 3.", "Section 4 introduces techniques for more efficient computation, e.g.", "branching rules and treatment on data which has linear dependency.", "We present numerical results in Section .", "In particular, we show the numerical comparison with MISOCP and MIQP formulations.", "In addition, we present numerical performances of branching rules proposed in subsection REF .", "We discuss future work of our proposed method in Section .", "This manuscript is a full paper version of [10]." ], [ "Preliminary on Akaike's information criterion in linear regression", "We explain how to select the best statistical model via AIC in linear regression analysis.", "Linear regression is a fundamental statistical tool which determines coefficients $\\beta _0, \\ldots , \\beta _p\\in \\mathbb {R}$ for the following equation from a given set of data: $y = \\beta _0 + \\sum _{j=1}^p \\beta _j x_{j}.$ Here $x_1, \\ldots , x_p$ and $y$ are called the explanatory variables and the response variable respectively.", "In fact, we adopt coefficients $\\beta _0, \\ldots , \\beta _p$ which minimize $\\sum _{i=1}^n\\epsilon _i^2$ for a given set of data $(x_{i1}, \\ldots , x_{ip}, y_i)\\in \\mathbb {R}^p\\times \\mathbb {R} \\ (i=1, \\ldots , n)$ , where $\\epsilon _i$ is the $i$ th residual and defined by $\\epsilon _i = y_i - \\beta _0 -\\sum _{j=1}^p \\beta _jx_{ij}$ .", "Variable selection in linear regression is the problem to select the best subset of explanatory variables based on a given criterion.", "In statistical applications, a preferred model keeps the goodness-of-fit for a given data set, and contains as a few unnecessary explanatory variable as possible.", "In fact, unnecessary explanatory variables may add the noise to the prediction based on the statistical model.", "As a result, the prediction performance of the model may get worse.", "In addition, we need to observe and/or monitor more data for unnecessary explanatory variables, and thus will spend more cost due to the unnecessary explanatory variables.", "Akaike's information criterion (AIC) is one of criteria for variable selection and proposed in [3].", "AIC is used as a measure to select the preferred statistical model in all candidates.", "The statistical model whose AIC value is the smallest is expected as the preferred statistical mode.", "In linear regression analysis, this selection corresponds to the selection of a subset of the set of explanatory variables in (REF ) via AIC.", "More precisely, for a set $S\\subseteq \\lbrace 1, \\ldots , p\\rbrace $ of candidates of explanatory variables in the statistical model (REF ), AIC is defined in [3] as follows: $\\mbox{AIC}(S) = -2\\max _{\\beta , \\sigma ^2}\\lbrace \\ell (\\beta , \\sigma ^2) : \\beta _j = 0 \\ (j\\in \\lbrace 1, \\ldots , p\\rbrace \\setminus S)\\rbrace + 2(\\#(S) + 2)$ where $\\beta = (\\beta _0, \\ldots , \\beta _p)\\in \\mathbb {R}^{p+1}$ , $\\#(S)$ stands for the number of elements in the set $S$ and $\\ell (\\beta , \\sigma ^2)$ is the log-likelihood function.", "Computing AIC values for all subsets $S$ of the explanatory variables in (REF ), we can obtain the best AIC-based subset.", "However, since the number of subsets is $2^p$ , the computation of all subsets is not practical.", "Under assumption that all the residual $\\epsilon _i$ are independent and normally distributed with the zero mean and variance $\\sigma ^2$ , the log-likelihood function can be formulated as $\\ell (\\beta , \\sigma ^2) = -\\frac{n}{2}\\log (2\\pi \\sigma ^2) -\\frac{1}{2\\sigma ^2}\\sum _{i=1}^n\\epsilon _i^2.$ We focus on the first term in (REF ) to simplify (REF ).", "Let $S$ be a set of candidates of explanatory variables in (REF ).", "By substituting $\\beta _j = 0 \\ (j\\in \\lbrace 1, \\ldots , p\\rbrace \\setminus S)$ to the objective function, the first term can be regarded as the unconstrained minimization.", "Thus minimum solutions satisfy the following equation $\\frac{d \\ell }{d (\\sigma ^2)} = -\\frac{n}{2\\sigma ^2} +\\frac{1}{2(\\sigma ^2)^2}\\sum _{i=1}^n\\epsilon _i^2=0.$ From this equation, we obtain $\\sigma ^2 = \\frac{1}{n}\\sum _{i=1}^n\\epsilon _i^2$ .", "Substituting this equation to (REF ), we simplify (REF ) as follows: $\\mbox{AIC}(S) &= \\min _{\\beta _j}\\left\\lbrace n\\log \\left(\\sum _{i=1}^n\\epsilon _i^2\\right): \\beta _j = 0 \\ (j\\in \\lbrace 1, \\ldots , p\\rbrace \\setminus S)\\right\\rbrace \\\\&+ 2(\\#(S) + 2)+ n\\left(\\log (2\\pi /n) + 1\\right).", "\\nonumber $ We use (REF ) to provide our MINLP formulation of the minimization of AIC in the next section.", "The following lemma ensures that the minimization in the first term of (REF ) has an optimal solution with a finite value.", "Lemma 2.1 For any subset $S\\subseteq \\lbrace 1, \\ldots , p\\rbrace $ , the minimization in the first term of (REF ) has an optimal solution with a finite value.", "Since the logarithm function has the monotonicity, the optimal solution of the minimization in the first term of (REF ) is also optimal for the following unconstrained quadratic problem: $&\\min _{\\beta _j}\\left\\lbrace \\sum _{i=1}^n \\left(y_i - \\beta _0 -\\sum _{j\\in S}\\beta _j x_{ij}\\right)^2: \\beta _j \\in \\mathbb {R} \\ (j\\in \\lbrace 0\\rbrace \\cup S)\\right\\rbrace .$ Since the objective function of (REF ) is bounded below, it follows from [7] that (REF ) has an optimal solution." ], [ "MINLP formulation for the minimization of AIC", "We provide the minimization of $\\mbox{AIC}(S)$ over $S\\subseteq \\lbrace 1, \\ldots , p\\rbrace $ by the following MINLP formulation: $\\displaystyle \\min _{\\beta _j, z_j, \\epsilon _i, k} \\left\\lbrace n\\displaystyle \\log \\left(\\sum _{i=1}^n \\epsilon _i^2\\right) + 2k : \\begin{array}{l}\\epsilon _i = y_i-\\beta _0 -\\displaystyle \\sum _{j=1}^p\\beta _j x_{ij} \\ (i=1, \\ldots , n), \\\\\\displaystyle \\sum _{j=1}^p z_j = k, \\beta _0, \\beta _j\\in \\mathbb {R} \\ (j=1, \\ldots , p), \\\\z_j\\in \\lbrace 0, 1\\rbrace , z_j=0\\Rightarrow \\beta _j=0 \\ (j=1, \\ldots , p)\\end{array}\\right\\rbrace $ Here the last constraints represent the logical relationships, i.e.", "$\\beta _j$ has to be zero if $z_j=0$ .", "This formulation is provided in [12].", "Next we provide a procedure to find a lower bound of the subproblem of (REF ) at each node in the branch-and-bound tree.", "Some variables $z_j$ in (REF ) are fixed to zero or one at each node of the tree.", "We define the sets $Z_0$ , $Z_1$ and $Z$ for a given node as follows: $Z_1 &= \\lbrace j\\in \\lbrace 1, \\ldots , p\\rbrace : z_j \\mbox{ is fixed to } 1\\rbrace , Z_0 = \\lbrace j\\in \\lbrace 1, \\ldots , p\\rbrace : z_j \\mbox{ is fixed to } 0\\rbrace , \\\\Z &= \\lbrace j\\in \\lbrace 1, \\ldots , p\\rbrace : z_j \\mbox{ is not fixed}\\rbrace .$ We remark that $Z_1\\cup Z_0\\cup Z = \\lbrace 1, \\ldots , p\\rbrace $ and that each set is disjoint with one another.", "In other words, we can uniquely specify a node in the branch-and-bound search tree by $Z_1, Z_0$ and $Z$ .", "We denote the node by $V(Z_1, Z_0, Z)$ .", "Then the subproblem at the node $V(Z_1, Z_0, Z)$ is formulated as follows: $\\left\\lbrace \\begin{array}{cl}\\displaystyle \\min _{\\beta _j, z_j} & n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\displaystyle \\sum _{j=1}^p z_j \\\\\\mbox{subject to} & z_j = 1 \\ (j\\in Z_1), z_j = 0 \\ (j\\in Z_0), z_j\\in \\lbrace 0, 1\\rbrace \\ (j\\in Z), \\\\& \\beta _0, \\beta _j\\in \\mathbb {R} \\ (j=1, \\ldots , p), \\beta _j=0 \\ (j\\in Z_0) \\ z_j=0\\Rightarrow \\beta _j=0 \\ (j\\in Z)\\end{array}\\right.$ By relaxing the integrality of variables $z_j$ in (REF ), we obtain the following relaxation problem: $\\left\\lbrace \\begin{array}{cl}\\displaystyle \\min _{\\beta _j, z_j} & n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\displaystyle \\sum _{j=1}^p z_j \\\\\\mbox{subject to} & z_j = 1 \\ (j\\in Z_1), z_j = 0 \\ (j\\in Z_0), 0\\le z_j\\le 1 \\ (j\\in Z), \\\\& \\beta _0, \\beta _j\\in \\mathbb {R} \\ (j=1, \\ldots , p), \\beta _j=0 \\ (j\\in Z_0) \\ z_j=0\\Rightarrow \\beta _j=0 \\ (j\\in Z)\\end{array}\\right.$ Moreover we consider the following problem by eliminating all the logical relationships and all the $z_j$ : $\\displaystyle \\min _{\\beta _j} \\left\\lbrace n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\#(Z_1) :\\begin{array}{l}\\beta _0, \\beta _j\\in \\mathbb {R} \\ (j\\in Z\\cup Z_1), \\\\\\beta _j = 0 \\ (j\\in Z_0)\\end{array}\\right\\rbrace .$ It should be noted that the optimal value of (REF ) is the same as the optimal value of (REF ).", "Hence we deal with(REF ) as the relaxation problem of (REF ).", "In fact, for the optimal solution $\\beta ^$ of (REF ), we construct a sequence $\\lbrace (\\beta ^N, z^N)\\rbrace _{N=1}^{\\infty }$ as follows: $\\beta ^N = \\beta ^ \\mbox{ and } z_j^N = \\left\\lbrace \\begin{array}{cl}1 & \\mbox{ if } j \\in Z_1, \\\\1/N & \\mbox{ if } j\\in Z \\mbox{ and } \\beta ^N_j\\ne 0,\\\\0 & \\mbox{ if } j\\in Z \\mbox{ and } \\beta ^N_j = 0, \\\\0 & \\mbox{ if } j\\in Z_0,\\end{array}\\right.", "\\ (j=1, \\ldots , p)$ for all $N\\ge 1$ .", "Clearly, $(\\beta ^N, z^N)$ is feasible for (REF ) for all $N\\ge 1$ .", "It is sufficient to prove that the objective value $\\theta ^N$ of (REF ) at $(\\beta ^N, z^N)$ converges the optimal value $\\theta ^$ of (REF ) as $N$ goes to $\\infty $ .", "Since we have $\\theta ^ \\le \\theta ^N \\le n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0^* -\\sum _{j=1}^p\\beta _j^* x_{ij}\\right)^2\\right) + 2\\#(Z_1) + \\frac{2}{N}\\#(Z),$ the right-hand side converges to $\\theta ^$ as $N$ goes to $\\infty $ .", "This implies that the optimal value of (REF ) is the same as the optimal value of (REF ).", "Although the objective function of (REF ) contains the logarithm function, we can freely remove the constant $2\\#(Z_1)$ and the logarithm by the monotonicity of the logarithm function in (REF ), and thus obtain the following problem from (REF ): $\\displaystyle \\min _{\\beta _j} \\left\\lbrace \\sum _{i=1}^n\\left(y_i-\\beta _0 -\\displaystyle \\sum _{j=1}^p\\beta _j x_{ij}\\right)^2 : \\begin{array}{l}\\beta _0, \\beta _j\\in \\mathbb {R} \\ (j\\in Z\\cup Z_1)\\\\\\beta _j = 0 \\ (j\\in Z_0)\\end{array}\\right\\rbrace .$ Since (REF ) is the unconstrained minimization of a quadratic function, we can obtain an optimal solution of (REF ) by solving a linear system.", "In our implementation, we call dposv, which is a built-in function of LAPACK [4] for solving the linear system.", "We denote the optimal value of (REF ) by $\\xi ^$ .", "The optimal value of (REF ) is $n\\log (\\xi ^) + 2\\#(Z_1)$ , which is used as a lower bound of the optimal value of (REF ).", "We provide a procedure that constructs a feasible solution of (REF ) and computes an upper bound of the optimal value of (REF ).", "For this we use an optimal solution $\\tilde{\\beta }\\in \\mathbb {R}^{p+1}$ obtained after solving (REF ).", "We define $\\tilde{z}_j = \\left\\lbrace \\begin{array}{cl}1 & (\\mbox{if } j\\in \\tilde{Z}\\cup Z_1), \\\\0 & (\\mbox{otherwise})\\end{array}\\right.", "\\ (j=1, \\ldots , p), \\tilde{\\epsilon }_i=y_i-\\tilde{\\beta }_0-\\sum _{j=1}^p\\tilde{\\beta }_jx_{ij} \\ (i=1, \\ldots , n) \\mbox{ and } \\tilde{k}=\\sum _{j=1}^p\\tilde{z}_j,$ where $\\tilde{Z}=\\lbrace j\\in Z : \\tilde{\\beta }_j\\ne 0\\rbrace $ .", "It is easy to see that $(\\tilde{\\beta }_j, \\tilde{z}_j, \\tilde{\\epsilon }_i, \\tilde{k})$ is feasible for (REF ) and the objective value is $n\\log (\\xi ^) + 2\\#(\\tilde{Z}\\cup Z_1)$ .", "If the objective value is smaller than the current best upper bound, then we update the current best upper bound.", "Finally, we give another understanding for our proposed formulation and propose an efficient computation based on this understanding.", "Since we can regard (REF ) as linear regression whose explanatory variables are in $Z_1\\cup Z$ , the computation of the lower bound from (REF ) corresponds to the computation of the value $\\mbox{AIC}(Z_1\\cup Z) - 2\\#(Z)$ , while the upper bound corresponds to the AIC value of the statistical model whose explanatory variables are in $Z_1\\cup Z$ , i.e.", "$\\mbox{AIC}(Z_1\\cup Z)$ .", "Therefore, our proposed method computes the AIC value of the the statistical model with $Z_1\\cup Z$ at each node $V(Z_1, Z_0, Z)$ , up to constant term $4 + n(\\log (2n\\pi ) + 1)$ of (REF ).", "In summary, we consider that our proposed method branches and prunes the branch-and-bound search tree efficiently by using this understanding.", "The statistical package leaps [11] in R [14] adopts the branch-and-bound scheme in a similar manner.", "A QR decomposition is exploited at each node in the branch-and-bound search tree.", "In particular, leaps solve a linear system effectively by using the QR decomposition obtained at its parent node.", "leaps finds the best statistical model much faster than our proposed method for data sets whose $p$ is less than or equal to 32 and which do not have linear dependency introduced in subsection REF .", "If the data set has linear dependency, leaps does not work effectively, while our proposed method works more efficiently by using the linear dependency in data sets.", "This technique will be discussed in Section REF .", "We provide an efficient computation of lower and upper bounds based on this understanding.", "We assume that we obtain the lower and upper bounds at a node $V(Z_1, Z_0, Z)$ .", "Then we do not need to solve (REF ) at its child node $V(Z_1\\cup \\lbrace j\\rbrace , Z_0, Z\\setminus \\lbrace j\\rbrace )$ , where $j\\in Z$ .", "This node is generated by branching $z_j=1$ at the node $V(Z_1, Z_0, Z)$ .", "In fact, since we have $(Z_1\\cup \\lbrace j\\rbrace )\\cup (Z\\setminus \\lbrace j\\rbrace ) = Z_1\\cup Z$ , the relaxation problem (REF ) at the child node $V(Z_1\\cup \\lbrace j\\rbrace , Z_0, Z\\setminus \\lbrace j\\rbrace )$ is equivalent to one at the node $V(Z_1, Z_0, Z)$ .", "Thus the upper bound at the child node is the same as one at the node $V(Z_1, Z_0, Z)$ , and the lower bound is the lower bound computed at the node $V(Z_1, Z_0, Z)$ plus two because of $2\\#(Z_1\\cup \\lbrace j\\rbrace ) = 2\\#(Z_1) + 2$ ." ], [ "Some techniques to improve the numerical performance", "We describe some techniques to improve numerical performance to solve (REF )." ], [ "SCIP", "In order to implement our proposed method, we use SCIP [2], [13], [17], which is a mathematical optimization software and a branch-and-bound framework.", "In fact, it has high user plug-in flexibility which helps to solve (REF ) efficiently.", "We implement a procedure, which is called relaxator or relaxation handler, to obtain lower bounds as in Section .", "In addition, we also implement procedures to compute upper bounds via a method based on stepwise methods discussed in subsection REF and to define branching rules described in subection REF ." ], [ "Handling the linear dependency in data", "We illustrate that we can efficiently compute the optimal value of (REF ) by using the linear dependency in data.", "Although linearly independent data is often the assumption in standard statistical textbooks, practical data has often linear dependency, e.g.", "servo and auto-mpg in UCI Machine Learning Repository [16].", "For a set of given data $(x_{i1}, \\ldots , x_{ip}, y_i)\\in \\mathbb {R}^p\\times \\mathbb {R} \\ (i=1, \\ldots , n)$ , we denote $x^0 =\\left(\\begin{array}{l}1 \\\\\\vdots \\\\1\\end{array}\\right), x^j =\\left(\\begin{array}{l}x_{1j} \\\\\\vdots \\\\x_{nj}\\end{array}\\right) \\ (j=1, \\ldots , p).$ We say that data has linear dependent variables if the vectors $x^0, x^1, \\ldots , x^p\\in \\mathbb {R}^n$ are linearly dependent.", "From the definition of the linear dependency in data, we can reduce the computational cost for solving (REF ) when the data has linearly dependency.", "At a node $V(Z_1, Z_0, Z)$ , if there exists a subset $S\\subseteq Z_1\\cup Z$ such that the vectors $\\lbrace x^k : k\\in S\\cup \\lbrace 0\\rbrace \\rbrace $ , we can fix one of variables $z_j$ in $j\\in S\\cap Z$ to zero.", "In fact, since we have $\\sum _{j\\in S\\cup \\lbrace 0\\rbrace }\\alpha _j x^j = 0$ for some $(\\alpha _j)_{j\\in S\\cup \\lbrace 0\\rbrace }\\ne 0$ , we can removes one variable $z_j$ by substituting this equation to (REF ).", "This implies that the number of variables in (REF ) decrease, and thus we solve the linear equation with a fewer variables.", "Moreover we can prune some nodes efficiently by using the linear dependency.", "The following lemma ensures that we do not need to branch $z_q=1$ for some $q\\in Z$ if the data has the linear dependency.", "Thus we need to handle only $z_q = 0$ in this case.", "Lemma 4.1 Assume that in (REF ), there exists $q\\in Z$ such that the vector $x^q$ and vectors $\\lbrace x^j : j\\in Z_1\\cup \\lbrace 0\\rbrace \\rbrace $ are linearly dependent.", "Then an optimal solution of (REF ) satisfies $z_q = 0$ .", "Let $(\\tilde{\\beta }_j, \\tilde{z}_j)$ be an optimal solution of (REF ), and $\\theta ^$ be the optimal value of (REF ).", "Suppose that $\\tilde{z}_q = 1$ .", "It follows from the assumption that there exists $\\alpha _j\\in \\mathbb {R} \\ (j\\in Z_1\\cup \\lbrace 0\\rbrace )$ such that $(\\alpha _j)_{j\\in Z_1\\cup \\lbrace 0\\rbrace }\\ne 0$ and $x^q = \\sum _{j\\in Z_1\\cup \\lbrace 0\\rbrace } \\alpha _j x^j.$ Then the following solution $(\\hat{\\beta }_j, \\hat{z}_j)$ is also feasible for (REF ): $\\hat{\\beta }_j =\\left\\lbrace \\begin{array}{cl}\\tilde{\\beta }_j + \\tilde{\\beta }_q\\alpha _j & (\\mbox{if }j\\in (Z\\setminus \\lbrace q\\rbrace )\\cup Z_1\\cup \\lbrace 0\\rbrace ), \\\\0 & (\\mbox{otherwise})\\end{array}\\right.\\mbox{ and }\\hat{z}_j = \\left\\lbrace \\begin{array}{cl}1 & (\\mbox{if }j\\ne q \\mbox{ and }\\tilde{z}_j = 1), \\\\0 & (\\mbox{otherwise})\\end{array}\\right.$ The objective value of (REF ) at $(\\hat{\\beta }_j, \\hat{z}_j)$ is $\\theta ^-2$ , which contradicts the optimal value $\\theta ^$ .", "A given set of data which has linear dependency satisfies the assumption of Lemma REF .", "In fact, there exists a subset $S\\subseteq \\lbrace 1, \\ldots , p\\rbrace $ such that the vectors $\\lbrace x^k : k\\in S\\cup \\lbrace 0\\rbrace \\rbrace $ are linearly dependent.", "Hence Lemma REF ensures that we do not need to generate a child node by branching $z_q = 1$ at a node $V(Z_1, Z_0, Z)$ when $q\\in S\\cap Z$ and $S\\setminus \\lbrace q\\rbrace \\subseteq Z_1$ .", "In addition, if there exists a subset $S\\subseteq \\lbrace 1, \\ldots , p\\rbrace $ such that for every $j\\in S$ , the vectors $\\lbrace x^k : k\\in \\lbrace 0\\rbrace \\cup (S\\setminus \\lbrace j\\rbrace )\\rbrace $ are linearly dependent, then we can prune some nodes before applying our proposed method to (REF ).", "In fact, it follows from the assumption on $S$ that for every $j\\in S$ we do not need to branch $z_j=1$ at the node $V(Z_1, Z_0, Z)$ .", "This implies that optimal solutions of (REF ) satisfy the following linear inequality: $\\sum _{j\\in S} z_j \\le \\#(S) -1.$ By adding this inequality in (REF ), we do not generate any nodes in which $S\\subseteq Z_1$ hold.", "We execute a greedy algorithm in Algorithm REF to find a collection $\\mathcal {C}$ of such sets $S$ .", "[H] Data $x^0$ , $x^1, x^2, \\ldots x^p\\in \\mathbb {R}^n$ A collection $\\mathcal {C}$ of sets of linearly dependent vectors $\\mathcal {C}\\longleftarrow \\emptyset $ , $S\\longleftarrow \\emptyset $ $j\\rightarrow 0$ $p$ the vectors $\\lbrace x^j : j\\in \\lbrace 0\\rbrace \\cup S\\cup \\lbrace j\\rbrace \\rbrace $ is linearly independent $S\\longleftarrow S\\cup \\lbrace j\\rbrace $ Solve the following linear equation: $\\sum _{k\\in S\\cup \\lbrace 0\\rbrace }\\alpha _k x^k&= x^j.$ $S^{\\prime }\\longleftarrow \\lbrace k\\in S : \\alpha _k\\ne 0\\rbrace $ , $\\mathcal {C}\\longleftarrow \\mathcal {C}\\cup \\lbrace S^{\\prime }\\rbrace $ $\\mathcal {C}$ A greedy algorithm to find a collection of sets of linearly dependent vectors We remark that the linear equation (REF ) has a unique solution because the matrix $(x^k)_{k\\in S\\cup \\lbrace 0\\rbrace }$ is of full column rank." ], [ "Computation of upper bounds based on stepwise methods", "Although we mainly use the procedure described in Section to compute upper bounds, we also use the stepwise methods with forward selection (SW$_+$ ) and backward elimination (SW$_-$ ).", "SW$_+$ starts with no explanatory variables and adds one explanatory variable at a time until the AIC value does not decrease.", "More precisely, for the current set $S$ of explanatory variables, we choose an explanatory variable whose the AIC value $\\mbox{AIC}(S\\cup \\lbrace j\\rbrace )$ is minimized over $j \\in \\lbrace 1, \\ldots , p\\rbrace \\setminus S$ .", "SW$_-$ is just the reverse of SW$_+$ .", "It starts with all explanatory variables and remove one explanatory variable at a time until the AIC value does not decrease.", "Note that since these methods add or remove one explanatory variable at a time, they may miss the best statistical model.", "In this sense, we can say that they are local search algorithms for variable selection.", "We describe our heuristics to computer an upper bound in more details in Algorithm REF .", "To this end, we define $S\\subseteq Z_1\\cup Z$ for subproblem (REF ) and consider the following problem: $& \\left\\lbrace \\begin{array}{cl}\\displaystyle \\min _{\\beta _j, z_j} & n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\displaystyle \\sum _{j=1}^p z_j \\\\\\mbox{subject to} & \\beta _0, \\beta _j\\in \\mathbb {R}, z_j = 1 \\ (j\\in S), \\beta _j=0, z_j = 0 \\ (j\\in \\lbrace 1, \\ldots , p\\rbrace \\setminus S)\\end{array}\\right.$ We denote the optimal value and an optimal solution of (REF ) by $\\bar{\\theta }_S$ and $(\\bar{\\beta }_S, \\bar{z}_S)$ , respectively.", "[H] $Z_1, Z_0$ and $Z$ A feasible solution $(\\beta , z)$ of (REF ) Stepwise method with forward selection $S\\longleftarrow Z_1$ , $v_f\\longleftarrow \\infty $ $\\bar{\\theta }_S < v_f$ $v_f\\longleftarrow \\bar{\\theta }_S$ , $(\\beta _f, z_f)\\longleftarrow (\\bar{\\beta }_S, \\bar{z}_S)$ Find $j\\in Z\\setminus S$ such that $\\bar{\\theta }_{S\\cup \\lbrace j\\rbrace }$ is minimized over all $j\\in Z\\setminus S$ $S\\longleftarrow S\\cup \\lbrace j\\rbrace $ Stepwise method with backward elimination $S\\longleftarrow Z_1\\cup Z$ , $v_b\\longleftarrow \\infty $ $\\bar{\\theta }_S < v_b$ $v_b\\longleftarrow \\bar{\\theta }_S$ , $(\\beta _b, z_b)\\longleftarrow (\\bar{\\beta }_S, \\bar{z}_S)$ Find $j\\in Z\\cap S$ such that $\\bar{\\theta }_{S\\setminus \\lbrace j\\rbrace }$ is minimized over all $j\\in Z\\cap S$ $S\\longleftarrow S\\setminus \\lbrace j\\rbrace $ $v_f < v_b$ $(\\beta _f, z_f)$ $(\\beta _b, z_b)$ Stepwise methods to compute an upper bound We remark that an optimal solution of (REF ) is feasible for the subproblem (REF ) if $Z_1\\subseteq S$ .", "Since $S$ always contains $Z_1$ in Algorithm REF , the returned solution $(\\beta , z)$ is feasible for (REF ).", "In addition, we set $Z_1$ as the initial set of SW$_+$ instead of the empty set because we execute Algorithm REF at the node $V(Z_1, Z_0, Z)$ .", "Similarly, we set $Z_1\\cup Z$ as the initial set of SW$_-$ .", "These are different from the original stepwise methods.", "In statistical applications, instead of finding the global minimum of (REF ), stepwise methods, which are local search algorithms, are commonly used in practice.", "In fact, they often find a better statistical model and work effectively in our implementation.", "However since stepwise methods spend more computational costs than the procedure described in Section , we apply Algorithm REF to only subproblem (REF ) at the node whose depth from the root node is less than or equal to 10 in our implementation." ], [ "Most frequent branching and Strong branching", "We define two branching rules for variables $z_j$ to improve the performance of our implementation.", "The first one is called most frequent branching and uses all stored feasible solutions in the procedure to compute upper bounds.", "The second one is called strong branching.", "This is based on the strong branching rule in [1].", "We propose a more efficient computation for the strong branching rule than [1].", "We will show the numerical comparison with branching rules implemented in SCIP in subsection REF .", "We will observe from the numerical results that most frequent branching is effective for a set data which has linear dependency, while strong branching is effective for a set data which does not have linear dependency.", "The most frequent branching is based on the tendency that some explanatory variables adopted for the best statistical model are also used in statistical models whose AIC value is close to the smallest AIC value.", "By branching variables $z_j$ in (REF ) which correspond to such explanatory variables, we can expect that (REF ) at the node generated by $z_j = 0$ is pruned as early as possible.", "To find such explanatory variables, we use feasible solution stored in our procedure to compute upper bounds.", "We describe the most frequent branching rule at the current node in Algorithm REF .", "[H] A positive integer $N$ , a set $Z$ of unfixed variables in the node and all feasible solutions of (REF ) found from the root node through the current node $J\\in Z$ Choose $N$ feasible solutions $(\\beta ^1, z^1), \\ldots , (\\beta ^N, z^N)$ out of all stored feasible solutions Here $(\\beta ^i, z^i)$ is a feasible solution of (REF ) whose objective value is the $i$ th smallest in all the stored solutions $j\\in Z$ Compute score value $s_j$ defined by $s_j = \\#(T_j)$ , where $T_j = \\lbrace \\ell \\in \\lbrace 1, \\ldots , N\\rbrace : z^{\\ell }_j = 1\\rbrace $ $J\\in Z$ with $s_{J}=\\displaystyle \\max _{j\\in Z }\\lbrace s_j\\rbrace $ Most frequent branching rule We observe in our preliminary numerical experiment that the obtained lower bound at the child node generated by $z_{J}=0$ tends to be relatively bigger and that the pruning process tends to work earlier in comparison to branching rules of SCIP.", "As a result, our proposed method with the most frequent branching rule often visits a fewer nodes in the branch-and-bound tree.", "In the strong branching rule, we compute lower bounds for all possible branching $z_k=1$ and $z_k=0$ , and choose index $k\\in Z$ so that the lower bound is maximized in all computed lower bounds.", "More precisely, for the subproblem (REF ) at a node $V(Z_1, Z_0, Z)$ and $k\\in Z$ , the relaxation problem of the subproblem branched by $z_k=1$ and $z_k=0$ can be formulated as (REF ) and () as follows, respectively.", "$& \\left\\lbrace \\begin{array}{cl}\\displaystyle \\min _{\\beta _j} & n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\#(Z_1\\cup \\lbrace k\\rbrace ) \\\\\\mbox{subject to} &\\beta _0, \\beta _j\\in \\mathbb {R} \\ (j\\in (Z\\setminus \\lbrace k\\rbrace )\\cup (Z_1\\cup \\lbrace k\\rbrace )), \\beta _j = 0 \\ (j\\in Z_0)\\end{array}\\right.", "\\\\& \\left\\lbrace \\begin{array}{cl}\\displaystyle \\min _{\\beta _j} & n\\displaystyle \\log \\left(\\sum _{i=1}^n\\left(y_i-\\beta _0 -\\sum _{j=1}^p\\beta _j x_{ij}\\right)^2\\right) + 2\\#(Z_1) \\\\\\mbox{subject to} & \\beta _0, \\beta _j\\in \\mathbb {R} \\ (j\\in (Z\\setminus \\lbrace k\\rbrace )\\cup Z_1), \\beta _j = 0 \\ (j\\in Z_0\\cup \\lbrace k\\rbrace )\\end{array}\\right.$ Since we have $(Z\\setminus \\lbrace k\\rbrace )\\cup (Z_1\\cup \\lbrace k\\rbrace )=Z\\cup Z_1$ , the optimal value of (REF ) for all $k\\in Z$ is $\\theta ^+2$ , where $\\theta ^$ is the optimal value of (REF ) at a node $V(Z_1, Z_0, Z)$ .", "Hence we select an index $k\\in Z$ only from all optimal values $\\theta ^_k$ of ().", "We describe the strong branching rule at the current node in Algorithm REF .", "[H] Subproblem (REF ) in the node $V(Z_1, Z_0, Z)$ $J\\in Z$ $k\\in Z$ Solve () with $k$ and obtain optimal value $\\theta ^_k$ $J\\in Z$ with $\\theta ^_J = \\displaystyle \\max _{k\\in Z}\\left\\lbrace \\theta ^_k\\right\\rbrace $ Strong branching rule" ], [ "Numerical experiments", "We implement our approach and procedures discussed in Sections and , and apply our implementationThis is available at [18].", "to benchmark data sets in [16].", "We apply our implementation to standardized data sets, i.e.", "the data is transformed to have the zero mean and unit variance.", "Note that the standardized data has also linear dependency even if we apply the standardization to the original data which has linear dependency.", "The specification of the computer is CPU : 3.5 GHz Intel Core i7, Memory : 16GB and OS : OS X 10.9.5.", "In subsections REF and REF , we adopt the most frequent branching rule for data which has linear dependency, while we adopt the strong branching for data which does not have linear dependency.", "In subsection REF , we discuss the reason why we use the different branching rules." ], [ "Comparison with stepwise methods and MISOCP approach", "We compare our proposed method with stepwise methods (SW$_+$ and SW$_-$ ) and the MISOCP approach proposed in [12] via CPLEX [9].", "This approach is also obtained from (REF ).", "Although the objective function of (REF ) is non-convex, the difficulty due to the non-convexity is overcome by using the identity $\\exp (\\log (x)) = x$ and the monotonicity of the exponential function $\\exp (x)$ .", "See [12] for the detail.", "The resulting problem is formulated as MISOCP and is tractable by CPLEX.", "Table REF shows the summary of numerical comparisons.", "The mark $\\bullet $ in the first column indicates that the data has linear dependency.", "The second, third, and sixth columns indicate the numbers of data, the explanatory variables in the statistical model (REF ), and the ones in the models found by using each method.", "The fifth column indicates the obtained AIC values by each method.", "The values with the bold font are the best among four values.", "The seventh column indicates the cpu time in seconds to compute the optimal value.", "“$>$ 5000\" means that the corresponded method cannot find the optimal value within 5000 seconds.", "The last column indicates the gap in the percent as follows: $\\mbox{gap} =\\displaystyle \\frac{\\mbox{upper bound} - \\mbox{lower bound}}{\\max \\lbrace 1, \|\\mbox{upper bound}\|\\rbrace }\\times 100.$ It should be noted that if the gap is sufficiently close to zero, then the obtained value is optimal.", "MINLP, MISOCP, SW$_+$ and SW$_-$ indicate the results obtained by our proposed method, MISOCP approach and the stepwise method with forward selection and backward elimination, respectively.", "We observe the following from Table REF .", "MINLP computes the optimal value much faster than MISOCP.", "MINLP finds smaller AIC values than MISOCP even when MINLP cannot find them within 5000 seconds.", "The AIC value obtained by SW$_+$ or SW$_-$ is equal to one by MINLP, i.e.", "crime and forestfires.", "In fact, as we mentioned in subsection REF , we use stepwise methods in some nodes in our implementation.", "This implies that our procedure to compute an upper bound discussed in Section cannot find better feasible solutions than ones by the stepwise methods.", "Table: Summary of numerical results by MINLP, MISOCP, SW + _+ and SW - _-" ], [ "Comparison of branching rules", "We compare the numerical performance of the most frequent branching and strong branching with branching rules implemented in SCIP.", "In Table REF , Std, MFB and SB stand for numerical results by the branching rules in SCIP, the most frequent branching rule and the strong branching rule.", "The sixth column indicates the number of visited nodes by our proposed method with the applied branching rule.", "The values with the bold font are the best among three values.", "We observe from Table REF : The most frequent branching rule works more effectively than other ones for sets of data which have linear dependency.", "In fact, the gap by the most frequent branching rule is the smallest and the computation time is the shortest.", "In addition, The number of the visited nodes by the most frequent branching is also smaller than other branching rules.", "In contrast, the strong branch is more efficient than other branching rules for data which do not have linear dependency.", "For $p\\le 32$ , the gap obtained by the best branching rule is the smallest in three branching rules, though it visits fewest nodes in the branch-and-bound tree.", "This means that the best branching computes tighter lower bounds than other branching rules.", "These are the reasons why we use different branching rules in Tables REF and REF .", "Table: Summary of numerical results by branching rules in SCIP (Std), the most frequent branching (MFB) and strong branching (SB)" ], [ "Comparison with MIQP formulation", "Bertsimas and Shioda [6] and Bertsimas et al [5] provide a mixed integer quadratic programming (MIQP) formulation with a cardinality constraint for linear regression.", "Their formulation is available to the minimization of AIC by fixing the number of explanatory numbers from 0 to $p$ .", "In fact, the minimization can be equivalently reformulated as follows: $& \\displaystyle \\min _{k=0, \\ldots , p}\\min _{S \\subseteq \\lbrace 1, \\ldots , p\\rbrace }\\left\\lbrace \\mbox{AIC}(S) : \\#(S) = k\\right\\rbrace .$ Since each inner optimization problem in (REF ) can be formulated as a MIQP problem, we can obtain the best statistical model by solving all $(p+1)$ optimization problems.", "In this subsection, we introduce a MIQP formulation by Bertsimas and Shioda [6] and Bertsimas et al [5] for the inner optimization problems in (REF ).", "In addition, we provide a more efficient algorithm than this naive algorithm and compare the algorithm with our proposed method.", "Each inner optimization problem in (REF ) can be reformulated as follows: $&\\displaystyle \\min _{\\beta _j}\\left\\lbrace n\\displaystyle \\log \\left(\\sum _{i=1}^n \\epsilon _i^2\\right) + 2k :\\begin{array}{l}\\epsilon _i = y_i-\\beta _0 -\\displaystyle \\sum _{j=1}^p\\beta _j x_{ij} \\ (i=1, \\ldots , n), \\\\\\displaystyle \\sum _{j=1}^p z_j = k, \\beta _0, \\beta _j\\in \\mathbb {R}\\ (j=1, \\ldots , p), \\\\z_j\\in \\lbrace 0, 1\\rbrace , z_j=0\\Rightarrow \\beta _j=0 \\ (j=1, \\ldots , p)\\end{array}\\right\\rbrace $ For any fixed $k$ , since the logarithm function in (REF ) has the monotonicity, we can find an optimal solution (REF ) by solving the following quadratic programming problem: $&\\displaystyle \\min _{\\beta _j}\\left\\lbrace \\displaystyle \\sum _{i=1}^n\\left(y_i-\\beta _0 -\\displaystyle \\sum _{j=1}^p\\beta _j x_{ij}\\right)^2 :\\begin{array}{l}\\displaystyle \\sum _{j=1}^p z_j = k, \\beta _0\\in \\mathbb {R}, \\\\z_j\\in \\lbrace 0, 1\\rbrace \\ (j=1, \\ldots , p), \\\\z_j=0\\Rightarrow \\beta _j=0 \\ (j=1, \\ldots , p)\\end{array}\\right\\rbrace $ (REF ) is a MIQP formulation.", "We denote the optimal value of (REF ) by $\\eta ^_k$ .", "If (REF ) is infeasible, we set $\\eta ^_k=+\\infty $ .", "Then the optimal value of inner problem (REF ) with $k$ is $n\\log (\\eta ^_k) + 2k$ .", "Therefore we obtain the optimal value and solution of (REF ) by computing all optimal values of (REF ) for $k=0, \\ldots , p$ .", "We describe the naive algorithm in Algorithm REF .", "[H] Minimization of AIC (REF ) An optimal solution of (REF ) $k\\rightarrow 0$ $p$ Find the optimal value $\\eta ^_k$ and an optimal solution $(\\beta ^_k, z^_k)$ of (REF ) with $k$ Find an index $K$ with $\\theta ^_{K} = \\displaystyle \\min _{k=0, \\ldots , p}\\lbrace n\\log (\\eta ^_k) + 2k\\rbrace $ $(\\beta ^_{K}, z^_{K})$ Naive algorithm for (REF ) via MIQP The following lemma ensures that we can find an upper bound of $k$ if we have a feasible solution of (REF ).", "Lemma 5.1 Let $\\hat{\\theta }\\in \\mathbb {R}^{p+1}$ be the optimal value of the following optimization problem: $& \\displaystyle \\min _{\\beta _j}\\left\\lbrace n\\log \\left(\\sum _{i=1}^n \\left(y_i - \\beta _0 - \\sum _{j=1}^p \\beta _{j}x_{ij}\\right)^2\\right) : \\beta _0, \\beta _j\\in \\mathbb {R} \\ (j=1, \\ldots , p)\\right\\rbrace .$ In addition, $\\bar{\\theta }$ is the objective value of (REF ) at a feasible solution of (REF ).", "Then any optimal solution $(\\beta ^, z^)$ of (REF ) satisfies $\\sum _{j=1}^p z_j^ \\le \\left\\lfloor \\frac{\\bar{\\theta }-\\hat{\\theta }}{2}\\right\\rfloor .$ Let $\\theta ^$ be the optimal value of (REF ) and $(\\beta ^, z^)$ be an optimal solution of (REF ).", "Then we have $\\bar{\\theta } \\ge \\theta ^= n\\log \\left(\\sum _{i=1}^n \\left(y_i - \\beta _0^* - \\sum _{j=1}^p \\beta _{j}^x_{ij}\\right)^2\\right) + 2\\sum _{j=1}^p z_j^\\ge \\hat{\\theta } + 2\\sum _{j=1}^p z_j^,$ and thus we have $\\sum _{j=1}^p z_j^ \\le (\\bar{\\theta }-\\hat{\\theta })/2$ .", "Since $z_j^$ is integer, we obtain the desired result.", "We describe an algorithm based on Lemma REF in Algorithm REF .", "[H] Minimization of AIC (REF ) An optimal solution of (REF ) Solve (REF ) and obtain $\\hat{\\theta }$ $\\bar{\\theta }\\longleftarrow +\\infty $ $k\\rightarrow 0$ $p$ $k> \\left\\lfloor \\frac{\\bar{\\theta }-\\hat{\\theta }}{2}\\right\\rfloor $ Stop Find the optimal value $\\eta ^_k$ and solution $(\\beta ^_k, z^_k)$ of (REF ) with $k$ $\\bar{\\theta }\\ge n\\log (\\eta ^_k) + 2k$ $\\bar{\\theta }\\longleftarrow n\\log (\\eta ^_k) + 2k$ , $(\\beta ^, z^)\\longleftarrow (\\beta ^_k, z^_k)$ $(\\beta ^, z^)$ Faster algorithm for (REF ) via MIQP We give details on our numerical experiment.", "We solve (REF ) by CPLEX.", "In particular, since the last constraints in (REF ) represent the logical relationship between $z_j$ and $\\beta _j$ , we use indicator implemented in CPLEX to represent these constraints.", "We add linear inequalities in (REF ) by applying Lemma REF to (REF ) when a given set of data has linear dependency.", "See subsection REF for the detail.", "We also solve optimization problems obtained by replacing the constraint $\\sum _{j=1}^p z_j = k$ by $\\sum _{j=1}^p z_j \\le k$ in (REF ).", "In Table REF , “Fast$\\le $ \" indicates that we solve those problems in Algorithm REF , while “Fast$=$ \" indicates that we solve (REF ) in Algorithm REF .", "By this replacement, we can use an optimal solution $(\\beta ^_k, z_k^)$ of the optimization problem with $k$ to compute an upper bound of the optimization problem with $k+1$ .", "We terminate if the corresponded method cannot find the best AIC value within 5000 seconds.", "In addition, the values with the bold font are the best among four values except for “$>$ 5000\" in the last column.", "We provide numerical results on our proposed method, Algorithms REF and REF in Table REF .", "We observe the following from Table REF : MINLP outperforms MIQP approaches.", "In particular, for larger $p$ , MINLP obtains much better AIC values than MIQP approaches although all approaches cannot solve within 5000 seconds.", "The performance of Fast$\\le $ is similar to Fast$=$ , though Fast$\\le $ uses an initial upper bound.", "Table: Summary of numerical results by MINLP, Naive (Algorithm ), Fast== and Fast≤\\le (Algorithm )" ], [ "Conclusion", "We propose the MINLP formulation (REF ) of AIC minimization for linear regression, and implement it by using SCIP.", "We formulate an unconstrained optimization problem (REF ) as the relaxation problem of the subproblem (REF ).", "As a result, a lower bound can be computed by solving a linear equation at each node.", "In addition, an upper bound is the lower bound plus a constant, and a feasible solution is generated from a solution after solving the relaxation problem (REF ).", "We implement this procedure with SCIP because it has the high flexibility in the user plugin.", "In fact, we implement a relaxator to compute lower and upper bounds, and two branching rules to prune subproblems efficiently.", "In addition, our implementation efficiently prunes and branches subproblems by using linear dependency in data set and two branching rules.", "As a result, we can obtain the best statistical models (REF ) for $p\\le 32$ .", "In addition, we observe that our implementation outperforms MISOCP approach [12] and MIQP approaches [6], [5] in our numerical experiments.", "Future work involves to apply our implementation to data sets with larger $p$ and/or $n$ .", "A possible choice to accomplish this involves the use of parallel computation via ParaSCIP and FiberSCIP [15].", "Secondly, various non-AIC information criterion, e.g.", "BIC and Hannan-Quinn information criteria are already proposed.", "By changing the objective function in (REF ), our proposed method can be applied to these information criteria as well." ], [ "Acknowledgements", "The second author was supported by JSPS KAKENHI Grant Numbers 26400203." ] ]	1606.05030
[ [ "The width of the Roper resonance in baryon chiral perturbation theory" ], [ "Abstract We calculate the width of the Roper resonance at next-to-leading order in a systematic expansion of baryon chiral perturbation theory with pions, nucleons, and the delta and Roper resonances as dynamical degrees of freedom.", "Three unknown low-energy constants contribute up to the given order.", "One of them can be fixed by reproducing the empirical value for the width of the Roper decay into a pion and a nucleon.", "Assuming that the remaining two couplings of the Roper interaction take values equal to those of the nucleon, the result for the width of the Roper decaying into a nucleon and two pions is consistent with the experimental value." ], [ "Introduction", "At low energies, chiral perturbation theory [1], [2] provides a successful description of the Goldstone boson sector of QCD.", "It turns out that a systematic expansion of loop diagrams in terms of small parameters in effective field theories (EFTs) with heavy degrees of freedom is a rather complicated issue.", "The problem of power counting in baryon chiral perturbation theory [3] may be solved by using the heavy-baryon approach [4], [5], [6] or by choosing a suitable renormalization scheme [7], [8], [9], [10].", "The $\\Delta $ resonance and (axial) vector mesons can also be included in EFT (see e.g.", "Refs.", "[11], [12], [13], [14], [15], [16], [17], [18], [19], [20]).", "On the other hand, the inclusion of heavier baryons such as the Roper resonance is more complicated.", "Despite the fact that the Roper resonance was found a long time ago in a partial wave analysis of pion-nucleon scattering data [21], a satisfactory theory of this state is still missing.", "The Roper is particularly interesting as it is the first nucleon resonance that exhibits a decay mode into a nucleon and two pions, besides the decay into a nucleon and a pion.", "Also, the Roper appears unexpectedly low in the spectrum, below the first negative parity nucleon resonance, the $S_{11}(1535)$ .", "It is therefore timely to address this state in a chiral EFT.", "First steps in this direction have been made in Refs.", "[22], [23], [24], [25], [26].", "In this work we calculate the width of the Roper resonance in a systematic expansion in the framework of baryon chiral perturbation theory with pions, nucleons, the delta and Roper resonances as explicit degrees of freedom.", "The paper is organised as follows: in Section we specify the effective Lagrangian, in Section the pole mass and the width of the Roper resonance are defined and the perturbative calculation of the width is outlined in Section .", "In Section we discuss the renormalization and the power counting applied to the decay amplitude of the Roper resonance, while Section contains the numerical results.", "We briefly summarize in Section ." ], [ "Effective Lagrangian", "We start by specifying the elements of the chiral effective Lagrangian which are relevant for the calculation of the width of the Roper at next-to-leading order in the power counting specified below.", "We consider pions, nucleons, the delta and Roper resonances as dynamical degrees of freedom.", "The corresponding most general effective Lagrangian can be written as ${\\cal L}_{\\rm eff}={\\cal L}_{\\pi \\pi }+{\\cal L}_{\\pi N}+{\\cal L}_{\\pi \\Delta }+{\\cal L}_{\\pi R}+{\\cal L}_{\\pi N\\Delta }+{\\cal L}_{\\pi NR}+{\\cal L}_{\\pi \\Delta R},$ where the subscripts indicate the dynamical fields contributing to a given term.", "From the purely mesonic sector we need the following structures [2], [27] ${\\cal L}_{\\pi \\pi }^{(2)}&=& \\frac{F^2}{4}\\langle \\partial _\\mu U \\partial ^\\mu U^\\dagger \\rangle +\\frac{F^2 M^2}{4}\\langle U^\\dagger + U\\rangle ,\\nonumber \\\\{\\cal L}_{\\pi \\pi }^{(4)}&=&\\frac{1}{8}l_4\\langle u^\\mu u_\\mu \\rangle \\langle \\chi _+\\rangle +\\frac{1}{16}(l_3+l_4)\\langle \\chi _+\\rangle ^2, $ where $\\langle ~~ \\rangle $ denotes the trace in flavor space, $F$ is the pion decay constant in the chiral limit and $M$ is the leading term in the quark mass expansion of the pion mass [2].", "The pion fields are contained in the unimodular unitary $2\\times 2$ matrix $U$ , with $u=\\sqrt{U}$ and $u_\\mu & = & i \\left[u^\\dagger \\partial _\\mu u -u \\partial _\\mu u^\\dagger \\right],\\nonumber \\\\\\chi ^+&=&u^\\dag \\chi u^\\dag + u\\chi ^\\dag u~,\\quad \\chi = \\left[ \\begin{array}{c c}M^2 & 0 \\\\0 & M^2 \\\\\\end{array}\\right].$ The terms of the Lagrangian with pions and baryons contributing to our calculation read: ${\\cal L}_{\\pi N}^{(1)}&=&\\bar{\\Psi }_N\\left\\lbrace i{D}-m+\\frac{1}{2}g \\,{u}\\gamma ^5\\right\\rbrace \\Psi _N\\, ,\\nonumber \\\\{\\cal L}_{\\pi R}^{(1)}&=&\\bar{\\Psi }_R\\left\\lbrace i{D}-m_R+\\frac{1}{2}g_R{u}\\gamma ^5\\right\\rbrace \\Psi _R\\, ,\\nonumber \\\\{\\cal L}_{\\pi R}^{(2)}&=&\\bar{\\Psi }_R\\left\\lbrace c_1^R\\langle \\chi ^+\\rangle \\right\\rbrace \\Psi _R\\, ,\\nonumber \\\\{\\cal L}_{\\pi NR}^{(1)}&=&\\bar{\\Psi }_R\\left\\lbrace \\frac{g_{\\pi NR}}{2}\\gamma ^\\mu \\gamma _5 u_\\mu \\right\\rbrace \\Psi _N+ {\\rm h.c.}\\, ,\\nonumber \\\\{\\cal L}^{(1)}_{\\pi \\Delta }&=&-\\bar{\\Psi }_{\\mu }^i\\xi ^{\\frac{3}{2}}_{ij}\\left\\lbrace \\left(i{D}^{jk}-m_\\Delta \\delta ^{jk}\\right)g^{\\mu \\nu }-i\\left(\\gamma ^\\mu D^{\\nu ,jk}+\\gamma ^\\nu D^{\\mu ,jk}\\right) +i \\gamma ^\\mu {D}^{jk}\\gamma ^\\nu +m_\\Delta \\delta ^{jk} \\gamma ^{\\mu }\\gamma ^\\nu \\right.\\nonumber \\\\&&\\left.+\\frac{g_1}{2}{u}^{jk}\\gamma _5g^{\\mu \\nu }+\\frac{g_2}{2} (\\gamma ^\\mu u^{\\nu ,jk}+u^{\\nu ,jk}\\gamma ^\\mu )\\gamma _5+\\frac{g_3}{2}\\gamma ^\\mu {u}^{jk}\\gamma _5\\gamma ^\\nu \\right\\rbrace \\xi ^{\\frac{3}{2}}_{kl}{\\Psi }_\\nu ^l\\, ,\\nonumber \\\\{\\cal L}^{(1)}_{\\pi N\\Delta }&=&h\\,\\bar{\\Psi }_{\\mu }^i\\xi _{ij}^{\\frac{3}{2}}\\Theta ^{\\mu \\alpha }(z_1)\\ \\omega _{\\alpha }^j\\Psi _N+ {\\rm h.c.}\\, ,\\nonumber \\\\{\\cal L}^{(1)}_{\\pi \\Delta R}&=&h_R\\,\\bar{\\Psi }_{\\mu }^i\\xi _{ij}^{\\frac{3}{2}}\\Theta ^{\\mu \\alpha }(\\tilde{z})\\ \\omega _{\\alpha }^j\\Psi _R+ {\\rm h.c.}\\, ,$ where $\\Psi _N$ and $\\Psi _R$ are isospin doublet fields with bare masses $m_{N 0}$ and $m_{R 0}$ , corresponding to the nucleon and the Roper resonance, respectively.", "The vector-spinor isovector-isospinor Rarita-Schwinger field $\\Psi _\\nu $ represents the $\\Delta $ resonance [28] with bare mass $m_{\\Delta 0}$ , $\\xi ^{\\frac{3}{2}}$ is the isospin-$3/2$ projector, $\\omega _\\alpha ^i=\\frac{1}{2}\\,\\langle \\tau ^i u_\\alpha \\rangle $ and $\\Theta ^{\\mu \\alpha }(z)=g^{\\mu \\alpha }+z\\gamma ^\\mu \\gamma ^\\nu $ , where $z$ is a so-called off-shell parameter.", "We fix the off-shell structure of the interactions involving the delta by adopting $g_1=-g_2=-g_3$ and $z_1=\\tilde{z}=0$ .", "Note that these off-shell parameters can be absorbed in LECs and are thus redundant [29], [30].", "Leaving out the external sources, the covariant derivatives are defined as follows: $D_\\mu \\Psi _{N/R} & = & \\left( \\partial _\\mu + \\Gamma _\\mu \\right) \\Psi _{N/R}\\,, \\nonumber \\\\\\left(D_\\mu \\Psi \\right)_{\\nu ,i} & = &\\partial _\\mu \\Psi _{\\nu ,i}-2\\,i\\,\\epsilon _{ijk}\\Gamma _{\\mu ,k} \\Psi _{\\nu ,j}+\\Gamma _\\mu \\Psi _{\\nu ,i}\\,,\\nonumber \\\\\\Gamma _\\mu & = &\\frac{1}{2}\\,\\left[u^\\dagger \\partial _\\mu u +u\\partial _\\mu u^\\dagger \\right]=\\tau _k\\Gamma _{\\mu ,k}~.", "$ Note that a mixing kinetic term of the form $i\\lambda _1\\bar{\\Psi }_R\\gamma _\\mu D^\\mu \\Psi _N-\\lambda _2\\bar{\\Psi }_R\\Psi _N+ {\\rm h.c}.$ can be dropped, since, using field transformations and diagonalising the nucleon-Roper mass matrix, it can be reduced to the form of operators of the Lagrangian presented above [22]." ], [ "The pole mass and the width of the Roper resonance", "The dressed propagator of the Roper resonance can be written as $i S_R(p) = \\frac{i}{p\\hspace{-4.49997pt}/\\hspace{1.00006pt}-m_{R0}-\\Sigma _R(p\\hspace{-4.49997pt}/\\hspace{1.00006pt})}\\,,$ where $-i\\,\\Sigma _R (p\\hspace{-4.49997pt}/\\hspace{1.00006pt})$ is the self-energy, i.e.", "the sum of all one-particle-irreducible diagrams contributing to the two-point function of the Roper resonance.", "The pole of the dressed propagator $S_R$ is obtained by solving the equation $S_R^{-1}(z) \\equiv z - m_{R0} -\\Sigma _R(z)=0\\,.", "$ We define the physical mass and the width of the Roper resonance by parameterizing the pole as $z = m_R -i\\,\\frac{\\Gamma _R}{2} \\,.$ The pertinent topologies of the one- and two-loop diagrams contributing to the self-energy of the Roper resonance are shown in Fig.", "REF .", "We use BPHZ renormalization by subtracting loop diagrams in their chiral limit and replace the parameters of the Lagrangian by their renormalized values, i.e.", "counterterm diagrams are not shown explicitly.", "Figure: One and two-loop self-energy diagrams of the Roper resonance up-to-and-includingfifth order according to the standard power counting.", "The dashed and thick solid linesrepresent the pions and the Roper resonances, respectively.", "The thin solid lines in theloops stand for either nucleons, Roper or delta resonances.", "The numbers in the circles givethe chiral order of the vertices.We solve Eq.", "(REF ) perturbatively order by order in the loop expansion.", "We parameterize the pole as $z=m_{2}+\\hbar \\delta z_1+\\hbar ^2 \\delta z_2 +{\\cal O}(\\hbar ^3),$ where $m_2=m_R^0+4 c_1^R M^2 $ , with $m_R^0$ the physical Roper mass in the chiral limit, and substitute in Eq.", "(REF ) in which we write the self-energy as an expansion in the number of loops $\\Sigma _R = \\hbar \\Sigma _1+\\hbar ^2 \\Sigma _2 +{\\cal O}(\\hbar ^3)\\, .$ By expanding in powers of $\\hbar $ , we get $\\hbar \\delta z_1+\\hbar ^2 \\delta z_2-\\hbar \\Sigma _1(m_2) -\\hbar ^2 \\delta z_1 \\Sigma _1^{\\prime }(m_{2})-\\hbar ^2 \\Sigma _2(m_{2}) +{\\cal O}(\\hbar ^3) =0\\,.", "$ Solving Eq.", "(REF ) we obtain $\\delta z_1 & = & \\Sigma _1(m_2),\\nonumber \\\\\\delta z_2 & = & \\Sigma _1(m_2) \\,\\Sigma _1^{\\prime }(m_{2})+\\Sigma _2(m_{2}).$ Eq.", "(REF ) leads to the following expression for the width $\\Gamma _R &=& \\hbar \\ 2 i \\,{\\rm Im} \\left[\\Sigma _1(m_2)\\right] \\nonumber \\\\&+& \\hbar ^2 \\ 2 i \\, \\biggl \\lbrace {\\rm Im} \\left[\\Sigma _1(m_2)\\right]{\\rm Re} \\left[\\Sigma _1^{\\prime }(m_2)\\right]+ {\\rm Re} \\left[\\Sigma _1(m_2)\\right]{\\rm Im} \\left[\\Sigma _1^{\\prime }(m_2)\\right] \\biggr \\rbrace \\nonumber \\\\&+& \\hbar ^2 \\ 2 i \\, {\\rm Im} \\left[\\Sigma _2(m_2)\\right] +{\\cal O}(\\hbar ^3).$ Using the power counting specified in section , it turns out that the contribution of the second term in Eq.", "(REF ) is of an order higher than the accuracy of our calculation, which is $\\delta ^5$ (where $\\delta $ is a small expansion parameter).", "In particular, ${\\rm Im} \\left[\\Sigma _1(m_2)\\right]$ is of order $\\delta ^3$ , ${\\rm Re} \\left[\\Sigma _1^{\\prime }(m_2)\\right]$ is of order $\\delta ^4$ , ${\\rm Re} \\left[\\Sigma _1(m_2)\\right]$ is of order $\\delta ^6$ and ${\\rm Im} \\left[\\Sigma _1^{\\prime }(m_2)\\right]$ is of order $\\delta ^2$ .", "Also, modulo higher order corrections, we can replace $m_2$ by the physical mass $m_R$ .", "To calculate the contributions of the one- and two-loop self-energy diagrams to the width of the Roper resonance, specified in the first and third terms of Eq.", "(REF ), respectively, we use the Cutkosky cutting rules.", "As shown in Ref.", "[31] in quantum field theories with unstable particles the scattering amplitude is unitary in the space of stable particles alone.", "Thus, to calculate the imaginary part of the self-energy of the Roper resonance at one loop order we need to take into account only the contribution of the diagrams with internal nucleon lines.", "At two-loop order only contributions obtained by cutting the lines, corresponding to stable particles, are needed.", "Details of the calculation of the Roper resonance width using the decay amplitudes are given in the next section." ], [ "The width of the Roper resonance obtained from the decay amplitudes", "By applying the cutting rules to the diagrams of Fig.", "REF we obtain the graphs contributing in the decay amplitudes of the Roper resonance into $\\pi N$ and $\\pi \\pi N$ systems, specified in Figs.", "REF and REF , respectively.", "The decay amplitude corresponding to $R(p)\\rightarrow N(p^\\prime )\\pi ^a(q)$ can be written as ${\\cal A}^{a}=\\bar{u}_N(p^{\\prime })\\left\\lbrace {A}\\,{q}\\gamma _5\\tau ^a\\right\\rbrace u_R(p)\\, ,$ where $a$ is an isospin index of the pion, and the $\\bar{u}, u$ are conventional spinors.", "The corresponding decay width reads $\\Gamma _{R\\rightarrow \\pi N}=\\frac{\\lambda ^{1/2}(m_R^2,m_N^2,M^2)}{16\\pi \\,m_R^3}\\left\|{\\cal M}\\right\|^2,$ with $\\lambda (x,y,z)=(x-y-z)^2-4 y z$ and the unpolarized squared amplitude has the form $\\left\|{\\cal M}\\right\|^2&=&3(m_N+m_R)^2\\left[(m_N-m_R)^2-M_\\pi ^2\\right]A^\\ast A\\, .$ Figure: Feynman diagrams contribution to the decay R→NπR\\rightarrow N\\pi up toleading one-loop order.", "Dashed, solid, double and thick solid linescorrespond to pions, nucleons, deltas and Roper resonances, respectively.The numbers in the circles give the chiral orders of the vertices.Next, we define the kinematical variables for the decay $R(p)\\rightarrow N(p^\\prime )\\pi ^a(q_1)\\pi ^b(q_2)$ via $&&s_1=(q_1+q_2)^2\\, ,\\qquad s_2=(p^\\prime +q_1)^2\\, ,\\qquad s_3=(p^\\prime +q_2)^2\\, ,$ subject to the contraint $s_1+s_2+s_3=m_R^2+m_N^2+2M_\\pi ^2 ~.$ The isospin and the Lorentz decomposition of the decay amplitude reads ${\\cal A}^{ab}&=&\\chi _N^\\dagger \\left\\lbrace \\delta ^{ab}F_++{i}\\epsilon ^{abc}\\tau ^cF_-\\right\\rbrace \\chi _R^{}\\ ,\\\\F_{\\pm }&=&\\bar{u}_N(p^\\prime )\\left\\lbrace F_{\\pm }^{(1)}-\\frac{1}{2(m_N+m_R)}\\left[{q}_1,{q}_2\\right] F_{\\pm }^{(2)}\\right\\rbrace u_R(p)\\, ,$ with the $\\chi $ being isospinors, $a$ and $b$ are isospin indices of the pions.", "The unpolarized squared invariant amplitude is given by $\|{\\cal M}\|^2&=& \\sum _{i,j=1}^2{\\cal Y}_{ij}\\left[\\frac{3}{2}\\,{F_{+}^{(i)}}^\\ast F_{+}^{(j)}+3\\,{F_{-}^{(i)}}^\\ast F_{-}^{(j)}\\right] ,\\nonumber \\\\{\\cal Y}_{11}&=&2\\left[(m_N+m_R)^2-s_1\\right],\\nonumber \\\\{\\cal Y}_{12}&=&{\\cal Y}_{21}=-s_1 \\nu \\,,\\nonumber \\\\{\\cal Y}_{22}&=&\\frac{1}{2}\\left[(4 M_\\pi ^2-s_1)(s_1-(m_R-m_N)^2)-s_1\\nu ^2\\right],$ with $\\nu $ given by $\\nu =\\frac{s_2-s_3}{m_N+m_R}\\,.$ The decay width corresponding to the $\\pi \\pi N$ final state is obtained by substituting $\|{\\cal M}\|^2$ from Eq.", "(REF ) in the following formula $\\Gamma _{R\\rightarrow \\pi \\pi N}=\\frac{1}{32m_R^3(2\\pi )^3}\\int _{4M_\\pi ^2}^{(m_R-m_N)^2}{\\rm d}s_1\\int _{s_{2-}}^{s_{2+}}{\\rm d}s_2\\,\|{\\cal M}\|^2\\, ,$ where the integration limits over $s_2$ are given by $s_{2\\pm }=\\frac{m_R^2+m_N^2+2M_\\pi ^2-s_1}{2}\\pm \\frac{1}{2s_1}\\lambda ^{1/2}(s_1,m_R^2,m_N^2)\\lambda ^{1/2}(s_1,M_\\pi ^2,M_\\pi ^2)\\, .$ Figure: Tree diagrams contributing to the R→ππNR\\rightarrow \\pi \\pi N decay.", "Crossed diagrams arenot shown.", "Dashed, solid, double and thick solid linescorrespond to pions, nucleons, deltas and Roper resonances, respectively.", "The numbers inthe circles give the chiral orders of the vertices.Let us emphasize here that to obtain the width of the Roper resonance we need to calculate the imaginary part of the self-energy in the complex region.", "However, within the accuracy of our calculation, we only need the imaginary parts of the one- and two-loop diagrams for the real mass of the Roper resonance.", "We can relate these to the decay amplitudes also calculated by putting the Roper external line on the real mass-shell.", "While this is an useful approximation well suited for our current accuracy, to define the physical properties of unstable particles one needs to use the complex on-shell conditions, see, e.g.", "Ref. [32].", "Thus, the contributions of the one- and two-loop self-energy diagrams in the width of the Roper resonance, specified in the first and third terms of Eq.", "(REF ) sum up to $\\Gamma _R = \\Gamma _{R\\rightarrow \\pi N} + \\Gamma _{R\\rightarrow \\pi \\pi N},$ where $\\Gamma _{R\\rightarrow \\pi N}$ and $\\Gamma _{R\\rightarrow \\pi \\pi N}$ are given by Eq.", "(REF ) and Eq.", "(REF ), respectively.", "The power counting and the diagrams contributing to each of these decay modes up to a given order of accuracy are discussed in the next section." ], [ "Renormalization and power counting", "By counting the mass differences $m_R-m_N$ , $m_\\Delta -m_N$ and $m_R-m_\\Delta $ as of the same order as the pion mass and the pion momenta, the standard power counting would apply to all tree and loop diagrams considered in this work.", "According to the rules of this counting a four-dimensional loop integration is of order $q^4$ , an interaction vertex obtained from an ${\\cal O}(q^n)$ Lagrangian counts as of order $q^n$ , a pion propagator as order $q^{-2}$ , and a nucleon propagator as order $q^{-1}$ .", "We would also assign the order $q^{-1}$ to the $\\Delta $ and the Roper resonance propagators for non-resonant kinematics.", "The propagators of the delta and the Roper resonance get enhanced for resonant kinematics when they appear as intermediate states outside the loop integration.", "In this case we would assign the order $q^{-3}$ to these propagators.", "As the mass difference $m_R-m_N\\sim 400$ MeV, the above mentioned power counting cannot be trusted.", "By considering $m_R-m_N$ as a small parameter of the order $\\delta ^1$ , it is more appropriate to count $M_\\pi \\sim \\delta ^2$ .", "To work out further details of the new counting, it is a more convenient to work with the kinematical variable $\\nu $ as defined in Eq.", "(REF ) for the $R\\rightarrow \\pi \\pi N$ decay.", "Within the range of integration specified by Eq.", "(REF ), $\\nu $ varies from $m_N-m_R$ to $m_R-m_N$ (for $M_\\pi =0$ ) and therefore we count $\\nu \\sim \\delta $ .", "As $s_1$ varies from $4 M_\\pi ^2$ to $(m_R-m_N)^2$ , we assign the order $\\delta ^2$ to it.", "We also count $m_R-m_\\Delta \\sim \\delta ^2$ .", "The $R\\rightarrow \\pi N$ width of Eq.", "(REF ) is of order $\\delta ^3 \\times {\\rm order \\ of \\ } A^* A$ .", "The tree and one loop diagrams, contributing to the $R\\rightarrow \\pi N$ decay are shown in Fig.", "REF .", "The tree order diagram (t2) is proportional to $q\\gamma _5 M_\\pi ^2$ and therefore it contributes at order $\\delta ^4$ to $A$ , while diagram (t1) gives an order $\\delta ^0$ contribution.", "All one loop diagrams are of the order $q^3$ in the standard counting.", "Thus they give order $q^2$ contributions in $A$ .", "Expanding these contributions in powers of $M_\\pi $ , we absorb the first, $M_\\pi $ -independent, term in the renormalization of the coupling of the tree diagram (t1), which becomes complex.", "However, the imaginary part is of the order $\\delta ^2$ and can be calculated explicitly.", "The next term in the expansion in powers of $M_\\pi $ is linear in $M_\\pi $ and hence, if non-vanishing, it does not violate the standard power counting (terms, non-analytic in $M_\\pi ^2$ do not violate the standard power counting).", "Therefore its coefficient must contain at least one power of $(m_R-m_N)$ .", "That is, the term linear in $M_\\pi $ is at least of order $\\delta ^3$ .", "Further terms are of even higher order.", "As a result, restricting ourselves to the order $\\delta ^2$ in $A$ , and thus to order $\\delta ^5$ in the width of the Roper resonance, the only contribution of one loop diagrams of Fig.", "REF which we might need is the $M_\\pi $ -independent imaginary part.", "However, this imaginary part starts contributing in $A^*A$ only at order $\\delta ^4$ .", "Thus all contributions of the one-loop diagrams are beyond the accuracy of our calculation.", "We have checked that for the numerical values of the couplings, as specified below, the individual contributions of the diagrams in Fig.", "REF in the decay amplitude are indeed small compared to the one of the tree order diagram.", "According to Eq.", "(REF ) the $R\\rightarrow \\pi \\pi N$ width is of order $\\delta ^3 \\times {\\rm order \\ of \\ } \|{\\cal M}\|^2 $ .", "The leading order tree diagrams contributing to the $R\\rightarrow \\pi \\pi N$ decay are shown in Fig.", "REF .", "The delta propagators in these diagrams are to be understood as dressed ones.", "Expanding these propagators around their pole, we observe that the non-pole parts start contributing at higher orders and therefore can be dropped.", "The contributions of the loop diagrams are suppressed by additional powers of $\\delta $ so that they do not contribute at order $\\delta ^5$ .", "Among these loop diagrams are those contributing to the decay of the Roper resonance to $\\pi \\pi N$ system with two final pions in the iso-singlet channel.", "Due to the presence of an iso-scalar scalar resonance $f_0(500)$ in this channel [33] an infinite number of pion-pion finite state interaction diagrams have to be summed up (see, e.g., Refs.", "[34], [35]).", "Alternatively one can include the $f_0(500)$ as an explicit degree of freedom in the effective Lagrangian [36].", "In both approaches it turns out that the corresponding contributions to $R\\rightarrow \\pi \\pi N$ amplitude are of higher order than $\\delta ^5$ , and hence estimated to be within the theoretical uncertainty due to higher order contributions given in Eq.", "(REF ) below." ], [ "Numerical results", "To calculate the full decay width of the Roper resonance we use the following standard values of the parameters [33] $&& M_\\pi = 139 \\ {\\rm MeV}, \\ \\ m_N=939 \\ {\\rm MeV}, \\ \\ m_\\Delta =1210\\ {\\rm MeV}, \\ \\ \\Gamma _\\Delta =100\\ {\\rm MeV}, \\nonumber \\\\&& \\ \\ m_R=1365 \\ {\\rm MeV}, F_\\pi =92.2 \\ {\\rm MeV},$ in Eqs.", "(REF ) and (REF ) and obtain $\\Gamma _{R\\rightarrow \\pi N}&=& 550 \\, g_{\\pi NR}^2\\ {\\rm MeV},\\nonumber \\\\\\Gamma _{R\\rightarrow \\pi \\pi N}&=& (1.49\\,g_A^2 \\,g_{\\pi NR}^2-2.76\\, g_A^{} \\, g_{\\pi NR}^2\\,g_R^{}+1.48\\,g_{\\pi NR}^2\\, g_R^2\\nonumber \\\\&+&2.96\\,g_A^{}\\, g_{\\pi NR}^{} \\,h^{} h_R^{} -3.79\\,g_{\\pi NR}^{}\\,g_R^{} \\,h^{} h_R^{} +9.93\\,h^2h_R^2) \\ {\\rm MeV}.$ Eq.", "(REF ) depends on five couplings in total for the two of which we substitute $g_A=1.27$ [33] and $h = 1.42\\pm 0.02$ .", "The latter value is the real part of this coupling taken from Ref. [37].", "As noted before, the imaginary part only contributes to orders beyond the accuracy of our calculations.", "As for the other unknown parameters, we choose to pin down $g_{\\pi NR}$ so that we reproduce the width $\\Gamma _{R\\rightarrow \\pi N}=(123.5\\pm 19.0)$ MeV from PDG [33], which yields $g_{\\pi NR}=\\pm (0.47\\pm 0.04)$ .", "In what follows we take both signs into account which contributes to the error budget.", "Further we assume $g_R=g_A$ and $h_{R}=h$ .", "With the values specified above, one can predict the decay width for the decay mode $R\\rightarrow \\pi \\pi N$ : $\\Gamma _{R\\rightarrow \\pi \\pi N}&=& \\left[0.53(9)-0.98(17)+0.53(9)\\pm 3.57(31)\\mp 4.57(40)+40.4(1.6)\\right]\\ {\\rm MeV}\\nonumber \\\\&=&[40.5(1.6)\\pm 1.0(0.5)]~{\\rm MeV}~,$ where the second term is due to the choice of the sign of $g_{\\pi NR}$ .", "If we incorporate the second term as error to the first term, then the decay width reads $\\Gamma _{R\\rightarrow \\pi \\pi N}=40.5(2.2)~{\\rm MeV}~,$ where the error is obtained in quadrature from Eq.", "(REF ).", "As is clearly seen from Eq.", "(REF ), the largest contribution in $\\Gamma _{R\\rightarrow \\pi \\pi N}$ width comes from the decay with the delta resonance as an intermediate state.", "Further, we estimate the theoretical error due to the omitting the higher order contributions using the approach of Ref.", "[38], which leads to $\\Gamma _{R\\rightarrow \\pi \\pi N}&=&(40.5\\pm 2.2\\pm 16.8) \\ {\\rm MeV}.$ Our estimation is consistent with $\\Gamma _{\\pi \\pi N}=(66.5\\pm 9.5)$ MeV quoted by PDG [33]." ], [ "Summary", "In current work we have calculated the width of the Roper resonance up to next-to-leading order in a systematic expansion of baryon chiral perturbation theory with pions, nucleons, delta and Roper resonances as dynamical degrees of freedom.", "We define the physical mass and the width of the Roper resonance by relating them to the real and imaginary parts of the complex pole of the dressed propagator.", "The next-to-leading order calculation of the width requires obtaining the imaginary parts of one- and two-loop self-energy diagrams.", "We employed the Cutkosky cutting rules and obtained the width up to given order accuracy by squaring the decay amplitudes.", "Three unknown coupling constants contribute in the corresponding expressions.", "One of them we fix by reproducing the PDG value for the width of the Roper decay in a pion and a nucleon.", "Assuming that the remaining two couplings of the Roper interaction take values equal to those of the nucleon, we obtain the result for the width of Roper resonance decaying in the two pions and a nucleon that is consistent with the PDG value, based on the the uncertainties within the given and from higher orders, cf.", "Eq.", "(REF ).", "To improve the accuracy of our calculation, three-loop contributions to the self-energy of the Roper resonance need to be calculated.", "Moreover, contributions of an infinite number of diagrams, corresponding to the scalar-isoscalar pion-pion scattering need to be re-summed either by solving pion-pion scattering equations or including the $f_0(500)$ as an explicit dynamical degree of freedom.", "Note also that new unknown low-energy coupling constants appear at higher orders, which need to be pinned down.", "This work was supported in part by Georgian Shota Rustaveli National Science Foundation (grant FR/417/6-100/14) and by the DFG (TR 16 and CRC 110).", "The work of UGM was also supported by the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (Grant No.", "2015VMA076)." ] ]	1606.04873
[ [ "Development of n-in-p pixel modules for the ATLAS Upgrade at HL-LHC" ], [ "Abstract Thin planar pixel modules are promising candidates to instrument the inner layers of the new ATLAS pixel detector for HL-LHC, thanks to the reduced contribution to the material budget and their high charge collection efficiency after irradiation.", "100-200 $\\mu$m thick sensors, interconnected to FE-I4 read-out chips, have been characterized with radioactive sources and beam tests at the CERN-SPS and DESY.", "The results of these measurements are reported for devices before and after irradiation up to a fluence of $14\\times10^{15}$ n$_{eq}$/cm$^2$.", "The charge collection and tracking efficiency of the different sensor thicknesses are compared.", "The outlook for future planar pixel sensor production is discussed, with a focus on sensor design with the pixel pitches (50x50 and 25x100 $\\mu$m$^2$) foreseen for the RD53 Collaboration read-out chip in 65 nm CMOS technology.", "An optimization of the biasing structures in the pixel cells is required to avoid the hit efficiency loss presently observed in the punch-through region after irradiation.", "For this purpose the performance of different layouts have been compared in FE-I4 compatible sensors at various fluence levels by using beam test data.", "Highly segmented sensors will represent a challenge for the tracking in the forward region of the pixel system at HL-LHC.", "In order to reproduce the performance of 50x50 $\\mu$m$^2$ pixels at high pseudo-rapidity values, FE-I4 compatible planar pixel sensors have been studied before and after irradiation in beam tests at high incidence angle (80$^\\circ$) with respect to the short pixel direction.", "Results on cluster shapes, charge collection and hit efficiency will be shown." ], [ "Introduction", "In this paper, different designs of n-in-p planar hybrid pixel modules are investigated and compared.", "The R&D activity is carried out in view of the upgrade of the ATLAS pixel system for the High Luminosity phase of the LHC (HL-LHC) [1], foreseen to start around 2025.", "The number of pile-up events per bunch crossing expected at the HL-LHC is 140-200 [2], [3].", "To keep the pixel occupancy at an acceptable level, smaller pixel cell dimensions are foreseen with respect to the ones presently implemented in the FE-I3 chip (50 $\\mu $ m x 400 $\\mu $ m) and in the FE-I4 chip (50 $\\mu $ m x 250 $\\mu $ m), developed for the ATLAS Insertable B-Layer (IBL) [4].", "The new readout chips for the ATLAS and CMS pixel systems at HL-LHC are being developed by the CERN RD53 Collaboration [5] with a pitch of 50 $\\mu $ m x 50 $\\mu $ m in the 65 nm CMOS technology.", "The feasibility of employing thin planar pixel detectors for the inner layers of the upgraded ATLAS pixel system is being investigated, comparing the performance of sensors in the thickness range of 100-200 $\\mu $ m. Using the SOI technology, n-in-p sensors with a full thickness of 100 and 200 $\\mu $ m were produced on 6\" wafers at VTT in the framework of a Multi-project Wafer run (MPW) [6].", "This production also implements activated edges to reduce the inactive area of these devices [7] [8] and the performance in terms of charge collection at the edges has been presented elsewhere [9].", "200 $\\mu $ m thick sensors were also produced at CIS, on 4\" wafers, with a standard guard ring structure and an inactive width of 450 $\\mu $ m [10].", "The sensors have been interconnected by bump-bonding to FE-I4 chips and characterized by means of radioactive sources in the laboratory and beam test experiment with 4 GeV electrons at DESY and 120 GeV pions at CERN-SPS." ], [ "Characterization of sensors with different thickness after irradiation", "The charge collection properties were measured by means of Strontium source scans using the ATLAS USBpix read-out system, developed by the University of Bonn [11].", "The pixel modules were investigated before and after irradiation, up to a fluence of $5\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ for the 100 $\\mu $ m thick sensors, corresponding to the integrated fluence at the end of the life-time for the second pixel layer at HL-LHC and for the 200 $\\mu $ m thick sensors up to a fluence of $14\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , the expected value for the inner layer after ten years of operation at HL-LHC.", "Fig.REF shows the collected charge for the 100 $\\mu $ m thick sensors, as a function of the applied bias voltage for different fluences.", "It can be observed that the collected charge starts to saturate at moderate bias voltages, between 200 and 300V, in the full fluence range explored, to values that are very close to those expected for not irradiated detectors, around 6500-7000 electrons [12].", "Figure: Charge collection of 100 μ\\mu m thick sensors as a function of the bias voltage for different fluencesFig.REF shows the results of the charge collection for the 200 $\\mu $ m thick sensors and of 300 $\\mu $ m thick sensors at the highest fluence only.", "After $2\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , no saturation of the charge is observed.", "At the maximum fluence of $14\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , the collected charge of the 200 and 300 $\\mu $ m thick sensors becomes very similar and at the highest measured voltage of 1000V is between 5 ke and 5.5 ke.", "Figure: Charge collection of 200 and 300 μ\\mu m thick sensors as a function of the bias voltage for different fluencesFE-I3 and FE-I4 modules of different thickness were also investigated in beam tests by using telescopes of the EUDET family [13].", "The irradiated modules were cooled with dry ice inside a box designed on purpose.", "With this setup typical temperatures between -50$^\\circ $ C and -40$^\\circ $ C are obtained during operations.", "Recent studies have shown the independence of the collected charge in a temperature range between -50$^\\circ $ C and -25$^\\circ $ C [14] and between -50$^\\circ $ C and -40$^\\circ $ C for the hit efficiency [15].", "Figure REF summarizes the results of the hit efficiency in a fluence range between 4 and $6\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ .", "The 150 $\\mu $ m thick sensors were obtained from an older production at MPG-HLL on SOI 6\" wafers while the 285 $\\mu $ m thick sensors were produced on 4\" wafers at CIS, as described in more details in [12].", "For all the values of the hit efficiencies quoted in the following, the dominant source of uncertainty is systematic and evaluated to be 0.3$\\%$ , as explained in [16].", "Figure: Hit efficiency as a function of the bias voltage for pixel modules of different thicknesses irradiatedto a fluence between 4 and 6×10 15 6\\times 10^{15} n eq / cm 2 \\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2.", "The modules were operated at a threshold of 1600 e.After an irradiation fluence between 4 and $6\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , FE-I4 modules with 150 and 200 $\\mu $ m thick sensors show similar performances reaching a hit efficiency of about 97$\\%$ at V$_{bias}$ =500V, while the same module type with a 100$\\mu $ m thick sensor starts to saturate to this value of the hit efficiency already at a bias voltage of 300V.", "These results suggest that a lower operational bias voltage is possible for thinner sensors, and a reduced power dissipation at these fluence levels.", "Fig.REF shows the expected sensor power density for these devices at a temperature of -25 $^\\circ $ C, at which it is foreseen that the sensors will be operated at HL-LHC, as a function of the applied bias voltage for different values of the sensor thickness.", "A high power consumption requires sufficient performance of the cooling system to dissipate the heat and avoid the thermal runaway of the sensor.", "Since this effect limits the achievable operational voltage of the modules, it has to be considered together with the hit efficiency results to determine the practical performance of the different sensor thicknesses.", "The power per area calculated for the thinner sensors of 100 and 150 $\\mu $ m at the optimal operational voltage of 300 and 500V estimated from the hit efficiency measurements in Fig.REF , is respectively 8 and 3.5 times lower than the runaway point of the worst case considered.", "The runaway point was calculated with the parameters descibed in [10], except for the higher chip power dissipation of 0.74 W/cm$^2$ assumed for the RD53 Collaboration read-out chip in 65 nm CMOS technology.", "Figure: Power density as a function of the bias voltage for sensors irradiated between 4 and 6×10 15 6\\times 10^{15} n eq / cm 2 \\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2estimated at a temperature of T=-25 ∘ ^\\circ C. The horizontal red line indicatesthe thermal runaway point as calculated in ." ], [ "Optimization of the pixel cell design", "Figure REF shows that at equal fluence and thickness, FE-I3 modules yield a hit efficiency 1$\\%$ higher with respect to the FE-I4.", "This is due to the lower fraction of area that the biasing structures occupy in the pixel cell, since the punch-through dot and the bias rail are implemented with the same design and the FE-I3 pixel lenght is 400 $\\mu $ m compared with the 250 $\\mu $ m in the FE-I4 case.", "It has found that these elements induce a decrease of the hit efficiency in the pixel cell [10].", "Given the reduced pitch for the future pixel read-out chips, an optimization of the biasing structures is mandatory, to avoid a large loss of hit effiiciency after irradiation, especially in the central pseudo-rapidity region.", "Different designs of the bias dot and rail were thus implemented in two different pixel sensors, both compatible with the FE-I4 chip, of a recent n-in-p production carried out at CIS on 6\" wafers, with a thickness of 270 $\\mu $ m. The first one includes three designs, repeated in every successive group of 30 rows, with a single bias dot for every pixel: the standard design (Fig.", "REF (a)) with the rail at the center between the short side of two neighbouring pixels, a slightly modified version with the rail running over the bias dot (b) and the last one (c), with the rail over the central part of the cell.", "After irradiation at a fluence of $3\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , the hit efficiencies of the different designs have been compared.", "The highest value of 98.7$\\%$ has been found for the geometry (b), while the standard geometry yields an efficiency of 97.7$\\%$ .", "The layout in Fig.", "REF (c) shows the lowest efficiency, especially in correspondence of the horizontal lines of the bias rail that runs in the inter-pixel region.", "It can instead be observed that where the bias rail is superimposed to the pixel implant, both in (b) and in (c), the aluminum line does not induce any efficiency loss because the effects on the electric field shapes are then screened by the pixel implant.", "Figure: Hit efficiency projected onto a pixel cell for different single bias dot designs after an irradiationfluence of 3×10 15 3\\times 10^{15} n eq / cm 2 \\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2.", "The module was operated at 800V.Higher hit efficiencies are obtained by implementing an external punch-through dot, common to four pixels, as shown by Fig.", "REF and REF .", "In this case the sensor pitch is 25 $\\mu $ m x 500 $\\mu $ m, still compatible with a FE-I4 chip.", "The overall area where a lower hit efficiency is observed is clearly reduced with respect to the standard FE-I4 design.", "The hit efficiency reaches 99.4$\\%$ at the highest measured voltage of 500V which is around 2$\\%$ higher than the 97.7$\\%$ measured for the standard FE-I4 design at the same fluence.", "Figure: Layout of the 25 x500 μ\\mu m 2 ^2 FE-I4 compatible sensor design with an external punch-through common to four pixel cells.", "The insert is a photograph of the produced device superimposed to the schematic design of the sensor.Figure: Comparison of the hit efficiency over the pixel cell for the FE-I4 modules of the CIS3production irradiated to a fluence of 3×10 15 3\\times 10^{15} n eq / cm 2 \\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2with different bias structures.", "(a) showsthe hit efficiency for the module with 25 x500 μ\\mu m 2 ^2 pitch and one common bias dot sharedamong four pixels.", "(b) shows the hit efficiency for the standard FE-I4 design with an internal punch-through dotand a pitch of 50 x 250 μ\\mu m 2 ^2.Two new pixel sensor productions at ADVACAM and CIS, with 100 and 150 $\\mu $ m thickness, have recently been completed and they include FE-I4 compatible devices with 50 $\\mu $ m x 250 $\\mu $ m pitch and the new external punch-through biasing structures.", "It is planned to repeat the hit efficiency measurements with these sensors in a fluence range up to $10^{16}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ to confirm the better performance observed with the new biasing design with thinner devices." ], [ "Performance of n-in-p planar pixels at high $\\phi $", "The smaller pixel cell dimensions foreseen at HL-LHC pose a severe challenge also for the tracking in high pseudo-rapidity regions (high $\\eta $ ) of the new trackers.", "To investigate the hit efficiency for a cell size of 50 $\\mu $ m x 50$\\mu $ m at $\\eta $ =2.5, FE-I4 modules were investigated with an electron beam at DESY crossing the sensor with an angle of $\\phi $ = 80$^{0}$ with respect to the pixel surface.", "In this set-up the particles cross the pixel cells along the 50 $\\mu $ m side, allowing to study the performance of a 50 $\\mu $ m x 50$\\mu $ m sensor at high $\\eta $ .", "Figure REF shows how the cluster size along $\\eta $ is strongly dependent on the sensor thickness, with thinner sensors yielding the smaller clusters and resulting in the lower pixel occupancy.", "The first module employed in this test is a not irradiated device, 100 $\\mu $ m thick, from the VTT production.", "Track reconstruction was not possible for the data sample recorded, and the analysis was performed using the hit information of the long clusters corresponding to a single particle traversing the sensor.", "The cluster size and hit efficiency were calculated under varying assumptions on the allowed number of holes in the cluster (defined as cluster distance).", "Given the fact that the particle path in silicon is only around 50 $\\mu $ m long, with an expected most probable value (MPV) of the collected charge of 3100 e, a dedicated tuning was performed, with a low target threshold of 1000 e. Fig.REF shows the single hit inefficiency as a function of the maximum value of the distance between two pixels allowed while building a cluster (cluster separation).", "The single hit inefficiency is defined as the number of holes ($h^{c}$$_{miss}$ ) divided by the cluster length $w^c_x$ , not including the entrance and exit pixels, since these are 100$\\%$ efficient by definition.", "It can be observed that values of the hit efficiency close to 100$\\%$ are obtained with this thin sensor module, for all the cluster distance hypothesis, suggesting the feasibility of employing this kind of detectors at high $\\eta $ values at HL-LHC.", "Figure: Mean cluster width along the short pixel cell side for a FE-I4 module placed at high φ\\phi in the beam, as a function ofbeam incidence angle.", "The relationship is also valid for a pixel sensor with 50 μ\\mu m x 50μ\\mu m pitch at high η\\eta with respectto the beam.Figure: Hit efficiency of single pixels as a function of the cluster separation for an FE-I4module employing a 100 μ\\mu m thick sensor and tilted by 80 ∘ ^\\circ around its x axis withrespect to the perpendicular beam incidence.", "The single hit inefficiency is defined as the number of holes (h c h^{c} miss _{miss}) divided by the cluster lenght w x c w^c_x, not including the entrance and exit pixels, since these are 100%\\% efficient by definition.The sum is over all the N reconstructed clusters for a given choice of the cluster separation value.A similar analysis was performed with a module assembled with a 200 $\\mu $ m thick sensor and irradiated to a fluence of $2\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ .", "The charge distribution in the depth of the silicon bulk is shown in Fig.REF , for a bias voltage range between 300 and 800V, in the case of cluster size equal to 24 where, according to Fig.REF the particle crosses all the depth of the sensor.", "Pixel number 0 corresponds to the front side and pixel 24 to the backside of the sensor.", "It can be observed that for lower bias voltages the collected charge decreases in the backside while at a bias voltage of 800V, resulting in a higher electric field throughout the bulk of the sensor, the charge collection is more uniform.", "Figure: Charge collection, expressed in units of Time over Threshold, as a function of the pixel number for a value of the cluster size equal to 24, obtained with aFE-I4 module, irradiated to a fluence of 2×10 15 2\\times 10^{15} n eq / cm 2 \\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2, employing a 200 μ\\mu m thick sensor and tilted by 80 ∘ ^\\circ around its x axis withrespect to the perpendicular beam incidence.", "The pixel number equal to 0 corresponds to the front side and 24 to the backside of the sensor.The hit efficiency for the single pixels of this irradiated module is shown in Fig.REF , also in this case only for a cluster size of 24.", "At a bias voltage of 800V, the range of the hit efficiency is (81.5-93.4)$\\%$ at the different depths.", "Figure: Hit efficiency as a function of the pixel number for a value of the cluster size equal to 24,module employing a 200 μ\\mu m thick sensor and tilted by 80 ∘ ^\\circ around its x axis withrespect to the perpendicular beam incidence.", "The x axis on the top indicates the corresponding value of the depthin the silicon bulk with 0 being the frontside and 200 μ\\mu m the backside.The efficiencies of the first and the last pixels are by construction 100%\\% since they define the cluster lenght." ], [ "Conclusions", "Pixel modules assembled with thin planar sensors were investigated for the upgrade of the ATLAS pixel system at HL-LHC.", "Charge collection properties and hit efficiency were analysed after irradiation for sensor thicknesses in the range from 100 to 200 $\\mu $ m. For integrated radiation fluences of $(4-6)\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ , as expected for the second pixel layer at HL-LHC, the best performance was observed for 100 $\\mu $ m thick sensors which reach the same hit efficiency as thicker sensors already with a bias voltage of 200-300V.", "The highest hit efficiency obtained for perpendicular incident tracks at these fluences is around (97-98)$\\%$ , with the main inefficiency regions corresponding to the bias dot and the bias rail areas of the pixel cell.", "New biasing structures with an external bias dot common to four pixel cells have been investigated for thicker sensors and found to yield an higher hit efficiency, when compared to the standard design.", "It is now planned to study the performance of this new layout when implemented in 100 and 150 $\\mu $ m thin sensors.", "Studies of cluster properties were performed for the pixel modules in the innermost layer at high pseudorapidity for the new pixel system at HL-LHC, where the particles traverse several pixels.", "In these conditions, thinner sensors have the advantage of a lower cluster size which results in a reduced occupancy and, after irradiation, are expected to perform better since the higher electric field counteracts trapping.", "For the ATLAS Phase II detector a smaller pitch in the z direction is foreseen, which together with an optimal single pixel efficiency would allow to increase the precision for measuring the entrance and the exit point of particles crossing the pixel modules at high pseudorapidity and thus obtaining a track seed with the standalone innermost layer [17].", "The performance of a 50 $\\mu $ m pitch along z was therefore investigated with FE-I4 modules placed at high $\\phi $ angle with respect to the beam direction, showing a single pixel efficiency more than 99.6$\\%$ before irradiation for 100 $\\mu $ m thin sensors.", "A hit efficiency of (81.5-93.4$)\\%$ was instead reconstructed for a module with a 200 $\\mu $ m thin sensor irradiated at a fluence of $2\\times 10^{15}$ $\\mathrm {n}_{\\mathrm {eq}}/\\mathrm {cm}^2$ .", "These measurements after irradiation will be also continued with thinner sensors in a wider fluence range." ], [ "Acknowledgements", "This work has been partially performed in the framework of the CERN RD50 Collaboration The authors thank V. Cindro for the irradiation at JSI and A. Dierlamm for the irradiation at KIT.", "Supported by the H2020 project AIDA-2020, GA no.", "654168. http://aida2020.web.cern.ch/\"" ] ]	1606.05246
[ [ "Proof Complexity Lower Bounds from Algebraic Circuit Complexity" ], [ "Abstract We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes.", "This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted boolean circuit classes.", "Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity).", "We give two general methods of converting certain algebraic lower bounds into proof complexity ones.", "Our methods require stronger notions of lower bounds, which lower bound a polynomial as well as an entire family of polynomials it defines.", "Our techniques are reminiscent of existing methods for converting boolean circuit lower bounds into related proof complexity results, such as feasible interpolation.", "We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof complexity results.", "We complement our lower bounds by giving short refutations of the previously-studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems." ], [ "Introduction", "Propositional proof complexity aims to understand and analyze the computational resources required to prove propositional tautologies, in the same way that circuit complexity studies the resources required to compute boolean functions.", "A typical goal would be to establish, for a given proof system, super-polynomial lower bounds on the size of any proof of some propositional tautology.", "The seminal work of Cook and Reckhow [15] showed that this goal relates quite directly to fundamental hardness questions in computational complexity such as the vs. question: establishing super-polynomial lower bounds for every propositional proof system would separate from (and thus also ¶ from ).", "We refer the reader to Krajíček [44] for more on this subject.", "Propositional proof systems come in a large variety, as different ones capture different forms of reasoning, either reasoning used to actually prove theorems, or reasoning used by algorithmic techniques for different types of search problems (as failure of the algorithm to find the desired object constitutes a proof of its nonexistence).", "Much of the research in proof complexity deals with propositional proof systems originating from logic or geometry.", "Logical proof systems include such systems as resolution (whose variants are related to popular algorithms for automated theory proving and SAT solving), as well as the Frege proof system (capturing the most common logic text-book systems) and its many subsystems.", "Geometric proof systems include cutting-plane proofs, capturing reasoning used in algorithms for integer programming, as well as proof systems arising from systematic strategies for rounding linear- or semidefinite-programming such as the lift-and-project or sum-of-squares hierarchies.", "In this paper we focus on algebraic proof systems, in which propositional tautologies (or rather contradictions) are expressed as unsatisfiable systems of polynomial equations and algebraic tools are used to refute them.", "This study originates with the work of Beame, Impagliazzo, Krajíček, Pitassi and Pudlák [8], who introduced the Nullstellensatz refutation system (based on Hilbert's Nullstellensatz), followed by the Polynomial Calculus system of Clegg, Edmonds, and Impagliazzo [11], which is a “dynamic” version of Nullstellensatz.", "In both systems the main measures of proof size that have been studied are the degree and sparsity of the polynomials appearing in the proof.", "Substantial work has lead to a very good understanding of the power of these systems with respect to these measures (see for example [9], [57], [35], [38], [7], [4] and references therein).", "However, the above measures of degree and sparsity are rather rough measures of a complexity of a proof.", "As such, Grochow and Pitassi [33] have recently advocated measuring the complexity of such proofs by their algebraic circuit size and shown that the resulting proof system can polynomially simulate strong proof systems such as the Frege system.", "This naturally leads to the question of establishing lower bounds for this stronger proof system, even for restricted classes of algebraic circuits.", "In this work we establish such lower bounds for previously studied restricted classes of algebraic circuits, and show that these lower bounds are interesting by providing non-trivial upper bounds in these proof systems for refutations of interesting sets of polynomial equations.", "This provides what are apparently the first examples of lower bounds on the algebraic circuit size of propositional proofs in the Ideal Proof System (IPS) framework of Grochow and Pitassi [33].", "We note that obtaining proof complexity lower bounds from circuit complexity lower bounds is an established tradition that takes many forms.", "Most prominent are the lower bounds for subsystems of the Frege proof system defined by low-depth boolean circuits, and lower bounds of Pudlák [56] on Resolution and Cutting Planes system using the so-called feasible interpolation method.", "We refer the reader again to Krajíček [44] for more details.", "Our approach here for algebraic systems shares features with both of these approaches.", "The rest of this introduction is arranged as follows.", "In sec:Nullstellensatz we give the necessary background in algebraic proof complexity, and explain the IPS system.", "In sec:intro:circuits we define the algebraic complexity classes that will underlie the subsystems of IPS we will study.", "In sec:results we state our results and explain our techniques, for both the algebraic and proof complexity worlds." ], [ "Algebraic Proof Systems", "We now describe the algebraic proof systems that are the subject of this paper.", "If one has a set of polynomials (called axioms) $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ over some field $\\mathbb {F}$ , then (the weak version of) Hilbert's Nullstellensatz shows that the system $f_1({\\overline{x}})=\\cdots =f_m({\\overline{x}})=0$ is unsatisfiable (over the algebraic closure of $\\mathbb {F}$ ) if and only if there are polynomials $g_1,\\ldots ,g_m\\in \\mathbb {F}[{\\overline{x}}]$ such that $\\sum _j g_j({\\overline{x}})f_j({\\overline{x}})=1$ (as a formal identity), or equivalently, that 1 is in the ideal generated by the $\\lbrace f_j\\rbrace _j$ .", "Beame, Impagliazzo, Krajíček, Pitassi, and Pudlák [8] suggested to treat these $\\lbrace g_j\\rbrace _j$ as a proof of the unsatisfiability of this system of equations, called a Nullstellensatz refutation.", "This is in particular relevant for complexity theory as one can restrict attention to boolean solutions to this system by adding the boolean axioms, that is, adding the polynomials $\\lbrace x_i^2-x_i\\rbrace _{i=1}^n$ to the system.", "As such, one can then naturally encode $$ -complete problems such as the satisfiability of 3CNF formulas as the satisfiability of a system of constant-degree polynomials, and a Nullstellensatz refutation is then an equation of the form $\\sum _{j=1}^m g_j({\\overline{x}})f_j({\\overline{x}})+\\sum _{i=1}^n h_i({\\overline{x}})(x_i^2-x_i)=1$ for $g_j,h_i\\in \\mathbb {F}[{\\overline{x}}]$ .", "This proof system is sound (only refuting unsatisfiable systems over $\\lbrace 0,1\\rbrace ^n$ ) and complete (refuting any unsatisfiable system, by Hilbert's Nullstellensatz).", "Given that the above proof system is sound and complete, it is then natural to ask what is its power to refute unsatisfiable systems of polynomial equations over $\\lbrace 0,1\\rbrace ^n$ .", "To understand this question one must define the notion of the size of the above refutations.", "Two popular notions are that of the degree, and the sparsity (number of monomials).", "One can then show (see for example Pitassi [55]) that for any unsatisfiable system which includes the boolean axioms, there exist a refutation where the $g_j$ are multilinear and where the $h_i$ have degree at most $O(n+d)$ , where each $f_j$ has degree at most $d$ .", "In particular, this implies that for any unsatisfiable system with $d=O(n)$ there is a refutation of degree $O(n)$ and involving at most $\\exp (O(n))$ monomials.", "This intuitively agrees with the fact that $$ is a subset of non-deterministic exponential time.", "Building on the suggestion of Pitassi [55] and various investigations into the power of strong algebraic proof systems ([28], [62], [63]), Grochow and Pitassi [33] have recently considered more succinct descriptions of polynomials where one measures the size of a polynomial by the size of an algebraic circuit needed to compute it.", "This is potentially much more powerful as there are polynomials such as the determinant which are of high degree and involve exponentially many monomials and yet can be computed by small algebraic circuits.", "They named the resulting system the Ideal Proof System (IPS) which we now define.", "[Ideal Proof System (IPS), Grochow-Pitassi [33]] Let $f_1({\\overline{x}}),\\ldots ,f_m({\\overline{x}})\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a system of polynomials.", "An IPS refutation for showing that the polynomials $\\lbrace f_j\\rbrace _j$ have no common solution in $\\lbrace 0,1\\rbrace ^n$ is an algebraic circuit $C({\\overline{x}},{\\overline{y}},{\\overline{z}})\\in \\mathbb {F}[{\\overline{x}},y_1,\\ldots ,y_m,z_1,\\ldots ,z_n]$ , such that $C({\\overline{x}},{\\overline{0}},{\\overline{0}}) = 0$ .", "$C({\\overline{x}},f_1({\\overline{x}}),\\ldots ,f_m({\\overline{x}}),x_1^2-x_1,\\ldots ,x_n^2-x_n)=1$ .", "The size of the IPS refutation is the size of the circuit $C$ .", "If $C$ is of individual degree $\\le 1$ in each $y_j$ and $z_i$ , then this is a linear IPS refutation (called Hilbert IPS by Grochow-Pitassi [33]), which we will abbreviate as IPS$_{\\text{LIN}}$ .", "If $C$ is of individual degree $\\le 1$ only in the $y_j$ then we say this is a $\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation.", "If $C$ comes from a restricted class of algebraic circuits $\\mathcal {C}$ , then this is a called a $\\mathcal {C}$ -IPS refutation, and further called a $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ refutation if $C$ is linear in ${\\overline{y}},{\\overline{z}}$ , and a $\\mathcal {C}$ -$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation if $C$ is linear in ${\\overline{y}}$ .", "Notice also that our definition here by default adds the equations $\\lbrace x_i^2-x_i\\rbrace _i$ to the system $\\lbrace f_j\\rbrace _j$ .", "For convenience we will often denote the equations $\\lbrace x_i^2-x_i\\rbrace _i$ as ${\\overline{x}}^2-{\\overline{x}}$ .", "One need not add the equations ${\\overline{x}}^2-{\\overline{x}}$ to the system in general, but this is the most interesting regime for proof complexity and thus we adopt it as part of our definition.", "The $\\mathcal {C}$ -IPS system is sound for any $\\mathcal {C}$ , and Hilbert's Nullstellensatz shows that $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ is complete for any complete class of algebraic circuits $\\mathcal {C}$ (that is, classes which can compute any polynomial, possibly requiring exponential complexity).", "We note that we will also consider non-complete classes such as multilinear-formulas (which can only compute multilinear polynomials, but are complete for multilinear polynomials), where we will show that the multilinear-formula-IPS$_{\\text{LIN}}$ system is not complete for the language of all unsatisfiable sets of multilinear polynomials (ex:multi-form:incomplete), while the stronger multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ version is complete (res:multilin:simulate-sparse).", "However, for the standard conversion of unsatisfiable CNFs into polynomial systems of equations, the multilinear-formula-IPS$_{\\text{LIN}}$ system is complete (thm:GrochowPitassi14).", "Grochow-Pitassi [33] proved the following theorem, showing that the IPS system has surprising power and that lower bounds on this system give rise to computational lower bounds.", "[Grochow-Pitassi [33]] Let $\\varphi =C_1\\wedge \\cdots \\wedge C_m$ be an unsatisfiable CNF on $n$ -variables, and let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_m]$ be its encoding as a polynomial system of equations.", "If there is a size-$s$ Frege proof (resp.", "Extended Frege) that $\\lbrace f_j\\rbrace _j,\\lbrace x_i^2-x_i\\rbrace _i$ is unsatisfiable, then there is a formula-IPS$_{\\text{LIN}}$ (resp.", "circuit-IPS$_{\\text{LIN}}$ ) refutation of size $(n,m,s)$ that is checkable in randomized $(n,m,s)$ time.We note that Grochow and Pitassi [33] proved this for Extended Frege and circuits, but essentially the same proof relates Frege and formula size.", "Further, $\\lbrace f_j\\rbrace _j,\\lbrace x_i^2-x_i\\rbrace _i$ has a IPS$_{\\text{LIN}}$ refutation, where the refutation uses multilinear polynomials in $$ .", "Thus, if every IPS refutation of $\\lbrace f_j\\rbrace _j,\\lbrace x_i^2-x_i\\rbrace _i$ requires formula (resp.", "circuit) size $\\ge s$ , then there is an explicit polynomial (that is, in $$ ) that requires size $\\ge s$ algebraic formulas (resp.", "circuits).", "One point to note is that the transformation from Extended Frege to IPS refutations yields circuits of polynomial size but without any guarantee on their degree.", "In particular, such circuits can compute polynomials of exponential degree.", "In contrast, the conversion from Frege to IPS refutations yields polynomial sized algebraic formulas and those compute polynomials of polynomially bounded degree.", "This range of parameters, polynomials of polynomially bounded degree, is the more common setting studied in algebraic complexity.", "The fact that $\\mathcal {C}$ -IPS refutations are efficiently checkable (with randomness) follows from the fact that we need to verify the polynomial identities stipulated by the definition.", "That is, one needs to solve an instance of the polynomial identity testing (PIT) problem for the class $\\mathcal {C}$ : given a circuit from the class $\\mathcal {C}$ decide whether it computes the identically zero polynomial.", "This problem is solvable in probabilistic polynomial time ($$ ) for general algebraic circuits, and there are various restricted classes for which deterministic algorithms are known (see sec:PIT).", "Motivated by the fact that PIT of non-commutative formulas These are formulas over a set of non-commuting variables.", "can be solved deterministically ([61]) and admit exponential-size lower bounds ([49]), Li, Tzameret and Wang [47] have shown that IPS over non-commutative polynomials (along with additional commutator axioms) can simulate Frege (they also provided a quasipolynomial simulation of IPS over non-commutative formulas by Frege; see Li, Tzameret and Wang [47] for more details).", "[Li, Tzameret and Wang [47]] Let $\\varphi =C_1\\wedge \\cdots \\wedge C_m$ be an unsatisfiable CNF on $n$ -variables, and let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_m]$ be its encoding as a polynomial system of equations.", "If there is a size-$s$ Frege proof that $\\lbrace f_j\\rbrace _j,\\lbrace x_i^2-x_i\\rbrace _i$ is unsatisfiable, then there is a non-commutative-IPS refutation of formula-size $(n,m,s)$ , where the commutator axioms $x_ix_j-x_jx_i$ are also included in the polynomial system being refuted.", "Further, this refutation is checkable in deterministic $(n,m,s)$ time.", "The above results naturally motivate studying $\\mathcal {C}$ -IPS for various restricted classes of algebraic circuits, as lower bounds for such proofs then intuitively correspond to restricted lower bounds for the Extended Frege proof system.", "In particular, as exponential lower bounds are known for non-commutative formulas ([49]), this possibly suggests that such methods could even attack the full Frege system itself." ], [ "Algebraic Circuit Classes", "Having motivated $\\mathcal {C}$ -IPS for restricted circuit classes $\\mathcal {C}$ , we now give formal definitions of the algebraic circuit classes of interest to this paper, all of which were studied independently in algebraic complexity.", "Some of them capture the state-of-art in our ability to prove lower bounds and provide efficient deterministic identity tests, so it is natural to attempt to fit them into the proof complexity framework.", "We define each and briefly explain what we know about it.", "As the list is long, the reader may consider skipping to the results (sec:results), and refer to the definitions of these classes as they arise.", "Algebraic circuits and formula (over a fixed chosen field) compute polynomials via addition and multiplication gates, starting from the input variables and constants from the field.", "For background on algebraic circuits in general and their complexity measures we refer the reader to the survey of Shpilka and Yehudayoff [75].", "We next define the restricted circuit classes that we will be studying in this paper." ], [ "Low Depth Classes", "We start by defining what are the simplest and most restricted classes of algebraic circuits.", "The first class simply represents polynomials as a sum of monomials.", "This is also called the sparse representation of the polynomial.", "Notationally we call this model $\\sum \\prod $ formulas (to capture the fact that polynomials computed in the class are represented simply as sums of products), but we will more often call these polynomials “sparse”.", "The class $\\mathcal {C}=\\sum \\prod $ compute polynomials in their sparse representation, that is, as a sum of monomials.", "The graph of computation has two layers with an addition gate at the top and multiplication gates at the bottom.", "The size of a $\\sum \\prod $ circuit of a polynomial $f$ is the multiplication of the number of monomials in $f$ , the number of variables, and the degree.", "This class of circuits is what is used in the Nullstellensatz proof system.", "In our terminology $\\sum \\prod $ -IPS$_{\\text{LIN}}$ is exactly the Nullstellensatz proof system.", "Another restricted class of algebraic circuits is that of depth-3 powering formulas (sometimes also called “diagonal depth-3 circuits”).", "We will sometimes abbreviate this name as a “$\\sum \\bigwedge \\sum $ formula”, where $\\bigwedge $ denotes the powering operation.", "Specifically, polynomials that are efficiently computed by small formulas from this class can be represented as sum of powers of linear functions.", "This model appears implicitly in Shpilka [70] and explicitly in the work of Saxena [67].", "The class of depth-3 powering formulas, denoted $\\sum \\bigwedge \\sum $ , computes polynomials of the following form $f({\\overline{x}})=\\sum _{i=1}^{s} \\ell _i({\\overline{x}})^{d_i},$ where $\\ell _i({\\overline{x}})$ are linear functions.", "The degree of this $\\sum \\bigwedge \\sum $ representation of $f$ is $\\max _i\\lbrace d_i\\rbrace $ and its size is $n\\cdot \\sum _{i=1}^{s}(d_i+1)$ .", "One reason for considering this class of circuits is that it is a simple, but non-trivial model that is somewhat well-understood.", "In particular, the partial derivative method of Nisan-Wigderson [51] implies lower bounds for this model and efficient polynomial identity testing algorithms are known ([67], [5], [24], [25], [27], as discussed further in sec:PIT).", "We also consider a generalization of this model where we allow powering of low-degree polynomials.", "The class $\\sum \\bigwedge \\sum \\prod ^{t} $ computes polynomials of the following form $f({\\overline{x}})=\\sum _{i=1}^{s} f_i({\\overline{x}})^{d_i}\\;,$ where the degree of the $f_i({\\overline{x}})$ is at most $t$ .", "The size of this representation is $\\binom{n+t}{t} \\cdot \\sum _{i=1}^{s}(d_i+1)$ .", "We remark that the reason for defining the size this way is that we think of the $f_i$ as represented as sum of monomials (there are $\\binom{n+t}{t}$ $n$ -variate monomials of degree at most $t$ ) and the size captures the complexity of writing this as an algebraic formula.", "This model is the simplest that requires the method of shifted partial derivatives of Kayal [42], [31] to establish lower bounds, and this has recently been generalized to obtain polynomial identity testing algorithms ([22], as discussed further in sec:PIT)." ], [ "Oblivious Algebraic Branching Programs", "Algebraic branching programs (ABPs) form a model whose computational power lies between that of algebraic circuits and algebraic formulas, and certain read-once and oblivious ABPs are a natural setting for studying the partial derivative matrix lower bound technique of Nisan [49].", "[Nisan [49]] An algebraic branching program (ABP) with unrestricted weights of depth $D$ and width $\\le r$ , on the variables $x_1,\\ldots ,x_n$ , is a directed acyclic graph such that: The vertices are partitioned in $D+1$ layers $V_0,\\ldots ,V_D$ , so that $V_0=\\lbrace s\\rbrace $ ($s$ is the source node), and $V_D=\\lbrace t\\rbrace $ ($t$ is the sink node).", "Further, each edge goes from $V_{i-1}$ to $V_{i}$ for some $0< i\\le D$ .", "$\\max \|V_i\|\\le r$ .", "Each edge $e$ is weighted with a polynomial $f_e\\in \\mathbb {F}[{\\overline{x}}]$ .", "The (individual) degree $d$ of the ABP is the maximum (individual) degree of the edge polynomials $f_e$ .", "The size of the ABP is the product $n\\cdot r\\cdot d\\cdot D$ , Each $s$ -$t$ path is said to compute the polynomial which is the product of the labels of its edges, and the algebraic branching program itself computes the sum over all $s$ -$t$ paths of such polynomials.", "There are also the following restricted ABP variants.", "An algebraic branching program is said to be oblivious if for every layer $\\ell $ , all the edge labels in that layer are univariate polynomials in a single variable $x_{i_\\ell }$ .", "An oblivious branching program is said to be a read-once oblivious ABP (roABP) if each $x_i$ appears in the edge label of exactly one layer, so that $D=n$ .", "That is, each $x_i$ appears in the edge labels in at exactly one layer.", "The layers thus define a variable order, which will be $x_1<\\cdots <x_n$ if not otherwise specified.", "An oblivious branching program is said to be a read-$k$ oblivious ABP if each variable $x_i$ appears in the edge labels of exactly $k$ layers, so that $D=kn$ .", "An ABP is non-commutative if each $f_e$ is from the ring $\\mathbb {F}{\\langle {\\overline{x}}\\rangle }$ of non-commuting variables and has $\\deg f_e\\le 1$ , so that the ABP computes a non-commutative polynomial.", "Intuitively, roABPs are the algebraic analog of read-once boolean branching programs, the non-uniform model of the class $$ , which are well-studied in boolean complexity.", "Algebraically, roABPs are also well-studied.", "In particular, roABPs are essentially equivalent to non-commutative ABPs ([25]), a model at least as strong as non-commutative formulas.", "That is, as an roABP reads the variables in a fixed order (hence not using commutativity) it can be almost directly interpreted as a non-commutative ABP.", "Conversely, as non-commutative multiplication is ordered, one can interpret a non-commutative polynomial in a read-once fashion by (commutatively) exponentiating a variable to its index in a monomial.", "For example, the non-commutative $xy-yx$ can be interpreted commutatively as $x^1y^2-y^1x^2=xy^2-x^2y$ , and one can show that this conversion preserves the relevant ABP complexity ([25]).", "The study of non-commutative ABPs dates to Nisan [49], who proved lower bounds for non-commutative ABPs (and thus also for roABPs, in any order).", "In a sequence of more recent papers, polynomial identity testing algorithms were devised for roABPs ([61], [23], [25], [27], [2], see also sec:PIT).", "In terms of proof complexity, Tzameret [76] studied a proof system with lines given by roABPs, and recently Li, Tzameret and Wang [47] (thm:LTW) showed that IPS over non-commutative formulas is essentially equivalent in power to the Frege proof system.", "Due to the close connections between non-commutative ABPs and roABPs, this last result suggests the importance of proving lower bounds for roABP-IPS as a way of attacking lower bounds for the Frege proof system (although our work obtains roABP-IPS$_{\\text{LIN}}$ lower bounds without obtaining non-commutative-IPS$_{\\text{LIN}}$ lower bounds).", "Finally, we mention that recently Anderson, Forbes, Saptharishi, Shpilka, and Volk [1] obtained exponential lower bounds for read-$k$ oblivious ABPs (when $k=o(\\log n/\\log \\log n)$ ) as well as a slightly subexponential polynomial identity testing algorithm." ], [ "Multilinear Formulas", "The last model that we consider is that of multilinear formulas.", "[Multilinear formula] An algebraic formula is a multilinear formula if the polynomial computed by each gate of the formula is multilinear (as a formal polynomial, that is, as an element of $\\,\\mathbb {F}[x_1,\\ldots ,x_n]$ ).", "The product depth is the maximum number of multiplication gates on any input-to-output path in the formula.", "Raz [59], [58] proved quasi-polynomial lower bounds for multilinear formulas and separated multilinear formulas from multilinear circuits.", "Raz and Yehudayoff proved exponential lower bounds for small depth multilinear formulas [65].", "Only slightly sub-exponential polynomial identity testing algorithms are known for small-depth multilinear formulas ([54])." ], [ "Our Results and Techniques", "We now briefly summarize our results and techniques, stating some results in less than full generality to more clearly convey the result.", "We present the results in the order that we later prove them.", "We start by giving upper bounds for the IPS (sec:results:upper).", "We then describe our functional lower bounds and the IPS$_{\\text{LIN}}$ lower bounds they imply (sec:results:funct).", "Finally, we discuss lower bounds for multiples and state our lower bounds for IPS (sec:intro:lbs-mult)." ], [ "Upper Bounds for Proofs within Subclasses of IPS", "Various previous works have studied restricted algebraic proof systems and shown non-trivial upper bounds.", "The general simulation (thm:GrochowPitassi14) of Grochow and Pitassi [33] showed that the formula-IPS and circuit-IPS systems can simulate powerful proof systems such as Frege and Extended Frege, respectively.", "The work of Li, Tzameret and Wang [47] (thm:LTW) show that even non-commutative-formula-IPS can simulate Frege.", "The work of Grigoriev and Hirsch [28] showed that proofs manipulating depth-3 algebraic formulas can refute hard axioms such as the pigeonhole principle, the subset-sum axiom, and Tseitin tautologies.", "The work of Raz and Tzameret [62], [63] somewhat strengthened their results by restricting the proof to depth-3 multilinear proofs (in a dynamic system, see sec:alg-proofs).", "However, these upper bounds are for proof systems (IPS or otherwise) for which no proof lower bounds are known.", "As such, in this work we also study upper bounds for restricted subsystems of IPS.", "In particular, we compare linear-IPS versus the full IPS system, as well as showing that even for restricted $\\mathcal {C}$ , $\\mathcal {C}$ -IPS can refute interesting unsatisfiable systems of equations arising from $$ -complete problems (and we will obtain corresponding proof lower bounds for these $\\mathcal {C}$ -IPS systems).", "Our first upper bound is to show that linear-IPS can simulate the full IPS proof system when the axioms are computationally simple, which essentially resolves a question of Grochow and Pitassi [33].", "[res:h-ipsvips] For $\|\\mathbb {F}\|\\ge (d)$ , if $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ are degree-$d$ polynomials computable by size-$s$ algebraic formulas (resp.", "circuits) and they have a size-$t$ formula-IPS (resp.", "circuit-IPS) refutation, then they also have a size-$(d,s,t)$ formula-IPS$_{\\text{LIN}}$ (resp.", "circuit-IPS$_{\\text{LIN}}$ ) refutation.", "This theorem is established by pushing the “non-linear” dependencies on the axioms into the IPS refutation itself, which is possible as the axioms are assumed to themselves be computable by small circuits.", "We note that Grochow and Pitassi [33] showed such a conversion, but only for IPS refutations computable by sparse polynomials.", "Also, we remark that this result holds even for circuits of unbounded degree, as opposed to just those of polynomial degree.", "We then turn our attention to IPS involving only restricted classes of algebraic circuits, and show that they are complete proof systems.", "This is clear for complete models of algebraic circuits such as sparse polynomials, depth-3 powering formulas Showing that depth-3 powering formulas are complete (in large characteristic) can be seen from the fact that any multilinear monomial can be computed in this model, see for example Fischer [19].", "and roABPs.", "The models of sparse-IPS$_{\\text{LIN}}$ and roABP-IPS$_{\\text{LIN}}$ can efficiently simulate the Nullstellensatz proof system measured in terms of number of monomials, as the former is equivalent to this system, and the latter follows as sparse polynomials have small roABPs.", "Note that depth-3 powering formulas cannot efficiently compute sparse polynomials in general (res:lbs-mult:sumpowsum) so cannot efficiently simulate the Nullstellensatz system.", "For multilinear formulas, showing completeness (much less an efficient simulation of sparse-IPS$_{\\text{LIN}}$ ) is more subtle as not every polynomial is multilinear, however the result can be obtained by a careful multilinearization.", "[ex:multi-form:incomplete, res:multilin:simulate-sparse] The proof systems of sparse-IPS$_{\\text{LIN}}$ , $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ (in large characteristic fields), and roABP-IPS$_{\\text{LIN}}$ are complete proof systems (for systems of polynomials with no boolean solutions).", "The multilinear-formula-IPS$_{\\text{LIN}}$ proof system is not complete, but the depth-2 multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ proof system is complete (for multilinear axioms) and can polynomially simulate sparse-IPS$_{\\text{LIN}}$ (for low-degree axioms).", "However, we recall that multilinear-formula-IPS$_{\\text{LIN}}$ is complete when refuting unsatisfiable CNF formulas (thm:GrochowPitassi14).", "We next consider the equation $\\sum _{i=1}^n \\alpha _i x_i-\\beta $ along with the boolean axioms $\\lbrace x_i^2-x_i\\rbrace _i$ .", "Deciding whether this system of equations is satisfiable is the $$ -complete subset-sum problem, and as such we do not expect small refutations in general (unless $=$ ).", "Indeed, Impagliazzo, Pudlák, and Sgall [38] (thm:IPS99) have shown lower bounds for refutations in the polynomial calculus system (and thus also the Nullstellensatz system) even when ${\\overline{\\alpha }}={\\overline{1}}$ .", "Specifically, they showed that such refutations require both $\\Omega (n)$ -degree and $\\exp (\\Omega (n))$ -many monomials, matching the worst-case upper bounds for these complexity measures.", "In the language of this paper, they gave $\\exp (\\Omega (n))$ -size lower bounds for refuting this system in $\\sum \\prod $ -IPS$_{\\text{LIN}}$ (which is equivalent to the Nullstellensatz proof system).", "In contrast, we establish here $(n)$ -size refutations for ${\\overline{\\alpha }}={\\overline{1}}$ in the restricted proof systems of roABP-IPS$_{\\text{LIN}}$ and depth-3 multilinear-formula-IPS$_{\\text{LIN}}$ (despite the fact that multilinear-formula-IPS$_{\\text{LIN}}$ is not complete).", "[res:ips-ubs:subset:roABP, res:ips-ubs:subset:mult-form] Let $\\mathbb {F}$ be a field of characteristic $\\operatorname{char}(\\mathbb {F})>n$ .", "Then the system of polynomial equations $\\sum _{i=1}^n x_i-\\beta $ , $\\lbrace x_i^2-x_i\\rbrace _{i=1}^n$ is unsatisfiable for $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ , and there are explicit $(n)$ -size refutations within roABP-IPS$_{\\text{LIN}}$ , as well as within depth-3 multilinear-formula-IPS$_{\\text{LIN}}$ .", "This theorem is proven by noting that the polynomial $p(t):=\\prod _{k=0}^n (t-k)$ vanishes on $\\sum _i x_i$ modulo $\\lbrace x_i^2-x_i\\rbrace _{i=1}^n$ , but $p(\\beta )$ is a nonzero constant.", "This implies that $\\sum _i x_i-\\beta $ divides $p(\\sum _i x_i)-p(\\beta )$ .", "Denoting the quotient by $f({\\overline{x}})$ , it follows that $\\frac{1}{-p(\\beta )}\\cdot f({\\overline{x}})\\cdot (\\sum _i x_i-\\beta )\\equiv 1\\mod {\\lbrace }x_i^2-x_i\\rbrace _{i=1}^n$ , which is nearly a linear-IPS refutation except for the complexity of establishing this relation over the boolean cube.", "We show that the quotient $f$ is easily expressed as a depth-3 powering circuit.", "Unfortunately, proving the above equivalence to 1 modulo the boolean cube is not possible in the depth-3 powering circuit model.", "However, by moving to more powerful models (such as roABPs and multilinear formulas) we can give proofs of this multilinearization to 1 and thus give proper IPS refutations." ], [ "Linear-IPS Lower Bounds via Functional Lower Bounds", "Having demonstrated the power of various restricted classes of IPS refutations by refuting the subset-sum axiom, we now turn to lower bounds.", "We give two paradigms for establishing lower bounds, the first of which we discus here, named a functional circuit lower bound.", "This idea appeared in the work of Grigoriev and Razborov [34] as well as in the recent work of Forbes, Kumar and Saptharishi [20].", "We briefly motivate this type of lower bound as a topic of independent interest in algebraic circuit complexity, and then discuss the lower bounds we obtain and their implications to obtaining proof complexity lower bounds.", "In boolean complexity, the primary object of interest are functions.", "Generalizing slightly, one can even seek to compute functions $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ for some field $\\mathbb {F}$ .", "In contrast, in algebraic complexity one seeks to compute polynomials as elements of the ring $\\mathbb {F}[x_1,\\ldots ,x_n]$ .", "These two regimes are tied by the fact that every polynomial $f\\in \\mathbb {F}[{\\overline{x}}]$ induces a function $\\hat{f}:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ via the evaluation $\\hat{f}:{\\overline{x}}\\mapsto f({\\overline{x}})$ .", "That is, the polynomial $f$ functionally computes the function $\\hat{f}$ .", "As an example, $x^2-x$ functionally computes the zero function despite being a nonzero polynomial.", "Traditional algebraic circuit lower bounds for the $n\\times n$ permanent are lower bounds for computing $\\operatorname{perm}_n$ as an element in the ring $\\mathbb {F}[\\lbrace x_{i,j}\\rbrace _{1\\le i,j\\le n}]$ .", "This is a strong notion of “computing the permanent”, while one can consider the weaker notion of functionally computing the permanent, that is, a polynomial $f\\in \\mathbb {F}[\\lbrace x_{i,j}\\rbrace ]$ such that $f=\\operatorname{perm}_n$ over $\\lbrace 0,1\\rbrace ^{n\\times n}$ , where $f$ is not required to equal $\\operatorname{perm}_n$ as a polynomial.", "As $\\operatorname{perm}_n:\\lbrace 0,1\\rbrace ^{n\\times n}\\rightarrow \\mathbb {F}$ is $\\#¶$ -hard (for fields of large characteristic), assuming plausible conjectures (such as the polynomial hierarchy being infinite) it follows that any polynomial $f$ which functionally computes $\\operatorname{perm}_n$ must require large algebraic circuits.", "Unconditionally obtaining such a result is what we term a functional lower bound.", "[Functional Circuit Lower Bound ([34], [20])] Obtain an explicit function $\\hat{f}:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ such that for any polynomial $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ satisfying $f({\\overline{x}})=\\hat{f}({\\overline{x}})$ for all ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , it must be that $f$ requires large algebraic circuits.", "Obtaining such a result is challenging, in part because one must lower bound all polynomials agreeing with the function $\\hat{f}$ (of which there are infinitely many).", "Prior work ([29], [34], [46]) has established functional lower bounds for functions when computing with polynomials over constant-sized finite fields, and the recent work of Forbes, Kumar and Saptharishi [20] has established some lower bounds for any field.", "While it is natural to hope that existing methods would yield such lower bounds, many lower bound techniques inherently use that algebraic computation is syntactic.", "In particular, techniques involving partial derivatives (which include the partial derivative method of Nisan-Wigderson [51] and the shifted partial derivative method of Kayal [42], [31]) cannot as is yield functional lower bounds as knowing a polynomial on $\\lbrace 0,1\\rbrace ^n$ is not enough to conclude information about its partial derivatives.", "We now explain how functional lower bounds imply lower bounds for linear-IPS refutations in certain cases.", "Suppose one considers refutations of the unsatisfiable polynomial system $f({\\overline{x}}),\\lbrace x_i^2-x_i\\rbrace _{i=1}^n$ .", "A linear-IPS refutation would yield an equation of the form $g({\\overline{x}})\\cdot f({\\overline{x}})+\\sum _i h_i({\\overline{x}})\\cdot (x_i^2-x_i)=1$ for some polynomials $g,h_i\\in \\mathbb {F}[{\\overline{x}}]$ .", "Viewing this equation modulo the boolean cube, we have that $g({\\overline{x}})\\cdot f({\\overline{x}})\\equiv 1\\mod {\\lbrace }x_i^2-x_i\\rbrace _i$ .", "Equivalently, since $f({\\overline{x}})$ is unsatisfiable over $\\lbrace 0,1\\rbrace ^n$ , we see that $g({\\overline{x}})={1}{f({\\overline{x}})}$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , as $f({\\overline{x}})$ is never zero so this fraction is well-defined.", "It follows that if the function ${\\overline{x}}\\mapsto {1}{f({\\overline{x}})}$ induces a functional lower bound then $g({\\overline{x}})$ must require large complexity, yielding the desired linear-IPS lower bound.", "Thus, it remains to instantiate this program.", "While we are successful, we should note that this approach as is seems to only yield proof complexity lower bounds for systems with one non-boolean axiom and thus cannot encode polynomial systems where each equation depends on $O(1)$ variables (such as those naturally arising from 3CNFs).", "Our starting point is to observe that the subset-sum axiom already induces a weak form of functional lower bound, where the complexity is measured by degree.", "[res:subsetsum:deg:ge] Let $\\mathbb {F}$ be a field of a characteristic at least $(n)$ and $\\beta \\notin \\lbrace 0,\\ldots ,n\\rbrace $ .", "Then $\\sum _i x_i-\\beta ,\\lbrace x_i^2-x_i\\rbrace _i$ is unsatisfiable and any polynomial $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ with $f({\\overline{x}})=\\frac{1}{\\sum _i x_i-\\beta }$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , satisfies $\\deg f\\ge n$ .", "A lower bound of ${\\lceil {\\frac{n}{2}}\\rceil }+1$ was previously established by Impagliazzo, Pudlák, and Sgall [38] (thm:IPS99), but the bound of `$n$ ' (which is tight) will be crucial for our results.", "We then lift this result to obtain lower bounds for stronger models of algebraic complexity.", "In particular, by replacing “$x_i$ ” with “$x_iy_i$ ” we show that the function $\\frac{1}{\\sum _i x_iy_i-\\beta }$ has maximal evaluation dimension between ${\\overline{x}}$ and ${\\overline{y}}$ , which is some measure of interaction between the variables in ${\\overline{x}}$ and those in ${\\overline{y}}$ (see sec:eval-dim).", "This measure is essentially functional, so that one can lower bound this measure by understanding the functional behavior of the polynomial on finite sets such as the boolean cube.", "Our lower bound for evaluation dimension follows by examining the above degree bound.", "Using known relations between this complexity measure and algebraic circuit classes, we can obtain lower bounds for depth-3 powering linear-IPS.", "[res:lbs-fn:lbs-ips:fixed-order] Let $\\mathbb {F}$ be a field of characteristic $\\ge (n)$ and $\\beta \\notin \\lbrace 0,\\ldots ,n\\rbrace $ .", "Then $\\sum _{i=1}^n x_iy_i-\\beta ,\\lbrace x_i^2-x_i\\rbrace _i,\\lbrace y_i^2-y_i\\rbrace _i$ is unsatisfiable and any $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ refutation requires size $\\ge \\exp (\\Omega (n))$ .", "The above axiom only gets maximal interaction between the variables across a fixed partition of the variables.", "By introducing auxiliary variables we can create such interactions in variables across any partition of (some) of the variables.", "By again invoking results showing such structure implies computational hardness we obtain more linear-IPS lower bounds.", "[res:lbs-fn:lbs-ips:vary-order] Let $\\mathbb {F}$ be a field of characteristic $\\ge (n)$ and $\\beta \\notin \\lbrace 0,\\ldots ,\\binom{2n}{2}\\rbrace $ .", "Then $\\sum _{i<j} z_{i,j}x_ix_j-\\beta ,\\lbrace x_i^2-x_i\\rbrace _{i=1}^n,\\lbrace z_{i,j}^2-z_{i,j}\\rbrace _{i<j}$ is unsatisfiable, and any roABP-IPS$_{\\text{LIN}}$ refutation (in any variable order) requires $\\exp (\\Omega (n))$ -size.", "Further, any multilinear-formula-IPS refutation requires $n^{\\Omega (\\log n)}$ -size, and any product-depth-$d$ multilinear-formula-IPS refutation requires $n^{\\Omega (({n}{\\log n})^{1/d}/d^2)}$ -size.", "Note that our result for multilinear-formulas is not just for the linear-IPS system, but actually for the full multilinear-formula-IPS system.", "Thus, we show that even though roABP-IPS$_{\\text{LIN}}$ and depth-3 multilinear formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ can refute the subset-sum axiom in polynomial size, slight variants of this axiom do not have polynomial-size refutations." ], [ "Lower Bounds for Multiples", "While the above paradigm can establish super-polynomial lower bounds for linear-IPS, it does not seem able to establish lower bounds for the general IPS proof system over non-multilinear polynomials, even for restricted classes.", "This is because such systems would induce equations such as $h({\\overline{x}}) f({\\overline{x}})^2+g({\\overline{x}}) f({\\overline{x}})\\equiv 1 \\mod {\\lbrace }x_i^2-x_i\\rbrace _{i=1}^n$ , where we need to design a computationally simple axiom $f$ so that this equation implies at least one of $h$ or $g$ is of large complexity.", "In a linear-IPS proof it must be that $h$ is zero, so that for any ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ we can solve for $g({\\overline{x}})$ , that is, $g({\\overline{x}})={1}{f({\\overline{x}})}$ .", "However, in general knowing $f({\\overline{x}})$ does not uniquely determine $g({\\overline{x}})$ or $h({\\overline{x}})$ , which makes this approach significantly more complicated.", "Further, even though we can efficiently simulate IPS by linear-IPS (res:h-ipsvips) in general, this simulation increases the complexity of the proof so that even if one started with a $\\mathcal {C}$ -IPS proof for a restricted circuit class $\\mathcal {C}$ the resulting IPS$_{\\text{LIN}}$ proof may not be in $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ .", "As such, we introduce a second paradigm, called lower bounds for multiples, which can yield $\\mathcal {C}$ -IPS lower bounds for various restricted classes $\\mathcal {C}$ .", "We begin by defining this question.", "[Lower Bounds for Multiples] Design an explicit polynomial $f({\\overline{x}})$ such that for any nonzero $g({\\overline{x}})$ we have that the multiple $g({\\overline{x}})f({\\overline{x}})$ is hard to compute.", "We now explain how such lower bounds yield IPS lower bounds.", "Consider the system $f,\\lbrace x_i^2-x_i\\rbrace _i$ with a single non-boolean axiom.", "An IPS refutation is a circuit $C({\\overline{x}},y,{\\overline{z}})$ such that $C({\\overline{x}},0,{\\overline{0}})=0$ and $C({\\overline{x}},f,{\\overline{x}}^2-{\\overline{x}})=1$ , where (as mentioned) ${\\overline{x}}^2-{\\overline{x}}$ denotes $\\lbrace x_i^2-x_i\\rbrace _i$ .", "Expressing $C({\\overline{x}},f,{\\overline{x}}^2-{\\overline{x}})$ as a univariate in $f$ , we obtain that $\\sum _{i\\ge 1} C_i({\\overline{x}},{\\overline{x}}^2-{\\overline{x}}) f^i = 1-C({\\overline{x}},0,{\\overline{x}}^2-{\\overline{x}})$ for some polynomials $C_i$ .", "For most natural measures of circuit complexity $1-C({\\overline{x}},0,{\\overline{x}}^2-{\\overline{x}})$ has complexity roughly bounded by that of $C$ itself.", "Thus, we see that a multiple of $f$ has a small circuit, as $\\left(\\sum _{i\\ge 1} C_i({\\overline{x}},{\\overline{x}}^2-{\\overline{x}}) f^{i-1}\\right)\\cdot f=1-C({\\overline{x}},0,{\\overline{x}}^2-{\\overline{x}})$ , and one can use the properties of the IPS refutation to show this is in fact a nonzero multiple.", "Thus, if we can show that all nonzero multiples of $f$ require large circuits then we rule out a small IPS refutation.", "We now turn to methods for obtaining polynomials with hard multiples.", "Intuitively if a polynomial $f$ is hard then so should small modifications such as $f^2+x_1f$ , and this intuition is supported by the result of Kaltofen [40] which shows that if a polynomial has a small algebraic circuit then so do all of its factors.", "As a consequence, if a polynomial requires super-polynomially large algebraic circuits then so do all of its multiples.", "However, Kaltofen's [40] result is about general algebraic circuits, and there are very limited results about the complexity of factors of restricted algebraic circuits ([17], [53]) so that obtaining polynomials for hard multiples via factorization results seems difficult.", "However, note that lower bound for multiples has a different order of quantifiers than the factoring question.", "That is, Kaltofen's [40] result speaks about the factors of any small circuit, while the lower bound for multiples speaks about the multiples of a single polynomial.", "Thus, it seems plausible that existing methods could yield such explicit polynomials, and indeed we show this is the case.", "We begin by noting that obtaining lower bounds for multiples is a natural instantiation of the algebraic hardness versus randomness paradigm.", "In particular, Heintz-Schnorr [36] and Agrawal [3] showed that obtaining deterministic (black-box) polynomial identity testing algorithms implies lower bounds (see sec:PIT for more on PIT), and we strengthen that connection here to lower bounds for multiples.", "We can actually instantiate this connection, and we use slight modifications of existing PIT algorithms to show that multiples of the determinant are hard in some models.", "[Informal Version of res:generatortolbs-mult, res:lbs-mult:pit:det] Let $\\mathcal {C}$ be a restricted class of $n$ -variate algebraic circuits.", "Full derandomization of PIT algorithms for $\\mathcal {C}$ yields a (weakly) explicit polynomial whose nonzero multiples require $\\exp (\\Omega (n))$ -size as $\\mathcal {C}$ -circuits.", "In particular, when $\\mathcal {C}$ is the class of sparse polynomials, depth-3 powering formulas, $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas (in characteristic zero), or “every-order” roABPs, then all nonzero multiples of the $n\\times n$ determinant are $\\exp (\\Omega (n))$ -hard in these models.", "The above statement shows that derandomization implies hardness.", "We also partly address the converse direction by arguing (sec:lbs-mult:hard-v-rand) that hardness-to-randomness construction of Kabanets and Impagliazzo [43] only requires lower bounds for multiples to derandomize PIT.", "Unfortunately, this direction is harder to instantiate for restricted classes as it requires lower bounds for classes with suitable closure properties.Although, we note that one can instantiate this connection with depth-3 powering formulas (or even $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas) using the lower bounds for multiples developed in this paper, building on the work of Forbes [22].", "However, the resulting PIT algorithms are worse than those developed by Forbes [22].", "Unfortunately the above result is slightly unsatisfying from a proof complexity standpoint as the (exponential-size) lower bounds for the subclasses of IPS one can derive from the above result would involve the determinant polynomial as an axiom.", "While the determinant is efficiently computable, it is not computable by the above restricted circuit classes (indeed, the above result proves that).", "As such, this would not fit the real goal of proof complexity which seeks to show that there are statements whose proofs must be super-polynomial larger than the length of the statement.", "Thus, if we measure the size of the IPS proof and the axioms with respect to the same circuit measure, the lower bounds for multiples approach cannot establish such super-polynomial lower bounds.", "However, we believe that lower bounds for multiples could lead, with further ideas, to proof complexity lower bounds in the conventional sense.", "That is, it seems plausible that by adding extension variables we can convert complicated axioms to simple, local axioms by tracing through the computation of that axiom.", "That is, consider the axiom $xyzw$ .", "This can be equivalently written as $\\lbrace a-xy,b-zw,c-ab,c\\rbrace $ , where this conversion is done by considering a natural algebraic circuit for $xyzw$ , replacing each gate with a new variable, and adding an axiom ensuring the new variables respect the computation of the circuit.", "While we are unable to understand the role of extension variables in this work, we aim to give as simple axioms as possible whose multiples are all hard as this may facilitate future work on extension variables.", "We now discuss the lower bounds for multiples we obtain.While we discussed functional lower bounds for multilinear formulas, this class is not interesting for the lower bounds for multiples question.", "This is because a multiple of a multilinear polynomial may not be multilinear, and thus clearly cannot have a multilinear formula.", "[Corollaries REF , REF , REF , REF , and REF ] We obtain the following lower bounds for multiples.", "All nonzero multiples of $x_1\\cdots x_n$ require $\\exp (\\Omega (n))$ -size as a depth-3 powering formula (over any field), or as a $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formula (in characteristic zero).", "All nonzero multiples of $(x_1+1)\\cdots (x_n+1)$ require $\\exp (\\Omega (n))$ -many monomials.", "All nonzero multiples of $\\prod _i (x_i+y_i)$ require $\\exp (\\Omega (n))$ -width as an roABP in any variable order where ${\\overline{x}}$ precedes ${\\overline{y}}$ .", "All nonzero multiples of $\\prod _{i<j} (x_i+x_j)$ require $\\exp (\\Omega (n))$ -width as an roABP in any variable order, as well as $\\exp (\\Omega (n))$ -width as a read-twice oblivious ABP.", "We now briefly explain our techniques for obtaining these lower bounds, focusing on the simplest case of depth-3 powering formulas.", "It follows from the partial derivative method of Nisan and Wigderson [50] (see Kayal [41]) that such formulas require exponential size to compute the monomial $x_1\\ldots x_n$ exactly.", "Forbes and Shpilka [24], in giving a PIT algorithm for this class, showed that this lower bound can be scaled down and made robust.", "That is, if one has a size-$s$ depth-3 powering formula, it follows that if it computes a monomial $x_{i_1}\\cdots x_{i_\\ell }$ for distinct $i_j$ then $\\ell \\le O(\\log s)$ (so the lower bound is scaled down).", "One can then show that regardless of what this formula actually computes the leading monomial $x_{i_1}^{a_{i_1}}\\cdots x_{i_\\ell }^{a_{i_\\ell }}$ (for distinct $i_j$ and positive $a_{i_j}$ ) must have that $\\ell \\le O(\\log s)$ .", "One then notes that leading monomials are multiplicative.", "Thus, for any nonzero $g$ the leading monomial of $g\\cdot x_1\\ldots x_n$ involves $n$ variables so that if $g\\cdot x_1\\ldots x_n$ is computed in size-$s$ then $n\\le O(\\log s)$ , giving $s\\ge \\exp (\\Omega (n))$ as desired.", "One can then obtain the other lower bounds using the same idea, though for roABPs one needs to define a leading diagonal (refining an argument of Forbes-Shpilka [23]).", "We now conclude our IPS lower bounds.", "[res:lbs-ips:mult:sumpowsum, res:lbs-ips:mult:roABP] We obtain the following lower bounds for subclasses of IPS.", "In characteristic zero, the system of polynomials $x_1\\cdots x_n,x_1+\\cdots +x_n-n,\\lbrace x_i^2-x_i\\rbrace _{i=1}^n$ is unsatisfiable, and any $\\sum \\bigwedge \\sum $ -IPS refutation requires $\\exp (\\Omega (n))$ -size.", "In characteristic $>n$ , the system of polynomials, $\\prod _{i<j}(x_i+x_j-1),x_1+\\cdots +x_n-n,\\lbrace x_i^2-x_i\\rbrace _i$ is unsatisfiable, and any roABP-IPS refutation (in any variable order) must be of size $\\exp (\\Omega (n))$ .", "Note that the first result is a non-standard encoding of $1=\\operatorname{AND}(x_1,\\ldots ,x_n)=0$ .", "Similarly, the second is a non-standard encoding of $\\operatorname{AND}(x_1,\\ldots ,x_n)=1$ yet $\\operatorname{XOR}(x_i,x_j)=1$ for all $i,j$ ." ], [ "Organization", "The rest of the paper is organized as follows.", "sec:notation contains the basic notation for the paper.", "In sec:background we give background from algebraic complexity, including several important complexity measures such as coefficient dimension and evaluation dimension (see sec:coeff-dim and sec:eval-dim).", "We present our upper bounds for IPS in sec:h-ips:ubs.", "In sec:lbs-fn we give our functional lower bounds and from them obtain lower bounds for IPS$_{\\text{LIN}}$ .", "sec:lbs-mult contains our lower bounds for multiples of polynomials and in sec:ips-mult we derive lower bounds for IPS using them.", "In sec:open-problems we list some problems which were left open by this work.", "In sec:alg-proofs we describe various other algebraic proof systems and their relations to IPS, such as the dynamic Polynomial Calculus of Clegg, Edmonds, and Impagliazzo [11], the ordered formula proofs of Tzameret [76], and the multilinear proofs of Raz and Tzameret [62].", "In sec:appendix we give an explicit description of a multilinear polynomial occurring in our IPS upper bounds." ], [ "Notation", "In this section we briefly describe notation used in this paper.", "We denote $[n]:=\\lbrace 1,\\ldots ,n\\rbrace $ .", "For a vector ${\\overline{a}}\\in \\mathbb {N}^n$ , we denote ${\\overline{x}}^{\\overline{a}}:=x_1^{a_1}\\cdots x_n^{a_n}$ so that in particular ${\\overline{x}}^{\\overline{1}}=\\prod _{i=1}^n x_i$ .", "The (total) degree of a monomial ${\\overline{x}}^{\\overline{a}}$ , denoted $\\deg {\\overline{x}}^{\\overline{a}}$ , is equal to ${\|{\\overline{a}}\|_1}:=\\sum _i a_i$ , and the individual degree, denoted $\\operatorname{ideg}{\\overline{x}}^{\\overline{a}}$ , is equal to ${\|{\\overline{a}}\|_\\infty }:=\\max \\lbrace a_i\\rbrace _i$ .", "A monomial ${\\overline{x}}^{\\overline{a}}$ depends on ${\|{\\overline{a}}\|_0}:=\|\\lbrace i:a_i\\ne 0\\rbrace \|$ many variables.", "Degree and individual degree can be defined for a polynomial $f$ , denoted $\\deg f$ and $\\operatorname{ideg}f$ respectively, by taking the maximum over all monomials with nonzero coefficients in $f$ .", "We will sometimes compare vectors ${\\overline{a}}$ and ${\\overline{b}}$ as “${\\overline{a}}\\le {\\overline{b}}$ ”, which is to be interpreted coordinate-wise.", "We will use $\\prec $ to denote a monomial order on $\\mathbb {F}[{\\overline{x}}]$ , see sec:mon-ord.", "Polynomials will often be written out in their monomial expansion.", "At various points we will need to extract coefficients from polynomials.", "When “taking the coefficient of ${\\overline{y}}\\!\\:^{\\overline{b}}$ in $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ ” we mean that both ${\\overline{x}}$ and ${\\overline{y}}$ are treated as variables and thus the coefficient returned is a scalar in $\\mathbb {F}$ , and this will be denoted ${\\mathrm {Coeff}}_{\\smash{{\\overline{y}}\\!\\:^{\\overline{b}}}}(f)$ .", "However, when “taking the coefficient of ${\\overline{y}}\\!\\:^{{\\overline{b}}}$ in $f\\in \\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ ” we mean that ${\\overline{x}}$ is now part of the ring of scalars, so the coefficient will be an element of $\\mathbb {F}[{\\overline{x}}]$ , and this coefficient will be denoted ${\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)$ .", "For a vector ${\\overline{a}}\\in \\mathbb {N}^n$ we denote ${\\overline{a}}_{\\le i}\\in \\mathbb {N}^i$ to be the restriction of ${\\overline{a}}$ to the first $i$ coordinates.", "For a set $S \\subseteq [n]$ we let $\\overline{S}$ denote the complement set.", "We will denote the size-$k$ subsets of $[n]$ by $\\binom{[n]}{k}$ .", "We will use $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]\\rightarrow \\mathbb {F}[{\\overline{x}}]$ to denote the multilinearization operator, defined by fact:multilinearization.", "We will use ${\\overline{x}}^2-{\\overline{x}}$ to denote the set of equations $\\lbrace x_i^2 -x_i\\rbrace _i$ .", "To present algorithms that are field independent, this paper works in a model of computation where field operations (such as addition, multiplication, inversion and zero-testing) over $\\mathbb {F}$ can be computed at unit cost, see for example Forbes [21].", "We say that an algebraic circuit is $t$ -explicit if it can be constructed in $t$ steps in this unit-cost model." ], [ "Algebraic Complexity Theory Background", "In this section we state some known facts regarding the algebraic circuit classes that we will be studying.", "We also give some important definitions that will be used later in the paper." ], [ "Polynomial Identity Testing", "In the polynomial identity testing (PIT) problem, we are given an algebraic circuit computing some polynomial $f$ , and we have to determine whether “$f\\equiv 0$ ”.", "That is, we are asking whether $f$ is the zero polynomial in $\\mathbb {F}[x_1,\\ldots ,x_n]$ .", "By the Schwartz-Zippel-DeMillo-Lipton Lemma [80], [69], [16], if $0\\ne f \\in \\mathbb {F}[{\\overline{x}}]$ is a polynomial of degree $\\le d$ and $S\\subseteq \\mathbb {F}$ , and ${\\overline{\\alpha }}\\in S^n$ is chosen uniformly at random, then $f({\\overline{\\alpha }}) =0$ with probability at most Note that this is non-trivial only if $d < \|S\| \\le \|\\mathbb {F}\|$ , which in particular implies that $f$ is not the zero function.", "$d/\|S\|$ .", "Thus, given the circuit, we can perform these evaluations efficiently, giving an efficient randomized procedure for deciding whether “$f\\equiv 0$ ?”.", "It is an important open problem to find a derandomization of this algorithm, that is, to find a deterministic procedure for PIT that runs in polynomial time (in the size of circuit).", "Note that in the randomized algorithm of Schwartz-Zippel-DeMillo-Lipton we only use the circuit to compute the evaluation $f({\\overline{\\alpha }})$ .", "Such algorithms are said to run in the black-box model.", "In contrast, an algorithm that can access the internal structure of the circuit runs in the white-box model.", "It is a folklore result that efficient deterministic black-box algorithms are equivalent to constructions of small hitting sets.", "That is, a hitting set is set of inputs so that any nonzero circuit from the relevant class evaluates to nonzero on at least one of the inputs in the set.", "For more on PIT we refer to the survey of Shpilka and Yehudayoff [75].", "A related notion to that of a hitting set is that of a generator, which is essentially a low-dimensional curve whose image contains a hitting set.", "The equivalence between hitting sets and generators can be found in the above mentioned survey.", "Let $\\mathcal {C}\\subseteq \\mathbb {F}[x_1,\\ldots ,x_n]$ be a set of polynomials.", "A polynomial ${\\overline{\\mathcal {G}}}:\\mathbb {F}^\\ell \\rightarrow \\mathbb {F}^n$ is a generator for $\\mathcal {C}$ with seed length $\\ell $ if for all $f\\in \\mathcal {C}$ , $f\\equiv 0 \\text{ iff } f\\circ {\\overline{\\mathcal {G}}}\\equiv 0$ .", "That is, $f({\\overline{x}})=0$ in $\\mathbb {F}[{\\overline{x}}]$ iff $f({\\overline{\\mathcal {G}}}({\\overline{y}}))=0$ in $\\mathbb {F}[{\\overline{y}}]$ .", "In words, a generator for a circuit class $\\mathcal {C}$ is a mapping ${\\overline{\\mathcal {G}}}:\\mathbb {F}^\\ell \\rightarrow \\mathbb {F}^n$ , such that for any nonzero polynomial $f$ , computed by a circuit from $\\mathcal {C}$ , it holds that the composition $f({\\overline{\\mathcal {G}}})$ is nonzero as well.", "By considering the image of ${\\overline{\\mathcal {G}}}$ on $S^\\ell $ , where $S \\subseteq \\mathbb {F}$ is of polynomial size, we obtain a hitting set for $\\mathcal {C}$ .", "We now list some existing work on derandomizing PIT for some of the classes of polynomials we study in this paper.", "paragraph41ex plus1ex minus.2ex-1emSparse Polynomials: There are many papers giving efficient black-box PIT algorithms for $\\sum \\prod $ formulas.", "For example, Klivans and Spielman [45] gave a hitting set of polynomial size.", "paragraph41ex plus1ex minus.2ex-1emDepth-3 Powering Formulas: Saxena [67] gave a polynomial time white-box PIT algorithm and Forbes, Shpilka, and Saptharishi [27] gave a $s^{O(\\lg \\lg s)}$ -size hitting set for size-$s$ depth-3 powering formulas.", "paragraph41ex plus1ex minus.2ex-1em$\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ Formulas: Forbes [22] gave an $s^{O(\\lg s)}$ -size hitting set for size-$s$ $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas (in large characteristic).", "paragraph41ex plus1ex minus.2ex-1emRead-once Oblivious ABPs: Raz and Shpilka [61] gave a polynomial time white-box PIT algorithm.", "A long sequence of papers calumniated in the work of Agrawal, Gurjar, Korwar, and Saxena [2], who gave a $s^{O(\\lg s)}$ -sized hitting set for size-$s$ roABPs.", "paragraph41ex plus1ex minus.2ex-1emRead-$k$ Oblivious ABPs: Recently, Anderson, Forbes, Saptharishi, Shpilka and Volk [1] obtained a white-box PIT algorithm running in time $2^{\\tilde{O}(n^{1-1/2^{k-1}})}$ for $n$ -variate $(n)$ -sized read-$k$ oblivious ABPs." ], [ "Coefficient Dimension and roABPs", "This paper proves various lower bounds on roABPs using a complexity measures known as coefficient dimension.", "In this section, we define this measures and recall basic properties.", "Full proofs of these claims can be found for example in the thesis of Forbes [21].", "We first define the coefficient matrix of a polynomial, called the “partial derivative matrix” in the prior work of Nisan [49] and Raz [59].", "This matrix is formed from a polynomial $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ by arranging its coefficients into a matrix.", "That is, the coefficient matrix has rows indexed by monomials ${\\overline{x}}^{\\overline{a}}$ in ${\\overline{x}}$ , columns indexed by monomials ${\\overline{y}}\\!\\:^{\\overline{b}}$ in ${\\overline{y}}$ , and the $({\\overline{x}}^{\\overline{a}},{\\overline{y}}\\!\\:^{\\overline{b}})$ -entry is the coefficient of ${\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ in the polynomial $f$ .", "We now define this matrix, recalling that ${\\mathrm {Coeff}}_{{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}}(f)$ is the coefficient of ${\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ in $f$ .", "Consider $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "Define the coefficient matrix of $f$ as the scalar matrix $(C_f)_{{\\overline{a}},{\\overline{b}}}:={\\mathrm {Coeff}}_{{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}}(f)\\;,$ where coefficients are taken in $\\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ , for ${\|{\\overline{a}}\|_1},{\|{\\overline{b}}\|_1}\\le \\deg f$ .", "We now give the related definition of coefficient dimension, which looks at the dimension of the row- and column-spaces of the coefficient matrix.", "Recall that ${\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)$ extracts the coefficient of ${\\overline{y}}\\!\\:^{\\overline{b}}$ in $f$ as a polynomial in $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ .", "Let ${\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{x}}]}$ be the space of $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ coefficients, defined by ${\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f):=\\left\\lbrace {\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)\\right\\rbrace _{{\\overline{b}}\\in \\mathbb {N}^n}\\;,$ where coefficients of $f$ are taken in $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ .", "Similarly, define ${\\mathbf {Coeff}}_{{\\overline{y}}\|{\\overline{x}}}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{y}}]}$ by taking coefficients in $\\mathbb {F}[{\\overline{y}}][{\\overline{x}}]$ .", "The following basic lemma shows that the rank of the coefficient matrix equals the coefficient dimension, which follows from simple linear algebra.", "[Nisan [49]] Consider $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "Then the rank of the coefficient matrix $C_f$ obeys $\\operatorname{rank}C_f=\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)=\\dim {\\mathbf {Coeff}}_{{\\overline{y}}\|{\\overline{x}}}(f)\\;.$ Thus, the ordering of the partition ($({\\overline{x}},{\\overline{y}})$ versus $({\\overline{y}},{\\overline{x}})$ ) does not matter in terms of the resulting dimension.", "The above matrix-rank formulation of coefficient dimension can be rephrased in terms of low-rank decompositions.", "Let $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "Then $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)$ equals the minimum $r$ such that there are ${\\overline{g}}\\in \\mathbb {F}[{\\overline{x}}]^r$ and ${\\overline{h}}\\in \\mathbb {F}[{\\overline{y}}]^r$ such that $f$ can be written as $f({\\overline{x}},{\\overline{y}})=\\sum _{i=1}^r g_i({\\overline{x}})h_i({\\overline{y}})$ .", "We now state a convenient normal form for roABPs (see for example Forbes [21]).", "A polynomial $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ is computed by width-$r$ roABP iff there exist matrices $A_i(x_{i})\\in \\mathbb {F}[x_{i}]^{r\\times r}$ of (individual) degree $\\le \\deg f$ such that $f=(\\prod _{i=1}^n A_i(x_{i}))_{1,1}$ .", "Further, this equivalence preserves explicitness of the roABPs up to $(n,r,\\deg f)$ -factors.", "By splitting an roABP into such variable-disjoint inner-products one can obtain a lower bound for roABP width via coefficient dimension.", "In fact, this complexity measure characterizes roABP width.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a polynomial.", "If $f$ is computed by a width-$r$ roABP then $r \\ge \\max _i\\dim {\\mathbf {Coeff}}_{{\\overline{x}}_{\\le i}\|{\\overline{x}}_{>i}}(f)$ .", "Further, $f$ is computable width-$\\left(\\max _i\\dim {\\mathbf {Coeff}}_{{\\overline{x}}_{\\le i}\|{\\overline{x}}_{>i}}(f)\\right)$ roABP.", "Using this complexity measure it is rather straightforward to prove the following closure properties of roABPs.", "If $f,g\\in \\mathbb {F}[{\\overline{x}}]$ are computable by width-$r$ and width-$s$ roABPs respectively, then $f+g$ is computable by a width-$(r+s)$ roABP.", "$f\\cdot g$ is computable by a width-$(rs)$ roABP.", "Further, roABPs are also closed under the follow operations.", "If $f({\\overline{x}},{\\overline{y}})\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ is computable by a width-$r$ roABP in some variable order then the partial substitution $f({\\overline{x}},{\\overline{\\alpha }})$ , for ${\\overline{\\alpha }}\\in \\mathbb {F}^{\|{\\overline{y}}\|}$ , is computable by a width-$r$ roABP in the induced order on ${\\overline{x}}$ , where the degree of this roABP is bounded by the degree of the roABP for $f$ .", "If $f(z_1,\\ldots ,z_n)$ is computable by a width-$r$ roABP in variable order $z_1<\\cdots <z_n$ , then $f(x_1y_1,\\ldots ,x_ny_n)$ is computable by a $(r,\\operatorname{ideg}f)$ -width roABP in variable order $x_1<y_1<\\cdots <x_n<y_n$ .", "Further, these operations preserve the explicitness of the roABPs up to polynomial factors in all relevant parameters.", "We now state the extension of these techniques which yield lower bounds for read-$k$ oblivious ABPs, as recently obtained by Anderson, Forbes, Saptharishi, Shpilka and Volk [1].", "[[1]] Let $f \\in \\mathbb {F}[x_1, \\ldots , x_n]$ be a polynomial computed by a width-$w$ read-$k$ oblivious ABP.", "Then there exists a partition ${\\overline{x}}=({\\overline{u}},{\\overline{v}},{\\overline{w}})$ such that $\|{\\overline{u}}\|, \|{\\overline{v}}\| \\ge n/k^{O(k)}$ .", "$\|{\\overline{w}}\| \\le n/10$ .", "$\\dim _{\\mathbb {F}({\\overline{w}})} {\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}(f_{\\overline{w}}) \\le w^{2k}$ , where $f_{\\overline{w}}$ is $f$ as a polynomial in $\\mathbb {F}({\\overline{w}})[{\\overline{u}},{\\overline{v}}]$ ." ], [ "Evaluation Dimension", "While coefficient dimension measures the size of a polynomial $f({\\overline{x}},{\\overline{y}})$ by taking all coefficients in ${\\overline{y}}$ , evaluation dimension is a complexity measure due to Saptharishi [66] that measures the size by taking all possible evaluations in ${\\overline{y}}$ over the field.", "This measure will be important for our applications as one can restrict such evaluations to the boolean cube and obtain circuit lower bounds for computing $f({\\overline{x}},{\\overline{y}})$ as a polynomial via its induced function on the boolean cube.", "We begin with the definition.", "[Saptharishi [66]] Let $S\\subseteq \\mathbb {F}$ .", "Let ${\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{x}}]}$ be the space of $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ evaluations over $S$, defined by ${\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}(f({\\overline{x}},{\\overline{y}})):=\\left\\lbrace f({\\overline{x}},{\\overline{\\beta }})\\right\\rbrace _{{\\overline{\\beta }}\\in S^{\|{\\overline{y}}\|}}\\;.$ Define ${\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}}}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{x}}]}$ to be ${\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}$ when $S=\\mathbb {F}$ .", "Similarly, define ${\\mathbf {Eval}}_{{\\overline{y}}\|{\\overline{x}},S}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{y}}]}$ by replacing ${\\overline{x}}$ with all possible evaluations ${\\overline{\\alpha }}\\in S^{\|{\\overline{x}}\|}$ , and likewise define ${\\mathbf {Eval}}_{{\\overline{y}}\|{\\overline{x}}}:\\mathbb {F}[{\\overline{x}},{\\overline{y}}]\\rightarrow 2^{\\mathbb {F}[{\\overline{y}}]}$ .", "The equivalence between evaluation dimension and coefficient dimension was shown by Forbes-Shpilka [25] by appealing to interpolation.", "[Forbes-Shpilka [25]] Let $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "For any $S\\subseteq \\mathbb {F}$ we have that ${\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}(f)\\subseteq \\operatorname{span}{\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)$ so that $\\dim {\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}(f)\\le \\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)$ .", "In particular, if $\|S\|>\\operatorname{ideg}f$ then $\\dim {\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},S}(f)=\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)$ .", "While evaluation dimension and coefficient dimension are equivalent when the field is large enough, when restricting our attention to inputs from the boolean cube this equivalence no longer holds (in particular, we have to consider all polynomials that obtain the same values on the boolean cube and not just one polynomial), but evaluation dimension will be still be helpful as it will always lower bound coefficient dimension." ], [ "Multilinear Polynomials and Multilinear Formulas", "We now turn to multilinear polynomials and classes that respect multilinearity such as multilinear formulas.", "We first state some well-known facts about multilinear polynomials.", "For any two multilinear polynomials $f,g\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ , $f=g$ as polynomials iff they agree on the boolean cube $\\lbrace 0,1\\rbrace ^n$ .", "That is, $f=g$ iff $f\|_{\\lbrace 0,1\\rbrace ^n}=g\|_{\\lbrace 0,1\\rbrace ^n}$ .", "Further, there is a multilinearization map $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]\\rightarrow \\mathbb {F}[{\\overline{x}}]$ such that for any $f,g\\in \\mathbb {F}[{\\overline{x}}]$ , $\\operatorname{ml}(f)$ is multilinear.", "$f$ and $\\operatorname{ml}(f)$ agree on the boolean cube, that is, $f\|_{\\lbrace 0,1\\rbrace ^n}=\\operatorname{ml}(f)\|_{\\lbrace 0,1\\rbrace ^n}$ .", "$\\deg \\operatorname{ml}(f)\\le \\deg f$ .", "$\\operatorname{ml}(fg)=\\operatorname{ml}(\\operatorname{ml}(f)\\operatorname{ml}(g))$ .", "$\\operatorname{ml}$ is linear, so that for any $\\alpha ,\\beta \\in \\mathbb {F}$ , $\\operatorname{ml}(\\alpha f+\\beta g)=\\alpha \\operatorname{ml}(f)+\\beta \\operatorname{ml}(g)$ .", "$\\operatorname{ml}(x_1^{a_1}\\cdots x_n^{a_n})=\\prod _i x_i^{\\max \\lbrace a_i,1\\rbrace }$ .", "If $f$ is the sum of at most $s$ monomials ($s$ -sparse) then so is $\\operatorname{ml}(f)$ .", "Also, if $\\hat{f}$ is a function $\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ that only depends on the coordinates in $S\\subseteq [n]$ , then the unique multilinear polynomial $f$ agreeing with $\\hat{f}$ on $\\lbrace 0,1\\rbrace ^n$ is a polynomial only in $\\lbrace x_i\\rbrace _{i\\in S}$ .", "One can also extend the multilinearization map $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]\\rightarrow \\mathbb {F}[{\\overline{x}}]$ to matrices $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]^{r\\times r}\\rightarrow \\mathbb {F}[{\\overline{x}}]^{r\\times r}$ by applying the map entry-wise, and the above properties still hold.", "Throughout the rest of this paper `$\\operatorname{ml}$ ' will denote the multilinearization operator.", "Raz [59], [58] gave lower bounds for multilinear formulas using the above notion of coefficient dimension, and Raz-Yehudayoff [64], [65] gave simplifications and extensions to constant-depth multilinear formulas.", "[Raz-Yehudayoff [59], [65]] Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_{2n},{\\overline{z}}]$ be a multilinear polynomial in the set of variables ${\\overline{x}}$ and auxiliary variables ${\\overline{z}}$ .", "Let $f_{\\overline{z}}$ denote the polynomial $f$ in the ring $\\mathbb {F}[{\\overline{z}}][{\\overline{x}}]$ .", "Suppose that for any partition ${\\overline{x}}=({\\overline{u}},{\\overline{v}})$ with $\|{\\overline{u}}\|=\|{\\overline{v}}\|=n$ that $\\dim _{\\mathbb {F}({\\overline{z}})} {\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}} f_{\\overline{z}}\\ge 2^n\\;.$ Then $f$ requires $\\ge n^{\\Omega (\\log n)}$ -size to be computed as a multilinear formula, and for $d=o({\\log n}{\\log \\log n})$ , $f$ requires $n^{\\Omega (({n}{\\log n})^{{1}{d}}/d^2)}$ -size to be computed as a multilinear formula of product-depth-$d$ ." ], [ "Depth-3 Powering Formulas", "In this section we review facts about depth-3 powering formulas.", "We begin with the duality trick of Saxena [67], which shows that one can convert a power of a linear form to a sum of products of univariate polynomials.", "[Saxena's Duality Trick [74], [67], [18]] Let $n\\ge 1$ , and $d\\ge 0$ .", "If $\|\\mathbb {F}\|\\ge nd+1$ , then there are $(n,d)$ -explicit univariates $f_{i,j}\\in \\mathbb {F}[x_i]$ such that $(x_1+\\cdots +x_n)^d=\\sum _{i=1}^s f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\;,$ where $\\deg f_{i,j}\\le d$ and $s=(nd+1)(d+1)$ .", "The original proof of Saxena [67] only worked over fields of large enough characteristic, and gave $s=nd+1$ .", "A similar version of this trick also appeared in Shpilka-Wigderson [74].", "The parameters we use here are from the proof of Forbes, Gupta, and Shpilka [18], which has the advantage of working over any large enough field.", "Noting that the product $f_{i,1}(x_1)\\cdots f_{i,n}(x_n)$ trivially has a width-1 roABP (in any variable order), it follows that $(x_1+\\cdots +x_n)^d$ has a $(n,d)$ -width roABP over a large enough field.", "Thus, size-$s$ $\\sum \\bigwedge \\sum $ formulas have $(s)$ -size roABPs over large enough fields by appealing to closure properties of roABPs (fact:roABP:closure).", "As it turns out, this result also holds over any field as Forbes-Shpilka [25] adapted Saxena's [67] duality to work over any field.", "Their version works over any field, but loses the above clean form (sum of product of univariates).", "[Forbes-Shpilka [25]] Let $f\\in \\mathbb {F}[{\\overline{x}}]$ be expressed as $f({\\overline{x}})=\\sum _{i=1}^s (\\alpha _{i,0}+\\alpha _{i,1}x_i+\\cdots +\\alpha _{i,n}x_n)^{d_i}$ .", "Then $f$ is computable by a $(r,n)$ -explicit width-$r$ roABP of degree $\\max _i\\lbrace d_i\\rbrace $ , in any variable order, where $r=\\sum _i (d_i+1)$ .", "One way to see this claim is to observe that for any variable partition, a linear function can be expressed as the sum of two variable-disjoint linear functions $\\ell ({\\overline{x}}_1,{\\overline{x}}_2)=\\ell _1({\\overline{x}}_1)+\\ell _2({\\overline{x}}_2)$ .", "By the binomial theorem, the $d$ -th power of this expression is a summation of $d+1$ variable-disjoint products, which implies a coefficient dimension upper bound of $d+1$ (res:nisandim-eq-width) and thus also an roABP-width upper bound (res:roABP-widtheqdim-coeffs).", "One can then sum over the linear forms.", "While this simulation suffices for obtaining roABP upper bounds, we will also want the clean form obtained via duality for application to multilinear-formula IPS proofs of the subset-sum axiom (res:ips-ubs:subset:mult-form)." ], [ "Monomial Orders", "We recall here the definition and properties of a monomial order, following Cox, Little and O'Shea [12].", "We first fix the definition of a monomial in our context.", "A monomial in $\\mathbb {F}[x_1,\\ldots ,x_n]$ is a polynomial of the form ${\\overline{x}}^{\\overline{a}}=x_1^{a_1}\\cdots x_n^{a_n}$ for ${\\overline{a}}\\in \\mathbb {N}^n$ .", "We will sometimes abuse notation and associate a monomial ${\\overline{x}}^{\\overline{a}}$ with its exponent vector ${\\overline{a}}$ , so that we can extend this order to the exponent vectors.", "Note that in this definition “1” is a monomial, and that scalar multiples of monomials such as $2x$ are not considered monomials.", "We now define a monomial order, which will be total order on monomials with certain natural properties.", "A monomial ordering is a total order $\\prec $ on the monomials in $\\mathbb {F}[{\\overline{x}}]$ such that For all ${\\overline{a}}\\in \\mathbb {N}^n\\setminus \\lbrace {\\overline{0}}\\rbrace $ , $1\\prec {\\overline{x}}^{{\\overline{a}}}$ .", "For all ${\\overline{a}},{\\overline{b}},{\\overline{c}}\\in \\mathbb {N}^n$ , ${\\overline{x}}^{\\overline{a}}\\prec {\\overline{x}}^{\\overline{b}}$ implies ${\\overline{x}}^{{\\overline{a}}+{\\overline{c}}}\\prec {\\overline{x}}^{{\\overline{b}}+{\\overline{c}}}$ .", "For nonzero $f\\in \\mathbb {F}[{\\overline{x}}]$ , the leading monomial of $f$ (with respect to a monomial order $\\prec $ ), denoted $\\operatorname{LM}(f)$ , is the largest monomial in $\\operatorname{Supp}(f):=\\lbrace {\\overline{x}}^{\\overline{a}}:{\\mathrm {Coeff}}_{{\\overline{x}}^{\\overline{a}}}(f)\\ne 0\\rbrace $ with respect to the monomial order $\\prec $ .", "The trailing monomial of $f$, denoted $\\operatorname{TM}(f)$ , is defined analogously to be the smallest monomial in $\\operatorname{Supp}(f)$ .", "The zero polynomial has neither leading nor trailing monomial.", "For nonzero $f\\in \\mathbb {F}[{\\overline{x}}]$ , the leading (resp.", "trailing) coefficient of $f$, denoted $\\operatorname{LC}(f)$ (resp.", "$(f)$ ), is ${\\mathrm {Coeff}}_{{\\overline{x}}^{\\overline{a}}}(f)$ where ${\\overline{x}}^{\\overline{a}}=\\operatorname{LM}(f)$ (resp.", "${\\overline{x}}^{\\overline{a}}=\\operatorname{TM}(f)$ ).", "Henceforth in this paper we will assume $\\mathbb {F}[{\\overline{x}}]$ is equipped with some monomial order $\\prec $ .", "The results in this paper will hold for any monomial order.", "However, for concreteness, one can consider the lexicographic ordering on monomials, which is easily seen to be a monomial ordering (see also Cox, Little and O'Shea [12]).", "We begin with a simple lemma about how taking leading or trailing monomials (or coefficients) is homomorphic with respect to multiplication.", "Let $f,g\\in \\mathbb {F}[{\\overline{x}}]$ be nonzero polynomials.", "Then the leading monomial and trailing monomials and coefficients are homomorphic with respect to multiplication, that is, $\\operatorname{LM}(fg)=\\operatorname{LM}(f)\\operatorname{LM}(g)$ and $\\operatorname{TM}(fg)=\\operatorname{TM}(f)\\operatorname{TM}(g)$ , as well as $\\operatorname{LC}(fg)=\\operatorname{LC}(f)\\operatorname{LC}(g)$ and $(fg)=(f)(g)$ .", "We do the proof for leading monomials and coefficients, the claim for trailing monomials and coefficients is symmetric.", "Let $f({\\overline{x}})=\\sum _{\\overline{a}}\\alpha _{\\overline{a}}{\\overline{x}}^{\\overline{a}}$ and $g({\\overline{x}})=\\sum _{\\overline{b}}\\beta _{\\overline{b}}{\\overline{x}}^{\\overline{b}}$ .", "Isolating the leading monomials, $f({\\overline{x}})=&\\operatorname{LC}(f)\\cdot \\operatorname{LM}(f)+\\sum _{{\\overline{x}}^{\\overline{a}}\\prec \\operatorname{LM}(f)} \\alpha _{\\overline{a}}{\\overline{x}}^{\\overline{a}},&g({\\overline{x}})=&\\operatorname{LC}(g)\\cdot \\operatorname{LM}(g)+\\sum _{{\\overline{x}}^{\\overline{b}}\\prec \\operatorname{LM}(g)} \\beta _{\\overline{b}}{\\overline{x}}^{\\overline{b}},$ with $\\operatorname{LC}(f)=\\alpha _{\\operatorname{LM}(f)}$ and $\\operatorname{LC}(g)=\\beta _{\\operatorname{LM}(g)}$ being nonzero.", "Thus, $f({\\overline{x}})g({\\overline{x}})=\\operatorname{LC}(f)\\operatorname{LC}(g) \\cdot \\operatorname{LM}(f)\\operatorname{LM}(g)+\\operatorname{LC}(f)\\operatorname{LM}(f)\\left(\\sum _{{\\overline{x}}^{\\overline{b}}\\prec \\operatorname{LM}(g)} \\beta _{\\overline{b}}{\\overline{x}}^{\\overline{b}}\\right)\\\\+\\operatorname{LC}(g)\\operatorname{LM}(g)\\left(\\sum _{{\\overline{x}}^{\\overline{a}}\\prec \\operatorname{LM}(f)} \\alpha _{\\overline{a}}{\\overline{x}}^{\\overline{a}}\\right)+\\left(\\sum _{{\\overline{x}}^{\\overline{a}}\\prec \\operatorname{LM}(f)} \\alpha _{\\overline{a}}{\\overline{x}}^{\\overline{a}}\\right)\\left(\\sum _{{\\overline{x}}^{\\overline{b}}\\prec \\operatorname{LM}(g)} \\beta _{\\overline{b}}{\\overline{x}}^{\\overline{b}}\\right).$ Using that ${\\overline{x}}^{\\overline{a}}{\\overline{x}}^{\\overline{b}}\\prec \\operatorname{LM}(f) \\operatorname{LM}(g)$ whenever ${\\overline{x}}^{\\overline{a}}\\prec \\operatorname{LM}(f)$ or ${\\overline{x}}^{\\overline{b}}\\prec \\operatorname{LM}(g)$ due to the definition of a monomial order, we have that $\\operatorname{LM}(f)\\operatorname{LM}(g)$ is indeed the maximal monomial in the above expression with nonzero coefficient, and as its coefficient is $\\operatorname{LC}(f)\\operatorname{LC}(g)$ .", "We now recall the well-known fact that for any set of polynomials the dimension of their span in $\\mathbb {F}[{\\overline{x}}]$ is equal to the number of distinct leading or trailing monomials in their span.", "Let $S\\subseteq \\mathbb {F}[{\\overline{x}}]$ be a set of polynomials.", "Then $\\dim \\operatorname{span}S=\\left\|\\operatorname{LM}(\\operatorname{span}S)\\right\|=\\left\|\\operatorname{TM}(\\operatorname{span}S)\\right\|$ .", "In particular, $\\dim \\operatorname{span}S\\ge \\left\|\\operatorname{LM}(S)\\right\|,\\left\|\\operatorname{TM}(S)\\right\|$ ." ], [ "Upper Bounds for Linear-IPS", "While the primary focus of this work is on lower bounds for restricted classes of the IPS proof system, we begin by discussing upper bounds to demonstrate that these restricted classes can prove the unsatisfiability of non-trivial systems of polynomials equations.", "In particular we go beyond existing work on upper bounds ([28], [62], [63], [33], [47]) and place interesting refutations in IPS subsystems where we will also prove lower bounds, as such upper bounds demonstrate the non-triviality of our lower bounds.", "We begin by discussing the power of the linear-IPS proof system.", "While one of the most novel features of IPS proofs is their consideration of non-linear certificates, we show that in powerful enough models of algebraic computation, linear-IPS proofs can efficiently simulate general IPS proofs, essentially answering an open question of Grochow and Pitassi [33].", "A special case of this result was obtained by Grochow and Pitassi [33], where they showed that IPS$_{\\text{LIN}}$ can simulate $\\sum \\prod $ -IPS.", "We then consider the subset-sum axioms, previously considered by Impagliazzo, Pudlák, and Sgall [38], and show that they can be refuted in polynomial size by the $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ proof system where $\\mathcal {C}$ is either the class of roABPs, or the class of multilinear formulas." ], [ "Simulating IPS Proofs with Linear-IPS", "We show here that general IPS proofs can be efficiently simulated by linear-IPS, assuming that the axioms to be refuted are described by small algebraic circuits.", "Grochow and Pitassi [33] showed that whenever the IPS proof computes sparse polynomials, one can simulate it by linear-IPS using (possibly non-sparse) algebraic circuits.", "We give here a simulation of IPS when the proofs use general algebraic circuits.", "To give our simulation, we will need to show that if a small circuit $f({\\overline{x}},y)$ is divisible by $y$ , then the quotient ${f({\\overline{x}},y)}{y}$ also has a small circuit.", "Such a result clearly follows from Strassen's [71] elimination of divisions in general, but we give two constructions for the quotient which tailor Strassen's [71] technique to optimize certain parameters.", "The first construction assumes that $f$ has degree bounded by $d$ , and produces a circuit for the quotient whose size depends polynomially on $d$ .", "This construction is efficient when $f$ is computed by a formula or branching program (so that $d$ is bounded by the size of $f$ ).", "In particular, this construction will preserve the depth of $f$ in computing the quotient, and as such we only present it for formulas.", "The construction proceeds via interpolation to decompose $f({\\overline{x}},y)=\\sum _i f_i({\\overline{x}})y^i$ into its constituent parts $\\lbrace f_i({\\overline{x}})\\rbrace _i$ and then directly constructs ${f({\\overline{x}},y)}{y}=\\sum _i f_i({\\overline{x}})y^{i-1}$ .", "Let $\\mathbb {F}$ be a field with $\|\\mathbb {F}\|\\ge d+1$ .", "Let $f({\\overline{x}},y)\\in \\mathbb {F}[x_1,\\ldots ,x_n,y]$ be a degree $\\le d$ polynomial expressible as $f({\\overline{x}},y)=\\sum _{0\\le i\\le d} f_i({\\overline{x}})y^i$ for $f_i\\in \\mathbb {F}[{\\overline{x}}]$ .", "Assume $f$ is computable by a size-$s$ depth-$D$ formula.", "Then for $a\\ge 1$ one can compute $\\sum _{i=a}^d f_i({\\overline{x}}) y^{i-a}\\;,$ by a $(s,a,d)$ -size depth-$(D+2)$ formula.", "Further, given $d$ and the formula for $f$ , the resulting formula is $(s,a,d)$ -explicit.", "In particular, if $y^a\|f({\\overline{x}},y)$ then the quotient ${f({\\overline{x}},y)}{y^a}$ has a formula of these parameters.", "Express $f({\\overline{x}},y)\\in \\mathbb {F}[{\\overline{x}}][y]$ by $f({\\overline{x}},y)=\\sum _{0\\le i\\le d} f_i({\\overline{x}})y^i$ .", "As $\|\\mathbb {F}\|\\ge 1+\\deg _y f$ , by interpolation there are $(d)$ -explicit constants $\\alpha _{i,j},\\beta _j\\in \\mathbb {F}$ such that $f_i({\\overline{x}})=\\sum _{j=0}^d \\alpha _{i,j}f({\\overline{x}},\\beta _j)\\;.$ It then follows that $\\sum _{i=a}^d f_i({\\overline{x}}) y^{i-a}=\\sum _{i=a}^d \\left(\\sum _{j=0}^d \\alpha _{i,j}f({\\overline{x}},\\beta _j)\\right) y^{i-a}=\\sum _{i=a}^d \\sum _{j=0}^d \\alpha _{i,j}f({\\overline{x}},\\beta _j) y^{i-a}\\;,$ which is clearly a formula of the appropriate size, depth, and explicitness.", "The claim about the quotient ${f({\\overline{x}},y)}{y^a}$ follows from seeing that if the quotient is a polynomial then ${f({\\overline{x}},y)}{y^a}=\\sum _{i=a}^d f_i({\\overline{x}}) y^{i-a}$ .", "The above construction suffices in the typical regime of algebraic complexity where the circuits compute polynomials whose degree is polynomially-related to their circuit size.", "However, the simulation of Extended Frege by general IPS proved by Grochow-Pitassi [33] (thm:GrochowPitassi14) yields IPS refutations with circuits of possibly exponential degree (see also rmk:EF-degree).", "As such, this motivates the search for an efficient division lemma in this regime.", "We now provide such a lemma, which is a variant of Strassen's [71] homogenization technique for efficiently computing the low-degree homogeneous components of an unbounded degree circuit.", "As weaker models of computation (such as formulas and branching programs) cannot compute polynomials of degree exponential in their size, we only present this lemma for circuits.", "Let $f({\\overline{x}},y)\\in \\mathbb {F}[x_1,\\ldots ,x_n,y]$ be a polynomial expressible as $f({\\overline{x}},y)=\\sum _i f_i({\\overline{x}})y^i$ for $f_i\\in \\mathbb {F}[{\\overline{x}}]$ , and assume $f$ is computable by a size-$s$ circuit.", "Then for $a\\ge 1$ there is an $O(a^2s)$ -size circuit with outputs gates computing $f_0({\\overline{x}}),\\ldots ,f_{a-1}({\\overline{x}}),\\sum _{i\\ge a} f_i({\\overline{x}}) y^{i-a}\\;.$ Further, given $a$ and the circuit for $f$ , the resulting circuit is $(s,a)$ -explicit.", "In particular, if $y^a\|f({\\overline{x}},y)$ then the quotient ${f({\\overline{x}},y)}{y^a}$ has a circuit of these parameters.", "The proof proceeds by viewing the computation in the ring $\\mathbb {F}[{\\overline{x}}][y]$ , and splitting each gate in the circuit for $f$ into its coefficients in terms of $y$ .", "However, to avoid a dependence on the degree, we only split out the coefficients of $y^0,y^1,\\ldots ,y^{a-1}$ , and then group together the rest of the coefficients together.", "That is, for a polynomial $g({\\overline{x}},y)=\\sum _{i\\ge 0} g_i({\\overline{x}})y^i$ , we can split this into $g=\\sum _{0\\le i<a} g_i({\\overline{x}})y^i+\\left(\\sum _{i\\ge a} g_i({\\overline{x}})y^{i-a}\\right)y^a$ to obtain the constituent parts $g_0({\\overline{x}}),\\ldots ,g_{a-1}({\\overline{x}}),\\sum _{i\\ge a}g_i({\\overline{x}})y^{i-a}$ .", "We can then locally update this split by appropriately keeping track of how addition and multiplication affects this grouping of coefficients.", "We note that we can assume without loss of generality that the circuit for $f$ has fan-in 2, as this only increases the size of the circuit by a constant factor (measuring the size of the circuit in number of edges) and simplifies the construction.", "construction: Let $\\Phi $ denote the circuit for $f$ .", "For a gate $v$ in $\\Phi $ , denote $\\Phi _v$ to be the configuration of $v$ in $\\Phi $ and let $f_v$ to be the polynomial computed by the gate $v$ .", "We will define the new circuit $\\Psi $ , which will be defined by the gates $\\lbrace (v,i): v\\in \\Phi , 0\\le i\\le a\\rbrace $ and the wiring between them, as follows.", "$\\Phi _v\\in \\mathbb {F}$ : $\\Psi _{(v,0)}:=\\Phi _v$ , $\\Psi _{(v,i)}:=0$ for $i\\ge 1$ .", "$\\Phi _v=x_i$ : $\\Psi _{(v,0)}:=x_i$ , $\\Psi _{(v,i)}:=0$ for $i\\ge 1$ .", "$\\Phi _v=y$ : $\\Psi _{(v,1)}:=1$ , $\\Psi _{(v,i)}:=0$ for $i\\ne 1$ .", "$\\Phi _v=\\Phi _u+\\Phi _w$ : $\\Psi _{(v,i)}:=\\Psi _{(u,i)}+\\Psi _{(w,i)}$ , all $i$ .", "$\\Phi _v=\\Phi _u\\times \\Phi _w$ , $0\\le i<a$ : $\\Psi _{(v,i)}:=\\sum _{0\\le j\\le i}\\Psi _{(u,j)}\\times \\Psi _{(w,i-j)}\\;.$ $\\Phi _v=\\Phi _u\\times \\Phi _w$ , $i=a$ : $\\Psi _{(v,a)}&:=\\sum _{\\ell =a}^{2(a-1)}y^{\\ell -a}\\sum _{\\begin{array}{c}i+j=\\ell \\\\0\\le i,j< a\\end{array}}\\Psi _{(u,i)}\\times \\Psi _{(w,j)}+ \\sum _{0\\le i<a} \\Psi _{(u,i)}\\times \\Psi _{(w,a)}\\times y^i\\\\&\\hspace{72.26999pt}+ \\sum _{0\\le j<a} \\Psi _{(u,a)}\\times \\Psi _{(w,j)}\\times y^j+\\Psi _{(u,a)}\\times \\Psi _{(w,a)}\\times y^a\\;.$ complexity: Split the gates in $\\Psi $ into two types, those gates $(v,i)$ where $i=a$ and $v$ is a multiplication gate in $\\Phi $ , and then the rest.", "For the former type, $\\Psi _{(v,a)}$ is computable by a size-$O(a^2)$ circuit in its children, and there are at most $s$ such gates.", "For the latter type, $\\Psi _{(v,i)}$ is computable by a size-$O(a)$ circuit in its children, and there are at most $O(as)$ such gates.", "As such, the total size is $O(a^2s)$ .", "correctness: We now establish correctness as a subclaim.", "For a gate $(v,i)$ in $\\Psi $ , let $g_{(v,i)}$ denote the polynomial that it computes.", "For each gate $v$ in $\\Phi $ , for $0\\le i<a$ we have that $g_{(v,i)}={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)$ and for $i=a$ we have that $g_{(v,a)}=\\sum _{i\\ge a} {\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)y^{i-a}$ .", "In particular, $f_v=\\sum _{i=0}^a g_{(v,i)}y^i$ .", "[Sub-Proof:]Note that the second part of the claim follows from the first.", "We now establish the first part by induction on the gates of the circuit.", "$\\Phi _v\\in \\mathbb {F}$ : By construction, $g_{(v,0)}=f_v={\\mathrm {Coeff}}_{{\\overline{x}}\|y^0}(f_v)$ , and for $i\\ge 1$ , $g_{(v,i)}=0={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)$ .", "$\\Phi _v=x_i$ : By construction, $g_{(v,0)}=f_v={\\mathrm {Coeff}}_{{\\overline{x}}\|y^0}(f_v)$ , and for $i\\ge 1$ , $g_{(v,i)}=0={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)$ .", "$\\Phi _v=y$ : By construction, $g_{(v,1)}=1={\\mathrm {Coeff}}_{{\\overline{x}}\|y^1}(f_v)$ , and for $i\\ne 1$ , $g_{(v,i)}=0={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)$ .", "$\\Phi _v=\\Phi _u+\\Phi _w$ : $g_{(v,i)}&=g_{(u,i)}+g_{(w,i)}\\\\&={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u)+{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_w)\\\\&={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u+f_w)={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)\\;.$ $\\Phi _v=\\Phi _u\\times \\Phi _w$ , $0\\le i<a$ : $g_{(v,i)}&=\\sum _{0\\le j\\le i} g_{(u,j)}\\cdot g_{(w,i-j)}\\\\&=\\sum _{0\\le j\\le i} {\\mathrm {Coeff}}_{{\\overline{x}}\|y^{j}}(f_u) \\cdot {\\mathrm {Coeff}}_{{\\overline{x}}\|y^{i-j}}(f_w)\\\\&={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u\\cdot f_w)={\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_v)\\;.$ $\\Phi _v=\\Phi _u\\times \\Phi _w$ , $i=a$ : $g_{(v,a)}&=\\sum _{\\ell =a}^{2(a-1)}y^{\\ell -a}\\sum _{\\begin{array}{c}i+j=\\ell \\\\0\\le i,j< a\\end{array}}g_{(u,i)}\\cdot g_{(w,j)}+ \\sum _{0\\le i<a} g_{(u,i)}\\cdot g_{(w,a)}\\cdot y^i\\\\&\\hspace{72.26999pt}+ \\sum _{0\\le j<a} g_{(u,a)}\\cdot g_{(w,j)}\\cdot y^j+g_{(u,a)}\\cdot g_{(w,a)}\\cdot y^a\\\\&=\\sum _{\\ell =a}^{2(a-1)}y^{-a}\\sum _{\\begin{array}{c}i+j=\\ell \\\\0\\le i,j< a\\end{array}}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u)y^i\\cdot {\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w)y^j\\\\&\\hspace{72.26999pt}+ \\sum _{0\\le i<a} {\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u)\\cdot \\left(\\sum _{j\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w)y^{j-a}\\right)\\cdot y^i\\\\&\\hspace{72.26999pt}+ \\sum _{0\\le j<a} \\left(\\sum _{i\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u)y^{i-a}\\right)\\cdot {\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w)\\cdot y^j\\\\&\\hspace{72.26999pt}+\\left(\\sum _{i\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u)y^{i-a}\\right)\\cdot \\left(\\sum _{j\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w)y^{j-a}\\right)\\cdot y^a\\\\&=\\sum _{\\begin{array}{c}i+j\\ge a\\\\0\\le i,j<a\\end{array}}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u){\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w) y^{i+j-a}+\\sum _{\\begin{array}{c}0\\le i<a\\\\ j\\ge a\\end{array}}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u){\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w) y^{i+j-a}\\\\&\\hspace{14.45377pt}+\\sum _{\\begin{array}{c}i\\ge a\\\\ 0\\le j<a\\end{array}}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u){\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w) y^{i+j-a}+\\sum _{i,j\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u){\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w) y^{i+j-a}\\\\&=\\sum _{i+j\\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^i}(f_u){\\mathrm {Coeff}}_{{\\overline{x}}\|y^j}(f_w)\\cdot y^{i+j-a}\\\\&=\\sum _{\\ell \\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^\\ell }(f_u \\cdot f_w)\\cdot y^{\\ell -a}\\\\&=\\sum _{\\ell \\ge a}{\\mathrm {Coeff}}_{{\\overline{x}}\|y^\\ell }(f_v)\\cdot y^{\\ell -a}\\;.$ The correctness then follows by examining $v_\\mathrm {out}$ , the output gate of $\\Phi $ , so that $f_{v_\\mathrm {out}}=f$ .", "The gates $(v_\\mathrm {out},0),\\ldots ,(v_\\mathrm {out},a)$ are then outputs of $\\Psi $ and by the above subclaim have the desired functionality.", "quotient: The claim about the quotient ${f({\\overline{x}},y)}{y^a}$ follows from seeing that if the quotient is a polynomial then ${f({\\overline{x}},y)}{y^a}=\\sum _{i\\ge a} f_i({\\overline{x}}) y^{i-a}$ which is one of the outputs of the constructed circuit.", "We now give our simulation of general IPS by linear-IPS.", "In the below set of axioms we do not separate out the boolean axioms from the rest, as this simplifies notation.", "Let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be unsatisfiable polynomials with an IPS refutation $C\\in \\mathbb {F}[{\\overline{x}},y_1,\\ldots ,y_m]$ .", "Then $f_1,\\ldots ,f_m$ have a linear-IPS refutation $C^{\\prime }\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ under the following conditions.", "Suppose $f_1,\\ldots ,f_m,C$ are computed by size-$s$ formulas, have degree at most $d$ , and $\|\\mathbb {F}\|\\ge d+1$ .", "Then $C^{\\prime }$ is computable by a $(s,d,m)$ -size formula of depth-$O(D)$ , and $C^{\\prime }$ is $(s,d,m)$ -explicit given $d$ and the formulas for $f_1,\\ldots ,f_m,C$ .", "Suppose $f_1,\\ldots ,f_m,C$ are computed by size-$s$ circuits.", "Then $C^{\\prime }$ is computable by a $(s,m)$ -size circuit, and $C^{\\prime }$ is $(s,m)$ -explicit given the circuits for $f_1,\\ldots ,f_m,C$ .", "Express $C({\\overline{x}},{\\overline{y}})$ as a polynomial in $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ , so that $C({\\overline{x}},{\\overline{y}})=\\sum _{{\\overline{a}}>{\\overline{0}}} C_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}$ , where we use that $C({\\overline{x}},{\\overline{0}})=0$ to see that we can restrict ${\\overline{a}}$ to ${\\overline{a}}>{\\overline{0}}$ .", "Partitioning the ${\\overline{a}}\\in \\mathbb {N}^n$ based on the index of their first nonzero value, and denoting ${\\overline{a}}_{<i}$ for the first $i-1$ coordinates of ${\\overline{a}}$ , we obtain $C({\\overline{x}},{\\overline{y}})&=\\sum _{{\\overline{a}}>{\\overline{0}}} C_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}\\\\&=\\sum _{i=1}^n\\; \\sum _{\\begin{array}{c}{\\overline{a}}: {\\overline{a}}_{<i}={\\overline{0}},\\\\ a_i>0\\end{array}} C_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}\\;.\\multicolumn{2}{l}{\\text{Now define $C_i({\\overline{x}},{\\overline{y}}):=\\sum _{\\begin{array}{c}{\\overline{a}}: {\\overline{a}}_{<i}={\\overline{0}},\\\\ a_i>0\\end{array}} C_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{a}}-{\\overline{e}}_i}$, where ${\\overline{e}}_i$ is the $i$-th standard basis vector.", "Note that this is a valid polynomial as in this summation we assume $a_i>0$ so that ${\\overline{a}}-{\\overline{e}}_i\\ge {\\overline{0}}$.", "Thus,}}\\\\C({\\overline{x}},{\\overline{y}})&=\\sum _{i=1}^n C_i({\\overline{x}},{\\overline{y}})y_i\\;.$ We now define $C^{\\prime }({\\overline{x}},{\\overline{y}}):=\\sum _{i=1}^n C_i({\\overline{x}},{\\overline{f}}({\\overline{x}}))y_i$ and claim it is the desired linear-IPS refutation, where note that we have only partially substituted in the $f_i$ for the $y_i$ .", "First, observe that it is a valid refutation, as $C^{\\prime }({\\overline{x}},{\\overline{0}})=\\sum _{i=1}^n C_i({\\overline{x}},{\\overline{f}}({\\overline{x}}))\\cdot 0=0$ , and $C^{\\prime }({\\overline{x}},{\\overline{f}}({\\overline{x}}))=\\sum _{i=1}^n C_i({\\overline{x}},{\\overline{f}}({\\overline{x}}))f_i({\\overline{x}})=C({\\overline{x}},{\\overline{f}}({\\overline{x}}))=1$ via the above equation and using that $C$ is a valid IPS refutation.", "We now argue that $C^{\\prime }$ can be efficiently computed in the two above regimes.", "(REF ): Up to constant-loss in the depth and polynomial-loss in the size, for bounding the complexity of $C^{\\prime }$ it suffices to bound the complexity of each $C_i({\\overline{x}},{\\overline{f}}({\\overline{x}}))$ .", "First, note that $C_i({\\overline{x}},{\\overline{y}})y_i=\\sum _{\\begin{array}{c}{\\overline{a}}: {\\overline{a}}_{<i}={\\overline{0}},\\\\ a_i>0\\end{array}} C_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}=C({\\overline{x}},{\\overline{0}},y_i,{\\overline{y}}_{> i})-C({\\overline{x}},{\\overline{0}},0,{\\overline{y}}_{> i})\\;,$ where each “${\\overline{0}}$ ” here is a vector of $i-1$ zeros.", "Clearly each of $C({\\overline{x}},{\\overline{0}},y_i,{\\overline{y}}_{> i})$ and $C({\\overline{x}},{\\overline{0}},0,{\\overline{y}}_{> i})$ have formula size and depth bounded by that of $C$ .", "From our division lemma for formulas (res:divide-by-y:black-box) it follows that $C_i({\\overline{x}},{\\overline{y}})=\\frac{1}{y_i}(C({\\overline{x}},{\\overline{0}},y_i,{\\overline{y}}_{> i})-C({\\overline{x}},{\\overline{0}},0,{\\overline{y}}_{> i}))$ has a $(s,d)$ -size depth-$O(D)$ formula of the desired explicitness (as $\\deg _y \\left(C({\\overline{x}},{\\overline{0}},y_i,{\\overline{y}}_{> i})-C({\\overline{x}},{\\overline{0}},0,{\\overline{y}}_{> i})\\right)\\le \\deg _y C\\le d\\le \|\\mathbb {F}\|-1$ , so that $\\mathbb {F}$ is large enough).", "We replace ${\\overline{y}}\\leftarrow {\\overline{f}}({\\overline{x}})$ to obtain $C_i({\\overline{x}},{\\overline{f}}({\\overline{x}}))$ of the desired size and explicitness, using that the $f_j$ themselves have small-depth formulas.", "(REF ): This follows as in (REF ), using now the division lemma for circuits (res:divide-by-y:white-box).", "Grochow and Pitassi [33] asked whether one can relate the complexity of IPS and linear-IPS, as they only established such relations for simulating $\\sum \\prod $ -IPS by (general) linear-IPS.", "Our above result essentially answers this question for general formulas and circuits, at least under the assumption that the unsatisfiable polynomial system $f_1=\\cdots =f_m=0$ can be written using small algebraic formulas or circuits.", "This is a reasonable assumption as it is the most common regime for proof complexity.", "However, the above result does not fully close the question of Grochow-Pitassi [33] with respect to simulating $\\mathcal {C}$ -IPS by $\\mathcal {D}$ -IPS$_{\\text{LIN}}$ for various restricted subclasses $\\mathcal {C},\\mathcal {D}$ of algebraic computation.", "That is, for such a simulation our result requires $\\mathcal {D}$ to at the very least contain $\\mathcal {C}$ composed with the axioms $f_1,\\ldots ,f_m$ , and the when applying this to the models considered in this paper (sparse polynomials, depth-3 powering formulas, roABPs, multilinear formulas) this seems to non-negligibly increase the complexity of the algebraic reasoning." ], [ "Multilinearizing roABP-IPS$_{\\text{LIN}}$", "We now exhibit instances where one can efficiently prove that a polynomial equals its multilinearization modulo the boolean axioms.", "That is, for a polynomial $f$ computed by a small circuit we wish to prove that $f\\equiv \\operatorname{ml}(f)\\mod {{\\overline{x}}}^2-{\\overline{x}}$ by expressing $f-\\operatorname{ml}(f)=\\sum _i h_i \\cdot (x_i^2-x_i)$ so that the $h_i$ also have small circuits.", "Such a result will simplify the search for linear-IPS refutations by allowing us to focus on the non-boolean axioms.", "That is, if we are able to find a refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ given by $\\sum _j g_j f_j \\equiv 1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;,$ where the $g_j$ have small circuits, multilinearization results of the above form guarantee that there are $h_i$ so that $\\sum _j g_j f_j +\\sum _i h_i\\cdot (x_i^2-x_i)=1\\;,$ which is a proper linear-IPS refutation.", "We establish such a multilinearization result when $f$ is an roABP in this section, and consider when $f$ the product of a low-degree multilinear polynomial and a multilinear formula in the next section.", "We will use these multilinearization results in our construction of IPS refutations of the subset-sum axiom (sec:subset:ub).", "We begin by noting that multilinearization for these two circuit classes is rather special, as these classes straddle the conflicting requirements of neither being too weak nor too strong.", "That is, some circuit classes are simply too weak to compute their multilinearizations.", "An example is the class of depth-3 powering formulas, where $(x_1+\\cdots +x_n)^n$ has a small $\\sum \\bigwedge \\sum $ formula, but its multilinearization has the leading term $n!x_1\\cdots x_n$ and thus requires exponential size as a $\\sum \\bigwedge \\sum $ formula (by appealing to res:lbs-mult:LM:sumpowsum).", "On the other hand, some circuit classes are too strong to admit efficient multilinearization (under plausible complexity assumptions).", "That is, consider an $n \\times n$ symbolic matrix $X$ where $(X)_{i,j}=x_{i,j}$ and the polynomial $f(X,{\\overline{y}}):=(x_{1,1} y_1+\\cdots +x_{1,n}y_n)\\cdots (x_{n,1} y_1+\\cdots +x_{n,n}y_n)$ , which is clearly a simple depth-3 ($\\prod \\sum \\prod $ ) circuit.", "Viewing this polynomial in $\\mathbb {F}[X][{\\overline{y}}]$ , one sees that ${\\mathrm {Coeff}}_{X\|y_1\\cdots y_n} f=\\operatorname{perm}(X)$ , where $\\operatorname{perm}(X)$ is the $n\\times n$ permanent.", "Viewing $\\operatorname{ml}(f)$ , the multilinearization of $f$ , in $\\mathbb {F}[X][{\\overline{y}}]$ one sees that $\\operatorname{ml}(f)$ is of degree $n$ and its degree $n$ component is the coefficient of $y_1\\cdots y_n$ in $\\operatorname{ml}(f)$ , which is still $\\operatorname{perm}(X)$ .", "Hence, by interpolation, one can extract this degree $n$ part and thus can compute a circuit for $\\operatorname{perm}(X)$ given a circuit for $\\operatorname{ml}(f)$ .", "Since we believe $\\operatorname{perm}(X)$ does not have small algebraic circuits it follows that the multilinearization of $f$ does not have small circuits.", "These examples show that efficient multilinearization is a somewhat special phenomenon.", "We now give our result for multilinearizing roABPs, where we multilinearize variable by variable via telescoping.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be computable by a width-$r$ roABP in variable order $x_1<\\cdots <x_n$ , so that $f({\\overline{x}})=\\left(\\prod _{i=1}^n A_i(x_i)\\right)_{1,1}$ where $A_i\\in \\mathbb {F}[x_i]^{r\\times r}$ have $\\deg A_i\\le d$ .", "Then $\\operatorname{ml}(f)$ has a $(r,n,d)$ -explicit width-$r$ roABP in variable order $x_1<\\cdots <x_n$ , and there are $(r,n,d)$ -explicit width-$r$ roABPs $h_1,\\ldots ,h_n\\in \\mathbb {F}[{\\overline{x}}]$ in variable order $x_1<\\cdots <x_n$ such that $f({\\overline{x}})=\\operatorname{ml}(f)+\\sum _{j=1}^n h_j\\cdot (x_j^2-x_j)\\;.$ Further, $\\operatorname{ideg}h_j\\le \\operatorname{ideg}f$ and the individual degree of the roABP for $\\operatorname{ml}(f)$ is $\\le 1$ .", "We apply the multilinearization map $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]\\rightarrow \\mathbb {F}[{\\overline{x}}]$ to matrices $\\operatorname{ml}:\\mathbb {F}[{\\overline{x}}]^{r\\times r}\\rightarrow \\mathbb {F}[{\\overline{x}}]^{r\\times r}$ by applying the map entry-wise (fact:multilinearization).", "It follows then that $A_i(x_i)-\\operatorname{ml}(A_i(x_i))\\equiv 0\\mod {x}_i^2-x_i$ , so that $A_i(x_i)-\\operatorname{ml}(A_i(x_i))=B(x_i)\\cdot (x_i^2-x_i)$ for some $B_i(x_i)\\in \\mathbb {F}[x_i]^{r\\times r}$ where $\\operatorname{ideg}B_i(x_i)\\le \\operatorname{ideg}A_i(x_i)$ .", "Now define $\\operatorname{ml}_{\\le i}$ be the map which multilinearizes the first $i$ variables and leaves the others intact, so that $\\operatorname{ml}_{\\le 0}$ is the identity map and $\\operatorname{ml}_{\\le n}=\\operatorname{ml}$ .", "Telescoping, $\\prod _{i=1}^n A_i(x_i)&=\\operatorname{ml}_{<1}\\left(\\prod _{i=1}^n A_i(x_i)\\right)\\\\&=\\operatorname{ml}_{\\le n}\\left(\\prod _{i=1}^n A_i(x_i)\\right)+\\sum _{j=1}^{n} \\left[\\operatorname{ml}_{<j}\\left(\\prod _{i=1}^n A_i(x_i)\\right)-\\operatorname{ml}_{\\le j}\\left(\\prod _{i=1}^n A_i(x_i)\\right)\\right]\\\\\\multicolumn{2}{l}{\\text{using that the identity $\\operatorname{ml}(gh)=\\operatorname{ml}(\\operatorname{ml}(g)\\operatorname{ml}(h))$ ({fact:multilinearization}) naturally extends from scalars to matrices, and also to partial-multilinearization (by viewing $\\operatorname{ml}_{\\le i}$ as multilinearization in $\\mathbb {F}[{\\overline{x}}_{>i}][{\\overline{x}}_{\\le i}]$),}}\\\\&=\\operatorname{ml}_{\\le n}\\left(\\prod _{i=1}^n \\operatorname{ml}_{\\le n} A_i(x_i)\\right)+\\sum _{j=1}^{n} \\left[\\operatorname{ml}_{<j}\\left(\\prod _{i< j} \\operatorname{ml}_{<j}(A_i(x_i))\\prod _{i\\ge j} A_i(x_i)\\right)\\right.\\\\&\\hspace{144.54pt}\\left.-\\operatorname{ml}_{\\le j}\\left(\\prod _{i\\le j} \\operatorname{ml}_{\\le j}(A_i(x_i))\\prod _{i>j} A_i(x_i)\\right)\\right]\\\\\\multicolumn{2}{l}{\\text{dropping the outside $\\operatorname{ml}_{<j}$ and $\\operatorname{ml}_{\\le j}$ as the inside polynomials are now multilinear in the appropriate variables, and replacing them with $\\operatorname{ml}$ as appropriate,}}\\\\&=\\prod _{i=1}^n \\operatorname{ml}(A_i(x_i))+\\sum _{j=1}^{n} \\left[\\prod _{i<j} \\operatorname{ml}(A_i(x_i))\\prod _{i\\ge j} A_i(x_i)\\right.\\\\&\\hspace{144.54pt}\\left.-\\prod _{i\\le j} \\operatorname{ml}(A_i(x_i))\\prod _{i>j} A_i(x_i)\\right]\\\\&=\\prod _{i=1}^n \\operatorname{ml}(A_i(x_i))+\\sum _{j=1}^{n} \\left[\\prod _{i<j} \\operatorname{ml}(A_i(x_i))\\cdot \\Big (A_j(x_j)-\\operatorname{ml}(A_j(x_j))\\Big )\\cdot \\prod _{i>j} A_i(x_i)\\right]\\\\&=\\prod _{i=1}^n \\operatorname{ml}(A_i(x_i))+\\sum _{j=1}^{n} \\left[\\prod _{i<j} \\operatorname{ml}(A_i(x_i))\\cdot B_j(x_j)\\cdot \\prod _{i>j} A_i(x_i)\\cdot (x_j^2-x_j)\\right]\\;.$ Taking the $(1,1)$ -entry in the above yields that $f({\\overline{x}})&=\\left(\\prod _{i=1}^n A_i(x_i)\\right)_{1,1}\\\\&=\\left(\\prod _{i=1}^n \\operatorname{ml}(A_i(x_i))\\right)_{1,1}+\\sum _{j=1}^{n} \\left(\\prod _{i<j} \\operatorname{ml}(A_i(x_i))\\cdot B_j(x_j)\\cdot \\prod _{i>j} A_i(x_i)\\right)_{1,1}\\cdot (x_j^2-x_j)\\;.$ Thus, define $\\hat{f}:=\\left(\\prod _{i=1}^n \\operatorname{ml}(A_i(x_i))\\right)_{1,1}\\;,\\qquad h_j :=\\left(\\prod _{i<j} \\operatorname{ml}(A_i(x_i))\\cdot B_j(x_j)\\cdot \\prod _{i>j} A_i(x_i)\\right)_{1,1}\\;.$ It follows by construction that $\\hat{f}$ and each $h_j$ are computable by width-$r$ roABPs of the desired explicitness in the correct variable order.", "Further, $\\operatorname{ideg}h_j\\le \\operatorname{ideg}f$ and $\\hat{f}$ has individual degree $\\le 1$ .", "Thus, the above yields that $f=\\hat{f}+\\sum _j h_j\\cdot (x_j^2-x_j)$ , from which it follows that $\\operatorname{ml}(f)=\\hat{f}$ , as desired.", "We now conclude that in designing an roABP-IPS$_{\\text{LIN}}$ refutation $\\sum _j g_j\\cdot f_j+\\sum _i h_i\\cdot (x_i^2-x_i)$ of $f_1({\\overline{x}}),\\ldots ,f_m({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ , it suffices to bound the complexity of the $g_j$ 's.", "Let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be unsatisfiable polynomials over ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ computable by width-$s$ roABPs in variable order $x_1<\\cdots <x_n$ .", "Suppose that there are $g_j\\in \\mathbb {F}[{\\overline{x}}]$ such that $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;,$ where the $g_j$ have width-$r$ roABPs in the variable order $x_1<\\cdots <x_n$ .", "Then there is an roABP-IPS$_{\\text{LIN}}$ refutation $C({\\overline{x}},{\\overline{y}},{\\overline{z}})$ of individual degree at most $1+\\operatorname{ideg}{\\overline{f}}$ and computable in width-$(s,r,n,m)$ in any variable order where $x_1<\\cdots <x_n$ .", "Furthermore, this refutation is $(s,r,\\operatorname{ideg}{\\overline{g}},\\operatorname{ideg}{\\overline{f}},n,m)$ -explicit given the roABPs for $f_j$ and $g_j$ .", "We begin by noting that we can multilinearize the $g_j$ , so that $\\sum _{i=1}^m \\operatorname{ml}(g_j({\\overline{x}}))f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ , and that $\\operatorname{ml}(g_j)$ are $(r,\\operatorname{ideg}{\\overline{g}},n,m)$ -explicit multilinear roABPs of width-$r$ by res:multilin:roABP.", "Thus, we assume going forward that the $g_j$ are multilinear.", "As $g_j,f_j$ are computable by roABPs, their product $g_jf_j$ is computable by a width-$rs$ roABP in the variable order $x_1<\\cdots <x_n$ (fact:roABP:closure) with individual degree at most $1+\\operatorname{ideg}f_j$ (res:roABP-normal-form).", "Thus, by the above multilinearization (res:multilin:roABP), there are $h_{j,i}\\in \\mathbb {F}[{\\overline{x}}]$ such that $g_j({\\overline{x}})f_j({\\overline{x}})=\\operatorname{ml}(g_jf_j)+\\sum _{i=1}^n h_{j,i}({\\overline{x}})\\cdot (x_i^2-x_i)\\;.$ where the $h_{j,i}$ are computable by width-$rs$ roABPs of individual degree at most $1+\\operatorname{ideg}f_j$ .", "We now define $C({\\overline{x}},{\\overline{y}},{\\overline{z}}):=\\sum _{j=1}^m g_j({\\overline{x}})y_j - \\sum _{i=1}^n\\left(\\sum _{j=1}^m h_{j,i}({\\overline{x}})\\right) z_i\\;.$ By the closure operations of roABPs (fact:roABP:closure) it follows that $C$ is computable the appropriately-sized roABPs in the desired variable orders, has the desired individual degree, and that $C$ has the desired explicitness.", "We now show that this is a valid IPS refutation.", "Observe that $C({\\overline{x}},{\\overline{0}},{\\overline{0}})=0$ and that $C({\\overline{x}},{\\overline{f}},{\\overline{x}}^2-{\\overline{x}})&=\\sum _{j=1}^m g_j({\\overline{x}})f_j({\\overline{x}}) - \\sum _{i=1}^n\\left(\\sum _{j=1}^m h_{j,i}({\\overline{x}})\\right) (x_i^2-x_i)\\\\&=\\sum _{j=1}^m \\left( g_j({\\overline{x}})f_j({\\overline{x}}) - \\sum _{i=1}^n h_{j,i}({\\overline{x}}) (x_i^2-x_i)\\right)\\\\&=\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)\\\\\\multicolumn{2}{l}{\\text{as $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ we have that$$\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)=\\operatorname{ml}\\left(\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\right)=1\\;,$$where we appealed to linearity of multilinearization ({fact:multilinearization}), so that}}\\\\C({\\overline{x}},{\\overline{f}},{\\overline{x}}^2-{\\overline{x}})&=\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)=1\\;,$ as desired." ], [ "Multilinear-Formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$", "We now turn to proving that $g\\cdot f\\equiv \\operatorname{ml}(g\\cdot f) \\mod {{\\overline{x}}}^2-{\\overline{x}}$ when $f$ is low-degree and $g$ is a multilinear formula.", "This multilinearization can be used to complete our construction of multilinear-formula-IPS refutations of subset-sum axiom (sec:subset:ub), though our actual construction will multilinearize more directly (res:ips-ubs:subset:mult-form).", "More importantly, the multilinearization we establish here shows that multilinear-formula-IPS can efficiently simulate sparse-IPS$_{\\text{LIN}}$ (when the axioms are low-degree and multilinear).", "Such a simulation holds intuitively, as multilinear formulas can efficiently compute any sparse (multilinear) polynomial, and as we work over the boolean cube we are morally working with multilinear polynomials.", "While this intuition suggests that such a simulation follows immediately, this intuition is false.", "Specifically, while sparse-IPS$_{\\text{LIN}}$ is a complete refutation system for any system of unsatisfiable polynomials over the boolean cube, multilinear-formula-IPS$_{\\text{LIN}}$ is incomplete as seen by the following example (though, by thm:GrochowPitassi14, multilinear-formula-IPS$_{\\text{LIN}}$ is complete for refuting unsatisfiable CNFs).", "Consider the unsatisfiable system of equations $xy+1,x^2-x,y^2-y$ .", "A multilinear linear-IPS proof is a tuple of multilinear polynomials $(f,g,h)\\in \\mathbb {F}[x,y]$ such that $f\\cdot (xy+1)+g\\cdot (x^2-x)+h\\cdot (y^2-y)=1$ .", "In particular, $f(x,y)=\\frac{1}{xy+1}$ for $x,y\\in \\lbrace 0,1\\rbrace $ , which implies by interpolation over the boolean cube that $f(x,y)=1\\cdot (1-x)(1-y)+1\\cdot (1-x)y+1\\cdot x(1-y)+\\frac{1}{2} \\cdot xy=1-\\frac{1}{2}\\cdot xy$ .", "Thus $f\\cdot (xy+1)$ contains the monomial $x^2y^2$ .", "However, as $g,h$ are multilinear we see that $x^2y^2$ cannot appear in $g\\cdot (x^2-x)+h\\cdot (y^2-y)-1$ , so that the equality $f\\cdot (xy+1)+g\\cdot (x^2-x)+h\\cdot (y^2-y)=1$ cannot hold.", "Thus, $xy+1,x^2-x,y^2-y$ has no linear-IPS refutation only using multilinear polynomials.", "Put another way, the above example shows that in a linear-IPS refutation $\\sum _j g_j f_j+\\sum _i h_i\\cdot (x_i^2-x_i)=1$ , while one can multilinearize the $g_j$ (with a possible increase in complexity) and still retain a refutation, one cannot multilinearize the $h_i$ in general.", "As such, to simulate sparse-IPS$_{\\text{LIN}}$ (a complete proof system) we must resort to using the more general $\\text{IPS}_{\\text{LIN}^{\\prime }}$ over multilinear formulas, where recall that the $\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation system considers refutes ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ with a polynomial $C({\\overline{x}},{\\overline{y}},{\\overline{z}})$ where $C({\\overline{x}},{\\overline{0}},{\\overline{0}})=0$ , $C({\\overline{x}},{\\overline{f}},{\\overline{x}}^2-{\\overline{x}})=1$ , with the added restriction that when viewing $C\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}}][{\\overline{y}}]$ that the degree of $C$ with respect to the ${\\overline{y}}$ -variables is at most 1, that is, $\\deg _{\\overline{y}}C\\le 1$ .", "In fact, we in fact establish such a simulation using the subclass of refutations of the form $C({\\overline{x}},{\\overline{y}},{\\overline{z}})=\\sum _j g_j y_j+C^{\\prime }({\\overline{x}},{\\overline{z}})$ where $C^{\\prime }({\\overline{x}},{\\overline{0}})=0$ .", "Note that such refutations are only linear in the non-boolean axioms, which allows us to circumvent ex:multi-form:incomplete.", "We now show how to prove $g\\cdot f\\equiv \\operatorname{ml}(g\\cdot f) \\mod {{\\overline{x}}}^2-{\\overline{x}}$ when $f$ is low-degree and $g$ is a multilinear formula, where the proof is supplied by the equality $g\\cdot f-\\operatorname{ml}(g\\cdot f)=C({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})$ where $C({\\overline{x}},{\\overline{0}})=0$ , and where we seek to show that $C$ has a small multilinear formula.", "We begin with the special case of $f$ and $g$ being the same monomial.", "Let ${\\overline{x}}^{\\overline{1}}=\\prod _{i=1}^n x_i$ .", "Then, $({\\overline{x}}^{{\\overline{1}}})^2-{\\overline{x}}^{{\\overline{1}}}=C({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\;,$ where $C({\\overline{x}},{\\overline{z}})\\in \\mathbb {F}[{\\overline{x}},z_1,\\ldots ,z_n]$ is defined by $C({\\overline{x}},{\\overline{z}}):=({\\overline{z}}+{\\overline{x}})^{\\overline{1}}-{\\overline{x}}^{\\overline{1}}=\\sum _{{\\overline{0}}<{\\overline{a}}\\le {\\overline{1}}} {\\overline{z}}\\!\\:^{\\overline{a}}{\\overline{x}}^{{\\overline{1}}-{\\overline{a}}}\\;,$ so that $C({\\overline{x}},{\\overline{0}})=0$ .", "Note that the first expression for $C$ is a $(n)$ -sized depth-3 expression, while the second is an $\\exp (n)$ -sized depth-2 expression.", "This difference will, going forward, show that we can multilinearize efficiently in depth-3, but can only efficiently multilinearize low-degree monomials in depth-2.", "We now give an IPS proof for showing how a monomial times a multilinear formula equals its multilinearization.", "Let $g\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_d]$ be a multilinear polynomial.", "Then there is a $C\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}},z_1,\\ldots ,z_d]$ such that $g({\\overline{x}},{\\overline{y}})\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}-\\operatorname{ml}(g({\\overline{x}},{\\overline{y}})\\cdot {\\overline{y}}\\!\\:^{\\overline{1}})=C({\\overline{x}},{\\overline{y}},{\\overline{y}}\\!\\:^2-{\\overline{y}})\\;,$ and $C({\\overline{x}},{\\overline{y}},{\\overline{0}})=0$ .", "If $g$ is $t$ -sparse, then $C$ is computable as a $(t,n,2^d)$ -size depth-2 multilinear formula (which is $(t,n,2^d)$ -explicit given the computation for $g$ ), as well as being computable by a $(t,n,d)$ -size depth-3 multilinear formula with a $+$ -output-gate (which is $(t,n,d)$ -explicit given the computation for $g$ ).", "If $g$ is computable by a size-$t$ depth-$D$ multilinear formula, then $C$ is computable by a $(t,2^d)$ -size depth-$(D+2)$ multilinear formula with a $+$ -output-gate (which is $(t,2^d)$ -explicit given the computation for $g$ ).", "defining $C$ : Express $g$ as $g=\\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}$ in the ring $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ , so that each $g_{\\overline{a}}$ is multilinear.", "Then $g({\\overline{x}},{\\overline{y}})\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}-\\operatorname{ml}(g({\\overline{x}},{\\overline{y}})\\cdot {\\overline{y}}\\!\\:^{\\overline{1}})&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}-\\operatorname{ml}\\left(\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}\\right)\\\\&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}})\\left({\\overline{y}}\\!\\:^{{\\overline{a}}+{\\overline{1}}}-{\\overline{y}}\\!\\:^{{\\overline{1}}}\\right)=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\left(({\\overline{y}}\\!\\:^{\\overline{a}})^2-{\\overline{y}}\\!\\:^{{\\overline{a}}}\\right)\\;,\\multicolumn{2}{l}{\\text{and appealing to {res:multilin:mon},}}\\\\&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\left((({\\overline{y}}\\!\\:^2-{\\overline{y}})+{\\overline{y}})^{\\overline{a}}-{\\overline{y}}\\!\\:^{\\overline{a}}\\right)\\\\&=C({\\overline{x}},{\\overline{y}},{\\overline{y}}\\!\\:^2-{\\overline{y}})\\;,$ where we define $C({\\overline{x}},{\\overline{y}},{\\overline{z}}):=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\left(({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-{\\overline{y}}\\!\\:^{{\\overline{a}}}\\right)\\;.$ As $({\\overline{z}}+{\\overline{y}})^{\\overline{a}}={\\overline{y}}\\!\\:^{{\\overline{a}}}$ under ${\\overline{z}}\\leftarrow {\\overline{0}}$ we have that $C({\\overline{x}},{\\overline{y}},{\\overline{0}})=0$ .", "$g$ is $t$ -sparse: As $g$ is $t$ -sparse and multilinear, so are each $g_{\\overline{a}}$ , so that $g_{\\overline{a}}=\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}$ , and thus $C({\\overline{x}},{\\overline{y}},{\\overline{z}})&=\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\left(({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-{\\overline{y}}\\!\\:^{{\\overline{a}}}\\right)\\\\&=\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{y}}\\!\\:^{\\overline{1}}\\;,\\multicolumn{2}{l}{\\text{where this is clearly an explicit depth-3 ($\\sum \\prod \\sum $) multilinear formula (as ${\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}$ is variable-disjoint from $({\\overline{z}}+{\\overline{y}})^{\\overline{a}}$), and the size of this formula is $(n,t,d)$ as there are at most $t$ such ${\\overline{a}}$ where $g_{\\overline{a}}\\ne 0$ as $g$ is $t$-sparse.", "Continuing the expansion, appealing to {res:multilin:mon},}}\\\\&=\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\sum _{0\\le {\\overline{b}}\\le {\\overline{a}}}{\\overline{z}}\\!\\:^{{\\overline{b}}}{\\overline{y}}\\!\\:^{{\\overline{a}}-{\\overline{b}}}-\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{y}}\\!\\:^{\\overline{1}}\\\\&=\\sum _{\\overline{a}}\\sum _{i=1}^t \\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}\\sum _{{\\overline{0}}<{\\overline{b}}\\le {\\overline{a}}}{\\overline{z}}\\!\\:^{{\\overline{b}}}{\\overline{y}}\\!\\:^{{\\overline{a}}-{\\overline{b}}}\\\\&=\\sum _{\\overline{a}}\\sum _{i=1}^t \\sum _{{\\overline{0}}<{\\overline{b}}\\le {\\overline{a}}}\\alpha _{{\\overline{a}},i}{\\overline{x}}^{{\\overline{c}}_{{\\overline{a}},i}}{\\overline{z}}\\!\\:^{{\\overline{b}}}{\\overline{y}}\\!\\:^{{\\overline{a}}-{\\overline{b}}}\\;,$ which is then easily seen to be explicit and $(n,t,2^d)$ -sparse appealing to the above logic.", "$g$ a multilinear formula: By interpolation, it follows that for each ${\\overline{a}}$ there are $(2^d)$ -explicit constants ${\\overline{\\alpha }}_{{\\overline{a}},{\\overline{\\beta }}}$ such that $g_{\\overline{a}}({\\overline{x}})=\\sum _{{\\overline{\\beta }}\\in \\lbrace 0,1\\rbrace ^d} \\alpha _{{\\overline{a}},{\\overline{\\beta }}}g({\\overline{x}},{\\overline{\\beta }})$ .", "From this it follows that $g_{\\overline{a}}$ is computable by a depth $D+1$ multilinear formula of size $(t,2^d)$ .", "Expanding the definition of $C$ we get that $C({\\overline{x}},{\\overline{y}},{\\overline{z}})&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}\\left(({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-{\\overline{y}}\\!\\:^{{\\overline{a}}}\\right)\\\\&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{1}}\\\\&=\\sum _{{\\overline{a}}} \\sum _{{\\overline{\\beta }}\\in \\lbrace 0,1\\rbrace ^d} \\alpha _{{\\overline{a}},{\\overline{\\beta }}}g({\\overline{x}},{\\overline{\\beta }}){\\overline{y}}\\!\\:^{{\\overline{1}}-{\\overline{a}}}({\\overline{z}}+{\\overline{y}})^{\\overline{a}}-\\sum _{{\\overline{a}}} \\sum _{{\\overline{\\beta }}\\in \\lbrace 0,1\\rbrace ^d} \\alpha _{{\\overline{a}},{\\overline{\\beta }}}g({\\overline{x}},{\\overline{\\beta }}){\\overline{y}}\\!\\:^{\\overline{1}}\\;,$ which is clearly an explicit depth $D+2$ multilinear formula of size $(t,2^d)$ , as $D\\ge 1$ so that the computation of the $z_i+y_i$ is parallelized with computing $g({\\overline{x}},{\\overline{\\beta }})$ , and we absorb the subtraction into the overall top-gate of addition.", "We can then straightforwardly extend this to multilinearizing the product of a low-degree sparse multilinear polynomial and a multilinear formula, as we can use that multilinearization is linear.", "Let $g\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a multilinear polynomial and $f\\in \\mathbb {F}[{\\overline{x}}]$ a $s$ -sparse multilinear polynomial of degree $\\le d$ .", "Then there is a $C\\in \\mathbb {F}[{\\overline{x}},z_1,\\ldots ,z_n]$ such that $g\\cdot f-\\operatorname{ml}(g\\cdot f)=C({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\;,$ and $C({\\overline{x}},{\\overline{0}})=0$ .", "If $g$ is $t$ -sparse, then $C$ is computable as a $(s,t,n,2^d)$ -size depth-2 multilinear formula (which is $(s,t,n,2^d)$ -explicit given the computations for $f,g$ ), as well as being computable by a $(s,t,n,d)$ -size depth-3 multilinear formula with a $+$ -output-gate (which is $(s,t,n,d)$ -explicit given the computations for $f,g$ ).", "If $g$ is computable by a size-$t$ depth-$D$ multilinear formula, then $C$ is computable by a $(s,t,2^d)$ -size depth-$(D+2)$ multilinear formula with a $+$ -output-gate (which is $(s,t,2^d)$ -explicit given the computations for $f,g$ ).", "We now show how to derive multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutations for ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ from equations of the form $\\sum _j g_jf_j\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ .", "Let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be degree at most $d$ multilinear $s$ -sparse polynomials which are unsatisfiable over ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Suppose that there are multilinear $g_j\\in \\mathbb {F}[{\\overline{x}}]$ such that $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Then there is a multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation $C({\\overline{x}},{\\overline{y}},{\\overline{z}})$ such that If the $g_j$ are $t$ -sparse, then $C$ is computable by a $(s,t,n,m,2^d)$ -size depth-2 multilinear formula (which is $(s,t,n,m,2^d)$ -explicit given the computations for the $f_j,g_j$ ), as well as being computable by a $(s,t,n,m,d)$ -size depth-3 multilinear formula (which is $(s,t,n,m,d)$ -explicit given the computations for $f_j,g_j$ ).", "If the $g_j$ are computable by size-$t$ depth-$D$ multilinear formula, then $C$ is computable by a $(s,t,m,2^d)$ -size depth-$(D+2)$ multilinear formula (which is $(s,t,m,2^d)$ -explicit given the computations for $f_j,g_j$ ).", "construction: By the above multilinearization (res:multilin:sparseformula), there are $C_j\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}}]$ such that $g_j({\\overline{x}})f_j({\\overline{x}})=\\operatorname{ml}(g_jf_j)+C_j({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\;.$ where $C_j({\\overline{x}},{\\overline{0}})=0$ .", "We now define $C({\\overline{x}},{\\overline{y}},{\\overline{z}}):=\\sum _{j=1}^m \\left(g_j({\\overline{x}})y_j - C_j({\\overline{x}},{\\overline{z}})\\right)\\;.$ We now show that this is a valid IPS refutation.", "Observe that $C({\\overline{x}},{\\overline{0}},{\\overline{0}})=0$ and that $C({\\overline{x}},{\\overline{f}},{\\overline{x}}^2-{\\overline{x}})&=\\sum _{j=1}^m \\left(g_j({\\overline{x}})f_j({\\overline{x}}) - C_j({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\right)\\\\\\ &=\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)\\\\\\multicolumn{2}{l}{\\text{as $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ we have that$$\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)=\\operatorname{ml}\\left(\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\right)=1\\;,$$where we appealed to linearity of multilinearization ({fact:multilinearization}), so that}}\\\\C({\\overline{x}},{\\overline{f}},{\\overline{x}}^2-{\\overline{x}})&=\\sum _{j=1}^m \\operatorname{ml}(g_jf_j)=1\\;.$ complexity: The claim now follows from appealing to res:multilin:sparseformula for bounding the complexity of the $C_j$ .", "That is, if the $g_j$ are $t$ -sparse then $\\sum _{j=1}^m g_j({\\overline{x}})y_j$ is $tm$ -sparse and thus computable by a $(t,m)$ -size depth-2 multilinear formula with $+$ -output-gate.", "As each $C_j$ is computable by a $(s,t,n,2^d)$ -size depth-2 or $(s,t,n,d)$ -size depth-3 multilinear formula (each having a $+$ -output-gate), it follows that $C({\\overline{x}},{\\overline{y}},{\\overline{z}}):=\\sum _{j=1}^m \\left(g_j({\\overline{x}})y_j - C_j({\\overline{x}},{\\overline{z}})\\right)$ can be explicitly computed by a $(s,t,n,m,2^d)$ -size depth-2 or $(s,t,n,m,d)$ -size depth-3 multilinear formula (collapsing addition gates into a single level).", "If the $g_j$ are computable by size-$t$ depth-$D$ multilinear formula then $\\sum _{j=1}^m g_j({\\overline{x}})y_j$ is computable by size $(m,t)$ -size depth-$(D+2)$ multilinear formula (with a $+$ -output-gate), and each $C_j$ is computable by a $(s,t,2^d)$ -size depth-$(D+2)$ multilinear formula with a $+$ -output-gate, from which it follows that $C({\\overline{x}},{\\overline{y}},{\\overline{z}}):=\\sum _{j=1}^m \\left(g_j({\\overline{x}})y_j - C_j({\\overline{x}},{\\overline{z}})\\right)$ can be explicitly computed by a $(s,t,m,2^d)$ -size depth-$(D+2)$ multilinear formula by collapsing addition gates.", "We now conclude by showing that multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ can efficiently simulate sparse-IPS$_{\\text{LIN}}$ when the axioms are low-degree.", "As this latter system is complete, this shows the former is as well.", "That is, we allow the refutation to depend non-linearly on the boolean axioms, but only linearly on the other axioms.", "Let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be degree at most $d$ $s$ -sparse polynomials unsatisfiable over ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Suppose they have an $\\sum \\prod $ -IPS$_{\\text{LIN}}$ refutation, that is, that there are $t$ -sparse polynomials $g_1,\\ldots ,g_m,h_1,\\ldots ,h_n\\in \\mathbb {F}[{\\overline{x}}]$ such that $\\sum _{j=1}^m g_j f_j+\\sum _{i=1}^n h_i \\cdot (x_i^2-x_i)=1$ .", "Then ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ can be refuted by a depth-2 multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ proof of $(s,t,n,m,2^d)$ -size, or by a depth-3 multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ proof of $(s,t,n,m,d)$ -size.", "This follows from res:multilin:multilin-lbIPS by noting that $\\sum _j \\operatorname{ml}(g_j)f_j\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ , and that the $\\operatorname{ml}(g_j)$ are multilinear and $t$ -sparse." ], [ "Refutations of the Subset-Sum Axiom", "We now give efficient IPS refutations of the subset-sum axiom, where these IPS refutations can be even placed in the restricted roABP-IPS$_{\\text{LIN}}$ or multilinear-formula-IPS$_{\\text{LIN}}$ subclasses.", "That is, we give such refutations for whenever the polynomial $\\sum _i \\alpha _ix_i-\\beta $ is unsatisfiable over the boolean cube $\\lbrace 0,1\\rbrace ^n$ , where the size of the refutation is polynomial in the size of the set $A:=\\lbrace \\sum _i \\alpha _ix_i: {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\rbrace $ .", "A motivating example is when ${\\overline{\\alpha }}={\\overline{1}}$ so that $A=\\lbrace 0,\\ldots ,n\\rbrace $ .", "To construct our refutations, we first show that there is an efficiently computable polynomial $f$ such that $f({\\overline{x}})\\cdot (\\sum _i \\alpha _ix_i-\\beta )\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ .", "This will be done by considering the univariate polynomial $p(t):=\\prod _{\\alpha \\in A}(t-\\alpha )$ .", "Using that for any univariate $p(x)$ that $x-y$ divides $p(x)-p(y)$ , we see that $p(\\sum _i\\alpha _ix_i)-p(\\beta )$ is a multiple of $\\sum _i \\alpha _i x_i-\\beta $ .", "As $\\sum _i \\alpha _ix_i-\\beta $ is unsatisfiable it must be that $\\beta \\notin A$ .", "This implies that $p(\\sum _i\\alpha _ix_i)\\equiv 0\\mod {{\\overline{x}}}^2-{\\overline{x}}$ while $p(\\beta )\\ne 0$ .", "Consequently, $p(\\sum _i\\alpha _ix_i)-p(\\beta )$ is equivalent to a nonzero constant modulo ${\\overline{x}}^2-{\\overline{x}}$ , yielding the Nullstellensatz refutation $\\frac{1}{-p(\\beta )}\\cdot \\frac{p(\\sum _i\\alpha _ix_i)-p(\\beta )}{\\sum _i \\alpha _i x_i-\\beta }\\cdot ({\\textstyle \\sum }_i \\alpha _i x_i-\\beta )\\equiv 1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ By understanding the quotient $\\frac{p(\\sum _i\\alpha _ix_i)-p(\\beta )}{\\sum _i \\alpha _i x_i-\\beta }$ we see that it can be efficiently computed by a small $\\sum \\bigwedge \\sum $ formula and thus an roABP, so that using our multilinearization result for roABPs (res:multilin:roABP) this yields the desired roABP-IPS$_{\\text{LIN}}$ refutation.", "However, this does not yield the desired multilinear-formula-IPS$_{\\text{LIN}}$ refutation.", "For this, we need to (over a large field) convert the above quotient to a sum of products of univariates using duality (res:sumpowsum:duality).", "We can then multilinearize this to a sum of products of linear univariates, which is a depth-3 multilinear formula.", "By appealing to our proof-of-multilinearization result for multilinear formula (res:multilin:multilin-lbIPS) one obtains a multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation, and we give a direct proof which actually yields the desired multilinear-formula-IPS$_{\\text{LIN}}$ refutation.", "We briefly remark that for the special case of ${\\overline{\\alpha }}={\\overline{1}}$ , one can explicitly describe the unique multilinear polynomial $f$ such that $f({\\overline{x}})(\\sum _i x_i-\\beta )\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ .", "This description (res:subsetsum:multlin) shows that $f$ is a linear combination of elementary symmetric polynomials, which implies the desired complexity upper bounds for this case via known bounds on the complexity of elementary symmetric polynomials ([51]).", "However, this proof strategy is more technical and thus we pursue the more conceptual outline given above to bound the complexity of $f$ for general $A$ .", "Let ${\\overline{\\alpha }}\\in \\mathbb {F}^n$ , $\\beta \\in \\mathbb {F}$ and $A:=\\lbrace \\sum _{i=1}^n \\alpha _i x_i : {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\rbrace $ be so that $\\beta \\notin A$ .", "Then there is a multilinear polynomial $f\\in \\mathbb {F}[{\\overline{x}}]$ such that $f({\\overline{x}})\\cdot \\left({\\textstyle \\sum }_i \\alpha _i x_i-\\beta \\right)\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ For any $\|\\mathbb {F}\|$ , $f$ is computable by a $(\|A\|,n)$ -explicit $(\|A\|,n)$ -width roABP of individual degree $\\le 1$ .", "If $\|\\mathbb {F}\|\\ge (\|A\|,n)$ , then $f$ is computable as a sum of product of linear univariates (and hence a depth-3 multilinear formula) $f({\\overline{x}})=\\sum _{i=1}^s f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\;,$ where each $f_{i,j}\\in \\mathbb {F}[x_i]$ has $\\deg f_{i,j}\\le 1$ , $s\\le (\|A\|,n)$ , and this expression is $(\|A\|,n)$ -explicit.", "computing $A$ : We first note that $A$ can be computed from ${\\overline{\\alpha }}$ in $(\|A\|,n)$ -steps (as opposed to the naive $2^n$ steps).", "That is, define $A_j:=\\lbrace \\sum _{i=1}^j \\alpha _i x_i : x_1,\\ldots ,x_j\\in \\lbrace 0,1\\rbrace \\rbrace $ , so that $A_0=\\emptyset $ and $A_n=A$ .", "It follows that for all $j$ , we have that $A_j\\subseteq A$ and thus $\|A_j\|\\le \|A\|$ , and that $A_{j+1}\\subseteq A_j\\cup (A_j+\\alpha _j)$ .", "It follows that we can compute $A_{j+1}$ from $A_j$ in $(\|A\|)$ time, so that $A=A_n$ can be computed in $(\|A\|,n)$ -time.", "defining $f$ : Define $p(t)\\in \\mathbb {F}[t]$ by $p:=\\prod _{\\alpha \\in A}(t-\\alpha )$ , so that $p(A)=0$ and $p(\\beta )\\ne 0$ .", "Express $p(t)$ in its monomial representation as $p(t)=\\sum _{k=0}^{\|A\|} \\gamma _k t^k$ , where the $\\gamma _k$ can be computed in $(\|A\|)$ time from ${\\overline{\\alpha }}$ by computing $A$ as above.", "Then observe that $p(t)-p(\\beta )&=\\left(\\sum _{k=0}^{\|A\|} \\gamma _k\\frac{t^k-\\beta ^k}{t-\\beta }\\right)(t-\\beta )\\\\&=\\left(\\sum _{k=0}^{\|A\|} \\gamma _k\\sum _{j=0}^{k-1} t^{j}\\beta ^{(k-1)-j}\\right)(t-\\beta )\\\\&=\\left(\\sum _{j=0}^{\|A\|-1}\\left(\\sum _{k=j+1}^{\|A\|} \\gamma _k\\beta ^{(k-1)-j} \\right)t^j\\right)(t-\\beta )\\;.$ Thus, plugging in $t\\leftarrow \\sum _i \\alpha _i x_i$ , we can define the polynomial $g({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ by $g({\\overline{x}})&:=\\frac{p(\\sum _i \\alpha _i x_i)-p(\\beta )}{\\sum _i \\alpha _i x_i -\\beta }\\nonumber \\\\&=\\sum _{j=0}^{\|A\|-1}\\left(\\sum _{k=j+1}^{\|A\|} \\gamma _k\\beta ^{(k-1)-j} \\right) \\left(\\sum _i \\alpha _i x_i\\right)^j\\;.$ Hence, $g({\\overline{x}}) ({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )&=p({\\textstyle \\sum }_i \\alpha _i x_i)-p(\\beta )\\;.$ For any ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ we have that $\\sum _i \\alpha _i x_i\\in A$ .", "As $p(A)=0$ it follows that $p(\\sum _i \\alpha _i x_i)=0$ for all ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "This implies that $p(\\sum _i \\alpha _i x_i)\\equiv 0 \\mod {{\\overline{x}}}^2-{\\overline{x}}$ , yielding $g({\\overline{x}}) ({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )&\\equiv -p(\\beta )\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ As $-p(\\beta )\\in \\mathbb {F}\\setminus \\lbrace 0\\rbrace $ , we have that $\\frac{1}{-p(\\beta )}\\cdot g({\\overline{x}}) \\cdot \\left({\\textstyle \\sum }_i \\alpha _ix_i-\\beta \\right)\\equiv 1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ We now simply multilinearize, and thus define the multilinear polynomial $f({\\overline{x}}):=\\operatorname{ml}\\left(\\frac{1}{-p(\\beta )}\\cdot g({\\overline{x}})\\right)\\;.$ First, we see that this has the desired form, using the interaction of multilinearization and multiplication (fact:multilinearization).", "$1&=\\operatorname{ml}\\left(\\frac{1}{-p(\\beta )} g({\\overline{x}}) \\cdot ({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )\\right)\\\\&=\\operatorname{ml}\\left(\\operatorname{ml}\\left(\\frac{1}{-p(\\beta )}\\cdot g({\\overline{x}})\\right)\\operatorname{ml}({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )\\right)\\\\&=\\operatorname{ml}\\Big (f \\cdot \\operatorname{ml}({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )\\Big )\\\\&=\\operatorname{ml}\\Big (f\\cdot ({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )\\Big )$ Thus, $f\\cdot ({\\textstyle \\sum }_i \\alpha _ix_i-\\beta )\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ as desired.", "computing $f$ as an roABP: By eq:subset we see that $g({\\overline{x}})$ is computable by a $(\|A\|,n)$ -size $\\sum \\bigwedge \\sum $ -formula, and by the efficient simulation of $\\sum \\bigwedge \\sum $ -formula by roABPs (res:sumpowsum:roABP) $g({\\overline{x}})$ and thus $\\frac{1}{-p(\\beta )}\\cdot g({\\overline{x}})$ are computable by $(\|A\|,n)$ -width roABPs of $(\|A\|,n)$ -degree.", "Noting that roABPs can be efficiently multilinearized (res:multilin:roABP) we see that $f=\\operatorname{ml}(\\frac{1}{-p(\\beta )}\\cdot g({\\overline{x}}))$ can thus be computed by such an roABP also, where the individual degree of this roABP is at most 1.", "Finally, note that each of these steps has the required explicitness.", "computing $f$ via duality: We apply duality (res:sumpowsum:duality) to see that over large enough fields there are univariates $g_{j,\\ell ,i}$ of degree at most $\|A\|$ , where $g({\\overline{x}})&=\\sum _{j=0}^{\|A\|-1}\\left(\\sum _{k=j+1}^{\|A\|} \\gamma _k\\beta ^{(k-1)-j} \\right) \\left(\\sum _i \\alpha _i x_i\\right)^j\\\\&=\\sum _{j=0}^{\|A\|-1}\\left(\\sum _{k=j+1}^{\|A\|} \\gamma _k\\beta ^{(k-1)-j} \\right) \\sum _{\\ell =1}^{(nj+1)(j+1)}g_{j,\\ell ,1}(x_1)\\cdots g_{j,\\ell ,n}(x_n)\\\\\\multicolumn{2}{l}{\\text{Absorbing the constant $\\left(\\sum _{k=j+1}^{\|A\|} \\gamma _k\\beta ^{(k-1)-j} \\right)$ into these univariates and re-indexing,}}\\\\&=\\sum _{i=1}^{s} g_{i,1}(x_1)\\cdots g_{i,n}(x_n)$ for some univariates $g_{i,j}$ , where $s\\le \|A\|(n\|A\|+1)(\|A\|+1)=(\|A\|,n)$ .", "We now obtain $f$ by multilinearizing the above expression, again appealing to multilinearization (fact:multilinearization).", "$f&=\\operatorname{ml}\\left(\\frac{1}{-p(\\beta )} g({\\overline{x}})\\right)\\\\&=\\operatorname{ml}\\left(\\frac{1}{-p(\\beta )} \\sum _{i=1}^{s} g_{i,1}(x_1)\\cdots g_{i,n}(x_n)\\right)\\\\\\multicolumn{2}{l}{\\text{absorbing the constant ${1}{-p(\\beta )}$ and renaming,}}\\\\&=\\operatorname{ml}\\left(\\sum _{i=1}^{s} g^{\\prime }_{i,1}(x_1)\\cdots g^{\\prime }_{i,n}(x_n)\\right)\\\\&=\\operatorname{ml}\\left(\\sum _{i=1}^{s} \\operatorname{ml}(g^{\\prime }_{i,1}(x_1))\\cdots \\operatorname{ml}(g^{\\prime }_{i,n}(x_n))\\right)\\\\\\multicolumn{2}{l}{\\text{defining $f_{i,j}(x_j):=\\operatorname{ml}(g_{i,j}(x_j))$, so that $\\deg f_{i,j}\\le 1$,}}\\\\&=\\operatorname{ml}\\left(\\sum _{i=1}^{s} f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\right)\\\\\\multicolumn{2}{l}{\\text{and we can drop the outside $\\operatorname{ml}$ as this expression is now multilinear,}}\\\\&=\\sum _{i=1}^{s} f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\;,$ showing that $f$ is computable as desired, noting that this expression has the desired explicitness.", "Note that computing $f$ via duality also implies an roABP for $f$ , but only in large enough fields $\|\\mathbb {F}\|\\ge (\|A\|,n)$ .", "Of course, the field must at least have $\|\\mathbb {F}\|\\ge \|A\|$ , but by using the field-independent conversion of $\\sum \\bigwedge \\sum $ to roABP (res:sumpowsum:roABP) this shows that $\\mathbb {F}$ need not be any larger than $A$ for the refutation to be efficient.", "The above shows that one can give an “IPS proof” $g({\\overline{x}})(\\sum _i \\alpha _ix_i-\\beta )+\\sum _i h_i({\\overline{x}})(x_i^2-x_i)=1$ , where $g$ is efficiently computable.", "However, this is not yet an IPS proof as it does not bound the complexity of the $h_i$ .", "We now extend this to an actual IPS proof by using the above multilinearization results for roABPs (res:multilin:roABP-lIPS), leveraging that $\\sum _i \\alpha _ix_i-\\beta $ is computable by an roABP in any variable order (and that the above result works in any variable order).", "Let ${\\overline{\\alpha }}\\in \\mathbb {F}^n$ , $\\beta \\in \\mathbb {F}$ and $A:=\\lbrace \\sum _{i=1}^n \\alpha _i x_i : {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\rbrace $ be so that $\\beta \\notin A$ .", "Then $\\sum _i \\alpha _ix_i-\\beta ,{\\overline{x}}^2-{\\overline{x}}$ has a $(\|A\|,n)$ -explicit roABP-IPS$_{\\text{LIN}}$ refutation of individual degree 2 computable in width-$(\|A\|,n)$ in any variable order.", "Note that while the above results give a small $\\sum \\bigwedge \\sum $ formula $g$ such that $g\\cdot (\\sum _i\\alpha _i x_i-\\beta )\\equiv -p(\\beta )\\mod {{\\overline{x}}}^2-{\\overline{x}}$ for nonzero scalar $-p(\\beta )$ , this does not translate to a $\\sum \\bigwedge \\sum $ -IPS refutation as $\\sum \\bigwedge \\sum $ formulas cannot be multilinearized efficiently (see the discussion in sec:multilinearization:roABP).", "We now turn to refuting the subset-sum axioms by multilinear-formula-IPS$_{\\text{LIN}}$ (which is not itself a complete proof system, but will suffice here).", "While one can use the multilinearization techniques for multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ of res:multilin:multilin-lbIPS it gives slightly worse results due to its generality, so we directly multilinearize the refutations we built above using that the subset-sum axiom is linear.", "Let ${\\overline{\\alpha }}\\in \\mathbb {F}^n$ , $\\beta \\in \\mathbb {F}$ and $A:=\\lbrace \\sum _{i=1}^n \\alpha _i x_i : {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\rbrace $ be so that $\\beta \\notin A$ .", "If $\|\\mathbb {F}\|\\ge (\|A\|,n)$ , then $\\sum _i \\alpha _ix_i-\\beta ,{\\overline{x}}^2-{\\overline{x}}$ has a $(\|A\|,n)$ -explicit $(\|A\|,n)$ -size depth-3 multilinear-formula-IPS$_{\\text{LIN}}$ refutation.", "By res:ips-ubs:subset, there is a multilinear polynomial $f\\in \\mathbb {F}[{\\overline{x}}]$ such that $f({\\overline{x}})\\cdot \\left({\\textstyle \\sum }_i \\alpha _i x_i-\\beta \\right)\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ , and $f$ is explicitly given as $f({\\overline{x}})=\\sum _{i=1}^s f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\;,$ where each $f_{i,j}\\in \\mathbb {F}[x_i]$ has $\\deg f_{i,j}\\le 1$ and $s\\le (\|A\|,n)$ .", "We now efficiently prove that $f({\\overline{x}})\\cdot (\\sum _{i=1}^n \\alpha _i x_i-\\beta )$ is equal to its multilinearization (which is 1) modulo the boolean cube.", "The key step is that for a linear function $p(x)=\\gamma x+\\delta $ we have that $(\\gamma x+\\delta )x=(\\gamma +\\delta )x+\\gamma (x^2-x)=p(1)x+(p(1)-p(0))(x^2-x)$ .", "Thus, $f({\\overline{x}})&\\cdot \\left({\\textstyle \\sum }_i\\alpha _i x_i-\\beta \\right)\\\\&=\\left( \\sum _{i=1}^s f_{i,1}(x_1)\\cdots f_{i,n}(x_n) \\right)\\cdot ({\\textstyle \\sum }_i\\alpha _i x_i-\\beta )\\\\&=\\sum _{i=1}^s -\\beta f_{i,1}(x_1)\\cdots f_{i,n}(x_n)\\\\&\\hspace{36.135pt}+\\sum _{i=1}^s\\sum _{j=1}^n \\alpha _j \\prod _{k\\ne j} f_{i,k}(x_k) \\cdot \\Big (f_{i,j}(1)x_j + (f_{i,j}(1)-f_{i,j}(0))(x_j^2-x_j)\\Big )\\\\&=\\sum _{i=1}^s -\\beta f_{i,1}(x_1)\\cdots f_{i,n}(x_n)+\\sum _{i=1}^s\\sum _{j=1}^n \\alpha _j \\prod _{k\\ne j} f_{i,k}(x_k) \\cdot f_{i,j}(1)x_j \\\\&\\hspace{36.135pt}+\\sum _{i=1}^s\\sum _{j=1}^n \\alpha _j \\prod _{k\\ne j} f_{i,k}(x_k) \\cdot (f_{i,j}(1)-f_{i,j}(0))\\cdot (x_j^2-x_j)\\multicolumn{2}{l}{\\text{absorbing constants and renaming, using $j=0$ to incorporate the above term involving $\\beta $,}}\\\\&=\\sum _{i=1}^s\\sum _{j=0}^n \\prod _{k=1}^n f_{i,j,k}(x_k)+\\sum _{j=1}^n \\left(\\sum _{i=1}^s\\prod _{k=1}^n h_{i,j,k}(x_k)\\right)(x_j^2-x_j)\\multicolumn{2}{l}{\\text{where each $f_{i,j,k}$ and $h_{i,j,k}$ are linear univariates.", "As $f({\\overline{x}})\\cdot (\\sum _{i=1}^n \\alpha _i x_i-\\beta )\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ it follows that $\\sum _i \\sum _j \\prod _k f_{i,j,k}(x_k)\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$, but as each $f_{i,j,k}$ is linear it must actually be that $\\sum _i \\sum _j\\prod _k f_{i,j,k}(x_k)=1$, so that,}}\\\\&=1+\\sum _{j=1}^n \\left(\\sum _{i=1}^s\\prod _{k=1}^n h_{i,j,k}(x_k)\\right)(x_j^2-x_j)\\;.$ Define $C({\\overline{x}},y,{\\overline{z}}):=f({\\overline{x}})y-\\sum _{j=1}^n h_j({\\overline{x}}) z_j$ , where $h_j({\\overline{x}}):=\\sum _{i=1}^s\\prod _{k=1}^n h_{i,j,k}(x_k)$ .", "It follows that $C({\\overline{x}},0,{\\overline{0}})=0$ and that $C({\\overline{x}},\\sum _i\\alpha _ix_i-\\beta ,{\\overline{x}}^2-{\\overline{x}})=1$ , so that $C$ is a linear-IPS refutation.", "Further, as each $f,h_j$ is computable as a sum of products of linear univariates, these are depth-3 multilinear formulas.", "By distributing the multiplication of the variables $y,z_1,\\ldots ,z_n$ to the bottom multiplication gates, we see that $C$ itself has a depth-3 multilinear formula of the desired complexity." ], [ "Lower Bounds for Linear-IPS via Functional Lower Bounds", "In this section we give functional circuit lower bounds for various measures of algebraic complexity, such as degree, sparsity, roABPs and multilinear formulas.", "That is, while algebraic complexity typically treats a polynomial $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ as a syntactic object, one can also see that it defines a function on the boolean cube $\\hat{f}:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ .", "If this function is particularly complicated then one would expect that this implies that the polynomial $f$ requires large algebraic circuits.", "In this section we obtain such lower bounds, showing that for any polynomial $f$ (not necessarily multilinear) that agrees with a certain function on the boolean cube must in fact have large algebraic complexity.", "Our lower bounds will proceed by first showing that the complexity of $f$ is an upper bound for the complexity of its multilinearization $\\operatorname{ml}(f)$ .", "While such a statement is known to be false for general circuits (under plausible assumptions, see sec:multilinearization:roABP), such efficient multilinearization can be shown for the particular restricted models of computation we consider.", "In particular, this multilinearization is easy for degree and sparsity, for multilinear formulas $f$ is already multilinear, and for roABPs this is seen in sec:multilinearization:roABP.", "As then $\\operatorname{ml}(f)$ is uniquely defined by the function $\\hat{f}$ (fact:multilinearization), we then only need to lower bound the complexity of $\\operatorname{ml}(f)$ using standard techniques.", "We remark that the actual presentation will not follow the above exactly, as for roABPs and multilinear formulas it is just as easy to just work directly with the underlying lower bound technique.", "We then observe that by deriving such lower bounds for carefully crafted functions $\\hat{f}:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ one can easily obtain linear-IPS lower bounds for the above circuit classes.", "That is, consider the system of equations $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ , where $f({\\overline{x}})$ is chosen so this system is unsatisfiable, hence $f({\\overline{x}})\\ne 0$ for all ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Any linear-IPS refutation yields an equation $g({\\overline{x}})\\cdot f({\\overline{x}})+\\sum _i h_i({\\overline{x}})(x_i^2-x_i)=1$ , which implies that $g({\\overline{x}})={1}{f({\\overline{x}})}$ for all ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ (that this system is unsatisfiable allows us to avoid division by zero).", "It follows that the polynomial $g({\\overline{x}})$ agrees with the function $\\hat{f}({\\overline{x}}):={1}{f({\\overline{x}})}$ on the boolean cube.", "If the function $\\hat{f}$ has a functional lower bound then this implies $g$ must have large complexity, giving the desired lower bound for the linear-IPS refutation.", "The section proceeds as follows.", "We begin by detailing the above strategy for converting functional lower bounds into lower bounds for linear-IPS.", "We then derive a tight functional lower bound of $n$ for the degree of ${1}{\\left(\\sum _i x_i+1\\right)}$ .", "We then extend this via random restrictions to a functional lower bound of $\\exp (\\Omega (n))$ on the sparsity of ${1}{\\left(\\sum _i x_i+1\\right)}$ .", "We can then lift this degree bound to a functional lower bound of $2^n$ on the evaluation dimension of ${1}{\\left(\\sum _i x_iy_i+1\\right)}$ in the ${\\overline{x}}\|{\\overline{y}}$ partition, which we then symmetrize to obtain a functional lower bound on the evaluation dimension in any partition of the related function ${1}{\\big (\\sum _{i<j} z_{i,j}x_ix_j+1\\big )}$ .", "In each case, the resulting linear-IPS lower bounds are immediate via the known relations of these measures to circuit complexity classes (sec:background)." ], [ "The Strategy", "We give here the key lemma detailing the general reduction from linear-IPS lower bounds to functional lower bounds.", "Let $\\mathcal {C}\\subseteq \\mathbb {F}[x_1,\\ldots ,x_n]$ be a set of polynomials closed under partial evaluation.", "Let $f\\in \\mathbb {F}[{\\overline{x}}]$ , where the system $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable.", "Suppose that for all $g\\in \\mathbb {F}[{\\overline{x}}]$ with $g({\\overline{x}})=\\frac{1}{f({\\overline{x}})},\\qquad \\forall {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\;,$ that $g\\notin \\mathcal {C}$ .", "Then $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ does not have $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ refutations.", "Suppose for contradiction that $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ has the $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ refutation $C({\\overline{x}},y,{\\overline{z}})=g({\\overline{x}})\\cdot y+\\sum _i h_i({\\overline{x}}) \\cdot z_i$ where $C({\\overline{x}},f,{\\overline{x}}^2-{\\overline{x}})=1$ (and clearly $C({\\overline{x}},0,{\\overline{0}})=0$ ).", "As $g=C({\\overline{x}},1,{\\overline{0}})$ , it follows that $g\\in \\mathcal {C}$ from the closure properties we assumed of $\\mathcal {C}$ .", "Thus, $1&=C({\\overline{x}},f,{\\overline{x}}^2-{\\overline{x}})\\\\&=g({\\overline{x}})\\cdot f({\\overline{x}})+\\sum _i h_i({\\overline{x}}) (x_i^2-x_i)\\\\&\\equiv g({\\overline{x}})\\cdot f({\\overline{x}})\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Thus, for any ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , as $f({\\overline{x}})\\ne 0$ , $g({\\overline{x}})={1}{f({\\overline{x}})}\\;.$ However, this yields the desired contradiction, as this contradicts the assumed functional lower bound for ${1}{f}$ .", "We now note that the lower bound strategy of using functional lower bounds actually produces lower bounds for $\\text{IPS}_{\\text{LIN}^{\\prime }}$ (and even for the full IPS system if we have multilinear polynomials), and not just IPS$_{\\text{LIN}}$ .", "This is because we work modulo the boolean axioms, so that any non-linear dependence on these axioms vanishes, only leaving a linear dependence on the remaining axioms.", "This slightly stronger lower bound is most interesting for multilinear-formulas, where the IPS$_{\\text{LIN}}$ version is not complete in general (ex:multi-form:incomplete) (though it is still interesting due to its short refutations of the subset-sum axiom (res:ips-ubs:subset:mult-form)), while the $\\text{IPS}_{\\text{LIN}^{\\prime }}$ version is complete (res:multilin:simulate-sparse).", "Let $\\mathcal {C}\\subseteq \\mathbb {F}[x_1,\\ldots ,x_n]$ be a set of polynomials closed under evaluation, and let $\\mathcal {D}$ be the set of differences of $\\mathcal {C}$ , that is, $\\mathcal {D}:=\\lbrace p({\\overline{x}})-q({\\overline{x}}): p,q\\in \\mathcal {C}\\rbrace $ .", "Let $f\\in \\mathbb {F}[{\\overline{x}}]$ , where the system $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable.", "Suppose that for all $g\\in \\mathbb {F}[{\\overline{x}}]$ with $g({\\overline{x}})=\\frac{1}{f({\\overline{x}})},\\qquad \\forall {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\;,$ that $g\\notin \\mathcal {D}$ .", "Then $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ does not have $\\mathcal {C}$ -$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutations.", "Furthermore, if $\\mathcal {C}$ (and thus $\\mathcal {D}$ ) are a set of multilinear polynomials, then $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ does not have $\\mathcal {C}$ -IPS refutations.", "Suppose for contradiction that $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ has the $\\mathcal {C}$ -$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation $C({\\overline{x}},y,{\\overline{z}})$ .", "That $\\deg _{y} C({\\overline{x}},y,{\\overline{z}})\\le 1$ implies there is the decomposition $C({\\overline{x}},y,{\\overline{z}})=C_1({\\overline{x}},{\\overline{z}})y+C_0({\\overline{x}},{\\overline{z}})$ .", "As $C_1({\\overline{x}},{\\overline{0}})=C({\\overline{x}},1,{\\overline{0}})-C({\\overline{x}},0,{\\overline{0}})$ , the assumed closure properties imply that $C_1({\\overline{x}},{\\overline{0}})\\in \\mathcal {D}$ .", "By the definition of an IPS refutation, we have that $0=C({\\overline{x}},0,{\\overline{0}})=C_1({\\overline{x}},{\\overline{0}})\\cdot 0+C_0({\\overline{x}},{\\overline{0}})$ , so that $C_0({\\overline{x}},{\\overline{0}})=0$ .", "By using the definition of an IPS refutation again, we have that $1=C({\\overline{x}},f,{\\overline{x}}^2-{\\overline{x}})=C_1({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\cdot f+C_0({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})$ , so that modulo the boolean axioms, $1&=\\textstyle C_1({\\overline{x}},{\\overline{x}}^2-{\\overline{x}})\\cdot f+C_0({\\overline{x}},{\\overline{x}}^2-{\\overline{x}}) \\\\&\\equiv \\textstyle C_1({\\overline{x}},{\\overline{0}})\\cdot f+C_0({\\overline{x}},{\\overline{0}}) \\mod {{\\overline{x}}}^2-{\\overline{x}}\\\\\\multicolumn{2}{l}{\\text{using that $C_0({\\overline{x}},{\\overline{0}})=0$,}}\\\\&\\equiv \\textstyle C_1({\\overline{x}},{\\overline{0}})\\cdot f \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Thus, for every ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ we have that $C_1({\\overline{x}},{\\overline{0}})={1}{f({\\overline{x}})}$ so that by the assumed functional lower bound $C_1({\\overline{x}},{\\overline{0}})\\notin \\mathcal {D}$ , yielding the desired contradiction to the above $C_1({\\overline{x}},{\\overline{0}})\\in \\mathcal {D}$ .", "Now suppose that $\\mathcal {C}$ is a set of multilinear polynomials.", "Any $\\mathcal {C}$ -IPS refutation $C({\\overline{x}},y,{\\overline{z}})$ of $f({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}}$ thus must have $\\deg _y C\\le 1$ as $C$ is multilinear, so that $C$ is actually a $\\mathcal {C}$ -$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation, thus the above lower bound applies." ], [ "Degree of a Polynomial", "We now turn to obtaining functional lower bounds, and deriving the corresponding linear-IPS lower bounds.", "We begin with a particularly weak form of algebraic complexity, the degree of a polynomial.", "While it is trivial to obtain such bounds in some cases (as any polynomial that agrees with the AND function on the boolean cube $\\lbrace 0,1\\rbrace ^n$ must have degree $\\ge n$ ), for our applications to proof complexity we will need such degree bounds for functions defined by $\\hat{f}({\\overline{x}})={1}{f({\\overline{x}})}$ for simple polynomials $f({\\overline{x}})$ .", "We show that any multilinear polynomial agreeing with ${1}{f({\\overline{x}})}$ , where $f({\\overline{x}})$ is the subset-sum axiom $\\sum _i x_i-\\beta $ , must have the maximal degree $n$ .", "We note that a degree lower bound of ${\\lceil {{n}{2}}\\rceil }+1$ was established by Impagliazzo, Pudlák, and Sgall [38] (thm:IPS99).", "They actually established this degree bound The degree lower bound of Impagliazzo, Pudlák, and Sgall [38] (thm:IPS99) actually holds for the (dynamic) polynomial calculus proof system (see section sec:alg-proofs), while we only consider the (static) Nullstellensatz proof system here.", "Note that for polynomial calculus Impagliazzo, Pudlák, and Sgall [38] also showed a matching upper bound of ${\\lceil {{n}{2}}\\rceil }+1$ for ${\\overline{\\alpha }}={\\overline{1}}$ .", "when $f({\\overline{x}})=\\sum _i \\alpha _i x_i-\\beta $ for any ${\\overline{\\alpha }}$ , while we only consider ${\\overline{\\alpha }}={\\overline{1}}$ here.", "However, we need the tight bound of $n$ here as it will be used crucially in the proof of res:lbs-fn:dim-eval.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a multilinear polynomial such that $f({\\overline{x}})\\left(\\sum _i x_i-\\beta \\right)=1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Then $\\deg f=n$ .", "$\\le n$ : This is clear as $f$ is multilinear.", "$\\ge n$ : Begin by observing that as $\\beta \\notin \\lbrace 0,\\ldots ,n\\rbrace $ , this implies that $\\sum _i x_i-\\beta $ is never zero on the boolean cube, so that the above functional equation implies that for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ the expression $f({\\overline{x}})= \\frac{1}{\\sum _i x_i-\\beta }\\;,$ is well defined.", "Now observe that this implies that $f$ is a symmetric polynomial.", "That is, define the multilinear polynomial $g$ by symmetrizing $f$ , $g(x_1,\\ldots ,x_n):=\\frac{1}{n!", "}\\sum _{\\sigma \\in \\mathfrak {S}_n} f(x_{\\sigma (1)},\\ldots ,x_{\\sigma (n)})\\;,$ where $\\mathfrak {S}_n$ is the symmetric group on $n$ symbols.", "Then we see that $f$ and $g$ agree on ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , as $g({\\overline{x}})&=\\frac{1}{n!", "}\\sum _{\\sigma \\in \\mathfrak {S}_n} f(x_{\\sigma (1)},\\ldots ,x_{\\sigma (n)})\\\\&=\\frac{1}{n!", "}\\sum _{\\sigma \\in \\mathfrak {S}_n} \\frac{1}{\\sum _i x_{\\sigma (i)}-\\beta }=\\frac{1}{n!", "}\\sum _{\\sigma \\in \\mathfrak {S}_n} \\frac{1}{\\sum _i x_i-\\beta }\\\\&=\\frac{1}{n!", "}\\cdot n!\\cdot \\frac{1}{\\sum _i x_i-\\beta }=\\frac{1}{\\sum _i x_i-\\beta }=f({\\overline{x}})\\;.$ It follows then that $g=f$ as polynomials, since they are multilinear and agree on the boolean cube (fact:multilinearization).", "As $g$ is clearly symmetric, so is $f$ .", "Thus $f$ can be expressed as $f=\\sum _{k=0}^d \\gamma _k S_{n,k}({\\overline{x}})$ , where $d:=\\deg f$ , $S_{n,k}:=\\sum _{S\\subseteq \\binom{[n]}{k}}\\prod _{i\\in S} x_i$ is the $k$ -th elementary symmetric polynomial, and $\\gamma _k\\in \\mathbb {F}$ are scalars with $\\gamma _d\\ne 0$ .", "Now observe that for $k<n$ , we can understand the action of multiplying $S_{n,k}$ by $\\sum _i x_i-\\beta $ .", "$\\textstyle (\\sum _i x_i-\\beta ) S_{n,k}({\\overline{x}})&=\\sum _{S\\in \\binom{[n]}{k}} \\textstyle (\\sum _i x_i-\\beta ) \\prod _{j\\in S} x_j\\\\&=\\sum _{S\\in \\binom{[n]}{k}} \\textstyle \\left(\\sum _{i\\notin S} x_i \\prod _{j\\in S} x_j+\\sum _{i\\in S} x_i \\prod _{j\\in S} x_j-\\beta \\prod _{j\\in S} x_j\\right)\\\\&=\\sum _{S\\in \\binom{[n]}{k}} \\left(\\sum _{\\begin{array}{c}\|T\|=k+1\\\\T\\supseteq S\\end{array}} \\prod _{j\\in T}x_j +(k-\\beta ) \\prod _{j\\in S} x_j\\right) \\mod {{\\overline{x}}}^2-{\\overline{x}}\\\\&=(k+1)S_{n,k+1}+(k-\\beta )S_{n,k}\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Note that we used that each subset of $[n]$ of size $k+1$ contains exactly $k+1$ subsets of size $k$ .", "Putting the above together, suppose for contradiction that $d<n$ .", "Then, $1&=f({\\overline{x}})\\left(\\sum _i x_i-\\beta \\right) \\mod {{\\overline{x}}}^2-{\\overline{x}}\\\\&=\\left(\\sum _{k=0}^d \\gamma _k S_{n,k}\\right)\\left(\\sum _i x_i-\\beta \\right) \\mod {{\\overline{x}}}^2-{\\overline{x}}\\\\&=\\left(\\sum _{k=0}^d \\gamma _k \\Big ((k+1)S_{n,k+1}+(k-\\beta )S_{n,k}\\Big )\\right) \\mod {{\\overline{x}}}^2-{\\overline{x}}\\\\&=\\gamma _d(d+1)S_{n,d+1} + (\\text{degree $\\le d$}) \\mod {{\\overline{x}}}^2-{\\overline{x}}$ However, as $\\gamma _d\\ne 0$ , $d+1\\le n$ (so that $d+1\\ne 0$ in $\\mathbb {F}$ and $S_{n,d+1}$ is defined) this shows that 1 (a multilinear degree 0 polynomial) equals $\\gamma _d(d+1)S_{n,d+1} + (\\text{degree $\\le d$})$ (a multilinear degree $d+1$ polynomial) modulo ${\\overline{x}}^2-{\\overline{x}}$ , which is a contradiction to the uniqueness of representation of multilinear polynomials modulo ${\\overline{x}}^2-{\\overline{x}}$ .", "Thus, we must have $d=n$ .", "To paraphrase the above argument, it shows that for multilinear $f$ of $\\deg f<n$ with $\\operatorname{ml}(f({\\overline{x}})\\cdot (\\sum _i x_i-\\beta ))=1$ it holds that $\\deg \\operatorname{ml}(f({\\overline{x}})\\cdot (\\sum _i x_i-\\beta ))=\\deg f+1$ .", "This contradicts the fact that $\\deg 1=0$ , so that $\\deg f=n$ .", "It is tempting to attempt to argue this claim without using that $\\operatorname{ml}(f({\\overline{x}})\\cdot (\\sum _i x_i-\\beta ))=1$ in some way.", "That is, one could hope to argue that $\\deg (\\operatorname{ml}(f({\\overline{x}})\\cdot (\\sum _i x_i-\\beta )))= \\deg f+1$ directly.", "Unfortunately this is false, as seen by the example $\\operatorname{ml}((x+y)(x-y))=\\operatorname{ml}(x^2-y^2)=x-y$ .", "However, one can make this approach work to obtain a degree lower bound of ${\\lceil {{n}{2}}\\rceil }+1$ , as shown by Impagliazzo, Pudlák, and Sgall [38].", "Putting the above together with the fact that multilinearization is degree non-increasing (fact:multilinearization) we obtain that any polynomial agreeing with $\\frac{1}{\\sum _i x_i-\\beta }$ on the boolean cube must be of degree $\\ge n$ .", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a polynomial such that $f({\\overline{x}})\\left(\\sum _i x_i-\\beta \\right)=1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Then $\\deg f\\ge n$ .", "Multilinearizing (fact:multilinearization) we see that $1=\\operatorname{ml}\\big (f({\\overline{x}})\\cdot \\left(\\sum _i x_i-\\beta \\right)\\big )=\\operatorname{ml}\\big (\\operatorname{ml}(f)\\cdot \\left(\\sum _i x_i-\\beta \\right)\\big )$ , so that $\\operatorname{ml}(f)\\cdot \\left(\\sum _i x_i-\\beta \\right)=1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ .", "Thus $\\deg f\\ge \\deg \\operatorname{ml}(f)$ (fact:multilinearization) and $\\deg \\operatorname{ml}(f)=n$ by the above res:subsetsum:deg, yielding the claim.", "The above proof shows that the unique multilinear polynomial $f$ agreeing with ${1}{\\left(\\sum _i x_i-\\beta \\right)}$ on the hypercube has degree $n$ , but does so without actually specifying the coefficients of $f$ .", "In res:subsetsum:multlin we compute the coefficients of this polynomial, giving an alternate proof that it has degree $n$ (res:subsetsum:multlin:deg-sparse).", "In particular, this computation yields a small algebraic circuit for $f$ , expressing it as an explicit linear combination of elementary symmetric polynomials (which have small algebraic circuits)." ], [ "Sparse polynomials", "We now use the above functional lower bounds for degree, along with random restrictions, to obtain functional lower bounds for sparsity.", "We then apply this to obtain exponential lower bounds for sparse-IPS$_{\\text{LIN}}$ refutations of the subset-sum axiom.", "Recall that sparse-IPS$_{\\text{LIN}}$ is equivalent to the Nullstellensatz proof system when we measure the size of the proof in terms of the number of monomials.", "While we provide the proof here for completeness, we note that this result has already been obtained by Impagliazzo-Pudlák-Sgall [38], who also gave such a lower bound for the stronger polynomial calculus proof system.", "We first recall the random restrictions lemma.", "This lemma shows that by randomly setting half of the variables to zero, sparse polynomials become sums of monomials involving few variables, which after multilinearization is a low-degree polynomial.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be an $s$ -sparse polynomial.", "Let $\\rho :\\mathbb {F}[{\\overline{x}}]\\rightarrow \\mathbb {F}[{\\overline{x}}]$ be the homomorphism induced by randomly and independently setting each variable $x_i$ to 0 with probability 12 and leaving $x_i$ intact with probability 12.", "Then with probability $\\ge {1}{2}$ , each monomial in $\\rho (f({\\overline{x}}))$ involves $\\le \\lg s+1$ variables.", "Thus, with probability $\\ge {1}{2}$ , $\\deg \\operatorname{ml}(\\rho (f))\\le \\lg s+1$ .", "Consider a monomial ${\\overline{x}}^{\\overline{a}}$ involving $\\ge t$ variables, $t\\in $ .", "Then the probability that $\\rho ({\\overline{x}}^{\\overline{a}})$ is nonzero is at most $2^{-t}$ .", "Now consider $f({\\overline{x}})=\\sum _{j=1}^s \\alpha _j {\\overline{x}}^{{\\overline{a}}_j}$ .", "By a union bound, the probability that any monomial ${\\overline{x}}^{{\\overline{a}}_j}$ involving at least $t$ variables survives the random restriction is at most $s2^{-t}$ .", "For $t=\\lg s+1$ this is at most 12.", "The claim about the multilinearization of $\\rho (f({\\overline{x}}))$ follows by observing that for a monomial ${\\overline{x}}^{\\overline{a}}$ involving $\\le \\lg s+1$ variables it must be that $\\deg \\operatorname{ml}(\\rho ({\\overline{x}}^{\\overline{a}}))\\le \\lg s+1$ (fact:multilinearization).", "We now give our functional lower bound for sparsity.", "This follows from taking any refutation of the subset-sum axiom and applying a random restriction.", "The subset-sum axiom will be relatively unchanged, but any sparse polynomial will become (after multilinearization) low-degree, to which our degree lower bounds (sec:lbs-fn:deg) can then be applied.", "Let $n\\ge 8$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a polynomial such that $f({\\overline{x}})=\\frac{1}{\\sum _i x_i-\\beta }\\;,$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Then $f$ requires $\\ge 2^{{n}{4}-1}$ monomials.", "Suppose that $f$ is $s$ -sparse so that $f({\\overline{x}})=\\sum _{j=1}^s \\alpha _j{\\overline{x}}^{{\\overline{a}}_j}$ .", "Take a random restriction $\\rho $ as in res:random-restriction, so that with probability at least 12 we have that $\\deg \\operatorname{ml}(\\rho (f))\\le \\lg s+1$ .", "By the Chernoff bound,For independent $[0,1]$ -valued random variables $\\mathsf {X}_1,\\ldots ,\\mathsf {X}_n$ , $\\Pr \\left[\\sum _i \\mathsf {X}_i -\\sum _i[\\mathsf {X}_i]\\le -\\epsilon n\\right]\\le \\mathrm {e}^{-2\\epsilon ^2n}$ .", "we see that $\\rho $ keeps alive at least ${n}{4}$ variables with probability at least $1-\\mathrm {e}^{-2\\cdot ({1}{4})^2\\cdot n}$ , which is $\\ge 1-\\mathrm {e}^{-1}$ for $n\\ge 8$ .", "Thus, by a union bound the probability that $\\rho $ fails to have either that $\\deg \\operatorname{ml}(\\rho (f))\\le \\lg s+1$ or that it keeps at least ${n}{4}$ variables alive is at most ${1}{2}+\\mathrm {e}^{-1}<1$ .", "Thus a $\\rho $ exists obeying both properties.", "Thus, the functional equation for $f$ implies that $f({\\overline{x}})\\left(\\sum _i x_i-\\beta \\right)=1+\\sum _i h_i({\\overline{x}}) (x_i^2-x_i)\\;,$ for some $h_i\\in \\mathbb {F}[{\\overline{x}}]$ .", "Applying the random restriction and multilinearization to both sizes of this equation, we obtain that $\\operatorname{ml}(\\rho (f))\\cdot \\left(\\sum _{\\rho (x_i)\\ne 0} x_i-\\beta \\right)\\equiv 1 \\mod {\\lbrace }x_i^2-x_i\\rbrace _{\\rho (x_i)\\ne 0}\\;.$ Thus, by appealing to the degree lower bound for this functional equation (res:subsetsum:deg) we obtain that $\\lg s+1\\ge \\deg \\operatorname{ml}(\\rho (f))$ is at least the number of variables which is $\\ge {n}{4}$ , so that $s\\ge 2^{{n}{4}-1}$ as desired.", "We remark that one can actually improve the sparsity lower bound to the optimal “$\\ge 2^n$ ” by computing the sparsity of the unique multilinear polynomial satisfying the above functional equation (res:subsetsum:multlin:deg-sparse).", "We now apply these functional lower bounds to obtain lower bounds for sparse-IPS$_{\\text{LIN}}$ refutations of $\\sum _i x_i-\\beta ,{\\overline{x}}^2-{\\overline{x}}$ via our reduction (res:lbs-fnlbs-ips:lIPS).", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Then $\\sum _{i=1}^n x_i-\\beta ,{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable and any sparse-IPS$_{\\text{LIN}}$ refutation requires size $\\exp (\\Omega (n))$ ." ], [ "Coefficient Dimension in a Fixed Partition", "We now seek to prove functional circuit lower bounds for more powerful models of computation such as roABPs and multilinear formulas.", "As recalled in sec:background, the coefficient dimension complexity measure can give lower bounds for such models.", "However, by definition it is a syntactic measure as it speaks about the coefficients of a polynomial.", "Unfortunately, knowing that a polynomial $f\\in \\mathbb {F}[{\\overline{x}}]$ agrees with a function $\\hat{f}:\\lbrace 0,1\\rbrace ^n\\rightarrow \\mathbb {F}$ on the boolean cube $\\lbrace 0,1\\rbrace ^n$ does not in general give enough information to determine its coefficients.", "In contrast, the evaluation dimension measure is concerned with evaluations of a polynomial (which is functional).", "Obtaining lower bounds for evaluation dimension, and leveraging the fact that the evaluation dimension lower bounds coefficient dimension (res:evalseq-coeffs) we can obtain the desired lower bounds for this complexity measure.", "We now proceed to the lower bound.", "It will follow from the degree lower bound for the subset-sum axiom (res:subsetsum:deg:ge).", "That is, this degree bound shows that if $f({\\overline{z}})\\cdot (\\sum _i z_i-\\beta )\\equiv 1\\mod {{\\overline{z}}}^2-{\\overline{z}}$ then $f$ must have degree $\\ge n$ .", "We can then “lift” this lower bound by the use of a gadget, in particular by replacing ${\\overline{z}}\\leftarrow {\\overline{x}}\\circ {\\overline{y}}$ , where `$\\circ $ ' is the Hadamard (entry-wise) product.", "Because the degree of $f$ is maximal, this gadget forces ${\\overline{x}}$ and ${\\overline{y}}$ to maximally “interact”, and hence the evaluation dimension is large in the ${\\overline{x}}$ versus ${\\overline{y}}$ partition.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ be a polynomial such that $f({\\overline{x}},{\\overline{y}})=\\frac{1}{\\sum _i x_iy_i-\\beta }\\;,$ for ${\\overline{x}},{\\overline{y}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Then $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f\\ge 2^n$ .", "By lower bounding coefficient dimension by the evaluation dimension over the boolean cube (res:evalseq-coeffs), $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f&\\ge \\dim {\\mathbf {Eval}}_{{\\overline{x}}\|{\\overline{y}},\\lbrace 0,1\\rbrace } f\\\\&=\\dim \\lbrace f({\\overline{x}},\\mathbb {1}_{S}) : S\\subseteq [n]\\rbrace \\\\&\\ge \\dim \\lbrace \\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S})) : S\\subseteq [n]\\rbrace \\;,$ where $\\mathbb {1}_{S}\\in \\lbrace 0,1\\rbrace ^n$ is the indicator vector for a set $S$ , and $\\operatorname{ml}$ is the multilinearization operator.", "Note that we used that multilinearization is linear (fact:multilinearization) and that dimension is non-increasing under linear maps.", "Now note that for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ , $f({\\overline{x}},\\mathbb {1}_{S})=\\frac{1}{\\sum _{i\\in S} x_i-\\beta }\\;,$ It follows then $\\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S}))$ is a multilinear polynomial only depending on ${\\overline{x}}\|_S$ (fact:multilinearization), and by its functional behavior it follows from res:subsetsum:deg that $\\deg \\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S}))=\|S\|$ .", "As $\\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S}))$ is multilinear it thus follows that the leading monomial of $\\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S}))$ is $\\prod _{i\\in S} x_i$ , which is distinct for each distinct $S$ .", "This is also readily seen from the explicit description of $\\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S}))$ given by res:subsetsum:multlin.", "Thus, we can lower bound the dimension of this space by the number of leading monomials (res:dim-eq-num-TM-spn), $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f&\\ge \\dim \\lbrace \\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S})) : S\\subseteq [n]\\rbrace \\\\&\\ge \\left\|\\operatorname{LM}\\Big (\\lbrace \\operatorname{ml}(f({\\overline{x}},\\mathbb {1}_{S})) : S\\subseteq [n]\\rbrace \\Big )\\right\|\\\\&=\\left\|\\left\\lbrace \\prod _{i\\in S} x_i : S\\subseteq [n]\\right\\rbrace \\right\|\\\\&=2^n\\;.$ Note that in the above proof we crucially leveraged that the degree bound of res:subsetsum:deg is exactly $n$ , not just $\\Omega (n)$ .", "This exact bound allows us to determine the leading monomials of these polynomials, which seems not to follow from degree lower bounds of $\\Omega (n)$ .", "As coefficient dimension lower bounds roABP-width (res:roABP-widtheqdim-coeffs) and depth-3 powering formulas can be computed by roABPs in any variable order (res:sumpowsum:roABP), we obtain as a corollary our functional lower bound for these models.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ be a polynomial such that $f({\\overline{x}},{\\overline{y}})=\\frac{1}{\\sum _i x_iy_i-\\beta }\\;,$ for ${\\overline{x}},{\\overline{y}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Then $f$ requires width $\\ge 2^n$ to be computed as an roABP in any variable order where ${\\overline{x}}$ precedes ${\\overline{y}}$ .", "In particular, $f$ requires $\\exp (\\Omega (n))$ -size as a depth-3 powering formula.", "We now conclude with a lower bound for linear-IPS over roABPs in certain variable orders, and thus also for depth-3 powering formulas, by appealing to our reduction to functional lower bounds (res:lbs-fnlbs-ips:lIPS).", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Then $\\sum _{i=1}^n x_iy_i-\\beta ,{\\overline{x}}^2-{\\overline{x}},{\\overline{y}}\\!\\:^2-{\\overline{y}}$ is unsatisfiable and any roABP-IPS$_{\\text{LIN}}$ refutation, where the roABP reads ${\\overline{x}}$ before ${\\overline{y}}$ , requires width $\\ge \\exp (\\Omega (n))$ .", "In particular, any $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ refutation requires size $\\ge \\exp (\\Omega (n))$ .", "That this system is unsatisfiable is clear from construction.", "The proof then follows from applying our functional lower bound (res:lbs-fn:roABP) to our reduction strategy (res:lbs-fnlbs-ips:lIPS), where we use that partial evaluations of small roABPs yield small roABPs in the induced variable order (fact:roABP:closure), and that depth-3 powering formulas are a subclass of roABPs (in any order) (res:sumpowsum:roABP).", "The above result shows an roABP-IPS$_{\\text{LIN}}$ lower bound for variable orders where ${\\overline{x}}$ precedes ${\\overline{y}}$ , and we complement this by giving an upper bound showing there are small roABP-IPS$_{\\text{LIN}}$ upper bounds for variable orders where ${\\overline{x}}$ and ${\\overline{y}}$ are tightly interleaved.", "This is achieved by taking the roABP-IPS$_{\\text{LIN}}$ upper bound of res:ips-ubs:subset:roABP for $\\sum _i z_i-\\beta ,{\\overline{z}}\\!\\:^2-{\\overline{z}}$ under the substitution $z_i\\leftarrow x_iy_i$ , and observing that such substitutions preserve roABP width in the $x_1<y_1<\\cdots <x_n<y_n$ order (fact:roABP:closure).", "In particular, as $\\sum \\bigwedge \\sum $ formulas are small roABPs in every variable order, this allows us to achieve an exponential separation between $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ and roABP-IPS$_{\\text{LIN}}$ .", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Then $\\sum _{i=1}^n x_iy_i-\\beta ,{\\overline{x}}^2-{\\overline{x}},{\\overline{y}}\\!\\:^2-{\\overline{y}}$ is unsatisfiable, has a $(n)$ -explicit $(n)$ -size roABP-IPS$_{\\text{LIN}}$ refutation in the variable order $x_1<y_1<\\cdots <x_n<y_n$ , and every $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ refutation requires size $\\ge \\exp (\\Omega (n))$ .", "res:lbs-fn:lbs-ips:fixed-order showed that this system is unsatisfiable and has the desired $\\sum \\bigwedge \\sum $ -IPS$_{\\text{LIN}}$ lower bound, so that it remains to prove the roABP upper bound.", "By res:ips-ubs:subset the unique multilinear polynomial $f\\in \\mathbb {F}[{\\overline{z}}]$ such that $f({\\overline{z}})\\cdot (\\sum _{i=1}^nz_i-\\beta )\\equiv 1\\mod {{\\overline{z}}}^2-{\\overline{z}}$ has a multilinear $(n)$ -size roABP in the variable order $z_1<\\cdots <z_n$ .", "Applying the variable substitution $z_i\\leftarrow x_iy_i$ , it follows that $f^{\\prime }({\\overline{x}},{\\overline{y}}):=f(x_1y_1,\\ldots ,x_ny_n)$ obeys $f^{\\prime }\\cdot (\\sum _{i=1}^nx_iy_i-\\beta )\\equiv 1 \\mod {{\\overline{x}}}^2-{\\overline{x}},{\\overline{y}}\\!\\:^2-{\\overline{y}}$ (as $z_i^2-z_i\\equiv 0 \\mod {{\\overline{x}}}^2-{\\overline{x}},{\\overline{y}}\\!\\:^2-{\\overline{y}}$ under the substitution $z_i\\leftarrow x_iy_i$ ) and that $f^{\\prime }$ is computable by a $(n)$ -size roABP in the variable order $x_1<y_1<\\cdots <x_n<y_n$ (fact:roABP:closure, using that $f$ has individual degree 1).", "Appealing to the efficient multilinearization of roABPs (res:multilin:roABP-lIPS) completes the claim as $\\sum _i x_iy_i -\\beta $ is computable by a $(n)$ -size roABP (in any order)." ], [ "Coefficient Dimension in any Variable Partition", "The previous section gave functional lower bounds for coefficient dimension, and thus roABP width, in the ${\\overline{x}}\|{\\overline{y}}$ variable partition.", "However, this lower bound fails for other variable orderings where ${\\overline{x}}$ and ${\\overline{y}}$ are interleaved because of corresponding upper bounds (res:lbs-fn:lbs-ips:order-sep).", "In this section we extend the lower bound to any variable ordering by using suitable auxiliary variables to plant the previous lower bound into any partition we desire by suitably evaluating the auxiliary variables.", "We begin by developing some preliminaries for how coefficient dimension works in the presence of auxiliary indicator variables.", "That is, consider a polynomial $f({\\overline{x}},{\\overline{y}},{\\overline{z}})$ where we wish to study the coefficient dimension of $f$ in the ${\\overline{x}}\|{\\overline{y}}$ partition.", "We can view this polynomial as lying in $\\mathbb {F}[{\\overline{z}}][{\\overline{x}},{\\overline{y}}]$ so that its coefficients are polynomials in ${\\overline{z}}$ and one studies the dimension of the coefficient space in the field of rational functions $\\mathbb {F}({\\overline{z}})$ .", "Alternatively one can evaluate ${\\overline{z}}$ at some point ${\\overline{z}}\\leftarrow {\\overline{\\alpha }}$ so that $f({\\overline{x}},{\\overline{y}},{\\overline{\\alpha }})\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ and study its coefficient dimension over $\\mathbb {F}$ .", "The following straightforward lemma shows the first dimension over $\\mathbb {F}({\\overline{z}})$ is lower-bounded by the second dimension over $\\mathbb {F}$ .", "Let $f\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}},{\\overline{z}}]$ .", "Let $f_{\\overline{z}}$ denote $f$ as a polynomial in $\\mathbb {F}[{\\overline{z}}][{\\overline{x}},{\\overline{y}}]$ , so that for any ${\\overline{\\alpha }}\\in \\mathbb {F}^{\|{\\overline{z}}\|}$ we have that $f_{\\overline{\\alpha }}\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "Then for any such ${\\overline{\\alpha }}$ , $\\dim _{\\mathbb {F}({\\overline{z}})}{\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f_{\\overline{z}}({\\overline{x}},{\\overline{y}}) \\ge \\dim _\\mathbb {F}{\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}f_{\\overline{\\alpha }}({\\overline{x}},{\\overline{y}})\\;.$ Let $f({\\overline{x}},{\\overline{y}},{\\overline{z}})$ be written in $\\mathbb {F}[{\\overline{x}},{\\overline{y}},{\\overline{z}}]$ as $f=\\sum _{{\\overline{a}},{\\overline{b}}} f_{{\\overline{a}},{\\overline{b}}} ({\\overline{z}}){\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ .", "By res:y-dimeq-x-dim we see that $\\dim _{\\mathbb {F}({\\overline{z}})}{\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f_{\\overline{z}}({\\overline{x}},{\\overline{y}})$ is equal to the rank (over $\\mathbb {F}({\\overline{z}})$ ) of the coefficient matrix $C_{f_{\\overline{z}}}$ , so that its entries $(C_{f_{\\overline{z}}})_{{\\overline{a}},{\\overline{b}}}=f_{{\\overline{a}},{\\overline{b}}}({\\overline{z}})$ are in $\\mathbb {F}[{\\overline{z}}]$ .", "Similarly, $\\dim _{\\mathbb {F}}{\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}} f_{\\overline{\\alpha }}({\\overline{x}},{\\overline{y}})$ is equal to the rank (over $\\mathbb {F}$ ) of the coefficient matrix $C_{f_{\\overline{\\alpha }}}$ , so that as $f({\\overline{x}},{\\overline{y}},{\\overline{\\alpha }})= \\sum _{{\\overline{a}},{\\overline{b}}} f_{{\\overline{a}},{\\overline{b}}}({\\overline{\\alpha }}){\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ we have that $(C_{f_{\\overline{\\alpha }}})_{{\\overline{a}},{\\overline{b}}}=f_{{\\overline{a}},{\\overline{b}}}({\\overline{\\alpha }})$ , which is in $\\mathbb {F}$ .", "Thus, it follows that $C_{f_{\\overline{z}}}\|_{{\\overline{z}}\\leftarrow {\\overline{\\alpha }}}=C_{f_{\\overline{\\alpha }}}$ .", "The claim then follows by noting that for a matrix $M({\\overline{w}})\\in \\mathbb {F}[{\\overline{w}}]^{r\\times r}$ it holds that $\\operatorname{rank}_{\\mathbb {F}({\\overline{w}})} M({\\overline{w}})\\ge \\operatorname{rank}_{\\mathbb {F}} M({\\overline{\\beta }})$ for any ${\\overline{\\beta }}\\in \\mathbb {F}^{\|{\\overline{w}}\|}$ .", "This follows as the rank of $M({\\overline{w}})$ is equal to the maximum size of a minor with a non-vanishing determinant.", "As such determinants are polynomials in ${\\overline{w}}$ , they can only further vanish when ${\\overline{w}}\\leftarrow {\\overline{\\beta }}$ .", "We now use auxiliary variables to embed the coefficient dimension lower bound from res:lbs-fn:dim-eval into any variable order.", "We do this by viewing the polynomial $\\sum _i u_iv_i-\\beta $ as using a matching between variables in ${\\overline{u}}$ and ${\\overline{v}}$ .", "We then wish to embed this matching graph-theoretically into a complete graph, where nodes are labelled with the variables ${\\overline{x}}$ .", "Any equipartition of this graph will induce many edges across this cut, and we can drop edges to find a large matching between the ${\\overline{x}}$ variables which we then identify as instance of $\\sum _i u_iv_i-\\beta $ .", "We introduce one new auxiliary variable $z_{i,j}$ per edge which, upon setting it to 0 or 1, allows us to have this edge (respectively) dropped from or kept in the desired matching.", "This leads to the new (symmetrized) equation $\\sum _{i<j} z_{i,j} x_ix_j-\\beta $ , for which we now give the desired lower bound.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>\\binom{2n}{2}$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,\\binom{2n}{2}\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_{2n},z_1,\\ldots ,z_{\\binom{2n}{2}}]$ be a polynomial such that $f({\\overline{x}},{\\overline{z}})=\\frac{1}{\\sum _{i<j} z_{i,j}x_ix_j-\\beta }\\;,$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^{2n}$ , ${\\overline{z}}\\in \\lbrace 0,1\\rbrace ^{\\binom{2n}{2}}$ .", "Let $f_{\\overline{z}}$ denote $f$ as a polynomial in $\\mathbb {F}[{\\overline{z}}][{\\overline{x}}]$ .", "Then for any partition ${\\overline{x}}=({\\overline{u}},{\\overline{v}})$ with $\|{\\overline{u}}\|=\|{\\overline{v}}\|=n$ , $\\dim _{\\mathbb {F}({\\overline{z}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}} f_{\\overline{z}}\\ge 2^n\\;.$ We wish to embed $\\sum _i u_iv_i-\\beta $ in this instance via a restriction of ${\\overline{z}}$ .", "Define the ${\\overline{z}}$ -evaluation ${\\overline{\\alpha }}\\in \\lbrace 0,1\\rbrace ^{\\binom{2n}{2}}$ to restrict $f$ to sum over those $x_ix_j$ in the natural matching between ${\\overline{u}}$ an ${\\overline{v}}$ , so that $\\alpha _{i.j}={\\left\\lbrace \\begin{array}{ll}1 & x_i=u_k, x_j=v_k\\\\0 & \\text{else}\\end{array}\\right.", "}\\;.$ It follows then that $f({\\overline{u}},{\\overline{v}},{\\overline{\\alpha }})=\\frac{1}{\\sum _{k=1}^n u_kv_k-\\beta }$ for ${\\overline{u}},{\\overline{v}}\\in \\lbrace 0,1\\rbrace ^{n}$ .", "Thus, by appealing to our lower bound for a fixed partition (res:lbs-fn:dim-eval) and the relation between the coefficient dimension in $f_{\\overline{z}}$ versus $f_{\\overline{\\alpha }}$ (res:coeff-dim:fraction-field), $\\dim _{\\mathbb {F}({\\overline{z}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}} f_{\\overline{z}}({\\overline{u}},{\\overline{v}})&\\ge \\dim _\\mathbb {F}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}f_{\\overline{\\alpha }}({\\overline{u}},{\\overline{v}})\\\\&\\ge 2^n\\;.$ We remark that this lower bound is only $\\exp (\\Omega (\\sqrt{m}))$ where $m=2n+\\binom{2n}{2}$ is the number of total variables, while one could hope for an $\\exp (\\Omega (m))$ lower bound as $2^m$ is the trivial upper bound for multilinear polynomials.", "One can achieve such a lower bound by replacing the above auxiliary variable scheme (which corresponds to a complete graph) with one derived from a constant-degree expander graph.", "That is because in such graphs any large partition of the vertices induces a large matching across that partition, where one can then embed the fixed-partition lower bounds of the previous section (sec:evaluation).", "However, we omit the details as this would not qualitatively change the results.", "We now obtain our desired functional lower bounds for roABPs and multilinear formulas.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>\\binom{2n}{2}$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,\\binom{2n}{2}\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_{2n},z_1,\\ldots ,z_{\\binom{2n}{2}}]$ be a polynomial such that $f({\\overline{x}},{\\overline{z}})=\\frac{1}{\\sum _{i<j} z_{i,j}x_ix_j-\\beta }\\;,$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^{2n}$ , ${\\overline{z}}\\in \\lbrace 0,1\\rbrace ^{\\binom{2n}{2}}$ .", "Then $f$ requires width $\\ge 2^n$ to be computed by an roABP in any variable order.", "Also, $f$ requires $n^{\\Omega (\\log n)}$ -size to be computed as a multilinear formula.", "For $d=o({\\log n}{\\log \\log n})$ , $f$ requires $n^{\\Omega (({n}{\\log n})^{{1}{d}}/d^2)}$ -size multilinear formulas of product-depth-$d$ .", "roABPs: Suppose that $f({\\overline{x}},{\\overline{z}})$ is computable by a width-$r$ roABP in some variable order.", "By pushing the ${\\overline{z}}$ variables into the fraction field, it follows that $f_{\\overline{z}}$ ($f$ as a polynomial in $\\mathbb {F}[{\\overline{z}}][{\\overline{x}}]$ ) is also computable by a width-$r$ roABP over $\\mathbb {F}({\\overline{z}})$ in the induced variable order on ${\\overline{x}}$ (fact:roABP:closure).", "By splitting ${\\overline{x}}$ in half along its variable order one obtains the lower bound by combining the coefficient dimension lower bound of res:lbs-fn:any-order:coeff-dim with its relation to roABPs (res:roABP-widtheqdim-coeffs).", "multilinear formulas: This follows immediately from our coefficient dimension lower bound (res:lbs-fn:any-order:coeff-dim) and the Raz [59] and Raz-Yehudayoff [65] results (thm:full-rank-lb).", "As before, this immediately yields the desired roABP-IPS$_{\\text{LIN}}$ and multilinear-formula-IPS lower bounds.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>\\binom{2n}{2}$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,\\binom{2n}{2}\\rbrace $ .", "Then $\\sum _{i<j} z_{i,j}x_ix_j-\\beta ,{\\overline{x}}^2-{\\overline{x}},{\\overline{z}}\\!\\:^2-{\\overline{z}}\\in \\mathbb {F}[x_1,\\ldots ,x_{2n},z_1,\\ldots ,z_{\\binom{2n}{2}}]$ is unsatisfiable, and any roABP-IPS$_{\\text{LIN}}$ refutation (in any variable order) requires $\\exp (\\Omega (n))$ -size.", "Further, any multilinear-formula-IPS refutation requires $n^{\\Omega (\\log n)}$ -size, and any product-depth-$d$ multilinear-formula-IPS refutation requires $n^{\\Omega (({n}{\\log n})^{{1}{d}}/d^2)}$ -size.", "The system is unsatisfiable as any setting of ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ yields a sum over at most $\\binom{2n}{2}$ $z$ -variables, which must be in $\\lbrace 0,\\ldots ,\\binom{2n}{2}\\rbrace $ which by hypothesis does not contain $\\beta $ .", "The roABP-IPS$_{\\text{LIN}}$ lower bound follows immediately from the above functional lower bound (res:lbs-fn:any-order) along with our reduction (res:lbs-fnlbs-ips:lIPS), just as in res:lbs-fn:lbs-ips:fixed-order.", "The multilinear-formula-IPS lower bound also follows immediately from the above functional lower bound (res:lbs-fn:any-order) along with our reduction from IPS lower bounds to functional lower bounds for multilinear polynomials (res:lbs-fnlbs-ips:lbIPS).", "In particular, this application uses that multilinear formulas are closed under partial evaluations, and that taking the difference of two formulas will only double its size and does not change the product depth." ], [ "Lower Bounds for Multiples of Polynomials", "In this section we consider the problem of finding explicit polynomials whose nonzero multiples are all hard.", "Such polynomials are natural to search for, as intuitively if $f$ is hard to compute then so should small modifications such as $x_1f^2+4f^3$ .", "This intuition is buttressed by Kaltofen's [40] result that if a polynomial has a small algebraic circuit then so do all of its factors (up to some pathologies in small characteristic).", "Taken in a contrapositive, this says that if a polynomial $f$ requires super-polynomial size algebraic circuits, then so must all of its nonzero multiples.", "Thus, for general circuits the question of lower bounds for multiples reduces to the standard lower bounds question.", "Unfortunately, for many restricted classes of circuits where lower bounds are known (depth-3 powering formulas, sparse polynomials, roABPs) Kaltofen's [40] result produces circuits for the factors which do not fall into (possibly stronger) restricted classes of circuits where lower bounds are still known.While some results ([17], [53]) can bound the depth of the factors in terms of the depth of the input circuit, there are only very weak lower bounds known for constant-depth algebraic circuits.", "As such, developing lower bounds for multiples against these restricted classes seems to require further work beyond the standard lower bound question.", "We will begin by discussing the applications of this problem to the hardness versus randomness paradigm in algebraic complexity.", "We then use existing derandomization results to show that multiples of the determinant are hard for certain restricted classes.", "However, this method is very rigidly tied to the determinant.", "Thus, we also directly study existing lower bound techniques for restricted models of computation (depth-3 powering formulas, sparse polynomials, and roABPs) and extend these results to also apply to multiples.", "We will show the applications of such polynomials to proof complexity in sec:ips-mult." ], [ "Connections to Hardness versus Randomness and Factoring Circuits", "To motivate the problem of finding polynomials with hard multiples, we begin by discussing the hardness versus randomness approach to derandomizing polynomial identity testing.", "That is, Kabanets and Impagliazzo [43] extended the hardness versus randomness paradigm of Nisan and Wigderson [50] to the algebraic setting, showing that sufficiently good algebraic circuit lower bounds for an explicit polynomial would qualitatively derandomize PIT.", "While much of the construction is similar (using combinatorial designs, hybrid arguments, etc.)", "to the approach of Nisan and Wigderson [50] for boolean derandomization, there is a key difference.", "In the boolean setting, obtaining a hardness versus randomness connection requires converting worst-case hardness (no small computation can compute the function everywhere) to average-case hardness (no small computation can compute the function on most inputs).", "Such a reduction (obtained by Impagliazzo and Wigderson [39]) can in fact be obtained using certain error-correcting codes based on multivariate polynomials (as shown by Sudan, Trevisan and Vadhan [72]).", "Such a worst-case to average-case reduction is also needed in the algebraic setting, but as multivariate polynomials are one source of this reduction in the boolean regime, it is natural to expect it to be easier in the algebraic setting.", "Specifically, the notion of average-case hardness for a polynomial $f({\\overline{x}})$ used in Kabanets-Impagliazzo [43] is that for any $g({\\overline{x}},y)$ satisfying $g({\\overline{x}},f({\\overline{x}}))=0$ , it must be that $g$ then requires large algebraic circuits (by taking $g({\\overline{x}},y):=y-f({\\overline{x}})$ this implies $f$ itself requires large circuits).", "This can be interpreted as average-case hardness because if such a $g$ existed with a small circuit, then for any value ${\\overline{\\alpha }}$ we have that $g({\\overline{\\alpha }},y)$ is a univariate polynomial that vanishes on $f({\\overline{\\alpha }})$ .", "By factoring this univariate (which can be done efficiently), we see that such $g$ give a small list (of size at most $\\deg g$ ) of possible values for $f({\\overline{\\alpha }})$ .", "By picking a random element from this list, one can correctly compute $f({\\overline{x}})$ with noticeable probability, which by an averaging argument one can convert to a (non-uniform) deterministic procedure to compute $f({\\overline{x}})$ on most inputs (over any fixed finite set).", "While this procedure (involving univariate factorization) is not an algebraic circuit, the above argument shows that the Kabanets-Impagliazzo [43] notion is a natural form of average case hardness.", "To obtain this form of average-case hardness from worst-case hardness, Kabanets and Impagliazzo [43] used a result of Kaltofen [40], who showed that (up to pathologies in low-characteristic fields), factors of small (general) circuits have small circuits.", "As $g({\\overline{x}},f({\\overline{x}}))=0$ iff $y-f({\\overline{x}})$ divides $g({\\overline{x}},y)$ , it follows that if $g({\\overline{x}},y)$ has a small circuit then so does $y-f({\\overline{x}})$ , and thus so does $f({\\overline{x}})$ .", "Taking the contrapositive, if $f$ requires large circuits (worst-case hardness) then any such $g({\\overline{x}},y)$ with $g({\\overline{x}},f({\\overline{x}}))=0$ also requires large circuits (average-case hardness).", "Note that this says that any worst-case hard polynomial is also average-case hard.", "In contrast, this is provably false for boolean functions, where such worst-case to average-case reductions thus necessarily modify the original function.", "Unfortunately, Kaltofen's [40] factoring algorithm does not preserve structural restrictions (such as multilinearity, homogeneousness, low-depth, read-once-ness, etc.)", "of the original circuit, so that obtaining average-case hardness for restricted classes of circuits requires worst-case hardness for much stronger classes.", "While follow-up work has reduced the complexity of the circuits resulting from Kaltofen's [40] algorithm (Dvir-Shpilka-Yehudayoff [17] and Oliveira [53] extended Kaltofen's [40] to roughly preserve the depth of the original computation) these works are limited to factoring polynomials of small individual degree and do not seem applicable to other types of computations such as roABPs.", "Indeed, it even remains an open question to show any non-trivial upper bounds on the complexity of the factors of sparse polynomials.", "In fact, we actually have non-trivial lower bounds.", "Specifically, von zur Gathen and Kaltofen [79] gave an explicit $s$ -sparse polynomial (over any field) which has a factor with $s^{\\Omega (\\log s )}$ monomials, and Volkovich [78] gave, for a prime $p$ , an explicit $n$ -variate $n$ -sparse polynomial of degree-$p$ which in characteristic $p$ has a factor with $\\binom{n+p-2}{n-1}$ monomials (an exponential separation for $p\\ge (n)$ ).", "We refer the reader to the survey of Forbes and Shpilka [26] for more on the challenges in factoring small algebraic circuits.", "While showing the equivalence of worst-case and average-case hardness for restricted circuit classes seems difficult, to derandomize PIT via Kabanets-Impagliazzo [43] only requires a single polynomial which is average case hard.", "To facilitate obtaining such hard polynomials, we now record an easy lemma showing that polynomials with only hard multiples are average-case hard.", "Let $f({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ and $g({\\overline{x}},y)\\in \\mathbb {F}[{\\overline{x}},y]$ both be nonzero, where $g({\\overline{x}},0)\\ne 0$ also.", "If $g({\\overline{x}},f({\\overline{x}}))=0$ then $g({\\overline{x}},0)$ is a nonzero multiple of $f({\\overline{x}})$ .", "Let $g({\\overline{x}},y)=\\sum _i g_i({\\overline{x}})y^i$ and $g_0({\\overline{x}}):=g({\\overline{x}},0)$ .", "That $g({\\overline{x}},f({\\overline{x}}))=0$ implies that $0=g({\\overline{x}},f({\\overline{x}}))=\\sum _i g_i({\\overline{x}})(f({\\overline{x}}))^i=g_0({\\overline{x}})+\\sum _{i\\ge 1} g_i({\\overline{x}})(f({\\overline{x}}))^i$ so that $g_0({\\overline{x}})=f({\\overline{x}})\\cdot \\left(-\\sum _{i\\ge 1} g_i({\\overline{x}})(f({\\overline{x}}))^{i-1}\\right)$ as desired.", "That is, saying that $f({\\overline{x}})$ is not average-case hard means that $g({\\overline{x}},f({\\overline{x}}))=0$ for a nonzero $g({\\overline{x}},y)$ .", "One can assume that $g({\\overline{x}},0)\\ne 0$ , as otherwise one can replace $g$ by ${g}{y^i}$ for some $i\\le \\deg g$ , as this only mildly increases the size for most measures of circuit size (see for example sec:h-ips:ips=h-ips).", "As then the complexity of $g({\\overline{x}},0)$ is bounded by that of $g({\\overline{x}},y)$ (for natural measures), the lemma shows then that $f$ has a nonzero multiple of low-complexity.", "Taken contrapositively, if $f$ only has hard nonzero multiples then it is average-case hard in the sense needed for Kabanets-Impagliazzo [43].", "This shows that lower bounds for multiples is essentially the lower bound needed for algebraic hardness versus randomness.However, it is not an exact equivalence between lower bounds for multiples and average case hardness, as the converse to res:lbs-multtononroot is false, as seen by considering $g(x,y):=y-x(x+1)$ , so that $x\|g(x,0)$ but $g(x,x)\\ne 0$ .", "While in the below sections we are able to give explicit polynomials with hard multiples for various restricted classes of algebraic circuits, some of these classes (such as sparse polynomials and roABPs) still do not have the required closure properties to use Kabanets-Impagliazzo [43] to obtain deterministic PIT algorithms.", "Even for classes with the needed closure properties (such as $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas, where the hard polynomial is the monomial), the resulting PIT algorithms are only worse than existing results (which for $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas is the result of Forbes [22]).", "However, it seems likely that future results establishing polynomials with hard multiples would imply new PIT algorithms." ], [ "Lower Bounds for Multiples via PIT", "This above discussion shows that obtaining lower bounds for multiples is sufficient for instantiating the hardness versus randomness paradigm.", "We now observe the converse, showing that one can obtain such polynomials with hard multiples via derandomizing (black-box) PIT, or equivalently, producing generators with small seed-length.", "That is, Heintz-Schnorr [36] and Agrawal [3] showed that one can use explicit generators for small circuits to obtain hard polynomials, and we observe here that the resulting polynomials also have only hard multiples.", "Thus the below claim shows that obtaining black-box PIT yields the existence of a polynomial with hard multiples, which yields average-case hardness, which (for general enough classes) will allow the Kabanets-Impagliazzo [43] reduction to again yield black-box PIT.", "Thus, we see that obtaining such polynomials with hard multiples is essentially what is needed for this hardness versus randomness approach.", "Note that we give the construction based on a non-trivial generator for a class of circuits.", "While one can analogously prove the hitting-set version of this claim, it is weaker.", "That is, it is possible to consider classes $\\mathcal {C}$ of unbounded degree and still have generators with small seed-length (see for example res:SVb-gen below), but for such classes one must have hitting sets with infinite size (as hitting univariate polynomials of unbounded degree requires an infinite number of points).", "Let $\\mathcal {C}\\subseteq \\mathbb {F}[{\\overline{x}}]$ be a class of polynomials and let ${\\overline{\\mathcal {G}}}:\\mathbb {F}^\\ell \\rightarrow \\mathbb {F}^{\\overline{x}}$ be a generator for $\\mathcal {C}$ .", "Suppose $0\\ne h\\in \\mathbb {F}[{\\overline{x}}]$ has $h\\circ {\\overline{\\mathcal {G}}}=0$ .", "Then for any nonzero $g\\in \\mathbb {F}[{\\overline{x}}]$ we have that $g\\cdot h\\notin \\mathcal {C}$ .", "By definition of ${\\overline{\\mathcal {G}}}$ , for any $f\\in \\mathcal {C}$ , $f=0$ iff $f\\circ {\\overline{\\mathcal {G}}}=0$ .", "Then for any nonzero $g$ , $g\\cdot h\\ne 0$ and $(g\\cdot h)\\circ {\\overline{G}}=(g\\circ {\\overline{\\mathcal {G}}})\\cdot (h\\cdot {\\overline{\\mathcal {G}}})=(g\\circ {\\overline{\\mathcal {G}}}) \\cdot 0=0$ .", "Thus, we must have that $g\\cdot h\\notin \\mathcal {C}$ .", "That is, if $\\ell <n$ then such an $h$ exists (as the coordinates of ${\\overline{\\mathcal {G}}}$ are algebraically dependent) and such an $h$ can be found in exponential time by solving an exponentially-large linear system.", "As such, $h$ is a weakly-explicit polynomial with only hard multiples, which is sufficient for instantiating hardness versus randomness.", "While there are now a variety of restricted circuit classes with non-trivial (black-box) PIT results, it seems challenging to find for any given generator $\\mathcal {G}$ an explicit nonzero polynomial $f$ with $f\\circ \\mathcal {G}=0$ .", "Indeed, to the best of our knowledge no such examples have ever been furnished for interesting generators.", "Aside from the quest for polynomials with hard multiples, this question is independently interesting as it demonstrates the limits of the generator in question, especially for generators that are commonly used.", "There is not even a consensus as to whether the generators currently constructed could suffice to derandomize PIT for general circuits.", "Agrawal [3] has even conjectured that a certain generator for depth-2 circuits (sparse polynomials) would actually suffice for PIT of constant-depth circuits.", "We consider here the generator of Shpilka-Volkovich [73].", "This generator has a parameter $\\ell $ , and intuitively can be seen as an algebraic analogue of the boolean pseudorandomness notion of a (randomness efficient) $\\ell $ -wise independent hash function.", "Just as $\\ell $ -wise independent hash functions are ubiquitous in boolean pseudorandomness, the Shpilka-Volkovich [73] generator has likewise been used in a number of papers on black-box PIT (for example [73], [6], [24], [27], [78], [22] is a partial list).", "As such, we believe it is important to understand the limits of this generator.", "However, $\\ell $ -wise independence is a property of a hash function and likewise the Shpilka-Volkovich [73] generator is really a family of generators that all share a certain property.", "Specifically, the map ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV}}_{\\ell ,n}:\\mathbb {F}^{r}\\rightarrow \\mathbb {F}^n$ has the property The most obvious algebraic analogue of an $\\ell $ -wise independent hash function would require that for a generator ${\\overline{G}}:\\mathbb {F}^r\\rightarrow \\mathbb {F}^n$ that any subset $S\\subseteq [n]$ with $\|S\|\\le \\ell $ the output of ${\\overline{G}}$ restricted to $S$ is all of $\\mathbb {F}^{S}$ .", "This property is implied by the Shpilka-Volkovich [73] property, but is strictly weaker, and is in fact too weak to be useful for PIT.", "That is, consider the map $(x_1,\\ldots ,x_n)\\mapsto (x_1,\\ldots ,x_n,x_1+\\cdots +x_n)$ .", "This map has this naive “algebraic $n$ -wise independence” property, but does not even fool linear polynomials (which the Shpilka-Volkovich [73] generator does).", "that the image ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV}}_{\\ell ,n}(\\mathbb {F}^r)$ contains all $\\ell $ -sparse vectors in $\\mathbb {F}^n$ .", "The most straightforward construction of a randomness efficient generator with this property (via Lagrange interpolation, given by Shpilka-Volkovich [73]) has that $r=2\\ell $ .", "Even this construction is actually a family of possibly constructions, as there is significant freedom to choose the finite set of points where Lagrange interpolation will be performed.", "As such, instead of studying a specific generator we seek to understand the power of the above property, and thus we are free to construct another generator ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n}$ with this property for which we can find an explicit nonzero $f$ where $f\\circ {\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n}=0$ for small $\\ell $ .", "We choose a variant of the original construction so that we can take $f$ as the determinant.", "In the original Shpilka-Volkovich [73] generator, one first obtains the $\\ell =1$ construction by using two variables, the control variable $y$ and another variable $z$ .", "By using Lagrange polynomials to simulate indicator functions, the value of $y$ can be set to choose between the outputs $z{\\overline{e}}_1,\\ldots ,z{\\overline{e}}_n\\in \\mathbb {F}[z]^n$ , where ${\\overline{e}}_i\\in \\mathbb {F}^n$ is the $i$ -th standard basis vector.", "By varying $z$ one obtains all 1-sparse vectors in $\\mathbb {F}^n$ .", "To obtain ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV}}_{\\ell ,n}$ one can sum $\\ell $ independent copies of ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV}}_{1,n}$ .", "In contrast, our construction will simply use a different control mechanism, where instead of using univariate polynomials we use bivariates.", "Let $n,\\ell \\ge 1$ .", "Let $\\mathbb {F}$ be a field of size $\\ge n$ .", "Let $\\Omega :=\\lbrace \\omega _1,\\ldots ,\\omega _n\\rbrace $ be distinct elements in $\\mathbb {F}$ .", "Define ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}:\\mathbb {F}^3\\rightarrow \\mathbb {F}^{n\\times n}$ by $\\left({\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}(x,y,z)\\right)_{i,j}=z\\cdot \\mathbb {1}_{\\omega _i,\\Omega }(x)\\cdot \\mathbb {1}_{\\omega _j,\\Omega }(y)\\;.$ where $\\mathbb {1}_{\\omega _i,\\Omega }(x)\\in \\mathbb {F}[x]$ is the unique univariate polynomial of degree $<n$ such that $\\mathbb {1}_{\\omega _i,\\Omega }(\\omega _j)={\\left\\lbrace \\begin{array}{ll}1 & j=i\\\\0 & \\text{else}\\end{array}\\right.", "}\\;.$ Define ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}:\\mathbb {F}^{3\\ell }\\rightarrow \\mathbb {F}^{n\\times n}$ by the polynomial map ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}(x_1,y_1,z_1,\\ldots ,x_\\ell ,y_\\ell ,z_\\ell ):={\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}(x_1,y_1,z_1)+\\cdots +{\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}(x_\\ell ,y_\\ell ,z_\\ell )\\;,$ working in the ring $\\mathbb {F}[{\\overline{x}},{\\overline{y}},{\\overline{z}}]$ .", "Note that this map has $n^2$ outputs.", "Now observe that it is straightforward to see that this map has the desired property.", "Assume the setup of const:SVb.", "Then the image of the generator, ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}(\\mathbb {F}^{3\\ell })$ , contains all $\\ell $ -sparse vectors in $\\mathbb {F}^{n\\times n}$ .", "To the best of the authors knowledge, existing works using the Shpilka-Volkovich [73] generator Note that for black-box PIT it is important that we use a generator that contains all sparse vectors, instead of just the set of sparse vectors.", "As an example, the monomial $x_1\\cdots x_n$ is zero on all $k$ -sparse vectors for $k<n$ , but is nonzero when evaluated on the Shpilka-Volkovich [73] generator for any $\\ell \\ge 1$ .", "only use the above property (and occasionally also the fact that a coordinate-wise sum of constantly-many such generators is a generator of the original form with similar parameters ([5], [27], [32], [22]), which our alternate construction also satisfies).", "As such, we can replace the standard construction with our variant in known black-box PIT results (such as [73], [5], [24], [27], [32], [22]), some of which we now state.", "Assume the setup of const:SVb.", "Then ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{O(\\log s),n^2}$ is a generator for the following classes of $n$ -variate polynomials, of arbitrary degree.", "Polynomials of sparsity $s$ ([73], [32], [22]).", "Polynomials computable as a depth-3 powering formula of top-fan-in $s$ ([5], [24]).", "Polynomials computable as a $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)}$ formula of top-fan-in $s$ ([22]), in characteristic zero.", "Polynomials computable by width-$s$ roABPs in every variable order, also known as commutative roABPs ([5], [27]).", "The above result shows the power of the ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}$ generator to hit restricted classes of circuits.", "We now observe that it is also limited by its inability to hit the determinant.", "Assume the setup of const:SVb.", "The output of ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}$ is an $n\\times n$ matrix of rank $\\le \\ell $ , when viewed as a matrix over the ring $\\mathbb {F}({\\overline{x}},{\\overline{y}},{\\overline{z}})$ .", "Thus, taking $\\det _n\\in \\mathbb {F}[X]$ to be the $n\\times n$ determinant, we have that $\\det _n\\circ \\;{\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}=0$ for $\\ell <n$ .", "$\\ell =1$ : For a field $$ , a (nonzero) matrix $M\\in ^{n\\times n}$ is rank-1 if it can be expressed as an outer-product, so that $(M)_{i,j}=\\alpha _i\\beta _j$ for ${\\overline{\\alpha }},{\\overline{\\beta }}\\in ^n$ .", "Taking ${\\overline{\\alpha }},{\\overline{\\beta }}\\in \\mathbb {F}({\\overline{x}},{\\overline{y}},{\\overline{z}})^n$ defined by $\\alpha _i:=z\\mathbb {1}_{\\omega _i,\\Omega }(x)$ and $\\beta _j:=\\mathbb {1}_{\\omega _j,\\Omega }(y)$ we immediately see that ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}(x,y,z)$ is rank-1.", "$\\ell >1$ : This follows as rank is subadditive, and ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}$ is the sum of $\\ell $ copies of ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{1,n^2}$ .", "$\\det _n$ vanishes: This follows as the $n\\times n$ determinant vanishes on matrices of rank $<n$ .", "Note that in the above we could hope to find an $f$ such that $f\\circ {\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}=0$ for all $\\ell <n^2$ , but we are only able to handle $\\ell <n$ .", "Given the above result, along with res:generatortolbs-mult, we obtain that the multiples of the determinant are hard.", "Let $\\det _n\\in \\mathbb {F}[X]$ denote the $n\\times n$ determinant.", "Then any nonzero multiple $f\\cdot \\det _n$ of $\\det _n$ , for $0 \\ne f\\in \\mathbb {F}[X]$ , has the following lower bounds.", "$f\\cdot \\det _n$ involves $\\exp (\\Omega (n))$ monomials.", "$f\\cdot \\det _n$ requires size $\\exp (\\Omega (n))$ to be expressed as a depth-3 powering formula.", "$f\\cdot \\det _n$ requires size $\\exp (\\Omega (n))$ to be expressed as a $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)}$ formula, in characteristic zero.", "$f\\cdot \\det _n$ requires width-$\\exp (\\Omega (n))$ to be computed as an roABP in some variable order.", "By res:SVb-gen, ${\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{O(\\log s),n^2}$ is a generator for the above size-$s$ computations in the above classes.", "However, following res:generatortolbs-mult, $(f\\cdot \\det _n)\\circ \\left({\\overline{\\mathcal {G}}}^{\\mathrm {\\scriptscriptstyle SV^{\\prime }}}_{\\ell ,n^2}\\right)=0$ for $\\ell <n$ .", "Thus, if $f\\cdot \\det _n$ (which is nonzero) is computable in size-$s$ it must be that $O(\\log s)\\ge n$ , so that $s\\ge \\exp (\\Omega (n))$ .", "Note that we crucially leveraged that the determinant vanishes on low-rank matrices.", "As such, the above results do not seem to imply similar results for the permanent, despite the fact that the permanent is a harder polynomial.", "That is, recall that because of $$ -completeness of the permanent the determinant $\\det _n(X)$ can be written as a projection of the permanent, so that $\\det _n(X)=\\operatorname{perm}_m(A(X))$ for an affine matrix $A(X)\\in \\mathbb {F}[X]^{m\\times m}$ with $m\\le (n)$ .", "Then, given a multiple $g(Y)\\cdot \\operatorname{perm}_m(Y)$ one would like to use this projection to obtain $g(A(X))\\operatorname{perm}_m( A(X))=g(A(X))\\det _n X$ , which is a multiple of $\\det _n$ .", "Unfortunately this multiple may not be a nonzero multiple: it could be that $g(A(X))=0$ , from which no lower bounds for $g(A(X))\\det _n(X)$ (and hence $g(Y)\\operatorname{perm}_m (Y)$ ) can be derived." ], [ "Lower Bounds for Multiples via Leading/Trailing Monomials", "We now use the theory of leading (and trailing) monomials to obtain explicit polynomials with hard multiples.", "We aim at finding as simple polynomials as possible so they will give rise to simple “axioms” with no small refutations for our proof complexity applications in sec:ips-mult.", "These results will essentially be immediate corollaries of previous work." ], [ "Depth-3 Powering Formulas", "Kayal [41] observed that using the partial derivative method of Nisan and Wigderson [51] one can show that these formulas require $\\exp (\\Omega (n))$ size to compute the monomial $x_1\\cdots x_n$ .", "Forbes and Shpilka [24] later observed that this result can be made robust by modifying the hardness of representation technique of Shpilka and Volkovich [73], in that similar lower bounds apply when the leading monomial involves many variables, as we now quote.", "[Forbes-Shpilka [24]] Let $f({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ be computed a $\\sum \\bigwedge \\sum $ formula of size $\\le s$ .", "Then the leading monomial ${\\overline{x}}^{\\overline{a}}=\\operatorname{LM}(f)$ involves ${\|{\\overline{a}}\|_0}\\le \\lg s$ variables.", "We now note that as the leading monomial is multiplicative (res:homLM-TMmult) this lower bound automatically extends to multiples of the monomial.", "Any nonzero multiple of $x_1\\cdots x_n$ requires size $\\ge 2^n$ to be computed as a $\\sum \\bigwedge \\sum $ formula.", "Consider any $0\\ne g({\\overline{x}})\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ .", "Then as the leading monomial is multiplicative (res:homLM-TMmult) we have that $\\operatorname{LM}(g\\cdot x_1\\cdots x_n)=\\operatorname{LM}(g)\\cdot x_1\\cdots x_n$ , so that $\\operatorname{LM}(g\\cdot x_1\\cdots x_n)$ involves $n$ variables.", "By the robust lower bound (res:lbs-mult:LM:sumpowsum) this implies $g({\\overline{x}})\\cdot x_1\\cdots x_n$ requires size $\\ge 2^n$ as a $\\sum \\bigwedge \\sum $ formula." ], [ "$\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ Formulas", "Kayal [42] introduced the method of shifted partial derivatives, and Gupta-Kamath-Kayal-Saptharishi [31] refined it to give exponential lower bounds for various sub-models of depth-4 formulas.", "In particular, it was shown that the monomial $x_1\\cdots x_n$ requires $\\exp (\\Omega (n))$ -size to be computed as a $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)}$ formula.", "Applying the hardness of representation approach of Shpilka and Volkovich [73], Mahajan-Rao-Sreenivasaiah [48] showed how to develop a deterministic black-box PIT algorithm for multilinear polynomials computed by $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas.", "Independently, Forbes [22] (following Forbes-Shpilka [24]) showed that this lower bound can again be made to apply to leading monomials The result there is stated for trailing monomials, but the argument equally applies to leading monomials.", "(which implies a deterministic black-box PIT algorithm for all $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas, with the same complexity as Mahajan-Rao-Sreenivasaiah [48]).", "This leading monomial lower bound, which we now state, is important for its applications to polynomials with hard multiples.", "[Forbes [22]] Let $f({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ be computed as a $\\sum \\bigwedge \\sum \\prod ^{t} $ formula of size $\\le s$ .", "If $\\operatorname{char}(\\mathbb {F})\\ge \\operatorname{ideg}({\\overline{x}}^{\\overline{a}})$ , then the leading monomial ${\\overline{x}}^{\\overline{a}}=\\operatorname{LM}(f)$ involves ${\|{\\overline{a}}\|_0}\\le O(t\\lg s)$ variables.", "As for depth-3 powering formulas (res:lbs-mult:sumpowsum), this immediately yields that all multiples (of degree below the characteristic) of the monomial are hard.", "All nonzero multiples of $x_1\\cdots x_n$ of degree $<\\operatorname{char}(\\mathbb {F})$ require size $\\ge \\exp (\\Omega ({n}{t}))$ to be computed as $\\sum \\bigwedge \\sum \\prod ^{t} $ formula." ], [ "Sparse Polynomials", "While the above approaches for $\\sum \\bigwedge \\sum $ and $\\sum \\bigwedge \\sum \\prod ^{\\mathcal {O}(1)} $ formulas focus on leading monomials, one cannot show that the leading monomials of sparse polynomials involve few variables as sparse polynomials can easily compute the monomial $x_1\\cdots x_n$ .", "However, following the translation idea of Agrawal-Saha-Saxena [5], Gurjar-Korwar-Saxena-Thierauf [32] showed that sparse polynomials under full-support translations have some monomial involving few variables, and Forbes [22] (using different techniques) showed that in fact the trailing monomial involving few variables (translations do not affect the leading monomial, so the switch to trailing monomials is necessary here).", "[Forbes [22]] Let $f({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ be $(\\le s)$ -sparse, and let ${\\overline{\\alpha }}\\in (\\mathbb {F}\\setminus \\lbrace 0\\rbrace )^n$ so that ${\\overline{\\alpha }}$ has full-support.", "Then the trailing monomial ${\\overline{x}}^{\\overline{a}}=\\operatorname{TM}(f({\\overline{x}}+{\\overline{\\alpha }}))$ involves ${\|{\\overline{a}}\|_0}\\le \\lg s$ variables.", "This result thus allows one to construct polynomials whose multiples are all non-sparse.", "All nonzero multiples of $(x_1+1)\\cdots (x_n+1)\\in \\mathbb {F}[{\\overline{x}}]$ require sparsity $\\ge 2^n$ .", "Similarly, all nonzero multiples of $(x_1+y_1)\\cdots (x_n+y_n)\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ require sparsity $\\ge 2^n$ .", "Define $f({\\overline{x}})=\\prod _{i=1}^n (x_i+1)$ .", "For any $0\\ne g({\\overline{x}})\\in \\mathbb {F}[{\\overline{x}}]$ the multiple $g({\\overline{x}})f({\\overline{x}})$ under the translation ${\\overline{x}}\\mapsto {\\overline{x}}-{\\overline{1}}$ is equal to $g({\\overline{x}}-{\\overline{1}})\\prod _i x_i$ .", "Then all monomials (in particular the trailing monomial) involves $n$ variables (as $g({\\overline{x}})\\ne 0$ implies $g({\\overline{x}}-{\\overline{1}})\\ne 0$ ).", "Thus, by res:transdiffTM-ubsparse it must be that $g({\\overline{x}})f({\\overline{x}})$ requires $\\ge 2^n$ monomials.", "The second part of the claim follows as the first, where we now work over the fraction field $\\mathbb {F}({\\overline{y}})[{\\overline{x}}]$ , noting that this can only decrease sparsity.", "Thus, using the translation ${\\overline{x}}\\mapsto {\\overline{x}}-{\\overline{y}}$ the above trailing monomial argument implies that the sparsity of nonzero multiples $\\prod _i (x_i+y_i)$ are $\\ge 2^n$ over $\\mathbb {F}({\\overline{y}})[{\\overline{x}}]$ and hence also over $\\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ .", "Note that it is tempting to derive the second part of the above corollary from the first, using that the substitution ${\\overline{y}}\\leftarrow {\\overline{1}}$ does not increase sparsity.", "However, this substitution can convert nonzero multiples of $\\prod _i (x_i+y_i)$ to zero multiples of $\\prod _i (x_i+1)$ , which ruins such a reduction, as argued in the discussion after res:lbs-mult:pit:det." ], [ "Lower Bounds for Multiples of Sparse Multilinear Polynomials", "While the previous section established that all multiples of $(x_1+1)\\cdots (x_n+1)$ are non-sparse, the argument was somewhat specific to that polynomial and fails to obtain an analogous result for $(x_1+1)\\cdots (x_n+1)+1$ .", "Further, while that argument can show for example that all multiples of the $n\\times n$ determinant or permanent require sparsity $\\ge \\exp (\\Omega (n))$ , this is the best sparsity lower bound obtainable for these polynomials with this method.Specifically, as the determinant and permanent are degree $n$ multilinear polynomials, and thus so are their translations, their monomials always involve $\\le n$ variables so no sparsity bound better than $2^n$ can be obtained by using res:transdiffTM-ubsparse.", "In particular, one cannot establish a sparsity lower bound of “$n!$ ” for the determinant or permanent (which would be tight) via this method.", "We now give a different argument, due to Oliveira [52] that establishes a much more general result showing that multiples of any multilinear polynomial have at least the sparsity of the original polynomial.", "While Oliveira [52] gave a proof using Newton polytopes, we give a more compact proof here using induction on variables (loosely inspired by a similar result of Volkovich [77] on the sparsity of factors of multi-quadratic polynomials).", "[Oliveira [52]] Let $f({\\overline{x}})\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a nonzero multilinear polynomial with sparsity exactly $s$ .", "Then any nonzero multiple of $f$ has sparsity $\\ge s$ .", "By induction on variables.", "$n=0$ : Then $f$ is a constant, so that $s=1$ as $f\\ne 0$ .", "All nonzero multiples are nonzero polynomials so have sparsity $\\ge 1$ .", "$n\\ge 1$ : Partition the variables ${\\overline{x}}=({\\overline{y}},z)$ , so that $f({\\overline{y}},z)=f_1({\\overline{y}})z+f_0({\\overline{y}})$ , where $f_i({\\overline{y}})$ has sparsity $s_i$ and $s=s_1+s_0$ .", "Consider any nonzero $g({\\overline{y}},z)=\\sum _{i=d_0}^{d_1} g_i({\\overline{y}})z^i$ with $g_{d_0}({\\overline{y}}),g_{d_1}({\\overline{y}})\\ne 0$ (possibly with $d_0=d_1$ ).", "Then $g({\\overline{y}},z) f({\\overline{y}},z)&=\\Big (f_1({\\overline{y}})z+f_0({\\overline{y}})\\Big )\\cdot \\left(\\sum _{i=d_0}^{d_1} g_i({\\overline{y}})z^i\\right)\\\\&=f_1({\\overline{y}})g_{d_1}({\\overline{y}})z^{d_1+1}+\\left[\\sum _{d_0<i\\le d_1} \\Big (f_1({\\overline{y}})g_{i-1}({\\overline{y}})+f_0({\\overline{y}})g_i({\\overline{y}})\\Big )z^i\\right]+f_0({\\overline{y}})g_{d_0}({\\overline{y}})z^{d_0}\\;.$ By partitioning this sum by powers of $z$ so that there is no cancellation, and then discarding the middle terms, $\\left\|\\operatorname{Supp}\\Big (g({\\overline{y}},z) f({\\overline{y}},z)\\Big )\\right\|&\\ge \\left\|\\operatorname{Supp}\\Big (f_1({\\overline{y}})g_{d_1}({\\overline{y}})\\Big )\\right\|+\\left\|\\operatorname{Supp}\\Big (f_0({\\overline{y}})g_{d_0}({\\overline{y}})\\Big )\\right\|\\multicolumn{2}{l}{\\text{so that appealing to the induction hypothesis, as $f_0$ and $f_1$ are multilinear polynomials of sparsity $s_0$ and $s_1$ respectively,}}\\\\&\\ge s_1+s_0=s\\;.$ We note that multilinearity is essential in the above lemma, even for univariates.", "This is seen by noting that the 2-sparse polynomial $x^n-1$ is a multiple of $x^{n-1}+\\cdots +x+1$ .", "Thus, the above not only gives a different proof of the non-sparsity of multiples of $\\prod _i (x_i+1)$ (res:lbs-mult:sparse-LM), but also establishes that nonzero multiples of $\\prod _i (x_i+1)+1$ are $\\ge 2^n$ sparse, and nonzero multiples of the determinant or permanent are $n!$ sparse, which is tight.", "Note further that this lower bound proof is “monotone” in that it applies to any polynomial with the same support, whereas the proof of res:lbs-mult:sparse-LM is seemingly not monotone as seen by contrasting $\\prod _i (x_i+1)$ and $\\prod _i (x_i+1)+1$ ." ], [ "Lower Bounds for Multiples by Leading/Trailing Diagonals", "In the previous sections we obtained polynomials with hard multiples for various circuit classes by appealing to the fact that lower bounds for these classes can be reduced to studying the number of variables in leading or trailing monomials.", "Unfortunately this approach is restricted to circuit classes where monomials (or translations of monomials) are hard to compute, which in particular rules out this approach for roABPs.", "Thus, to develop polynomials with hard multiples for roABPs we need to develop a different notion of a “leading part” of a polynomial.", "In this section, we define such a notion called a leading diagonal, establish its basic properties, and obtain the desired polynomials with hard multiples.", "The ideas of this section are a cleaner version of the techniques used in the PIT algorithm of Forbes and Shpilka [23] for commutative roABPs." ], [ "Leading and Trailing Diagonals", "We begin with the definition of a leading diagonal, which is a generalization of a leading monomial.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ be nonzero.", "The leading diagonal of $f$, denoted $\\operatorname{LD}(f)$ , is the leading coefficient of $f({\\overline{x}}\\circ {\\overline{z}},{\\overline{y}}\\circ {\\overline{z}})$ when this polynomial is considered in the ring $\\mathbb {F}[{\\overline{x}},{\\overline{y}}][z_1,\\ldots ,z_n]$ , and where ${\\overline{x}}\\circ {\\overline{z}}$ denotes the Hadamard product $(x_1z_1,\\ldots ,x_nz_n)$ .", "The trailing diagonal of $f$ is defined analogously.", "The zero polynomial has no leading or trailing diagonal.", "As this notion has not explicitly appeared prior in the literature, we now establish several straightforward properties.", "The first is that extremal diagonals are homomorphic with respect to multiplication.", "Let $f,g\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ be nonzero.", "Then $\\operatorname{LD}(fg)=\\operatorname{LD}(f)\\operatorname{LD}(g)$ and $\\operatorname{TD}(fg)=\\operatorname{TD}(f)\\operatorname{TD}(g)$ .", "As $\\operatorname{LD}(f)=\\operatorname{LC}_{{\\overline{x}},{\\overline{y}}\|{\\overline{z}}}(f({\\overline{x}}\\circ {\\overline{z}},{\\overline{y}}\\circ {\\overline{z}}))$ , where this leading coefficient is taken in the ring $\\mathbb {F}[{\\overline{x}},{\\overline{y}}][{\\overline{z}}]$ , this automatically follows from the fact that leading coefficients are homomorphic with respect to multiplication (res:homLM-TMmult).", "The result for trailing diagonals is symmetric.", "We now show how to relate the leading monomials of the coefficient space of $f$ to the respective monomials associated to the leading diagonal of $f$ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ .", "For any ${\\overline{b}}$ , if ${\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(\\operatorname{LD}(f))\\ne 0$ , then $\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(\\operatorname{LD}(f))\\right)=\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)\\right)\\;.$ The respective trailing statement also holds.", "We prove the leading statement, the trailing version is symmetric.", "Let $f=\\sum _{{\\overline{a}},{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ .", "We can then expand $f({\\overline{x}}\\circ {\\overline{z}},{\\overline{y}}\\circ {\\overline{z}})$ as follows.", "$f({\\overline{x}}\\circ {\\overline{z}},{\\overline{y}}\\circ {\\overline{z}})&=\\textstyle \\sum _{{\\overline{c}}}\\left(\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}\\right){\\overline{z}}\\!\\:^{\\overline{c}}\\\\\\multicolumn{2}{l}{\\text{choose ${\\overline{c}}_0$ so that $\\operatorname{LC}_{{\\overline{x}},{\\overline{y}}\|{\\overline{z}}}(f)={\\mathrm {Coeff}}_{{\\overline{x}},{\\overline{y}}\|{\\overline{z}}\\!\\:^{{\\overline{c}}_0}}(f)$, we get that}}\\\\&=\\textstyle \\left(\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}_0} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}\\right){\\overline{z}}\\!\\:^{{\\overline{c}}_0}+\\sum _{{\\overline{c}}\\prec {\\overline{c}}_0}\\left(\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}\\right){\\overline{z}}\\!\\:^{\\overline{c}}\\;,$ where $\\operatorname{LD}(f)=\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}_0} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ and $\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}=0$ for ${\\overline{c}}\\succ {\\overline{c}}_0$ .", "In particular, this means that for any ${\\overline{b}}$ we have that $\\alpha _{{\\overline{a}},{\\overline{b}}}=0$ for ${\\overline{a}}\\succ {\\overline{c}}_0-{\\overline{b}}$ .", "Thus we have that $\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(\\operatorname{LD}(f))\\right)&=\\textstyle \\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}\\left(\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}_0} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}\\right)\\right)\\\\&=\\operatorname{LM}\\left(\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}{\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}\\right)\\\\&={\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}\\;,$ as we assume this leading monomial exists, which is equivalent here to $\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}\\ne 0$ .", "In comparison, $\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)\\right)&=\\textstyle \\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}\\left(\\sum _{{\\overline{a}},{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}\\right)\\right)\\\\&=\\textstyle \\operatorname{LM}\\left(\\sum _{{\\overline{a}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}\\right)\\\\&=\\textstyle \\operatorname{LM}\\left(\\sum _{{\\overline{a}}\\succ {\\overline{c}}_0-{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}+ \\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}{\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}+\\sum _{{\\overline{a}}\\prec {\\overline{c}}_0-{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}\\right)\\multicolumn{2}{l}{\\text{as $\\alpha _{{\\overline{a}},{\\overline{b}}}=0$ for ${\\overline{a}}\\succ {\\overline{c}}_0-{\\overline{b}}$,}}\\\\&=\\textstyle \\operatorname{LM}\\left(\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}{\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}+\\sum _{{\\overline{a}}\\prec {\\overline{c}}_0-{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}\\right)\\\\&={\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}\\;,$ where in the last step we again used that $\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}\\ne 0$ .", "This establishes the desired equality.", "We now relate the extremal monomials of the coefficient space of $f$ to the monomials of the coefficient space of the extremal diagonals of $f$ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ .", "Then $\\operatorname{LM}({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f))\\supseteq \\operatorname{LM}({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(\\operatorname{LD}(f)))\\;.$ The respective trailing statement also holds.", "This follows as $\\operatorname{LM}({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(\\operatorname{LD}(f)))$ is equal to $\\left\\lbrace \\left.\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}\\big (\\operatorname{LD}(f)\\big )\\right) \\, \\right\|\\, {\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}\\big (\\operatorname{LD}(f)\\big )\\ne 0\\right\\rbrace \\;,$ but by res:diagonal:tm this set equals $\\left\\lbrace \\left.\\operatorname{LM}\\left({\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(f)\\right) \\, \\right\|\\, {\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}\\big (\\operatorname{LD}(f)\\big )\\ne 0\\right\\rbrace \\;,$ which is clearly contained in $\\operatorname{LM}({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f))$ .", "We now observe that the number of leading monomials of the coefficient space of a leading diagonal is equal to its sparsity.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ .", "For a polynomial $g$ , let ${\|g\|_0}$ denotes its sparsity.", "Then $\\left\|\\operatorname{LM}\\left({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}\\big (\\operatorname{LD}(f)\\big )\\right)\\right\|={\|\\operatorname{LD}(f)\|_0}\\;.$ The respective trailing statement also holds.", "We prove the claim for the leading diagonal, the trailing statement is symmetric.", "Note that the claim is a vacuous “0=0” if $f$ is zero.", "For nonzero $f$ , express it as $f=\\sum _{{\\overline{a}},{\\overline{b}}} \\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}$ so that $\\operatorname{LD}(f)=\\sum _{{\\overline{a}}+{\\overline{b}}={\\overline{c}}_0}\\alpha _{{\\overline{a}},{\\overline{b}}}{\\overline{x}}^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{b}}=\\sum _{{\\overline{b}}}\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}{\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}{\\overline{y}}\\!\\:^{\\overline{b}}$ for some ${\\overline{c}}_0\\in \\mathbb {N}^n$ .", "Then ${\\mathrm {Coeff}}_{{\\overline{x}}\|{\\overline{y}}\\!\\:^{\\overline{b}}}(\\operatorname{LD}(f))=\\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}{\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}$ .", "As the monomials ${\\overline{x}}^{{\\overline{c}}_0-{\\overline{b}}}$ are distinct and hence linearly independent for distinct ${\\overline{b}}$ , it follows that $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(\\operatorname{LD}(f))=\|\\lbrace {\\overline{b}}\| \\alpha _{{\\overline{c}}_0-{\\overline{b}},{\\overline{b}}}\\ne 0\\rbrace \|$ , which is equal the sparsity ${\|\\operatorname{LD}(f)\|_0}$ .", "Finally, we now lower bound the coefficient dimension of a polynomial by the sparsity of its extremal diagonals.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ .", "Then $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)\\ge {\|\\operatorname{LD}(f)\|_0},{\|\\operatorname{TD}(f)\|_0}\\;,$ where for a polynomial $g$ , ${\|g\|_0}$ denotes its sparsity.", "We give the proof for the leading diagonal, the trailing diagonal is symmetric.", "By the above, $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)\\ge \\left\|\\operatorname{LM}\\left({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(f)\\right)\\right\|\\ge \\left\|\\operatorname{LM}\\left({\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}\\big (\\operatorname{LD}(f)\\big )\\right)\\right\|= {\|\\operatorname{LD}(f)\|_0}\\;,$ where we passed from span to number of leading monomials (res:dim-eq-num-TM-spn), and then passed to the leading monomials of the leading diagonal (res:diagonal:lm-coeffs), and then passed to sparsity of the leading diagonal (lem:LM-by-LD)." ], [ "Lower Bounds for Multiples for Read-Once and Read-Twice ABPs", "Having developed the theory of leading diagonals in the previous section, we now turn to using this theory to obtain explicit polynomials whose nonzero multiples all require large roABPs.", "We also generalize this to read-$O(1)$ oblivious ABPs, but only state the results for $k=2$ as this has a natural application to proof complexity (sec:ips-mult).", "As the restricted computations considered above ($\\sum \\bigwedge \\sum $ formulas and sparse polynomials) have small roABPs, the hard polynomials in this section will also have multiples requiring large complexity in these models as well and thus qualitatively reprove some of the above results.", "However, we included the previous sections as the hard polynomials there are simpler (being monomials or translations of monomials), and more importantly we will need those results for the proofs below.", "The proofs will use the characterization of roABPs by their coefficient dimension (res:roABP-widtheqdim-coeffs), the lower bound for coefficient dimension in terms of the sparsity of the extremal diagonals (res:diagonal:coeff-dim-lb), and polynomials whose multiples are all non-sparse (res:lbs-mult:sparse-LM).", "Let $f({\\overline{x}},{\\overline{y}}):=\\prod _{i=1}^n (x_i+y_i+\\alpha _i)\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_n]$ , for $\\alpha _i\\in \\mathbb {F}$ .", "Then for any $0\\ne g\\in \\mathbb {F}[{\\overline{x}},{\\overline{y}}]$ , $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(g\\cdot f)\\ge 2^n\\;.$ In particular, all nonzero multiples of $f$ require width at least $2^n$ to be computed by an roABP in any variable order where ${\\overline{x}}\\prec {\\overline{y}}$ .", "Observe that leading diagonal of $f$ is insensitive to the $\\alpha _i$ .", "That is, $\\operatorname{LD}(x_i+y_i+\\alpha _i)=x_i+y_i$ , so by multiplicativity of the leading diagonal (res:diagonals:mult-hom) we have that $\\operatorname{LD}(f)=\\prod _i (x_i+y_i)$ .", "Thus, appealing to res:diagonal:coeff-dim-lb and res:lbs-mult:sparse-LM, $\\dim {\\mathbf {Coeff}}_{{\\overline{x}}\|{\\overline{y}}}(g\\cdot f)&\\ge {\|\\operatorname{LD}(g\\cdot f)\|_0}\\\\&={\|\\operatorname{LD}(g)\\cdot \\operatorname{LD}(f)\|_0}\\\\&=\\textstyle {\|\\operatorname{LD}(g)\\cdot \\prod _i (x_i+y_i)\|_0}\\\\&\\ge 2^n\\;.$ The claim about roABP width follows from res:roABP-widtheqdim-coeffs.", "Note that this lower bound actually works in the “monotone” setting (if we replace res:lbs-mult:sparse-LM with the monotone res:lbs-mult:sparse:newton), as the result only uses the zero/nonzero pattern of the coefficients.", "The above result gives lower bounds for coefficient dimension in a fixed variable partition.", "We now symmetrize this construction to get lower bounds for coefficient dimension in any variable partition.", "We proceed as in sec:lbs-fn:every-partition, where we plant the fixed-partition lower bound into an arbitrary partition.", "Note that unlike that construction, we will not need auxiliary variables here.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be defined by $f({\\overline{x}}):=\\prod _{i<j}(x_i+x_j+\\alpha _{i,j})$ for $\\alpha _{i,j}\\in \\mathbb {F}$ .", "Then for any partition ${\\overline{x}}=({\\overline{u}},{\\overline{v}},{\\overline{w}})$ with $m:=\|{\\overline{u}}\|=\|{\\overline{v}}\|$ , and any $0\\ne g\\in \\mathbb {F}[{\\overline{x}}]$ , $\\dim _{\\mathbb {F}({\\overline{w}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}(g\\cdot f)\\ge 2^m\\;,$ where we treat $g\\cdot f$ as a polynomial in $\\mathbb {F}({\\overline{w}})[{\\overline{u}},{\\overline{v}}]$ .", "In particular, all nonzero multiples of $f$ require width $\\ge 2^n$ to be computed by an roABP in any variable order.", "We can factor $f$ into a copy of the hard polynomial from prop:good-f-roabp, and the rest.", "That is, $f({\\overline{x}})=\\prod _{i<j} (x_i+x_j+\\alpha _{i,j})=\\prod _{i=1}^m(u_i+v_i+\\beta _i)\\cdot f^{\\prime }({\\overline{u}},{\\overline{v}},{\\overline{w}})\\;,$ for some $\\beta _i\\in \\mathbb {F}$ and nonzero $f^{\\prime }({\\overline{u}},{\\overline{v}},{\\overline{w}})\\in \\mathbb {F}[{\\overline{u}},{\\overline{v}},{\\overline{w}}]$ .", "Thus, $g\\cdot f=\\left(g({\\overline{u}},{\\overline{v}},{\\overline{w}})\\cdot f^{\\prime }({\\overline{u}},{\\overline{v}},{\\overline{w}})\\right)\\cdot \\prod _{i=1}^m(u_i+v_i+\\beta _i)\\;.$ Noting that $g,f^{\\prime }$ are nonzero in $\\mathbb {F}[{\\overline{u}},{\\overline{v}},{\\overline{w}}]$ , they are also nonzero in $\\mathbb {F}({\\overline{w}})[{\\overline{u}},{\\overline{v}}]$ , so that $g\\cdot f$ is nonzero multiple of $\\prod _{i=1}^m(u_i+v_i+\\beta _i)$ in $\\mathbb {F}({\\overline{w}})[{\\overline{u}},{\\overline{v}}]$ .", "Appealing to our lower bound for (nonzero) multiples of coefficient dimension (prop:good-f-roabp), we have that $\\dim _{\\mathbb {F}({\\overline{w}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}(g\\cdot f)=\\dim _{\\mathbb {F}({\\overline{w}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}\\left(g\\cdot f^{\\prime }\\cdot \\prod _{i=1}^m(u_i+v_i+\\beta _i)\\right)\\ge 2^m\\;.$ The statement about roABPs follows from res:roABP-widtheqdim-coeffs.", "We briefly remark that the above bound does not match the naive bound achieved by writing the polynomial $\\prod _{i<j}(x_i+x_j+\\alpha _{i,j})$ in its sparse representation, which has $2^{\\Theta (n^2)}$ terms.", "The gap between the lower bound ($2^{\\Omega (n)}$ ) and the upper bound ($2^{O(n^2)}$ ) is explained by our use of a complete graph to embed the lower bounds of prop:good-f-roabp into an arbitrary partition.", "As discussed after res:lbs-fn:any-order:coeff-dim one can use expander graphs to essentially close this gap.", "We now observe that the above lower bounds for coefficient dimension suffices to obtain lower bounds for read-twice oblivious ABPs, as we can appeal to the structural result of Anderson, Forbes, Saptharishi, Shpilka and Volk [1] (thm:read-k-eval-dim).", "This result shows that for any read-twice oblivious ABP that (after discarding some variables) there is a partition of the variables across which has small coefficient dimension, which is in contrast to the above lower bound.", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be defined by $f({\\overline{x}}):=\\prod _{i<j}(x_i+x_j+\\alpha _{i,j})$ for $\\alpha _{i,j}\\in \\mathbb {F}$ .", "Then for any $0\\ne g\\in \\mathbb {F}[{\\overline{x}}]$ , $g\\cdot f$ requires width-$2^{\\Omega (n)}$ as a read-twice oblivious ABP.", "Suppose that $g\\cdot f$ has a read-twice oblivious ABP of width-$w$ .", "By the lower-bound of Anderson, Forbes, Saptharishi, Shpilka and Volk [1] (thm:read-k-eval-dim), there exists a partition ${\\overline{x}}=({\\overline{u}},{\\overline{v}},{\\overline{w}})$ where $\|{\\overline{u}}\|,\|{\\overline{v}}\|\\ge \\Omega (n)$ , and such that $\\dim _{\\mathbb {F}({\\overline{w}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}(g\\cdot f)\\le w^4$ (where we treat $g\\cdot f$ as a polynomial in $\\mathbb {F}({\\overline{w}})[{\\overline{u}},{\\overline{v}}]$ ).", "Note that we can take enforce that the partition obeys $m:=\|{\\overline{u}}\|=\|{\\overline{v}}\|\\ge \\Omega (n)$ , as we can balance ${\\overline{u}}$ and ${\\overline{v}}$ by pushing variables into ${\\overline{w}}$ , as this cannot increase the coefficient dimension (fact:roABP:closure).", "However, appealing to our coefficient dimension bound (cor:lbs-mult:every-partition) $w^4\\ge \\dim _{\\mathbb {F}({\\overline{w}})}{\\mathbf {Coeff}}_{{\\overline{u}}\|{\\overline{v}}}(g\\cdot f)\\ge 2^m\\ge 2^{\\Omega (n)}\\;,$ so that $w\\ge 2^{\\Omega (n)}$ as desired." ], [ "IPS Lower Bounds via Lower Bounds for Multiples", "In this section we use the lower bounds for multiples of sec:lbs-mult to derive lower bounds for $\\mathcal {C}$ -IPS proofs for various restricted algebraic circuit classes $\\mathcal {C}$ .", "The advantage of this approach over the functional lower bounds strategy of sec:lbs-fn is that we derive lower bounds for the general IPS system, not just its subclass linear-IPS.", "While our equivalence (res:h-ipsvips) of $\\mathcal {C}$ -IPS and $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ holds for any strong-enough class $\\mathcal {C}$ , the restricted classes we consider here (depth-3 powering formulas and roABPs) As in sec:lbs-mult, we will not treat multilinear formulas in this section as they are less natural for the techniques under consideration.", "Further, IPS lower bounds for multilinear formulas can be obtained via functional lower bounds (res:lbs-fn:lbs-ips:vary-order).", "are not strong enough to use res:h-ipsvips to lift the results of sec:lbs-fn to lower bounds for the full IPS system.", "However, as discussed in the introduction, the techniques of this section can only yield lower bounds for $\\mathcal {C}$ -IPS refutations of systems of equations which are hard to compute within $\\mathcal {C}$ (though our examples are computable by small (general) circuits).", "We begin by first detailing the relation between IPS refutations and multiples.", "We then use our lower bounds for multiples (sec:lbs-mult) to derive as corollaries lower bounds for $\\sum \\bigwedge \\sum $ -IPS and roABP-IPS refutations.", "Let $f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}}\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be an unsatisfiable system of equations, where ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ is satisfiable.", "Let $C\\in \\mathbb {F}[{\\overline{x}},y,{\\overline{z}},{\\overline{w}}]$ be an IPS refutation of $f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ .", "Then $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is a nonzero multiple of $f$ .", "That $C$ is an IPS refutation means that $C({\\overline{x}},f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})=1,\\qquad C({\\overline{x}},0,{\\overline{0}},{\\overline{0}})=0\\;.$ We first show that $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is a multiple of $f$ , using the first condition on $C$ .", "Expand $C({\\overline{x}},y,{\\overline{z}},{\\overline{w}})$ as a univariate in $y$ , so that $C({\\overline{x}},y,{\\overline{z}},{\\overline{w}})=\\sum _{i\\ge 0} C_i({\\overline{x}},{\\overline{z}},{\\overline{w}})y^i\\;,$ for $C_i\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}},{\\overline{w}}]$ .", "In particular, $C_0({\\overline{x}},{\\overline{z}},{\\overline{w}})=C({\\overline{x}},0,{\\overline{z}},{\\overline{w}})$ .", "Thus, $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})&=C({\\overline{x}},f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})\\\\&=\\textstyle \\left(\\sum _{i\\ge 0} C_i({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})f^i\\right)-C_0({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})\\\\&=\\textstyle \\sum _{i\\ge 1} C_i({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})f^i\\\\&=\\textstyle \\left(\\sum _{i\\ge 1} C_i({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})f^{i-1}\\right)\\cdot f\\;.$ Thus, $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is a multiple of $f$ as desired.", "We now show that this is a nonzero multiple, using the second condition on $C$ and the satisfiability of ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ .", "That is, the second condition implies that $0=C({\\overline{x}},0,{\\overline{0}},{\\overline{0}})=C_0({\\overline{x}},{\\overline{0}},{\\overline{0}})$ .", "If $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is zero, then by the above we have that $C_0({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})=1$ , so that $C_0({\\overline{x}},{\\overline{z}},{\\overline{w}})$ is an IPS refutation of ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ , which contradicts the satisfiability of ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ as IPS is a sound proof system.", "So it must then be that $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is nonzero.", "That is, take an ${\\overline{\\alpha }}$ satisfying ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ so that ${\\overline{g}}({\\overline{\\alpha }})={\\overline{0}},{\\overline{\\alpha }}^2-{\\overline{\\alpha }}={\\overline{0}}$ .", "Substituting this ${\\overline{\\alpha }}$ into $C_0({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ , we have that $C_0({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})\|_{{\\overline{x}}\\leftarrow {\\overline{\\alpha }}}=C_0({\\overline{\\alpha }},{\\overline{0}},{\\overline{0}})$ , and because $C_0({\\overline{x}},{\\overline{0}},{\\overline{0}})\\equiv 0$ in $\\mathbb {F}[{\\overline{x}}]$ via the above we have that $C_0({\\overline{\\alpha }},{\\overline{0}},{\\overline{0}})=0$ .", "Thus, we have that $1-C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})=1-C_0({\\overline{x}},{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ is a nonzero polynomial as its evaluation at ${\\overline{x}}\\leftarrow {\\overline{\\alpha }}$ is 1.", "The above lemma thus gives a template for obtaining lower bounds for IPS.", "First, obtain a “hard” polynomial $f$ whose nonzero multiples are hard for $\\mathcal {C}$ , where $f$ is hopefully also computable by small (general) circuits.", "Then find additional (simple) polynomials ${\\overline{g}}$ such that ${\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ is satisfiable yet $f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable.", "By the above lemma one then has the desired IPS lower bound for refuting $f,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}}$ , assuming that $\\mathcal {C}$ is sufficiently general.", "However, for our results we need to more careful as even though $C({\\overline{x}},y,{\\overline{z}},{\\overline{w}})$ is from the restricted class $\\mathcal {C}$ , the derived polynomial $C({\\overline{x}},0,{\\overline{g}},{\\overline{x}}^2-{\\overline{x}})$ may not be, and as such we will need to appeal to lower bounds for stronger classes.", "We now instantiate this template, first for depth-3 powering formulas, where we use lower bounds for multiples of the stronger $\\sum \\bigwedge \\sum \\prod ^{2}$ model.", "Let $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})=0$ .", "Let $f:=x_1\\cdots x_n$ and $g:=x_1+\\cdots +x_n-n$ with $f,g\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ .", "Then $f,g,{\\overline{x}}^2-{\\overline{x}}$ are unsatisfiable and any $\\sum \\bigwedge \\sum $ -IPS refutation requires size at least $\\exp (\\Omega (n))$ .", "The hypothesis $\\operatorname{char}(\\mathbb {F})=0$ implies that $\\lbrace 0,\\ldots ,n\\rbrace $ are distinct numbers.", "In particular, the system $g({\\overline{x}})=0$ and ${\\overline{x}}^2-{\\overline{x}}={\\overline{0}}$ is satisfiable and has the unique satisfying assignment ${\\overline{1}}$ .", "However, this single assignment does not satisfy $f$ as $f({\\overline{1}})=\\prod _{i=1}^n 1=1\\ne 0$ , so the entire system is unsatisfiable.", "Thus, applying our strategy (res:lbs-mult:strategy), we see that for any $\\sum \\bigwedge \\sum $ -IPS refutation $C({\\overline{x}},y,z,{\\overline{w}})$ of $f,g,{\\overline{x}}^2-{\\overline{x}}$ that $1-C({\\overline{x}},0,g,{\\overline{x}}^2-{\\overline{x}})$ is a nonzero multiple of $f$ .", "Let $s$ be the size of $C$ as a $\\sum \\bigwedge \\sum $ formula.", "As $g$ is linear and the boolean axioms ${\\overline{x}}^2-{\\overline{x}}$ are quadratic, it follows that $1-C({\\overline{x}},0,g,{\\overline{x}}^2-{\\overline{x}})$ is a sum of powers of quadratics ($\\sum \\bigwedge \\sum \\prod ^{2} $ ) of size $(s)$ .", "As nonzero multiples of $f$ requires $\\exp (\\Omega (n))$ -size as a $\\sum \\bigwedge \\sum \\prod ^{2}$ formula (res:lbs-mult:sumpowt) it follows that $(s)\\ge \\exp (\\Omega (n))$ , so that $s\\ge \\exp (\\Omega (n))$ as desired.", "We similarly obtain a lower bound for roABP-IPS, where here we use lower bounds for multiples of read-twice oblivious ABPs.", "Let $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Let $f:=\\prod _{i<j}(x_i+x_j-1)$ and $g:=x_1+\\cdots +x_n-n$ with $f,g\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ .", "Then $f,g,{\\overline{x}}^2-{\\overline{x}}$ are unsatisfiable and any roABP-IPS refutation (in any variable order) requires size $\\ge \\exp (\\Omega (n))$ .", "The hypothesis $\\operatorname{char}(\\mathbb {F})>n$ implies that $\\lbrace 0,\\ldots ,n\\rbrace $ are distinct numbers.", "In particular, the system $g({\\overline{x}})=0$ and ${\\overline{x}}^2-{\\overline{x}}={\\overline{0}}$ is satisfiable and has the unique satisfying assignment ${\\overline{1}}$ .", "However, this single assignment does not satisfy $f$ as $f({\\overline{1}})=\\prod _{i<j}(1+1-1)=1\\ne 0$ , so the entire system is unsatisfiable.", "Thus, applying our strategy (res:lbs-mult:strategy), we see that for any roABP-IPS refutation $C({\\overline{x}},y,z,{\\overline{w}})$ of $f,g,{\\overline{x}}^2-{\\overline{x}}$ that $1-C({\\overline{x}},0,g,{\\overline{x}}^2-{\\overline{x}})$ is a nonzero multiple of $f$ .", "Let $s$ be the size of $C$ as an roABP, and we now argue that $1-C({\\overline{x}},0,g,{\\overline{x}}^2-{\\overline{x}})$ has a small read-twice oblivious ABP.", "First, note that we can expand $C({\\overline{x}},0,z,{\\overline{w}})$ into powers of $z$ , so that $C({\\overline{x}},0,z,{\\overline{w}})=\\sum _{0\\le i\\le s} C_i({\\overline{x}},{\\overline{w}})z^i$ (where we use that $s$ bounds the width and degree of the roABP $C$ ).", "Each $C_i({\\overline{x}},{\\overline{w}})$ has a $(s)$ -size roABP (in the variable order of $C$ where $z$ is omitted) as we can compute $C_i$ via interpolation over $z$ , using that each evaluation preserves roABP size (fact:roABP:closure).", "Further, as $g$ is linear, for any $i$ we see that $g^i$ can be computed by a $(n,i)$ -size roABP (in any variable order) (res:sumpowsum:roABP).", "Combining these facts using closure properties of roABPs under addition and multiplication (fact:roABP:closure), we see that $C({\\overline{x}},0,g,{\\overline{w}})$ , and hence $1-C({\\overline{x}},0,g,{\\overline{w}})$ , has a $(s,n)$ -size roABP in the variable order that $C$ induces on ${\\overline{x}},{\\overline{w}}$ .", "Next observe, that as each boolean axiom $x_i^2-x_i$ only refers to a single variable, substituting ${\\overline{w}}\\leftarrow {\\overline{x}}^2-{\\overline{x}}$ in the roABP for $1-C({\\overline{x}},0,g,{\\overline{w}})$ will preserve obliviousness of the ABP, but now each variable will be read twice, so that $1-C({\\overline{x}},0,g,{\\overline{x}}^2-{\\overline{x}})$ has a $(s,n)$ -size read-twice oblivious ABP.", "Now, using that nonzero multiples of $f$ requires $\\exp (\\Omega (n))$ -size to be computed as read-twice oblivious ABPs (cor:r2abp-multiples) it follows that $(s,n)\\ge \\exp (\\Omega (n))$ , so that $s\\ge \\exp (\\Omega (n))$ as desired.", "We note that the above lower bound is for the size of the roABP.", "One can also obtain the stronger result (for similar but less natural axioms) showing that the width (and hence also the size) of the roABP must be large, but we do not pursue this as it does not qualitatively change the result." ], [ "Discussion", "In this work we proved new lower bounds for various natural restricted versions of the Ideal Proof System (IPS) of Grochow and Pitassi [33].", "While existing work in algebraic proof complexity showed limitations of weak measures of complexity such as the degree and sparsity of a polynomial, our lower bounds are for stronger measures of circuit size that match many of the frontier lower bounds in algebraic circuit complexity.", "However, our work leaves several open questions and directions for further study, which we now list.", "Can one obtain proof complexity lower bounds from the recent techniques for lower bounds for depth-4 circuits, such as the results of Gupta, Kamath, Kayal and Saptharishi [31]?", "Neither of our approaches (functional lower bounds or lower bounds for multiples) currently extend to their techniques.", "Many proof complexity lower bounds are for refuting unsatisfiable $k$ -CNFs, where $k=O(1)$ , which can be encoded as systems of polynomial equations where each equation involves $O(1)$ variables.", "Can one obtain interesting IPS lower bounds for such systems?", "Our techniques only establish exponential lower bounds where there is at least one axiom involving $\\Omega (n)$ variables.", "Given an equation $f({\\overline{x}})=0$ where $f$ has a size-$s$ circuit, there is a natural way to convert this equation to $(s)$ -many equations on $O(1)$ extension variables by tracing through the computation of $f$ .", "Can one understand how introducing extension variables affects the complexity of refuting polynomial systems of equations?", "This seems a viable approach to the previous question when applied to our technique of using lower bounds for multiples.", "We have shown various lower bounds for multiples by invoking the hardness of the determinant (res:lbs-mult:pit:det), but this does not lead to satisfactory proof lower bounds as the axioms are complicated.", "Can one implicitly invoke the hardness of the determinant?", "For example, consider the hard matrix identities suggested by Cook and Rackoff (see for example the survey of Beame and Pitassi [10]) and later studied by Soltys and Cook [68].", "That is, consider unsatisfiable equations such as $XY-{\\mathbf {I}}_n,YX-2\\cdot {\\mathbf {I}}_n$ , where $X$ and $Y$ are symbolic $n\\times n$ matrices and ${\\mathbf {I}}_n$ is the $n\\times n$ identity matrix.", "The simplest refutations known involve the determinant (see Hrubeš-Tzameret [37], and the discussion in Grochow-Pitassi [33]), can one provide evidence that computing the determinant is intrinsic to such refutations?", "The lower bounds of this paper are for the static IPS system, where one cannot simplify intermediate computations.", "There are also dynamic algebraic proof systems (see sec:alg-proofs), can one extend our techniques to that setting?" ], [ "Acknowledgments", "We would like to thank Rafael Oliviera for telling us of res:lbs-mult:sparse:newton, Mrinal Kumar and Ramprasad Saptharishi for conversations [20] clarifying the roles of functional lower bounds in this work, as well as Avishay Tal for pointing out how res:subsetsum:multlin implies an optimal functional lower bound for sparsity (res:subsetsum:multlin:deg-sparse).", "We would also like to thank Joshua Grochow for helpful discussions regarding this work.", "We are grateful for the anonymous reviewers for their careful read of the paper and for their comments." ], [ "Relating IPS to Other Proof Systems", "In this section we summarize some existing work on algebraic proof systems and how these other proof systems compare to IPS.", "In particular, we define the (dynamic) Polynomial Calculus refutation system over circuits (related to but slightly different than the system of Grigoriev and Hirsch [28]) and relate it to the (static) IPS system ([55], [33]) considered in this paper.", "We then examine the roABP-PC system, essentially considered by Tzameret [76], and its separations from sparse-PC.", "Finally, we consider multilinear-formula-PC as studied by Raz and Tzameret [62], [63] and show that its tree-like version simulates multilinear-formula-IPS, and is hence separated from sparse-PC." ], [ "Polynomial Calculus Refutations", "A substantial body of prior work considers dynamic proof systems, which are systems that allow simplification of intermediate polynomials in the proof.", "In contrast, IPS is a static system where the proof is single object with no “intermediate” computations to simplify.", "We now define the principle dynamic system of interest, the Polynomial Calculus system.", "We give a definition over an arbitrary circuit class, which generalizes the definition of the system as introduced by Clegg, Edmonds, and Impagliazzo [11].", "Let $f_1({\\overline{x}}),\\ldots ,f_m({\\overline{x}})\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a system of polynomials.", "A Polynomial Calculus (PC) proof for showing that $p\\in \\mathbb {F}[{\\overline{x}}]$ is in the ideal generated by ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ is a directed acyclic graph with a single sink, where Leaves are labelled with an equation from ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ .", "An internal node $v$ with children $u_1,\\ldots ,u_k$ for $k>1$ is labelled with a linear combination $v=\\alpha _1 u_1+\\cdots +\\alpha _ku_k$ for $\\alpha _i\\in \\mathbb {F}$ .", "An internal node $v$ with a single child $u$ is labelled with the product $g\\cdot u$ for some $g\\in \\mathbb {F}[{\\overline{x}}]$ .", "The value of a node in the proof is defined inductively via the above labels interpreted as equations, and the value of the output node is required to be the desired polynomial $p$ .", "The proof is tree-like if the underlying graph is a tree, and is otherwise dag-like.", "A PC refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ is a proof that 1 is in the ideal of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ so that ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable.", "The size of each node is defined inductively as follows.", "The size of a leaf $v$ is the size of the minimal circuit agreeing with the value of $v$ .", "The size of an addition node $v=\\alpha _1 u_1+\\cdots +\\alpha _ku_k$ is $k$ plus the size of each child $u_i$ , plus the size of the minimal circuit agreeing with the value of $v$ .", "The size of a product node $v=g\\cdot u$ is the size of the child $u$ plus the size of the minimal circuit agreeing with the value of $v$ .", "The size of the proof is the sum of the sizes of each node in the proof.", "For a restricted algebraic circuit class $\\mathcal {C}$ , a $\\mathcal {C}$ -PC proof is a PC proof where the circuits are measured as their size coming from the restricted class $\\mathcal {C}$ .", "As with IPS, one can show this is a sound and complete proof system for unsatisfiability of equations.", "Also as with IPS, in our definition of PC we included the boolean axioms ${\\overline{x}}^2-{\\overline{x}}$ as this in the most common regime.", "An important aspect of the above proof system is that it is semantic, as the polynomials derived in the proof are simplified to their smallest equivalent algebraic circuit.", "This is a valid in that such simplifications can be efficiently verified (with randomness) using polynomial identity testing (which can sometimes be derandomized, see sec:background).", "In contrast, one could instead require a syntactic proof system, which would have to provide a proof via syntactic manipulation of algebraic circuits that such simplifications are valid.", "We will focus on semantic systems as they more naturally compare with IPS, which also requires polynomial identity testing for verification.", "While many priors work ([55], [62], [63], [76], [33]) considered algebraic proof systems whose verification relied on polynomial identity testing (because of semantic simplification or otherwise), we note that the system of Grigoriev and Hirsch [28] (which they called “formula-$\\mathcal {PC}$ ”) is actually a syntactic system and as such is deterministically checkable.", "Despite their system being restricted to being syntactic, it is still strong enough to simulate Frege and obtain low-depth refutations of the subset-sum axiom, the pigeonhole principle, and Tseitin tautologies.", "Note that our definition here varies slightly from the definition of Clegg, Edmonds, and Impagliazzo [11], in that we allow products by an arbitrary polynomial $g$ instead of only allowing products of a single variable $x_i$ .", "For some circuit classes $\\mathcal {C}$ these two definitions are polynomially equivalent (see for example the discussion in Raz and Tzameret [62]).", "In general however, using the product rule $f\\vdash x_i\\cdot f$ in a tree-like proof can only yield $g\\cdot f$ where $g$ is a small formula.", "However, we will be interested in algebraic circuit classes not known to be simulated by small formulas (such as roABPs, which can compute iterated matrix products which are believed to require super-polynomial-size formulas) and as such will consider this stronger product rule.", "We now observe that tree-like $\\mathcal {C}$ -PC can simulate $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ for natural restricted circuit classes $\\mathcal {C}$ .", "Let $\\mathcal {C}$ be a restricted class of circuits computing polynomials in $\\mathbb {F}[x_1,\\ldots ,x_n]$ , and suppose that $\\mathcal {C}$ -circuits grow polynomially in size under multiplication and addition, that is, $\\operatorname{size}_\\mathcal {C}(f\\cdot g)\\le (\\operatorname{size}_\\mathcal {C}(f),\\operatorname{size}_\\mathcal {C}(g))$ .", "$\\operatorname{size}_\\mathcal {C}(f+g)\\le (\\operatorname{size}_\\mathcal {C}(f))+(\\operatorname{size}_\\mathcal {C}(g))$ .", "In particular, one can take $\\mathcal {C}$ to be sparse polynomials, depth-3 powering formulas (in characteristic zero), or roABPs.", "Then if ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ are computable by size-$t$ $\\mathcal {C}$ -circuits and have a $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ refutation of size-$s$ , then ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ have a tree-like $\\mathcal {C}$ -PC refutation of size-$(s,t,n)$ , which is $(s,t,n)$ -explicit given the IPS refutation.", "That the relevant classes obey these closure properties is mostly immediate.", "See for example fact:roABP:closure for roABPs.", "For depth-3 powering formulas, the closure under addition is immediate and for multiplication it follows from Fischer [19].", "Turning to the simulation, such an IPS refutation is an equation of the form $\\sum _j g_jf_j+\\sum _i h_i\\cdot (x_i^2-x_i)=1$ .", "Using the closure properties of $\\mathcal {C}$ , one can compute the expression $\\sum _j g_jf_j+\\sum _i h_i\\cdot (x_i^2-x_i)$ in the desired size, which yields the required (explicit) derivation of 1.", "Note that the above claim does not work for multilinear formulas, as multilinear polynomials are not closed under multiplication.", "That tree-like multilinear-formula-PC simulates multilinear-formula-IPS$_{\\text{LIN}}$ is more intricate, and is given in res:mult-form-PC:mult-form-lbIPS.", "The Polynomial Calculus proof system has received substantial attention since its introduction by Clegg, Edmonds, and Impagliazzo [11], typically when the complexity of the proofs are measured in terms of the number of monomials.", "In particular, Impagliazzo, Pudlák and Sgall [38] showed an exponential lower bound for the subset-sum axiom.", "[Impagliazzo, Pudlák and Sgall [38]] Let $\\mathbb {F}$ be a field of characteristic zero.", "Let ${\\overline{\\alpha }}\\in \\mathbb {F}^n$ , $\\beta \\in \\mathbb {F}$ and $A:=\\lbrace \\sum _{i=1}^n \\alpha _i x_i : {\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n\\rbrace $ be so that $\\beta \\notin A$ .", "Then $\\alpha _1x_1+\\cdots +\\alpha _nx_n-\\beta ,{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable and any PC refutation requires degree $\\ge {\\lceil {{n}{2}}\\rceil }+1$ and $\\exp (\\Omega (n))$ -many monomials." ], [ "roABP-PC", "The class of roABPs are a natural restricted class of algebraic computation that non-trivially goes beyond sparse polynomials.", "In proof complexity, roABP-PC was explored by Tzameret [76] (under the name of ordered formulas, a formula-variant of roABPs, but the results there apply to roABPs as well).", "In particular, Tzameret [76] observed that roABP-PC can be deterministically checked using the efficient PIT algorithm for roABPs due to Raz and Shpilka [61].", "Given the Impagliazzo, Pudlák and Sgall [38] lower bound for the subset-sum axiom (thm:IPS99), our roABP-IPS upper bound for this axiom (res:ips-ubs:subset:roABP), and the relation between IPS$_{\\text{LIN}}$ and tree-like PC (res:ips-vs-pc), we can conclude the following exponential separation.", "Let $\\mathbb {F}$ be a field of characteristic zero.", "Then $x_1+\\cdots +x_n+1,{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable, requires sparse-PC refutations of size-$\\exp (\\Omega (n))$ , but has $(n)$ -explicit $(n)$ -size roABP-IPS$_{\\text{LIN}}$ and tree-like roABP-PC refutations.", "This strengthens a result of Tzameret [76], who separated dag-like roABP-PC from sparse-PC.", "However, we note that it is not clear whether sparse-PC can be efficiently simulated by roABP-IPS$_{\\text{LIN}}$ ." ], [ "Multilinear Formula PC", "We now proceed to study algebraic proofs defined in terms of multilinear formulas, as explored by Raz and Tzameret [62], [63].", "We seek to show that the tree-like version of this system can simulate multilinear-formula-IPS$_{\\text{LIN}}$ .", "While tree-like $\\mathcal {C}$ -PC can naturally simulate $\\mathcal {C}$ -IPS$_{\\text{LIN}}$ if $\\mathcal {C}$ is closed under multiplication (res:ips-vs-pc), the product of two multilinear polynomials may not multilinear.", "As such, the simulation we derive is more intricate, and is similar to the efficient multilinearization results for multilinear formulas from sec:multilinearization:mult-form.", "We first define the Raz-Tzameret [62], [63] system (which they called fMC).", "Let $f_1({\\overline{x}}),\\ldots ,f_m({\\overline{x}})\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a system of polynomials.", "A multilinear-formula-PC$\\,^\\lnot $ refutation for showing that the system ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable is a multilinear-formula-PC refutation of ${\\overline{f}}({\\overline{x}}),{\\overline{x}}^2-{\\overline{x}},{\\overline{{\\lnot x}}}^2-{\\overline{{\\lnot x}}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ in the ring $\\mathbb {F}[x_1,\\ldots ,x_n,{\\lnot x}_1,\\ldots ,{\\lnot x}_n]$ , where `$\\circ $ ' denotes the entry-wise product so that ${\\overline{x}}\\circ {\\overline{{\\lnot x}}}=(x_1{\\lnot x}_1,\\ldots ,x_n{\\lnot x}_n)$ .", "That is, a multilinear-formula-PC$^\\lnot $ refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ is a multilinear-formula-PC refutation with the additional variables ${\\overline{{\\lnot x}}}:=({\\lnot x}_1,\\ldots ,{\\lnot x}_n)$ which are constrained so that ${\\lnot x}_i=1-x_i$ (so that `$\\lnot $ ' here is simply a modifier of the symbol `$x$ ' as opposed to being imbued with mathematical meaning).", "These additional variables are important, as without them the system is not complete.", "For example, in attempting to refute the subset-sum axiom $\\sum _i x_i+1,{\\overline{x}}^2-{\\overline{x}}$ in multilinear-formula-PC alone, one can never multiply the axiom $\\sum _i x_i+1$ by another (non-constant) polynomial as it would ruin multilinearity.", "However, in multilinear-formula-PC$^\\lnot $ one can instead multiply by polynomials in ${\\overline{{\\lnot x}}}$ and appropriately simplify.", "We now formalize this by showing that tree-like multilinear-formula-PC$^\\lnot $ can simulate multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ (which is complete (res:multilin:simulate-sparse)).", "We begin by proving a lemma on how the ${\\overline{{\\lnot x}}}$ variables can help multilinearize products.", "In particular, if we have a monomial $({\\overline{1}}-{\\overline{{\\lnot x}}})^{\\overline{a}}$ (which is meant to be equal to ${\\overline{x}}^{\\overline{a}}$ ) and multiply by ${\\overline{x}}^{\\overline{1}}$ we should be able to prove that this product equals ${\\overline{x}}^{\\overline{1}}$ modulo the axioms.", "Working in the ring $\\mathbb {F}[x_1,\\ldots ,x_d,{\\lnot x}_1,\\ldots ,{\\lnot x}_d]$ , and for ${\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}$ , $({\\overline{1}}-{\\overline{{\\lnot x}}})^{\\overline{a}}{\\overline{x}}^{\\overline{1}}-{\\overline{x}}^{\\overline{1}}=C({\\overline{x}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\;,$ for $C({\\overline{x}},{\\overline{z}})\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}}]$ where $C({\\overline{x}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})$ can be $(2^d)$ -explicitly derived from the axioms ${\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ in $(2^d)$ steps using tree-like multilinear-formula-PC.", "$({\\overline{1}}-{\\overline{{\\lnot x}}})^{\\overline{a}}{\\overline{x}}^{\\overline{1}}&={\\overline{x}}^{{\\overline{1}}-{\\overline{a}}}\\cdot ({\\overline{x}}-{\\overline{x}}\\circ {\\overline{{\\lnot x}}})^{\\overline{a}}\\\\&={\\overline{x}}^{{\\overline{1}}-{\\overline{a}}}\\cdot \\left(\\sum _{{\\overline{0}}\\le {\\overline{b}}\\le {\\overline{a}}} {\\overline{x}}^{{\\overline{a}}-{\\overline{b}}}(-{\\overline{x}}\\circ {\\overline{{\\lnot x}}})^{{\\overline{b}}}\\right)\\\\&={\\overline{x}}^{{\\overline{1}}-{\\overline{a}}}\\cdot \\left({\\overline{x}}^{\\overline{a}}+\\sum _{{\\overline{0}}<{\\overline{b}}\\le {\\overline{a}}} {\\overline{x}}^{{\\overline{a}}-{\\overline{b}}}(-{\\overline{x}}\\circ {\\overline{{\\lnot x}}})^{{\\overline{b}}}\\right)\\\\&={\\overline{x}}^{\\overline{1}}+\\sum _{{\\overline{0}}<{\\overline{b}}\\le {\\overline{a}}} {\\overline{x}}^{{\\overline{1}}-{\\overline{b}}}(-{\\overline{x}}\\circ {\\overline{{\\lnot x}}})^{{\\overline{b}}}\\\\&={\\overline{x}}^{\\overline{1}}+C({\\overline{x}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\;,$ where $C({\\overline{x}},{\\overline{z}})$ is defined by $C({\\overline{x}},{\\overline{z}}):=\\sum _{{\\overline{0}}<{\\overline{b}}\\le {\\overline{a}}} {\\overline{x}}^{{\\overline{1}}-{\\overline{b}}}(-{\\overline{z}})^{{\\overline{b}}}\\;.$ Now note that $C({\\overline{x}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})$ can be derived by tree-like multilinear-formula-PC.", "That is, the expression ${\\overline{x}}^{{\\overline{1}}-{\\overline{b}}}(-{\\overline{x}}\\circ {\\overline{{\\lnot x}}})^{{\\overline{b}}}$ is multilinear (as the product is variable disjoint) and in the ideal of ${\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ as ${\\overline{b}}>{\\overline{0}}$ , and is clearly a $(d)$ -size explicit multilinear formula.", "Summing over the $2^d-1$ relevant ${\\overline{b}}$ gives the result.", "We now show how to prove the equivalence of $g({\\overline{x}})$ and $g({\\overline{1}}-{\\overline{{\\lnot x}}})$ modulo ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}}$ , if $g$ is computable by a small multilinear formula, where we proceed variable by variable.", "Working in the ring $\\mathbb {F}[x_1,\\ldots ,x_n,{\\lnot x}_1,\\ldots ,{\\lnot x}_n]$ , if $g\\in \\mathbb {F}[{\\overline{x}}]$ is computable by a size-$t$ multilinear formula, than $g({\\overline{x}})-g({\\overline{1}}-{\\overline{{\\lnot x}}})=C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})\\;,$ for $C({\\overline{x}},{\\overline{z}})\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}}]$ where $C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})$ is derivable from ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}}$ in size-$(t,n)$ tree-like multilinear-formula-PC, which is $(t,n)$ -explicit given the formula for $g$ .", "We proceed to replace ${\\overline{1}}-{\\overline{{\\lnot x}}}$ with ${\\overline{x}}$ one variable at a time.", "Using $({\\overline{x}}_{\\le i},({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})$ to denote $(x_1,\\ldots ,x_i,1-{\\lnot x}_{i+1},\\ldots ,1-{\\lnot x}_{n})$ , we see that via telescoping that $g({\\overline{x}})-g({\\overline{1}}-{\\overline{{\\lnot x}}})&=g({\\overline{x}}_{\\le n},({\\overline{1}}-{\\overline{{\\lnot x}}})_{>n})-g({\\overline{x}}_{<1},({\\overline{1}}-{\\overline{{\\lnot x}}})_{\\ge 1})\\\\&=\\sum _{i=1}^n \\Big (g({\\overline{x}}_{\\le i},({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i}) - g({\\overline{x}}_{<i},({\\overline{1}}-{\\overline{{\\lnot x}}})_{\\ge i})\\Big )\\\\&=\\sum _{i=1}^n \\Big (g({\\overline{x}}_{<i},x_i,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i}) - g({\\overline{x}}_{<i},1-{\\lnot x}_i,({\\overline{1}}-{\\overline{{\\lnot x}}})_{> i})\\Big )\\;.$ Now note that $g({\\overline{x}}_{<i},y,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})$ is a multilinear polynomial, which as it is linear in $y$ can be written as $g({\\overline{x}}_{<i},y,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})=\\Big (g({\\overline{x}}_{<i},1,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\Big )y+g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\;.$ Thus, plugging in $x_i$ and $1-{\\lnot x}_i$ , $g({\\overline{x}}_{<i},x_i&,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},1-{\\lnot x}_i,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\\\&=\\left(\\Big (g({\\overline{x}}_{<i},1,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\Big )x_i+g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\right)\\\\&\\hspace{21.68121pt}-\\left(\\Big (g({\\overline{x}}_{<i},1,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\Big )(1-{\\lnot x}_i)+g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\right)\\\\&=\\Big (g({\\overline{x}}_{<i},1,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\Big )(x_i+{\\lnot x}_i-1)\\;.$ Plugging this into the above telescoping equation, $g({\\overline{x}})-g({\\overline{1}}-{\\overline{{\\lnot x}}})&=\\sum _{i=1}^n \\Big (g({\\overline{x}}_{<i},x_i,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i}) - g({\\overline{x}}_{<i},1-{\\lnot x}_i,({\\overline{1}}-{\\overline{{\\lnot x}}})_{> i})\\Big )\\\\&=\\sum _{i=1}^n \\Big (g({\\overline{x}}_{<i},1,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})-g({\\overline{x}}_{<i},0,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})\\Big )(x_i+{\\lnot x}_i-1)\\\\&=:C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})\\;.$ Clearly each $g({\\overline{x}}_{<i},b,({\\overline{1}}-{\\overline{{\\lnot x}}})_{>i})$ for $b\\in \\lbrace 0,1\\rbrace $ has a $(t)$ -size multilinear algebraic formula, so the entire expression $C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})$ can be computed by tree-like multilinear-formula-PC from ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}}$ explicitly in $(t,n)$ steps.", "Using the above lemma, we now show how to multilinearize a multilinear-formula times a low-degree multilinear monomial.", "Let $g,f\\in \\mathbb {F}[x_1,\\ldots ,x_n,y_1,\\ldots ,y_d]$ , where $g$ is computable by a size-$t$ multilinear formula and $y=\\prod _{i=1}^d y_i$ .", "Then $g({\\overline{1}}-{\\overline{{\\lnot x}}},{\\overline{1}}-{\\overline{{\\lnot y}}}){\\overline{y}}\\!\\:^{\\overline{1}}- \\operatorname{ml}(g({\\overline{x}},{\\overline{y}}){\\overline{y}}\\!\\:^{\\overline{1}})=C({\\overline{x}},{\\overline{y}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\;,$ where $C({\\overline{x}},{\\overline{y}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})$ can be derived from the axioms ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}}$ in $(2^d,t,n)$ steps using tree-like multilinear-formula-PC.", "Express $g$ as $g({\\overline{x}},{\\overline{y}})=\\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}}g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}$ in the ring $\\mathbb {F}[{\\overline{x}}][{\\overline{y}}]$ , so that each $g_{\\overline{a}}$ is multilinear.", "Then, $g({\\overline{1}}-{\\overline{{\\lnot x}}},{\\overline{1}}-{\\overline{{\\lnot y}}})\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}&=\\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}} g_{\\overline{a}}({\\overline{1}}-{\\overline{{\\lnot x}}})({\\overline{1}}-{\\overline{{\\lnot y}}})^{\\overline{a}}\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}\\\\\\multicolumn{2}{l}{\\text{appealing to {res:mult-form-PC:mon-mon} to obtain $({\\overline{1}}-{\\overline{{\\lnot y}}})^{\\overline{a}}{\\overline{y}}\\!\\:^{\\overline{1}}={\\overline{y}}\\!\\:^{\\overline{1}}+C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})$ for $C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})$ derivable in tree-like multilinear-formula-PC from ${\\overline{y}}\\circ {\\overline{{\\lnot y}}}$ in $(2^d)$ steps,}}\\\\&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{1}}-{\\overline{{\\lnot x}}})\\left({\\overline{y}}\\!\\:^{\\overline{1}}+C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\right)\\multicolumn{2}{l}{\\text{appealing to {res:mult-form-PC:form-form} to obtain $g_{\\overline{a}}({\\overline{1}}-{\\overline{{\\lnot x}}})=g_{\\overline{a}}({\\overline{x}})+B_{\\overline{a}}({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})$ for $B_{\\overline{a}}({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})$ derivable in tree-like multilinear-formula-PC from ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}}$ in $(t,n)$ steps,}}\\\\&=\\sum _{{\\overline{a}}} \\left(g_{\\overline{a}}({\\overline{x}})+B_{\\overline{a}}({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})\\right)\\cdot \\left({\\overline{y}}\\!\\:^{\\overline{1}}+C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\right)\\\\&=\\sum _{{\\overline{a}}} g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{1}}+\\sum _{\\overline{a}}\\Big ( B_{\\overline{a}}({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}}){\\overline{y}}\\!\\:^{\\overline{1}}+ g_{\\overline{a}}({\\overline{x}})C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\\\&\\hspace{72.26999pt}+B_{\\overline{a}}({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}})C_{\\overline{a}}({\\overline{y}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\Big )\\\\&=\\operatorname{ml}(g({\\overline{x}},{\\overline{y}}){\\overline{y}}\\!\\:^{\\overline{1}})+C({\\overline{x}},{\\overline{y}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})\\;,$ by defining $C$ appropriately, and as $\\operatorname{ml}(g({\\overline{x}},{\\overline{y}}){\\overline{y}}\\!\\:^{\\overline{1}})&=\\textstyle \\operatorname{ml}\\left(\\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}}g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{\\overline{a}}\\cdot {\\overline{y}}\\!\\:^{\\overline{1}}\\right)\\\\&=\\textstyle \\operatorname{ml}\\left(\\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}}g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{a}}+{\\overline{1}}}\\right)\\\\&=\\textstyle \\sum _{{\\overline{0}}\\le {\\overline{a}}\\le {\\overline{1}}}g_{\\overline{a}}({\\overline{x}}){\\overline{y}}\\!\\:^{{\\overline{1}}}\\;.$ By interpolation, it follows that for each exponent ${\\overline{a}}$ there are constants ${\\overline{\\alpha }}_{{\\overline{a}},{\\overline{\\beta }}}$ such that $g_{\\overline{a}}({\\overline{x}})=\\sum _{{\\overline{\\beta }}\\in \\lbrace 0,1\\rbrace ^d} \\alpha _{{\\overline{a}},{\\overline{\\beta }}}g({\\overline{x}},{\\overline{\\beta }})$ .", "From this it follows that $g_{\\overline{a}}$ is computable by a multilinear formula of size-$(t,2^d)$ .", "It thus follows that $C({\\overline{x}},{\\overline{y}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})$ is a sum of $2^d$ terms, each of which is explicitly derivable in $(2^d,t,n)$ steps in tree-like multilinear-formula-PC from ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}}$ (as the multiplications are variable-disjoint), and thus $C({\\overline{x}},{\\overline{y}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{y}}\\circ {\\overline{{\\lnot y}}})$ is similar derived by tree-like multilinear-formula-PC.", "By linearity we can extend the above to multilinearization of a multilinear-formula times a sparse low-degree multilinear polynomial.", "Let $g,f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be multilinear, where $g$ is computable by a size-$t$ multilinear formula and $f$ is $s$ -sparse and $\\deg f\\le d$ .", "Then $g({\\overline{1}}-{\\overline{{\\lnot x}}})\\cdot f({\\overline{x}}) - \\operatorname{ml}(g({\\overline{x}})\\cdot f({\\overline{x}}))=C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\;,$ where $C({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})$ can be derived from the axioms ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ in $(2^d,s,t,n)$ steps using tree-like multilinear-formula-PC.", "We now conclude by showing that tree-like multilinear-formula-PC$^\\lnot $ can efficiently simulate multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ .", "Recall that this proof system simply requires an IPS refutation that is linear in the non-boolean axioms, so that in particular $\\sum _j g_jf_j\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ for efficiently computable $g_j$ .", "Let $f_1,\\ldots ,f_m\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be degree at most $d$ multilinear $s$ -sparse polynomials which are unsatisfiable over ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Suppose that there are multilinear $g_j\\in \\mathbb {F}[{\\overline{x}}]$ computable by size-$t$ multilinear formula such that $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Then there is a tree-like multilinear-formula-PC$\\,^\\lnot $ refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ of size $(2^d,s,t,n,m)$ , which is $(2^d,s,t,n,m)$ -explicit given the formulas for the $f_j,g_j$ .", "In particular, if there is a size-$t$ multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ , then there is a tree-like multilinear-formula-PC$\\,^\\lnot $ refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ of size $(2^d,s,t,n,m)$ which is $(2^d,s,t,n,m)$ -explicit given the refutation of ${\\overline{f}},{\\overline{x}}^2-{\\overline{x}}$ and formulas for the $f_j$ .", "By the above multilinearization (res:mult-form-PC:form-sparse), there are $C_j\\in \\mathbb {F}[{\\overline{x}},{\\overline{z}},{\\overline{w}}]$ such that $g_j({\\overline{1}}-{\\overline{{\\lnot x}}})f_j({\\overline{x}})=\\operatorname{ml}(g_j({\\overline{x}})f_j({\\overline{x}}))+C_j({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\;.$ where $C_j({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})$ is derivable from ${\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ in $(2^d,s,t,n)$ steps of tree-like multilinear-formula-PC.", "Thus, as $g_j({\\overline{1}}-{\\overline{{\\lnot x}}})$ has a $(t)$ -size multilinear formula, in $(2^d,s,t,n,m)$ steps we can derive from ${\\overline{f}}({\\overline{x}}),{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}}$ , $\\sum _{j=1}^m \\left(g_j({\\overline{1}}-{\\overline{{\\lnot x}}})f_j({\\overline{x}}) - C_j({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\right)&=\\sum _{j=1}^m \\operatorname{ml}(g_j({\\overline{x}})f_j({\\overline{x}}))\\\\\\multicolumn{2}{l}{\\text{as $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ we have that$$\\sum _{j=1}^m \\operatorname{ml}(g_j({\\overline{x}})f_j({\\overline{x}}))=\\operatorname{ml}\\left(\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\right)=1\\;,$$where we appealed to linearity of multilinearization ({fact:multilinearization}), so that}}\\\\\\sum _{j=1}^m \\left(g_j({\\overline{1}}-{\\overline{{\\lnot x}}})f_j({\\overline{x}}) - C_j({\\overline{x}},{\\overline{x}}+{\\overline{{\\lnot x}}}-{\\overline{1}},{\\overline{x}}\\circ {\\overline{{\\lnot x}}})\\right)&=1\\;,$ yielding the desired refutation, where the explicitness is clear.", "The claim about multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ follows, as such a refutation induces the equation $\\sum _{i=1}^m g_j({\\overline{x}})f_j({\\overline{x}})\\equiv 1\\mod {{\\overline{x}}}^2-{\\overline{x}}$ with the appropriate size bounds.", "Given this efficient simulation of multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ by tree-like multilinear-formula-PC$^\\lnot $ (res:mult-form-PC:mult-form-lbIPS), our multilinear-formula-IPS$_{\\text{LIN}}$ refutation of the subset-sum axiom (res:ips-ubs:subset:mult-form), and the lower bound for sparse-PC of the subset-sum axiom (thm:IPS99), we obtain the following separation result.", "Let $\\mathbb {F}$ be a field of characteristic zero.", "Then $x_1+\\cdots +x_n+1,{\\overline{x}}^2-{\\overline{x}}$ is unsatisfiable, requires sparse-PC refutations of size-$\\exp (\\Omega (n))$ , but has $(n)$ -explicit $(n)$ -size multilinear-formula-IPS$_{\\text{LIN}}$ and tree-like multilinear-formula-PC$\\,^\\lnot $ refutations.", "This strengthens a results of Raz and Tzameret [62], [63], who separated dag-like multilinear-formula-PC$^\\lnot $ from sparse-PC.", "However, we note that it is not clear whether sparse-PC can be efficiently simulated by multilinear-formula-$\\text{IPS}_{\\text{LIN}^{\\prime }}$ ." ], [ "Explicit Multilinear Polynomial Satisfying a Functional Equation", "In sec:lbs-fn:deg we showed that any polynomial that agrees with the function ${\\overline{x}}\\mapsto {1}{\\left(\\sum _i x_i-\\beta \\right)}$ on the boolean cube must have degree $\\ge n$ .", "However, as there is a unique multilinear polynomial obeying this functional equation it is natural to ask for an explicit description of this polynomial, which we now give.", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be the unique multilinear polynomial such that $f({\\overline{x}})=\\frac{1}{\\sum _i x_i-\\beta }\\;,$ for ${\\overline{x}}\\in \\lbrace 0,1\\rbrace ^n$ .", "Then $f({\\overline{x}})=-\\sum _{k=0}^n \\frac{k!", "}{\\prod _{j=0}^{k}(\\beta -j)} S_{n,k}\\;,$ where $S_{n,k}:=\\sum _{S\\subseteq \\binom{[n]}{k}}\\prod _{i\\in S} x_i$ is the $k$ -th elementary symmetric polynomial.", "It follows from the uniqueness of the evaluations of multilinear polynomials over the boolean cube that $f({\\overline{x}})&=\\sum _{T\\subseteq [n]} f(\\mathbb {1}_{T})\\prod _{i\\in T} x_i \\prod _{i\\notin T} (1-x_i)\\multicolumn{2}{l}{\\text{where $\\mathbb {1}_{T}\\in \\lbrace 0,1\\rbrace ^n$ is the indicator vector of the set $T$, so that}}\\\\&=\\sum _{T\\subseteq [n]} \\frac{1}{\|T\|-\\beta }\\prod _{i\\in T} x_i \\prod _{i\\notin T} (1-x_i)\\;.$ Using this, let us determine the coefficient of $\\prod _{i\\in S} x_i$ in $f({\\overline{x}})$ , for $S\\subseteq [n]$ with $\|S\|=k$ .", "First observe that setting $x_i=0$ for $i\\notin S$ preserves this coefficient, so that ${\\mathrm {Coeff}}_{\\prod _{i\\in S} x_i}\\Big (f({\\overline{x}})\\Big )&={\\mathrm {Coeff}}_{\\prod _{i\\in S} x_i}\\left.\\left(\\sum _{T\\subseteq [n]} \\frac{1}{\|T\|-\\beta }\\prod _{i\\in T} x_i \\prod _{i\\notin T} (1-x_i)\\right)\\right\|_{x_i\\leftarrow 0, i\\in S}\\\\\\multicolumn{2}{l}{\\text{and thus those sets $T$ with $T\\lnot \\subseteq S$ are zeroed out,}}\\\\&={\\mathrm {Coeff}}_{\\prod _{i\\in S} x_i}\\left(\\sum _{T\\subseteq S} \\frac{1}{\|T\|-\\beta }\\prod _{i\\in T} x_i \\prod _{i\\in S\\setminus T} (1-x_i)\\right)\\\\&=\\sum _{T\\subseteq S} \\frac{1}{\|T\|-\\beta }{\\mathrm {Coeff}}_{\\prod _{i\\in S} x_i}\\left(\\prod _{i\\in T} x_i \\prod _{i\\in S\\setminus T} (1-x_i)\\right)\\\\&=\\sum _{T\\subseteq S} \\frac{1}{\|T\|-\\beta } (-1)^{k-\|T\|}\\\\&=\\sum _{j=0}^k \\binom{k}{j} \\frac{1}{j-\\beta } (-1)^{k-j}\\\\&=-\\frac{k!", "}{\\prod _{j=0}^k (\\beta -j)}\\;,$ where the last step uses the below subclaim.", "$\\sum _{j=0}^k \\binom{k}{j} \\frac{1}{j-\\beta } (-1)^{k-j}=-\\frac{k!", "}{\\prod _{j=0}^k (\\beta -j)}\\;.$ [Sub-Proof:]Clearing denominators, $\\prod _{i=0}^k (i-\\beta )\\cdot \\sum _{j=0}^k \\binom{k}{j} \\frac{1}{j-\\beta } (-1)^{k-j}&=\\sum _{j=0}^k \\binom{k}{j}(-1)^{k-j} \\prod _{i\\ne j} (i-\\beta )\\;.$ Note that the right hand side is a univariate degree $\\le k$ polynomial in $\\beta $ , so it is determined by its value on $\\ell \\in \\lbrace 0,\\ldots ,k\\rbrace $ (that $\\mathbb {F}$ has large characteristic implies that these values are distinct).", "Note that on these values, $\\sum _{j=0}^k \\binom{k}{j}(-1)^{k-j} \\prod _{i\\ne j} (i-\\ell )&=\\binom{k}{\\ell }(-1)^{k-\\ell } \\prod _{0\\le i<\\ell } (i-\\ell ) \\cdot \\prod _{\\ell <i\\le k} (i-\\ell )\\\\&=\\binom{k}{\\ell }(-1)^{k-\\ell } \\cdot (-1)^{\\ell } \\ell !", "\\cdot (k-\\ell )!\\\\&=(-1)^k k!\\;.$ Thus by interpolation $\\sum _{j=0}^k \\binom{k}{j}(-1)^{k-j} \\prod _{i\\ne j} (i-\\beta )=(-1)^k k!$ for all $\\beta $ , and thus dividing by $\\prod _{i=0}^k (i-\\beta )$ and clearing $-1$ 's yields the claim.", "This then gives the claim as the coefficient of $\\prod _{i\\in S}x_i$ only depends on $\|S\|=k$ .", "Noting that the above coefficients are all nonzero because $\\operatorname{char}(\\mathbb {F})>n$ .", "Thus, we obtain the following corollary by observing that degree and sparsity are non-increasing under multilinearization (fact:multilinearization).", "Let $n\\ge 1$ and $\\mathbb {F}$ be a field with $\\operatorname{char}(\\mathbb {F})>n$ .", "Suppose that $\\beta \\in \\mathbb {F}\\setminus \\lbrace 0,\\ldots ,n\\rbrace $ .", "Let $f\\in \\mathbb {F}[x_1,\\ldots ,x_n]$ be a polynomial such that $f({\\overline{x}})\\left(\\sum _i x_i-\\beta \\right)=1 \\mod {{\\overline{x}}}^2-{\\overline{x}}\\;.$ Then $\\deg f\\ge n$ , and $f$ requires $\\ge 2^n$ monomials." ] ]	1606.05050
[ [ "The distribution of minimum-weight cliques and other subgraphs in graphs\n with random edge weights" ], [ "Abstract We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions.", "For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form.", "The direct form extends to other subgraphs and other edge-weight distributions.", "We illustrate the clique results for various values of $k$ and $n$.", "The results may be applied to evaluate whether an observed minimum-weight copy of a graph $H$ in a network provides statistical evidence that the network's edge weights are not independently distributed but have some structure." ], [ "Introduction", "In this note we consider the distribution of the minimum weight copy of a fixed subgraph $H$ in a randomly edge weighted complete graph $K_n$ ; a natural special case that may be of particular interest is when $H$ is a $k$ -clique.", "This seemingly natural problem seems to be absent from the literature thus far.", "It can be viewed as a fixed-size version of the NP-complete Maximum Weighted Clique Problem (MWCP); the review article [3] on Maximum Clique includes discussion of MWCP algorithms (Section 5.3) and the performance of heuristics on random graphs (in Section 6.5).", "The same article reviews applications, including in telecommunications, fault diagnosis, and computer vision and pattern recognition.", "Wikipedia's article [5] on the Clique problem includes applications in chemistry, bioinformatics, and social networks, and many of these seem more naturally modelled as weighted than unweighted problems.", "Research on fast algorithms for MWCP in “massive graphs” arising in practice is ongoing; a recent example, [4], notes applications in telecommunications and biology (specifically, the study [11] of finding a 5-protein gene marker for Alzheimer's disease).", "The weight distribution of a minimum-weight subgraph $H$ of a randomly weighted network is a natural question in discrete mathematics and probability, and in addition has statistical ramifications that we think may be of importance in applied settings.", "Taking the clique for purposes of discussion, the distribution we seek would allow one to judge whether the weight of the smallest-weight clique in a given network provides statistical evidence that the network's edge weights are not independently, uniformly random.", "Because uniformity is not essential to our analysis (see Section ), an anomalous weight is evidence that the edge weights are not independent, i.e., that the network's weights have some structure.", "The statistics could alternatively be performed by repeated simulations of random networks matching the null hypothesis but our approach is preferable for the usual reasons that simulation is cumbersome, it does not provide rigorous results, and the number of simulations must be more than the reciprocal of the desired significance level (potentially quite large), and additionally because finding a minimum-weight clique in a graph is an NP-hard problem that each simulation would have to solve.", "We focus on the case where each edge $e$ in $G$ has an independent weight $X_e$ which is uniform in $[0,1]$ , and $H$ is a complete graph $K_k$ , and here we derive explicit (non-asymptotic) bounds.", "We also obtain asymptotic results for other graphs $H$ , and we extend both sets of results to other distributions.", "We only consider finding subgraphs $H$ of the complete graph $K_n$ , but the same approach may be applicable to other networks of interest.", "The density of a graph $H$ is defined as $\\mathrm {den}(H)=e(H)/v(H)$ where $e(H)$ and $v(H)$ denote the number of edges and vertices of $H$ , and $H$ is strictly balanced if ${\\mathrm {den}(H)>\\mathrm {den}(H^{\\prime })\\text{ for all strict subgraphs }H^{\\prime } \\subset H.}$ Theorem 1 Let $H$ be a fixed strictly balanced graph with $v$ vertices, $m$ edges, $$ automorphisms, and density $d=m/v$ .", "Let the edges of $K_n$ be given independent uniform $[0,1]$ edge weights, and let $W$ denote the minimum weight of a subgraph isomorphic to $H$ .", "Then, for any non-negative $z=z(n)$ asymptotically in $n$ , ${\\sf P}\\left(W\\ge \\frac{z}{n^{1/d}}\\right)&=\\exp \\left\\lbrace -\\frac{z^{m}}{m!\\,}\\right\\rbrace +o(1) .\\multicolumn{2}{l}{\\text{Also,}}\\\\{\\sf E}(W)&\\sim \\hat{\\mu }:= n^{-1/d} \\frac{(m!", "\\, )^{1/m}}{m}\\Gamma \\left(\\frac{1}{m}\\right) .$ Here $f(n) \\sim g(n)$ means that $f(n)/g(n) \\rightarrow 1$ as $n\\rightarrow \\infty $ , which we may also write as $f(n) = g(n) \\, (1+o(1))$ .", "In the case when $H$ is a clique we have made some effort to control the $o(1)$ error, to demonstrate that our approach is useful for reasonable problem sizes.", "Theorem 2 Fix $k \\ge 3$ , let the edges of $K_n$ be given independent uniform $[0,1]$ edge weights, and let $W$ denote the minimum weight of a clique $K_k$ .", "Let $w=\\frac{z}{n^{2/(k-1)}}\\text{ and }\\lambda =\\binom{n}{k}\\frac{w^{\\binom{k}{2}}}{{\\binom{k}{2}}!", "}\\sim \\frac{z^{\\binom{k}{2}}}{k!{\\binom{k}{2}}!}", "\\ \\ \\text{(as in (\\ref {1x}))}.$ Then, for $z \\le \\min \\left\\lbrace n^{2/{\\binom{k}{2}}},n^{2/(k-1)} \\exp (-\\frac{k-1}{k-2})\\right\\rbrace $ (equivalently, $w \\le \\min \\left\\lbrace n^{-2/k} , \\exp (-\\frac{k-1}{k-2}) \\right\\rbrace $ ), $\\left\| {\\sf P}\\left(W\\ge w =\\frac{z}{n^{2/(k-1)}}\\right) - \\exp (-\\lambda ) \\right\|& \\le \\, \\frac{8}{7} \\, \\frac{(k-2)}{\\binom{k}{2}!", "(k-1)!^2} \\:\\: \\frac{z^{\\binom{k}{2}+k-1}}{n} .$ Theorem REF derives tighter bounds for the clique than those of Theorem REF , but is presented later because it requires additional notation.", "The proofs use the Stein-Chen method.", "Section presents the method as applied to finding a low-weight clique, and establishes the explicit probability bounds of Theorem REF .", "Section outlines how explicit bounds could be obtained for other distributions, and other subgraphs $H$ .", "Section simplifies (but loosens) the bounds to give Theorem REF .", "Section applies the Stein-Chen method to strictly balanced graphs $H$ to obtain the asymptotic probability bounds and asymptotic expectation of Theorem REF .", "Section generalizes Theorem REF to non-uniform edge weight distributions, as Theorem REF , Section presents plots and tables for the application of Theorem REF to various clique sizes $k$ and graph sizes $n$ , giving an indication of where our results are effective and where they need improvement.", "The Conclusions in Section 8 recapitulate the results achieved and discuss where they might be applied and how they might be extended." ], [ "Stein-Chen Method", "We use a version of the Stein-Chen method given in [1].", "We will summarize the general formulation, which may become clearer when we then show how it applies in our case.", "The formulation characterizes the distribution of a value $X=\\sum _{\\alpha \\in I} X_\\alpha $ where $I$ is an arbitrary index set and each $X_\\alpha $ is a Bernoulli random variable, $X_\\alpha \\sim \\operatorname{Be}(p) , $ that is, $X_\\alpha $ is 1 with probability $p$ and 0 otherwise.", "For each $z \\in I$ there is a “neighborhood of dependence” $B_\\alpha \\subseteq I$ with the property that $X_\\alpha $ is independent of all the $X_\\beta $ for $\\beta $ outside of $B_\\alpha $ .", "With $\\lambda = {\\sf E}X$ , $Z \\sim \\operatorname{Po}(\\lambda )$ a Poisson random variable with mean $\\lambda $ , and $b_1 & := \\sum _{\\alpha \\in I} \\sum _{\\beta \\in B_\\alpha } p_\\alpha p_\\beta ,\\\\b_2 & := \\sum _{\\alpha \\in I} \\sum _{\\alpha \\ne \\beta \\in B_\\alpha } p_{\\alpha \\beta }\\text{, where } p_{\\alpha \\beta }= {\\sf E}(X_\\alpha X_\\beta ) ,$ the conclusion of [1] is that ${\\operatorname{TVD}(X,Z) \\le b_1+b_2 ,}$ where $\\operatorname{TVD}(X,Z)$ denotes the total variation distance between the two distributions, i.e., the maximum, over events $E$ , of the difference in the probabilities of $E$ under the two distributions.", "(The full theorem involves an additional term $b_3$ if there are weak dependencies, but we do not need this.", "Also, we have adjusted for our use of the standard definition of $\\operatorname{TVD}$ , where [1] defines $\\operatorname{TVD}$ as twice this.)", "In our case, with $G=K_n$ , we are interested in ${X=\\text{number of $k$-cliques of $G$ weighing $\\le w$.", "}}$ We are specifically interested in ${\\sf P}(X=0)$ i.e.", "the probability that there is no such clique.", "We have $X=\\sum _{\\alpha \\in I} X_\\alpha $ where the index set $I$ is the set of all $k$ -cliques of $G$ , ${I=\\binom{[n]}{k}}$ and ${X_\\alpha ={\\left\\lbrace \\begin{array}{ll}1, & \\mbox{if clique $\\alpha $ has weight $\\le w$}\\\\0, & \\mbox{otherwise} .\\end{array}\\right.", "}}$ Denoting the number of edges in a $k$ -clique by ${m= \\binom{k}{2} ,}$ each $X_\\alpha $ satisfies $ X_\\alpha \\sim \\operatorname{Be}(p) $ where, for $w \\le 1$ , ${p= p(w) := E(X_\\alpha )= \\frac{w^m}{m!}", ",}$ the probability that the sum of $m$ i.i.d.", "uniform $[0,1]$ variables is at most $w$ .", "The sum of i.i.d.", "uniform random variables has an Irwin-Hall distribution, whose CDF is well known (more on this in Section REF ), but it is not hard to see that when $w \\le 1$ this probability is given by (REF ).", "For a clique to have total weight $\\le w$ , first, each of its $m$ edges must have weight $\\le w$ .", "With i.i.d.", "uniform $[0,1]$ random edge weights, conditioned on the vector of $m$ edge weights lying in $[0,w]^m$ , the vector is a uniformly random point in this $m$ -dimensional cube, the event that the sum of its coordinates is $\\le w$ is the event that the point lies in a standard $m$ -dimensional simplex scaled by $w$ , and the volume of this simplex is $w^m/m!$ .", "Immediately, $\\lambda &= \\lambda (w) := {\\sf E}X = \\binom{n}{k} p= \\binom{n}{k} \\frac{w^m}{m!", "}\\in \\frac{n^k}{k!}", "\\frac{w^m}{m!}", "\\left[1-\\frac{k^2}{2n},1\\right] .$ The neighborhood of dependence $B_\\alpha $ is the set of cliques that share an edge with clique $\\alpha $ .", "Breaking this down more finely, into cliques sharing $\\ell $ vertices with $\\alpha $ , we see that $\\left\|B_\\alpha \\right\| &= \\sum _{\\ell =2}^{k} u_\\ell \\text{ where }u_\\ell =\\binom{k}{\\ell }\\binom{n-k}{k-\\ell } .$ Then, $b_1&=\\binom{n}{k} \\sum _{\\ell =2}^{k}u_\\ell p^2 .$ To calculate $b_2$ , suppose that two cliques (or indeed copies of any graph $H$ ) $\\alpha $ and $\\beta $ have $a$ edges in common and each has $b$ edges not in the other (with $a+b=m$ ).", "The event that $X_\\alpha X_\\beta =1$ , i.e., that both cliques weigh $\\le w$ , is equivalent to the shared edges weighing some $W_a \\le w$ , and both sets of unshared edges weighing $\\le w-W_a$ .", "As in (REF ), the CDF for the sum of $s$ edges to be at most $r$ is $F_s(r)=r^s/s!$ , so the corresponding density is $f_s(r)=r^{s-1}/(s-1)!$ .", "Thus, $p_{\\alpha \\beta }=p_{\\alpha \\beta }(\\alpha ,\\beta ,w)&= \\int _0^w f_a(w_a) (F_b(w-w_a))^2 \\, d {w_a} \\\\ &= \\int _0^w \\frac{1}{(a-1)!}", "(w_a)^{a-1}\\left( \\frac{1}{b!}", "(w-w_a)^{b} \\right)^2 \\, d {w_a} \\multicolumn{2}{l}{\\text{which by a change of variable to $t=w_a/w$ is}}\\\\&= \\frac{w^{a+2b}}{(a-1)!", "\\, (b!", ")^2} \\int _0^1 t^{a-1} (1-t)^{2b} \\, dt .\\multicolumn{2}{l}{\\text{By Euler's integral of the first kind,$B(x,y) = \\int _{0}^1 t^{x-1} (1-t)^{y-1}dt= \\left.", "{\\Gamma (x)\\Gamma (y)} \\right\\bad.", "{\\Gamma (x+y)}$,this is}}\\\\&= \\frac{w^{a+2b}}{(a-1)!", "\\, (b!", ")^2} \\cdot \\frac{(a-1)!", "(2b)!}{(a+2b)!}", "\\\\ &= \\frac{w^{a+2b}}{(a+2b)!}", "\\binom{2b}{b} .$ Cliques $\\alpha $ and $\\beta $ sharing $\\ell $ vertices share $a=\\binom{\\ell }{2}$ edges, while the number of edges unique to $\\beta $ is $ b = {m_\\ell }:= \\binom{k}{2}-\\binom{\\ell }{2} .$ Thus, from (), with () and (REF ), $b_2&= \\binom{n}{k} \\sum _{\\ell =2}^{k-1} u_\\ell \\, p_{\\alpha \\beta }\\big ( \\tbinom{\\ell }{2}, \\tbinom{k}{2}-\\tbinom{\\ell }{2}, w \\big ) \\\\ &= \\binom{n}{k} \\sum _{\\ell =2}^{k-1} u_\\ell \\, \\frac{w^{m+{m_\\ell }}}{(m+{m_\\ell })!}", "\\binom{2{m_\\ell }}{{m_\\ell }} .$ With ${Z \\sim \\operatorname{Po}(\\lambda )}$ the conclusion from [1], per (REF ), is that $\\left\|{\\sf P}(W \\ge w)- \\exp (-\\lambda )\\right\|&= \\left\|{\\sf P}(X=0)-{\\sf P}(Z=0)\\right\|\\le \\operatorname{TVD}(X,Z)\\le b_1+b_2.$ We have thus proved the following theorem: Theorem 3 Fix $k \\ge 3$ , let the edges of $K_n$ be given independent uniform $[0,1]$ edge weights, and let $W$ denote the minimum weight of a clique $K_k$ .", "Let $w=z/{n^{2/(k-1)}}$ , $\\lambda =\\binom{n}{k}\\frac{w^{\\binom{k}{2}}}{{\\binom{k}{2}}!", "}\\sim \\frac{z^{\\binom{k}{2}}}{k!{\\binom{k}{2}}!", "}$ (as in (REF )), and $b_1$ and $b_2$ be as given in (REF ) and () (calling in turn on (REF ) and (REF )).", "Then, for $z \\le n^{2/(k-1)}$ (equivalently, $w \\le 1$ ), $\\left\| {\\sf P}\\left(W\\ge w =\\frac{z}{n^{2/(k-1)}}\\right) - \\exp (-\\lambda )\\right\|& \\le b_1+b_2.", "$" ], [ "Extensions of Theorem ", "In this section, we illustrate how we could, with the aid of a computer, extend Theorem REF and obtain precise values for $b_1,b_2$ in more general circumstances.", "Given any edge weight distribution, we can at least in principle know the CDF $F_m(w)$ for a set of $m$ edges to have total weight at most $w$ , and the corresponding PDF $f_m(w)$ .", "We may then generalise (REF ) to $p=F_m(w)$ , where as usual $m=\\binom{k}{2}$ , as before define $\\lambda $ by (REF ) and $b_1$ by (REF ), and compute $b_2$ through (REF ), with $p_{\\alpha \\beta }$ given by (REF )." ], [ "Uniform edge weights", "With edge weights uniformly distributed as before, but removing Theorem REF 's restriction to $w \\le 1$ , the sum of $m$ uniform weights has Irwin-Hall distribution, with distribution and density functions $F_m(w) &= \\frac{1}{m!}", "\\sum _{i=0}^{\\left\\lfloor w \\right\\rfloor } (-1)^i \\binom{m}{i} (w-i)^m\\\\f_m(w) &= \\frac{1}{(m-1)!}", "\\sum _{i=0}^{\\left\\lfloor w \\right\\rfloor } (-1)^i \\binom{m}{i} (w-i)^{m-1} ;$ see [7], [8], [10].", "Given $k$ (thus $m$ ) and $w$ it is straightforward to calculate $F_m(w)$ and thus $p$ , $\\lambda $ , and $b_1$ .", "Calculating $b_2$ reduces, for each $\\ell $ in (REF ), to calculating $p_{\\alpha \\beta }$ through the integral in (REF ).", "Break the range of integration into intervals within which neither $w$ nor $w_a$ takes an integral value, by splitting at the points where either does take an integral value; since $w \\le m$ there are at most $2m$ such points.", "Within each subintegral, the integrand is the product of a polynomial in $w_a$ (from $f_a(w_a)$ ) and a polynomial in $(w-w_a)$ (from the square of $F_b(w-w_a)$ ).", "Each term of this can be integrated as an Euler integral of the first kind, as was done in going from () to (), or indeed expanded to a polynomial in $w_a$ (the powers are all bounded in terms of $k$ ) where each term can be integrated straightforwardly (even as an indefinite integral).", "In principle, then, we can extend Theorem (REF ) to all $w \\le m$ .", "Indeed, for each $w$ we can produce $p$ and $p_{\\alpha \\beta }$ , thus giving $b_1$ and $b_2$ as explicit functions of $n$ .", "(We cannot get an explicit function of $w$ , at least by the method above, because the partition of the integral into sub-integrals is different for each $w$ .)", "In practice, a naive implementation in Maple struggles with $k=10$ both in computation time and numerical stability." ], [ "Exponential edge weights", "We may also consider exponentially distributed edge weights.", "Without loss of generality we assume rate 1; anything else is a simple rescaling.", "The sum of $m$ rate-1 exponentials has Erlang distribution with well known distribution and density functions that can be stated in a variety of forms including $F_m(w) &= 1- e^{-w} \\, \\sum _{i=0}^{m-1} \\frac{w^i}{i!", "}\\\\f_m(w) &= \\frac{w^{m-1} e^{-w}}{(m-1)!", "};$ see for example [6] and [9].", "Again given $w$ it is straightforward to compute $p$ , $\\lambda $ , and $b_1$ , while computing $b_2$ requires computing $p_{\\alpha \\beta }$ for each $\\ell $ in the sum in (REF ) and the key is to evaluate (REF ).", "In this case the form of $F_b(w)$ means that the integrand is a finite sum (with length a function of $k$ ), each term of which has form a constant (with respect to $w$ ) times $(w_a)^a e^{-w_a}$ (coming from $f_a(w_a)$ ) times $(w-w_a)^r e^{-s (w-w_a)}$ (from the square of $F_b(w-w_a)$ ), for some integers $r$ and $s$ .", "These may in turn be expanded to terms of form $(w_a)^r e^{s \\, w_a}$ .", "Each of these is integrable (even as an indefinite integral); alternatively, each definite integral (over $w_a$ from 0 to $w$ ) is a lower incomplete gamma function.", "It thus seems feasible, if unenviable, to extend Theorem REF to provide explicit bounds for exponential edge weights.", "In practice, Maple has little difficulty with the calculations through $k=6$ but they quickly get more difficult: at $k=10$ is is challenging to calculate even a single $\\ell $ the corresponding function $p_{\\alpha \\beta }(w)$ ." ], [ "Subgraphs $H$ other than cliques", "Generalising Theorem REF to subgraphs $H$ other than cliques appears straightforward.", "The neighborhood of dependence of a given copy of $H$ needs more careful treatment, but this can be done in this non-asymptotic setting precisely as presented in Section for asymptotic calculations.", "The explicit calculations here do not even require that $H$ be strictly balanced, but it can be expected that the error term $b_2$ will be large if it is not (for the same reasons that the strict balance is generally required in application of the second moment method)." ], [ "Calculating bounds", "Given values of $k$ and $w$ , in practice one would apply Theorem REF using (REF ), calculating $b_1+b_2$ exactly from (REF ) and (), as indeed we do in Section .", "However, to characterize the quality of the estimate of ${\\sf P}(W \\ge w)$ we derive an upper bound on $b_1+b_2$ as a relatively simple (summation-free) function of $k$ and $w$ .", "We start with $b_2$ , the more difficult and (as we will see) larger of these two parameters.", "First, in lieu of (), we observe that for clique $\\beta $ to have weight at most $w$ , the ${m_\\ell }$ edges unique to it must have total weight $\\le w$ , and therefore ${\\sf P}(X_\\beta =1 \\mid X_\\alpha =1) \\le \\frac{w^{m_\\ell }}{{m_\\ell }!", "}\\le \\frac{w^{{m_\\ell }}}{(k-1)!}.", "$ In the first inequality we have used that since these edges are unique to $\\beta $ , conditioning on the weight of $\\alpha $ being at most $w$ is irrelevant.", "The second is simply because, over the range of $\\ell $ from 2 to $k-1$ , ${m_\\ell }\\ge \\binom{k}{2} - \\binom{k-1}{2} = k-1$ .", "Also, $u_\\ell & \\le \\binom{k}{\\ell }\\, \\frac{n^{k-\\ell }}{(k-\\ell )!}", ".$ Substituting (REF ) and (REF ) into (REF ), it follows that $ b_2 & \\le b_2^{\\prime } := \\binom{n}{k} \\sum _{\\ell =2}^{k-1} v_\\ell \\text{ where }v_\\ell = \\binom{k}{\\ell }\\, \\frac{n^{k-\\ell }}{(k-\\ell )!", "}\\, p\\, \\frac{w^{m_\\ell }}{(k-1)!}", ".$ Claim 4 Assuming that $w \\le \\min \\left\\lbrace n^{-2/k}, \\exp (-\\frac{k-1}{k-2}) \\right\\rbrace $ , over $2 \\le l \\le k-1$ , $v_\\ell $ is maximized at $\\ell =k-1$ .", "Proof We first claim that $v_\\ell $ is log-convex over $2 \\le \\ell \\le k-1$ , so that the maximum occurs either at $\\ell =2$ or $\\ell =k-1$ .", "For $k=3$ and 4 this is trivial.", "Otherwise, for $3\\le \\ell \\le k-2$ , ${\\frac{v_{\\ell +1}}{v_\\ell }=\\frac{(k-\\ell )^2}{w^\\ell n (\\ell +1)}}$ and $\\frac{v_\\ell ^2}{v_{\\ell -1}v_{\\ell +1}}&=\\frac{\\ell +1}{\\ell }\\cdot \\left(\\frac{k-\\ell +1}{k-\\ell }\\right)^2 w.$ To establish that $v_\\ell $ is log-convex over $2 \\le l \\le k-1$ it suffices to show that (REF ) is $\\le 1$ over $3 \\le l \\le k-2$ .", "Using $1+x \\le \\exp (x)$ , from (REF ) we have $\\frac{v_\\ell ^2}{v_{\\ell -1}v_{\\ell +1}}&\\le w\\exp \\left\\lbrace \\frac{1}{\\ell }+ \\frac{2}{k-\\ell }\\right\\rbrace \\le w \\exp \\left\\lbrace \\frac{k-1}{k-2}\\right\\rbrace \\le 1 ,$ where the final inequality is by hypothesis and the previous one because $\\frac{1}{\\ell }+ \\frac{2}{k-\\ell }$ is convex, so its maximum occurs at one of the extremes, either $\\ell =3$ or (in fact) $\\ell =k-2$ .", "Thus, $v_\\ell $ is log-convex and its maximum occurs either at $v_2$ or $v_{k-1}$ .", "However, $\\frac{v_2}{v_{k-1}} =\\frac{\\binom{k}{2} n^{k-2} w^{\\binom{k}{2}-\\binom{2}{2}} / (k-2)!", "}{\\binom{k}{k-1} n^1 w^{\\binom{k}{2}-\\binom{k-1}{2}} / (1!", ")}= \\frac{k-1}{2(k-2)!}", "(n w^{\\frac{1}{2} k})^{k-3}\\le 1,$ by the hypothesis that $w \\le n^{-2/k}$ .", "Thus, $v_2 \\le v_{k-1}$ , proving the claim.", "$\\Box $ It follows from (REF ) and Claim REF that $b_2 \\le b_2^{\\prime }& \\le \\binom{n}{k} (k-2) \\cdot v_{k-1}\\le \\binom{n}{k} (k-2) \\cdot \\binom{k}{k-1} \\, \\frac{n^1}{1!}", "\\, \\frac{w^{\\binom{k}{2}}}{\\binom{k}{2}!}", "\\, \\frac{w^{k-1}}{(k-1)!", "}\\\\& \\le \\frac{(k-2)}{\\binom{k}{2}!", "(k-1)!^2} \\: n^{k+1} w^{\\binom{k}{2}+k-1} .$ Recalling (REF ), using (REF ), and by analogy with (REF ), $ b_1 & \\le b_1^{\\prime } := \\binom{n}{k} \\sum _{\\ell =2}^{k} v^{\\prime }_\\ell \\text{ where }v^{\\prime }_\\ell = \\binom{k}{\\ell }\\, \\frac{n^{k-\\ell }}{(k-\\ell )!}", "\\, p^2 .$ From the definitions of $v_\\ell $ and $v^{\\prime }_\\ell $ in (REF ) and (REF ), for $2 \\le \\ell \\le k-1$ , $\\frac{v^{\\prime }_\\ell }{v_\\ell }= \\frac{p}{\\left.", "{w^{m_\\ell }} \\right\\bad.", "{(k-1)!", "}}= \\frac{\\left.", "{w^m} \\right\\bad.", "{m!}}{\\left.", "{w^{m_\\ell }} \\right\\bad.", "{(k-1)!", "}}\\le \\frac{1}{3} w^{\\binom{\\ell }{2}}\\le \\frac{1}{3} w\\le \\frac{1}{3}\\exp \\left(-\\frac{k-1}{k-2}\\right)\\le \\frac{1}{3e} .$ It follows (referring now to the definitions of $b^{\\prime }_2$ and $b^{\\prime }_1$ in (REF ) and (REF )) that the sum of all but the $\\ell =k$ term in $b^{\\prime }_1$ , which has no counterpart in $b^{\\prime }_2$ , is at most $\\frac{1}{3e} b^{\\prime }_2$ .", "Also, from (REF ), the $\\ell =k$ term of $b^{\\prime }_1$ is is small compared with its $\\ell =k-1$ term: $\\frac{v^{\\prime }_k}{v^{\\prime }_{k-1}}= \\frac{\\binom{k}{k} n^{0} / 0!", "}{\\binom{k}{k-1} n^1/1!", "}= \\frac{1}{k n}\\le \\frac{1}{9} .$ That is, the last term in the summation for $b^{\\prime }_1$ is at most $1/9$ times the second-last, therefore it is at most $1/9$ times the sum of all but the last, so that $ b_1 \\le b^{\\prime }_1 \\le \\left(1+\\frac{1}{9}\\right) \\frac{1}{3e} b^{\\prime }_2< \\frac{b^{\\prime }_2}{7} .$ Equation (REF ) of Theorem REF follows from (REF ), (REF ), and (REF ) of Theorem REF , since substituting $w = z/ n^{2/(k-1)}$ into the term $n^{k+1} w^{\\binom{k}{2}+k-1}$ of (REF ) gives $z^{\\binom{k}{2} +k-1}/{n}$ ." ], [ "Strictly balanced $H$", "Let $H$ be a strictly balanced graph with $v$ vertices, $m$ edges, and automorphism group $\\mathrm {aut}(H)$ of cardinality $$ .", "We apply the Stein-Chen method in parallel with Section .", "A bit more care is needed in defining the index set $I$ .", "Think of a copy of $H$ in $G$ as defined by a set of $v$ vertices of $G$ , together with a 1-to-1 mapping from these vertices to those of $H$ .", "The set $S$ of vertices is drawn from the collection $\\binom{[n]}{v}$ , the set of all $v$ -element subsets of $[n]$ , with $\\left\|\\binom{[n]}{v}\\right\| = \\binom{n}{v}$ .", "Taking the elements of $S$ in lexicographic order, the mapping into $V(H)$ is given by a permutation $\\pi $ of the values $1 \\ldots v$ , taken modulo the automorphism group of $H$ .", "Thus we may draw $\\pi $ from a set $L$ of permutations, with $ \\left\|L\\right\| = \\frac{v!", "}{} , $ $L$ consisting of one permutation from each equivalence class.", "Then, in analogy with Section 's equation (REF ), here ${I=\\binom{[n]}{v}\\times L .", "}$ With no change from before, the number of copies of $H$ of weight $\\le w$ is given by a random variable $X = \\sum _{\\alpha \\in I} X_\\alpha $ , the $X_\\alpha $ Bernoulli random variables.", "In parallel with (REF ), each $X_\\alpha $ has expectation ${p:= E(X_\\alpha ) = \\frac{w^m}{m!}", ",}$ assuming $w \\le 1$ , and in parallel with (REF ), ${\\lambda = \\lambda (w) := {\\sf E}X = \\left\|I\\right\| \\cdot p= \\binom{n}{v}\\frac{v!", "}{} \\frac{w^m}{m!}", ".", "}$ As before, we focus on $b_2$ and then treat $b_1$ .", "The structure of dependent events here is a bit subtle, and an example may be useful.", "Suppose $H$ is the 2-path with edges $\\left\\lbrace 1,2\\right\\rbrace $ and $\\left\\lbrace 2,3\\right\\rbrace $ .", "Its only automorphism is relabeling 123 to 321.", "On vertices, say $S=\\left\\lbrace 9, 11, 15\\right\\rbrace $ of $G$ , there are then 3 index sets: the set $\\left\\lbrace 9,11,15\\right\\rbrace $ in combination with any permutation chosen from $L=\\left\\lbrace 123, 231, 312\\right\\rbrace $ , corresponding respectively to paths 9–11–15, 11–15–9, and 15–9–11.", "(For instance in 231, we take the 2nd, 3rd, and 1st elements of the set in that order.)", "The permutations 321, 132, and 213 are eliminated from $L$ by automorphism, and correspond respectively to paths 15–11–9, 9–15–11, and 11–9–15 already listed.", "Suppose $\\alpha $ is vertex set $\\left\\lbrace 9,11,15\\right\\rbrace $ with permutation 123, giving path 9–11–15.", "Consider all possible dependent indices $\\beta $ on vertices $\\left\\lbrace 11,15,18\\right\\rbrace $ .", "With $\\pi =$ 123, $\\beta $ gives path 11–15–18, sharing edge 11–15 with $\\alpha $ .", "With $\\pi =$ 312, $\\beta $ gives path 18–11–15, again sharing edge 11–15 with $\\alpha $ .", "Finally, with $\\pi =$ 231, $\\beta $ gives path 15–18–11, sharing no edges with $\\alpha $ , and thus $\\beta \\notin B(\\alpha )$ .", "For $\\beta \\in B(\\alpha )$ , the pair $(\\alpha ,\\beta )$ describes an overlapping pair of copies of $H$ : a pair of labeled graphs $(V_1,E_1)$ and $(V_2,E_2)$ , each isomorphic to $H$ , $V_1,V_2 \\in \\binom{[n]}{v}$ , with $\\left\|E_1 \\cap E_2\\right\| \\ge 1$ and thus $\\ell :=\\left\|V_1 \\cap V_2\\right\| \\ge 2$ .", "Their union is a graph $F= (V_1 \\cup V_2, E_1 \\cup E_2)$ , and for strictly balanced $H$ , $\\mathrm {den}(F) > \\mathrm {den}(H)$ .", "(This is easy to show, well known, and the reason for introducing strict balance; one early reference is [2].", "Any such graph $F$ has at most $2v-2$ vertices, so up to the labeling of the vertices there are only finitely many possibilities.", "Let $ d^{\\prime }=d^{\\prime }(H)$ be the minimum density of all such graphs $F$ .", "For example, if $H=K_k$ then $d^{\\prime }$ is obtained for two $k$ -cliques sharing $k-1$ vertices, and $d^{\\prime }=\\frac{2\\binom{k}{2}-\\binom{k-1}{2}}{2k-(k-1)}=\\frac{(k-1)(k+2)}{2(k+1)}>d=\\frac{k-1}{2}$ .", "For copies $\\alpha $ and $\\beta $ both to have weight $\\le w$ , their union graph $F$ must have weight $\\le 2w$ , and as in (REF ) this event has probability $(2w)^{m(F)}/m(F)!$ , assuming $2w \\le 1$ .", "Thus, if the two copies overlap in $\\ell $ vertices, implying that $m(F) \\ge d^{\\prime }(2v-\\ell )$ , we have ${\\sf E}(X_\\alpha X_\\beta )& \\le \\frac{(2w)^{m(F)}}{m(F)!", "}\\le \\frac{(2w)^{d^{\\prime }(2v-\\ell )}}{(d^{\\prime }(2v-\\ell ))!}", ".$ It follows that $b_2&\\le \\left\|I\\right\| \\sum _{\\ell =2}^v\\binom{v}{\\ell }\\binom{n-v}{v-\\ell } \\left\|L\\right\|\\frac{(2w)^{d^{\\prime }(2v-\\ell )}}{(d^{\\prime }(2v-\\ell ))!", "}\\\\&= O\\left( n^v\\sum _{\\ell =2}^vn^{v-\\ell } w^{d^{\\prime }(2v-\\ell )}\\right)\\\\&= O\\left( \\sum _{\\ell =2}^v(n \\, w^{d^{\\prime }})^{2v-\\ell } \\right)\\\\&= O\\left( (n \\, w^{d^{\\prime }})^v\\right) = o(1),$ the last pair of inequalities holding subject to the condition that $ w=o(n^{-1/d^{\\prime }}) \\quad \\text{or equivalently} \\quad z = {w} \\, {n^{1/d}} = o(n^{1/d-1/d^{\\prime }}) .$ Note that unlike in () the sum here includes $\\ell =v$ but nonetheless capitalizes on $\\beta \\ne \\alpha $ from the definition of $b_2$ (see ()): the vertex sets of $\\beta $ and $\\alpha $ may be equal but the index sets themselves are different, so that the union graph $F$ is not isomorphic to $H$ , and therefore $\\mathrm {den}(F) \\ge d^{\\prime } >d$ .", "Now compare $b_1$ and $b_2$ from their definitions in (REF ) and ().", "For each term $(\\alpha ,\\beta )$ common to both sums, the summand in $b_1$ is $p_\\alpha p_\\beta = p^2 = O(w^{2m})$ , and (with $w \\le 1$ ) this is of smaller order than the corresponding summand in $b_2$ (see () and (REF )), which is of order $w^{m(F)}$ .", "($F$ is formed of two copies of $H$ sharing at least one edge, thus $m(F) \\le 2m-1$ .)", "Only the terms $(\\alpha ,\\alpha )$ are unique to $b_1$ , within $b_1$ they are fewer than the other terms, and all terms are equal, so they do not change the order of $b_1$ .", "It follows that $b_1=O(b_2) .", "$ We have established that, subject to (REF ), $ \|{\\sf P}(W\\ge w) - e^{-\\lambda }\| \\le b_1 + b_2 = O\\left( (n \\, w^{d^{\\prime }})^v\\right) = o(1) .$ With $z=w \\, n^{1/d}$ , and using the usual falling-factorial notation, observe from (REF ) that ${\\lambda = \\frac{(n)_{v}}{a} \\frac{w^m}{m!", "}\\sim \\frac{n^v}{} \\frac{w^m}{m!}", "= \\frac{z^m}{m!", "\\, } =: \\lambda ^{\\prime } .", "}$ By the intermediate value theorem, there is a point $\\lambda ^{\\prime \\prime } \\in [\\lambda ,\\lambda ^{\\prime }]$ at which $\\frac{d}{d\\lambda } e^{-\\lambda } = (\\exp (-\\lambda )-\\exp (-\\lambda ^{\\prime }))/(\\lambda -\\lambda ^{\\prime })$ .", "It follows that ${e^{-\\lambda ^{\\prime }}-e^{-\\lambda }}& = { (\\lambda ^{\\prime }-\\lambda ) \\cdot \\frac{d}{d \\lambda } \\exp (-\\lambda ) \\big \|_{\\lambda =\\lambda ^{\\prime \\prime }}}\\\\&= o(1) \\lambda ^{\\prime \\prime } \\cdot \\exp (-\\lambda ^{\\prime \\prime })=o(1) ,$ using that both $\\lambda $ and $\\lambda $ are $\\lambda ^{\\prime \\prime }(1+o(1))$ and that $\\lambda ^{\\prime \\prime } \\exp (-\\lambda ^{\\prime \\prime }) \\le 1/e$ for any $\\lambda ^{\\prime \\prime } \\ge 0$ .", "Now, in (REF ) substitute $w=z/n^{1/d}$ , yielding $ \\left\|{\\sf P}\\left(W\\ge \\frac{z}{n^{1/d}}\\right)-e^{-\\lambda ^{\\prime }}\\right\|&\\le \\left\|{\\sf P}\\left(W\\ge \\frac{z}{n^{1/d}}\\right)-e^{-\\lambda }\\right\| + \\left\|e^{-\\lambda }-e^{-\\lambda ^{\\prime }}\\right\| = o(1) .$ This completes the proof of (REF ) subject to (REF ), i.e., for $z=o(n^{1/d-1/d^{\\prime }})$ .", "To extend this to all $z=z(n)$ , we will observe that there is a weight threshold $w_0$ (see (REF ) below) where $w_0$ is large compared with $n^{-1/d}$ so that a cheap copy of $H$ (cheaper than $w_0$ ) is almost certainly present, but $w_0$ is small compared with $n^{-1/d^{\\prime }}$ so that an overlapping pair of cheap copies is almost certainly not present and thus the error bound $b_1+b_2$ is small.", "Values $w<w_0$ are controlled by the previous case, while for values $w>w_0$ , (REF ) holds trivially because all its terms are $o(1)$ .", "The same thresholding around $w_0$ will be used shortly in proving ().", "We now implement this idea.", "For any $0 < \\alpha < 1$ (throughout, $\\alpha =1/2$ will do), define $ w_0 &= n^{-(1-\\alpha )(1/d)-(\\alpha )(1/d^{\\prime })}\\quad \\text{and} \\quad z_0 = {w_0} \\, {n^{1/d}} = n^{\\alpha (1/d-1/d^{\\prime } )} .$ By construction, $ w_0 \\, n^{1/d^{\\prime }} = n^{(1-\\alpha )(1/d^{\\prime }-1/d)} = o(1)$ so that (REF ) is satisfied, while at the same time $ z_0 = w_0 \\, n^{1/d} = n^{(\\alpha )(1/d-1/d^{\\prime })}= \\omega (1) .$ A putative counterexample to Theorem REF equation (REF ) consists of an infinite sequence $z=z(n)$ for which the error terms are not $o(1)$ .", "Divide such a sequence into two subsequences according to whether $z \\le z_0$ or $z>z_0$ .", "We have just established that the subsequence with $z \\le z_0$ must give error terms $o(1)$ , so consider the subsequence with $z>z_0$ .", "Here, $\\left\|{\\sf P}\\left(W\\ge \\frac{z}{n^{1/d}}\\right)-\\exp \\left\\lbrace -\\frac{z^{m}}{m!\\,}\\right\\rbrace \\right\|\\le {\\sf P}\\left(W\\ge \\frac{z_0}{n^{1/d}}\\right)+\\exp \\left\\lbrace -\\frac{z^{m}}{m!\\,}\\right\\rbrace \\\\ \\le \\left( \\exp \\left\\lbrace -\\frac{z_0^m}{m!\\,}\\right\\rbrace +o(1) \\right) +\\exp \\left\\lbrace -\\frac{z^{m}}{m!\\,}\\right\\rbrace =o(1),$ where the application of (REF ) is justified by (REF ) and both exponential terms are small because of (REF ) and $z \\ge z_0$ .", "This completes the proof of (REF ) for all $z$ .", "We now turn to Theorem REF , Equation ().", "We have ${\\sf E}(W)&= \\int _{0}^{m}{\\sf P}(W\\ge w) \\, dw= \\int _{0}^{w_0}{\\sf P}(W\\ge w) \\, dw+ \\int _{w_0}^{m}{\\sf P}(W\\ge w) \\, dw .$ Setting $c &= \\frac{(n)_{v}}{a \\, m!", "}$ and substituting (REF ) into the first integral (as justified by (REF ) and (REF )) gives ${\\sf E}(W)&=\\int _{0}^{w_0}\\left( { {e^{-c w^m} +O\\left( { {(n\\,w^{d^{\\prime }})}^v } \\right) }} \\right) \\, dw+ \\int _{w_0}^{m} {\\sf P}(W\\ge w) \\, dw\\\\&=\\int _{0}^{w_0} e^{-c w^m} \\, dw+ \\int _{0}^{w_0}O\\left( { {(n\\,w^{d^{\\prime }})}^v } \\right) \\, dw+ \\int _{w_0}^{m} {\\sf P}(W\\ge w) \\, dw\\\\&= (1+o(1))n^{-1/d} \\frac{(a \\, m!", ")^{-1/m}}{m}\\; \\Gamma \\left( {\\frac{1}{m}} \\right)+ o(n^{-1/d}) + O(\\exp (-n^{\\Omega (1)})) ;$ we will prove () by considering each of the three integrals in (REF ) in turn.Landau notation does not normally presume the sign of the quantity in question, but in error expressions like (REF ) and () we mean for $\\Omega $ to denote a positive quantity.", "From (), Theorem REF , Equation () follows immediately.", "The first integral in (REF ) is the principal one.", "Let $x=c w^m$ so that $w=(x/c)^{1/m}$ and $dw=c^{-1/m} \\frac{1}{m} x^{\\frac{1}{m}-1} dx$ .", "Then $\\int _0^{w_0} e^{-cw^m} dw&= \\frac{c^{-1/m}}{m} \\:\\int _0^{c w_0^m} e^{-x} \\: x^{\\frac{1}{m}-1} dx\\\\& \\sim n^{-1/d} \\frac{(a \\, m!", ")^{-1/m}}{m}\\; \\Gamma \\left( {\\frac{1}{m}} \\right) .$ The asymptotic equality above follows from considering the two multiplicands separately.", "For the first multiplicand, $c^{-1/m}/m$ , we just observe that $c$ 's term $((n)_{v})^{-1/m} \\sim n^{-v/m} = n^{-1/d}$ .", "For the second multiplicand, the integral, the upper limit of integration is tending to infinity: $c \\, w_0^m$ is of order $n^v w_0^m= (n^{1/d} w_0)^m\\rightarrow \\infty $ , by (REF ).", "Thus the integral is asymptotic to $ \\int _0^{\\infty } e^{-x} \\: x^{\\frac{1}{m}-1} dx$ , which is equal to $ \\Gamma \\left( {\\frac{1}{m}} \\right) $ : it is an example of Euler's integral of the second kind, $\\Gamma (t)=\\int _{0}^\\infty x^{t-1}e^{-x}dx$ .", "For the second integral in (REF ), $\\int _{0}^{w_0} O\\left( { {(n\\,w_0^{d^{\\prime }})}^v } \\right) \\, dw&= O\\left( { w_0 \\cdot (n^{1/d^{\\prime }} w_0)^{d^{\\prime } v} } \\right)\\multicolumn{2}{l}{\\text{which, from $n^{1/d^{\\prime }} w_0 = o(1)$ by (\\ref {w0small}) and $v \\ge 2$ is}}\\\\&= O\\left( { w_0 \\cdot (n^{1/d^{\\prime }} w_0)^{2 d^{\\prime }} } \\right)= O\\left( { w_0 \\cdot (n \\, w_0^{d^{\\prime }})^2 } \\right) .$ To show that this is $o(n^{-1/d})$ as claimed in () means showing that, when multiplied by $n^{1/d}$ , it is $o(1)$ .", "This follows from $ n^{1/d} \\cdot w_0 \\cdot (n \\, w_0^{d^{\\prime }})^2&=n^{(1/d-1/d^{\\prime }) \\, (2 \\alpha d^{\\prime }+\\alpha -2 d^{\\prime })}= n^{-\\Omega (1)} = o(1) ,$ the final two inequalities holding if $2 \\alpha d^{\\prime }+\\alpha -2 d^{\\prime }<0$ , i.e., if $\\alpha < 2d^{\\prime }/(2d^{\\prime }+1)$ .", "Recall from (REF ) that $d^{\\prime }$ is the density of a graph $F$ describing an overlapping pair of copies of $H$ ; say $F$ has $m^{\\prime }$ edges and $v^{\\prime }$ vertices.", "Since $F$ is connected, $v^{\\prime } \\le m^{\\prime }+1$ and $d^{\\prime }=m^{\\prime }/v^{\\prime } \\ge m^{\\prime }/(m^{\\prime }+1)$ .", "Since there is at least one edge shared between the two copies and one edge unique to each copy, $m^{\\prime } \\ge 3$ , so $d^{\\prime } \\ge 3/4$ .", "Thus (REF ) holds for any $\\alpha < (2 \\cdot 3/4)/(2 \\cdot 3/4+1) = 3/5$ .", "Fixing for example $\\alpha =1/2$ (in all other parts of the proof, any $\\alpha $ strictly between 0 and 1 will do), the integrated error term is indeed $o(n^{-1/d})$ .", "For the third integral in (REF ), while ${\\sf P}(W \\ge w)$ of course decreases with $w$ , it is difficult for us to capitalize on this since our estimates cannot be applied for $w>n^{1/d^{\\prime }}$ where condition (REF ) is violated.", "If as for the second integral we reason through ${\\sf P}(W \\ge w) \\le {\\sf P}(W \\ge w_0)$ , the estimate is not good enough: we get an expression like (REF ) but with its integration range of $w_0$ replaced by $\\Theta (1)$ , giving $n^{-\\frac{1}{d} [(d^{\\prime }-d)(2-2\\alpha )-1]}$ , and if $d^{\\prime }$ and $d$ are nearly equal the exponent is not negative for any $\\alpha $ between 0 and 1.", "Claim 5 For any $0 < \\alpha < 1$ , with $w_0$ given by (REF ), ${\\sf P}(W > w_0) \\le \\exp (-n^{\\Omega (1)})$ .", "Proof First, we claim that an Erdős–Rényi random graph $G \\sim \\mathcal {G}(n, w_0)$ contains a copy of $H$ w.p.", "$>1/2$ .", "This can be obtained as a classical application of the second-moment method, but it also follows trivially from (REF ): The set of edges of weight $\\le w_0$ forms a random graph $G$ , the claim is that this subgraph includes a copy of $H$ w.p.", "$>1/2$ , and (REF ) says that with even higher probability (namely $1-o(1)$ ) there exists such a copy with additional properties (not only is each edge weight $\\le w_0$ , but the total is also $\\le w_0$ ).", "In particular, the set of edges of weight $\\le w_0$ forms a random graph $G_1 \\sim \\mathcal {G}(n, w_0)$ , and $G_1$ contains a copy of $H$ w.p.", "$>1/2$ .", "Now form a second, independent random graph $G_2 \\sim \\mathcal {G}(n, w_0)$ , each of whose edges has weight $\\le 2 w_0$ , by the following standard trick.", "For an edge appearing in $G_1$ , accept it into $G_2$ w.p.", "$w_0$ .", "For an edge not appearing in $G_1$ , accept it into $G_2$ with probability $1-w_0$ if its weight is between $w_0$ and $2w_0$ , and otherwise reject it.", "Note that in the second case, the weight is in the range $(w_0,2w_0)$ with probability $w_0/(1-w_0)$ , and then we take it only w.p.", "$1-w_0$ , for a net probability of $w_0$ .", "Thus, each edge appears in $G_2$ with probability exactly $w_0$ independent of $G_1$ and the other edges.", "As an Erdős–Rényi random graph, $G_2$ contains a copy of $H$ w.p.", "$>1/2$ , independently of $G_1$ , and in such a copy every edge has weight $\\le 2w_0$ .", "Repeat this process for graphs $G_3$ , ..., $G_k$ .", "With probability $\\ge 1-2^{-k}$ at least one of these graphs contains a copy of $H$ , and if so all its edges have weight $\\le k w_0$ for total weight $\\le k \\, m \\, w_0$ .", "To get the claim, given $\\alpha $ , choose smaller constants $0 < \\alpha ^{\\prime \\prime } < \\alpha ^{\\prime } < \\alpha $ .", "These give rise to corresponding values $w_0^{\\prime \\prime } < w_0 ^{\\prime } < w_0$ , and the ratios $\\Delta ^{\\prime \\prime } = w_0^{\\prime }/w_0^{\\prime \\prime }$ and $\\Delta ^{\\prime }=w_0/w_0^{\\prime }$ are both of order $n^{\\Omega (1)}$ .", "Given $G_1,G_2,\\ldots G_k$ with $k=\\Delta ^{\\prime \\prime }$ , we look for $H$ in the $k$ copies of $G(n,w_0^{\\prime \\prime })$ : we find such a copy of $H$ w.p.", "$1-2^{-\\Delta ^{\\prime \\prime }} = 1-\\exp (-n^{\\Omega (1)})$ , and any such copy has weight at most $k \\cdot m \\cdot w_0^{\\prime \\prime } \\le \\Delta ^{\\prime \\prime } \\cdot \\Delta ^{\\prime } \\cdot w_0^{\\prime \\prime } = w_0$ .", "$\\Box $ From Claim REF it is immediate that $0 <\\int _{w_0}^{m} {\\sf P}(W\\ge w) \\, dw&\\le m \\cdot {\\sf P}(W\\ge w_0)= m \\cdot \\exp (-n^{\\Omega (1)}) ,$ and we absorb the constant $m$ into the $\\Omega $ .", "This concludes analysis of the third integral in (REF ), and thus concludes the proof of Theorem REF , Equation ()." ], [ "Extension of Theorem ", "Theorem REF extends to distributions other than uniform on $[0,1]$ ; such extensions are common in situations where, intuitively, only edges with very small weights are relevant.", "Theorem 6 The conclusions of Theorem REF hold under the same hypotheses except that now the edge weights are i.i.d.", "copies of any non-negative random variable $X$ with finite expectation and a continuous distribution function $F$ that is differentiable from the right at 0, with slope $F^{\\prime }(0)=1$ .", "The assumption that $F^{\\prime }(0)=1$ is without loss of generality.", "As is standard, it can be extended to a variable $X$ for which $F^{\\prime }(0)=c$ , for any $c>0$ , simply by rescaling: applying the theorem to $c \\, X$ .", "To prove the theorem, couple $X$ with a random variable $U=F(X)$ .", "As the quantile of $X$ , $U$ is distributed uniformly on $[0,1]$ .", "As in (REF ), fix $0 < \\alpha < 1/2$ and define $w_0$ accordingly; recall that $w_0 \\rightarrow 0$ as $n \\rightarrow \\infty $ .", "Any edge of weight $w_U \\le 2 w_0 \\rightarrow 0$ in model $U$ has weight $w_X = F^{-1}(w_U)=w_U \\, (1+o(1))$ in model $X$ , since $F^{\\prime }(w_0) \\rightarrow 1$ .", "Symmetrically, any edge with weight $w_X \\le 2 w_0$ in model $X$ has weight $w_U = F(w_X)=w_X \\, (1+o(1))$ in model $U$ .", "Let $H_U$ and $H_X$ denote the lowest-weight copies of $H$ in the two models, and $W_U$ and $W_X$ the corresponding optimal weights.", "If $W_U \\le w_0$ then $W_X \\le W_U (1+o(1))$ , since $H_U$ would give such an $X$ -weight (each of its constituent edges has weight $w_U \\le w_0$ , thus asymptotically equal weight $w_X$ ) and the weight of $H_X$ may be even smaller.", "Taking the same hypothesis not the symmetric one as might be expected, if $W_U \\le w_0$ , then $W_X \\le 2 \\, w_0$ (this is why we introduced the factor of 2), in which case (now symmetrically) $W_U \\le W_X (1+o(1))$ .", "Thus, if $W_U \\le w_0$ — call this “event $E$ ” — then $ W_X/L \\le W_U \\le W_X L$ for some $L = L(w_0) = 1+o(1))$ in the limit $n \\rightarrow \\infty $ and thus $w_0 \\rightarrow 0$ .", "Let ${\\bar{E}}$ be the complementary event and recall from Claim REF that ${\\sf P}(E) = 1-\\exp (n^{-\\Omega (1)})$ .", "We now prove the distributional result (REF ) for $X \\sim F$ .", "Rewrite (REF ) as $ {\\sf P}(W_U \\ge w) &= f(w) + o(1) ,$ where $f(w) =\\exp \\left\\lbrace -\\frac{(w n^{1/d})^{m}}{m!\\,}\\right\\rbrace $ .", "Note that if we change $w$ by a factor $L=1+o(1)$ then the argument of the exponential changes by a factor $L^m=1+o(1)$ and thus, by (REF ), $f$ changes by an additive $o(1)$ , i.e., $f(wL)=f(w)+o(1)$ .", "Given any $w$ , and conditioning on event $E$ , ${\\sf P}(W_X \\ge w \\mid E)& \\ge {\\sf P}(W_U/L \\ge w) \\quad \\text{from (\\ref {wRatios})}\\\\& = {\\sf P}(W_U \\ge w \\, L)\\\\ &= f(w \\, L) + o(1) \\quad \\text{from (\\ref {Hasymp}) and (\\ref {Hasympf})}\\\\ &= f(w) + o(1) \\quad \\text{by the argument below (\\ref {Hasympf})} .$ Symmetrically, ${\\sf P}(W_X \\ge w \\mid E)\\le {\\sf P}(W_U \\ge w/L)= f(w) + o(1)$ and thus ${\\sf P}(W_X \\ge w \\mid E)= f(w) + o(1) .$ For any event $A$ , and any event $E$ of probability $1-o(1)$ it holds that ${\\sf P}(A) = {\\sf P}(A \\mid E) + o(1)$ , so here it follows that ${\\sf P}(W_X \\ge w)= {\\sf P}(W_X \\ge w \\mid E) + o(1)= f(w) + o(1) .$ This completes the proof of the distributional result.", "We now prove the expectation result () for $X \\sim F$ .", "Let $w_0$ and event $E$ be as above.", "Recall from Claim REF that ${\\sf P}({\\bar{E}}) = \\exp (-n^{\\Omega (1)})$ , and from Theorem REF that ${\\sf E}(W_U) = \\Theta (n^{-1/d})$ .", "By the law of total expectation, ${\\sf E}(W_U)&= {\\sf E}(W_U \\mid E) {\\sf P}(E) + {\\sf E}(W_U \\mid {\\bar{E}}) {\\sf P}({\\bar{E}})$ which in this case gives ${\\sf E}(W_U) = {\\sf E}(W_U \\mid E) (1+o(1))$ and thus $ {\\sf E}(W_U \\mid E) &= {\\sf E}(W_U) (1+o(1)) .$ Also by the law of total expectation, $ {\\sf E}(W_X)&= {\\sf E}(W_X \\mid E) {\\sf P}(E) + {\\sf E}(W_X \\mid {\\bar{E}}) {\\sf P}({\\bar{E}}) .$ As previously established, under event $E$ , $W_X = W_U (1+o(1))$ .", "It follows that $ {\\sf E}(W_X \\mid E)= {\\sf E}(W_U \\mid E) \\, (1+o(1))= {\\sf E}(W_U) \\, (1+o(1)),$ where in the second equality we have used (REF ).", "In the event ${\\bar{E}}$ , $W_X$ lies between 0 and the weight of a prescribed copy of $H$ (say, on vertices $1,\\ldots ,v$ , in that order).", "We may test for event $E$ by revealing edge weights up to $w_0$ in model $U$ , so for each edge we know either the exact weight (at most $w_0$ ), or know that the weight is $\\ge w_0$ .", "In the $X$ model, correspondingly, in the prescribed copy we know the edge weights up to $x_0 = F^{-1}(w_0)$ , or that the weight is $\\ge x_0$ .", "The expected weight of each edge is larger in the case that it is known to be $\\ge x_0$ and, even making this pessimistic assumption for every edge in the prescribed copy, we have $ 0 < {\\sf E}(W_X \\mid {\\bar{E}}) \\le m \\cdot {\\sf E}(X \\mid X>x_0)) = O(1) .$ (The conditional expectation ${\\sf E}(X \\mid X>x_0)$ cannot be infinite, as then by the law of total expectation the expectation of $X$ itself would be infinite, contradicting our hypothesis.)", "Substituting (REF ) and (REF ) into (REF ) gives ${\\sf E}(W_X)&= {\\sf E}(W_U) \\, (1+o(1)) + O(1) \\exp (-n^{\\Omega (1)})\\sim {\\sf E}(W_U) .$ This completes the proof of the extended expectation result." ], [ "Sample Results and Discussion", "In this section we discuss the quality of the results provided by Theorem REF , the lower and upper bounds — call them respectively $F^-(w)$ and $F^+(w)$ — on the CDF $F(w)$ of the weight $W$ of a minimum-weight $k$ -clique in a randomly edge-weighted complete graph of order $n$ .", "Figure REF shows $F^-$ and $F^+$ for $k=3$ , $n=100$ (left) and $k=3$ , $n=1,000$ (right).", "The vertical axis indicates cumulative probability; the horizontal axis indicates $w$ and is given in units of the estimated mean $\\hat{\\mu }$ given by () of Theorem REF .", "Here $m=\\binom{k}{2}$ , $d=m/k=(k-1)/2$ , and $a=k!$ , and by () of Theorem REF , ${\\sf E}(W) \\sim \\hat{\\mu }= \\frac{1}{n^{2/(k-1)}}\\frac{\\left( { \\binom{k}{2}}!", "k!", "\\right)^{1/\\binom{k}{2}}}{\\binom{k}{2}}\\: \\Gamma \\left( \\frac{1}{\\tbinom{k}{2}} \\right).$ For $k=3$ this gives ${\\sf E}(W)\\sim \\hat{\\mu }= \\frac{1}{n} \\frac{36^{1/3}}{3} \\Gamma (1/3)= (1.2878\\ldots ) \\, n^{-1}$ .", "Looking at Figure REF , for $k=3$ , $n=100$ we are getting good estimates in the lower tail, mediocre estimates for values of $w$ near the (estimated) mean, and poor estimates in the upper tail.", "For $k=3$ , $n=1,000$ we get good results in the lower tail and through the mean, but still poor results in the upper tail.", "Figure: CDFs fork=3k=3, n=100n=100 (left) andk=3k=3, n=1,000n=1,000 (right).The probability bounds from the theorem can be less than 0 or greater than 1.", "In the following discussion and in Figure REF , we have truncated both bounds to the range $[0,1]$ , in particular capping $F^+$ at 1.", "Also, because the error terms increase with $W$ , $F^-$ is not monotone increasing.", "In the following discussion we artificially force it to be (weakly) increasing, by replacing $F^-(w)$ with $\\max \\left\\lbrace w^{\\prime } \\le w \\colon F^-(w^{\\prime }) \\right\\rbrace $ .", "(To show the nature of the calculated bounds, though, this was not done in Figure REF .)", "Table: Measurements of the quality of lower and upper bounds on the CDF for variousvalues of kk and nn.The column “0.05” gives the value of F - (w)F^-(w) at the ww where F + (w)=0.05F^+(w)=0.05.The next three columns give the estimated mean μ ^\\hat{\\mu } of WWalong with F - (μ ^)F^-(\\hat{\\mu }) and F + (μ ^)F^+(\\hat{\\mu }).The column “0.95” gives F + (w)F^+(w) where F - (w)=0.95F^-(w)=0.95.The column “max gap” is the largest difference, over all ww, of F + (w)-F - (w)F^+(w)-F^-(w);in all cases, it was equal to the gap between the maximum of F - F^- and 1.These preliminary observations on Figure REF suggest a few measures of interest, compiled in Table REF .", "Let us explain the table and the results observed.", "Lower tail tests The most natural application of our results is to perform lower tail tests.", "It is easy to imagine contexts which would result in smaller-weight cliques than i.i.d.", "edge weights would produce, for example social networks in which if there is an affinity (modeled as a small weight) between A and B, and an affinity between B and C, then there is likely also to be an affinity between A and C. If we wish to show that values of $W$ as small as one observed occur with probability less than (say) 5% under the null hypothesis (that weights are i.i.d.", "uniform $(0,1)$ random variables), then that observation must be at or below the point $w$ where $F^+(w)=0.05$ .", "If at this point the lower bound is, say, $F^-(w)=0.02$ , and if the latter happens to be the truth (if $F^-$ rather than $F^+$ is a good approximation to $F$ here), then we require an observation at the 2% level to demonstrate significance at the 5% level, and this may prevent our doing so.", "We therefore take as a measure of lower-tail performance the value of $F^-(w)$ at the point $w$ where $F^+(w)=0.05$ , that is, $F^-(w)\|_{F^+(w)=0.05}$ .", "Show in the table column “$0.05$ ”, if this value is close to $0.05$ our estimates have given up little, and this is seen largely to be the case throughout the table.", "Of course we may be interested in one-tail significance tests at confidence levels $\\alpha < 0.05$ ($0.05$ being the largest threshold in common use).", "In this case we would hope for a small gap $(F^+(w)-F^-(w))\|_{F^+(w)=\\alpha }$ or small ratio $F^+(w)/F^-(w)\|_{F^+(w)=\\alpha }$ .", "Experimentally, both of these measures appear to be increasing functions of $\\alpha $ (i.e., decreasing as $\\alpha $ decreases), and thus the high quality of our bounds at $\\alpha =0.05$ implies the same for any $\\alpha \\le 0.05$ .", "Our bounds thus appear to be quite useful for lower tail tests.", "Mid-range values It is natural to check how good our probability bounds are for typical values of $W$ .", "Taking the estimated mean $\\hat{\\mu }$ of $W$ to stand in for a typical value, the table reports $\\hat{\\mu }$ and the lower and upper bounds $F^-(\\hat{\\mu })$ and $F^+(\\hat{\\mu })$ on the CDF $F(w)$ at this point.", "It can be seen that the quality of this mid-range estimate is poor for $k=3$ , $n=100$ , where the gap $F^+(\\hat{\\mu })-F^-(\\hat{\\mu })$ is above $0.1$ , but considerably better for $n=1,000$ and $n=10,000$ , where the gap is only about $0.01$ or $0.001$ respectively.", "Upper tail estimates Our results might also be applied to perform upper tail tests.", "This would be appropriate for contexts that would produce larger-weight cliques than i.i.d.", "edge weights would produce, perhaps a social “enmity” network in which if there is enmity (modeled as a small weight) between A and B, and enmity between B and C, then (on the basis that “the enemy of my enemy is my friend”) there is likely to be less enmity (larger weight) between A and C. Our first measure here is the obvious analogue of the lower-tail one: the value of $F^+$ at the point where $F^-$ is 0.95, $F^+(w)\|_{F^-(w)=0.95}$ .", "This is shown in the table in the column “0.95”.", "In many cases $F^-$ never even reaches $0.95$ , this measure is undefined, and the implication is that it is impossible to establish upper-tail significance with our method even at the 5% confidence level.", "However, once $n$ is large enough that the measure is defined, larger values of $n$ quickly lead to a small gap $(Fp(w)-F^-(w))\|_{F^+(w)=0.95}$ : if we can in principle establish upper-tail significance, we can often do so fairly efficiently.", "A second measure relevant here is the maximum gap between our lower and upper bounds, $\\max _{w >0}(F^+(w)-F^-(w))$ .", "Typically the maximum gap is achieved at the smallest point where $F^+(w)=1$ , and thus if the gap is larger than 0.05 (as for instance for $k=3$ , $n=1,000$ , for which the maximum gap is around $0.11$ ) an upper-tail significance at the 5% level cannot possibly be established.", "We chose this measure rather than the maximum of $F^-$ because this has the stronger interpretation that our probability estimates are this accurate across the range: for any observed $W$ , we can report lower and upper bounds on the corresponding CDF value (under the null hypothesis) no further apart than this gap.", "Parameter values and potential improvements For $k=3$ , values of $n$ as small as 100 give good estimates in the lower tail, and modestly good ones for typical values of $W$ , but no upper tail results.", "By $n=10,000$ , results are good across the range, with a maximum gap of around $0.02$ ; with $n=100,000$ this decreases to $0.004$ .", "For $k=4$ there is a similar pattern, with only slightly less sharp results for the same $n$ .", "For $k=10$ the picture is significantly different.", "Recall that our methods restrict us to estimating $F(w)$ for $w \\le 1$ and here that leaves us hopelessly far into the left tail.", "With $k=10$ and $n=100,000$ , the estimated mean of $W$ is $\\hat{\\mu }= 1.8856$ , while $F^+(1)$ is less than $10^{-12}$ .", "For $n=1,000,000$ , with $w\\le 1$ we still cannot access the estimated mean $\\hat{\\mu }= 1.13036$ , and $F^+(1) \\sim 0.00231$ : our methods would be useful for observations anomalously small at the 0.1% confidence level (to name a standard value near 0.00231), but nothing much above that.", "However, with $k=10$ , $W$ is concentrated near $\\hat{\\mu }$ , and thus, once $n$ is large enough that $\\hat{\\mu }$ falls below 1, our methods give good results across the range, as shown in the table.", "So, for $k=10$ , the problem we observe is with the range of validity of our estimates rather than their quality.", "If Theorem REF is extended as outlined in Section REF , the results might well be adequately tight.", "One other weakness of our methods is that the lower bound $F^-$ falls significantly short of 1 for $k=3$ and $k=4$ with $n=100$ and $n=1,000$ , revealed in the table's large “maximum gap” measures.", "It might be possible to improve this by applying the method used in proving Claim REF , but we have not attempted this." ], [ "Conclusions", "The object studied in this work is the distribution of a minimum-weight clique, or copy of a strictly balanced graph $H$ , in a complete graph $G$ with i.i.d.", "edge weights.", "Theorem REF provides asymptotic characterizations of the distribution and its mean for any strictly balanced graph $H$ , while Theorems REF and REF provide explicit (non-asymptotic) descriptions of the distribution for cliques.", "This distribution is a natural object of mathematical study, but also likely to have practical relevance, particularly for statistical determination that a given network's weights are not i.i.d.", "We look forward to seeing such applications of the work.", "Some potential applications would involve networks that are not complete graphs, and extending our results to such cases seems challenging, whether by extending to other infinite graph classes of graphs or by including a particular graph as part of the input.", "As presented, our explicit methods are for cliques, the uniform distribution and clique weights at most 1.", "However, as discussed in Section , it is easy to write down calculations for the uniform distribution and all clique weights, the exponential distribution, and probably other common distributions.", "In doing so we encountered computational challenges, but these seem surmountable if the incentive is more than just fleshing out a table.", "As noted in Section REF , extending explicit results to subgraphs $H$ other than cliques is straightforward.", "The quality of the results from our methods, as discussed in Section , is largely good, especially when we are interested in lower-tail results and relatively small cliques, or larger cliques in very large graphs.", "To improve the probability bounds in the middle range, near the mean, would seem to require an approach other than Stein-Chen, but we have no concrete alternative suggestions.", "Improving the bounds in the upper tail might be done, as suggested earlier, by applying the ideas of Claim REF ." ] ]	1606.04925
[ [ "A priori error estimates and computational studies for a Fermi\n pencil-beam equation" ], [ "Abstract We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three dimensional Fokker-Planck equation in space ${\\mathbf x}=(x,y,z)$ and velocity $\\tilde {\\mathbf v}=(\\mu, \\eta, \\xi)$ variables.", "The Fokker-Planck term appears as a Laplace-Beltrami operator in the unit sphere.", "The diffusion term in the Fermi equation is obtained as a projection of the FP operator onto the tangent plane to the unit sphere at the pole $(1,0,0)$ and in the direction of $ {\\mathbf v}_0=(1,\\eta, \\xi)$.", "Hence the Fermi equation, stated in three dimensional spatial domain ${\\mathbf x}=(x,y,z)$, depends only on two velocity variables ${\\mathbf v}=(\\eta, \\xi)$.", "Since, for a certain number of cross-sections, there is a closed form analytic solution available for the Fermi equation, hence an a posteriori error estimate procedure is unnecessary and in our adaptive algorithm for local mesh refinements we employ the a priori approach.", "Different numerical examples, in two space dimensions are justifying the theoretical results.", "Implementations show significant reduction of the computational error by using our adaptive algorithm." ], [ "Introduction", "This work is a further development of studies in [2]-[5] where adaptive finite element method was proposed for a reduction of computational cost in numerical approximation for pencil-beam equations.", "However, focusing in theoretical convergence and stability aspects, except some special cases with limited amount of implementation in [3] and [5], the detailed numerical tests were postponed to future works.", "Here, first we construct and analyze fully discrete schemes using both standard Galerkin and flux correcting streamline diffusion finite element methods for the Fermi pencil beam equation in three dimensions.", "We consider the direction of $x$ -axis as the penetration direction of the beam particles, in the two-dimensional transverse spatial domain $\\Omega _\\perp :=\\lbrace x_\\perp \\vert x_\\perp :=(y,z)\\rbrace $ , moving with velocities in $\\Omega _{\\mathbf {v}}:\\lbrace (\\eta , \\xi )\\rbrace $ , (where we have assumed $\\mu \\equiv 1$ ).", "We have derived our error estimates, in this geometry, while in numerical implementations we have also considered examples in lower dimensions.", "More specifically, our study concerns a “pencil beam” of neutral or charged particles that are normally incident on a slab of finite thickness at the spatial origin $(0,0,0)$ and in the direction of the positive $x$ -axis.", "The governing equation for the pencil beam problem is the Fermi equation which is obtained by two equivalent approaches (see [7]): either as an asymptotic limit of the linear Boltzmann equation as the transport cross-section $\\sigma _{tr}\\rightarrow 0$ and the total cross-section $\\sigma _t\\rightarrow \\infty $ , or as an asymptotic limit in Taylor expansion of angular flux with respect to the velocity where the terms with derivatives of order three or higher are ignored.", "This procedure rely on a approach that follows the Fokker-Planck development.", "The Boltzmann transport equation modeling the energy independent pencil beam process can be written as a two-point boundary value problem viz, $\\begin{split}\\mu \\frac{\\partial u}{\\partial x}+\\eta \\frac{\\partial u}{\\partial y}+\\xi \\frac{\\partial u}{\\partial z}=\\int _{S^2}\\sigma _s({\\mathbf {v}}\\cdot {\\mathbf {v}}^\\prime )[u({\\mathbf {x}},{\\mathbf {v}}^\\prime )-u({\\mathbf {x}},{\\mathbf {v}})]\\,d^2 {\\mathbf {v}}^\\prime , \\quad 0<x<1,\\end{split}$ where ${\\mathbf {x}}=(x,y,z)$ and ${\\mathbf {v}}=(\\mu , \\eta , \\xi )$ are the space and velocity vectors, respectively.", "The model problem concerns sharply forward peaked beam of particles entering the spatial domain at $x=0$ : $u(0, y, z, {\\mathbf {v}})=\\delta (y)\\delta (z)\\frac{\\delta (1-\\mu )}{2\\pi },\\qquad 0<\\mu \\le 1,$ which are demising leaving the domain at $x=1$ (or $x=L$ ), viz, e.g.", "$u(1, y, z, {\\mathbf {v}})=0,\\qquad -1\\le \\mu < 0.$ In the realm of the Boltzmann transport equation (REF ) an overview of the transport theory of charged particles can be found in [13].", "In this setting a few first coefficients in a Legendre polynomial expansion for $\\sigma _{s}$ and its integral $\\sigma _t$ are parameters corresponding to some physical quantities of vital importance.", "For instance, the slab width in the unit of mean free path: $\\sigma _t^{-1}$ , is the reciprocal of the total cross-section $\\sigma _t=2\\pi \\int _{-1}^1\\sigma _s (\\omega )\\, d\\omega .$ In the absorptionless case, the differential scattering cross section is given by $\\sigma _s(\\omega )=\\sigma _t\\sum _{k=0}^\\infty \\frac{2k+1}{4\\pi }c_kP_k(\\omega ),\\quad c_0=1,\\quad c_1=\\omega ,$ with $P_k(\\omega )$ being the Legendre polynomial of degree $k$ .", "The Fokker-Planck approximation to problem (REF ), is based on using spherical harmonics expansions and yields the following, degenerate type partial differential equation $\\begin{split}\\mu \\frac{\\partial u}{\\partial x}+\\eta \\frac{\\partial u}{\\partial y}+\\xi \\frac{\\partial u}{\\partial z}=\\frac{\\sigma _{tr}}{2}\\Delta _{\\mathbf {V}}u({\\mathbf {x}},{\\mathbf {v}}),\\qquad 0<x<1,\\end{split}$ associated with the same boundary data as (REF ) and (REF ), and with $\\Delta _{\\mathbf {V}}$ denoting the Laplace-Beltrami operator $\\Delta _{\\mathbf {V}}:=\\Big [\\frac{\\partial }{\\partial \\mu }(1-\\mu ^2) \\frac{\\partial }{\\partial \\mu }+\\frac{1}{1-\\mu ^2}\\frac{\\partial ^2}{\\partial \\phi ^2}\\Big ].$ Here, $\\phi $ is the angular variable appearing in the polar representation $\\eta =\\sqrt{1-\\mu ^2}\\cos \\phi $ , $\\xi =\\sqrt{1-\\mu ^2}\\sin \\phi $ .", "Further, $\\sigma _{tr}$ is the transport cross-section defined by $\\sigma _{tr}=\\sigma _t(1-\\omega ).$ A thorough exposition of the Fokker-Planck operator as an asymptotic limit is given by Pomraning in [14].", "Due to successive asymptotic limits used in deriving the Fokker-Planck approximation, it is not obvious that this approximation is sufficiently accurate to be considered as a model for the pencil beams.", "However, for sufficiently small transport cross-section $\\sigma _{tr}<<1$ , Fermi proposed the following form of, projected, Fokker-Planck model: $\\frac{\\partial u}{\\partial x}+\\eta \\frac{\\partial u}{\\partial y}+\\xi \\frac{\\partial u}{\\partial z}=\\frac{\\sigma _{tr}}{2}\\Big (\\frac{\\partial ^2}{\\partial \\eta ^2} +\\frac{\\partial ^2}{\\partial \\xi ^2} \\Big )u({\\mathbf {x}},{\\mathbf {v}}),\\qquad 0<x<1,$ with $u(0,y,z,\\eta ,\\xi )=\\delta (y) \\delta (z) \\delta (\\eta )\\delta (\\xi ).$ Fermi's approach is different from the asymptotic ones and uses physical reasoning based on modeling cosmic rays.", "Note that the Fokker-Planck operator on the right hand side of (REF ), i.e.", "(REF ), is the Laplacian on the unit sphere.", "The tangent plane to the unit sphere $S^2$ at the point $\\mu _0:=(1,0,0)$ is an ${\\mathcal {O}}(\\eta ^2+\\xi ^2)$ approximation to the $S^2$ at the vicinity of $\\mu _0$ .", "Extending $(\\eta , \\xi )$ to ${\\mathbb {R}}^2$ , the Fourier transformation with respect to $y, z, \\eta $ , and $\\xi $ , assuming constant $\\sigma _{tr}$ , yields the following exact solution for the angular flux $u(x,y,z,\\eta ,\\xi )=\\frac{3}{\\pi ^2\\sigma _{tr}^2x^4}\\exp \\Big [-\\frac{2}{\\sigma _{tr}}\\Big (\\frac{\\eta ^2+\\xi ^2}{x}-3 \\frac{y\\eta +z\\xi }{x^2}+3\\frac{y^2+z^2}{x^3}\\Big )\\Big ] .$ The closed form solution (REF ) was first derived by Fermi as referred in [16].", "Eyges [10] has extended this exact solution to the case of an $x$ -depending $\\sigma _{tr}=\\sigma _{tr}(x)$ .", "However, for the general case of $\\sigma _{tr}({\\mathbf {x}})=\\sigma _{tr}(x,y,z)$ , the closed-form analytic solution is not known.", "To obtain the scalar flux we integrate (REF ) over $(\\eta , \\xi )\\in {\\mathbb {R}}^2$ : $\\widetilde{U}(x,y,z)=\\int _{{\\mathbb {R}}^2}u(x,y,z,\\eta ,\\xi )\\, d\\eta d\\xi =\\frac{3}{2\\pi \\sigma _{tr}x^3}\\exp \\Big [-\\frac{3}{2\\sigma _{tr}}\\Big ( \\frac{y^2+z^2}{x^3}\\Big )\\Big ].$ Equation (REF ) satisfies the transverse diffusion equation $\\frac{\\partial \\widetilde{U}}{\\partial x}=\\frac{\\sigma _{tr} x^2}{2}\\Big (\\frac{\\partial ^2 \\widetilde{U}}{\\partial y^2}+\\frac{\\partial ^2\\widetilde{U}}{\\partial z^2}\\Big ),$ with $\\widetilde{U}(0,y,z)=\\delta (y)\\delta (z).$ Restricted to bounded phase-space domain, Fermi equation (REF ) can be written as the following “initial” boundary value problem $\\left\\lbrace \\begin{array}{ll}u_x+{\\mathbf {v}}\\cdot \\nabla {_\\perp }u=\\frac{\\sigma _{tr}}{2}\\Delta _{\\mathbf {V}}u \\qquad & \\mbox{in} \\quad \\Omega :=\\Omega _{\\mathbf {x}}\\times \\Omega _{\\mathbf {v}},\\\\\\nabla _{\\mathbf {v}} u(x, x_\\perp , {\\mathbf {v}})=0 & \\mbox{for} \\quad (x, x_\\perp , {\\mathbf {v}})\\in \\Omega _{\\mathbf {x}}\\times \\partial \\Omega _{\\mathbf {v}}, \\\\u(0, x_\\perp , {\\mathbf {v}})=u_0(x_\\perp , {\\mathbf {v}}) & \\mbox{for} \\quad (x_\\perp , {\\mathbf {v}})\\in \\Omega _{x_\\perp }\\times \\Omega _{\\mathbf {v}}=: \\Omega _\\perp , \\\\u(x, x_\\perp , {\\mathbf {v}})=0 & \\mbox{on} \\quad \\Gamma ^-_{\\tilde{\\beta }}\\setminus \\lbrace (0, x_\\perp , {\\mathbf {v}})\\rbrace ,\\end{array}\\right.$ where ${\\mathbf {v}}=(\\eta , \\xi )$ , $\\nabla {_\\perp }=(\\partial /\\partial y, \\partial /\\partial z)$ and $\\Gamma ^-_{\\tilde{\\beta }}:=\\lbrace (x, x_\\perp ,{\\mathbf {v}})\\in \\partial \\Omega ,{\\mathbf {n}}\\cdot \\tilde{\\beta }<\\,0 \\rbrace $ is the inflow boundary with respect to the characteristic line $\\tilde{\\beta }:=(1, {\\mathbf {v}}, 0,0)$ and ${\\mathbf {n}}$ is the outward unit normal to the boundary $\\partial \\Omega $ .", "Note that, to derive energy estimates, the associated boundary data (viewed as a replacement for the initial data) at $x=0$ is, in a sense, approximating the product of the Dirac's delta functions on the right hand side of (REF ).", "Assuming that we can use separation of variables, we may write the data function $u_0$ as product of two functions $f(x_\\perp )$ and $g({\\mathbf {v}})$ , $u_0(x_\\perp , {\\mathbf {v}})=f(x_\\perp ) g({\\mathbf {v}}).$ The regularity of these functions have substantial impact in deriving theoretical stabilities and are essential in robustness of implemented results." ], [ "The phase-space standard Galerkin procedure", "Below we introduce a framework that concerns a standard Galerkin discretization based on a quasi-uniform triangulation of the phase-space domain $\\Omega _\\perp :=\\Omega _{x_\\perp }\\times \\Omega _{\\mathbf {v}}:=I_\\perp \\times \\Omega _{\\mathbf {v}}$ , where $ I_\\perp : = I_y \\times I_z $ .", "This is an extension of our studies in two-dimensions in a flatland model [3].", "Previous numerical approaches are mostly devoted to the study of the one-dimensional problem see, e.g.", "[12] and [15].", "Here we consider triangulations of the rectangular domains $I_\\perp $ and $\\Omega _{\\mathbf {v}}:=I_\\eta \\times I_\\xi $ into triangles $\\tau _\\perp $ and $\\tau _{\\mathbf {v}}$ , and with the corresponding mesh parameters $h_\\perp $ and $h_{\\mathbf {v}}$ , respectively.", "Then a general polynomial approximation of degree $\\le r$ can be formulated in ${\\mathbb {P}}_r (\\tau ):={\\mathbb {P}}_r (\\tau _\\perp )\\otimes {\\mathbb {P}}_r (\\tau _{\\mathbf {v}})$ .", "These polynomial spaces are more specified in the implementation section.", "We will assume a minimal angle condition on the triangles $ \\tau _\\perp $ and $ \\tau _{\\mathbf {v}} $ (see e.g.", "[8]).", "Treating the beams entering direction $x$ similar to a time variable, we let ${\\mathbf {n}}:={\\mathbf {n}}(y,z, {\\mathbf {v}})$ be the outward unit normal to the boundary of the phase-space domain $\\Omega _{x_\\perp }\\times \\Omega _{\\mathbf {v}}$ at $(y,z, {\\mathbf {v}})\\in \\partial \\Omega _\\perp $ where $\\partial \\Omega _\\perp :=(\\partial \\Omega _{x_\\perp }\\times \\Omega _{\\mathbf {v}} )\\cup (\\Omega _{x_\\perp }\\times \\partial \\Omega _{\\mathbf {v}})$ .", "Now set $\\beta := ({\\mathbf {v}} ,0,0)$ and define the inflow (outflow) boundary as $\\Gamma ^{-(+)}_\\beta :=\\lbrace (x_\\perp ,{\\mathbf {v}})\\in \\Gamma :=\\partial \\Omega _{\\perp }:{\\mathbf {n}}\\cdot \\beta <0 \\, (>0)\\rbrace .$ We shall also need an abstract finite element space as a subspace of a function space of Sobolev type, viz: ${\\mathcal {V}}_{h,\\beta }\\subset H^1_\\beta :=\\lbrace w\\in H^1(I_\\perp \\times \\Omega _{\\mathbf {v}}): w=0\\, \\,\\mbox{on}\\,\\,\\Gamma ^{-}_\\beta \\rbrace .$ Now for all $w\\in H^1(I_\\perp \\times \\Omega _{\\mathbf {v}})\\cap H^r(I_\\perp \\times \\Omega _{\\mathbf {v}})$ a classical standard estimate reads as $\\inf _{\\chi \\in {\\mathcal {V}}_{h,\\beta }}\\vert \\vert w-\\chi \\vert \\vert _j\\le Ch^{\\alpha -j}\\vert \\vert w\\vert \\vert _\\alpha ,\\quad j=1,2, \\quad 1\\le \\alpha \\le r\\quad \\mbox{ and } \\,h=\\max (h_\\perp , h_{\\mathbf {v}}).$ To proceed let $\\tilde{u}$ be an auxiliary interpolant of the solution $u$ for the equation (REF ) defined by ${\\mathcal {A}}( u-\\tilde{u} , \\chi )_\\perp = 0,\\,\\, \\; \\; \\forall \\;\\chi \\in {\\mathcal {V}}_{h,\\beta },$ where ${\\mathcal {A}}( u, w)_\\perp =(u_x, w)_{\\Omega _{\\perp }}+({\\mathbf {v}}\\cdot \\nabla _\\perp u, w)_{\\Omega _{\\perp }},$ and $(\\cdot , \\cdot )_\\perp := (\\cdot , \\cdot )_{\\Omega _{\\perp }}=(\\cdot , \\cdot )_{I_\\perp \\times \\Omega _ { \\mathbf {v}}}$ .", "With these notation the weak formulation for the problem (REF ) can be written as follows: for each $ x \\in (0, L] $ , find $ u(x, \\cdot ) \\in H^1_{\\beta }$ such that, $\\left\\lbrace \\begin{array}{ll}{\\mathcal {A}}( u, \\chi )_\\perp +\\frac{1}{2} (\\sigma _{tr}\\nabla _{\\mathbf {v}} u, \\nabla _{\\mathbf {v}}\\chi )_\\perp =0 \\qquad &\\forall \\chi \\in H^1_{\\beta }, \\\\u(0,x_\\perp ,{\\mathbf {v}})=u_0(x_\\perp ,{\\mathbf {v}}) & \\,\\mbox{for}\\,\\,(x_\\perp ,{\\mathbf {v}})\\in \\Gamma ^+_\\beta , \\\\u(x,x_\\perp ,{\\mathbf {v}})=0 & \\,\\mbox{on}\\,\\,\\Gamma ^{-}_\\beta \\setminus \\lbrace (0,x_\\perp ,{\\mathbf {v}})\\rbrace .\\end{array}\\right.$ Our objective is to solve the following finite element approximation for the problem (REF ): for each $ x \\in (0, L] $ , find $u_h (x, \\cdot )\\in {\\mathcal {V}}_{h,\\beta }$ such that, $\\left\\lbrace \\begin{array}{ll}{\\mathcal {A}}( u_h, \\chi )_\\perp +\\frac{1}{2}(\\sigma _{tr}\\nabla _{\\mathbf {v}} u_h, \\nabla _{\\mathbf {v}}\\chi )_\\perp =0 \\qquad &\\forall \\chi \\in {\\mathcal {V}}_{h,\\beta }, \\\\u_h(0,x_\\perp ,{\\mathbf {v}})=u_{0,h}(x_\\perp ,{\\mathbf {v}})& \\,\\mbox{for}\\,\\,(x_\\perp ,{\\mathbf {v}})\\in \\Gamma ^+_\\beta , \\\\u_h(x,x_\\perp ,{\\mathbf {v}})=0 & \\,\\mbox{on}\\,\\,\\Gamma ^{-}_\\beta \\setminus \\lbrace (0,x_\\perp ,{\\mathbf {v}})\\rbrace ,\\end{array}\\right.$ where $ u_{0,h} (x_\\perp , \\mathbf {v} ) = \\tilde{u} (0, x_\\perp , \\mathbf {v} ) $ ." ], [ "A fully discrete scheme", "For a partition of the interval $[0, L]$ into the subintervals $I_{m}:=(x_{m-1}, x_m), m=1, 2, \\ldots , M$ with $k_m:=\\vert I_m\\vert := x_m-x_{m-1}$ , a finite element approximation $U$ with continuous linear functions $ \\psi _m (x) $ on $I_m$ can be written as: $u_h (x, x_\\perp , {\\mathbf {v}})=U_{m-1}(x_\\perp , {\\mathbf {v}})\\psi _{m-1}(x)+U_{m}(x_\\perp , {\\mathbf {v}})\\psi _{m}(x),$ where $x_\\perp :=(y, z)$ and $\\psi _{m-1}(x)=\\frac{x_m-x}{k_m}, \\qquad \\psi _m (x)=\\frac{x-x_{m-1}}{k_m}.$ Hence, the setting (REF )-(REF ) may be considered for an iterative, e.g.", "backward Euler, scheme with continuous piecewise linear or discontinuous (with jump discontinuities at grid points $x_m$ ) piecewise linear functions for whole $I_x=[0,L]$ .", "To proceed we consider a normalized, rectangular domain $\\Omega _{\\mathbf {V}}$ for the velocity variable ${\\mathbf {v}}$ , as $(\\eta ,\\xi )\\in [-1,1]\\times [-1,1]$ and assume a uniform, “central adaptive” discretization mesh viz: $\\Omega _{\\mathbf {v}}^N:=\\left\\lbrace {\\mathbf {v}}_{i,j}\\subset \\Omega _{\\mathbf {v}}\\Big \\vert {\\mathbf {v}}_{i,j}=(\\eta _i, \\xi _j):=\\Big (\\sin \\frac{i \\pi }{ 2n},\\sin \\frac{j\\pi }{ 2n}\\Big ),\\,\\, i,j=0, \\pm 1,\\ldots \\pm n \\right\\rbrace ,$ where $N=(2n+1)^2$ .", "Further we assume that $U$ has compact support in $\\Omega _{\\mathbf {V}}$ .", "By a standard approach one can show that, for each $m=1, 2,\\ldots , M$ , a finite element or finite difference solution $U_m^N$ obtained using the discretization (REF ) of the velocity domain $\\Omega _{\\mathbf {v}}$ , satisfies the $L_2(\\Omega _{\\mathbf {v}})$ error estimate $\\vert \\vert {U_m-U_m^N}\\vert \\vert _{L_2(\\Omega _{\\mathbf {v}})}\\le \\frac{C}{N^2}\\vert \\vert {D^2_{\\mathbf {V}}U_m(x_\\perp , \\cdot )}\\vert \\vert _{L_2(\\Omega _{\\mathbf {v}})}.$ Now we introduce a final, finite element, discretization using continuous piecewise linear basis functions $\\varphi _j(x_\\perp )$ , on a partition ${\\mathcal {T}}_h$ of the spatial domain $\\Omega _{x_\\perp }$ , on a quasi-uniform triangulation with the mesh parameter $h$ and obtain the fully discrete solution $U_m^{N,h}$ .", "We introduce discontinuities on the direction of entering beam (on the $x$ -direction).", "We also introduce jumps appearing in passing a collision site; say $x_m$ , as the difference between the values at $x_m^-$ and $x_m^+$ : $\\Big [U_m\\Big ]:=U_m^+-U_m^-, \\qquad U_m^{\\pm }:=\\lim _{s\\rightarrow 0}U(x_m\\pm s, x_\\perp , {\\mathbf {v}}).$ Due to the hyperbolic nature of the problem in $x_\\perp $ , for the solutions in the Sobolev space $H^{k+1}(\\cdot , x_\\perp , {\\mathbf {v}})$ , (see Adams [1] for the exact definitions of the Sobolev norms and spaces) the final finite element approximation yields an $L_2(\\Omega _{x_\\perp })$ error estimate viz, $\\vert \\vert {U_m^N-U_m^{N,k}}\\vert \\vert _{L_2(\\Omega _{x_\\perp })}\\le C h^{k+1/2}\\vert \\vert {U_m^N(x_\\perp , \\cdot )}\\vert \\vert _{H^{k+1}(\\Omega _{x_\\perp })}.$ To be specific, for each $m$ and each ${\\mathbf {v}}_{i,j}\\in \\Omega _{\\mathbf {v}}$ we obtain a spatially continuous version of the equations system (REF ) where, for $u$ , we insert $U_m(x_\\perp , {\\mathbf {v}}_{i,j})=\\sum _{k=1}^K U_{m,k}({\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ).$ Thus for each ${\\mathbf {v}}_{i,j}\\in \\Omega _{\\mathbf {v}}$ a variational formulation for a space-time like discretization in $(x, x_\\perp )$ of (REF ) reads as follows: find $U\\in {\\mathcal {V}}_{h,\\beta }$ such that $\\begin{split}\\int _{I_m}\\int _{\\Omega _{x_\\perp }} & U_x(x,x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp dx+\\int _{I_m}\\int _{\\Omega _{x_\\perp }}{\\mathbf {v}}_{i,j}\\cdot \\nabla _\\perp U(x,x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp dx\\\\&= \\int _{I_m}\\int _{\\Omega _{x_\\perp }} \\frac{\\sigma _{tr}}{2}\\Delta _ {\\mathbf {V}}U(x,x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp dx, \\qquad \\forall \\,\\,\\varphi _k\\in {\\mathcal {V}}_{h,\\beta },\\end{split}$ where ${\\mathcal {V}}_{h,\\beta }:=\\lbrace w\\in {\\mathcal {V}}_{\\beta }\\vert w_{ \\vert _\\tau } \\in \\mathbb {P}_1 ( \\tau ) , w \\,\\, \\mbox{is continuous} \\rbrace .$ This yields $\\begin{split}\\int _{\\Omega _{x_\\perp }} &\\Big (U_m(x_\\perp , {\\mathbf {V}}_{i,j})-U_{m-1}(x_\\perp , {\\mathbf {v}}_{i,j})\\Big )\\varphi _k(x_\\perp ) dx_\\perp +\\frac{k_m}{2} \\int _{\\Omega _{x_\\perp }}{\\mathbf {v}}_{i,j}\\cdot \\nabla _\\perp U_m(x_\\perp , {\\mathbf {V}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp \\\\&+\\frac{k_m}{2} \\int _{\\Omega _{x_\\perp }}{\\mathbf {v}}_{i,j}\\cdot \\nabla _\\perp U_{m-1}(x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp \\\\&=\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\int _{\\Omega _{x_\\perp }}\\Delta _ {\\mathbf {v}}U_m(x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp +\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\int _{\\Omega _{x_\\perp }}\\Delta _ {\\mathbf {v}}U_{m-1}(x_\\perp , {\\mathbf {v}}_{i,j})\\varphi _k(x_\\perp ) dx_\\perp .\\end{split}$ Such an equation would lead to a linear system of equations which in compact form can be written as the following matrix equation $\\begin{split}MU_m({\\mathbf {v}}_{i,j})& -MU_{m-1}({\\mathbf {v}}_{i,j})+\\frac{k_m}{2} C_{{\\mathbf {v}}_{i,j}} U_m({\\mathbf {v}}_{i,j})+\\frac{k_m}{2} C_{{\\mathbf {v}}_{i,j}} U_{m-1}({\\mathbf {v}}_{i,j})\\\\&=\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Delta _ {\\mathbf {v}}MU_m({\\mathbf {v}}_{i,j})+\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Delta _ {\\mathbf {v}}MU_{m-1}({\\mathbf {v}}_{i,j})\\end{split}$ Now considering ${\\mathbf {v}}$ -continuous version of (REF ): $\\begin{split}MU_m({\\mathbf {v}})& -MU_{m-1}({\\mathbf {v}})+\\frac{k_m}{2} C_{{\\mathbf {V}}} U_m({\\mathbf {v}})+\\frac{k_m}{2} C_{{\\mathbf {V}}} U_{m-1}({\\mathbf {v}})\\\\&=\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Delta _ {\\mathbf {V}}MU_m({\\mathbf {v}})+\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Delta _ {\\mathbf {V}}MU_{m-1}({\\mathbf {v}}),\\end{split}$ we may write $U_m(x_\\perp , {\\mathbf {v}})=\\sum _{k=1}^K \\sum _{j=1}^J U_{m,k, j}\\chi _{j}({\\mathbf {v}})\\varphi _k(x_\\perp ).$ Then a further variational form is obtained by multiplying (REF ) by $\\chi _{j}$ , $j=1,2, \\ldots , J$ and integrating over $\\Omega _{\\mathbf {V}}$ : $\\begin{split}&\\Big (M_{x_\\perp }\\otimes M_{\\mathbf {v}}\\Big ) U_m+\\frac{k_m}{2} \\Big (C_{x_\\perp }\\otimes \\widetilde{M}_{\\mathbf {v}} \\Big ) U_m+\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Big (S_{\\mathbf {v}}\\otimes M_{x_\\perp }\\Big )U_m\\\\&\\qquad =\\Big (M_{x_\\perp }\\otimes M_{\\mathbf {v}} \\Big )U_{m-1}-\\frac{k_m}{2} \\Big (C_{x_\\perp }\\otimes \\widetilde{M}_{\\mathbf {v}} \\Big )U_{m-1}-\\frac{\\sigma _{tr}}{2}\\frac{k_m}{2} \\Big (S_ {\\mathbf {v}}\\otimes M_{x_\\perp }\\Big )U_{m-1},\\end{split}$ where $\\otimes $ represents tensor products with the obvious notations for the coefficient matrices $M_{x_\\perp }$ , $M_{\\mathbf {v}}$ being the mass-matrices in spatial and velocity variables, $C_{x_\\perp }$ is the convection matrix in space, $\\widetilde{M}_{\\mathbf {v}}:={\\mathbf {v}}\\otimes M_{\\mathbf {v}}$ corresponds to the spatial convection terms with the coefficient ${\\mathbf {v}}$ : ${\\mathbf {v}}\\cdot \\nabla _\\perp $ , and finally $S_ {\\mathbf {v}}$ is the stiffness matrix in ${\\mathbf {v}}$ .", "Now, given an initial beam configuration, $U_0=u_0$ , our objective is to use an iteration algorithm as the finite element version above or the corresponding equivalent backward Euler (or Crank-Nicolson) approach for discretization in the $x_\\perp $ variable, and obtain successive $U_m$ -values at the subsequent discrete $x_\\perp $ -levels.", "To this end the delicate issues of an initial data viz (REF ), as a product of Dirac delta functions, as well as the desired dose to the target that imposes the model to be transferred to a case having an inverse problem nature are challenging practicalities." ], [ "Standard stability estimates", "We use the notion of the scalar products over a domain ${\\mathbf {D}}$ and its boundary $\\partial {\\mathbf {D}}$ as $(\\cdot , \\cdot )_{\\mathbf {D}}$ and $\\langle \\cdot , \\cdot \\rangle _{\\partial {\\mathbf {D}}}$ , respectively.", "Here, ${\\mathbf {D}}$ can be $\\Omega :=I_x\\times \\Omega _{\\mathbf {x}}\\times \\Omega _{\\mathbf {v}}$ , $ I_x\\times \\Omega _{\\mathbf {x}}$ , $\\Omega _{\\mathbf {x}}\\times \\Omega _{\\mathbf {v}}$ , or possibly other relevant domains in the problem.", "Below we state and prove a stability lemma which, in some adequate norms, guarantees the control of the solution for the continuous problem by the data.", "The lemma is easily extended to the case of approximate solution.", "We derive the stability estimate using the triple norm $\\vert \\vert \\vert w\\vert \\vert \\vert _\\beta ^2 =\\int _0^L \\int _{\\Gamma _\\beta ^+} w^2({\\mathbf {n}}\\cdot \\beta )\\, d\\Gamma dx+\\vert \\vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} w\\vert \\vert ^2_{L_2 (\\Omega )}.$ Lemma 2.1 For $u$ satisfying (REF ) we have the stability estimates $\\sup _{x\\in I_x}\\Vert u(x,\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})}\\le \\Vert u_0(\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})},$ and $\\vert \\vert \\vert u\\vert \\vert \\vert _\\beta \\le \\Vert u_0(\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})}.$ We let $\\chi =u$ in (REF ) and use (REF )-(REF ) to obtain $\\frac{1}{2} \\frac{d}{dx}\\Vert u\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2 +({\\mathbf {v}}\\cdot \\nabla _\\perp u, u)_{I_\\perp \\times \\Omega _{\\mathbf {v}} } +\\frac{1}{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}u\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2 = 0,$ where using Green's formula and with $\\beta =({\\mathbf {v}},0)$ we have $\\begin{split}({\\mathbf {v}}\\cdot \\nabla _\\perp u, u)_{I_\\perp \\times \\Omega _{\\mathbf {v}} }& =\\int _{\\Omega _\\mathbf {v}}\\Big (\\int _{I_\\perp }({\\mathbf {v}}\\cdot \\nabla _\\perp u)u\\, \\Big ) dx_\\perp \\, d{\\mathbf {v}}\\\\&=\\frac{1}{2} \\int _{\\Omega _{\\mathbf {v}}} ({\\mathbf {n}}\\cdot {\\mathbf {v}}) u^2\\, d{\\mathbf {v}}=\\frac{1}{2} \\int _{\\Gamma _\\beta ^+} u^2({\\mathbf {n}}\\cdot \\beta )\\, d\\Gamma \\ge 0.\\end{split}$ Thus $\\frac{d}{dx}\\Vert u\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2 \\le 0,$ which yields (REF ) after integration over $ (0, x)$ and taking supremum over $ x \\in I_x $ .", "Integrating (REF ) over $x\\in (0, L)$ and using (REF ) together with the definition of the triple norm $ \|\|\| \\cdot \|\|\|_\\beta $ we get $\\Vert u(L,\\cdot ,\\cdot )\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2+ \|\|\| u \|\|\|_{\\beta }^2=\\Vert u(0,\\cdot ,\\cdot )\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2$ and the estimate (REF ) follows.", "Using the same argument as above we obtain the semi-discrete version of the stability Lemma REF : Corollary 2.2 The semi-discrete solution $u_h$ with $h=\\max (h_\\perp , h_{\\mathbf {v}})$ and standard Galerkin approximation in phase-space $I_\\perp \\times \\Omega _{\\mathbf {v}}$ satisfies the semidiscrete stability estimates: $\\sup _{x\\in I_x}\\Vert u_h(x,\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})}\\le \\Vert u_{0,h}(\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})},$ $\\vert \\vert \\vert u_h\\vert \\vert \\vert _\\beta \\le \\Vert u_{0,h}(\\cdot , \\cdot )\\Vert _{L_2(\\Omega _\\perp \\times \\Omega _{\\mathbf {v}})}.$" ], [ "Convergence", "Below we state and prove an a priori error estimate for the finite element approximation $ u_h $ satisfying (REF ).", "The a priori error estimate will be stated in the triple norm defined by (REF ).", "Lemma 2.3 [An a priori error estimate in the triple norm] Assume that $u$ and $u_h$ satisfy the continuous and discrete problems (REF ) and (REF ), respectively.", "Let $u\\in H^r(\\Omega )= H^r( \\Omega _{\\mathbf {x}}\\times \\Omega _{\\mathbf {v}})$ , $r\\ge 2$ , then there is a constant $C$ independent of $ v$ , $u$ and $h$ such that $\\vert \\vert \\vert u-u_h\\vert \\vert \\vert _{\\tilde{\\beta }}\\le Ch^{r-1/2}\\Vert u\\Vert _{H^r(\\Omega )}.$ Taking the first equations in (REF ) and (REF ) and using (REF ) we end up with $\\begin{split}\\Big ((u_h-\\tilde{u})_x, \\chi \\Big )_{I_\\perp \\times \\Omega _{\\mathbf {v}}} &+\\Big ({\\mathbf {v}}\\cdot \\nabla _{\\perp }(u_h-\\tilde{u}), \\chi \\Big )_{I_\\perp \\times \\Omega _{\\mathbf {v}}}+\\frac{1}{2} \\Big (\\sigma _{tr}\\nabla _{\\mathbf {v}}(u_h-\\tilde{u}), \\nabla _{\\mathbf {v}}\\chi \\Big )_{I_\\perp \\times \\Omega _{\\mathbf {v}}}\\\\& =\\frac{1}{2}\\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}}(u-\\tilde{u}), \\nabla _{\\mathbf {v}}\\chi \\Big )_{I_\\perp \\times \\Omega _{\\mathbf {v}}}.\\end{split}$ Let now $\\chi =u_h-\\tilde{u}$ , then by the same argument as in the proof of Lemma REF we get $\\begin{split}& \\frac{d}{dx}\\Vert u_h-\\tilde{u}\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}+\\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) (u_h-\\tilde{u})^2\\, d\\Gamma +\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u_h-\\tilde{u})\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}\\\\&\\quad \\le \\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u_h-\\tilde{u})\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}+\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-\\tilde{u})\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})},\\end{split}$ which yields $\\begin{split}\\frac{d}{dx}\\Vert u_h-\\tilde{u}\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})} &+\\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) (u_h-\\tilde{u})^2\\, d\\Gamma +\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u_h-\\tilde{u})\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}\\\\&\\le \\frac{1}{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-\\tilde{u})\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}.\\end{split}$ Hence, integrating over $x\\in (0,L)$ we obtain $\\begin{split}&\\Vert (u_h-\\tilde{u})(L,\\cdot ,\\cdot )\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})} +\\int _{\\Gamma _\\beta ^+\\setminus \\Gamma _L}(\\tilde{\\mathbf {n}}\\cdot \\tilde{\\beta }) (u_h-\\tilde{u})^2\\, d\\Gamma +\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u_h-\\tilde{u})\\Vert ^2_{L_2(I_x\\times I_\\perp \\times \\Omega _{\\mathbf {v}})}\\\\&\\qquad \\le \\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-\\tilde{u})\\Vert ^2_{L_2(I_x\\times I_\\perp \\times \\Omega _{\\mathbf {v}})}+\\Vert (u_h-\\tilde{u})(0,\\cdot ,\\cdot )\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})},\\end{split}$ where $\\Gamma _\\beta ^+\\setminus \\Gamma _L:=\\lbrace \\lbrace L\\rbrace \\times I_\\perp \\times \\Omega _{\\mathbf {v}}\\rbrace $ .", "Now recalling that $u_h(0,\\cdot ,\\cdot )=\\tilde{u}(0,\\cdot ,\\cdot )=u_{0,h}$ and the definition of $\\vert \\vert \\vert \\cdot \\vert \\vert \\vert _{\\tilde{\\beta }}$ we end up with $\\vert \\vert \\vert u_h-\\tilde{u}\\vert \\vert \\vert _{\\tilde{\\beta }}\\le \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-\\tilde{u})\\Vert ^2_{L_2(I_x\\times I_\\perp \\times \\Omega _{\\mathbf {v}})}.$ Finally using the identity $u_h-u=(u_h-\\tilde{u})+ (\\tilde{u}-u)$ and the interpolation estimate below we obtain the desired result.", "Proposition 2.4 (See [9]) Let $h^2\\le \\sigma _{tr}({\\mathbf {x}})\\le h$ , then there is an interpolation constant $\\tilde{C}$ such that $\\vert \\vert \\vert u-\\tilde{u}\\vert \\vert \\vert _{\\tilde{\\beta }}\\le \\tilde{C} h^{r-1/2}\\Vert u\\Vert _r.$ We rely on classical interpolation error estimates (see [8] and [9]): Let $u\\in H^r(\\Omega )$ , then there exists interpolation constants $C_1$ and $C_2$ such that for the nodal interpolant $\\pi _h\\in {\\mathcal {V}}_{h, \\tilde{\\beta }}$ of $u$ we have the interpolation error estimates $\\Vert u-\\pi _hu\\Vert _s & \\le C_1h^{r-s}\\Vert u\\Vert _r,\\qquad s=0,\\,\\,1 \\\\\\vert u-\\pi _hu\\vert _{\\tilde{\\beta }} & \\le C_2h^{r-1/2}\\Vert u\\Vert _r,$ where $\\vert \\varphi \\vert _{\\tilde{\\beta }}:=\\Big (\\int _{\\Gamma _{\\tilde{\\beta }}}\\varphi ^2 ({\\mathbf {n}}\\cdot \\tilde{\\beta })\\, d\\Gamma \\Big )^{1/2}.$ Using the definition of the triple-norm we have that $\\begin{split}\\vert \\vert \\vert u-\\pi _hu\\vert \\vert \\vert _{\\tilde{\\beta }}^2 = &\\vert u-\\pi _hu\\vert _{\\tilde{\\beta }}^2+\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-\\pi _hu)\\Vert ^2\\\\&\\le \\vert u-\\pi _hu\\vert _{\\tilde{\\beta }}^2+\\Vert \\sigma _{tr}^{1/2}\\Vert _\\infty ^2\\Vert u-\\pi _hu\\Vert _{H^1(\\Omega )}^2\\\\&\\le C_2^2 h^{2r-1}\\Vert u\\Vert _r^2+C_1^2\\sup _{{\\mathbf {x}}}\\vert \\sigma _{tr}\\vert h^{2r-1}\\Vert u\\Vert _r^2\\\\=&\\Big ( C_2^2 +C_1^2\\sup _{{\\mathbf {x}}}\\vert \\sigma _{tr}\\vert \\Big )h^{2r-1}\\Vert u\\Vert _r^2\\end{split}$ where in the last inequality we have used (REF ) and ().", "Now choosing the constant $\\tilde{C}=\\Big ( C_2^2 +C_1^2\\sup _{{\\mathbf {x}}}\\vert \\sigma _{tr}\\vert \\Big )^{1/2}$ we get the desired result.", "This proposition yields the $L_2$ error estimate viz: Theorem 2.5 ($L_2$ error estimate) For $u\\in H^r(\\Omega )$ and $u_h\\in {\\mathcal {V}}_{h,\\tilde{\\beta }}$ satisfying (REF ) and (REF ), respectively, and with $h^2\\le \\sigma _{tr}\\le h$ , we have that there is a constant $C=C(\\Omega , f)$ such that $\\Vert u-u_h\\Vert _{L_2(\\Omega )}\\le C h^{r-3/2}\\Vert u\\Vert _r.$ Using the Poincaré inequality $\\Vert u-u_h\\Vert _{L_2(\\Omega )}\\le C \\Vert \\nabla _{\\mathbf {v}}( u-u_h)\\Vert _{L_2(\\Omega )}\\le \\frac{C}{\\min \\sigma _{tr}^{1/2}}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}( u-u_h)\\Vert _{L_2(\\Omega )}$ Further using Lemma REF $\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}}(u-u_h)\\Vert \\le \\vert \\vert \\vert u-u_h\\vert \\vert \\vert _{\\tilde{\\beta }}\\le C h^{r-1/2} \\Vert u\\Vert _r.$ Combining (REF ), (REF ) and recalling that $\\sigma _{tr}$ is in the interval $[h^2, h]$ we end up with $\\Vert u-u_h\\Vert _{L_2(\\Omega )}\\le C h^{r-3/2} \\Vert u\\Vert _r,$ and the proof is complete." ], [ "Petrov-Galerkin approaches", "Roughly speaking, in the Petrov-Galerkin method one adds a streaming term to the test function.", "The raison dé etre of such approach is described, motivated and analyzed in the classical SD methods.", "Here, our objective is to briefly introduce a few cases of Petrov-Galerkin approaches in some lower dimensional geometry and implement them in both direct and adaptive settings.", "Some specific form of the Petrov-Galerkin methods are studied in [11] where the method of exact transport + projection is introduced.", "Also both the semi-streamline diffusion as well as the Characteristic streamline diffusion methods, which in their simpler forms are implemented here, are studied in [6]." ], [ "A semi-streamline diffusion scheme (SSD)", "Here the main difference with the standard approach is that we employ modified test functions of the form $w+\\delta {\\mathbf {v}}\\cdot \\nabla _\\perp w$ with $\\delta \\ge \\sigma _{tr}$ .", "Further, we assume that $w$ satisfies the vanishing inflow boundary condition of (REF ).", "Hence, multiplying the differential equation in (REF ) by $w+\\delta ({\\mathbf {v}}\\cdot \\nabla _\\perp w)$ and integrating over $\\Omega _\\perp =\\Omega _{{\\mathbf {x}}_\\perp }\\times \\Omega _{\\mathbf {v}}$ we have a variational formulation, viz $\\begin{split}\\Big (u_x & +{\\mathbf {v}}\\cdot \\nabla _\\perp u-\\frac{1}{2}\\sigma _{tr}\\Delta _{\\mathbf {v}}u ,w+\\delta ({\\mathbf {v}}\\cdot \\nabla _\\perp w)\\Big )_\\perp =(u_x, w)_\\perp +\\delta (u_x, {\\mathbf {v}}\\cdot \\nabla _\\perp w)_\\perp +({\\mathbf {v}}\\cdot \\nabla _\\perp u, w)_\\perp \\\\&+\\delta ( {\\mathbf {v}}\\cdot \\nabla _\\perp u,{\\mathbf {v}}\\cdot \\nabla _\\perp w)_\\perp +\\frac{1}{2}(\\sigma _{tr} \\nabla _{\\mathbf {v}} u, \\nabla _{\\mathbf {v}}w)_\\perp +\\frac{\\delta }{2}( \\sigma _{tr} \\nabla _{\\mathbf {v}} u,\\nabla _{\\mathbf {v}}({\\mathbf {v}}\\cdot \\nabla _\\perp w))_\\perp =0.\\end{split}$" ], [ "The SSD stability estimate", "We let in (REF ) $w=u$ and obtain the following identity $\\begin{split}\\frac{1}{2} \\frac{d}{dx} \\Vert u\\Vert _\\perp ^2 & +\\delta (u_x, {\\mathbf {v}}\\cdot \\nabla _\\perp u)_\\perp +\\frac{1}{2} \\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) u^2\\, d\\Gamma +\\delta \\Vert {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _\\perp ^2 \\\\&+\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2} \\nabla _{\\mathbf {v}} u\\Vert _\\perp ^2+\\frac{\\delta }{2} \\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}} u,\\nabla _{\\mathbf {v}}({\\mathbf {v}}\\cdot \\nabla _\\perp u)\\Big )_\\perp =0.\\end{split}$ Now it is easy to verify that the last term above can be written as $\\delta \\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}} u,\\nabla _{\\mathbf {v}}({\\mathbf {v}}\\cdot \\nabla _\\perp u)\\Big )_\\perp = \\delta \\int _{\\Omega _\\perp }\\sigma _{tr}\\Big (\\nabla _{\\mathbf {v}}u\\cdot \\nabla _\\perp u+\\frac{1}{2} {\\mathbf {v}}\\cdot \\nabla _\\perp (\\vert \\nabla _{\\mathbf {v}} u\\vert )^2\\Big )\\,dx_\\perp \\, d{\\mathbf {v}}.$ Due to symmetry the second term in the integral above vanishes.", "Hence we end up with $\\begin{split}\\frac{1}{2} \\frac{d}{dx} \\Vert u\\Vert _\\perp ^2 & +\\delta (u_x, {\\mathbf {v}}\\cdot \\nabla _\\perp u)_\\perp +\\frac{1}{2} \\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) u^2\\, d\\Gamma +\\delta \\Vert {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _\\perp ^2 \\\\&+\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2} \\nabla _{\\mathbf {v}} u\\Vert _\\perp ^2+\\frac{\\delta }{2} \\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}} u, \\nabla _\\perp u)\\Big )_\\perp =0.\\end{split}$ Next, we multiply the differential equation (REF ) by $\\delta u_x$ and integrate over $I_\\perp \\times \\Omega _{{\\mathbf {v}}}$ to get $\\delta \\Vert u_x\\Vert ^2+(\\delta u_x, {\\mathbf {v}}\\cdot \\nabla _\\perp u)_\\perp +\\frac{\\delta }{2} (\\sigma _{tr} \\nabla _{{\\mathbf {v}}}u, \\nabla _{{\\mathbf {v}}}u_x)_\\perp =0.$ The last inner product on the left hand side of (REF ) can be written as $(\\sigma _{tr} \\nabla _{{\\mathbf {v}}}u, \\nabla _{{\\mathbf {v}}}u_x)_\\perp =\\frac{1}{2}\\frac{d}{dx}\\int _{I_\\perp \\times \\Omega _{{\\mathbf {v}}}}\\sigma _{tr}\\vert \\nabla _{{\\mathbf {v}}}\\vert ^2\\, dx_\\perp \\, d {\\mathbf {v}}-\\frac{1}{2}\\int _{I_\\perp \\times \\Omega _{\\mathbf {v}}}\\frac{\\partial \\sigma _{tr}}{\\partial x}\\Big (\\vert \\nabla _{\\mathbf {v}}\\vert ^2\\Big )\\, dx_\\perp \\, d {\\mathbf {v}}.$ Now inserting (REF ) in (REF ) and adding the result to (REF ) we end up with $\\begin{split}\\frac{1}{2} \\frac{d}{dx} \\Vert u\\Vert _\\perp ^2 & +\\delta \\Vert u_x+ {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _\\perp ^2+\\frac{1}{2} \\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) u^2\\, d\\Gamma +\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2} \\nabla _{\\mathbf {v}} u\\Vert _\\perp ^2\\\\&+\\frac{\\delta }{2} \\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}} u, \\nabla _\\perp u \\Big )_\\perp +\\frac{\\delta }{4}\\frac{d}{dx}\\int _{I_\\perp \\times \\Omega _{{\\mathbf {v}}}}\\sigma _{tr}\\vert \\nabla _{{\\mathbf {v}}}\\vert ^2\\, dx_\\perp \\, d {\\mathbf {v}}\\\\&-\\frac{\\delta }{4}\\int _{I_\\perp \\times \\Omega _{\\mathbf {v}}}\\frac{\\partial \\sigma _{tr}}{\\partial x}\\Big (\\vert \\nabla _{\\mathbf {v}}\\vert ^2\\Big )\\, dx_\\perp \\, d {\\mathbf {v}}=0.\\end{split}$ Further we use the Cauchy-Schwarz inequality to get $\\vert \\Big (\\sigma _{tr} \\nabla _{\\mathbf {v}} u, \\nabla _\\perp u\\Big )_\\perp \\vert \\le \\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _\\perp u\\Vert _\\perp ^2+\\frac{1}{2}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u\\Vert _\\perp ^2.$ Finally with an additional symmetry assumption on $x_\\perp $ and ${\\mathbf {v}}$ convections as (this is motivated by forward peakedness assumption in angle and energy which is used in deriving the Fokker-Plank/Fermi equations) $\\Vert \\sigma _{tr}^{1/2}\\nabla _\\perp u\\Vert _\\perp \\sim \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u\\Vert _\\perp ,$ and the fact that $\\sigma _{tr}$ is decreasing in the beams penetration direction, i.e.", "$\\frac{\\partial \\sigma _{tr}}{\\partial x}\\le 0$ , we may write (REF ) as $\\begin{split}\\frac{1}{2}\\frac{d}{dx}\\Big (\\Vert u\\Vert _\\perp ^2 +\\frac{1}{2} \\delta \\int _{I_\\perp \\times \\Omega _{{\\mathbf {v}}}}\\sigma _{tr}\\vert \\nabla _{{\\mathbf {v}}}\\vert ^2\\, dx_\\perp \\, d {\\mathbf {v}}\\Big ) & +\\frac{1}{2} \\int _{\\Gamma _\\beta ^+}({\\mathbf {n}}\\cdot \\beta ) u^2\\, d\\Gamma +\\delta \\Vert u_x+ {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _\\perp ^2\\\\&+\\frac{1}{2}(1-\\delta )\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u\\Vert _\\perp ^2\\le 0.\\end{split}$ As a consequence for sufficiently small $\\delta $ (actually $\\delta \\approx \\sigma _{tr}^{1/2}\\ll 1$ ) we have, e.g.", "$\\frac{d}{dx}\\Big (\\Vert u\\Vert _\\perp ^2 + \\frac{1}{2}\\delta \\int _{I_\\perp \\times \\Omega _{{\\mathbf {v}}}}\\sigma _{tr}\\vert \\nabla _{{\\mathbf {v}}}\\vert ^2\\, dx_\\perp \\, d {\\mathbf {v}}\\Big ) <0 ,$ and hence $\\Vert u\\Vert _\\perp ^2 +\\frac{\\delta }{2}\\int _{I_\\perp \\times \\Omega _{{\\mathbf {v}}}}\\sigma _{tr}\\vert \\nabla _{{\\mathbf {v}}}\\vert ^2\\, dx_\\perp \\, d {\\mathbf {v}}$ is strictly decreasing in $x$ .", "Consequently, for each $x^\\prime \\in [0,L]$ we have that $\\Vert u(x^\\prime , \\cdot ,\\cdot )\\Vert ^2_{\\perp }+\\frac{\\delta }{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u(x^\\prime , \\cdot ,\\cdot )\\Vert ^2_\\perp \\le \\Vert u(0, \\cdot ,\\cdot )\\Vert ^2_{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}+\\frac{\\delta }{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u(0, \\cdot ,\\cdot )\\Vert ^2_\\perp .$ Thus, summing up we have proved the following stability estimates Proposition 3.1 Under the assumption (REF ) the following $L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})$ stability holds true $\\Vert u(L, \\cdot ,\\cdot )\\Vert ^2_{\\perp }+\\frac{\\delta }{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u(L, \\cdot ,\\cdot )\\Vert ^2_\\perp \\le \\Vert u_0\\Vert _{\\perp }^2+\\frac{\\delta }{2} \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u_0\\Vert _{\\perp }^2.$ Moreover, we have also the second stability estimate $\\vert \\vert \\vert u\\vert \\vert \\vert _{\\tilde{\\beta }}^2+\\delta \\Vert u_x+ {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _{L_2(\\Omega )}^2\\le \\tilde{C}\\Big (\\Vert u_0\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2+\\delta \\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u_0\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}^2\\Big ).$ Remark 3.2 Due to the size of smallness parameters $\\delta $ and $\\sigma _{tr}$ we can easily verify the second stability estimate (REF ) which also yields $\\Vert u_x+ {\\mathbf {v}}\\cdot \\nabla _\\perp u\\Vert _{L_2(\\Omega )}\\le \\tilde{C}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u_0\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}.$ Hence, using the equation (REF ), we get $\\Vert \\sigma _{tr}\\Delta _{\\mathbf {v}} u\\Vert _{L_2(\\Omega )}\\le \\tilde{C}\\Vert \\sigma _{tr}^{1/2}\\nabla _{\\mathbf {v}} u_0\\Vert _{L_2(I_\\perp \\times \\Omega _{\\mathbf {v}})}.$ The estimate (REF ) indicates the regularizing effect of the diffusive term $\\Delta _{\\mathbf {v}} u$ in the sense that $u_0\\in H_{\\sqrt{\\sigma _{tr}}}^r(I_\\perp \\times \\Omega _{\\mathbf {v}})$ implies that $u\\in H_{\\sigma _{tr}}^{r+1}(I_\\perp \\times \\Omega _{\\mathbf {v}})$ .", "However this regularizing effect will decrease by the size of $\\sigma _{tr}$ ." ], [ "Model problems in lower dimensions", "We consider now a forward peaked narrow radiation beam entering into the symmetric domain $I_y \\times I_\\eta = [-y_0 , y_0] \\times [-\\eta _0, \\eta _0]$ ; $(y_0 , \\eta _0) \\in \\mathbb {R}_+^2$ at $(0,0)$ and penetrating in the direction of the positive $x$ -axis.", "Then the computational domain $ \\Omega $ of our study is a three dimensional slab with $(x, y, \\eta )\\in \\Omega = I_x \\times I_y \\times I_\\eta $ where $ I_x = [0, L]$ .", "In this way, the problem (REF ) will be transformed into the following lower dimensional model problem $\\left\\lbrace \\begin{array}{rl}u_x + \\eta u_y = \\frac{1}{2} \\sigma _{tr} u_{\\eta \\eta } \\quad &(x,y,\\eta ) \\in \\Omega , \\\\u_\\eta (x, y, \\pm \\eta _0) = 0 \\quad &(x,y) \\in I_x \\times I_y, \\\\u(0, y, \\eta ) = f(y, \\eta ) \\quad &(y, \\eta ) \\in I_y \\times I_\\eta , \\\\u(x, y, \\eta ) = 0 \\quad & \\mbox{ on } \\Gamma _\\beta ^- \\setminus \\lbrace (0, y, \\eta )\\rbrace .\\end{array}\\right.$ For this problem we implement two different versions of the streamline diffusion method: the semi-streamline diffusion and the characteristic streamline diffusion.", "Both cases are discretized using linear polynomial approximations." ], [ "The semi-streamline diffusion method", "In this version we derive a discrete scheme for computing the approximate solution $ u_h $ of the exact solution $ u $ using the SD-method for discretizing the $ (y, \\eta ) $ -variables (corresponding to multiply the equation by test functions of the form $w+\\delta \\eta w_y$ ) combined with the backward Euler method for the $ x $ -variable.", "We start by introducing the bilinear forms $a(\\cdot , \\cdot )$ and $b(\\cdot , \\cdot )$ for the problem (REF ) as: $\\begin{split}a(u, w) = & \\, (\\eta u_y , w)_\\perp + \\delta (\\eta u_y ,\\eta w_y)_\\perp +\\frac{1}{2} (\\sigma _{tr} u_\\eta , w_\\eta )_\\perp \\\\& +\\frac{1}{2} \\delta (\\sigma _{tr}u_\\eta , w_y+\\eta w_{y\\eta })_\\perp -\\frac{1}{2}\\delta \\int _{I_y}\\sigma _{tr} \\eta u_\\eta w_y\\Big \|_{\\eta =-\\eta _0}^{\\eta =\\eta _0} dy, \\\\b(u, w) =& \\, ( u , w)_\\perp + \\delta (u , \\eta w_y)_\\perp ,\\end{split}$ where $(\\cdot , \\cdot )_\\perp :=(\\cdot , \\cdot )_{I_y\\times I_\\eta }$ .", "Then the continuous problem reads as: for each $ x \\in (0, L] $ , find $ u(x, \\cdot ) \\in H^1_\\beta $ such that $b(u_x, w)+a(u, w)=0, \\qquad \\forall w\\in H^1_\\beta ,$ where $H^1_\\beta :=\\lbrace w\\in H^1 (I_y\\times I_\\eta );w=0 \\mbox{ on } \\Gamma _\\beta ^- \\rbrace ,$ and $\\Gamma _\\beta ^- :=\\lbrace (y,\\eta )\\in \\Gamma :=\\partial ( I_y\\times I_\\eta ), \\,\\,\\,\\mbox{with}\\,\\, {\\mathbf {n}}\\cdot \\beta <0\\rbrace ,$ with $\\beta =(\\eta , 0)$ .", "Then the semi-streamline diffusion method for the continuous problem (REF ) reads as follows: for each $ x \\in (0, L] $ , find $ u_h (x, \\cdot ) \\in \\mathcal {V}_{h,\\beta } $ such that, $b(u_{h, x} ,w) + a(u_h, w) = 0, \\quad \\forall w \\in \\mathcal {V}_{h,\\beta },$ where $\\mathcal {V}_{h,\\beta } \\subset H^1_\\beta $ consists of continuous piecewise linear functions.", "Next, we write the global discrete solution by separation of variables as $u_h(x,y,\\eta ) = \\sum _{j=1}^{N} U_j(x) \\phi _j(y,\\eta ),$ where $ N $ is the number of nodes in the mesh.", "Letting $ w = \\phi _i $ for $ i = 1, 2, \\ldots , N $ and inserting (REF ) into (REF ) we get the following discrete system of equations, $\\sum _{j=1}^{N} U^{\\prime }_j(x) b(\\phi _j, \\phi _i) + \\sum _{j=1}^{N} U_j(x) a(\\phi _j, \\phi _i) = 0, \\quad i = 1, 2, \\ldots , N.$ Equation (REF ) in matrix form can be written as $B U^{\\prime }(x) +A U(x) = 0,$ with $ U=[U_1, ..., U_N]^T, B = (b_{ij}), b_{ij} =b(\\phi _j , \\phi _i)$ and $A = (a_{ij}), a_{ij} =a(\\phi _j , \\phi _i), i, j = 1, 2, \\ldots , N$ .", "We apply now the backward Euler method for further discretization of the equation (REF ) in variable $ x $ , and with the step size $ k_m $ , to obtain an iterative form viz $B(U^{m+1} - U^{m}) + k_m A U^{m+1} =0.$ The equation above can be rewritten as a system of equations for finding the solution $ U^{m+1}$ (at “time” level $x=x_{m+1}$ ) on iteration $ m+1 $ from the known solution $ U^{m} $ from the previous iteration step $ m$ : $[ B + k_m A] U^{m+1} = B U^m.$" ], [ "Characteristic Streamline Diffusion Method", "In this part we construct an oriented phase-space mesh to obtain the characteristic streamline diffusion method.", "Before formulating this method, we need to construct a new subdivision of $ \\Omega = I_x \\times I_y \\times I_\\eta $ .", "To this end and for $ m = 1, 2, \\ldots , M $ , we define a subdivision of $\\Omega _m = [ x_{m-1}, x_m ] \\times I_y \\times I_\\eta :=I_m\\times I_y \\times I_\\eta $ into elements $\\hat{\\tau }_m = \\lbrace (x, y + (x - x_m) \\eta , \\eta ) :(y, \\eta ) \\in \\tau \\in \\mathcal {T}_h, \\,\\, x \\in I_m \\rbrace ,$ where $ \\mathcal {T}_h $ is a previous triangulation of $ I_\\perp $ .", "Then we introduce, slabwise, the function spaces $\\hat{\\mathcal {V}}_m = \\lbrace \\hat{w} \\in C (\\Omega _m ) :\\hat{w}(x, y, \\eta ) = w(y + (x - x_m)\\eta , \\eta ), w \\in \\mathcal {V}_{h,\\beta } \\rbrace .$ In other words $ \\hat{\\mathcal {V}}_m $ consists of continuous functions $ \\hat{w} (x, y, \\eta ) $ on $ \\Omega _m $ such that $ \\hat{w} $ is constant along characteristics $ (\\hat{y}, \\hat{\\eta }) = (y + x\\eta , \\eta ) $ parallel to the sides of the elements $ \\hat{\\tau }_m $ , meaning that the derivative in the characteristic direction: $ \\hat{w}_x + \\eta \\hat{w}_y = 0 $ .", "The streamline diffusion method can now be reduced to the following formulation (where only the $\\sigma _{tr}$ -term survives): find $ \\hat{u}_h $ such that, for each $ m = 1, 2, \\ldots , M,$ $ \\hat{u}_h\|_{\\Omega _m} \\in \\hat{\\mathcal {V}}_m $ and $\\begin{split}\\frac{1}{2} \\int _{\\Omega _m}\\sigma _{tr} \\hat{u}_{h, \\eta } w_\\eta \\, dx dy d\\eta + \\int _{I_\\perp } \\hat{u}_{h,+} (x_{m-1}, y, \\eta ) w_+ (x_{m-1}, y, \\eta )\\, dy d\\eta \\\\= \\int _{I_\\perp } \\hat{u}_{h,-} (x_{m-1}, y, \\eta ) w_+ (x_{m-1}, y, \\eta ) \\, dy d\\eta ,\\quad \\forall w \\in \\hat{\\mathcal {V}}_m.\\end{split}$ Here, for definition of $ \\hat{u}_{h,+},\\hat{u}_{h,-}, w_+ $ we refer to (REF )." ], [ "Adaptive algorithm", "In this section we formulate an adaptive algorithm, which is used in computations of the numerical examples studied in Section .", "This algorithm improves the accuracy of the computed solution $ u_h $ of the model problem (REF ).", "In the sequel for simplicity we denote $I_y\\times I_\\eta $ also by $\\Omega _\\perp $ (this however, should not be mixed with the notation in the theoretical Sections 1-3).", "The Mesh Refinement Recommendation We refine the mesh in neighborhoods of those points in $I_y\\times I_\\eta $ where the error $ \\varepsilon _n = \| u - u_h^n \| $ attains its maximal values.", "More specifically, we refine the mesh in such subdomains of $I_y\\times I_\\eta $ where $\\varepsilon _n \\ge \\widetilde{\\gamma } \\max \\limits _{\\Omega _\\perp } \\varepsilon _n.$ Here $ \\widetilde{\\gamma } \\in (0,1)$ is a number which should be chosen computationally and $ u_h^n $ denotes the computed solution on the $ n $ -th refinement of the mesh.", "The steps in adaptive algorithm Step 0.", "Choose an initial mesh $ I_m^0 \\times \\tau ^0 $ in $ I_x \\times I_y\\times I_\\eta $ and obtain the numerical solution $ u_h^n,\\, n >0 $ , where $n$ is number of the mesh refinements, in the following steps: Step 1.", "Compute the numerical solution $ u_h^n$ on $ \\tau ^n$ using any of the finite element methods introduced in section .", "Step 2.", "Refine those elements in the mesh $ \\tau ^n $ for which $ \\varepsilon _n \\ge \\widetilde{\\gamma } \\max \\limits _{\\Omega _\\perp } \\varepsilon _n.$ Here, the values for the tolerance $ \\widetilde{\\gamma } \\in \\left(0,1\\right) $ are chosen by the user.", "Step 3.", "Define a new refined mesh as $ \\tau ^{n+1} $ .", "Construct a new partition $ I_m^{n+1}$ if needed.", "Perform steps 1-3 on the mesh $ I_m^{n+1} \\times \\tau ^{n+1} $ .", "Stop mesh refinements when $\\Vert u_h^n - u_h^{n-1} \\Vert _{L_2(\\Omega _\\perp )} < tol$ , where $tol$ is a total tolerance chosen by the user.", "Table: Test 1-a).", "Computed errors e n =∥u-u h n ∥ L 2 (Ω ⊥ ) e_n = \\Vert u -u_h^n \\Vert _{L_2(\\Omega _\\perp )} and e n /e n+1 e_n/e_{n+1} on the adaptivelyrefined meshes.", "Here, the solution u h n u_h^n is computed usingsemi-streamline diffusion method of section withγ ˜=0.5\\widetilde{\\gamma } = 0.5 in the adaptive algorithm and α=0.1\\alpha =0.1 in ().Table: Test 1-b).", "Computed errors e n =∥u-u h n ∥ L 2 (Ω ⊥ ) e_n = \\Vert u -u_h^n \\Vert _{L_2(\\Omega _\\perp )} and e n /e n+1 e_n/e_{n+1} on the adaptivelyrefined meshes.", "Here, the solution u h n u_h^n is computed usingsemi-streamline diffusion method of section with γ ˜=0.7 \\widetilde{\\gamma } = 0.7 in the adaptive algorithm and α=0.1\\alpha =0.1 in ()." ], [ "Numerical examples", "In this section we present numerical examples which show the performance of an adaptive finite element method for the solution of the model problem (REF ).", "Here, all computations are performed in Matlab COMSOL Multiphysics using module LIVE LINK MATLAB.", "We choose the domain $ \\Omega _\\perp = I_y \\times I_\\eta $ as $\\Omega _\\perp = \\left\\lbrace (y, \\eta ) \\in (-1.0,1.0) \\times (-1.0,1.0)\\right\\rbrace .$ Our tests are performed with a fixed diffusion coefficient $\\sigma _{tr} = 0.002$ .", "Further, due to smallness of the parameters $\\delta $ and $\\sigma _{tr}$ , the terms that involve the product $\\delta \\sigma _{tr}$ are assumed to be negligible.", "In the backward Euler scheme, used discretization in $x$ -variable, we solve the system of equations (REF ) which ends up with a discrete (computed) solution $U^{m+1}$ of (REF ) at the time iteration $m+1$ and with the time step $k_m$ which has been chosen to be $ k_m = 0.01$ .", "Previous computational studies, e.g.", "[3], have shown oscillatory behavior of the solution $ u_h $ when the semi-streamline diffusion method was used, and layer behavior when the standard Galerkin method was applied to solve the model problem (REF ).", "In this work we significantly improve results of [3] by using the adaptive algorithm of section on the locally adaptively refined meshes.", "All our computations are compared with the closed form analytic solution for the model problem (REF ) given by $u (x,y, \\eta ) = \\frac{ \\sqrt{3}}{ \\pi \\sigma _{tr} x^{2}}\\exp \\left[ - \\frac{2}{\\sigma _{tr}} \\left( \\frac{3 y^2}{x^3} - \\frac{3 y \\eta }{x^2}+ \\frac{\\eta ^2}{x} \\right) \\right],$ when the initial data is given by $ u (0, y, \\eta ) = \\delta (y) \\delta (\\eta ) $ .", "We have performed the following computational tests: Test 1.", "Solution of the model problem (REF ) with a “Dirac type” initial condition $u(0, y,\\eta ) = f(y,\\eta ) = 1/( y^2 + \\eta ^2 +\\alpha ),\\quad (y, \\eta ) \\in \\Omega _\\perp ,$ for different values of the parameter $\\alpha \\in (0,1)$ .", "Test 2.", "Solution of the model problem (REF ) with “Maxwellian type” initial condition $u(0, y, \\eta ) = f(y, \\eta ) = \\exp ^{-(y^2 + \\eta ^2 + \\alpha )},\\quad (y, \\eta ) \\in \\Omega _\\perp ,$ for different values of $\\alpha \\in (0,1)$ .", "Test 3.", "Solution of the model problem (REF ) with a “hyperbolic type” initial condition $u (0,y, \\eta ) = f(y, \\eta ) = \\frac{1}{\\sqrt{y^2+ \\eta ^2+ \\alpha }},\\quad (y, \\eta ) \\in \\Omega _\\perp ,$ for $\\alpha = 0.19$ .", "Figure: Test 1-a.", "a)-d) Locally adaptively refined meshes of Table; e) Computed solution on the 4 times adaptively refined mesh d).Figure: Test 1-a).", "Computed errors ℰ n (x,y)=u(x,y)-u h n (x,y) \\mathcal {E}_n (x,y) = u (x,y) - u_h^n (x,y) on the locally adaptivelyrefined meshes of Table on the meshesof Figure -a)-d).Figure: Test 1-b.", "a)-d) Locally adaptively refined meshes of Table; e) Computed solution on the 4 times adaptively refined mesh d)." ], [ "Test 1", "In this test we compute numerical simulations for the problem (REF ) with a “Dirac type” initial condition (REF ) and for different values of the parameter $\\alpha \\in (0,1)$ in (REF ), where we use adaptive algorithm of Section on the locally adaptively refined meshes.", "These meshes were refined according to the error indicator (REF ) in the adaptive algorithm.", "For computation of the finite element solution we employ semi-streamline diffusion method of Section REF .", "We performed two set of numerical experiments: Test 1-a).", "We take $\\widetilde{\\gamma } = 0.5$ in (REF ).", "This choice of the parameter allows to refine the mesh $ \\tau $ not only at the center of the domain $ \\Omega _\\perp $ , but also at the boundaries of $ \\Omega _\\perp $ .", "Test 1-b).", "We take $\\widetilde{\\gamma } = 0.7$ in (REF ).", "Such choice of the parameter allows to refine the mesh $ \\tau $ only at the middle of the domain $ \\Omega _\\perp $ .", "Our computational tests have shown that the values for $\\alpha \\in (0.05,0.1)$ give smaller computational errors $e_n = \\Vert u -u_h^n \\Vert _{L_2(\\Omega _\\perp )}$ than the other $\\alpha $ -values.", "The results of the computations for $\\alpha =0.1$ are presented in Tables REF and REF for Test 1-a) and Test 1-b), respectively.", "Using these tables and Figures REF and REF we observe that we have obtained significant reduction of the computational error $e_n = \\Vert u - u_h^n \\Vert _{L_2(\\Omega _\\perp )}$ on the adaptively refined meshes.", "These errors in the form $ \\mathcal {E}_n (x,y) = u (x,y) - u_h^n (x,y) $ , on the adaptively refined meshes, are shown on Figure REF .", "Using Tables REF , REF we observe that the reduction of the computational error is faster and more significant in the case a) than in the case b).", "Thus, choosing the parameter $\\widetilde{\\gamma } = 0.5$ in (REF ) gives a better computational result and smaller error $e_n$ than $\\widetilde{\\gamma } = 0.7$ .", "This allows us to conclude that the refinement of the mesh $ \\tau $ not only at the center of the domain $ \\Omega _\\perp $ , but also at the boundaries of $ \\Omega _\\perp $ give significantly smaller computational error $ e_n = \\Vert u - u_h^n \\Vert _{L_2(\\Omega _\\perp )} $ .", "We present the final solution $ u_h^4 $ computed on the 4 times adaptively refined mesh on the Figure REF -f) for Test 1-a) and on the Figure REF -f) for Test 1-b).", "We note that in both cases we have obtained smoother computed solution $ u_h^4 $ without any oscillatory behavior.", "This is a significant improvement of the result of [3] where mainly oscillatory solution could be obtained." ], [ "Test 2", "In this test we perform numerical simulations for the problem (REF ) with Maxwellian initial condition (REF ) and for different values of the parameter $\\alpha \\in (0,1)$ .", "Again we use the error indicator (REF ) in the adaptive algorithm for local refinement of meshes and perform two set of tests as in the case of Test 1 and with the same values on the parameter $\\widetilde{\\gamma }$ .", "For finite element discretization we use semi-streamline diffusion method as in the Test 1.", "To be able to control the formation of the layer which appears at the central point $ (y, \\eta ) = (0, 0) $ we use different values of $\\alpha \\in (0,1)$ inside the function (REF ).", "Our computational tests show that the value of the parameter $\\alpha = 0.19$ is optimal one.", "We present results of our computations for $\\alpha = 0.19$ in Tables REF and REF for Test 2-a) and Test 2-b), respectively.", "Using these Tables and Figures REF and REF , once again, we observe significant reduction of the computational error $e_n = \\Vert u - u_h^n \\Vert _{L_2(\\Omega _\\perp )}$ on the adaptively refined meshes.", "Using Tables REF and REF again we observe more significant reduction of the computational error in the case a) then in the case b).", "Thus, choosing the parameter $\\widetilde{\\gamma } = 0.5$ in (REF ) yields better computational results.", "Final solution $ u_h^4$ computed on the 4 times adaptively refined mesh is shown on the Figure REF -f) for Test 2-a) and on the Figure REF -f) for Test 2-b).", "Again we observe that, with the above numerical values for the parameters, $\\alpha $ and $\\widetilde{\\gamma }$ we have avoided the formation of layers and in both tests we have obtained smooth computed solution $ u_h^4$ ." ], [ "Test 3", "In this test we perform numerical simulations of the problem (REF ) with hyperbolic initial condition (REF ) on the locally adaptively refined meshes.", "Taking into account results of our previous Tests 1,2 we take fixed value of $\\alpha =0.19$ in (REF ).", "For finite element discretization we used the semi-streamline diffusion method of Section .", "We again perform two set of tests with different values of $\\tilde{\\gamma }$ in (REF ): in the Test 3-a) we choose $\\tilde{\\gamma }=0.5$ , and in the Test 3-b) we assign this parameter to be $\\tilde{\\gamma }=0.7$ .", "We present results of our computations in Tables REF and REF .", "Using these tables and Figures REF and REF we observe significant reduction of the computational error $e_n = \\Vert u - u_h^n \\Vert _{L_2(\\Omega _\\perp )}$ on the adaptively refined meshes.", "Final solutions $ u_h^4 $ computed on the 4 times adaptively refined meshes are shown on the Figure REF -f) for the Test 3-a) and on the Figure REF -f) for the Test 3-b), respectively." ], [ "Conclusion", "Finite element method (FEM) is commonly used as numerical method for solution of PDEs.", "In this work FEM is applied to compute approximate solution of a, degenerate type, convection dominated convection-diffusion problem.", "We consider linear polynomial approximations and study different finite element discretizations for the solutions for pencil-beam models based on Fermi and Fokker-Planck equations.", "First we have derived stability estimates and proved optimal convergence rates (due to the maximal available regularity of the exact solution) in a more general setting in physical domain.", "Then we have specified some “goal oriented” numerical schemes.", "These numerical schemes are derived using a variety Galerkin methods such as Standard Galerkin, Semi-Streamline Diffusion, Characteristic Galerkin and Characteristic Streamline Diffusion methods.", "Our focus has been in two of these approximation schemes: (i) the Semi-Streamline Diffusion and (ii) the Characteristic Streamline Diffusion methods.", "For these two setting, we derived a priori error estimates and formulated the adaptive algorithm.", "Since in our numerical tests we have used a closed form of the analytic solution, therefore it suffices to use a priori error estimates for the local mesh refinements.", "Numerically we tested our adaptive algorithm for different type of initial data in (REF ) in three tests with different mesh refinement parameter $\\widetilde{\\gamma }$ in the mesh refinement criterion (REF ).", "The goal of our numerical experiments was to remove oscillatory behavior of the computational solution as well as removing of the formation of the artificial layer.", "Using Tables and Figures of section we can conclude that the oscillatory behavior of the computed solutions which were obtained in [3] are appearing for problems with non-smooth initial data and on non-refined meshes.", "In this work we removed these oscillations by adaptive mesh refinement and decreasing the dominance of the coefficient in the convection term." ], [ "Acknowledgments", "The research of first and second authors is supported by the Swedish Research Council (VR)." ] ]	1606.05085
[ [ "Duality invariance in Fayet-Iliopoulos gauged supergravity" ], [ "Abstract We propose a geometric method to study the residual symmetries in $N=2$, $d=4$ $\\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity.", "It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory.", "In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed.", "We illustrate the procedure for finding this group in general and then show how it works in some specific models.", "For the prepotential $F=-iX^0X^1$, we use our method to add one more parameter to the rotating Chow-Comp\\`ere solution, representing scalar hair." ], [ "Introduction", "Duality transformations have played, and continue to play, an important role in fundamental developments in string theory, supergravity, quantum field theory as well as in the physics of black holes.", "Perhaps the most relevant example for this is the fact that the five known string theories are actually all related by a web of dualities, and correspond just to perturbative expansions of a single underlying theory about a distinct point in the moduli space of quantum vacua, cf. e.g.", "[1] for a review.", "This web contains in particular weak/strong coupling dualities, of which the celebrated AdS/CFT correspondence [2] is another famous example.", "Duality transformations have been instrumental also in the construction of black hole solutions in string theory.", "Typically one reduces a higher-dimensional theory (in presence of Killing directions) to lower dimensions, in particular to $d=3$ , where all vector fields can be dualized to become scalars.", "One gets then three-dimensional gravity coupled to a nonlinear sigma model, and employs the global symmetries of the latter to obtain new black holes from a given seed.", "This technique was used by Cvetič and Youm [3] to construct the most general rotating five-dimensional black hole solution to toroidally compactified heterotic string theory, specified by 27 charges, two rotational parameters and the ADM mass.", "In a similar way, Chow and Compère [4] obtained the most general asymptotically flat, stationary, rotating, nonextremal, dyonic black hole of four-dimensional $N=2$ supergravity coupled to 3 vector multiplets (the so-called stu model).", "It generates through U-dualities the most general asymptotically flat, stationary black hole of $N=8$ supergravity.", "Note that this typical structure of getting, after a Kaluza-Klein reduction, three-dimensional gravity coupled to a nonlinear sigma model, is also crucial to prove full integrability in some particular cases, cf. e.g.", "[5], [6].", "When (part of the) global symmetries of some given supergravity theory are gauged, as it typically happens in AdS supergravity, the sigma model target space isometries are generically broken by the presence of a scalar potential, so that the powerful solution-generating techniques described above seem to break down.", "An instructive example is the timelike dimensional reduction of four-dimensional Einstein-Maxwell gravity down to three dimensions, which gives Euclidean gravity coupled to an $\\text{SU}(2,1)/\\text{S}(\\text{U}(1,1)\\times \\text{U}(1))$ sigma model [7], [8].", "Adding a cosmological constant to the Einstein-Maxwell theory leads to a scalar potential in three dimensions, that breaks three of the eight $\\text{SU}(2,1)$ generators, corresponding to the generalized Ehlers and the two Harrison transformations.", "This leaves merely a semidirect product of a one-dimensional Heisenberg group and a translation group $\\mathbb {R}^2$ as residual symmetry [9].", "Although in this concrete example the surviving symmetries cannot be used to generate new solutions from known ones, they may nevertheless be useful in more general settings.", "The aim of this paper is thus to provide a systematical and thorough investigation of the residual symmetries in $N=2$ , $d=4$ $\\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity, elaborating on [10], where a particular stu model was considered.", "To this end, we shall use a geometric method, whose underlying idea is the following: The on-shell global symmetry group of the ungauged theory is called U-duality, and consists of the isometries of the special Kähler non-linear sigma model that act linearly also on the field strengths via the symplectic embedding [8].", "For purely electric gaugings, the scalar potential generically spoils this invariance, but allowing also for dyonic gaugings one can recover the whole U-duality invariance, at the price of changing the vector of gauge couplings and so the physical theory.", "We will call this group $U_{\\text{fi}}$ , that stands for fake internal symmetry group, which acts on a solution by mapping it to other solutions of other theories.", "Given $U_{\\text{fi}}$ , we fix a generic choice of the coupling constants $\\mathcal {G}$ .", "The true internal symmetry group $U_{\\text{i}}$ of the gauged supergravity theory is then $S_{\\mathcal {G}}$ , the stabilizer of $\\mathcal {G}$ under the action of $U_{\\text{fi}}$As we will see later, this is true up to possible $\\text{U}(1)$ factors..", "The remainder of this paper is organized as follows: In the next section, we briefly review the theory we are interested in, namely $N=2$ , $d=4$ $\\text{U}(1)$ FI-gauged supergravity, and explain more in detail the general idea outlined above.", "In section we explicitely determine the residual symmetry group for four different prepotentials that are frequently used, but we stress that our method is general, and can be applied to arbitrary prepotentials and extended to $N=4$ and $N=8$ gauged supergravity theories as well.", "After that, in section , it is shown how to apply the residual symmetries to generate new black hole solutions from a given seed in each of the four cases.", "In section we comment on a possible extension of our work to include also gauged hypermultiplets.", "Section contains our conclusions and some final remarks.", "Some supplementary material is deferred to two appendices." ], [ "$N=2$ , {{formula:84f3e983-33ef-4d78-a9ac-0543e5c7c4bf}} FI-gauged supergravity", "The bosonic sector of $N=2$ , $d=4$ supergravity coupled to $n_{\\text{V}}$ vector multiplets consists of the vierbein $e^a{}_\\mu $ , $n_{\\text{V}}+1$ vector fields $A^\\Lambda _\\mu $ with $\\Lambda =0,\\dots n_{\\text{V}}$ (the graviphoton plus $n_{\\text{V}}$ other fields from the vector multiplets), and $n_{\\text{V}}$ complex scalar fields $z^i$ ($i=1,\\dots ,n_{\\text{V}}$ ).", "The latter parametrize an $n_{\\text{V}}$ -dimensional special Kähler manifold, i.e., a Kähler-Hodge manifold, with Kähler metric $g_{i\\bar{\\jmath }}(z,\\bar{z})$ , which is the base of a symplectic bundle with the covariantly holomorphic sectionsWe use the conventions of [11].", "$\\mbox{$\\mathcal {V}$}=\\left(\\begin{array}{c}L^\\Lambda \\\\M_\\Lambda \\end{array}\\right), \\qquad D_{\\bar{\\imath }}\\mbox{$\\mathcal {V}$}\\equiv \\partial _{\\bar{\\imath }}\\mbox{$\\mathcal {V}$}-\\frac{1}{2}\\left(\\partial _{\\bar{\\imath }}\\mbox{$\\mathcal {K}$}\\right)\\mbox{$\\mathcal {V}$}=0\\,,$ where $\\mbox{$\\mathcal {K}$}$ is the Kähler potential.", "$\\mbox{$\\mathcal {V}$}$ obeys the constraint $\\left\\langle \\mbox{$\\mathcal {V}$}\|\\mbox{$\\mathcal {\\bar{V}}$}\\right\\rangle \\equiv \\bar{L}^\\Lambda M_\\Lambda -L^\\Lambda \\bar{M}_\\Lambda =-i\\,.", "$ Alternatively one can introduce the explicitly holomorphic sections of a different symplectic bundle, $v \\equiv e^{-\\mathcal {K}/2}\\mbox{$\\mathcal {V}$}\\equiv \\left(\\begin{array}{c}X^\\Lambda \\\\F_\\Lambda \\end{array}\\right)\\,.$ In appropriate symplectic frames it is possible to choose a homogeneous function $F(X)$ of second degree, called prepotential, such that $F_\\Lambda =\\partial _\\Lambda F$ .", "In terms of the sections $v$ the constraint (REF ) becomes $\\left\\langle v\|\\bar{v}\\right\\rangle \\equiv \\bar{X}^\\Lambda F_\\Lambda -X^\\Lambda {\\bar{F}}_\\Lambda =-i e^{-\\mathcal {K}}.$ The couplings of the vector fields to the scalars are determined by the $(n_{\\text{V}}+1)\\times (n_{\\text{V}}+1)$ period matrix $\\mathcal {N}$, defined by the relations $M_\\Lambda = \\mbox{$\\mathcal {N}$}_{\\Lambda \\Sigma }\\, L^\\Sigma \\,,\\qquad D_{\\bar{\\imath }}\\bar{M}_\\Lambda =\\mbox{$\\mathcal {N}$}_{\\Lambda \\Sigma }\\,D_{\\bar{\\imath }}\\bar{L}^\\Sigma \\,.$ If the theory is defined in a frame in which a prepotential exists, $\\mathcal {N}$ can be obtained from $\\mbox{$\\mathcal {N}$}_{\\Lambda \\Sigma }=\\bar{F}_{\\Lambda \\Sigma }+ 2i\\frac{(N_{\\Lambda \\Gamma }X^\\Gamma )(N_{\\Sigma \\Delta }X^\\Delta )}{X^\\Omega N_{\\Omega \\Psi }X^\\Psi }\\,,$ where $F_{\\Lambda \\Sigma }=\\partial _\\Lambda \\partial _\\Sigma F$ and $N_{\\Lambda \\Sigma }\\equiv \\mathrm {Im}(F_{\\Lambda \\Sigma })$ .", "Introducing the matrixWe defined $R=\\mathrm {Re}\\,\\mbox{$\\mathcal {N}$}$ and $I=\\mathrm {Im}\\,\\mbox{$\\mathcal {N}$}$ .", "$\\mbox{$\\mathcal {M}$}=\\left(\\begin{array}{cc}I +R I^{-1} R & \\,\\,- R I ^{-1} \\\\- I ^{-1} R & I ^{-1} \\\\\\end{array}\\right), $ we have the important relation between the symplectic sections and their derivatives, $\\frac{1}{2} (\\mathcal {M} - i\\Omega ) = \\Omega \\bar{\\mathcal {V}}\\mathcal {V}\\Omega + \\Omega D_i\\mathcal {V}g^{i\\bar{\\jmath }}D_{\\bar{\\jmath }}\\bar{\\mathcal {V}}\\Omega \\,, $ with $\\Omega = \\left(\\begin{array}{cc} 0 & -1 \\\\ 1 & 0 \\end{array}\\right)\\,.$ The bosonic Lagrangian reads $\\sqrt{-g}^{-1}\\!", "{L} = \\frac{R}{2} - g_{i\\bar{\\jmath }}\\,\\partial _{\\mu }z^i\\partial ^{\\mu }\\bar{z}^{\\bar{\\jmath }} + \\frac{1}{4} I_{\\Lambda \\Sigma }F^{\\Lambda \\mu \\nu }F^{\\Sigma }{}_{\\mu \\nu }+ \\frac{1}{4} R_{\\Lambda \\Sigma }F^{\\Lambda \\mu \\nu }\\star \\!", "F^{\\Sigma }{}_{\\mu \\nu } - V(z,\\bar{z})\\,.$ In the case of dyonic $\\text{U}(1)$ FI-gauging, the scalar potential has the form [12] $V = g^{i\\bar{\\jmath }}D_i{\\mathcal {L}} D_{\\bar{\\jmath }}\\bar{\\mathcal {L}} - 3{\\mathcal {L}}\\bar{\\mathcal {L}}\\,,$ where $\\mathcal {L} = \\langle \\mathcal {G},\\mathcal {V}\\rangle $ , and $\\mathcal {G}=(g^\\Lambda ,g_\\Lambda )^t$ denotes the symplectic vector of gauge couplings (FI parameters)." ], [ "Fake internal symmetries, stabilization and solutions", "The kinetic part of (REF ) corresponds to the action of the ungauged theory, whose on-shell global symmetry group is called U-duality, consisting of the isometries of the non-linear sigma model that act linearly also on the field strengths via the symplectic embedding [8].", "For purely electric gaugings, the scalar potential generically spoils this invariance, but, as is clear from (REF ), for dyonic gauging one recovers the whole U-duality invariance, at the price of changing the vector of gauge couplings and so the physical theory.", "We will call this group $U_{\\text{fi}}$ , that stands for fake internal symmetry groupWhen the special Kähler manifold is symmetric we define the Lie algebra $\\mathfrak {u}_{\\text{fi}}$ of $U_{\\text{fi}}$ through the equations (REF ).", "The corresponding definition for nonsymmetric special Kähler manifolds requires more care..", "The action of $U_{\\text{fi}}$ on a solution is the mapping to other solutions of other theories, in the same way in which some elements of the symplectic group map solutions of theories with different prepotential into each other [12], cf. e.g.", "(REF ), (REF ).", "Given $U_{\\text{fi}}$ , we fix a choice of the coupling constants $\\mathcal {G}$ and, at least at the beginning, we suppose that they are generic.", "We want to underline that for abelian dyonic gaugings, the Maxwell equations remain homogeneous and so the action (REF ) doesn't have topological terms [13].", "The true internal symmetry group $U_{\\text{i}}$ of the gauged supergravity theory is $S_{\\mathcal {G}}$ , the stabilizer of $\\mathcal {G}$ under the action of $U_{\\text{fi}}$ , up to possible $\\text{U}(1)$ factors.", "This is obvious from the definition of the stabilizer, $S_{\\mathcal {G}} = \\lbrace g \\in U_{\\text{fi}}\\,\|\\, g\\mathcal {G} = \\mathcal {G}\\rbrace \\,,$ which means that we impose to stay in the same theory, and this restricts of course the group of internal symmetries.", "By acting with $S\\in S_{\\mathcal {G}}$ on a given seed solution $(\\mbox{$\\mathcal {V}$},\\mbox{$\\mathcal {G}$},\\mbox{$\\mathcal {F}$}_{\\mu \\nu })$Actually we should write $(\\mbox{$\\mathcal {V}$},\\mbox{$\\mathcal {G}$},\\mbox{$\\mathcal {F}$}_{\\mu \\nu },g_{\\mu \\nu })$ , but since $S_{\\cal G}$ does not act on the metric, we shall suppress the dependence on $g_{\\mu \\nu }$ .", "of the equations of motion, we can generate another configuration via the map $(\\mbox{$\\mathcal {V}$},\\mbox{$\\mathcal {G}$},\\mbox{$\\mathcal {F}$}_{\\mu \\nu }) \\mapsto (\\tilde{\\mbox{$\\mathcal {V}$}},\\tilde{\\mbox{$\\mathcal {G}$}},\\tilde{\\mbox{$\\mathcal {F}$}}_{\\mu \\nu }) :=(S\\mbox{$\\mathcal {V}$},S\\mbox{$\\mathcal {G}$},S\\mbox{$\\mathcal {F}$}_{\\mu \\nu }) = (S\\mbox{$\\mathcal {V}$},\\mbox{$\\mathcal {G}$},S\\mbox{$\\mathcal {F}$}_{\\mu \\nu } )\\,.$ The transformed fields solve the field equations by constructionAs is clear from the formalism introduced in [12], the application of $S\\in S_{\\mathcal {G}}$ on a static solution of the BPS flow preserves the same amount of supersymmetry as the original configuration.", "In the rotating case, the same is true if one considers electric gaugings only [14]..", "In general, the scalars transform nonlinearly under the corresponding isometry, the field strengths are rotated and the metric is functionally invariant.", "Technically, in order to determine $S_{\\mathcal {G}}$ , it is simpler to work with the corresponding algebra $\\mathfrak {s}_{\\mathcal {G}} = \\lbrace a \\in \\mathfrak {u} _{\\text{fi}} \\, \| \\,a\\mathcal {G} = 0\\rbrace \\,.$ There are some cases in which $U_{\\text{i}}$ strictly contains $S_{\\mathcal {G}}$ , and this depends on some particular symmetric structures of the model under consideration.", "Typically, this happens because the symmetry of the model allows to act with some symplectic matrices in a more general way than (REF ), leaving nevertheless the theory invariant." ], [ "Stabilization and symmetries for some prepotentials", "Now we want to apply these techniques to some specific prepotentials.", "Each of them exhibits different peculiar features related to the geometry of the underlying special Kähler manifold, namely to the symplectic embedding of the isometry group of the non-linear sigma model (cf. app.", ")." ], [ "Prepotential $F=-iX^0X^1$", "This prepotential encodes a particular special Kähler structure on the symmetric manifold $\\text{SU}(1,1)/\\text{U}(1)$ .", "The symplectic section is $\\mathcal {V} = (X^0,X^1,-iX^1,-iX^0 )^t$ , and we fix the couplings in a completely electric frame, $\\mbox{$\\mathcal {G}$} = (0,0,g_0,g_1)^t$ .", "The solution to (REF ) defines the algebra $\\mathfrak {u}_{\\text{fi}}$ , $b_1 t_1 + b_2 t_2 + b_3 t_3 + b_4 t_4 =\\left(\\begin{array}{cccc}b_4 & 0 & b_1 & b_2 \\\\0 & -b_4 & b_2 & b_3 \\\\-b_3 & -b_2 & -b_4 & 0 \\\\-b_2 & -b_1 & 0 & b_4 \\\\\\end{array}\\right)\\,,$ to be the U-duality $\\text{su}(1,1)$ plus a $\\text{u}(1)$ , generated by $t_2$ , which acts trivially on the $z^i$ , as we will see shortly.", "From the stability equation (REF ) one finds that $\\mathfrak {s}_{\\mathcal {G}}$ is generated by $s = t_2 - \\frac{g_1}{g_0} t_1 - \\frac{g_0}{g_1} t_3\\,,$ so that $S_{\\mathcal {G}}\\subseteq \\text{U}(1,1)$ is the 1-parameter subgroup $S = e^{\\beta s} = \\left(\\begin{array}{cccc}\\cos ^2\\!\\beta &\\frac{g_1}{g_0} \\sin ^2\\!\\beta & - \\frac{g_1}{g_0}\\cos \\beta \\sin \\beta & \\cos \\beta \\sin \\beta \\\\\\frac{g_0}{g_1}\\sin ^2\\!\\beta & \\cos ^2\\!\\beta & \\cos \\beta \\sin \\beta & - \\frac{g_0}{g_1}\\cos \\beta \\sin \\beta \\\\\\frac{g_0}{g_1}\\sin \\beta \\cos \\beta & -\\cos \\beta \\sin \\beta & \\cos ^2\\!\\beta & \\frac{g_0}{g_1}\\sin ^2\\!\\beta \\\\-\\cos \\beta \\sin \\beta & \\frac{g_1}{g_0}\\cos \\beta \\sin \\beta & \\frac{g_1}{g_0}\\sin ^2\\!\\beta & \\cos ^2\\!\\beta \\end{array}\\right)\\,.", "$ On the other hand, the $\\text{U}(1)$ generated by $t_2$ is given by $T_{\\alpha } = e^{\\alpha t_2}=\\left(\\begin{array}{cccc}\\cos \\alpha & 0 & 0 & \\sin \\alpha \\\\0 & \\cos \\alpha & \\sin \\alpha & 0 \\\\0 & -\\sin \\alpha & \\cos \\alpha & 0 \\\\-\\sin \\alpha & 0 & 0 & \\cos \\alpha \\end{array}\\right)\\,,$ and it transforms the section $\\mathcal {V}$ according to $T_{\\alpha } \\mathcal {V} = e^{-i\\alpha }\\mathcal {V}\\,.", "$ The projective special Kähler coordinates are thus insensible to its action.", "The matrix $\\mathcal {M}$ defined in (REF ) transforms as $T_{\\alpha }^t\\mathcal {M} T_{\\alpha } = \\mathcal {M}\\,.$ One can thus act with $T_{\\alpha }$ on $\\mbox{$\\mathcal {F}$}_{\\mu \\nu }$ only, leaving the equations of motion still invariant.", "$T_\\alpha $ is an example for a `field rotation matrix' that is commonly used to generate non-BPS solutions, a technique first introduced in [15], [16] and subsequently applied to gauged supergravity in [17], [18].", "In conclusion, the internal symmetry group of this model is $U_{\\text{i}}=\\text{U}(1)\\times \\text{U}(1)\\supset S_{\\mathcal {G}}$ , with the two $\\text{U}(1)$ factors identified respectively with $S$ and $T_\\alpha $ ." ], [ "Prepotential $F=\\frac{i}{4} X^\\Lambda \\eta _{\\Lambda \\Sigma }X^\\Sigma $", "The prepotential $F=\\frac{i}{4} X^\\Lambda \\eta _{\\Lambda \\Sigma }X^\\Sigma $ , with $\\eta _{\\Lambda \\Sigma }=\\text{diag}(-1,1,...,1)$ , describes a special Kähler structure on the symmetric manifolds $\\text{SU}(1,n_{\\text{V}})/(\\text{U}(1)\\times \\text{SU}(n_{\\text{V}}))$ .", "The symplectic section reads $\\mathcal {V} = (X^\\Lambda , \\frac{i}{2}\\eta _{\\Lambda \\Sigma }X^\\Sigma )^t\\,.$ Due to the linearity of $\\mathcal {V}$ in the coordinates $X^\\Lambda $ , one can easily construct the one-parameter subgroup $L_{\\alpha } = \\left(\\begin{array}{cccc}\\cos \\alpha & 0 & 2\\sin \\alpha & 0 \\\\0 & I_{n_{\\text{V}}}\\cos \\alpha & 0 & - 2I_{n_{\\text{V}}}\\sin \\alpha \\\\-\\frac{1}{2}\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & \\frac{1}{2} I_{n_{\\text{V}}}\\sin \\alpha & 0 & I_{n_{\\text{V}}}\\cos \\alpha \\end{array}\\right) $ of $\\text{Sp}(2n_{\\text{V}}+2,\\mathbb {R})$ , under which the section $\\mathcal {V}$ transforms as $L_{\\alpha } \\mathcal {V} = e^{-i\\alpha } \\mathcal {V}\\,.", "$ Since $L_{\\alpha }^t\\mathcal {M} L_{\\alpha } = \\mathcal {M}\\,, $ we can add a new parameter to all the solutions of this model by acting with $L_{\\alpha }$ on $\\mbox{$\\mathcal {F}$}_{\\mu \\nu }$ only.", "The stability equation is slightly more involved.", "Notice that the case with only one vector multiplet is symplectically equivalent to $F=-iX^0X^1$ , and thus the results for $n_{\\text{V}}=1$ can be obtained from the previous subsection by an appropriate symplectic rotation, cf. app. .", "Let us discuss the general case of $n_{\\text{V}}=n$ vector multiplets.", "Eq.", "(REF ) defining the algebra $\\mathfrak {u}_{\\text{fi}}$ is equivalent to $Q^t = -\\eta Q\\eta \\,, \\qquad S = -\\frac{1}{4}\\eta R\\eta \\,.", "$ These equations define an embedding of $\\text{U}(1,n)$ into $\\text{Sp}(2n+2,\\mathbb {R})$ .", "To see this, let $z=A+iB\\in \\mathfrak {u}(1,n)$ .", "Then, $z^t\\eta +\\eta z=0$ implies $A^t = -\\eta A \\eta \\,, \\qquad B^t\\eta = \\eta B\\,,$ so $\\eta B$ is symmetric.", "This suggests an embedding $\\iota _\\alpha :\\, \\mathfrak {u}(1,n) \\longrightarrow \\mathfrak {sp}(2n+2,\\mathbb {R})\\,, \\qquad A+iB \\longmapsto \\begin{pmatrix}A & \\alpha B\\eta \\\\ -\\frac{1}{\\alpha }\\eta B & -A^t\\end{pmatrix},$ for any real $\\alpha \\ne 0$ .", "This is indeed an injective Lie algebra morphism, and its image consists of the elements of $\\mathfrak {sp}(2n+2,\\mathbb {R})$ which solve (REF ) with $F_\\Lambda =\\frac{i}{\\alpha }\\eta _{\\Lambda \\Sigma } X^\\Sigma $ .", "In particular, (REF ) selects $\\iota _2$ .", "A basis for $\\mathfrak {u}(1,n)$ is given by the matrices {Aa}a=1n(n+1)/2 , {iBk}k=0n(n+3)/2 , where $A_a$ are a basis for the space of $(n+1)\\times (n+1)$ real matrices $A$ such that $\\eta A$ is antisymmetric, and $B_k$ generate the space of $(n+1)\\times (n+1)$ real matrices $B$ such that $\\eta B$ is symmetric, with $B_0=I$ , the identity matrix.", "The embedding extends obviously to the group level via the exponential map, and, in particular, notice that $\\exp (\\alpha \\iota _2(iB_0)) = L_\\alpha \\,.$ Let us now consider the symmetry group $S_{\\mathcal {G}}$ .", "If we set $\\mathcal {G}=(\\underline{0},\\underline{g})^t=(0,\\vec{0}, g_0,\\vec{g})^t\\,,$ with $\\vec{g}=(g_1,\\ldots ,g_n)$ , then we see that the invariance of $\\mathcal {G}$ is defined by the equations $A^t\\underline{g} = 0\\,, \\qquad B\\eta \\underline{g} = 0\\,,$ which define a maximal compact subgroupTo be precise, this is the subgroup $\\text{S}(\\text{U}(1)\\times \\text{U}(n))$ .", "$\\text{U}(n)$ of $\\text{U}(1,n)$ .", "To see this, let us first putWe assume $\\underline{g}$ to be timelike future-directed, i.e., $\\eta ^{\\Lambda \\Sigma }g_\\Lambda g_\\Sigma <0$ , $g_0>0$ .", "$\\hat{g}:=\\sqrt{-\\underline{g}^2}\\,,$ and define $\\Lambda _{\\underline{g}}\\in \\text{SO}(1,n)$ by $(g_0,\\vec{g}) = (\\hat{g},\\vec{0})\\Lambda _{\\underline{g}}\\,.$ Thus, $A$ (or $\\eta B^t$ ) has $\\underline{g}$ in the cokernel if and only if $\\Lambda _{\\underline{g}}A\\Lambda ^{-1}_{\\underline{g}}$ (or $\\Lambda _{\\underline{g}}\\eta B^t\\Lambda _{\\underline{g}}^{-1}$ ) has $(\\hat{g},\\vec{0})$ in the cokernel.", "From this we immediately get that $\\mathfrak {s}_{\\mathcal {G}}$ is generated by the elements of $\\mathfrak {u}(1,n)$ of the form $z_{\\underline{g}} = \\Lambda _{\\underline{g}}^{-1}z\\Lambda _{\\underline{g}}\\,,$ where $z\\in \\mathfrak {u}(1,n)$ has vanishing first row and first column.", "Thus, $z_{\\underline{g}}\\in \\text{U}(n)$ .", "This provides also a way to realize an explicit construction of the group elements of $S_{\\mathcal {G}}$ .", "One can choose e.g.", "a generalized Gell-Mann basis [19] for $\\mathfrak {su}(n)$ , add the identity matrix $I_n$ and then embed the basis into $\\mathfrak {u}(1,n)$ by adding a first row and column of zeros.", "If we call $\\lbrace z_I\\rbrace _{I=0}^{n^2-1}$ such a basis for the compact subalgebra $\\mathfrak {u}(n)$ of $\\mathfrak {su}(1,n)$ , then $\\lbrace \\iota _2(z_I)\\rbrace _{I=0}^{n^2-1}$ is a basis for $\\mathfrak {s}_{\\mathcal {G}_0}$ , where $\\mathcal {G}_0\\equiv (0,\\vec{0},\\hat{g},\\vec{0})$ .", "Then we can explicitly construct the group elements by means of the Euler construction of $S_{\\mathcal {G}_0}$In a similar way one can use the Iwasawa construction to obtain the whole group $U_{\\text{fi}}$ , whose compact part is just $S_{\\mathcal {G}}$ [20]., as in [19], [21].", "Finally we have $S_{\\mathcal {G}} = \\tilde{\\Lambda }_{\\underline{g}}^{-1} S_{\\mathcal {G}_0}\\tilde{\\Lambda }_{\\underline{g}}\\,,$ with $\\tilde{\\Lambda }_{\\underline{g}} =\\begin{pmatrix}\\Lambda _{\\underline{g}} & 0 \\\\ 0 & \\Lambda _{\\underline{g}}^{-1}\\end{pmatrix}\\,.$ For practical purposes we can take $\\Lambda _{\\underline{g}}$ defined by ${\\Lambda _{\\underline{g}}}^0{}_0 = \\frac{g_0}{\\hat{g}}\\,, \\qquad {\\Lambda _{\\underline{g}}}^i{}_0 = {\\Lambda _{\\underline{g}}}^0{}_i = \\frac{g_i}{\\hat{g}}\\,, \\qquad {\\Lambda _{\\underline{g}}}^i{}_j = \\frac{g_0 - \\hat{g}}{\\hat{g}\\vec{g}^2} g_i g_j + \\delta ^i{}_j\\,,$ whose inverse is obtained by the replacement $\\vec{g}\\rightarrow -\\vec{g}$ .", "Let us focus on the first nontrivial case $\\text{SU}(1,2)/(\\text{U}(1)\\times \\text{SU}(2))$ .", "We fix the couplings in a completely electric frame, $\\mbox{$\\mathcal {G}$} = (0,0,0,g_0,g_1,g_2)^t$ .", "A basis for $\\mathfrak {u}(2)$ (relative to the vector $\\mathcal {G}_0=(0,\\vec{0},\\hat{g},\\vec{0})$ ) is $t_0 =\\begin{pmatrix}0 & 0 & 0 \\\\ 0 & i & 0 \\\\ 0 & 0 & i\\end{pmatrix}\\,, \\quad t_1 =\\begin{pmatrix}0 & 0 & 0 \\\\ 0 & 0 & i \\\\ 0 & i & 0\\end{pmatrix}\\,, \\quad t_2 =\\begin{pmatrix}0 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0\\end{pmatrix}\\,, \\quad t_3 =\\begin{pmatrix}0 & 0 & 0 \\\\ 0 & i & 0 \\\\ 0 & 0 & -i\\end{pmatrix}\\,,$ which, by means of $\\iota _2$ , defines the basis of $\\mathfrak {s}_{\\mathcal {G}_0}$ T0= 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 -12 0 0 0 0 0 0 -12 0 0 0 , T1= 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 -12 0 0 0 0 -12 0 0 0 0 , T2= 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 , T3= 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 -12 0 0 0 0 0 0 12 0 0 0 .", "Note that $T_0^2 = -\\Delta \\,, \\qquad [T_i,T_j]_+ = -\\delta _{ij}\\Delta \\,, \\quad 1\\le i\\le j\\le 3\\,,$ with $\\Delta =\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1\\end{pmatrix}\\,,$ from which we immediately get the expression for a generic element of $S_{\\mathcal {G}_0}$ , S0(x0,x)=ex0 T0exT =(I6-22 x02 +x0 T0)(I6-22 \|x\|2 + \|x\| x T) , where $\\vec{x}=(x^1, x^2, x^3)$ , $\|\\vec{x}\|=\\sqrt{\\vec{x}\\cdot \\vec{x}}$ , $\\vec{T}=(T_1, T_2, T_3)$ and $\\vec{x} \\cdot \\vec{T}=\\sum _{i=1}^3 x^i T_i$ .", "Finally, after setting $T^g_\\mu = \\tilde{\\Lambda }_{\\underline{g}}^{-1} T_\\mu \\tilde{\\Lambda }_{\\underline{g}}\\,, \\quad \\mu =0,1,2,3\\,, \\qquad \\Delta _{\\underline{g}} = \\tilde{\\Lambda }_{\\underline{g}}^{-1} \\Delta \\tilde{\\Lambda }_{\\underline{g}}\\,,$ we get for a generic element of $S_{\\mathcal {G}}$ $S_{\\underline{g}}(x^0,\\vec{x}) &=& \\tilde{\\Lambda }_{\\underline{g}}^{-1} S_0(x^0,\\vec{x})\\tilde{\\Lambda }_{\\underline{g}} \\\\&=& (I_6 - 2\\sin ^2\\frac{x^0}{2}\\Delta _{\\underline{g}} + \\sin x^0 T^g_0)(I_6 - 2\\sin ^2\\frac{\|\\vec{x}\|}{2}\\Delta _{\\underline{g}} + \\sin \|\\vec{x}\|\\ \\vec{x}\\cdot \\vec{T}^g)\\,.", "\\nonumber $ In order to have even more manageable expressions for the matrices, it may be convenient to change to the basis $R_\\mu $ defined by $R_0 = T^g_0\\,, \\quad R_1 = \\frac{g_1^2 - g_2^2}{g_1^2 + g_2^2} T^g_1 - \\frac{2g_1g_2}{g_1^2 + g_2^2} T^g_3\\,, \\quad R_2 = T^g_2\\,, \\quad R_3 = \\frac{g_1^2 - g_2^2}{g_1^2 + g_2^2} T^g_3 + \\frac{2g_1g_2}{g_1^2 + g_2^2} T^g_1\\,.$" ], [ "Prepotential $F=-X^1X^2X^3/X^0$", "This prepotential describes a special Kähler structure on the symmetric manifold $\\left(\\text{SU}(1,1)/\\text{U}(1)\\right)^3$ , the well-known stu model.", "This is symplectically equivalent to the model with $F=-2i(X^0X^1X^2X^3)^{1/2}$ , for which supersymmetric black holes with purely electric gaugings are known analytically [22].", "After a symplectic transformation to $F=-X^1X^2X^3/X^0$ , the electric gaugings considered in [22] become $\\mathcal {G} = (0,g^1,g^2,g^3,g_0,0,0,0)^t$ , so we shall concentrate on this case in what follows.", "The symplectic section reads $\\mathcal {V} = (X^0, X^1, X^2, X^3, X^1 X^2 X^3/(X^0)^2, -X^2 X^3/X^0, -X^1 X^3/X^0,-X^2 X^1/X^0 )^t\\,.$ Let us now look at the solutions of (REF ).", "To this end, we define X X03 X02 X1 X02 X2 X02 X3 , F X1X2X3 -X0 X2 X3 -X0 X1X3 -X0 X1 X2 , so that (REF ) becomes $X S X -F R F -2 X Q^t F = 0\\,.", "$ Since the lhs is a homogeneous polynomial of degree 6 in $(X^0,X^1,X^2,X^3)$ , the coefficients of each monomial must be zero.", "The simplest way to get the general solutions is then to look at the powers of $X^0$ .", "The possible powers of $X^0$ in $p_S\\equiv X SX$ , $p_R\\equiv F RF$ and $p_Q\\equiv X Q^tF$ are $(6,5,4)$ , $(2,1,0)$ and $(4,3,2)$ respectively.", "Since $S$ and $R$ are symmetric, $p_S$ and $p_R$ can vanish only if $S$ and $R$ are zero.", "Thus, we are left with the following three possibilities: $R=0$ and $p_Q$ cancels $p_S$ .", "The only common power for $X^0$ is 4, so we have to take matrices which generate only this power and equal degrees for the remaining variables.", "A quick inspection gives the solutionsTo avoid confusion, note that $S$ denotes the $4\\times 4$ matrix in (REF ), while $S_1$ , $S_2$ and $S_3$ defined below are $8\\times 8$ matrices.", "S1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , T1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 , U1= 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 .", "$S=0$ and $p_Q$ cancels $p_R$ .", "The only common power for $X^0$ is 2, so we have to take matrices generating only this and equal degrees for the remaining variables.", "The solution is $S_2 = S_1^t\\,, \\qquad T_2 = T_1^t\\,, \\qquad U_2 = U_1^t\\,.$ $R=S=0$ and $Q$ satisfies $p_Q=0$ .", "This implies that $Q$ must be diagonal and that the space of such solutions is 3-dimensional.", "The simplest way to fix a basis of this space is to choose $S_3 = [S_1,S_2]\\,, \\qquad T_3 = [T_1,T_2]\\,, \\qquad U_3 = [U_1,U_2]\\,.$ In this way the nine matrices $\\vec{S}$ , $\\vec{T}$ and $\\vec{U}$ generate the group $U_{\\text{fi}}=(\\text{SL}(2,\\mathbb {R}))^3$ .", "In order to determine the symmetry algebra $\\mathfrak {s}_{\\mathcal {G}}$ we have to consider the equation (using the same notation as in the previous subsection) $(\\vec{x}\\cdot \\vec{S} + \\vec{y}\\cdot \\vec{T} + \\vec{z}\\cdot \\vec{U})\\mathcal {G} = 0\\,,$ whose general solution is given by ${\\cal U}(x,z) = g_0 g^3 x S_1 + g^1 g^2 x S_2 - g_0 g^2 (x+z) T_1 - g^1 g^3 (x+z) T_2 + g_0 g^1 z U_1+ g^2 g^3 z U_2\\,,$ for arbitrary $x,z\\in \\mathbb {R}$ .", "A convenient basis is ${\\cal U}_1 = {\\cal U}(1,-1)\\,, \\qquad {\\cal U}_2 = {\\cal U}(1,0)\\,, $ which defines a two-dimensional abelian algebra.", "Notice that ${\\rm tr}\\,{\\cal U}_1^2 = {\\rm tr}\\,{\\cal U}_2^2 = 8 g_0 g^1 g^2 g^3\\,,$ so that the algebra is compact (and thus defines the group $\\text{U}(1)\\times \\text{U}(1)$ ) if and only if $g_0g^1g^2g^3<0$ .", "One can easily verify that, unfortunately, none of these continuous symmetries survives for the truncation to the $t^3$ model [23], [24] with prepotential $F=-(X^1)^3/X^0$ .", "It is worth noting that a particular situation arises for $g^1=g^2=g^3=-g_0\\equiv g$ .", "As was shown in [10], there is an enhancement of the internal symmetry group in this case.", "This happens because the scalar potential $V$ can be written in terms of fundamental objects that define the nonlinear sigma model of the non-homogeneous projective coordinates $z^i=x^i+ iy^i$ [10], [8], namely $V = g^2\\sum _{i=1}^3\\mathrm {tr}M_i\\,, \\qquad M_i = \\left(\\begin{array}{cc}y^i + \\frac{x^i{}^2}{y^i} & \\frac{x^i}{y^i} \\\\\\frac{x^i}{y^i} & \\frac{1}{y^i}\\end{array}\\right)\\,.$ In fact, the transformation property of $M_i$ , $M_i \\longmapsto {\\cal T}^t M_i\\,{\\cal T}\\,,$ implies the invariance of the potential only if ${\\cal T}{\\cal T}^t=1$ .", "Going back to the symplectic formalism we see that this condition is equivalent to require for the symmetry group to be orthogonal, which, in terms of the elements of $\\mathfrak {u}_{\\text{fi}}$ amounts to consider just the subspace of antisymmetric matrices.", "Thus, the symmetry algebra is generated by $W_1 = S_1 - S_2\\,, \\qquad W_2 = T_1 - T_2\\,, \\qquad W_3 = U_1 - U_2\\,,$ while the subalgebra leaving $\\mathcal {G}$ fixed is generated by $W_2-W_1$ and $W_3-W_2$ .", "The full symmetry group is therefore an extension $U_{\\text{i}}=\\text{U}(1)^3$ of $S_{\\mathcal {G}}=\\text{U}(1)^2$ ." ], [ "Prepotential $F=X^1X^2X^3/X^0-\\frac{A}{3}(X^3)^3/X^0$", "The base manifold for this prepotential is neither symmetric nor homogeneous and it has been studied in [25].", "The symplectic section is given by $\\mathcal {V}=(X^\\Lambda ,F_\\Lambda )^t$ , with $X^\\Lambda {}^t = \\begin{pmatrix} X^0 \\\\ X^1 \\\\ X^2 \\\\ X^3 \\end{pmatrix}\\,,\\qquad F_\\Lambda ^t = \\begin{pmatrix} -X^1 X^2 X^3/(X^0)^2 + \\frac{A}{3}(X^3)^3/(X^0)^2 \\\\X^2 X^3/X^0 \\\\ X^1 X^3/X^0 \\\\ X^1 X^2/X^0 - A(X^3)^2/X^0 \\end{pmatrix}\\,.$ The solution to (REF ) is obtained by proceeding exactly like in the previous subsection.", "After introducing the vectors $X=\\begin{pmatrix}{X^0}^3 \\\\ {X^0}^2 X^1 \\\\ {X^0}^2 X^2 \\\\ {X^0}^2 X^3\\end{pmatrix}\\,, \\qquad F=\\begin{pmatrix}\\frac{A}{3} {X^3}^3-X^1X^2X^3 \\\\ X^0 X^2 X^3 \\\\ X^0 X^1X^3 \\\\ X^0 X^1 X^2-AX^0 {X^3}^2\\end{pmatrix}\\,,$ we reduce the equations to a polynomial identity, and looking at the coefficients we get a five-dimensional space of solutions generated by the symplectic matrices $S_1\\!", "=\\!\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -2A & 0 & 0 & 0 & 0\\end{pmatrix}, \\quad S_2\\!", "=\\!\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\end{pmatrix}, \\quad S_3\\!", "=\\!\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\end{pmatrix},$ $D_1 =\\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\end{pmatrix}, \\qquad D_2 =\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{pmatrix}.$ A direct comparison with the results of [25] shows that this algebra strictly contains the U-duality algebra.", "This is due to the fact that the group of symmetries of the scalar potential is larger than the symmetry group of the whole Lagrangian.", "Indeed the generator $D_2$ does not leave the metric invariant.", "Thus, the U-duality group is generated by the algebra $\\langle S_1, S_2, S_3, D_1\\rangle _{\\mathbb {R}}\\,.$ Notice that the $S_i$ are nilpotent of order 4 for $i=1$ and order 2 for $i=2,3$ .", "They are indeed eigenmatrices for the adjoint action of $D_1$ , all with eigenvalue $-2$ .", "The stability equation (REF ) has a nontrivial solution only if $A=-g^1g^2/(g^3)^2$ .", "With this choice for $A$ one gets a one-dimensional algebra $\\mathfrak {s}_{\\mathcal {G}} $ generated by $s = S_1 - \\frac{g^1}{g^3} S_3 - \\frac{g^2}{g^3} S_2\\,.$ It is nilpotent of order 4 so that $U_{\\text{i}}=\\mathcal {S}_{\\mathcal {G}}$ is a unipotent group of order 4.", "It is worthwhile to note that for $g^1=g^2=g^3$ one gets $A= -1$ , which is the physically most interesting case, since the corresponding prepotential arises in the context of type IIA string theory compactifed on Calabi-Yau manifolds [26]." ], [ "Scalar hair and dyonic solutions", "We shall now use the results of the previous section in order to generate new supergravity solutions from a given seed.", "The transformations in $U_{\\text{i}}$ add new parameters to a given solution and leave not only the equations of motion invariant, but also some potential first-order flow equations (if these are satisfied by the seed).", "The transformed field configuration preserves thus the same amount of supersymmetry as the one from which we started.", "As was stressed in [10], the latter statement is not true in the stu model for the additional $\\text{U}(1)$ that arises for equal couplings, whose action generically leads to a non-BPS solution.", "The same story holds also in the quadratic models for $T_{\\alpha }$ and $L_{\\alpha }$ , due to the properties (REF ) and (REF ) [18].", "In what follows we will consider several relevant examples for some well-studied prepotentials, but there is no obstacle to extending this method to other solutions and prepotentials as well.", "We underline that in the static case, owing to the existence of the black hole potential $V_{\\text{BH}}$ [27], [28], one can directly rotate the charges $\\mathcal {Q}$ instead of the field strengths $\\mathcal {F}_{\\mu \\nu }$ ." ], [ "Prepotential $F=-iX^0X^1$", "For this prepotential, we have $U_{\\text{i}}=\\text{U}(1)^2$ , whose action on the static and magnetic BPS seed solution of [22] is $(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q} ) \\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}}, \\tilde{\\mathcal {G}},\\tilde{\\mathcal {Q}}) = (S\\mathcal {V}, \\mathcal {G}, T_{\\alpha }S\\mathcal {Q})\\,.$ Using the results of section REF and the constraints on the seed parameters (cf.", "[22]), one gets $\\begin{split}&\\tilde{\\mathcal {Q}} = (p^0\\cos \\alpha , p^1\\cos \\alpha , -p^1\\sin \\alpha , -p^0\\sin \\alpha )^t\\,, \\\\&\\tilde{z} = \\frac{\\tilde{X}^1}{\\tilde{X}^0} = \\frac{g_0}{g_1}\\cdot \\frac{g_1 z\\cos \\beta +i g_0\\sin \\beta }{g_0\\cos \\beta + i g_1 z\\sin \\beta }\\,, \\qquad z \\equiv \\frac{X^1}{X^0}\\,.\\end{split} $ The parameter $\\beta $ does not modify the supersymmetry of the solution; for $\\alpha =0$ the new configuration satisfies again the BPS flow equations of [22], [12].", "For $\\alpha \\ne 0$ one gets a solution that still obeys a first-order flow, but this time a non-BPS one [18], driven by the fake superpotential $W = e^U\|\\langle T_{-\\alpha }\\tilde{\\mathcal {Q}}, \\tilde{\\mathcal {V}}\\rangle - i e^{2(\\psi - U)}\\tilde{\\mathcal {L}}\|\\,, $ where $U(r)$ and $\\psi (r)$ are functions appearing in the metric $ds^2 = -e^{2U}dt^2 + e^{-2U}dr^2 + e^{2(\\psi -U)}(d\\theta ^2 + \\sinh ^2\\!\\theta d\\phi ^2)\\,,$ and $\\mathcal {L}$ was defined in section REF .", "The first-order equations following from (REF ) imply the equations of motion provided the Dirac-type charge quantization condition $\\langle \\mathcal {G}, \\mathcal {Q}\\rangle = 1$ holds [18].", "From (REF ) we see that for $\\alpha \\ne 0$ one generates a dyonic solution from a purely magnetic one, while $\\beta $ adds scalar hair to the seed.", "Note that this result was first obtained in [10].", "As another example for the action of $U_{\\text{i}}$ we consider the Chow-Compère solution [29], that solves the equations of motion following from the Lagrangian (2.12) of [29], ${L} &=& R\\star \\!", "1 -\\frac{1}{2}\\star \\!", "d\\varphi \\wedge d\\varphi -\\frac{1}{2} e^{2\\varphi }\\star \\!", "d\\chi \\wedge d\\chi - e^{-\\varphi }\\star \\!", "F^1\\wedge F^1 + \\chi F^1\\wedge F^1 \\\\&& -\\frac{1}{1+\\chi ^2 e^{2\\varphi }}\\left(e^{\\varphi }\\star \\!", "F^2\\wedge F^2 + \\chi e^{2\\varphi }F^2\\wedge F^2\\right) + g^2\\left(4 + e^{\\varphi } + e^{-\\varphi } + \\chi ^2 e^{\\varphi }\\right)\\star \\!", "1\\,, \\nonumber $ which is obtained from (REF ) by setting $z = \\frac{g_0}{g_1}\\left(e^{-\\varphi } - i\\chi \\right)\\,, \\qquad g_0 g_1 = g^2\\,,$ and redefiningWe assume $g_0/g_1>0$ .", "$F^0\\longrightarrow \\sqrt{\\frac{g_1}{g_0}}F^1\\,, \\qquad F^1\\longrightarrow \\sqrt{\\frac{g_0}{g_1}}F^2\\,.$ The dyonic rotating black hole solution of [29] is given by $ds^2 = -\\frac{R}{W}\\left(dt - \\frac{a^2 - u_1u_2}{a} d\\phi \\right)^2 + \\frac{W}{R} dr^2 + \\frac{U}{W}\\left(dt -\\frac{r_1r_2 + a^2}{a} d\\phi \\right)^2 + \\frac{W}{U} du^2\\,,$ where $R(r) &=& r^2 - 2mr + a^2 + g^2 r_1 r_2(r_1 r_2 + a^2)\\,, \\nonumber \\\\U(u) &=& -u^2 + 2nu + a^2 + g^2 u_1 u_2 (u_1 u_2 - a^2)\\,, \\\\W(r,u) &=& r_1 r_2 + u_1 u_2\\,, \\qquad r_{1,2} = r + \\Delta r_{1,2}\\,, \\qquad u_{1,2} = u +\\Delta u_{1,2}\\,, \\nonumber $ and $\\Delta r_{1,2}$ , $\\Delta u_{1,2}$ are constants defined by $\\Delta r_1 &=& m[\\cosh (2\\delta _1)\\cosh (2\\gamma _2) - 1] + n\\sinh (2\\delta _1)\\sinh (2\\gamma _1)\\,,\\nonumber \\\\\\Delta r_2 &=& m[\\cosh (2\\delta _2)\\cosh (2\\gamma _1) - 1] + n\\sinh (2\\delta _2)\\sinh (2\\gamma _2)\\,,\\nonumber \\\\\\Delta u_1 &=& n[\\cosh (2\\delta _1)\\cosh (2\\gamma _2) - 1] - m\\sinh (2\\delta _1)\\sinh (2\\gamma _1)\\,,\\nonumber \\\\\\Delta u_2 &=& n[\\cosh (2\\delta _2)\\cosh (2\\gamma _1) - 1] - m\\sinh (2\\delta _2)\\sinh (2\\gamma _2)\\,.$ Below we shall also use the linear combinations $\\Sigma _{\\Delta r} &=& \\frac{1}{2}(\\Delta r_1 + \\Delta r_2)\\,, \\qquad \\Delta _{\\Delta r} = \\frac{1}{2}(\\Delta r_2- \\Delta r_1)\\,, \\nonumber \\\\\\Sigma _{\\Delta u} &=& \\frac{1}{2}(\\Delta u_1 + \\Delta u_2)\\,, \\qquad \\Delta _{\\Delta u} = \\frac{1}{2}(\\Delta u_2- \\Delta u_1)\\,.", "$ The complex scalar field has the very simple form $z = \\frac{g_0}{g_1}\\frac{r_1 - i u_1}{r_2 - i u_2}\\,,$ while the gauge fields and their duals read $A^1 &=& \\zeta ^1(dt - a d\\phi ) + \\frac{r_2 u_2\\tilde{\\zeta }_1}{a}d\\phi \\,, \\qquad A^2 = \\zeta ^2(dt - a d\\phi ) + \\frac{r_1 u_1\\tilde{\\zeta }_2}{a}d\\phi \\,, \\nonumber \\\\\\tilde{A}_1 &=& \\tilde{\\zeta }_1(dt - a d\\phi ) - \\frac{r_1 u_1\\zeta ^1}{a}d\\phi \\,, \\qquad \\tilde{A}_2 = \\tilde{\\zeta }_2(dt - a d\\phi ) - \\frac{r_2 u_2\\zeta ^2}{a}d\\phi \\,, $ where the three-dimensional electromagnetic scalars are $\\zeta ^1 &=& \\frac{1}{2W}\\frac{\\partial W}{\\partial \\delta _1} = \\frac{Q_1r_2 - P^1u_2}{W}\\,, \\qquad \\tilde{\\zeta }_1 = \\frac{Q_1u_1 + P^1r_1}{W}\\,, \\nonumber \\\\\\zeta ^2 &=& \\frac{1}{2W}\\frac{\\partial W}{\\partial \\delta _2} = \\frac{Q_2r_1 - P^2u_1}{W}\\,, \\qquad \\tilde{\\zeta }_2 = \\frac{Q_2u_2 + P^2r_2}{W}\\,.$ Here, $Q_{1,2}$ and $P^{1,2}$ denote respectively the electric and magnetic charges given by [29] $Q_1 = \\frac{1}{2}\\frac{\\partial r_1}{\\partial \\delta _1}\\,, \\qquad Q_2 = \\frac{1}{2}\\frac{\\partial r_2}{\\partial \\delta _2}\\,, \\qquad P^1 = -\\frac{1}{2}\\frac{\\partial u_1}{\\partial \\delta _1}\\,, \\qquad P^2 =-\\frac{1}{2}\\frac{\\partial u_2}{\\partial \\delta _2}\\,.", "$ The solution is thus specified by the 7 parameters $m$ , $n$ , $a$ , $\\gamma _{1,2}$ and $\\delta _{1,2}$ that are related to the mass, NUT charge, angular momentum, two electric and two magnetic charges.", "Notice that a similar class of rotating black holes containing one parameter less was constructed in [30].", "Let us now consider the action of $S$ defined in (REF ).", "For the transformed scalar we get $\\tilde{z} = \\frac{\\tilde{X}^1}{\\tilde{X}^0} = \\frac{g_0}{g_1}\\frac{r + \\Delta r_1^{\\prime } - i(u + \\Delta u_1^{\\prime })}{r +\\Delta r_2^{\\prime } - i(u + \\Delta u_2^{\\prime })}\\,,$ where $\\left(\\begin{array}{c} \\Delta r_1^{\\prime } \\\\ \\Delta r_2^{\\prime } \\\\ \\Delta u_1^{\\prime } \\\\ \\Delta u_2^{\\prime }\\end{array}\\right) =\\left(\\begin{array}{cccc} \\cos ^2\\!\\beta & \\sin ^2\\!\\beta & -\\cos \\beta \\sin \\beta & \\cos \\beta \\sin \\beta \\\\\\sin ^2\\!\\beta & \\cos ^2\\!\\beta & \\cos \\beta \\sin \\beta & -\\cos \\beta \\sin \\beta \\\\\\cos \\beta \\sin \\beta & -\\cos \\beta \\sin \\beta & \\cos ^2\\!\\beta & \\sin ^2\\!\\beta \\\\-\\cos \\beta \\sin \\beta & \\cos \\beta \\sin \\beta & \\sin ^2\\!\\beta & \\cos ^2\\!\\beta \\end{array}\\right)\\left(\\begin{array}{c} \\Delta r_1 \\\\ \\Delta r_2 \\\\ \\Delta u_1 \\\\ \\Delta u_2\\end{array}\\right)\\,.$ Note that the quantities $\\Sigma _{\\Delta r}$ and $\\Sigma _{\\Delta u}$ defined in (REF ) remain invariant under (REF ), while $\\Delta _{\\Delta r}$ and $\\Delta _{\\Delta u}$ transform as $\\left(\\begin{array}{c} \\Delta _{\\Delta r}^{\\prime } \\\\ \\Delta _{\\Delta u}^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{cc}\\cos 2\\beta & -\\sin 2\\beta \\\\ \\sin 2\\beta & \\cos 2\\beta \\end{array}\\right)\\left(\\begin{array}{c}\\Delta _{\\Delta r} \\\\ \\Delta _{\\Delta u}\\end{array}\\right)\\,.$ The transformed gauge fields can be easily inferred from $\\left(\\begin{array}{c} A^1 + A^2 \\\\ \\frac{g_1}{g_0}\\tilde{A}_1 + \\frac{g_0}{g_1}\\tilde{A}_2 \\\\ A^2 - A^1 \\\\\\frac{g_0}{g_1}\\tilde{A}_2 - \\frac{g_1}{g_0}\\tilde{A}_1\\end{array}\\right)^\\prime = \\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\cos 2\\beta & -\\sin 2\\beta \\\\ 0 & 0 & \\sin 2\\beta & \\cos 2\\beta \\end{array}\\right)\\left(\\begin{array}{c} A^1 + A^2 \\\\ \\frac{g_1}{g_0}\\tilde{A}_1 + \\frac{g_0}{g_1}\\tilde{A}_2 \\\\A^2 - A^1 \\\\ \\frac{g_0}{g_1}\\tilde{A}_2 - \\frac{g_1}{g_0}\\tilde{A}_1\\end{array}\\right)\\,.$ In conclusion, $S$ adds one more parameter $\\beta $ to the solution of [29].", "Under the action of $T_\\alpha $ (cf.", "(REF )) the scalar $z$ does not change.", "It turns out that the new gauge fields can again be written in the form (REF ), but with the three-dimensional electromagnetic scalars replaced by $\\left(\\begin{array}{c} \\sqrt{\\frac{}{}}{g_1}{g_0}\\zeta ^1 \\\\ \\sqrt{\\frac{}{}}{g_0}{g_1}\\zeta ^2 \\\\\\sqrt{\\frac{}{}}{g_1}{g_0}\\tilde{\\zeta }_1 \\\\ \\sqrt{\\frac{}{}}{g_0}{g_1}\\tilde{\\zeta }_2\\end{array}\\right)\\longmapsto \\left(\\begin{array}{cccc} \\cos \\alpha & 0 & 0 & \\sin \\alpha \\\\ 0 & \\cos \\alpha & \\sin \\alpha & 0 \\\\0 & -\\sin \\alpha & \\cos \\alpha & 0 \\\\ -\\sin \\alpha & 0 & 0 & \\cos \\alpha \\end{array}\\right)\\left(\\begin{array}{c} \\sqrt{\\frac{}{}}{g_1}{g_0}\\zeta ^1 \\\\ \\sqrt{\\frac{}{}}{g_0}{g_1}\\zeta ^2 \\\\\\sqrt{\\frac{}{}}{g_1}{g_0}\\tilde{\\zeta }_1 \\\\ \\sqrt{\\frac{}{}}{g_0}{g_1}\\tilde{\\zeta }_2\\end{array}\\right)\\,.$ In other words, they transform (up to prefactors) with the same matrix $T_\\alpha $ .", "This invariance can be used to generate additional charges by starting from a given seed.", "Set e.g.", "$\\gamma _2=\\delta _2=0$ in (REF ), which by (REF ) implies $P^2=Q_2=0$ .", "After acting with $T_\\alpha $ one gets a solution with all four charges nonvanishing, namely $Q_1^{\\prime } = Q_1\\cos \\alpha \\,, \\qquad {P^1}^{\\prime } = P^1\\cos \\alpha \\,, \\qquad Q_2^{\\prime } = \\frac{g_1}{g_0}P^1\\sin \\alpha \\,,\\qquad {P^2}^{\\prime } = -\\frac{g_1}{g_0}Q_1\\sin \\alpha \\,.$" ], [ "Prepotential $F=\\frac{i}{4}((X^1)^2+(X^2)^2-(X^0)^2)$", "In this case the most interesting feature of $U_{\\text{i}}$ is the non-abelianity of $\\mathcal {S}_{\\mathcal {G}}$ , cf. sec.", "REF .", "As far as $L_{\\alpha }$ is concerned, its effect is the same as the one of $T_\\alpha $ for $F=-iX^0X^1$ , namely the transformed configuration solves non-BPS first-order flow equations.", "The nonabelian part acts nontrivially on the special scalars.", "With the 1-parameter subgroups $\\exp (\\alpha _\\mu R_\\mu )$ ($\\mu =0,\\ldots ,3$ , no summation over $\\mu $ ), where the $R_\\mu $ are defined in section REF , one can describe the action of $\\mathcal {S}_{\\mathcal {G}}$ on a static seed solution with charge vector $\\mathcal {Q}$ as $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q})\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}},\\tilde{\\mathcal {G}}, \\tilde{\\mathcal {Q}}) = (e^{\\alpha _0 R_0}\\mathcal {V}, \\mathcal {G},e^{\\alpha _0 R_0}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1=\\frac{-g_1 (g_0 + g_1 z^1 + g_2 z^2) + e^{i\\alpha _0} (g_0 g_1 +(g_0^2 - g_2^2) z^1 + g_1 g_2 z^2)}{g_0 (g_0 + g_1 z^1 + g_2 z^2) - e^{i\\alpha _0}(g_1^2 + g_2^2 +g_0 g_2 z^2 + g_0 g_1 z^1)}\\,, \\\\&\\tilde{z}^2=\\frac{-g_2 (g_0 + g_1 z^1 + g_2 z^2) + e^{i\\alpha _0} (g_0 g_2 +(g_0^2 - g_1^2) z^2 + g_1 g_2 z^1)}{g_0 (g_0 + g_1 z^1 + g_2 z^2) - e^{i\\alpha _0} (g_1^2 + g_2^2 +g_0 g_2 z^2 + g_0 g_1 z^1)}\\,,\\end{split}$ $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q})\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}},\\tilde{\\mathcal {G}},\\tilde{\\mathcal {Q}}) = (e^{\\alpha _1 R_1}\\mathcal {V}, \\mathcal {G},e^{\\alpha _1 R_1}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1=\\frac{-g_1 (g_0 +g_1 z^1 +g_2 z^2) + (g_0 g_1 +g_0^2 z^1 -g_2^2 z^1 + g_1 g_2 z^2) \\cos \\alpha _1 - \\hat{g} (g_2 + g_0 z^2) \\sin \\alpha _1}{g_0(g_0 +g_1 z^1 + g_2 z^2) - (g_1^2 + g_0 g_1 z^1 + g_2^2 +g_0 g_2 z^2 )\\cos \\alpha _1 + \\hat{g} (g_1 z^2 - g_2 z^1)\\sin \\alpha _1}\\,, \\\\&\\tilde{z}^2=\\frac{-g_2 (g_0 +g_1 z^1 +g_2 z^2) + (g_0 g_2 + g_0^2 z^2 -g_1^2 z^2 + g_2 g_1 z^1)\\cos \\alpha _1 + \\hat{g} (g_1 + g_0 z^1)\\sin \\alpha _1}{ g_0 (g_0 +g_1 z^1 + g_2 z^2) - (g_1^2 +g_0 g_1 z^1 + g_2^2 +g_0 g_2 z^2 )\\cos \\alpha _1 + \\hat{g} (g_1 z^2 - g_2 z^1)\\sin \\alpha _1}\\,,\\end{split}$ $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q})\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}},\\tilde{\\mathcal {G}}, \\tilde{\\mathcal {Q}}) = (e^{\\alpha _2 R_2}\\mathcal {V}, \\mathcal {G},e^{\\alpha _2 R_2}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1=\\frac{-g_1 (g_0+g_1 z^1+g_2 z^2)+f(g_1,g_2,z^1,z^2)\\cos \\alpha _2 - h(g_1,g_2,z^1,z^2)\\sin \\alpha _2}{g_0(g_0 +g_1 z^1 +g_2 z^2)-(g_1^2 + g_0 g_1 z^1 + g_2 (g_2 + g_0 z^2))\\cos \\alpha _2 +i \\hat{g}(g_2 z^1 - g_1 z^2)\\sin \\alpha _2}\\,, \\\\&\\tilde{z}^2=\\frac{-g_2 (g_0+g_1 z^1+g_2 z^2)+f(g_2,g_1,z^2,z^1)\\cos \\alpha _2 + h(g_2,g_1,z^2,z^1)\\sin \\alpha _2}{g_0(g_0+g_1 z^1+g_2 z^2)-(g_1^2 + g_0 g_1 z^1+g_2(g_2 + g_0 z^2))\\cos \\alpha _2 +i \\hat{g}(g_2 z^1 - g_1 z^2)\\sin \\alpha _2}\\,, \\\\\\end{split}$ $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q})\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}},\\tilde{\\mathcal {G}}, \\tilde{\\mathcal {Q}}) = (e^{\\alpha _3 R_3}\\mathcal {V}, \\mathcal {G},e^{\\alpha _3 R_3}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1 = \\frac{-g_1(g_1^2+g_2^2)(g_0+g_1 z^1+g_2 z^2) + e^{i\\alpha _3} k(g_1,g_2,z^1,z^2)+e^{-i\\alpha _3}g_2 \\hat{g}^2 (g_2 z^1 - g_1 z^2)}{(g_1^2 + g_2^2)\\left(g_0(g_0 +g_1 z^1+g_2 z^2) - e^{i\\alpha _3}(g_1^2+g_0 g_1 z^1 + g_2^2 + g_0 g_2 z^2 )\\right)}\\,, \\\\&\\tilde{z}^2=\\frac{-g_2(g_1^2+g_2^2)(g_0+g_1 z^1+g_2 z^2) + e^{i\\alpha _3} k(g_2,g_1,z^2,z^1) +e^{-i\\alpha _3}g_1 \\hat{g}^2 (g_1 z^2 - g_2 z^1)}{(g_1^2 + g_2^2)\\left(g_0(g_0+g_1 z^1+g_2 z^2)- e^{i\\alpha _3}(g_1^2+g_0 g_1 z^1 + g_2^2 + g_0 g_2 z^2 )\\right) }\\,, \\\\\\end{split}$ where we used the definitions $\\begin{split}&\\hat{g} = \\sqrt{g_0^2 - g_1^2 - g_2^2}\\,, \\qquad f(g_1,g_2,z^1,z^2) = g_0 g_1 + g_0^2 z^1 +g_1 g_2 z^2 - g_2^2 z^1\\,, \\\\&h(g_1,g_2,z^1,z^2) = \\frac{i\\hat{g}}{g_1^2 + g_2^2}(2 g_0 g_1 g_2 z^1 + g_1^2(g_2 - g_0 z^2) +g_2^2(g_2 + g_0 z^2))\\,, \\\\&k(g_1,g_2,z^1,z^2) = g_0 g_1 (g_1^2 + g_0 g_1 z^1 + g_2^2 + g_0 g_2 z^2)\\,.\\end{split}$ The explicit expressions for $\\tilde{\\mathcal {Q}}$ are not particularly enlightening, so we don't report them here.", "One may apply the above transformations to the static and magnetic BPS seed given by eqns.", "(3.100) and (3.101) of [22] to generate dyonic and axionic solutions.", "Note that the form of (REF ) splits the dependence of the group coordinates from the couplings.", "Defining the section $\\mathcal {V}_{\\underline{g}}=({X}_{\\underline{g}},{F}_{\\underline{g}})^t\\equiv \\tilde{\\Lambda }_{\\underline{g}}\\mathcal {V}$ , the action of $\\mathcal {S}_{\\mathcal {G}}$ becomes $\\tilde{\\mathcal {V}}_{\\underline{g}}=S_0(x^0,\\vec{x})\\mathcal {V}_{\\underline{g}}$ that more explicitly reads $\\tilde{X}_{\\underline{g}} =\\begin{pmatrix}X_{\\underline{g}}^0 \\\\e^{i x^0}\\left( X_{\\underline{g}}^1\\cos \|\\vec{x}\| + i ((x^1 + i x^2) X_{\\underline{g}}^2 + i x^3X_{\\underline{g}}^1)\\sin \|\\vec{x}\|\\right)\\\\e^{i x^0}\\left( X_{\\underline{g}}^2\\cos \|\\vec{x}\| + i ((x^1 - i x^2) X_{\\underline{g}}^1 - i x^3X_{\\underline{g}}^2)\\sin \|\\vec{x}\|\\right)\\end{pmatrix}\\,.$ This split is independent of the parametrization of the group and so one can also use that of [21], [19]." ], [ "Prepotential $F=-X^1X^2X^3/X^0$", "This model is related to the one with $F=-2i(X^0X^1X^2X^3)^{1/2}$ by a symplectic rotation with the matrix (REF ).", "As a seed solution we shall thus take the static magnetic BPS black holes given by eqns.", "(3.31)-(3.34) of [22], transformed to $F=-X^1X^2X^3/X^0$ .", "In this new frame, the vectors of charges and couplings are respectively given by $\\mathcal {Q} = (p^0,0,0,0,0,q_1,q_2,q_3)^t\\,, \\qquad \\mathcal {G} = (0,g^1,g^2,g^3,g_0,0,0,0)^t\\,.$ Assuming $g_0g^1g^2g^3<0$ and defining $A\\equiv (-g_0g^1g^2g^3)^{1/2}$ , the finite transformations $\\exp (\\alpha _1 {\\cal U}_1)$ and $\\exp (\\alpha _2 {\\cal U}_2)$ generated by (REF ) act as $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q} )\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}}, \\tilde{\\mathcal {G}},\\tilde{\\mathcal {Q}}) = (e^{\\alpha _1 {\\cal U}_1}\\mathcal {V}, \\mathcal {G}, e^{\\alpha _1 {\\cal U}_1}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1 = \\frac{A z^1\\cos (A\\alpha _1) + g_0 g^1\\sin (A\\alpha _1)}{A\\cos (A\\alpha _1) + z^1 g^2 g^3\\sin (A\\alpha _1)}\\,, \\\\&\\tilde{z}^2 = z^2\\,, \\\\&\\tilde{z}^3 = \\frac{A z^3\\cos (A\\alpha _1) - g_0 g^3\\sin (A\\alpha _1)}{A\\cos (A\\alpha _1) - z^3 g^1 g^2\\sin (A\\alpha _1)}\\,,\\end{split} $ $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q} )\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}}, \\tilde{\\mathcal {G}},\\tilde{\\mathcal {Q}}) = (e^{\\alpha _2 {\\cal U}_2}\\mathcal {V}, \\mathcal {G}, e^{\\alpha _2 {\\cal U}_2}\\mathcal {Q})\\,, \\\\&\\tilde{z}^1 = z^1\\,, \\\\&\\tilde{z}^2 = \\frac{A z^2\\cos (A\\alpha _2) + g_0 g^2\\sin (A\\alpha _2)}{A\\cos (A\\alpha _2) + z^2 g^1 g^3\\sin (A\\alpha _2)}\\,, \\\\&\\tilde{z}^3 = \\frac{A z^3\\cos (A\\alpha _2) - g_0 g^3\\sin (A\\alpha _2)}{A\\cos (A\\alpha _2) - z^3 g^1 g^2\\sin (A\\alpha _2)}\\,.\\end{split} $ Again, the expressions for $\\tilde{\\mathcal {Q}}$ are not particularly enlightening, so we shall not report them here.", "Notice that the transformations (REF ), (REF ) preserve the supersymmetry of the seed.", "As we pointed out in section REF , in the special case $\\mathcal {G} = (0,g,g,g,-g,0,0,0)^t$ there is an enhancement of the symmetry group to $\\text{U}(1)^3$ generated by (REF ).", "If we define $T=\\exp [\\frac{\\alpha _3}{3}(W_1+W_2+W_3)]$ , the action of the extra $\\text{U}(1)$ is $\\begin{split}&(\\mathcal {V}, \\mathcal {G}, \\mathcal {Q})\\quad \\longmapsto \\quad (\\tilde{\\mathcal {V}}, \\tilde{\\mathcal {G}},\\tilde{\\mathcal {Q}}) = (T\\mathcal {V}, \\mathcal {G}, T\\mathcal {Q} )\\,, \\\\&\\tilde{z}^1 = \\frac{z^1\\cos \\alpha _3 - \\sin \\alpha _3}{z^1\\sin \\alpha _3 + \\cos \\alpha _3}\\,, \\\\&\\tilde{z}^2 = \\frac{z^2\\cos \\alpha _3 - \\sin \\alpha _3}{z^2\\sin \\alpha _3 + \\cos \\alpha _3}\\,, \\\\&\\tilde{z}^3 = \\frac{z^3\\cos \\alpha _3 - \\sin \\alpha _3}{z^3\\sin \\alpha _3 + \\cos \\alpha _3}\\,,\\end{split} $ plus an expression for the charges $\\tilde{\\mathcal {Q}}$ .", "(REF ), (REF ) and (REF ) where first obtained in [10].", "Note that $T$ breaks supersymmetry, since it does not belong to the stabilizer $\\mathcal {S}_{\\mathcal {G}}$ .", "In fact, $T\\mathcal {G}\\equiv \\mathcal {G}_{\\alpha _3} = g(\\sin \\alpha _3, \\cos \\alpha _3, \\cos \\alpha _3, \\cos \\alpha _3,-\\cos \\alpha _3, \\sin \\alpha _3, \\sin \\alpha _3, \\sin \\alpha _3)^t\\,.$ However, the transformed solution still satisfies first-order non-BPS flow equations driven by the fake superpotential [18]Notice that this flow is a BPS flow for a theory with gaugings given by $\\mathcal {G}_{\\alpha _3}$ .", "$W = e^U\|\\langle \\tilde{Q}, \\tilde{\\mathcal {V}}\\rangle - i e^{2(\\psi - U)}\\langle \\mathcal {G}_{\\alpha _3},\\tilde{\\mathcal {V}}\\rangle \|\\,,$ provided the charge quantization condition $\\langle \\mathcal {G},\\mathcal {Q}\\rangle =-\\kappa $ holds, where $\\kappa =0,1,-1$ for flat, spherical or hyperbolic horizons respectively." ], [ "Prepotential $F=X^1X^2X^3/X^0 +\\frac{g^1g^2}{3(g^3)^2}(X^3)^3/X^0$", "In this case the only known solution with running scalars is that of [25], with static metric and purely imaginary scalar fields, $X^1 /X^0 = z^1 = -i\\lambda ^1\\,, \\qquad X^2/X^0 = z^2 = -i\\lambda ^2\\,, \\qquad X^3/X^0 = z^3= -i\\lambda ^3\\,.$ The charges and coupling constants are given by $\\mathcal {Q} = (p^0,0,0,0,0,q_1,q_2,q_3)^t\\,, \\qquad \\mathcal {G} = (0,g^1,g^2,g^3,g_0,0,0,0)^t\\,.$ Applying the finite transformation generated by (REF ) yields for the scalars $\\tilde{z}^1 = -i\\lambda ^1 - \\frac{g^1}{g^3} c\\,, \\qquad \\tilde{z}^2 = -i\\lambda ^2 - \\frac{g^2}{g^3} c\\,,\\qquad \\tilde{z}^2 = -i\\lambda ^3 +c \\,,$ and for the charges $\\tilde{\\mathcal {Q}}=\\left(\\begin{array}{c}p^0 \\\\-(c g^1 p^0)/g^3 \\\\-(c g^2 p^0)/g^3 \\\\c p^0 \\\\- (4 c^3 g^1 g^2 p^0)/(3 {g^3}^2) + (g^1 q_1 +g^2 q_2 -g^3 q_3)/g^3\\\\q_1 - c^2 g^2 p^0/g^3 \\\\q_2 - c^2 g^1 p^0/g^3 \\\\q_3 +2 c^2 g^1 g^2 p^0/{g^3}^2\\end{array}\\right)\\,,$ where $c$ is a group parameter.", "This solution is again BPS but has also nontrivial (constant) axions turned on and all charges are nonvanishing." ], [ "Extension to hypermultiplets", "In this section we briefly comment on a possible generalization of our work to include also hypermultiplets.", "In this case the situation is more involved, since the coupling constants are replaced by the moment maps $\\mbox{$\\mathcal {P}$}^x$ .", "However, when only abelian isometries of the quaternionic hyperscalar target space are gauged, the scalar potential can be cast into the form [31] $V = \\mathbb {G}^{AB}\\mathbb {D}_A\\mbox{$\\mathcal {L}$}\\,\\mathbb {D}_B\\bar{\\mbox{$\\mathcal {L}$}} - 3\|\\mbox{$\\mathcal {L}$}\|^2\\,,$ where we defined $\\mathbb {G}^{AB} = \\left(\\begin{array}{cc}g^{i\\bar{\\jmath }} & 0 \\\\ 0 & h^{uv}\\end{array}\\right), \\quad \\mathbb {D}_A = \\left(\\begin{array}{l} D_i \\\\\\mbox{$\\mathsf {D}$}_u\\end{array}\\right), \\quad \\mbox{$\\mathcal {L}$} = \\mbox{$\\mathcal {Q}$}^x\\mbox{$\\mathcal {W}$}^x, \\quad \\mathcal {Q}^x = \\langle \\mbox{$\\mathcal {P}$}^x, \\mbox{$\\mathcal {Q}$}\\rangle \\,, \\quad \\mbox{$\\mathcal {W}$}^x = \\langle \\mbox{$\\mathcal {P}$}^x, \\mbox{$\\mathcal {V}$}\\rangle \\,.$ Here, $h_{uv}$ denotes the metric on the quaternionic manifold, and $\\mbox{$\\mathsf {D}$}_u$ is the covariant derivative acting on the hyperscalars.", "The most general symmetry transformation of the nonlinear sigma model is a linear combination of the isometries of the quaternionic and the special Kähler manifold.", "Let us define the formal operator $\\delta = k^u\\mbox{$\\mathsf {D}$}_u + U \\mathcal {V} \\frac{\\delta }{\\delta \\mathcal {V}} + U \\bar{\\mathcal {V}} \\frac{\\delta }{\\delta \\bar{\\mathcal {V}}}+ U \\mathcal {A}_{\\mu } \\frac{\\delta }{\\delta \\mathcal {A}_{\\mu }}+ k^i\\partial _i + k^{\\bar{\\imath }}\\partial _{\\bar{\\imath }}\\,,$ where $k^u$ is a Killing vector of the quaternionic manifold, $U$ an element of the U-duality algebra, $k^i$ the corresponding holomorphic special Kähler Killing vector, and $ \\mathcal {A}_{\\mu } $ is the symplectic vector of the gauge potentials [31].", "Then it is clear from (REF ) that a sufficient condition for $\\delta V=0$ is $\\delta \\mathcal {L}=0$Note that, as in the FI case, $\\delta \\mathcal {L}=0$ is in general sufficient but not necessary., that holds if and only $k^u\\mbox{$\\mathsf {D}$}_u\\hat{\\mathcal {P}}^x = U \\hat{\\mathcal {P}}^x\\,, $ where we added a hat to the quaternionic quantities that define the gaugings.", "Moreover the invariance of the kinetic term of the hyperscalars [11] leads to $({\\cal L}_k \\hat{k})^v = U \\hat{k}^v\\,, $ where $\\cal L$ denotes the Lie-derivative.", "After choosing a specific model, these equations can in principle be solved for the parameters that define the linear combination of Killing vectors (REF ).", "In practice, (REF ) and (REF ) represent a highly constrained and very model-dependent system, and it is a priori not guaranteed that a nontrivial solution exists in general.", "In the FI limit, (REF ) boils down to the stabilization equation for the coupling constants $\\mathcal {G}$ and (REF ) is trivially satisfied, as it must be.", "An interesting class of these models are the $N=2$ truncations of M-theory described in [32], [33].", "In this case the solution of (REF ) and (REF ) could simplify the study of the attractor equations [31], necessary to work along the lines of [34], namely to compare the gravity side with the recent field theory results of [35], [36], [37]." ], [ "Conclusions", "In this paper we presented a geometric method to determine the residual symmetries in $N=2$ , $d=4$ $\\text{U}(1)$ Fayet-Iliopoulos gauged supergravity.", "It involves the stabilization of the symplectic vector of gauge couplings, i.e., the FI parameters, under the action of the U-duality symmetry of the ungauged theory.", "We then applied this to obtain the surviving symmetry group for a number of prepotentials frequently used in the string theory literature, and showed how this group can be used to produce hairy and dyonic black holes from a given seed solution.", "Moreover, we pointed out how our method may be extended to a more general setting including also gauged hypermultiplets.", "It would be very interesting to combine our results with dimensional reduction or oxidation as a solution-generating technique much like in the ungauged case discussed in the introduction.", "For instance one might think of starting from five-dimensional $N=2$ gauged supergravity coupled to vector multiplets and then reduce to $d=4$ along a Killing direction to get one of the models discussed here.", "One can then apply the residual symmetry group of the four-dimensional theory and subsequently lift back to $d=5$ to generate new solutions.", "Notice that, for a timelike dimensional reduction, the scalar manifold of the resulting Euclidean four-dimensional theory is para-Kähler rather than Kähler [38], so that our results can not be applied straightforwardly, but require some modifications.", "Another direction for future work could be to reduce gauged supergravity theories to three dimensions and study in general the surviving symmetry preserved by the scalar potential.", "Work along these directions is in progress [39]." ], [ "Acknowledgements", "This work was supported partly by INFN.", "We would like to thank A. Marrani, M. Nozawa, N. Petri and A. Santambrogio for useful discussions." ], [ "Reparametrization and invariances", "A symplectic reparametrization of the section $\\mathcal {V}$ for a prepotential $F=F(X)$ is a transformation $\\mathcal {V} = (X^\\Lambda , F_\\Lambda )^t\\longmapsto \\tilde{\\mathcal {V}} = (\\tilde{X}^\\Lambda ,\\tilde{F}_\\Lambda )^t\\,.$ In the new frame a prepotential does not necessarily exist.", "We are interested in the subgroup of $\\text{Sp}(2n_{\\text{V}}+2,\\mathbb {R})$ that leaves the prepotential invariant [40], [41], [42], $F(\\tilde{X}) = \\tilde{F}(\\tilde{X})\\,.$ Its algebra is determined by the equation $X^\\Lambda S_{\\Lambda \\Sigma } X^\\Sigma - F_\\Lambda R^{\\Lambda \\Sigma } F_\\Sigma -2 X^\\Lambda Q^t{}_\\Lambda {}^\\Sigma F_\\Sigma = 0\\,, $ where $Q$ , $R$ and $S$ parametrize the symplectic algebra, $U = \\left(\\begin{array}{cc} Q & R \\\\ S & -Q^t\\end{array}\\right)\\,, \\qquad R = R ^t\\,, \\qquad S = S^t\\,.$ A reparametrization of this type, in special projective coordinates, leaves $\\mbox{$\\mathcal {V}$}$ invariant up to a Kähler transformation." ], [ "Symplectic embedding", "The choice of the symplectic embedding of the non-linear sigma model isometry group is necessary to completely specify the special Kähler structure over a manifold [11], [23], [41], [40], [20].", "In what follows we shall summarize some properties used in the bulk of our paper." ], [ "Symplectically equivalent embeddings", "The way in which the isometry group is embedded in the symplectic group is fixed by supersymmetry, and in particular for $\\text{SU}(1,n_{\\text{V}})/(\\text{U}(1)\\times \\text{SU}(n_{\\text{V}}))$ and $\\text{SU}(1,1)/\\text{U}(1)\\times \\text{SO}(2,2)/(\\text{SO}(2)\\times \\text{SO}(2))$ one has respectively [23] $(\\mathbf {n_{\\text{V}} + 1})\\oplus (\\mathbf {n_{\\text{V}} +1})\\qquad \\mathrm {and}\\qquad \\mathbf {2}\\otimes (\\mathbf {4}\\oplus \\mathbf {4})\\,.", "$ This embedding is not unique since one can always act by conjugation with a symplectic matrix to construct a symplectically equivalent embedding.", "There are choices for the section $\\mathcal {V}$ such that the isometry group sits in the symplectic group in a simple way, but the existence of a prepotential in that frame is in general not guaranteed.", "On the other hand, many symplectically equivalent embeddings are encoded by different prepotentials.", "Two physically interesting examples are [43], [44] $\\mathcal {S}_1 = \\left(\\begin{array}{cccc}1 & 1 & 0 & 0 \\\\1 & -1 & 0 & 0 \\\\0 & 0 & \\frac{1}{2} & \\frac{1}{2} \\\\0 & 0 & \\frac{1}{2} & - \\frac{1}{2}\\end{array}\\right)\\,, \\qquad -i X^0 X^1\\longmapsto \\frac{i}{4} ({X^1}^2 - {X^0}^2)\\,, $ $\\mathcal {S}_2 = \\left(\\begin{array}{cccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\end{array}\\right)\\,, \\qquad -\\frac{X^1 X^2 X^3}{X^0}\\longmapsto -2i\\sqrt{X^0 X^1 X^2 X^3}\\,.$ A physically less important transformation, which is nevertheless useful for practical purposes, is for instance $\\mathcal {S}_a = \\left(\\begin{array}{cc}a & 0 \\\\ 0 & \\frac{1}{a}\\end{array}\\right)\\,, \\qquad \\frac{i}{4} X^\\Lambda \\eta _{\\Lambda \\Sigma } X^\\Sigma \\longmapsto \\frac{i}{4a^2} X^\\Lambda \\eta _{\\Lambda \\Sigma } X^\\Sigma \\,.$ One can also construct inequivalent embeddings over the same manifold, the simplest example being $\\text{SU}(1,1)/\\text{U}(1)$ [23].", "Notice finally that symplectic equivalence does not mean physical equivalence.", "Even if it is possible to construct maps between the solutions of symplectically equivalent models, in general the solutions are physically different." ], [ "Special Kähler structure over $\\text{SU}(1,n_{\\text{V}})/(\\text{U}(1)\\times \\text{SU}(n_{\\text{V}}))$", "For this noncompact version of $\\mathbb {C}\\text{P}^n$ a simple way to embed $\\text{SU}(1,n_{\\text{V}})$ into $\\text{Sp}(2n_{\\text{V}}+2,\\mathbb {R})$ is obtained from the fact that $\\text{Sp}(2n_{\\text{V}} + 2,\\mathbb {R})\\cong \\text{Usp}(1 + n_{\\text{V}}, 1 + n_{\\text{V}}) =\\text{Sp}(2n_{\\text{V}} + 2,\\mathbb {C})\\cap \\text{U}(1 + n_{\\text{V}}, 1 + n_{\\text{V}})\\,.$ This isomorphism is provided by conjugation with the Cayley matrix, $C_\\alpha :\\,\\text{Sp}(2n_{\\text{V}} + 2,\\mathbb {R})\\longrightarrow \\text{Usp}(1 + n_{\\text{V}}, 1 +n_{\\text{V}})\\,, \\qquad U\\longmapsto \\hat{\\mathcal {C}}_\\alpha U\\hat{\\mathcal {C}}_\\alpha ^{-1}\\,,$ where $\\hat{\\mathcal {C}}_\\alpha = \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc}\\frac{1}{\\sqrt{\\alpha }} I_{n_{\\text{V}} + 1} & i\\sqrt{\\alpha }\\eta \\\\\\frac{1}{\\sqrt{\\alpha }} I_{n_{\\text{V}} + 1} & -i\\sqrt{\\alpha }\\eta \\end{array}\\right)\\,,$ and $\\eta $ is the Minkowski metric in $n_{\\text{V}}+1$ dimensions.", "In fact $\\text{Usp}(1+n_{\\text{V}},1+n_{\\text{V}})$ is defined by the conditions ${\\mathcal {U}}\\mathbb {H}\\,{\\mathcal {U}}^\\dagger = \\mathbb {H}\\,,\\qquad {\\mathcal {U}}\\tilde{\\Omega }\\,{\\mathcal {U}}^t = \\tilde{\\Omega }\\,.", "$ If the invariant bilinear forms are chosen as $\\mathbb {H} = \\left(\\begin{array}{cc} \\eta & 0 \\\\ 0 & -\\eta \\end{array}\\right)\\,, \\qquad \\tilde{\\Omega } = \\left(\\begin{array}{cc} 0 & -\\eta \\\\ \\eta & 0\\end{array}\\right)\\,,$ (REF ) becomes ${\\mathcal {U}} = \\left(\\begin{array}{cc} A & C^* \\\\ C & A^\\end{array}\\right)\\,, \\qquad A\\eta A^\\dagger - C^\\eta C^t = \\eta \\,, \\qquad A^*\\eta C^t - C\\eta A^\\dagger = 0\\,.", "$ The first of (REF ) is obtained by restricting the action of $\\iota _\\alpha \\equiv C_\\alpha ^{-1}$ to the subgroup with $C=0$ .", "One can also explicitly verify that in this frame the prepotential exists and is given by $F=-\\frac{i}{2\\alpha }X^\\Lambda \\eta _{\\Lambda \\Sigma }X^\\Sigma $ ." ], [ "Special Kähler structure over\n$\\text{SU}(1,1)/\\text{U}(1)\\times \\text{SO}(2,2)/(\\text{SO}(2)\\times \\text{SO}(2))$", "This manifold belongs to the infinite sequence $\\text{SU}(1,1)/\\text{U}(1)\\times \\text{SO}(2,n)/(\\text{SO}(2)\\times \\text{SO}(n))$ , which for $n=2$ is isomorphic to $(\\text{SL}(2,\\mathbb {R})/\\text{SO}(2))^3$ .", "To find the symplectic embedding it is useful to choose a frame [23], [45], [46], [47] in which the symplectic section cannot be integrated to have a prepotential.", "In this frame the Calabi-Visentini parametrization appears in a natural way.", "The embedding problem is solved by $\\text{SO}(2,2)\\ni L\\longmapsto \\left(\\begin{array}{cc} L & 0 \\\\ 0 & {L^{-1}}^t\\end{array}\\right)\\in \\text{Sp}(8,\\mathbb {R})\\,,$ $\\text{SL}(2,\\mathbb {R})\\ni \\left(\\begin{array}{cc} a & b \\\\ c & d\\end{array}\\right)\\longmapsto \\left(\\begin{array}{cc} a & b \\hat{\\eta }\\\\ c \\hat{\\eta }& d\\end{array}\\right)\\in \\text{Sp}(8,\\mathbb {R})\\,,$ where $\\hat{\\eta }$ is the metric preserved by $\\text{SO}(2,2)$ .", "A symplectic transformation that leads to a frame in which a prepotential exists is highly nontrivial to find [23]." ] ]	1606.05160
[ [ "Chern insulating state in laterally patterned semiconductor\n heterostructures" ], [ "Abstract Hexagonally patterned two-dimensional $p$-type semiconductor systems are quantum simulators of graphene with strong and highly tunable spin-orbit interactions.", "We show that application of purely in-plane magnetic fields, in combination with the crystallographic anisotropy present in low-symmetry semiconductor interfaces, allows Chern insulating phases to emerge from an originally topologically insulating state after a quantum phase transition.", "These phases are characterized by pairs of co-propagating edge modes and Hall conductivities $\\sigma_{xy} = +\\frac{2 e^2}{h}, -\\frac{2 e^2}{h}$ in the absence of Landau levels or cyclotron motion.", "With current lithographic technology, the Chern insulating transitions are predicted to occur in GaAs heterostructures at magnetic fields of $\\sim 5\\text{T}$." ], [ "Calculation of edge states for a hard-wall potential", "The edge states shown in Fig.", "1 were evaluated for a hard-wall potential along the $x$ -direction, $V(y > 0) = 0 \\ \\ , \\ \\ V(y < 0) = \\infty $ .", "The momentum along the edge $q_x$ is a good quantum number, and the wavefunction has the form $\\psi = C_+ e^{i q_x x - \\kappa _+ y} \\psi _+ + C_- e^{i q_x x - \\kappa _- y} \\psi _-$ where $\\kappa _+, \\kappa _-$ are generally complex, with the wavefunction being exponentially localized at the edge when the hard-wall condition $\\psi (x, y= 0)$ is satisfied and the energy lies in a range where $\\text{Re} \\kappa _\\pm > 0$ .", "The states $\\psi _\\pm $ are eigenvectors of the Hamiltonian (6) with energy $E$ and complex $q_y = i \\kappa _\\pm $ satisfying $\\kappa _\\pm =\\nonumber \\\\( v^2 q_x^2 - E^2 - \\gamma ^2 - \\eta ^2 \\pm 2i \\sqrt{ -\\gamma ^2 E^2 - b^2 ((E - \\eta )^2 - \\gamma ^2) })^{\\frac{1}{2} } \\ \\ .$ Taking into account the real-space structure of the basis states (5), the boundary condition implies that $( \\langle a, s_z\| \\psi _+ \\rangle + \\langle b, s_z\| \\psi _+ \\rangle ) C_+ + ( \\langle a, s_z \| \\psi _- \\rangle + \\langle b, s_z \| \\psi _-\\rangle ) C_-\\nonumber \\\\= 0$ for $s_z = \\pm \\frac{1}{2}$ .", "Solutions $C_+, C_-$ exist when $f(q_x, E) =\\tau _z \\gamma E(E - \\eta + v \\tau _z q_x)(\\kappa _+ - \\kappa _-) +\\nonumber \\\\i \\sqrt{- \\gamma ^2 E^2 - \\beta ^2((E - \\eta )^2 - \\gamma ^2)}\\times \\nonumber \\\\ ( (E - \\eta + v \\tau _z q_x)(E - \\eta + v \\tau _z q_x) \\nonumber \\\\- \\kappa _+ \\kappa _-+ \\tau _z \\gamma (\\kappa _+ + \\kappa _-) - \\gamma ^2 ) = 0 \\ \\ .$ Regarding $\\kappa _\\pm $ as a function of $q_x, E$ , the equation $f(q_x, E) = 0 $ provides the implicit dispersion relation of the one-dimensional edge states.", "There are two solutions for $\\tau _z = +1$ which close the gap and propagate to the left, $\\frac{dE}{dq_x} < 0$ , in addition to a pair of solutions for $\\tau _z= -1$ corresponding to counter-propagating modes which disappear when the energy becomes sufficiently close to the lower ($E_3(\\mathbf {q})$ ) band.", "The energy can be chosen to intersect only the co-propagating modes.", "The edge states in Fig.", "1 correspond to modes which propagate along the bottom edge of the sample.", "We may calculate the edge states for the top edge in the same way, taking the edge potential $V(y < L_y = 0), V(y > L_y) = \\infty $ where $L_y= \\frac{ \\sqrt{3} na}{2}, n = 1, 2, \\dots $ .", "In order for these states to be exponentially localized, we require $q_y = -i \\kappa _\\pm $ with $\\text{Re} \\kappa _\\pm > 0$ .", "This implies that the condition for the existence of edge states on the top edge is identical to Eq.", "(REF ) after substitution $\\kappa _\\pm \\rightarrow - \\kappa _\\pm $ .", "Equivalently, we may perform a substitution $q_x \\rightarrow - q_x, \\tau _z \\rightarrow - \\tau _z$ to obtain the dispersion of edge states along the top of the system.", "This implies that the 1D modes at the top of the system will exist at opposite valleys and propagate in the opposite direction to the modes along the bottom of the system.", "The modes along the left and right edges, corresponding to hard-wall potentials $V_L(x < 0) = +\\infty , V_L(x > 0) = 0$ and $V_R(x > L_x) = +\\infty , V_R(x < L_x ) = 0$ with $L_x = \\frac{ n a}{2}, n=1, 2, \\dots $ , may be calculated in a similar fashion using the boundary conditions $\\psi (x = 0) = \\psi (x = L_x) = 0$ .", "The edge states have identical structure to those along the top and bottom, with the left mode propagating along the $+y$ direction and the right mode propagating along the $-y$ direction." ] ]	1606.05098
[ [ "A Complete Characterization of Determinantal Quadratic Polynomials" ], [ "Abstract The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems.", "In this paper we provide a necessary and sufficient condition for the existence of \\textit{monic Hermitian determinantal representation} as well as \\textit{monic symmetric determinantal representation} of size $2$ for a given quadratic polynomial.", "Further we propose a method to construct such a monic determinantal representtaion (MDR) of size $2$ if it exists.", "It is known that a quadratic polynomial $f(\\x)=\\x^{T}A\\x+b^{T}\\x+1$ has a symmetric MDR of size $n+1$ if $A$ is \\textit{negative semidefinite}.", "We prove that if a quadratic polynomial $f(\\x)$ with $A$ which is not negative semidefinite has an MDR of size greater than $2$, then it has an MDR of size $2$ too.", "We also characterize quadratic polynomials which exhibit diagonal MDRs." ], [ "Introduction", "This paper deals with characterization of quadratic real multivariate polynomials which admit monic Hermitian (symmetric) determinantal representations, that is polynomials which can be written as the determinant of a monic linear matrix polynomial (LMP) whose coefficient matrices are Hermitian (symmetric) and the constant coefficient matrix is the identity matrix.", "Note that the coefficient matrices of the LMP could be of any order greater than two.", "In particular, in this paper, we focus on the existence and determination of a monic LMP whose coefficient matrices are Hermitian (symmetric) and of order 2 for a given quadratic polynomial.", "Besides, we identify the class of quadratic polynomials for which an MDR of size greater than 2 ensures the existence of an MDR of size 2 respectively.", "Determinantal representations of polynomials have generated a lot of interest due to its connection with the problem of determining (definite) LMI representable sets, also known as spectrahedra [2] which play a crucial role in optimization problems.", "Indeed, if the feasible set of an optimization problem is a definite LMI representable set, the problem can be transformed into a semidefinite programming (SDP) problem which in turn can be efficiently solved by SDP solvers.", "It is important to recall that any polynomial can be expressed as the determinant of a symmetric LMP [9].", "It is also known that the algebraic interior defined by a real zero (RZ) quadratic polynomial is always a spectrahedron, since a Hermitian determinantal representation can be obtained for higher powers of RZ quadratic polynomials using Clifford algebra [7] and sum of squares (SOS) decomposition of a parametrized Hermite matrix [6].", "To the best of authors' knowledge, characterization of quadratic polynomials that have an MDR of size 2 has not been done before.", "In this paper, we provide a necessary and sufficient condition for the existence of MDRs of size 2 for any quadratic polynomial.", "We also propose a constructive method to determine these MDRs.", "We show that for a certain sub-class of quadratic polynomials that have MDRs of size 2, there are precisely two non-equivalent classes of MDRs, whereas for all other quadratic polynomials that have MDRs of size 2, all MDRs are unitarily equivalent.", "Recall that a quadratic polynomial $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T} {\\mathbf {x}}+ 1$ admits a symmetric MDR of size $n+1$ if the corresponding matrix $A$ is negative semidefinite [10], [1].", "This need not imply the existence of MDR of size 2 for the same polynomial.", "Therefore, it is natural to ask whether quadratic polynomials which have an MDR of size 2 can be characterized.", "We further characterize all quadratic polynomials that have an MDR of any size [Sec ].", "We show that if a quadratic polynomial $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T} {\\mathbf {x}}+ 1$ has an MDR, then either $A$ is negative semidefinite or $f({\\mathbf {x}})$ admits an MDR of size 2.", "In other words, if $f({\\mathbf {x}})$ has an MDR, but $A$ is not negative semidefinite, then $f({\\mathbf {x}})$ has an MDR of size 2." ], [ "Preliminaries", "We begin with the concept of definite LMI representable sets and its relation to monic determinantal representations.", "A set $S \\subseteq {\\mathbb {R}}^{n}$ is said to be LMI representable if $ S= \\lbrace {\\mathbf {x}}\\in {\\mathbb {R}}^{n} : A_{0} +x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n} \\succeq 0 \\rbrace $ for some real symmetric matrices $A_{i}, i=0,\\dots ,n$ and ${\\mathbf {x}}=(x_{1}, \\dots , x_{n})^{T}$ .", "If $A_{0}\\succ 0$ , the set $S$ is called a definite LMI representable set whereas if $A_{0}=I$ , $S$ is known to be a monic LMI representable set.", "By $A \\succ 0 (\\succeq 0)$ we mean that the matrix $A$ is positive (semi)-definite.", "A spectrahedron is another name used for an LMI representable set.", "It is evident that a spectrahedron is both convex and basic closed semialgebraic set.", "Moreover, if a spectrahedron has a nonempty interior, it is a definite LMI representable set [[10],section $1.4$ ], [7] – without loss of generality, the origin may be considered as an interior point of the set.", "It is also known that a definite LMI representable set is always monic LMI representable [5].", "A polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ is said to have a determinantal representation if $f({\\mathbf {x}})$ is the determinant of a linear matrix polynomial (LMP) i.e., $ f({\\mathbf {x}}) = \\det (A_{0}+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}), \\mbox{ where } A_{i} \\in {\\mathbb {H}}^{k \\times k}({\\mathbb {C}}) \\ \\mbox{for some} \\ k.$ If the matrices $A_{i} \\in S{\\mathbb {R}}^{k \\times k}$ (symmetric matrices of order $k$ ), then the polynomial has a symmetric determinantal representation.", "Note that as $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ , the matrices $A_{i}$ could have been Hermitian matrices too.", "Therefore, if the matrices $A_{i} \\in {\\mathbb {H}}^{k \\times k}({\\mathbb {C}})$ (Hermitian matrices of order $k$ ), then the polynomial is said to have a Hermitian determinantal representation.", "When the matrices $A_{i}$ s are of size $k$ , we call $k$ the size of the determinantal representation.", "The determinantal representation is definite if $A_{0} \\succ 0$ .", "Further if $A_{0} = I_{k}$ , the identity matrix of order $k$ , then we have a monic determinantal representation (MDR).", "Throughout the paper, we are interested in monic determinantal representations of polynomials using either symmetric or Hermitian matrices.", "Therefore, we use the acronyms MSDR and MHDR for monic symmetric determinantal representation and monic Hermitian determinantal representation, respectively.", "If all $A_{i} \\in S{\\mathbb {R}}^{k \\times k}$ are diagonal, then the polynomial is said to have a diagonal determinantal representation.", "It is obvious that a polynomial which admits a definite determinantal representation can be scaled in order to admit a monic determinantal representation.", "Without loss of generality, throughout this paper we consider only problems dealing with monic determinantal representations and hence we consider only those polynomials whose constant coefficient is 1, unless stated otherwise.", "If a polynomial $f({\\mathbf {x}})$ also admits an MDR, i.e., $f({\\mathbf {x}})=\\det (I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n})$ , then $f({\\mathbf {x}})>0$ when ${\\mathbf {x}}\\in \\mbox{Int}(S)$ where the spectrahedron $S=\\lbrace {\\mathbf {x}}: I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}\\succeq 0\\rbrace $ and $f({\\mathbf {x}})=0$ when ${\\mathbf {x}}\\in \\partial S$ .", "On the other hand, given $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ , a closed subset $C_{f}$ of ${\\mathbb {R}}^{n}$ is called an algebraic interior associated with $f$ if it is the closure of a (arcwise) connected component of $\\lbrace {\\mathbf {x}}\\in {\\mathbb {R}}^{n}: f({\\mathbf {x}}) > 0\\rbrace $ .", "The polynomial $f$ is called a defining polynomial for $C_{f}.$ Consequently, if $f({\\mathbf {x}})$ has an MDR, $C_{f}=S$ [5].", "But the converse of this statement need not be true [5].", "One way to characterize monic (definite) LMI representable sets is by identifying polynomials which have monic (definite) symmetric or Hermitian determinantal representations [8], [5].", "A recent literature survey in this area can be found in [11].", "It is to be noted that amongst all spectrahedra, those defined by a LMI (REF ) which have $A_{0} \\succ 0$ are special, as problems related to these spectrahedra can be connected to semidefinite programming.", "A multivariate polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ is said to be a real zero (RZ) polynomial if the polynomial has only real zeros when restricted to any line passing through origin i.e., for any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ , all the roots of the univariate polynomial $f_{{\\mathbf {x}}}(t):=f(t\\cdot {\\mathbf {x}})$ are real (and $f(0) \\ne 0$ ).", "If a polynomial $f({\\mathbf {x}})$ admits an MDR, say $f({\\mathbf {x}})=\\det (I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n})$ then it is indeed a RZ polynomial.", "This follows from the fact that the univariate polynomial $f_{{\\mathbf {x}}}(t):=\\det (I+t(x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}))$ has only real zeros which are in fact the negatives of the reciprocals of non-zero eigenvalues of the Hermitian or symmetric matrix $x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}$ for any given ${\\mathbf {x}}\\in {\\mathbb {R}}^n.$ It has been proved that any RZ bivariate polynomial always has an MDR [5].", "However, if the number of variables of a RZ polynomial is more than 2, it may not have an MDR at all.", "For example, dehomogenized polynomial of Vamos cube $V_{8}$ is a RZ polynomial without a definite determinantal representation [3]." ], [ "Polynomials with Monic Determinantal Representations of size 2", "In this section, our aim is to characterize all sets which are $2 \\times 2$ monic LMI representable and to identify all quadratic polynomials which admit an MDR of size 2.", "As RZ property is a necessary condition for a polynomial to have an MDR, we begin with RZ quadratic polynomials." ], [ "RZ property for Quadratic Polynomials", "It is well known that any quadratic polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ (where ${\\mathbf {x}}=(x_{1}, \\dots , x_{n})$ ) can be written as $f = Z^{T}[{\\mathbf {x}}] Q Z[{\\mathbf {x}}]$ where $Q \\in {\\mathbb {R}}^{(n+1) \\times (n+1)}$ and $Z[{\\mathbf {x}}] =\\left[ \\begin{matrix} 1 & x_{1} & \\dots & x_{n} \\end{matrix} \\right]^{T}$ .", "Such a matrix $Q$ associated with $f$ is unique if $Q \\in S{\\mathbb {R}}^{(n+1) \\times (n+1)}$ , – this matrix $Q$ is referred to as the matrix representation of the polynomial $f({\\mathbf {x}})$ .", "The following proposition provides a necessary and sufficient condition for a quadratic polynomial to be a RZ polynomial that shall be used in sequel.", "Proposition 3.1 Let $f({\\mathbf {x}})=Z^{T}[{\\mathbf {x}}] Q Z[{\\mathbf {x}}] \\in {\\mathbb {R}}[{\\mathbf {x}}]$ be a quadratic polynomial with nonzero constant term $c$ , and let $Q \\in S{\\mathbb {R}}^{(n+1) \\times (n+1)}$ be the matrix representation of $f({\\mathbf {x}})$ .", "Then $f({\\mathbf {x}})$ is a RZ polynomial if and only if the Schur complement of $Q$ with respect to the $(1,1)$ element $c$ of $Q$ is negative semidefinite.", "Proof: Note that $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ can be written as $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T}{\\mathbf {x}}+c,$ where $c=f(0) \\ne 0$ .", "Then $ f({\\mathbf {x}})= \\left[\\begin{array} {cc}1 & {\\mathbf {x}}\\end{array}\\right] \\left[\\begin{array} {cc}c & b^{T}/2 \\\\b/2 & A\\end{array}\\right] \\left[\\begin{array} {c}1 \\\\{\\mathbf {x}}\\\\\\end{array}\\right] = Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}].$ The Schur complement of $(1,1)$ element $c$ in the matrix representation $Q$ is $A- \\frac{b}{2}(1/c)\\frac{b^{T}}{2}$ .", "As $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T}{\\mathbf {x}}+c$ , so $f_{{\\mathbf {x}}}(t)=f(t{\\mathbf {x}}) = t^{2}{\\mathbf {x}}^{T}A{\\mathbf {x}}+ t b^{T}{\\mathbf {x}}+c$ , $t\\in {\\mathbb {R}}$ .", "Therefore, for any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n},$ $&\\mbox{the roots of polynomial} \\ f_{{\\mathbf {x}}}(t) \\ \\mbox{ are real} \\\\\\Leftrightarrow &(b^{T}{\\mathbf {x}})^{2} - 4c {\\mathbf {x}}^{T}A{\\mathbf {x}}\\ge 0 \\\\\\Leftrightarrow &{\\mathbf {x}}^{T}(bb^{T}- 4Ac){\\mathbf {x}}\\ge 0 \\\\&\\Leftrightarrow 4Ac - bb^{T} \\preceq 0.$ $\\Box $ Any polynomial admitting an MDR is a RZ polynomial, but the converse need not be true.", "We give an example below where the converse does not hold.", "Example 3.2 [7] The (shifted hyperbolic) polynomial $f({\\mathbf {x}})=(x_{1}+1)^{2} - x_{2}^{2} - x_{3}^{2} - x_{4}^{2}$ is a quadratic RZ polynomial which has no MSDR, but it has an MHDR of size 2.", "On the other hand the polynomial $f({\\mathbf {x}})=1-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}$ has no MHDR (so it can not have an MSDR too).", "We now provide a necessary and sufficient condition for the existence of MDR of size 2 for RZ quadratic polynomials." ], [ "Quadratic polynomials having MDR of size 2", "In this subsection, we provide a necessary and sufficient condition for a quadratic polynomial to have an MDR of size 2.", "We further provide an algorithm to determine MDRs for a quadratic polynomial when they exist.", "Note that since we are interested in monic representations, therefore the constant term of the quadratic polynomial must be one.", "Henceforth, for a quadratic polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ we denote the Schur complement of its matrix representation $Q$ with respect to the $(1,1)$ element by $Q/(1,1)$ .", "Theorem 3.3 A quadratic polynomial $f({\\mathbf {x}})=Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}] \\in {\\mathbb {R}}[{\\mathbf {x}}]$ with $f(0)=1$ has an MDR of size 2 if and only if $Q/(1,1)$ is negative semidefinite and the rank$(Q/(1,1)) \\le 3$ .", "If $Q/(1, 1)$ is negative semidefinite and the rank$(Q/(1,1)) = 3$ , then the polynomial has Hermitian MDR but no symmetric MDR.", "For symmetric MDR, $Q/(1,1)$ must be negative semidefinite and rank$(Q/(1, 1)) \\le 2$ .", "Proof: Suppose $f({\\mathbf {x}})=\\det (I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n})$ where $ \\nonumber A_{j}= \\left[\\begin{array} {cc}r_{j} & t_{j}-iu_{j} \\\\t_{j}+iu_{j} & s_{j}\\end{array}\\right] \\in {\\mathbb {H}}^{2 \\times 2}({\\mathbb {C}}), j=1,\\dots ,n.$ Let ${\\mathbf {r}}=\\left[ \\begin{matrix} 1 & r_{1} & \\dots & r_{n} \\end{matrix} \\right]^{T}, {\\mathbf {s}}=\\left[ \\begin{matrix} 1 & s_{1} & \\dots & s_{n} \\end{matrix} \\right]^{T}, {\\mathbf {t}}=\\left[ \\begin{matrix} 0 & t_{1} & \\dots & t_{n} \\end{matrix} \\right]^{T}, {\\mathbf {u}}=\\left[ \\begin{matrix} 0 & u_{1} & \\dots & u_{n} \\end{matrix} \\right]^{T}$ Consider the truncated vectors $\\tilde{{\\mathbf {r}}}=\\left[ \\begin{matrix} r_{1} & \\dots & r_{n} \\end{matrix} \\right]^{T}, \\tilde{{\\mathbf {s}}}=\\left[ \\begin{matrix} s_{1} & \\dots & s_{n} \\end{matrix} \\right]^{T}, \\tilde{{\\mathbf {t}}}=\\left[ \\begin{matrix} t_{1} & \\dots & t_{n} \\end{matrix} \\right]^{T}, \\tilde{{\\mathbf {u}}}=\\left[ \\begin{matrix} u_{1} & \\dots & u_{n} \\end{matrix} \\right]^{T}$ of ${\\mathbf {r}},{\\mathbf {s}},{\\mathbf {t}}$ and ${\\mathbf {u}}$ respectively.", "Note that $ \\nonumber Q = (1/2)({\\mathbf {r}}{\\mathbf {s}}^{T} + {\\mathbf {s}}{\\mathbf {r}}^{T}) + (-{\\mathbf {t}}{\\mathbf {t}}^{T})+ (-{\\mathbf {u}}{\\mathbf {u}}^{T}).$ Therefore, rank$(Q)\\le 4$ , as $Q$ is the sum of four rank one matrices.", "Consequently, $Q/(1,1)$ is given by $&(1/2)(\\tilde{{\\mathbf {r}}}\\tilde{{\\mathbf {s}}}^{T} + \\tilde{{\\mathbf {s}}}\\tilde{{\\mathbf {r}}}^{T}) + (-\\tilde{{\\mathbf {t}}} \\tilde{{\\mathbf {t}}}^{T})+(-\\tilde{{\\mathbf {u}}} \\tilde{{\\mathbf {u}}}^{T})+ [- (\\frac{\\tilde{{\\mathbf {r}}}+\\tilde{{\\mathbf {s}}}}{2}) (\\frac{\\tilde{{\\mathbf {r}}}+\\tilde{{\\mathbf {s}}}}{2})^{T}]\\\\&= -((\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2}) (\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2})^{T})+ (-\\tilde{{\\mathbf {t}}} \\tilde{{\\mathbf {t}}}^{T})+ (-\\tilde{{\\mathbf {u}}} \\tilde{{\\mathbf {u}}}^{T}).$ Thus rank$(Q/(1,1)) \\le 3$ follows from the fact that $Q/(1,1)$ is the sum of three rank one matrices.", "It is known that if a quadratic polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ has an MDR, then $f({\\mathbf {x}})$ is a RZ polynomial.", "By Proposition REF , $Q/(1,1)$ is negative semidefinite.", "Hence, if a quadratic polynomial $f({\\mathbf {x}})$ with $f(0)=1$ has an MDR of size 2, then rank$(Q/(1,1))\\le 3$ and $Q/(1,1)$ is a negative semidefinite matrix.", "Conversely, suppose $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ is a quadratic polynomial with $f(0)=1$ and $Q/(1,1)$ is a negative semidefinite matrix such that rank$(Q/(1,1)) \\le 3$ .", "Since $Q/(1,1)$ is the Schur complement with respect to the $(1, 1)$ element of the matrix $Q=\\left[ \\begin{matrix} 1 & b^{T}/2 \\\\ b/2 & A \\end{matrix} \\right]$ which represents the quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A {\\mathbf {x}}+b^{T} {\\mathbf {x}}+1$ , so we have $Q/(1,1)= A-(1/4)bb^{T}$ .", "Thus to obtain an MDR we need to find the vectors $\\tilde{{\\mathbf {r}}},\\tilde{{\\mathbf {s}}},\\tilde{{\\mathbf {t}}}$ and $\\tilde{{\\mathbf {u}}}$ , which were defined earlier.", "Since rank of $(Q/(1,1))$ is at most 3, we can obtain $-(Q/(1,1))$ as sum of three rank one matrices $\\alpha _{1}\\alpha _{1}^{T}+\\alpha _{2}\\alpha _{2}^{T}+\\alpha _{3}\\alpha _{3}^{T}$ (for example, by using the Cholesky decomposition).", "In fact, $ \\nonumber -(Q/(1,1))=(1/4)b b^{T}-A=((\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2}) (\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2})^{T}) +(\\tilde{{\\mathbf {t}}} \\tilde{{\\mathbf {t}}}^{T})+(\\tilde{{\\mathbf {u}}} \\tilde{{\\mathbf {u}}}^{T})$ Note that $\\tilde{{\\mathbf {r}}}+\\tilde{{\\mathbf {s}}}=b$ where $b$ is defined in the matrix representation $Q$ .", "Therefore one can obtain a Hermitian MDR by setting $(1/2)(\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}})=\\alpha _{1}, \\tilde{{\\mathbf {t}}}=\\alpha _{2}$ , and $\\tilde{{\\mathbf {u}}}=\\alpha _{3}$ , and solving this system of linear equations along with $\\tilde{{\\mathbf {r}}}+\\tilde{{\\mathbf {s}}}=b$ .", "Note that for a symmetric MDR, the coefficient matrices $A_{j}$ are of the form $\\left[ \\begin{matrix} r_{j} & t_{j}\\\\t_{j} & s_{j} \\end{matrix} \\right]$ which make $A_{j}$ s symmetric.", "Hence from the proof above, it is clear that the vector $\\tilde{{\\mathbf {u}}}$ must now be the zero vector.", "Thus $Q/(1,1) $ must be the sum of two rank one matrices and therefore the rank of matrix $Q/(1,1) \\le 2$ .", "$\\square $ As a consequence of the above theorem we have the following corollary which provides a necessary and sufficient condition for a quadratic polynomial to have a diagonal MDR of size 2.", "Corollary 3.4 A quadratic polynomial $f({\\mathbf {x}})=Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}], f(0)=1$ has a diagonal MDR of size 2 if and only if $Q/(1,1)$ is negative semidefinite and of rank at most 1.", "Proof: The proof follows from the proof of the above theorem by setting $\\tilde{{\\mathbf {t}}}=0$ and $\\tilde{{\\mathbf {u}}}=0$ .", "${\\square }$ Based on the constructive proof of Theorem REF , we now provide an algorithm to determine MDR of size 2 for a quadratic polynomial, whenever it exists.", "Algorithm to find MHDR (MSDR) of size 2 Input: a quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1$ Output: $A_{j}=\\left[ \\begin{matrix} r_{j} & t_{j}-iu_{j} \\\\ t_{i}+iu_{j} & s_{i} \\end{matrix} \\right], j=1,\\dots ,n$ such that $f({\\mathbf {x}})=\\det \\left(I+x_{1}A_{1}+\\dots +x_{n}A_{n}\\right)$ .", "Form the matrix representation $Q$ of $f({\\mathbf {x}})$ Calculate $Q/(1,1)=A-\\frac{1}{4} bb^{T}$ Compute the Cholesky factor of $-Q/(1,1)$ .", "If Cholesky factor does not exist, no MHDR (MSDR) of size 2 possible.", "If rank$(Q/(1,1)) > 3$ , then exit – no MDR of size 2 possible Otherwise, $Q/(1,1)=\\alpha _{1}\\alpha _{1}^{T}+\\alpha _{2}\\alpha _{2}^{T}+\\alpha _{3}\\alpha _{3}^{T}$ .", "If rank $(Q/(1,1)) = 3$ , then construct a Hermitian MDR of size 2 by setting $r_{j}=(\\frac{(2\\alpha _{1}+b)}{2})_{j}, s_{j}=(\\frac{(b-2\\alpha _{1})}{2})_{j}, t_{j}=(\\alpha _{2})_{j}, u_{j}=(\\alpha _{3})_{j}$ .", "Construct $A_{j}=\\left[ \\begin{matrix} r_{j} & t_{j}-iu_{j} \\\\ t_{j}+iu_{j} & s_{j} \\end{matrix} \\right]$ for $j=1, \\dots ,n$ .", "If rank $(Q/(1,1)) \\le 2$ , then obtain a symmetric MDR by setting $u_{j}=0, r_{j}=(\\frac{(2\\alpha _{2}+b)}{2})_{j}, s_{j}=(\\frac{(b-2\\alpha _{2})}{2})_{j}$ , and $t_{j}=(\\alpha _{1})_{j}$ Assign $A_{j}=\\left[ \\begin{matrix} r_{j} & t_{j} \\\\ t_{j} & s_{j} \\end{matrix} \\right]$ for $j=1, \\dots ,n$ .", "We demonstrate the algorithm in the following examples.", "Example 3.5 We provide three examples below.", "Consider $f({\\mathbf {x}})= 1-8 x_{1} x_{2} -4 x_{1}x_{3} -100 x_{2}^{2}-12 x_{2} x_{3} -x_{3}^{2} -5 x_{1}^{2}$ .", "Then $f({\\mathbf {x}})=Z^T[{\\mathbf {x}}]Q Z[{\\mathbf {x}}]$ where $ Q =\\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\ 0 & -5 & -4 & -2 \\\\0 & -4 & -100 & -6 \\\\ 0 & -2 & -6 & -1 \\end{matrix} \\right].$ Then it is easy to verify that $Q/(1,1)$ is negative semidefinite and the rank of $Q/(1,1)$ is $2.$ Indeed, $-Q/(1,1) &=\\left[ \\begin{matrix} 5 & 4 & 2 \\\\4 & 100 & 6 \\\\2 & 6 & 1 \\end{matrix} \\right]=\\left[ \\begin{matrix} 2/5 \\\\ 10 \\\\ 3/5 \\end{matrix} \\right] \\left[ \\begin{matrix} 2/5 & 10 & 3/5 \\end{matrix} \\right] + \\left[ \\begin{matrix} 11/5 \\\\ 0 \\\\ 4/5 \\end{matrix} \\right]\\left[ \\begin{matrix} 11/5 & 0 & 4/5 \\end{matrix} \\right]$ Hence $f({\\mathbf {x}})$ admits an MSDR.", "By applying Algorithm 1, one obtains $f({\\mathbf {x}}) =\\det \\left( I+x_{1}\\left[ \\begin{matrix} 11/5 & 2/5 \\\\ 2/5 & -11/5 \\end{matrix} \\right] + x_{2}\\left[ \\begin{matrix} 0 & 10 \\\\ 10 & 0 \\end{matrix} \\right] + x_{3} \\left[ \\begin{matrix} 4/5 & 3/5 \\\\ 3/5 & -4/5 \\end{matrix} \\right] \\right),$ Consider $f({\\mathbf {x}})= 1+4x_{1}-10x_{2}-x_{1}^{2}-2x_{1}x_{2}-x_{2}^{2}$ which has a Hermitian determinantal representation provided in the chapter [4].", "In fact $f({\\mathbf {x}})$ admits a symmetric MDR, since $Q=\\left[ \\begin{matrix} 1 & 2 & 5 \\\\ 2 & -1 & -1 \\\\5 & -1 & -1 \\end{matrix} \\right]$ and $Q/(1,1)$ is negative semidefinite of rank 2.", "$-Q/(1,1)=\\left[ \\begin{matrix} 5 & 11 \\\\ 11 & 26 \\end{matrix} \\right]=\\left[ \\begin{matrix} 2.2316 \\\\4.9193 \\end{matrix} \\right]\\left[ \\begin{matrix} 2.2316 & 4.9193 \\end{matrix} \\right] + \\left[ \\begin{matrix} 0 \\\\ 1.3416 \\end{matrix} \\right]\\left[ \\begin{matrix} 0 & 1.3416 \\end{matrix} \\right].$ Then by Algorithm 1, $f({\\mathbf {x}}) =\\det \\left(I+x_{1}\\left[ \\begin{matrix} 2 & 2.2361 \\\\ 2.2361 & 2 \\end{matrix} \\right]+x_{2}\\left[ \\begin{matrix} 6.3416 & 4.9193 \\\\4.9193 & 3.6584 \\end{matrix} \\right]\\right).$ Consider the quadratic polynomial $f({\\mathbf {x}})=(x_{1}+1)^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}$ .", "Here matrix representation $Q=\\left[ \\begin{matrix} 1 & 1 & 0 & 0 &0\\\\1 & 1 & 0 & 0 & 0\\\\0 & 0 & -1 & 0 & 0\\\\0 & 0 & 0 & -1 & 0\\\\0 & 0 & 0 & 0 & -1 \\end{matrix} \\right], -Q/(1,1)=\\left[ \\begin{matrix} 0 & 0 & 0 & 0\\\\0 & 1 & 0 &0\\\\0& 0 & 1 & 0\\\\0 & 0 & 0 &1 \\end{matrix} \\right]=\\alpha _{1}\\alpha _{1}^{T}+\\alpha _{2}\\alpha _{2}^{T}+\\alpha _{3}\\alpha _{3}^{T}.$ where one could take $\\alpha _{1}=\\left[ \\begin{matrix} 0\\\\0\\\\0\\\\1 \\end{matrix} \\right],\\alpha _{2}=\\left[ \\begin{matrix} 0\\\\0\\\\1\\\\0 \\end{matrix} \\right],\\alpha _{3}=\\left[ \\begin{matrix} 0\\\\1\\\\0\\\\0 \\end{matrix} \\right].$ Since the rank of the matrix is 3, there is no symmetric MDR, but then Hermitian MDR does exist.", "Coefficient matrices for a Hermitian MDR are as follows.", "$A_{1}=\\left[ \\begin{matrix} 1 & 0\\\\0 & 1 \\end{matrix} \\right], A_{2}=\\left[ \\begin{matrix} 0 & -i\\\\i & 0 \\end{matrix} \\right], A_{3}= \\left[ \\begin{matrix} 0 & 1\\\\1 & 0 \\end{matrix} \\right], A_{4}=\\left[ \\begin{matrix} 1 & 0\\\\0 & -1 \\end{matrix} \\right].$" ], [ "Equivalent and Non-equivalent MSDRs and MHDRs", "Note that in the algorithm given above, there is considerable freedom in constructing the MDR.", "For one, the choice of the vectors $\\alpha _{1}, \\alpha _{2},\\alpha _{3}$ to express $Q/(1,1)$ as the sum of three rank one matrices is immense.", "Each such choice, leads to a different MDR.", "Two linear matrix polynomials $A_{0}+x_{1}A_{1}+\\dots +x_{n}A_{n}$ and $B_{0}+x_{1}B_{1}+\\dots +x_{n}B_{n}$ of size $k$ are said to be unitarily (orthogonally) equivalent if there exists an unitary (orthogonal) matrix $U$ of order $k$ such that $U(A_{0} + x_{1}A_{1} + \\dots + x_{n}A_{n})U^{\\ast } = B_{0} + x_{1}B_{1} + \\dots + x_{n}B_{n}$ .", "Note that for an orthogonal matrix $U^{\\ast } = U^{T}$ .", "Note that the determinants of two equivalent linear matrix polynomials are the same.", "Using this equivalence, we can declare two different MDRs of a given polynomial as equivalent, if the corresponding linear matrix polynomials are equivalent.", "The equivalence class of representations is said to be definite (monic) if it contains a definite (monic) representative.", "Naturally, one would be interested in determining how many non-equivalent classes of MDR exists for a given polynomial.", "Observe that the matrices $A_{j}$ obtained in the algorithm given above, can be re-written as the sum of a diagonal matrix and a traceless matrix.", "Thus $A_{j}=\\frac{b_{j}}{2}\\left[ \\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix} \\right]+\\left[ \\begin{matrix} (\\alpha _{1})_{j} & (\\alpha _{2})_{j}-i(\\alpha _{3})_{j} \\\\ (\\alpha _{2})_{j}+i(\\alpha _{3})_{j} & -(\\alpha _{1})_{j} \\end{matrix} \\right]$ Here $b_{j}, (\\alpha _{1})_{j}, (\\alpha _{2})_{j},(\\alpha _{3})_{j}$ are the $j$ -th entries of the vectors $b, \\alpha _{1}, \\alpha _{2},\\alpha _{3}$ respectively.", "Similarly, for the symmetric case, $A_{j}=\\frac{b_{j}}{2}\\left[ \\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix} \\right]+\\left[ \\begin{matrix} (\\alpha _{1})_{j} & (\\alpha _{2})_{j} \\\\ (\\alpha _{2})_{j} & -(\\alpha _{1})_{j} \\end{matrix} \\right]$ .", "Note further that similarity transforms using unitary (orthogonal) matrices does not affect the diagonal matrix, i.e., $U^{\\ast }A_{j}U=\\frac{b_{j}}{2}\\left[ \\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix} \\right]+U^{\\ast }\\left[ \\begin{matrix} (\\alpha _{1})_{j} & (\\alpha _{2})_{j}-i(\\alpha _{3})_{j} \\\\ (\\alpha _{2})_{j}+i(\\alpha _{3})_{j} & -(\\alpha _{1})_{j} \\end{matrix} \\right]U$ .", "Thus, it is enough to consider unitary (orthogonal) equivalence of traceless Hermitian (symmetric) matrices.", "Traceless Hermitian matrices form a three dimesional real vector space spanned by the Pauli matrices $\\sigma _{z}=\\left[ \\begin{matrix} 1 & 0\\\\0 & -1 \\end{matrix} \\right], \\sigma _{x}=\\left[ \\begin{matrix} 0 & 1\\\\1 & 0 \\end{matrix} \\right], \\sigma _{y}=\\left[ \\begin{matrix} 0 & -i\\\\i & 0 \\end{matrix} \\right]$ .", "Similarly, traceless symmetric matrices form a two dimensional real vector space spanned by $\\sigma _{z}=\\left[ \\begin{matrix} 1 & 0\\\\0 & -1 \\end{matrix} \\right]$ and $\\sigma _{x}=\\left[ \\begin{matrix} 0 & 1\\\\1 & 0 \\end{matrix} \\right]$ .", "Therefore, to figure out equivalent Hermitian MDRs, one needs to consider only equivalence of Hermitian matrices of the form $(\\alpha _{1})_{j}\\sigma _{z}+(\\alpha _{2})_{j}\\sigma _{x}+(\\alpha _{3})_{j}\\sigma _{y}$ .", "Similarly, for symmetric MDRs, it is enough to consider equivalence of symmetric matrices of the form $(\\alpha _{1})_{j}\\sigma _{z}+(\\alpha _{2})_{j}\\sigma _{x}$ .", "We first consider the case of symmetric MDR.", "Consider an orthogonal matrix $V=\\left[ \\begin{matrix} v_{11} & v_{12}\\\\v_{21} & v_{22} \\end{matrix} \\right]$ .", "One can compute that $ V^{T}(k \\sigma _{z}+l \\sigma _{x})V=((v_{11}^{2}-v_{21}^{2})k+2v_{11}v_{21}l)\\sigma _{z}+((v_{11}v_{12}-v_{21}v_{22})k+(v_{11}v_{22}+v_{12}v_{21})l)\\sigma _{x}$ We have shown that a symmetric MDR of size 2 for a quadratic polynomial exists if and only if the Schur complement matrix $Q/(1, 1)$ is negative semidefinite with rank less than or equal to 2.", "Let $-Q/(1,1) = R^{T}R = \\alpha _{1}\\alpha _{1}^{T}+\\alpha _{2}\\alpha _{2}^{T}$ which can be used to construct the coefficient matrices $A_{j}$ s as outlined in the earlier section.", "Here $\\alpha _{1}^{T},\\alpha _{2}^{T}$ are the rows of the matrix $R \\in {\\mathbb {R}}^{2 \\times n}$ .", "Note that we can construct two alternate symmetric MDRs by either assigning $\\alpha _{1}=\\tilde{{\\mathbf {t}}}, \\alpha _{2}=\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2}$ or $\\alpha _{1}=\\frac{\\tilde{{\\mathbf {r}}}-\\tilde{{\\mathbf {s}}}}{2}, \\alpha _{2}=\\tilde{{\\mathbf {t}}}$ .", "Proposition 3.6 All symmetric MDRs of size 2 of a quadratic polynomial are orthogonally equivalent.", "Proof: Let $-Q/(1, 1) = R^{T}R$ where $R \\in {\\mathbb {R}}^{2 \\times n}$ .", "If the $j$ -th column of $R$ is given by the vector $\\left[ \\begin{matrix} k \\\\l \\end{matrix} \\right]$ , then the matrix $A_{j}$ of the corresponding MDR is by construction equal to $\\frac{b_{j}}{2} I + k\\sigma _{z} + l \\sigma _{x}$ .", "If $-Q/(1,1)= (OR)^{T}OR$ where $O$ is any orthogonal $2 \\times 2$ matrix, then the corresponding matrix of the new MDR is equal to $\\frac{b_{j}}{2} I + k_{1}\\sigma _{z} + l_{1} \\sigma _{x}$ where $\\left[ \\begin{matrix} k_{1}\\\\ l_{1} \\end{matrix} \\right]=O\\left[ \\begin{matrix} k\\\\l \\end{matrix} \\right]$ .", "Observe that if $O =\\left[ \\begin{matrix} v_{11}^{2}-v_{21}^{2} & 2v_{11}v_{21}\\\\v_{11}v_{12}-v_{21}v_{22} & v_{11}v_{22}+v_{12}v_{21} \\end{matrix} \\right]$ , then by equation (REF ) above, the matrices associated to the new MDR are equal to $V^{T}A_{j}V$ .", "Thus, it is enough to demonstrate that for every orthogonal matrix $O$ , there exists another orthogonal matrix $V$ such that the elements of $O$ are related to those of $V$ in the manner described above.", "If the orthogonal matrix $O$ is a rotation matrix: $O=\\left[ \\begin{matrix} \\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{matrix} \\right]$ , then $V=\\left[ \\begin{matrix} \\cos \\frac{\\theta }{2} & -\\sin \\frac{\\theta }{2}\\\\\\sin \\frac{\\theta }{2} & \\cos \\frac{\\theta }{2} \\end{matrix} \\right]$ satisfies the required relations.", "On the other hand, if the orthogonal matrix $O$ is a reflection matrix $O=\\left[ \\begin{matrix} -\\cos \\theta & -\\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{matrix} \\right]$ , then $V=\\left[ \\begin{matrix} -\\sin \\frac{\\theta }{2} & \\cos \\frac{\\theta }{2}\\\\ \\cos \\frac{\\theta }{2} & \\sin \\frac{\\theta }{2} \\end{matrix} \\right]$ satisfies the required relations.", "$\\square $ We now consider the case of Hermitian MDRs.", "We begin with some remarks about unitary matrices.", "The set of unitary matrices $U(2)$ is a Lie group and consists of matrices $U$ such that $U^{\\ast }U = I_{2}$ .", "Clearly $\\det (U) = e^{i \\phi }$ , for $U \\in U(2)$ .", "The set of $2 \\times 2$ unitary matrices with determinant 1 is also a Lie group, denoted by $SU(2)$ .", "Given $U \\in U(2)$ with determinant $e^{i\\phi }$ , observe that the matrix $U_{1} = e^{\\frac{-i \\phi }{2}}U \\in SU(2)$ .", "Further, for any Hermitian matrix $A$ , $U^{\\ast }AU = U_{1}^{\\ast }AU_{1}$ .", "Thus, for unitary equivalence of Hermitian MDRs, it is enough to consider unitary matrices from $SU(2)$ .", "Any $U \\in SU(2)$ can be written as $U = \\left[ \\begin{matrix} a+ib & -c + id \\\\ c+ id & a- ib \\end{matrix} \\right]$ where $a^{2} + b^{2} + c^{2} + d^{2} = 1$ .", "If $H$ is a traceless Hermitian matrix, then $H_{1} = U^{\\ast }HU$ is also a traceless Hermitian matrix.", "Further, if one expresses these traceless Hermitian matrices in terms of the Pauli matrices as $H = k\\sigma _{z} + l\\sigma _{x} + m\\sigma _{y}$ and $H_{1} = k_{1}\\sigma _{z} + l_{1}\\sigma _{x} + m_{1}\\sigma _{y}$ , then $ \\left[ \\begin{matrix} k_{1}\\\\l_{1}\\\\m_{1} \\end{matrix} \\right]=\\left[ \\begin{matrix} a^{2}+b^{2}-c^{2}-d^{2} & 2ac+2bd & 2ad-2bc \\\\2bd-2ac & a^{2}-b^{2}-c^{2}+d^{2} & -2ab-2cd\\\\-2ad-2bc & 2ab-2cd & a^{2}-b^{2}+c^{2}-d^{2} \\end{matrix} \\right] \\left[ \\begin{matrix} k\\\\l\\\\m \\end{matrix} \\right]$ We have shown that a Hermitian MDR of size 2 exists for a quadratic polynomial if and only if $Q/(1, 1)$ is negative semidefinite with rank $(Q/(1, 1)) \\le 3$ .", "Let $-Q/(1,1) = R^{T}R =\\alpha _{1}\\alpha _{1}^{T}+\\alpha _{2}\\alpha _{2}^{T}+\\alpha _{3}\\alpha _{3}^{T}$ where $\\alpha _{1}^{T},\\alpha _{2}^{T},\\alpha _{3}^{T}$ are the rows of $R \\in {\\mathbb {R}}^{3 \\times n}$ .", "Proposition 3.7 Quadratic polynomials that have a Hermitian MDR of size 2 but no symmetric MDR have two classes of unitarily equivalent MDRs.", "All other quadratic polynomials that have a MDR of size 2 have only one class of unitarily equivalent MDRs.", "Proof: By Theorem REF , if a quadratic polynomial has a Hermitian MDR of size 2 but no symmetric MDR, then $-Q/(1, 1) = R^{T}R$ where the full row rank matrix $R \\in {\\mathbb {R}}^{3 \\times n}$ .", "Observe that if the $j$ -th column of $R$ is given by $\\left[ \\begin{matrix} k\\\\l\\\\m \\end{matrix} \\right]$ , then the matrix $A_{j}$ of the MDR is given by $\\frac{b_{j}}{2} I+k \\sigma _{z}+l \\sigma _{x}+m \\sigma _{y}$ .", "One can obtain another factorization of $-Q/(1, 1)$ as $-Q/(1, 1) = (OR)^{T}OR$ where $O$ is a $3 \\times 3$ orthogonal matrix.", "The MDR obtained from this new factorization would have $A_{j}=\\frac{b_{j}}{2} I+k_{1} \\sigma _{z}+l_{1} \\sigma _{x}+m_{1} \\sigma _{y}$ where $\\left[ \\begin{matrix} k_{1}\\\\l_{1}\\\\m_{1} \\end{matrix} \\right]=O \\left[ \\begin{matrix} k\\\\l\\\\m \\end{matrix} \\right]$ .", "If $O$ has the form of the matrix in equation REF , then the MDRs obtained by the two factorizations are unitarily equivalent.", "The determinant of the matrix in equation REF is 1 and therefore determinant of $O$ must be 1 for the two MDRs to be unitarily equivalent.", "As an orthogonal matrix can have determinant equal to $\\pm 1$ , therefore if one uses an orthogonal matrix $O$ with determinant equal to $-1$ , then the two MDRs obtained from $R$ and $OR$ are not unitarily equivalent.", "Thus there are two classes of unitarily equivalent MDRs.", "Now we consider the case of a quadratic polynomial where rank$(Q/(1, 1)) < 3$ .", "In this case, $-Q/(1, 1) = R^{T}R$ where the full row rank matrix $R \\in {\\mathbb {R}}^{s \\times n}$ with $s < 3$ .", "We can also view this as $-Q/(1, 1) = R^{T}_{1} R_{1}$ where $R_{1} \\in {\\mathbb {R}}^{3 \\times n}$ has been obtained from $R$ by appending all zero row(s).", "In this particular case, one can obtain equivalent Hermitian MDRs by modifying $R_{1}$ to $OR_{1}$ , where $O$ is a $3 \\times 3$ orthogonal matrix where the all-zero rows are preserved as all-zero rows of the new matrix.", "Such a transformation ensures that $-Q/(1, 1) = R^{T}_{1} R_{1} =(OR_{1})^{T}OR_{1}$ .", "We therefore explore what happens when one equates an $O$ that preserves the all-zero rows of $R_{1}$ to the matrix from equation REF $\\left[ \\begin{matrix} a^{2}+b^{2}-c^{2}-d^{2} & 2ac+2bd & 2ad-2bc \\\\2bd-2ac & a^{2}-b^{2}-c^{2}+d^{2} & -2ab-2cd\\\\-2ad-2bc & 2ab-2cd & a^{2}-b^{2}+c^{2}-d^{2} \\end{matrix} \\right]$ Let us assume that the rank$(-Q/(1, 1)) = 2$ and $R_{1}$ is a matrix whose third row is the all zero row.", "Therefore one of the rows of the orthogonal matrix $O$ must be $\\pm e_{3} = (0, 0,\\pm 1)$ to preserve the all zero row.", "If the third row is $e_{3}$ , then $a^{2} + c^{2} = 1$ and $b = d = 0$ .", "This automatically ensures that all the other elements of the third row and third column are zero.", "Thus we obtain $O$ to have the form $\\left[ \\begin{matrix} a^{2}-c^{2} & 2ac & 0\\\\ -2ac & a^{2}-c^{2} & 0\\\\0 & 0 & a^{2}+c^{2} \\end{matrix} \\right]$ Thus we observe that a rotation matrix is applied to the first two rows of the matrix $R_{1}$ .", "On the other hand, if the third row is $\\pm e_{3}$ , then $b^{2} + d^{2} = 1$ and $a = c = 0$ giving the matrix $\\left[ \\begin{matrix} b^{2}-d^{2} & 2bd & 0\\\\ 2bd & -b^{2}+d^{2} & 0\\\\ 0 & 0 & -b^{2}-d^{2} \\end{matrix} \\right]$ This corresponds to the reflection matrix being applied to the first two rows of $R_{1}$ .", "Now consider the case where the first row of the matrix $O$ is $\\pm e_{3}$ .", "For such a $O$ to be of the form given by equation (REF , equating the expressions from the first row and the third column, one obtains either $a = d, b = -c$ with $a^{2} + b^{2} = 1/2$ when the first row is $e_{3}$ or $a = -d,b = c$ with $a^{2} + b^{2} = 1/2$ when the first row is $-e_{3}$ .", "For the first case, the $2 \\times 2$ submatrix of $O$ acting on the nontrivial rows of $R_{1}$ is a rotation matrix whereas for the second case, this submatrix is a reflection matrix.", "Similarly, if one assumes the second row of $O$ is $\\pm e_{3}$ , then one can show that the relevant $2 \\times 2$ submatrix of $O$ that acts on the nontrivial rows of $R_{1}$ is a reflection matrix when the second row of $O$ is $e_{3}$ whereas it is a rotation matrix when the second row of $O$ is $-e_{3}$ .", "This clearly shows that all unitarily equivalent MDRs for this case are indeed obtained from the original $R_{1}$ by a $3 \\times 3$ orthogonal matrix having determinant equal to 1.", "The case where $R_{1}$ has only one nontrivial row is trivial, since that row or its negative should be the only nontrivial row of $OR_{1}$ and this is easily obtained with $O$ having determinant equal to 1.", "$\\square $ We can demonstrate this by using the earlier Example REF .", "Example 3.8 Recall 1 of Example REF .", "The sets of coefficient matrices $\\lbrace \\left[ \\begin{matrix} 11/5 & 2/5 \\\\ 2/5 & -11/5 \\end{matrix} \\right],\\left[ \\begin{matrix} 0 & 10 \\\\ 10 & 0 \\end{matrix} \\right], \\left[ \\begin{matrix} 4/5 & 3/5 \\\\ 3/5 & -4/5 \\end{matrix} \\right]\\rbrace \\mbox{and }\\lbrace \\left[ \\begin{matrix} 11/5 & 2i/5 \\\\ -2i/5 & -11/5 \\end{matrix} \\right],\\left[ \\begin{matrix} 0 & 10i \\\\ -10i & 0 \\end{matrix} \\right], \\left[ \\begin{matrix} 4/5 & 3i/5 \\\\ -3i/5 & -4/5 \\end{matrix} \\right]\\rbrace $ are unitarily equivalent by matrix $U=\\left[ \\begin{matrix} \\frac{1-i}{\\sqrt{2}} & 0 \\\\ 0 & \\frac{1+i}{\\sqrt{2}} \\end{matrix} \\right]$ .", "Another unitarily equivalent MDR is given by the coefficient matrices $\\left[ \\begin{matrix} 4/5 & 3/5 \\\\ 3/5 & -4/5 \\end{matrix} \\right]\\rbrace \\mbox{and} \\lbrace \\left[ \\begin{matrix} 2/5 & 11i/5 \\\\ -11i/5 & -2/5 \\end{matrix} \\right],\\left[ \\begin{matrix} 10 & 0 \\\\ 0 & -10 \\end{matrix} \\right], \\left[ \\begin{matrix} 3/5 & 4i/5 \\\\ -4i/5 & -3/5 \\end{matrix} \\right]\\rbrace $ and these are obtained by using unitary matrix $U=\\left[ \\begin{matrix} \\frac{1+i}{2} & -\\frac{1-i}{2} \\\\ \\frac{1+i}{2} & \\frac{1-i}{2} \\end{matrix} \\right]$ on the original coefficient matrices.", "From part 3 of Example REF , recall that coefficient matrices for a Hermitian MDR for the polynomial $f({\\mathbf {x}})=(x_{1}+1)^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}$ were $A_{1}=\\left[ \\begin{matrix} 1 & 0\\\\0 & 1 \\end{matrix} \\right], A_{2}=\\left[ \\begin{matrix} 0 & -i\\\\i & 0 \\end{matrix} \\right], A_{3}= \\left[ \\begin{matrix} 0 & 1\\\\1 & 0 \\end{matrix} \\right], A_{4}=\\left[ \\begin{matrix} 1 & 0\\\\0 & -1 \\end{matrix} \\right].$ Observe that another Hermitian MDR for the same polynomial is given by the coefficient matrices $A_{1}=\\left[ \\begin{matrix} 1 & 0\\\\0 & 1 \\end{matrix} \\right], A_{2}=\\left[ \\begin{matrix} 0 & 1\\\\1 & 0 \\end{matrix} \\right], A_{3}= \\left[ \\begin{matrix} 0 & -i\\\\i & 0 \\end{matrix} \\right], A_{4}=\\left[ \\begin{matrix} 1 & 0\\\\0 & -1 \\end{matrix} \\right].$ These two MDRs are not unitarily equivalent.", "This is because the vectors $\\alpha _{2}$ and $\\alpha _{3}$ were swapped in the original factorization to obtain the second MDR from the first one.", "This swapping of vectors arises out of the action of a permutation matrix whose determinant is $-1$ ." ], [ "Complete characterization of quadratic polynomials that admit MDRs", "We now completely characterize all quadratic polynomials which are determinants of monic linear matrix polynomials of any size." ], [ "Spectrahedra", "We assume that a spectrahedron has a nonempty interior and therefore without loss of generality, we assume that the spectrahedron contains origin as an interior point.", "Therefore it is determined by a definite LMP [10], [7].", "Now we define what is meant by the expression “a spectrahedron contains a full dimensional cone” [7].", "Definition 4.1 Consider a spectrahedron $S =\\lbrace {\\mathbf {x}}\\in {\\mathbb {R}}^{n}: I+x_{1}A_{1}+\\dots +x_{n}A_{n} \\succeq 0\\rbrace $ .", "Let $f({\\mathbf {x}}) =\\det (I+x_{1}A_{1}+\\dots +x_{n}A_{n})$ and $d$ is the degree of the polynomial $f({\\mathbf {x}})$ .", "Then the spectrahedron $S$ contains a full dimensional cone if and only if the half ray through some point ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ is contained within the spectrahedron $S$ and rank$(x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}) = d$ .", "Given a point ${\\mathbf {x}}\\in {\\mathbb {R}}^n$ , the half ray through the point ${\\mathbf {x}}$ is the set of points obtained as $\\lbrace \\lambda {\\mathbf {x}}: \\lambda \\ge 0 \\rbrace $ .", "Observe that $x_{1}A_{1}+\\dots +x_{n}A_{n} \\succeq 0$ if and only if $\\lambda x_{1}A_{1}+\\dots + \\lambda x_{n}A_{n} \\succeq 0$ for every $\\lambda \\ge 0$ .", "So, a spectrahedron $S=\\lbrace {\\mathbf {x}}\\in {\\mathbb {R}}^{n}: I+x_{1}A_{1}+\\dots +x_{n}A_{n} \\succeq 0\\rbrace $ contains a full dimensional cone if and only if there exists some ${\\mathbf {x}}=(x_{1},\\dots ,x_{n}) \\in {\\mathbb {R}}^{n}$ such that $x_{1}A_{1}+\\dots +x_{n}A_{n} \\succeq 0$ with rank$(x_{1}A_{1}+\\dots +x_{n}A_{n})=d$ .", "The following theorem [7] illustrates the connection between a spectrahedron $S$ containing a full dimensional cone and an MDR of the polynomial $f({\\mathbf {x}}).$ Theorem 4.2 Let $f({\\mathbf {x}}) =\\det (I+x_{1}A_{1}+\\dots +x_{n}A_{n}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ where $A_{i} \\in S {\\mathbb {R}}^{k \\times k}$ or $A_{i} \\in {\\mathbb {H}}^{k \\times k}({\\mathbb {C}})$ , for some $k$ , and the degree of $f({\\mathbf {x}})$ be $d$ .", "If the spectrahedron $S:=\\lbrace {\\mathbf {x}}\\in {\\mathbb {R}}^{n}: I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n} \\succeq 0\\rbrace $ contains a full dimensional cone, then the polynomial $f({\\mathbf {x}}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ admits an MDR of size $d$ .", "Note that this theorem guarantees the existence of an MDR of size $d$ for a polynomial $f({\\mathbf {x}})$ of degree $d$ if there exists an MDR of some size $k$ for the polynomial $f({\\mathbf {x}}).$ Also note that the converse of the Theorem REF need not be true [7].", "Notice the polynomial $f({\\mathbf {x}})=1-x_{1}^{2}-x_{2}^{2}$ has a symmetric MDR of size 2, though the spectrahedron defined by this polynomial does not contain a full dimensional cone." ], [ "Quadratic Polynomials with MDR of Any Size", "In Section , a necessary and sufficient condition for a quadratic polynomial to have an MHDR (MSDR) of size 2 was provided.", "Using that result, we now derive a necessary and sufficient condition for the existence of an MDR of any size, for a given quadratic polynomial.", "Given a quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1$ , where $A$ is negative semidefinite, it is easy to construct a symmetric MDR – a result well known in literature.", "On the other hand, the case of $A$ not being negative semidefinite is not well known.", "We shall throw some light on this case.", "We recall the following proposition.", "Proposition 4.3 Let $f:=\\det (I + x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}) \\in {\\mathbb {R}}[{\\mathbf {x}}]$ .", "Then for each ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ the nonzero eigenvalues of $x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n}$ are in one to one correspondence with the zeros of the univariate polynomial $f_{{\\mathbf {x}}}(t)=f(t {\\mathbf {x}})$ .", "The correspondence is given by the rule $t \\rightarrow -1 / t$ .", "We prove the main theorem of this section based on the following lemmas that deal with the cases of $A$ not being negative semidefinite.", "Lemma 4.4 Consider the spectrahedron $S$ defined by a quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1=Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}]$ , with $f({\\mathbf {x}})=\\det (I+x_{1}A_{1} +x_{2}A_{2} + \\dots + x_{n}A_{n})$ , where $A_{i} \\in {\\mathbb {H}}^{k \\times k}({\\mathbb {C}})$ or $A_{i} \\in S{\\mathbb {R}}^{k \\times k}$ for some $k \\ge 2$ .", "Then $S$ contains a full dimensional cone if $A$ is not a negative semidefinite matrix.", "Proof: If $A$ is not a negative semidefinite matrix, then $A$ has at least one positive eigenvalue, say $\\lambda $ .", "Let $v = (v_1,v_2, \\cdots , v_n)$ be an eigenvector of $A$ corresponding to that positive eigenvalue $\\lambda $ .", "Consider the univariate polynomials $f_{v}(t)=t^{2}v^{T}Av+tb^{T}v+1=t^{2} \\lambda \|\|v\|\|^{2}+tb^{T}v+1 \\\\f_{-v}(t)=t^{2}v^{T}Av-tb^{T}v+1=t^{2} \\lambda \|\|v\|\|^{2}-tb^{T}v+1$ All the coefficients of either $f_{v}(t)$ or $f_{-v}(t)$ are positive.", "Without loss of generality, let us assume that $b^{T}v >0$ and so all the coefficients of $f_{v}(t)$ are positive.", "Therefore the polynomial $f_{v}(t)$ has two negative real roots.", "By Proposition REF , $v_{1}A_{1} + v_{2}A_{2} + \\dots + v_{n}A_{n}$ is therefore positive semidefinite (its nonzero eigenvalues are the negative reciprocals of the roots of $f_v(t)$ ).", "As $f_v(t)$ has only two roots, therefore rank$(v_{1}A_{1} +v_{2}A_{2} + \\dots + v_{n}A_{n}) = 2$ .", "Thus the spectrahedron $S$ contains a full dimensional cone.", "${\\square }$ Theorem 4.5 Consider a quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1=Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}]$ which an MDR such that $f({\\mathbf {x}})=\\det (I+L({\\mathbf {x}}))$ , where $L({\\mathbf {x}}):=x_{1}A_{1}+\\dots +x_{n}A_{n}$ .", "Then the spectrahedron $S$ defined by polynomial $f({\\mathbf {x}})$ does not contain a full dimensional cone if and only if $A$ is negative semidefinite.", "Proof: The `only if' part of this lemma follows from Lemma REF .", "For proving the `if' part, let us assume that $A$ is negative definite.", "So, ${\\mathbf {x}}^{T}A{\\mathbf {x}}< 0$ for any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}\\setminus \\lbrace 0\\rbrace $ .", "Consider the univariate polynomial $f_{{\\mathbf {x}}}(t)=t^{2}{\\mathbf {x}}^{T}A {\\mathbf {x}}+t b^{T} {\\mathbf {x}}+1.$ As the coefficient of $t^{2}$ is always negative, the two roots of $f_{{\\mathbf {x}}}(t)$ are real and have opposite signs.", "Therefore using one-to-one correspondence (Proposition REF ) the non-zero eigenvalues of $L({\\mathbf {x}})$ are of opposite signs.", "Thus, $L({\\mathbf {x}}) 0$ for any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ .", "Therefore, spectrahedron $S$ does not contain a full dimensional cone.", "If $A$ is negative semidefinite, then the way this case differs from the earlier case is that there exists some $u \\in {\\mathbb {R}}^{n}\\setminus \\lbrace 0\\rbrace $ for which $u^{T}Au=0$ .", "This implies the coefficient of $t^{2}$ in $f_{u}(t)$ vanishes, so $f_{u}(t)$ becomes a linear polynomial.", "Due to Proposition REF , the number of non-zero eigenvalues of $L(u)$ is one and so rank$(L(u)) = 1$ .", "Therefore, the spectrahedron does not contain a full dimensional cone, even though it may contain the half ray along $u$ .", "$\\Box $ We demonstrate the construction of a symmetric MDR for a quadratic function $f({\\mathbf {x}})$ where $A$ is negative semidefinite.", "Consider any decomposition (for example, the Cholesky decomposition) of $-A=C^{T}C$ .", "This yields $ f({\\mathbf {x}})=-{\\mathbf {x}}^{T}C^{T}C {\\mathbf {x}}+b^{T} {\\mathbf {x}}+1 =\\det \\left(\\left[ \\begin{matrix} I & C {\\mathbf {x}}\\\\ {\\mathbf {x}}^{T} C^{T} & 1+b^{T}{\\mathbf {x}} \\end{matrix} \\right]\\right).$ Here the identity matrix $I$ is $r \\times r$ matrix, where $r$ is the rank of $A$ .", "Using Schur complement determinant formula for a partitioned matrix $L({\\mathbf {x}}):= \\left[ \\begin{matrix} \\mathbf {0} & C {\\mathbf {x}}\\\\ (C {\\mathbf {x}})^{T} & b^{T} {\\mathbf {x}} \\end{matrix} \\right]$ the characteristic polynomial of $L({\\mathbf {x}})$ is given by $\\det \\left( \\left[ \\begin{array}{c:c}\\lambda I_{r \\times r} & C{\\mathbf {x}}_{r \\times 1} \\\\ (C{\\mathbf {x}})^{T}_{1 \\times r} & (\\lambda - b^{T} {\\mathbf {x}})_{1 \\times 1}\\end{array} \\right]\\right) &=& \\det (\\lambda I) \\det (\\lambda -b^{T}{\\mathbf {x}}-(C {\\mathbf {x}})^{T} (\\lambda I)^{-1} (C {\\mathbf {x}})) \\nonumber \\\\ &=& \\lambda ^{r}(\\lambda -b^{T}{\\mathbf {x}}-\\frac{1}{\\lambda }\\Vert C{\\mathbf {x}}\\Vert ^{2}).", "\\nonumber $ Observe that at most two of $r+1$ eigenvalues are non-zero irrespective of the choice of ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ .", "Further if there are exactly two non-zero eigenvalues (which implies rank($L({\\mathbf {x}}))=2$ ), these two non-zero eigenvalues are of opposite signs.", "This implies that there does not exist any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ such that $x_{1}A_{1}+\\dots +x_{n}A_{n} \\succeq 0$ i.e., $L({\\mathbf {x}}) 0$ for any ${\\mathbf {x}}\\in {\\mathbb {R}}^{n}$ .", "Therefore, spectrahedron $S$ does not contain a full dimensional cone.", "Observe that the above construction gives a symmetric MDR for a quadratic polynomial $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T} {\\mathbf {x}}+ 1$ , where the matrix $A$ is negative semidefinite.", "We can also get a whole set of Hermitian MDRs from the above construction, by combining pairs of rows of the matrix $C$ where $A = -C^{T}C$ .", "We illustrate this with an example.", "Example 4.6 Consider the polynomial $1-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}-x_{5}^{2}$ .", "Clearly, this polynomial does not have an MDR of size 2, since the rank of $Q/(1, 1)$ is 5.", "On the other hand, using the construction given above one obtains a size 6 symmetric MDR given by the linear matrix polynomial $\\left[ \\begin{matrix} 1 & 0 & 0 & 0 & 0 & x_{1}\\\\0 & 1 & 0 & 0 & 0 & x_{2}\\\\0 & 0 & 1 & 0 & 0 & x_{3}\\\\0 & 0 & 0 & 1 & 0 & x_{4}\\\\0 & 0 & 0 & 0 & 1 & x_{5}\\\\x_{1}& x_{2}& x_{3}& x_{4}& x_{5} & 1 \\end{matrix} \\right]$ One can now combine the first two rows of the $C$ matrix in this case and obtain a size 5 MDR given by the linear matrix polynomial $\\left[ \\begin{matrix} 1 & 0 & 0 & 0 & x_{1}+i x_{2}\\\\0 & 1 & 0 & 0 & x_{3}\\\\0 & 0 & 1 & 0 & x_{4}\\\\0 & 0 & 0 & 1 & x_{5}\\\\x_{1}-i x_{2}& x_{3}& x_{4}& x_{5} & 1 \\end{matrix} \\right]$ Combining two other rows of the C matrix, one can go down to a size 4 Hermitian MDR.", "For example, $\\left[ \\begin{matrix} 1 & 0 & 0 & x_{1}+i x_{2}\\\\0 & 1 & 0 & x_{3}+ix_{4}\\\\0 & 0 & 1 & x_{5}\\\\x_{1}-i x_{2}& x_{3}-ix_{4}& x_{5} & 1 \\end{matrix} \\right]$ Thus essentially using the same construction, one can build both symmetric and Hermitian MDRs.", "Further observe that this process gives a whole range of sizes for the MDRs.", "We now characterize all quadratic polynomials that exhibit an MDR.", "Theorem 4.7 A quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1=Z^{T}[{\\mathbf {x}}]QZ[{\\mathbf {x}}] \\in {\\mathbb {R}}[{\\mathbf {x}}]$ admits an MDR if and only if either one (possibly both) of the following two conditions is true.", "$A$ is negative semidefinite $Q/(1,1)$ is negative semidefinite and $\\mathrm {rank}(Q/(1,1)) \\le 3$ where $Q/(1,1)=A-\\frac{1}{4}bb^{T}$ .", "Proof: If $A$ is negative semidefinite, then one can construct a symmetric MDR as demonstrated above to obtain $\\left[ \\begin{matrix} I & C{\\mathbf {x}}\\\\ (C {\\mathbf {x}})^{T} & 1+b^{T}{\\mathbf {x}} \\end{matrix} \\right]$ where $A=-C^{T}C$ .", "On the other hand if $Q/(1,1)$ is negative semidefinite and $\\mathrm {rank}(Q/(1,1)) \\le 3$ , then by the Theorem REF $f({\\mathbf {x}})$ has an MDR of size 2.", "Conversely, if $f({\\mathbf {x}})$ has an MDR, then $f({\\mathbf {x}})$ is a RZ polynomial.", "By Proposition REF , $Q/(1,1)=A-\\frac{1}{4}bb^{T}$ is certainly negative semidefinite.", "So either $A$ is negative semidefinite or $A$ is not negative semidefinite.", "If $A$ is not negative semidefinite, then by Lemma REF we know that the spectrahedron $S$ contains a full dimensional cone.", "So, in this case if the quadratic polynomial $f({\\mathbf {x}})$ has an MDR of some size, then $f({\\mathbf {x}})$ has an MDR of size 2 by the Theorem REF .", "On the other hand, by the Theorem REF if quadratic polynomial $f({\\mathbf {x}})$ has an MDR of size 2 then $Q/(1,1)$ is negative semidefinite and $\\mathrm {rank}(Q/(1,1)) \\le 3$ .", "Therefore, in the case of $A$ being not negative semidefinite, if $f({\\mathbf {x}})$ has an MDR, then $Q/(1,1)$ is negative semidefinite and $\\mathrm {rank}(Q/(1,1)) \\le 3$ .", "$\\square $ Remark 4.8 The above theorems characterizes all quadratic polynomials that have an MDR.", "The size of MDR for a quadratic polynomial $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T} {\\mathbf {x}}+ 1$ with $A$ being negative semidefinite can range from $\\lceil \\frac{r}{2}+1 \\rceil $ to $r + 1$ , where rank $(A) = r$ .", "Of course, even larger sizes MDRs are possible, but these MDRs are such that the intersection of kernels of all the matrices Aj would be non-trivial.", "Factoring out this common kernel, would reduce the situation to one of the sizes outlined above.", "On the other hand, if the matrix $A$ is not negative semidefinite, then $f({\\mathbf {x}})$ has an MDR guarantees that $f(x)$ has an MDR of size 2.", "This restricts the rank of $A$ which is not negative semidefinite to a maximum of 4, for an MDR to exist.", "In other words, a polynomial $f({\\mathbf {x}}) = {\\mathbf {x}}^{T}A{\\mathbf {x}}+ b^{T} {\\mathbf {x}}+ 1$ with rank$(A) > 4$ , has an MDR if and only if $A$ is negative semidefinite.", "Example 4.9 We once again invoke Example REF .", "Recall that condition 2 of Theorem REF is satisfied by all three examples, whereas condition 1 of Theorem REF is only satisfied by the first two cases.", "Thus for case 1, the polynomial $1-8x_{1}x_{2}-4x_{1}x_{3}-100x_{2}^{2}-12x_{2}x_{3}-x_{3}^{2}-5x_{1}^{2}$ has a size 3 symmetric MDR given by the linear matrix polynomial $\\left[ \\begin{matrix} 1 & 0 & 11x_{1}/5+4x_{3}/5\\\\0 & 1 & 2x_{1}/5+10x_{2}+3x_{3}/5\\\\11x_{1}/5+4x_{3}/5 &2x_{1}/5+10x_{2}+3x_{3}/5 & 1 \\end{matrix} \\right]$ Combining the two rows of the C matrix in this case, one can also obtain a size 2 Hermitian MDR given by $\\left[ \\begin{matrix} 1 & \\frac{11+2i}{5}x_{1}+10ix_{2}+\\frac{4+3i}{5}x_{3}\\\\\\frac{11-2i}{5}x_{1}-10ix_{2}+\\frac{4-3i}{5}x_{3} & 1 \\end{matrix} \\right]$ It is instructive to note that this is precisely one of the size 2 linear matrix polynomials obtained for case 1, in the follow-up Example REF .", "Similarly, for case 2, the polynomial $1 + 4x_{1} + 10x_{2}-x_{1}^{2}- 2x_{1}x_{2}- x_{2}^{2}$ has another size 2 MDR given by the linear matrix polynomial $\\left[ \\begin{matrix} 1 & x_{1}+x_{2}\\\\x_{1}+x_{2} &1+4x_{1}+10x_{2} \\end{matrix} \\right]$ which is orthogonally equivalent to MDR obtained in Example REF .", "As a result of the Theorem REF we have the following corollary.", "Corollary 4.10 A quadratic multivariate polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1 \\in {\\mathbb {R}}[{\\mathbf {x}}]$ has a diagonal MSDR of any size if and only if Rank of $Q/(1,1) \\le 1$ and $Q/(1,1)$ is a negative semidefinite matrix.", "Proof: It follows from the equation (REF ) that when $A$ is negative semidefinite, there can not exist any diagonal MSDR, otherwise $f({\\mathbf {x}})$ can not be a quadratic polynomial.", "When $A$ is not negative semidefinite, but if quadratic polynomial has an MSDR of some size $>2$ , it is proved in the Theorem REF that it has an MSDR of size 2 too.", "It is shown in Corollary REF that a quadratic polynomial has a diagonal MSDR of size 2 if and only if $Q/(1,1)$ is a negative semidefinite matrix and of rank $\\le 1$ .", "$\\square $" ], [ "Conclusion", "It is well known that a quadratic polynomial $f({\\mathbf {x}})={\\mathbf {x}}^{T}A{\\mathbf {x}}+b^{T}{\\mathbf {x}}+1$ has an MDR with $(n+1) \\times (n+1)$ LMP, if the matrix $A$ is negative semidefinite.", "On the other hand, not much seems to be available in literature about quadratic polynomials having MDR of size 2.", "In this chapter, we provide necessary and sufficient conditions for a quadratic multivariate polynomial to have an MDR of size 2.", "We have provided an algorithm which can be used to construct MDRs of size 2, when they exist.", "It has been shown that quadratic polynomials having MDRs of size 2 are of two kinds – those that have exactly two unitarily non-equivalent Hermitian MDRs of size 2 with none of the MDRs being a symmetric one and those that have exactly one unitarily equivalent Hermitian MDR which includes symmetric MDRs.", "It is further shown that all possible symmetric MDRs are orthogonally equivalent.", "Further, we have shown that if a quadratic polynomial having $A$ which is not negative semidefinite has no MSDR of size 2, it cannot have MSDR of any size greater than 2.", "Consequently, we have completely characterized quadratic polynomials which have an MHDR (MSDR) of any size.", "The class of such quadratic polynomials belongs to any one of the following two categories: quadratic polynomials having $A$ which is negative semidefinite or quadratic polynomials which have an MDR of size 2.", "As a consequence of this result, we have effectively characterized spectrahedra with non-empty interior defined by $2 \\times 2$ linear matrix inequalities.", "Furthermore, if the mentioned conditions in Theorem REF are true, quadratic optimization problems (including (QCQP) and trust-region subproblems can be converted into an SDP relaxation problem irrespective of the fact that quadratic functions (objective or constraints) are convex or not." ] ]	1606.05184
[ [ "The detection and X-ray view of the changing look AGN HE 1136-2304" ], [ "Abstract We report the detection of high-amplitude X-ray flaring of the AGN HE 1136-2304, which is accompanied by a strong increase in the flux of the broad Balmer lines, changing its Seyfert type from almost type 2 in 1993 down to 1.5 in 2014.", "HE 1136-2304 was detected by the XMM-Newton slew survey at >10 times the flux it had in the ROSAT all-sky survey, and confirmed with Swift follow-up after increasing in X-ray flux by a factor of 30.", "Optical spectroscopy with SALT shows that the AGN has changed from a Seyfert 1.95 to a Seyfert 1.5 galaxy, with greatly increased broad line emission and an increase in blue continuum AGN flux by a factor of > 4.", "The X-ray spectra from XMM-Newton and NuSTAR reveal moderate intrinsic absorption and a high energy cutoff at 100 keV.", "We consider several different physical scenarios for a flare, such as changes in obscuring material, tidal disruption events, and an increase in the accretion rate.", "We find that the most likely cause of the increased flux is an increase in the accretion rate, although it could also be due to a change in obscuration." ], [ "Introduction", "Outbursts in radio-quiet active galactic nuclei (AGN) are a relatively unexplored area of AGN physics.", "However, observations of these unusual events have the potential to trace accretion physics in detail, particularly when multiple instruments can be used to observe at different wavelengths.", "In recent years, there have been several detections of AGN in anomalously low or high X-ray states using Swift monitoring or XMM-Newton surveys , , , , , , , , , , , .", "Despite the high-amplitude X-ray variability (factors $>$ 20–100) of AGN, these events rarely come with strong changes in the optical emission lines, implying that the observed changes mostly happen along our line-of-sight, and do not affect the bulk of the emission-line regions.", "An interesting example is GSN 069, which was detected during an XMM-Newton slew observation in 2010 at a soft X-ray flux level at least 240 times that which it had when ROSAT failed to detect it in 1994 .", "The X-ray spectrum is extremely soft and unobscured, and it shows rapid soft X-ray variability, which is consistent with Seyfert 1 behaviour.", "However, it has no detected broad line emission, so can be classified as a Seyfert 2.", "The term `changing look' was coined to refer to X-ray observations of Compton-thick AGN becoming Compton-thin, and vice-versa , , , , , .", "This transition can be extremely rapid: found NGC 1365 switching from Compton thick to Compton thin in under 6 weeks.", "There are also well documented cases of changing-look behaviour in the optical band, where the optical spectra of AGN change so as to move to a different Seyfert classification.", "Cases include NGC 3515 , NGC 7306 , NGC 4151 , Fairall 9 , Mrk 1018 , Mrk 99 , NGC 1097 , NGC 7582 , NGC 2617 , Mrk 590 , and the quasars SDSS J015957.64+003310.5 and SDSS J101152.98+544206.4 .", "The most extreme examples known come with dramatic changes not only in their broad Balmer lines, but also in their more narrow high-ionization lines like [FeVII] and [FeXIV].", "Cases include IC 3599 , , , SDSS J095209.56+214313.3 and SDSS J074820.67+471214.3 .", "More recent studies have begun to look at small samples of changing look AGN , , .", "There are several potential causes for rapid changes in the (apparent) brightness of AGN.", "In the unified model of AGN, the differences between Seyfert 1 and Seyfert 2 galaxies are interpreted as being largely due to the effect of inclination, with the inclination of Seyfert 2 AGN being such that the torus obscures the inner regions, strongly absorbing the X-ray emission and blocking the broad line region (BLR) from the observer.", "When the observer's line of sight skims the edge of the torus, rapid changes in the observed flux can occur when obscuring clouds at the edge of the torus move to allow a clear view of the central regions of the AGN , [1], .", "However, it is challenging to attenuate a large fraction of the light from the BLR with this mechanism, as this requires different parts of the extended torus to act simultaneously.", "Alternatively, these events could be caused by large changes in the accretion rate of the AGN, with a large increase in the accretion rate correspondingly increasing the AGN luminosity , , , .", "Finally, stellar tidal disruption events , could cause a sudden increase in brightness.", "Systematic searches for changing look AGN in quasars suggest that the majority are related to changes in the accretion rate and $>15$ % of strongly-variable luminous quasars show such variability on time-scales of 8–10 years , although the sample sizes in these studies are relatively small.", "In this work we present XMM-Newton, NuSTAR, and SALT observations of the flaring AGN HE 1136-2304, with the aim of establishing the cause of the outburst and its change in Seyfert type.", "HE 1136-2304 is a nearby, relatively unknown AGN at redshift $z$ =0.027 .", "It was detected in X-rays in the ROSAT All-Sky Survey and is a faint radio source .", "Throughout this paper, we assume a cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _\\Lambda =0.73$ and $\\Omega _\\textrm {M}=0.27$ .", "The X-ray observations presented here are based off XMM-Newton proposal ID 74126 (PI N. Schartel), `Outbursts of Radio-Quiet AGN'.", "This proposal was intended to obtain an observation of a flaring AGN, and as such had a triggering condition that the source flux had to be in excess of 15 times the flux observed with ROSAT.", "HE 1136-2304 was initially detected in outburst using the XMM-Newton Slew Survey in 2010, at a flux ratio of $13.3\\pm 2.0$ .", "While significantly above the ROSAT level, this did not meet the required flux level.", "However, follow up observations with Swift found that the source reached a flux level of $\\sim 16$ times that found by ROSAT, representing a genuine outburst, and a peak flux ratio of $\\sim 30$ was later found with Swift.", "Based on this detection, we triggered the simultaneous observations with XMM-Newton and NuSTAR .", "The flux ratios for all observations are given in Table REF .", "We note that the source was also detected in the Swift BAT 70-month catalogue, with an average 14–195 keV flux of $17_{-11}^{+22}\\times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$ .", "Table: ROSAT PSPC, XMM-Newton EPIC-pn and Swift XRT fluxes for HE 1136-2304.We used the XMM-Newton Science Analysis Software (SAS) version 14.0.0 to reduce the XMM-Newton data, extracting EPIC-pn and MOS event files using epproc and emproc respectively.", "Both the EPIC-pn and MOS detectors were operated in small window mode.", "We filter for background flares, and extract source and background spectra from $30^{\\prime \\prime }$ circular extraction regions, choosing the background region so as to avoid contaminating photons from HE 1136-2304 or other objects in the field of view.", "We use the specgroup tool to rebin the spectra to a signal-to-noise ratio of 6 and oversampling the instrumental resolution by a factor of 3, ensuring the applicability of $\\chi ^2$ statistics.", "A preliminary investigation of the reflection grating spectrometer (RGS) spectra reveals no highly significant features, so we restrict our analysis to the lower resolution, higher sensitivity detectors.", "The NuSTAR data were reduced using the NuSTAR Data Analysis Software (nustardas) version 1.4.1 and the CALDB version 20150316 (more recent versions at the time of writing only update the NuSTAR clock correction file, so will not affect our results).", "Spectra and lightcurves were extracted with the nuproducts tool, using 60$^{\\prime \\prime }$ circular extraction regions.", "As with the XMM-Newton spectra, the NuSTAR spectra are binned to oversample the data by a factor of 3 and a signal-to-noise ratio of 6.", "We fit the NuSTAR spectra over the whole band (3–79 keV).", "The details of these observations, and those of XMM-Newton, are given in Table REF , and the fluxes are plotted in Fig.", "REF .", "Table: Observation details for the XMM-Newton and NuSTAR detectorsFigure: Long-term lightcurve showing the fluxes from Table .", "The 1990 ROSAT flux is shown by the horizontal dashed line and the time of the joint XMM-Newton/NuSTAR observation is shown by the vertical dashed line.", "The SALT spectrum was taken 3 days after the end of the NuSTAR observation.All X-ray spectra are fit using Xspec version 12.8.2l, and all errors are reported at 1$\\sigma $ unless otherwise stated.", "We assume Galactic absorption of $3.82\\times 10^{20}$ cm$^{-2}$ , and use the abundances of ." ], [ "Optical spectroscopy with the SALT telescope", "To determine if the X-ray flare corresponded to a changing look in the optical band, we observed HE 1136-2304 with the 10m Southern African Large Telescope (SALT) nearly simultaneously to the XMM observations on July 7, 2014 under photometric conditions.", "The optical spectrum was taken with the Robert Stobie Spectrograph attached to the telescope using the PG0900 grating with a 2.0$^{\\prime \\prime }$ wide slit.", "We covered the wavelength range from 4203 to 7261 Å at a spectral resolution of 6.5 Å (FWHM) and a reciprocal dispersion of 0.98 Å pixel$^{-1}$ .", "The observed wavelength range corresponds to a wavelength range from 4078–7050 Å in the rest frame of the galaxy.", "There are two gaps in the spectrum caused by the gaps between the three CCDs: one between the blue and the central CCD chip as well as one between the central and red CCD chip covering the wavelength ranges 5206.5–5262.7 Å and 6254.4–6309.1 Å (5079–5135 Å and 6100–6150 Å in the rest frame).", "The seeing was 1.1$^{\\prime \\prime }$ , and the exposure time was $2\\times 600$ seconds (20 minutes).", "In addition to the galaxy spectrum, necessary flat-field and Xe arc frames were also observed, as well as a spectrophotometric standard star for flux calibration (LTT4364).", "The spectrophotometric standard star was used to correct the measured counts for the combined transmission of the instrument, telescope and atmosphere as a function of wavelength.", "Flat-field frames were used to correct for differences in sensitivity both between detector pixels and across the field.", "The spatial resolution per binned pixel is 0.2534 for our SALT spectrum.", "We extracted 7 columns from our object spectrum corresponding to 1.77.", "The reduction of the spectra (bias subtraction, cosmic ray correction, flat-field correction, 2D-wavelength calibration, night sky subtraction, and flux calibration) was done in a homogeneous way with IRAF reduction packages .", "We corrected the optical spectra of HE1136-2304 for Galactic extinction.", "We used the reddening value E(B-V) = 0.03666 deduced from the re-calibration of the infrared-based dust map.", "The reddening law of with R$_{V}$ = 3.1 was applied to our spectra.", "All wavelengths were converted to the rest frame of the galaxy (z=0.027)." ], [ "X-ray spectrum", "We run a series of tests on the X-ray spectrum of HE 1136-2304, using the excellent broad-band data to try and shed light on the nature of the changes in the optical spectrum.", "We look for changes in the flux and spectral shape within the observations, which may indicate what dominates the variability on short time-scales, we search for the signatures of absorption (either ionized or neutral) in the spectrum, and fit the spectrum with various models that can be used to compare it to other, similar, AGN.", "In Fig.", "REF we show the XMM-Newton and NuSTAR light curves over the whole observation.", "A $\\sim 30$ per cent drop in flux is immediately obvious in the XMM-Newton EPIC-pn light curve, which declines smoothly over around 25–75 ks.", "A similar, though less strong, drop is also observed in the NuSTAR lightcurve, which declines by around 25 per cent.", "The high and low flux intervals are (coincidentally) neatly divided by the two NuSTAR observation IDs, so the high energy spectral evolution, if present, should be visible in the difference between the two spectra.", "Figure: Left: Background subtracted XMM-Newton EPIC-pn and NuSTAR FPMA light curves over the whole energy range, binned to 1 and 3 ks, respectively.", "The two NuSTAR observation IDs are plotted separately, in red and blue.", "Right: Ratio of the EPIC-pn data to an absorbed power law for the two halves (high and low flux) of the XMM-Newton observation, showing the absence of spectral changes with flux.While there is very little short-term variability in the lightcurve (Fig.", "REF , left), there is a $\\sim 30$ % drop in flux in the 0.3–10 keV band over the course of the XMM-Newton observation.", "In the right panel of Fig.", "REF we show the ratio of the EPIC-pn spectrum to an absorbed power law for the two halves of the XMM-Newton observation, allowing for a difference in normalization between the two.", "It is apparent from this that there is no significant change in spectral shape corresponding to the drop in flux.", "The only potential difference is a slight increase in the iron line residuals, as would be expected if the narrow line originates far from the X-ray continuum source and therefore responds slowly.", "Because of the lack of spectral evolution, we combine the two NuSTAR observations into a single spectrum for each of the two FPMs, and fit all the data simultaneously.", "Figure: Unfolded XMM-Newton and NuSTAR spectrum of HE 1136-2304 to a power law with Γ=0\\Gamma =0.", "The blue line shows the EPIC-pn points after correcting for Galactic absorption.", "It is obvious from this that there is additional absorption intrinsic to the source.In Fig.", "REF we show the EPIC-pn and focal plane module (FPM) A/B spectra, unfolded to a $\\Gamma =0$ power law.", "The spectrum is fairly hard, and shows significant line emission around 6.4 keV.", "We also show the pn spectrum after correcting for Galactic absorption.", "A decrease in flux at low energies is still apparent, implying that there is a significant absorbing column intrinsic to the source.", "We next examine the residuals after taking into account this additional absorption.", "In Fig.", "REF we show the ratio of the broad-band X-ray spectrum to the same absorbed power law, including both Galactic and intrinsic absorption (tbabs$\\times $ ztbabs$\\times $ powerlaw as an Xspec model, i.e.", "Galactic absorption$\\times $ intrinsic absorption$\\times $ power law).", "A strong narrow line is clearly visible at 6.4 keV, along with a soft excess and a turnover in the NuSTAR band.", "Figure: Ratio of the X-ray spectrum to an absorbed power law, fit from 0.3–80 keV.", "The soft excess, iron line and hard excess/cutoff are all clearly visible.", "NuSTAR FPMA and FPMB data are grouped in Xspec and all spectra are rebinned for clarity.From the EPIC-pn spectrum alone, it appears that there may be some additional complexity around the 6.4 keV iron line, most noticeably an apparent emission line at $\\sim 7$ keV.", "This could potentially be due to the iron K$\\beta $ line, or could be due to an absorption feature superimposed on a broader emission line.", "To probe the iron line region in more detail, we use a simple line search over the 4–9 keV band, following the method outlined in and assuming a power law continuum.", "The results are shown in Fig.", "REF .", "We perform the search separately for each set of detectors, then in a combined fit, so that detections can be compared between instruments.", "In all cases, we find the 6.4 keV line to be highly significant, with the only other feature significant at the 3$\\sigma $ level being a spurious feature at $\\sim 5.5$ keV in the EPIC-pn, which does not correspond to any features in the other instruments.", "There is a general trend towards an excess at the high energy side of the iron line, which may correspond to presence of an additional line or lines, a broad component, or be due to the presence of a 7 keV absorption edge.", "There are various features significant at the 2$\\sigma $ level above 7 keV in the combined analysis (including NuSTAR, and MOS), however they are not consistent in size or energy and most likely correspond to the continuum over-predicting the flux in this band .", "The absence of any apparent ionized absorption (either warm absorber or ultra-fast outflow) in this source is interesting, and may be relevant to the change in Seyfert classification.", "In particular, it implies that either there is a weak or absent disk wind or that we are observing the source at such an angle that the wind does not intercept the line of sight.", "Figure: Results of the search for emission and absorption lines, over the 4–9 keV band.", "The first plot shows the results for the EPIC-pn individually, and the combined plot shows the results from fitting all five instruments (pn, MOS1/2 and FPMA/B) simultaneously.", "Contour lines show the 1, 2 and 3σ\\sigma confidence levels.", "Normalizations are in units of photons cm -2 ^{-2} s -1 ^{-1} and energy values are in the rest frame.", "The 7 keV line that appears in the EPIC-pn spectrum is not significant when the other data is taken into account.We fit the combined X-ray spectrum with a variety of models, with the aim of measuring the level of the intrinsic absorption and for easy comparison with other sources.", "The fit results are shown in Table REF .", "In all fits we allow for a normalization offset between the 5 detectors.", "The following models are used: Model 1: Fitting XMM-Newton data from 0.3–2 keV with an absorbed (Galactic and intrinsic absorption) power law.", "This model is for comparison with the ROSAT and XMM-Newton slew surveys, which fit models over similar bands.", "Model 2: As model 1, but fit up to 10 keV, again only using XMM-Newton data and including a narrow emission line at 6.4 keV.", "This model allows for more general comparison with XMM-Newton (and other) observed AGN.", "Model 3: As model 2, but now including a soft excess, modelled with a black body (bbody in xspec).", "We test adding both neutral (modelled using zpcfabs) and ionized partially-covering absorbers, and find no improvement to the fit from adding either.", "Model 4: As model 3, but now including the NuSTAR data from 3–79 keV, and replacing the power law with a cut-off power law (cutoffpl).", "Again, we try including additional partial covering absorption and find no improvement to the fit.", "This is our best-fit phenomenological model, which includes spectral components to describe all major features of the spectrum.", "Model 5: A more physical model, where we replace the narrow Gaussian line with a distant reflection component modelled with xillver , and use a combination of Comptonization and relativistic reflection to fit the soft excess and broad-band spectral curvature.", "We use a broken power law emissivity profile, fixing the outer index at 3, and the break radius at 6 $R_\\textrm {G}$ .", "Removing either of the reflection and Comptonization components worsens the fit significantly ($\\Delta \\chi ^2\\sim 50$ ), although it is not entirely clear what spectral features this is due to in the reflection case.", "It is most likely required to fit subtle spectral curvature in the high-quality broad-band spectrum.", "As with the other models, additional absorbers do not improve the fit significantly.", "Table: X-ray spectral fit parameters for the different models.", "Models 1–4 are phenomenological, and model 5 is made up of physical components.", "Models 1–3 are fit to different subsets of the data.Models 1–4 are phenomenological, and are intended for comparison purposes.", "We show the residuals to the best-fit models 4 and 5 in Fig.", "REF , and the models themselves in Fig.", "REF .", "For model 4, the main residual features are at low energies, and presumably arise from our phenomenological model of the soft excess.", "There is a small disagreement in $\\Gamma $ between the XMM-Newton and NuSTAR instruments from 3–10 keV, which also contributes to the relatively high $\\chi ^2$ values we find.", "For model 5, the main residuals are around 1 keV, where there is a disagreement between the MOS and pn detectors, and from 3–10 keV, where the NuSTAR and XMM-Newton spectra overlap.", "Figure: χ 2 \\chi ^2 residuals for the best-fit phenomenological and physical models (top and bottom, respectively).", "Different colours correspond to the five different instruments.", "A disagreement between the MOS and pn detectors is seen just above 1 keV, and a slightly different slope is found between the FPMs and the EPIC instruments between 3–10 keV.", "Aside from these instrumental effects, there are no significant residuals in the physical model.", "The phenomenological model shows strong residuals around the soft excess.", "The corresponding models are shown in Fig.", "Figure: Best-fit broad-band spectral models, showing the different spectral components used.", "Left and right plots correspond to the phenomenological and physical models, respectively.", "In both cases, the spectrum is dominated by the continuum power-law, with a strong narrow iron line.Figure: XMM-Newton and NuSTAR SED.", "Green points show the EPIC-pn spectrum after correcting for both Galactic and intrinsic absorption.We can estimate from the best fit model (model 5) the column density that would be needed to reduce the X-ray flux to the level observed in 1990.", "By adjusting the column density of the neutral absorber at the source redshift (ztbabs), we find that the column density needed is $\\sim 2.45\\times 10^{22}$ cm$^{-2}$ , compared to the observed $1.5\\times 10^{21}$ cm$^{-2}$ in 2014.", "However, the column density measured in the 2014 observations is not sufficient to explain the rise in flux to the highest flux Swift XRT observation.", "The XRT maximum 0.2–2 keV flux is more than twice that found with XMM-Newton, and removing the absorber completely only produces a 50% increase in flux.", "This suggests that the majority of the short term X-ray variability is intrinsic to the source.", "In Fig.", "REF we show the SED of HE 1136-2304 from 2014-07-02 with the XMM-Newton EPIC-pn and optical monitor (OM), and NuSTAR.", "We also show the effect of the absorption, by correcting the EPIC-pn spectrum for both intrinsic and Galactic absorption.", "It is clear from this that a large fraction of the flux is being emitted in the X-rays." ], [ "Optical spectrum", "The optical spectrum of HE1136-2304 taken with the SALT telescope in 2014 is presented in Fig.", "REF .", "This spectrum is shown in the rest frame of the AGN.", "Based on this observed spectrum we calculate a blue magnitude m$_{B}$ of 17.3$\\pm {}$ 0.1 for HE 1136-2304 Figure: Optical spectrum of HE1136-2304 taken with the SALT telescopeon July 7th, 2014 (black line).A further spectrum of this AGN taken in 1993 (blue line) is overlayedfor comparison.", "The broad Hα\\alpha and Hβ\\beta lines are clearly much stronger in the 2014 spectrum than in the 1993 spectrum.We derived the flux of the emission line intensities by integrating the flux above a linearly interpolated continuum, locally defined by regions just blueward and redward of the individual emission lines.", "The fluxes were integrated for the wavelength ranges given in column 3.", "We present in Table REF the observed emission line intensities as well as those corrected for Galactic extinctionNote that we have neglected a possible weak Feii component in our optical spectra which will overlap with the [Oiii] and H$\\beta $ lines.", "Given how bright these lines are relative to the Feii component, this contamination is negligible.", "Additionally, the relative flux contribution of the [Nii] lines with respect to the H$\\alpha $ -[Nii] line complex is far stronger in the faint state, introducing additional uncertainty into the measurements of the H$\\alpha $ fluxes.. With respect to the relative intensities of the broad and narrow Balmer line components, for example H$\\beta $ , HE 1136-2304 has to be classified as Seyfert 1.5 type based on its spectrum taken in July 2014.", "This is based on the definition of that Seyfert galaxies with intermediate-type HI profiles in which both components can easily be recognized are of Seyfert type 1.5.", "Table: Rest-frame optical emission line intensities of HE1136-2304 from 2014:observed values (2) and corrected for Galactic extinction (3).The relative intensities of the highly ionized lines are quite strong.", "The [Fex]$\\lambda $ 6374 line intensity, for example, is stronger in HE 1136-2304 than in the prototype Seyfert 1.5 galaxy NGC 5548 .", "The narrow lines, for example [Oiii]$\\lambda $ 5007, hold line widths (FWHM) of 510 $\\pm {}$ 10 km s$^{-1}$ .", "The broad H$\\beta $ component shows a width of 4200 $\\pm {}$ 200 km s$^{-1}$ .", "These values are typical for Seyfert galaxies.", "We determined the line widths in units of km s$^{-1}$ by converting our spectrum into velocity space with the IRAF task `disptrans'.", "This broad component of H$\\beta $ is redshifted with respect to the narrow component by 250$\\pm {}$ 50 km s$^{-1}$ .", "We use the flux variation gradient (FVG) method , , to estimate the relative contributions of the host and AGN to the continuum flux (shown in Fig.", "REF ).", "We calculate B and R-band fluxes for each spectrum by convolving them with B and R filter curves.", "From 1993 to 2014 the R-band flux increases from $0.73\\pm 0.04$ to $0.89\\pm 0.02$ and the B-band from $0.26\\pm 0.06$ to $0.56\\pm 0.03$ .", "The AGN and host flux lines intercept at R and B band fluxes of 0.73 mJy and 0.25 mJy, respectively.", "From this, we can infer that the maximum contribution of the AGN to the blue continuum in 1993 was $\\sim $ 20%, and that it must have increased by a factor of at least 4 in 2014.", "Similarly, the 1993 broad H$\\alpha $ and H$\\beta $ line fluxes were $(55\\pm 10)\\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ and $(15\\pm 5) \\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ , a factor of 3–4 below those measured in 2014.", "A caveat here is that the two spectra were taken with different aperture sizes, so may contain slightly different host galaxy contributions.", "Figure: B versus R flux variations of He1136-2304.", "The two measurements of He1136-2304in the bright state (2014) and in the low state (1993) are used to determinethe AGN slope (blue).", "The red dashed lines give the range of host slopes fornearby AGN as determined by .", "The intersections between theAGN and host galaxy slopes gives the possible range of host galaxy fluxes in the B and R bands (grey lines indicate the central value of R=0.73=0.73 mJy, B=0.25=0.25 mJy).Based on the column density needed to lower the X-ray flux to the level of 1990 ($\\sim 2.45\\times 10^{22}$ cm$^{-2}$ , see § REF ), we can estimate the corresponding effect that the associated reddening should have had on the blue continuum flux, in the case that a drop in the absorbing column is the cause of the X-ray flare.", "Converting the column density to extinction using the relation from and assuming $A_V=3.1E_{B-V}$ , we find that the flux at 5500 Å should be lower by a factor of $\\sim 10^5$ with respect to the 2014 flux.", "We note that in AGN the ratio of $A_V$ to $N_\\textrm {H}$ is generally significantly lower than that from Galactic absorption: find that most AGN have $A_V/N_\\textrm {H}$ a factor of 10 lower than Galactic, with some up to 100 times lower.", "Lowering the $A_V$ values by 10 and 100 corresponds to flux ratios ($F(2014)/F(1993)$ ) of 46 and 1.5, respectively.", "The high-amplitude variability of HE 1136-2304 makes BH mass estimates more uncertain.", "For a first order of magnitude estimate, we use the optical continuum luminosity, which varied very little.", "Using the width of the broad H$\\beta $ line, continuum luminosity ($\\log [\\lambda L_\\lambda (5100\\textrm {Å})]=42.72\\pm 0.05$ ) and the mass scaling relation of we find a black hole mass of $\\log (M_\\textrm {BH}/M_\\odot )=7.3\\pm 0.4$ in 2014 and $6.9\\pm 0.5$ in 1993, assuming 50% and 10% of the luminosity at 5100Å is due to the AGN, respectively, based on the FVG results.", "Assuming a bolometric luminosity of $L_\\textrm {bol}\\approx 9\\lambda L_\\lambda (5100\\textrm {Å})=2.3\\times 10^{43}$ erg s$^{-1}$ in 2014 we find an Eddington fraction of $\\sim 0.01$ .", "This may be an underestimate, however, as the 0.5–10 keV X-ray luminosity is $\\sim 2.8\\times 10^{43}$ erg s$^{-1}$ , which would suggest the bolometric luminosity should be significantly larger.", "From the 2–10 keV luminosity (Table REF ) we can calculate an alternative estimate of the bolometric luminosity of $3.4\\times 10^{44}$ erg s$^{-1}$ , assuming a bolometric correction factor of $\\kappa _\\textrm {2--10~keV}=20$ .", "This corresponds to an Eddington fraction of $\\sim 0.09$ , which is significantly higher than that found from the continuum luminosity.", "Note that this range of inferred Eddington fractions is on the same order as the critical accretion rate ($\\dot{m}/\\dot{m}_\\textrm {Edd}=0.01$ ) required in the model of for the onset of a significant BLR.", "As seen in the SED (Fig.", "REF ) the X-ray flux is relatively high, which explains the difference in results found here.", "For an independent estimate of the historical X-ray emission, we have used the [Oiii] emission from the narrow-line region Table REF .", "Based on the correlation between [Oiii] luminosity and (2-10) keV X-ray luminosity of , and the observed [Oiii] luminosity of HE 1136-2304, we predict an X-ray luminosity of 1.2 $\\times 10^{42}$ erg s$^{-1}$ , which is a factor of 14 lower than the one measured during the deep XMM-Newton observation (Table REF )Note that we did not apply any extinction correction when estimating $L_\\textrm {[{O}{iii}]}$ .", "While we cannot estimate the BLR reddening from the optical spectrum because broad H$\\alpha $ is blended with [NII], the narrow Balmer-line ratio does not indicate any NLR reddening..", "This provides independent evidence that HE 1136-2304 is currently undergoing an epoch of enhanced X-ray emission." ], [ "Discussion", "While the 2014 optical and X-ray data are simultaneous within a few days, an important caveat for the following discussion is that the earlier archival data are not simultaneous.", "There is a three year gap between the two, which is long enough for dramatic changes to occur in both the X-ray and optical bands, meaning that the observed fluxes and spectra are not necessarily identical to those in the corresponding band at the time of the other observation.", "We fit the X-ray spectra with a variety of phenomenological models, all of which agree that a significant absorbing column (around $10^{21}$ cm$^{-2}$ ) is required in addition to Galactic absorption, although in no case does it greatly exceed this value.", "The spectrum is fairly conventional for a Seyfert 1.5.", "It shows a moderate soft excess, a narrow iron line at 6.4 keV and a high energy cut-off at $\\sim 100$ keV , .", "This cut-off energy is fairly low, but not exceptional .", "Based on the two EPIC-pn spectra shown in the right panel of Fig.", "REF , it is unlikely that absorption variability can be responsible for all of the variations in flux seen, as the $\\sim 30$ % drop in flux seen by XMM-Newton appears to be uniform across the bandpass, inconsistent with a change in the absorption column.", "Notably, there is a lack of ionized absorption in the form of either a warm absorber or an outflow, which may indicate a viewing angle that prevents the line of sight being intercepted by this material.", "While the physical model gives the best overall fit to the X-ray spectrum, we caution that there are problems with this model.", "Several of the parameters are hitting their limits, particularly in the reflection model.", "Since no broad iron line is apparent in the spectrum it is difficult to be certain about the measured values.", "While the reflection component greatly improves the fit quality this may be for more complex reasons, such as a more complex continuum model being needed to properly reproduce the data.", "Using SALT spectroscopy we have determined that HE 1136-2304 changed its spectral type from nearly Seyfert 2 type (i.e.", "Seyfert 1.95) in March 1993 to a Seyfert 1.5 type in July 2014, coinciding with a huge increase in X-ray flux measured with XMM-Newton and Swift.", "There was only a very weak broad H$\\alpha $ component apparent in the optical spectrum taken in March 1993 and there was no indication for a broad H$\\beta $ component in 1993.", "The definition for a Seyfert 1.9 type is based on that of , i.e.", "that no broad H$\\beta $ emission can be seen in the spectrum.", "We scaled the spectrum taken in the year 1993 with respect to the intensity of the [Oiii] $\\lambda $ 5007 line taken in 2014.", "In contrast to the Balmer line and X-ray flux we did not see much change in the optical continuum flux when comparing the spectra taken in 1993 and 2014.", "While the general optical continuum flux strength is similar at both epochs there is a difference in the gradient, which is bluer in 2014 relative to 1993.", "We give in Table 5 the derived continuum intensities at three optical continuum wavelengths for the years 1993 and 2014.", "Table: Optical continuum fluxes for the years 1993 and 2014.There are several potential explanations for the observed increase in X-ray flux and appearance of broad lines, which we will now consider: Some inactive galaxies have been seen to flare due to stellar tidal disruption events .", "HE 1136-2304 itself is a classical AGN, judged from its optical emission line ratios ($\\log $ ([Oiii]5007/H$\\beta $ ) = 0.96, $\\log $ ([Sii]6724/H$\\beta $ ) = –0.35) which place it well within the AGN regime in diagnostic diagrams.", "While TDEs can also occur in AGN, it is more difficult to make a positive case for them in systems which permanently harbour an accretion disk, as the increase in flux is less dramatic relative to the non-flaring flux in an AGN.", "Further, we note that the high-state X-ray spectrum below 2 keV (Tab.", "2), is rather flat, unlike the majority of soft X-ray TDEs which showed ultra-soft X-ray spectra .", "In addition, the detection of the source in XMM-Newton slews from 2010 and 2011 is inconsitent with a single TDE, which would follow a well established decay profile on shorter timescales , .", "Another idea is that there is in fact no change in the intrinsic brightness – instead, the broad lines and X-ray emission were obscured by dusty absorbing material, which left the line of sight, causing the apparent X-ray outburst and revealing the broad line region.", "The presence of mild absorption in the X-ray spectrum may support this, and the high inclination suggested by the reflection model is also consistent with this picture.", "suggest a model where the width of the broad lines is controlled by the accretion rate .", "In this model the BLR is associated with a disc wind, the radius of which increases with accretion rate.", "Thus, as the accretion rate increases the breadth of the lines decreases, as it is determined by the Keplerian velocity at the wind radius.", "Below a limiting accretion rate, however, no such wind is produced, as evaporation inhibits its formation.", "This model could therefore explain our observations if, as seems plausible, the accretion rate of HE 1136-2304 has crossed this threshold value, causing the X-ray outburst and BLR simultaneously.", "Finally, arguably the simplest explanation is a flare in the emission from the EUV and X-rays due to an increase in the accretion rate, which then excited a larger amount of broad line emission.", "This can potentially be triggered by disk instabilities, without the need for any external cause , .", "In practice this model is difficult to distinguish from the previous scenario, but makes no assumptions on the nature of the BLR." ], [ "Conclusions", "We have presented X-ray and optical observations of the AGN HE 1136-2304 with XMM-Newton, NuSTAR, Swift, and SALT.", "The AGN was found to have increased in X-ray flux by a factor of $\\sim 30$ , coinciding with the appearance of broad lines in the optical spectrum (and hence a change in classification from Seyfert 1.95 to Seyfert 1.5).", "We find that the X-ray spectrum requires significant absorption in excess of the Galactic column, with a column density of $\\sim 10^{21}$ cm$^{-2}$ .", "An increase in column density of approximately 1 order of magnitude (to 2–3$\\times 10^{22}$ cm$^{-2}$ ) would be enough to explain the lower flux observed in 1993.", "However, changes in the absorption cannot explain the increased flux seen by Swift, meaning the majority of the short-term variability must be intrinsic to the source.", "While this is not conclusive, as the long-term variability could be driven by a different mechanism, it favours the interpretation of the long-term change in flux as also intrinsic to the source, caused by an increase in the accretion rate.", "Sources like HE 1136-2304, which show simultaneous high-amplitude variability in X-rays and optical emission lines, provide us with tight constraints on the physics behind AGN classifications and accretion variability.", "If other sources where the optical continuum does not respond to the X-ray changes can be identified this may help us understand the nature of disk instabilities, particularly if the relevant timescales can be constrained.", "In general, radio-quiet AGN flaring in X-rays remain a relatively unexplored area, with great potential for shedding light on accretion physics." ], [ "Acknowledgements", "We thank the anonymous referee for their detailed and constructive feedback.", "We thank Lutz Wisotzki for making available the optical spectrum of HE1136-2304 taken in 1993, and we thank Vassilis Karamanavis for a careful reading of the manuscript.", "MLP acknowledges financial support from the Science and Technology Facilities Council (STFC).", "This paper is based on observations taken with the SALT telescope.", "This work has been supported by DFG grant Ko 857/32-2.", "Based on observations with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.", "This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration.", "This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).", "We would also like to thank Neil Gehrels for approving the Swift ToO requests, and the Swift science operation team for performing the observations." ] ]	1606.04955
[ [ "Robust Active Perception via Data-association aware Belief Space\n planning" ], [ "Abstract We develop a belief space planning (BSP) approach that advances the state of the art by incorporating reasoning about data association (DA) within planning, while considering additional sources of uncertainty.", "Existing BSP approaches typically assume data association is given and perfect, an assumption that can be harder to justify while operating, in the presence of localization uncertainty, in ambiguous and perceptually aliased environments.", "In contrast, our data association aware belief space planning (DA-BSP) approach explicitly reasons about DA within belief evolution, and as such can better accommodate these challenging real world scenarios.", "In particular, we show that due to perceptual aliasing, the posterior belief becomes a mixture of probability distribution functions, and design cost functions that measure the expected level of ambiguity and posterior uncertainty.", "Using these and standard costs (e.g.~control penalty, distance to goal) within the objective function, yields a general framework that reliably represents action impact, and in particular, capable of active disambiguation.", "Our approach is thus applicable to robust active perception and autonomous navigation in perceptually aliased environments.", "We demonstrate key aspects in basic and realistic simulations." ], [ "Introduction", "In the context of partially observable Markovian systems, planning over belief space (BSP) under some simplifying assumptions, provides scalable applications including autonomous navigation, object grasping and manipulation, active SLAM, and robotic surgery.", "In presence of uncertainty, such as in robot motion and sensing, the true state of variables of interest (e.g.", "robot poses), is unknown and can only be represented by a probability distribution over possible states, given available data.", "This distribution, the belief space, is inferred using probabilistic approaches based on incoming sensor observations and prior knowledge.", "The corresponding problem is an instantiation of a partially observable Markov decision problem (POMDP) [16].", "Apart from simplifying structural assumptions – such as Gaussian noise around a given observation and motion model – state-of-the-art BSP approaches typically assume data association to be given and perfect (see Figure REF ), i.e.", "the robot is assumed to correctly perceive the environment to be observed by its sensors, given a candidate action.", "For brevity, we shall call it DAS.", "In reality, the world is often full of ambiguity, that together with other sources of uncertainty, make perception a challenging task.", "As an example, matching images from two different but similar in appearance places, or attempting to recognise an object that is similar in appearance, from the current viewpoint, to another object.", "Both cases are examples of ambiguous situations, where naïve and straightforward approaches using DAS are likely to yield incorrect results, i.e.", "mistakenly considering the two places as same, and incorrectly associating the observed object.", "Figure: (a) Generative graphical model.", "Standard BSP approaches assume data association (DA) is given and perfect (DAS).", "We incorporate data association aspects within BSP and thus can reason about ambiguity (e.g.", "perceptual aliasing) at a decision-making level.", "(b) Schematic representation of pose, scene and observation spaces.", "Scenes A 1 A_1 and A 3 A_3 when viewed from perspective xx and x ' x^{\\prime } respectively, produce the same nominal observation z ^\\hat{z}, giving rise to perceptual aliasing.Thus, in presence of ambiguity, DAS may lead to incorrect posterior beliefs and as a result, to sub-optimal actions.", "More advanced approaches are therefore required to enable reliable operation in ambiguous conditions, approaches often referred to as (active) robust perception.", "These approaches typically involve probabilistic data association and hypothesis tracking given available data.", "Thus, for the object detection example, each hypothesis may represent a candidate object from a given database that the current observation (e.g.", "image or point-cloud) is successfully registered to.", "Similarly, one might reason probabilistically regarding perceptual aliasing, as in the first example above, which would also involve probabilistic data association.", "Yet, existing robust perception approaches focus on the passive case, where robot actions are externally determined and given, while the closely related approaches for active object detection and classification consider the robot to be perfectly localised.", "In this work we develop a general data association aware belief space planning (DA-BSP) framework capable of better handling complexities arising in real world, possibly perceptually aliased, scenarios.", "We rigorously incorporate reasoning about data association within belief space planning, while also considering other sources of uncertainty (motion, sensing and environment).", "In particular, we show our framework can be used for active disambiguation by determining appropriate actions, e.g.", "future viewpoints, for increasing confidence in a certain data association hypothesis.", "Organization of the paper: After discussing related work and stating our contributions, we formulate the considered problem in Section .", "In Section we provide concept overview and then discuss in detail the proposed approach, while demonstrating key aspects in simulated basic and realistic scenarios in Section .", "Finally, in Section we conclude the discussion and suggest potential directions for future research.", "Calculating optimal solutions to POMDP is computationally intractable (PSPACE-complete) [22] for all but the smallest problems.", "The vast research area of approximate approaches (with reduced computational complexity) can be roughly segmented into point-based value iteration methods [26], [19], simulation based [30] and sampling based approaches [27], [6], [2], and direct trajectory optimization [33], [25], [11] methods.", "In all cases, finding the (locally) optimal actions involves evaluating a given objective function while considering future observations to be acquired as a result of each candidate action.", "They all assume DAS.", "For example, it is typically assumed that the robot can be localised by making observations of known landmarks or beacons (see, e.g.", "[27], [2]), while assuming to correctly associate each future measurement with an appropriate landmark.", "Though reasonable in certain scenarios, DAS becomes unrealistic in the presence of perceptually aliased environments (two scenes that look alike) and localisation uncertainty, as in this work.", "The issue of perceptual aliasing has been considered in the earlier works on POMDP planning, though again with highly simplified scenarios, since the data-association further complicates the problem.", "In a slightly separate line of research, the approaches that study the issue were in the context of multiple hypothesis tracking(see [28] for earliest work on MHT) or more recently, of active robust perception.", "Both these approaches rely on passive and often non-parametric approaches, through various filtering techniques; we refer an interested reader to the book [15] and tutorial [3] for further details.", "For example, [34] proposed using Gaussian mixture probability hypothesis density (PHD) filter.", "To the best of our knowledge, such approaches are not considered in the context of active planning.", "Coming back to scalable planning methods such as BSP, we note that while the traditional BSP approaches had typically assumed the environment to be accurately known (e.g.", "a given map), recent works, including [8], [9], [35], [18], [11], relax this assumption and model the uncertainty of the environment mapped thus far within the belief.", "The corresponding framework is thus tightly related to active SLAM, with the well known trade-off between exploration and exploitation.", "Recent work [18], [11], [9], [35] in this branch focused in particular on probabilistically modelling what future observations will be obtained given a candidate action.", "Though none of them relax DAS assumption.", "In the last few years, the SLAM research community has investigated approaches to be resilient to false data association (outliers) overlooked by front-end algorithms (e.g.", "image matching), see e.g.", "[31], [21], [7], [14], [13].", "However these approaches, also known as robust graph optimization approaches, are developed only for the passive problem setting, i.e.", "robot actions are given and externally determined.", "In contrast, we consider a complimentary active framework that incorporates data association aspects within BSP.", "Our approach is also tightly related with recent work on active hypothesis disambiguation in the context object detection and classification [4], [29], [20], [36], [32].", "Given hypotheses regarding object class and pose, these approaches aim to find a sequence future viewpoints that will lead to disambiguation, i.e.", "identifying the correct hypothesis.", "However, these approaches assume the sensor is perfectly localized and can be shown to be a specific case of DA-BSP.", "Probably the closest work to our approach is by Agarwal et al.", "[1], where the authors also consider hypotheses due to ambiguous data association and develop a BSP approach for active disambiguation.", "However, unlike them, DA-BSP considers ambiguous data association also in posterior and thus does not require a guarantee of fully disambiguating action in the future." ], [ "Contributions", "To summarize, our main contributions in this paperEarlier versions of this paper appeared in [24] and [23].", "are as follows: (a) relaxing the data-association-is-solved assumption for a general data-association aware BSP framework (DA-BSP) with GMM priors (b) considering active data-association aspect for both planning and inference, hence providing a closed-loop framework (c) reducing some of the known recent BSP approaches to a degenerate cases of DA-BSP (d) demonstrating empirical results in support of two claims: data-association is crucial for a robust BSP and the principled approach of DA-BSP can be scalable enough to be applied on practical problems.", "Consider a robot operating in a partially known or pre-mapped environment which can be ambiguous and perceptually aliased.", "The robot takes observations of different scenes and objects in the environment, and uses these observations to infer application-dependent random variables of interest (e.g.", "past and current robot poses).", "The following three spaces are involved in the considered problem, as shown in Figure REF : pose-space, scene-space and observation-space.", "Pose-space involves all possible perspectives a robot can take with respect to a given environment model and in the context of task at hand.", "We denote the robot pose at time step $k$ by $x_k$ and a sequence of poses from 0 up to $k$ by $X_k\\doteq \\lbrace x_0,\\ldots ,x_k\\rbrace $ .", "Given all controls $u_{0:k-1}\\doteq \\lbrace u_0,\\ldots ,u_{k-1}\\rbrace $ and observations $Z_{0:k}\\doteq \\lbrace Z_0,\\ldots ,Z_k\\rbrace $ up to time step $k$ , the posterior probability distribution functionStrictly speaking, this is either the probability mass function or the probability density function for a discrete or a continuous random variable, respectively.", "is defined as $\\mathbb {P}({X_k\|u_{0:k-1},Z_{0:k}})$ .", "For notational convenience, we define below histories $\\mathcal {H}_k$ and $\\mathcal {H}_{k+1}^-$ and rewrite the posterior pdf (belief), at time $k$ as $b[X_k] \\doteq \\mathbb {P}({X_k\|\\mathcal {H}_k})$ .", "$\\mathcal {H}_k\\doteq \\lbrace u_{0:k-1},Z_{0:k}\\rbrace \\ \\ , \\ \\ \\mathcal {H}_{k+1}^- \\doteq \\mathcal {H}_k \\cup \\lbrace u_k\\rbrace .$ The scene-space involves a discrete set of objects or scenes, denoted by the set $\\lbrace A_{\\mathbb {N}} \\rbrace $ , in the given world model, and which can be detected through the sensors of the robot.", "We will use symbols $A_{i}$ and $A_{j}$ to denote such typical scenes.", "Note that even if the objects are identical, they are distinct in scene space.", "This will be important when we shall consider the cases where the objects look similar from some perspectives.", "Finally, observation-space is the set of all possible observations that the robot is capable of obtaining when considering its mission and sensory capabilities.", "We consider probabilistic motion and observation models $x_{k+1} = f(x_k,u_k)+w_k \\ \\ , \\ \\ z_{k} = h(x_k,A_i)+v_k,$ and denote them by $\\mathbb {P}({x_{k+1}\|x_k,u_k})$ and $\\mathbb {P}({z_k\|x_k,A_i})$ , respectively.", "As common in literature, we consider Gaussian zero-mean process and measurement noise $w_i \\sim \\mathcal {N}(0,\\Sigma _w)$ and $v_k \\sim \\mathcal {N}(0,\\Sigma _v)$ , with known noise covariance matrices $\\Sigma _w$ and $\\Sigma _v$ .", "Here, $h(x_k, A_{i})$ is a noise-free observation which we would refer as nominal or predicted observation $\\hat{z}$ , that corresponds to observing scene $A_i$ from pose $x_k$ .", "Given a prior $\\mathbb {P}({x_0})$ and motion and observation models (REF ), the joint posterior pdf at the current time $k$ can be written as $\\mathbb {P}({X_k\|\\mathcal {H}_k}) = \\mathbb {P}({x_0})\\prod _{i=1}^k \\mathbb {P}({x_i\|x_{i-1},u_{i-1}}) \\mathbb {P}({Z_i\|x_i,A_{i}}).$ Note that DAS is the underlying assumption in the above equation.", "If the prior $\\mathbb {P}({x_0})$ is Gaussian, it is not difficult to show that $b[X_k]$ is also a Gaussian with some mean $\\hat{X}_k$ and covariance $\\Sigma _k$ that can be efficiently calculated via maximum a posteriori (MAP) inference, see e.g. [17].", "It is also valid in case where the environment model is given but uncertain, and when this model is unknown a priori and instead is constructed on-line within SLAM framework.", "However, in this paper we consider a more general case where the prior belief is modeled by a Gaussian mixture model (GMM).", "Such a situation can arise, for example, in the kidnapped robot problem in a perceptually aliased environment (e.g.", "different similar in appearance rooms), where matching sensor observations against a given map would indicate several most probable robot locations.", "In such a case the belief at time $k$ can be represented by a GMM, $b[X_k] = \\sum _{j=1}^{M_{k}} \\xi _{k}^j \\mathbb {P}({X_{k}\|\\mathcal {H}_{k}, \\gamma =j}),$ where $M_{k}$ is the number of components (or modes), the $j$ th component is represented by the weight $\\xi _{k}^j \\doteq \\mathbb {P}({\\gamma =j\|\\mathcal {H}_{k}})$ , modeling the probability of the robot being in that component, and by the conditional Gaussian $b[X_{k}^j]\\doteq \\mathbb {P}({X_{k}\|\\mathcal {H}_{k}, \\gamma =j})=\\mathcal {N}(\\hat{X}_{k}^j,\\Sigma _{k}^j),$ with appropriate mean $\\hat{X}_{k}^j$ and covariance $\\Sigma _{k}^j$ .", "Here, $\\gamma $ is an indicator variable denoting the component number.", "Given the belief at time $k$ , one can reason about the robot's best future actions that would minimize (or maximize) an objective function $J$ .", "$J(u_{k}) = \\mathbb {E}\\left\\lbrace c \\left(b[X_{k+1}], u_{k} \\right) \\right\\rbrace ,$ where the expectation is over the (unknown) future observation $z_{k+1}$ , and $c(.", ")$ is the immediate cost.", "The posterior belief at time $t_{k+1}$ is a function of control $u_{k}$ and observation $z_{k+1}$ , i.e.", "$b[X_{k+1}] \\doteq \\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}}) \\equiv \\mathbb {P}(X_{k+1}\| \\mathcal {H}_{k}, z_{k+1}, u_{k}).$ Note that, according to Eq.", "(REF ), we need to calculate the posterior belief (REF ) for each possible value of $z_{k+1}$ .", "Similarly, we define the propagated joint belief as $b[X_{k+1}^-]\\doteq \\mathbb {P}({X_{k}\|\\mathcal {H}_{k}}) \\mathbb {P}({x_{k+1}\|x_{k},u_{k}}),$ from which the marginal belief over the future pose $x_{k+1}$ can be calculated as $b[x_{k+1}^-] \\doteq \\int _{\\lnot x_{k+1}} b[X_{k+1}^-]$ .", "In particular, the propagated belief at the first look ahead step, given the GMM belief (REF ) at time $k$ is $b[X_{k+1}^-] = \\sum _{j=1}^{M_{k}} \\xi _{k+1}^{j-} b[X_{k+1}^{j-}],$ with $\\xi _{k+1}^{j-} \\doteq \\mathbb {P}({\\gamma _{k+1}=j\|\\mathcal {H}_{k+1}^-})\\equiv \\xi _{k}^j$ , and $b[X_{k+1}^{j-}] \\doteq \\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-, \\gamma _{k}=j})= b[X_{k}^j] \\mathbb {P}({x_{k+1}\|x_k,u_k})$ .", "As earlier, with DAS assumption, one can consider for each specific value of $z_{k+l}$ the corresponding observed scene $A_i$ , and express the posterior (REF ) recursively as $b[X_{k+1}] \\!=\\!", "\\eta b[X_{k+1}^-] \\mathbb {P}({z_{k+1}\|x_{k+1},A_i}),$ which can be represented as $b[X_{k+1}]=\\mathcal {N}(\\hat{X}_{k+1}, \\Sigma _{k+1})$ with appropriate mean $\\hat{X}_{k+1}$ and covariance $\\Sigma _{k+1}$ .", "The optimal control is then defined as: $u_{k}^{\\star } \\doteq \\operatornamewithlimits{arg\\,min}_{u_{k}} J(u_{k})$ .", "DAS assumption simplifies greatly the above formulation.", "Yet, in practice, determining data association reliably is often a non trivial task by itself, especially when operating in perceptually aliased environments.", "An incorrect data association (wrong scene $A_i$ in Eq.", "(REF )) can lead to catastrophic results, see, e.g.", "[14], [12], [13].", "In this work we relax this restricting assumption and rigorously incorporate data association aspects within belief space planning and inference considering the underlying distributions are GMMs." ], [ "Approach", "Given some candidate action (or sequence of actions) and the belief at planning time $k$ , we can reason about a future observation $z_{k+1}$ (e.g.", "an image) to be obtained once this action is executed; its actual value is unknown.", "All the possible values such an observation can assume should be taken into account while evaluating the objective function; hence, the expectation operator in Eq.", "(REF ).", "When written explicitly it transforms to $J(u_k) \\!\\!", "\\doteq \\!\\!", "\\int _{z_{k+1}} \\!\\!\\!\\!\\!\\!\\!", "\\overbrace{\\mathbb {P}({z_{k+1} \\mid \\mathcal {H}_{k+1}^-})}^{(a)}\\;\\; \\!\\!", "c\\left(\\overbrace{ \\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}}) }^{(b)} \\!", "\\right) $ The two terms $(a)$ and $(b)$ in the above equation have intuitive meaning: for each considered value of $z_{k+1}$ , $(a)$ represents how likely is it to get such an observation when both the history ${\\cal H}$ and control $u_k$ are known, while $(b)$ corresponds to the posterior belief given this specific $z_{k+1}$ .", "Considering DAS means we can correctly associate each possible measurement $z_{k+1}$ with the corresponding scene $A_i$ it captures, as in Eq.", "(REF ).", "Yet, it is unknown from what future robot pose $x_{k+1}$ the actual observation $z_{k+1}$ will be acquired, since the actual robot pose $x_k$ at time $k$ is unknown and the control is stochastic.", "Indeed, as a result of action $u_k$ , the robot actual (true) pose $x_{k+1}$ can be anywhere within the propagated belief $b[x_{k+1}^-]$ .", "In inference, we have a similar situation with the key difference that the observation $z$ has been acquired.", "We must first associate the captured measurement $z$ with the scene or object $A_i$ it describes, i.e.", "write the appropriate measurement likelihood term in the posterior (REF ).", "In BSP framework, solved data association means that for each such observation $z\\in \\lbrace z\\rbrace $ the corresponding observed scene $A_i\\in \\mathcal {A}$ is known.", "In contrast, we do not assume this, and instead reason about possible scenes or objects that the future observation $z_{k+1}$ could be generated from, see Figures REF and REF ." ], [ "Parsimonious data association:", "Incorporating data-association is expensive.", "However, if the environment has only distinct scenes or objects, then for each specific value of $z_{k+1}$ , there will be only one scene $A_i$ that can generate such an observation according to the model (REF ).", "In case of perceptually aliased environments, there could be also several scenes (or objects) that are either completely identical, or have a similar visual appearance when observed from appropriate viewpoints.", "They could equally well explain the considered observation $z_{k+1}$ .", "Thus, there are several possible associations $\\lbrace A_i\\rbrace $ and due to localisation uncertainty determining which association is the correct one is not trivial.", "As we show in the sequel, in these cases the posterior $b[X_{k+1}]$ (term $(b)$ in Eq.", "(REF )) becomes a Gaussian mixture with appropriate weights that we rigorously compute.", "Additionally, the weight updates are capable of discriminating against unlikely data-associations, during the planning steps." ], [ "Perceptual aliasing:", "Intuitively speaking, perceptual aliasing occurs when an object different from the actual one, produces the same observation and thereby is an alias, in the sense of perception, to the true object.", "Consider two notions of perceptual aliasing: exact and probabilistic.", "Exact perceptual aliasing of scenes $A_{i}$ and $A_{j}$ is defined as $\\exists x, x^{\\prime },\\; h(x,A_{i}) = h(x^{\\prime },A_{j})$ , and will be denoted in this paper by $\\lbrace {A_i,A_j}\\rbrace _{\\textbf {aliased}}$ .", "In other words, the same nominal (noise-free) observation $\\hat{z}$ can be generated by observing different scenes, possibly from different viewpoints.", "Such a situation is depicted in Figure REF .", "A probabilistic perceptual aliasing is a more general form of aliasing, which can be defined as $\\exists x, x^{\\prime },\\; \| \\mathbb {P}({z \| A_{i},x}) - \\mathbb {P}({z \| A_{j},x^{\\prime }})\|<\\epsilon $ for some small threshold $\\epsilon $ ." ], [ "Computing the term (a) : $\\mathbb {P}(z_{k+1}\|\\mathcal {H}_{k+1}^-)$", "Applying total probability over non-overlapping scene space $\\lbrace A_{\\mathbb {N}} \\rbrace $ and marginalizing over all possible robot poses, yields $\\mathbb {P}({z_{k+1} \| \\mathcal {H}_{k+1}^-}) \\!\\!", "\\equiv \\!\\!", "\\sum _i^{\|A_{\\mathbb {N}}\|} \\!", "\\int _x \\!", "\\mathbb {P}({z_{k+1}, x, A_i \\!", "\\mid \\!", "\\mathcal {H}_{k+1}^-}) \\!\\!", "\\doteq \\!\\!", "\\sum _i^{\|A_{\\mathbb {N}}\|} \\!\\!", "w_{k+1}^i.$ As seen from the above equation, to calculate the likelihood of obtaining some observation $z_{k+1}$ , we consider separately, for each scene $A_i \\in \\lbrace A_{\\mathbb {N}} \\rbrace $ , the likelihood that this observation was generated by scene $A_i$ .", "This probability is captured for each scene $A_i$ by a corresponding weight $w_{k+1}^i$ ; these weights are then summed to get the actual likelihood of observation $z_{k+1}$ .", "As will be seen below, these weights naturally account for perceptual aliasing aspects for each considered $z_{k+1}$ .", "In practice, instead of considering the entire scene space $\\lbrace A_{\\mathbb {N}} \\rbrace $ that could be huge, the availability of the belief $b[X_{k+1}^-]$ makes it possible to consider only those scenes that could be actually observed from viewpoints with non-negligible probability according $b[X_{k+1}^-]$ , e.g.", "within 3 standard deviations of uncertainty for each GMM component.", "In the following, however, we proceed while reasoning about the entire scene space $\\lbrace A_{\\mathbb {N}} \\rbrace $ .", "Proceeding with the derivation further, using the chain rule we compute $\\sum _i \\int _x \\mathbb {P}({z_{k+1} \\mid x, A_i, \\mathcal {H}_{k+1}^-}) \\mathbb {P}({A_i, x \\mid \\mathcal {H}_{k+1}^-})$ However, since $\\mathbb {P}({A_i, x \\mid \\mathcal {H}_{k+1}^-}) = \\mathbb {P}({A_i \| x \\mid \\mathcal {H}_{k+1}^-}) b[x_{k+1}^- = x]$ , we get $\\sum _i^{\|A_{\\mathbb {N}}\|} \\int _x \\mathbb {P}({z_{k+1} \| x, A_i, \\mathcal {H}_{k+1}^-}) \\mathbb {P}({A_i \| \\mathcal {H}_{k+1}^-, x }) b[x_{k+1}^-=x].$ Thus, $w_{k+1}^i \\!", "\\doteq \\!\\!", "\\int _x \\!", "\\!\\mathbb {P}({z_{k+1} \| x, A_i, \\mathcal {H}_{k+1}^-}) \\mathbb {P}({A_i \| \\mathcal {H}_{k+1}^-, x }) b[x_{k+1}^-\\!\\!=\\!x].$ Since the propagated belief (REF ), from which $b[x_{k+1}^-]$ is calculated, is a GMM, we can replace $b[x_{k+1}^-=x]$ with $\\sum _{j=1}^{M_k} \\xi _{k+1,j}^- b[x_{k+1,j}^-=x]$ .", "Here, $\\mathbb {P}({z_{k+1} \\mid A_{i}, x, \\mathcal {H}_{k+1}^-})\\equiv \\mathbb {P}({z_{k+1} \\mid A_{i}, x})$ is the standard measurement likelihood term, while $\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-,x})$ represents the event likelihood, which denotes the probability of scene $A_i$ to be observed from viewpoint $x$ .", "In other words, this scenario-dependent term encodes from what viewpoints each scene $A_i$ is observable and could also model occlusion and additional aspects.", "As such, this term can be determined given a model of the environment and thus, in this work, we consider this term to be given.", "The weights $w_{k+1}^i$ (REF ) naturally capture perceptual aliasing aspects: consider some observation $z_{k+1}$ and the corresponding generative model $z_{k+1}=h(x^{tr},A^{tr})+v$ with appropriate unknown true robot pose $x^{tr}$ and scene $A^{tr}\\in \\lbrace A_{\\mathbb {N}} \\rbrace $ .", "Clearly, the measurement likelihood $\\mathbb {P}({z_{k+1} \\mid x, A_i, \\mathcal {H}_{k+1}^-})$ will be high when evaluated for $A_i=A^{tr}$ and in vicinity of $x^{tr}$ .", "Note that we will necessarily consider such a case, since according to Eq.", "(REF ) we separately consider each scene $A_i$ in $\\lbrace A_{\\mathbb {N}} \\rbrace $ , and, given $A_i$ , we reason about all poses $x$ in Eq.", "(REF ).", "In case of perceptual aliasing, however, there will be also another scene(s) $A_j$ which could generate the same observation $z_{k+1}$ from appropriate robot pose $x^{\\prime }$ .", "Thus, the corresponding measurement likelihood term to $A_j$ will also be high for $x^{\\prime }$ .", "However, the actual value of $w_i$ (for each $A_i \\in \\lbrace A_{\\mathbb {N}} \\rbrace $ ) depends, in addition to the measurement likelihood, also on the mentioned-above event likelihood and on the GMM belief $b[x_{k+1}^-]$ , with the latter weighting the probability of each considered robot pose $x$ .", "This correctly captures the intuition that those observations $z$ with low-probability poses $b[x_{k+1}^-=x^{tr}]$ will be unlikely to be actually acquired, leading to low value of $w_i$ with $A_i=A^{tr}$ .", "Since $b[x_{k+1}^-]$ is a GMM with $M_k$ components, low-probability pose $x^{tr}$ corresponds to low probabilities $b[x_{k+1}^{j-}=x^{tr}]$ for each component $j\\in \\lbrace 1,\\ldots ,M_k\\rbrace $ .", "However, the likelihood term (REF ) could still go up in case of perceptual aliasing, where the aliased scene $A_j$ generates a similar observation to $z_{k+1}$ from viewpoint $x^{\\prime }$ with latter being more probable, i.e.", "high probability $b[x_{k+1}^-=x^{\\prime }]$ .", "In practice, calculating the integral in Eq.", "(REF ) can be done efficiently considering separately each component of the GMM $b[x_{k+1}^-]$ .", "Each such component is a Gaussian that is multiplied by the measurement likelihood $\\mathbb {P}({z_{k+1} \\mid A_{i}, x, {\\cal H}})$ which is also a Gaussian and it is known that a product of Gaussians remains a Gaussian.", "The integral can then be only calculated for the window where event likelihood is non-zero i.e $\\mathbb {P}({A_{i} \\mid x, {\\cal H}}) > 0$ .", "For general probability distributions, the integral in Eq.", "(REF ) should be computed numerically.", "Since in practical applications $\\mathbb {P}({A_{i} \\mid x, {\\cal H}})$ is sparse w.r.t.", "$x$ , this computational cost is not severe." ], [ "Computing the term (b) : $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}})$", "The term $(b)$ , $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}})$ , represents the posterior probability conditioned on observation $z_{k+1}$ .", "This term can be similarly calculated, with a key difference: since the observation $z_{k+1}$ is given, it must have been generated by one specific (but unknown) scene $A_i$ according to measurement model (REF ).", "Hence, also here, we consider all possible such scenes and weight them accordingly, with weights $\\tilde{w}_{k+1}^i$ representing the probability of each scene $A_i$ to have generated the observation $z_{k+1}$ .", "As will be seen next, the posterior $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}})$ is a GMM with $M_{k+1}$ components.", "Applying total probability over non-overlapping $\\lbrace A_{\\mathbb {N}} \\rbrace $ and chain-rule, we get: $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}}) = \\sum _{i=1}^{\|A_{\\mathbb {N}}\|} \\mathbb {P}({X_{k+1} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}, A_{i}}) \\cdot \\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}}).$ The first term, $\\mathbb {P}({X_{k+1} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}, A_{i}})$ , is the posterior belief conditioned on observation $z_{k+1}$ , history $\\mathcal {H}_{k+1}^-$ , as well as a candidate scene $A_{i}$ that supposedly generated the observation $z_{k+1}$ .", "It is not difficult to show that this posterior is actually the GMM $\\mathbb {P}({X_{k+1} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}, A_{i}}) = \\sum _{j=1}^{M_k}\\xi _{k}^j b[X_{k+1}^{j+}\|A_i],$ where $b[X_{k+1}^{j+}\|A_i]\\doteq \\mathbb {P}(X_{k+1}\|\\mathcal {H}_{k+1}^-,\\gamma =j, A_i, z_{k+1})$ is the posterior of the $j$ th GMM component of the propagated belief $b[X_{k+1}^{-}]$ , see Eq.", "(REF ).", "Plugging in Eq.", "(REF ) back into $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}})$ yields from Eq.", "(REF ): $b[X_{k+1}]\\equiv \\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}}) = \\sum _{i=1}^{\|A_{\\mathbb {N}}\|} \\sum _{j=1}^{M_k}\\xi _{k}^j \\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}}) b[X_{k+1}^{j+}\|A_i].$ The term, $\\mathbb {P}({A_{i} \\mid {\\cal H} _k, u_k, z_{k+1}})$ , is merely the likelihood of $A_{i}$ being actually the one which generated the observation $z_{k+1}$ .", "This term can be evaluated, in a similar fashion to Section REF , accounting for $b[x_{k+1}^{j-}]$ for each considered $j$ th component as $\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-, z_{k+1}}) = \\int _x \\mathbb {P}({A_{i}, x \\mid \\mathcal {H}_{k+1}^-, z_{k+1}})$ , and applying Bayes' rule yields $\\tilde{w}_{k+1}^{ij} \\!\\!", "\\doteq \\!", "\\eta ^{\\prime } \\!\\!\\!", "\\int _x \\!\\!", "\\mathbb {P}({\\!", "z_{k+1} \| A_i, x, \\mathcal {H}_{k+1}^-\\!})", "\\mathbb {P}({A_i \| \\mathcal {H}_{k+1}^-, x \\!})", "b[x_{k+1}^{j-} \\!\\!=\\!", "x],$ with $\\eta ^{\\prime } = 1/ \\mathbb {P}({z_{k+1} \\mid \\mathcal {H}_{k+1}^-})$ .", "Note that for each component $j$ , $\\sum _i \\tilde{w}_{k+1}^{ij}=1$ .", "Finally, we can re-write Eq.", "(REF ) as $\\mathbb {P}({X_{k+1}\|\\mathcal {H}_{k+1}^-,z_{k+1}\\!})", "\\!= \\!\\!", "\\sum _{r=1}^{M_{k+1}} \\!\\!", "\\xi _{k+1}^r \\mathbb {P}(\\!X_{k+1}\|\\mathcal {H}_{k+1},\\gamma =r\\!", "),$ or in short, $b[X_{k+1}] = \\sum _{r=1}^{M_{k+1}} \\xi _{k+1}^r b[X_{k+1}^{r+}]$ , where $\\xi _{k+1}^r \\doteq \\xi _{k+1}^{ij} \\equiv \\xi _{k}^j \\tilde{w}_{k+1}^{ij} \\ \\ , \\ \\ b[X_{k+1}^{r+}]\\doteq b[X_{k+1}^{j+}\|A_i].$ As seen, we got a new GMM with $M_{k+1}$ components, where each component $r\\in [1,M_{k+1}]$ , with appropriate mapping to indices $(i,j)$ from Eq.", "(REF ), is represented by weight $\\xi _{k+1}^r$ and posterior conditional belief $b[X_{k+1}^{r+}]$ .", "The latter can be evaluated as the Gaussian $b[X_{k+1}^{r+}] \\propto b[X_{k+1}^{j-}] \\mathbb {P}({z_{k+1} \\mid x_{k+1}, A_i})=\\mathcal {N}(\\hat{X}^{r}_{k+1}, \\Sigma ^{r}_{k+1})$ , where the mean $\\hat{X}^{r}_{k+1}$ and covariance $\\Sigma ^{r}_{k+1}$ can be efficiently recovered via MAP inference." ], [ "Summary thus Far", "To summarize the discussion thus far, we have shown that for the myopic case, the objective function (REF ) can be re-written as $J(u_k) = \\int _{z_{k+1}} (\\sum _i^{\|A_{\\mathbb {N}}\|} w_{k+1}^i) \\cdot c\\left(\\sum _r^{M_{k+1}} \\xi _{k+1}^r b[X_{k+1}^{r+}] \\right).$ One can observe that according to Eq.", "(REF ), each of the $M_k$ components from the belief at a previous time, is split into $\|A_{\\mathbb {N}}\|$ new components with appropriate weights.", "This would imply an explosion in the number of components, making the proposed framework hardly applicable.", "However, in practice, the majority of the weights will be negligible, and therefore can be pruned, while the remaining number of components is denoted by $M_{k+1}$ in Eq.", "(REF ).", "Depending on the scenario and the degree of perceptual aliasing, this can correspond to full or partial disambiguation.", "Having shown incorporating data association within belief space planning leads to Eq.", "(REF ), we now proceed with the exposition of our approach." ], [ "Simulating Future Observations $\\lbrace z_{k+1}\\rbrace $ given {{formula:cc318ccd-7954-4119-a463-391e2c4ff97a}}", "Calculating the objective function (REF ) for each candidate action $u_k$ involves considering all possible realizations of $z_{k+1}$ .", "One approach to perform this in practice, is to simulate future observations $\\lbrace z_{k+1}\\rbrace $ given propagated GMM belief $b[X_{k+1}^-]$ , scenes $\\lbrace A_{\\mathbb {N}} \\rbrace $ and observation model (REF ).", "One can then evaluate Eq.", "(REF ) considering all observations in $\\lbrace z_{k+1}\\rbrace $ .", "We now briefly describe how this concept can be realised.", "First, viewpoints $\\lbrace x\\rbrace $ are sampled from $b[X_{k+1}^-]$ .", "For each viewpoint $x \\in \\lbrace x\\rbrace $ , an observed scene $A_i$ is determined according to event likelihood $\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k},x})$ .", "Together, $x$ and $A_i$ are then used to generate nominal $\\hat{z}=h(x,A_i)$ and noise-corrupted observations $\\lbrace z\\rbrace $ according to observation model (REF ): $z=h(x,A_i)+v$ .", "The set $\\lbrace z_{k+1}\\rbrace $ is then the union of all such generated observations $\\lbrace z\\rbrace $ .", "Note that while generating $\\lbrace z_{k+1}\\rbrace $ , the true association is known (scene $A_i$ ), it is unknown to our algorithm, i.e.", "while evaluating Eq.", "(REF )." ], [ "Computing Mixture of Posterior Beliefs $\\sum _i \\tilde{w}_i b[X_{k+1}^{i+}]$", "As seen from Eq.", "(REF ), reasoning about data association aspects resulted in a mixture of posteriors within the cost $c(.", ")$ , i.e.", "$\\sum _i \\tilde{w}_i b[X_{k+1}^{i+}]$ , for each possible observation $z_{k+1}\\in \\lbrace z_{k+1}\\rbrace $ .", "In this section we briefly describe how one can actually calculate the corresponding posterior distributions, given some specific observation $z_{k+1}\\in \\lbrace z_{k+1}\\rbrace $ .", "For simplicity, we consider the belief at planning time $k$ is a Gaussian $b[X_k]=\\mathcal {N}(\\hat{X}_k, \\Sigma _k)$ .", "However, our approach could be applied also to more general cases (e.g.", "mixture of Gaussians) with a certain price in terms of computational complexity.", "Further investigation of these aspects is left to future research.", "Under this setting, each of the components $b[X_{k+1}^{i+}]$ in the mixture pdf can be written as $b[X_{k+1}^{i+}] \\propto b[X_{k}] \\mathbb {P}({x_{k+1} \\mid x_k, u_k}) \\mathbb {P}({z_{k+1} \\mid x_{k+1}, A_i})$ .", "It is then not difficult to show that the above belief is a Gaussian $b[X_{k+1}^{i+}] = \\mathcal {N}(\\hat{X}^{i}_{k+1}, \\Sigma ^{i}_{k+1})$ and to find its first two moments via MAP inference.", "Obviously, the mixture of posterior beliefs in the cost $c(.", ")$ from Eq.", "(REF ) is now a mixture of Gaussians: $\\sum _i \\tilde{w}_i b[X_{k+1}^{i+}] = \\sum _i \\tilde{w}_i \\mathcal {N}(\\hat{X}^{i}_{k+1}, \\Sigma ^{i}_{k+1}).$" ], [ "Designing a Specific Cost Function", "The treatment so far has been agnostic to the structure of the cost function $c(.", ")$ .", "Recalling Eq.", "(REF ) we see that the belief over which the cost function is defined, is multimodal in general.", "Standard cost functions in literature, typically include terms such as control usage $c_u$ , distance to goal $c_G$ and uncertainty $c_{\\Sigma }$ , see e.g.", "[33], [11].", "In our case, however, the specific form of the latter should be re-examined and an additional term quantifying ambiguity level can be introduced.", "In this section we thus briefly discuss these two terms, starting with the cost over posterior uncertainty.", "Since, unlike in usual BSP, the posterior belief in our case is multimodal and represented as mixture of Gaussians $\\sum _i \\tilde{w}_i \\mathcal {N}(\\hat{X}^{i}_{k+1}, \\Sigma ^{i}_{k+1})$ , see Eq.", "(REF ), we could define several different cost structures depending on how we treat the different modes.", "Two particular such costs are taking the worst-case covariance among all covariances $\\Sigma ^{i}_{k+1}$ in the mixture, e.g.", "$\\Sigma =\\max _i \\lbrace tr(\\Sigma _i)\\rbrace $ , or to collapse the mixture into a single Gaussian $\\mathcal {N}(., \\Sigma )$ , see e.g. [5].", "In both cases, we can define the cost due to uncertainty as $c_\\Sigma = trace(\\hat{\\Sigma })$ .", "The cost due to ambiguity, $c_w$ , should penalise ambiguities such as those arising out of perceptual aliasing.", "Here, we note that non-negligible weights $w_i$ in Eq.", "(REF ) arise due to perceptual aliasing, whereas in case of no aliasing, all but one of these weights are zero.", "In most severe case of aliasing (all scenes or objects $A_i$ are identical), all of these weights are comparable among each other.", "Thus we take Kullback-Leibler divergence $KL_u(\\lbrace \\tilde{w}_i\\rbrace )$ of these weights $\\lbrace \\tilde{w}_i\\rbrace $ from a uniform distribution to penalise higher aliasing, and define $c_w(\\lbrace \\tilde{w}_i\\rbrace )\\doteq \\frac{1}{ KL_u(\\lbrace \\tilde{w}_i\\rbrace )+\\epsilon }$ , where $\\epsilon $ is a small number to avoid division-by-zero in case of extreme perceptual aliasing.", "With user-defined weights $M_u, M_G, M_{\\Sigma }$ and $M_w$ , the overall cost then can be defined as a combination $c \\doteq M_uc_u + M_Gc_G + M_{\\Sigma } c_\\Sigma + M_w c_w,$" ], [ "Formal Algorithm for DA-BSP", "We now have all the ingredients to present the overall framework of data-association aware belief space planning, calling it DA-BSP for brevity.", "It is summarised in Algorithm REF and briefly described below.", "Given a GMM belief $b[X_k]$ and candidate action $u_k$ , we first propagate the belief to get $b[X_{k+1}^-]$ and then simulate future observations $\\lbrace z_{k+1}\\rbrace $ (line REF ).", "The algorithm then calculates the contribution of each observation $z_{k+1}\\in \\lbrace z_{k+1}\\rbrace $ to the objective function (REF ).", "In particular, on lines REF and REF we calculate the weights $w_{k+1}^i$ that are used in evaluating the likelihood $w_s$ of obtaining observation $z_{k+1}$ .", "On lines REF -REF we compute the posterior belief: this involves updating each $j$ th component from the propagated belief $b[X_{k+1}^{j-}]$ with observation $z_{k+1}$ , considering each of the possible scenes $A_i$ .", "After pruning (line REF ), this yields a posterior GMM with $M_{k+1}$ components.", "We then evaluate the cost $c(.", ")$ (line REF ) and use $w_s$ to update the value of the objective function $J$ with the weighted cost for measurement $z_{k+1}$ (line REF ).", "Data association aware belief-space planning [1] Current GMM belief $b[X_k]$ at step-$k$ , history ${\\cal H} _k$ , action $u_k$ , scenes $\\lbrace A_{\\mathbb {N}} \\rbrace $ , event likelihood $\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k},x})$ for each $A_i\\in \\lbrace A_{\\mathbb {N}} \\rbrace $ $b[X_{k+1}^-]$ $\\leftarrow $ $b[X_k] \\mathbb {P}({x_{k+1} \\mid x_k,u_k})$ Eq.", "(REF ) $\\lbrace z_{k+1}\\rbrace $ $\\leftarrow $ SimulateObservations($b[X_{k+1}^-]$ , $\\lbrace A_{\\mathbb {N}} \\rbrace $ ) $J$ $\\leftarrow $ 0 $\\forall z_{k+1} \\in \\lbrace z_{k+1}\\rbrace $ $w_s$ $\\leftarrow $ 0 $i \\in [1\\dots \|A_{} \|]$ compute weight, Eq.", "(REF ) $w^i_{k+1}$ $\\leftarrow $ CalcWeights($z_{k+1},\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-,x}), b[X_{k+1}^-]$ ) $w_s$ $\\leftarrow $ $w_s + w_i$ $\\forall j \\in [1,\\ldots ,M_k]$ compute weight $\\tilde{w}^{ij}_{k+1}$ for each GMM component, Eq.", "(REF ) $\\tilde{w}^{ij}_{k+1}$ $\\leftarrow $ CalcWeights($z_{k+1},\\mathbb {P}({A_{i} \\mid \\mathcal {H}_{k+1}^-,x}), b[X_{k+1}^{j-}]$ ) $\\xi _{k+1}^{ij}$ $\\leftarrow $ $\\xi _{k}^j \\tilde{w}_{k+1}^{ij}$ Eq.", "(REF ) Calculate posterior of $b[X_{k+1}^{j-}]$ , given $A_i$ $b[X_{k+1}^{ij+}]$ $\\leftarrow $ UpdateBelief($b[X_{k+1}^{j-}], z_{k+1}, A_i$ ) Prune components with weights $\\xi _{k+1}^{ij}$ below a threshold Construct $b[X_{k+1}^{+}]$ from the remaining $M_{k+1}$ components via Eq.", "(REF ) $c$ $\\leftarrow $ CalcCost($b[X_{k+1}^{+}]$ ) Eq.", "REF $J$ $\\leftarrow $ $J+w_s\\cdot c$ $J$" ], [ "An Abstract Example for DA-BSP", "Consider the problem of robotic manipulation of objects in the kitchen.", "For simplicity, let us abstract it to a simpler domain of three objects, $\|\\lbrace A_{\\mathbb {N}} \\rbrace \| = 3$ .", "We consider a single step control at time step $k$ , from a given belief $b[X_k]$ , as well as that of one step ahead $b[X_{k+1}^-]$ , and assume the following motion and observation models $f$ and $h$ $\\begin{split}f(x,u) = \\left( \\begin{array}{cc} 1 & 0\\\\0 & 1 \\end{array}\\right) \\cdot x + d \\; \\bigg \\lbrace \\begin{array}{cr} [0, 1]^T & \\text{if ${u}$ = \\emph {up}} \\\\ [1, 0]^T & \\text{if $u$ = \\emph {right}} \\\\ \\end{array},\\\\h(x, A_i) = h_i(x) = \\left( \\begin{array}{cc} 1 & 0\\\\0 & 1 \\end{array}\\right) \\cdot (x - x_i) + s_i.\\end{split}$ where observations as well as the shift $s_i$ is in an object-centric frame, with $x_i$ representing location of $A_i$ .", "Intuitively, $s_i$ is a simple mechanism to model perceptual aliasing between objects; e.g., identical objects $A_i$ would have the same $s_i$ .", "Figure REF illustrates the process of simulating future observations $\\lbrace z_{k+1}\\rbrace $ for ${u}_k$ = up, considering unique and perceptually aliased scenes (Figures REF -REF ).", "In particular, a sampled pose $x^{tr}$ used to generate an observation $z_{k+1}\\in \\lbrace z_{k+1}\\rbrace $ is shown in Figure REF .", "Figure REF demonstrates key aspects in our approach, considering each time a single observation $z_{k+1}$ .", "Our approach reasons about data association and hence we consider each $z_{k+1}$ could have been generated by one of the 3 objects; each such association would fetch us a conditional posterior belief $b[X^{i+}_{k+1}]$ as denoted by small ellipses.", "Finally, we compute the total cost according to Algorithm REF .", "Figure: Pose and observation space.", "(a) black-colored samples {x k }\\lbrace x_{k}\\rbrace are drawn from b[X k ]≐𝒩([0,0] T ,Σ k )b[X_k]\\doteq {\\cal N}([0,0]^T, \\Sigma _k), from which, given control u k u_k, samples {x x+1 }\\lbrace x_{x+1}\\rbrace are computed, colored according to different scenes A i A_{i} being observed, and used to generate observations {z k+1 }\\lbrace z_{k+1}\\rbrace .", "(b) Stripes represent locations from which each scene A i A_i is observable, histogram represents distribution of {x x+1 }\\lbrace x_{x+1}\\rbrace , which corresponds to b[X k+1 - ]b[X^-_{k+1}].", "(c)-(d) distributions of {z k+1 }\\lbrace z_{k+1}\\rbrace without aliasing and when {A 1 ,A 3 } 𝐚𝐥𝐢𝐚𝐬𝐞𝐝 \\lbrace {A_{1},A_{3}}\\rbrace _{\\textbf {aliased}}.Figures REF -REF denote the situation when the true pose $x^{tr}$ is close to center and observe $A_{2}$ , while in Figures REF -REF it is at the left side and observe $A_{1}$ .", "Different degrees of aliasing are considered.", "Both weights $w_i$ and $\\tilde{w}_i$ are shown in the inset histograms.", "Note that the unnormalised weight $w_i$ is higher when the object is at the centre, because the overall likelihood of the observation is higher.", "Also, with no aliasing, for any other scene $A_{j}$ than the true one, the normalised weight $w_j$ is small irrespective of where $x^{tr}$ is.", "In other words, weights are also related to how likely the objects are to be the causes behind an observation; in case of no aliasing, this can be negligibly small.", "This is crucial since it implies that DA-BSP in practical applications with infrequent aliasing, would not require any significant additional computational effort w.r.t.", "usual BSP.", "Figure: DA-BSP for a single observation z k+1 z_{k+1}.", "Red-dotted ellipse denotes b[X k+1 - ]b[X^-_{k+1}], while the true pose that generated z k+1 z_{k+1} is shown by inverted triangle.", "Smaller ellipses are the posterior beliefs b[X k+1 i+ ]b[X^{i+}_{k+1}].", "(a-d) x tr x^{tr} is near center, observing A 2 A_{2} ; (e-f) x tr x^{tr} is on the left, observing A 1 A_{1} .", "Weights w i w_i and w ˜ i \\tilde{w}_i, corresponding to each scene A i A_{i} are shown in the inset bar-graphs.Figures REF -REF depict $\\lbrace {A_{1},A_{2}}\\rbrace _{\\textbf {aliased}}$ , $\\lbrace {A_{1},A_{3}}\\rbrace _{\\textbf {aliased}}$ and $\\lbrace {A_{1},A_{2}, A_{3}}\\rbrace _{\\textbf {aliased}}$ .", "When $\\lbrace {A_{1},A_{3}}\\rbrace _{\\textbf {aliased}}$ , the weights $w_i$ are similar, and indeed our cost $c_w$ of weights (in Eq.", "(REF )) is high.", "For similar uncertainty in pose, this cost would remain constant.", "Hence, in the presence of identical objects placed similarly within the current belief, optimization of general cost function would be guided towards active localization.", "On the other hand, if one object $j$ lies closer to the current nominal pose, it will have slightly higher $w_j$ .", "In case $\\lbrace {A_{1},A_{2}, A_{3}}\\rbrace _{\\textbf {aliased}}$ , i.e.", "all objects are identical, the weights $w_i$ are simply an indication of the prior.", "This is reasonable since in such a case, considering different data association does not yield any new information.", "Finally, in Table REF , we present the numerical analysis of cost computation (see Eq.", "(REF )) of these configurations, as well as a metric $\\lbrace \\epsilon _{BSP}, \\epsilon _{DA} \\rbrace $ quantifying estimation error, defined over incorrect (w.r.t ground-truth) associations through random sampling of various modes.", "Intuitively $\\epsilon _{BSP} \\text{ and } \\epsilon _{DA}$ evaluate how good the posterior mean is w.r.t.", "ground-truth $x^{tr}$ for usual BSP and DA-BSP respectively (lower is better).", "Recall that unlike action $u_1$ , action $u_2$ leads to fully unambiguous observations, around most-likely value (see Fig.", "REF ) and consequently, $\\epsilon _{BSP} \\simeq \\epsilon _{DA}$ .", "Table: Evaluating different cost functions for various configurations (see Fig. )" ], [ "Gazebo World", "To demonstrate generality of DA-BSP, we compared (in simulation) it with current state of the art (denoted as BSP) and the approach proposed in [1].", "For the latter case (see Fig.", "REF ), we have a simulated environment of rectangular corridors with shelfs($s_i$ ) and elevator($e$ ), where a pioneer robot has a non-Gaussian belief prior (shown with $p_1$ , $p_2$ ); as it can be in either of the two corridors, with localization as its objective.", "We evaluate our algorithm for inference (infer) as well as active planning (plan) (see Table REF ).", "The absence of pose uncertainty, which is the case in [1], would lead us to a wrong inference and estimate the robot to be at pose $p_1$ (corridor 1) initially, whereas our approach which considers pose uncertainty will lead to correct inference with high weight for being in corridor 2 (Fig.", "REF ).", "Figure: Using Pioneer robot in Gazebo simulation.", "(a) four-floor aliased world.", "(b) counter-example for hypothesis reduction in absence of pose-uncertainty in prior.", "The inference incorrectly deduces that the robot is in mode 1.To compare with the current state of the art, we consider another scenario with 4 floors.", "Floor 1 has the same configuration as in Fig.", "REF , floor 2 and floor 3 have the left and the right shelves (w.r.t floor 1) removed respectively and floor 4 has no shelves at all.", "We use the metric $\\eta _{da}$ as one of the possible ways to quantify data association performance by computing the probability of picking the right mode in the posterior GMM (which corresponds to ground-truth position of robot).", "Thus, for two equally weighted GMM components, $\\eta _{da}=0.5\\, \\text{or} \\,0$ if the correct component is one of them or if it has been pruned.", "The number of modes in the posterior is also indicated (for planning, the total number of modes for all the considered future observations is shown).", "As seen, many of the data associations considered within BSP are wrong ($\\eta _{da}=0.29$ ), while also in inference an incorrect association is made ($\\eta _{da}=0$ ).", "This can lead to (possibly catastrophic) mission failure; in this case, failure to associate to the correct corridor.", "We also show a counter-example (Fig.", "REF )) for [1] ($\\eta _{da}=0$ ) since that approach does not model uncertainty within each of the GMM components.", "In contrast, DA-BSP outperforms both approaches within planning and inference.", "Table: Evaluating DA-BSPIn order to evaluate DA-BSP, we consider a simplistic set of actions, namely $\\lbrace \\texttt {$ fwd1$},\\texttt {$ fwd2$},\\texttt {$ bwd1$}\\rbrace $ for a one-step forward, two-step forward and one-step backward movements, respectively (see Table REF ).", "These actions highlight the challenges of data-association aware planning, even in the context of a simplistic scenario.", "Note that number of modes (which signifies different associations planner is considering at that step) is significantly higher in the planning, than in the inference.", "However, when there exists a disambiguating action, such as $fwd_2$ is, the planner is able to associate to the correct association all the time (demonstrated by $\\eta _{da} = 1$ ).", "Table: DA-BSP for candidate actions" ], [ "Aliased Multi-Floor Environment", "Since, incorporating data-association implies that (at least theoretically) an exponential blow-up of number of unimodal beliefs maintained in the posterior, we therefore evaluated DA-BSP in simple but real-world scene by deploying a real Pioneer robot in a 3-floor aliased environment (see Fig.", "REF ), with similar objective of floor and position disambiguation.", "When not reasoning about perceptual aliasing, the robot takes greedy (w.r.t.", "control cost and position uncertainty) action and fails to disambiguate, whereas DA-BSP successfully tackles planning with data-association, as shown in Figure REF .", "Although not witnessed here, BSP can not guarantee global optimum solution (w.r.t.", "control and uncertainty cost) and DA-BSP similarly can not provide a global least cost path for full disambiguation - a problem known to be NP-hard [10].", "Figure: Using Pioneer robot in the real-world.", "(a) two (of three) severely-aliased floors, and belief space planning for it (b) DA-BSP can plan for fully disambiguating path (otherwise sub-optimal, due to path-length) while usual BSP with maximum likelihood assumption can not (as shown by almost equi-probable modes in the histogram)." ], [ "Conclusions", "State-of-the-art belief space planning (BSP) approaches typically consider data association to be given and perfect.", "However, such an assumption is less appropriate in presence of localisation uncertainty while operating in ambiguous environments, where two scenes could be similar in appearance when observed from appropriate viewpoints.", "In this work, we developed a data association aware belief space planning (DA-BSP) approach that relaxes the aforementioned assumption.", "Our framework rigorously incorporates data association aspects within BSP, while considering different sources of uncertainty (uncertainty in robot motion, sensing and possibly in the observed environment).", "As such, it is capable of better coping with ambiguous, perceptually aliased, situations by appropriately calculating belief evolution and expected cost due to candidate actions, and in particular, could be used for active disambiguation.", "Thanks to this association being inherent in planning, DA-BSP considers data-association parsimoniously and a simple thresholding is enough for a scalable application of data-association aware belief space planning.", "We demonstrated key aspects of DA-BSP in abstract example as well as Gazebo simulations.", "We also applied it on a real-world problem with Pioneer robot lost in a multi-storied building.", "One of the major contributions of this work is in proposing a data-association-aware robust perception in a unified framework of plan-infer-execute.", "This is in contrast with passive approaches known in robust perception literature as well as that of multi-hypothesis tracking.", "Consequently, we are currently looking into extending the approach to non-myopic setting such that the generality of the framework becomes further explicit.", "Additionally, proving the general theoretical properties of DA-BSP, such as probabilistic completeness under uncertainty along the lines proposed by [2], is another interesting direction of research.", "Apart from this, evaluating the approach in a more complex real-world scenarios is also an avenue for future research." ] ]	1606.05124
[ [ "Meta-analysis of two studies in the presence of heterogeneity with\n applications in rare diseases" ], [ "Abstract Random-effects meta-analyses are used to combine evidence of treatment effects from multiple studies.", "Since treatment effects may vary across trials due to differences in study characteristics, heterogeneity in treatment effects between studies must be accounted for to achieve valid inference.", "The standard model for random-effects meta-analysis assumes approximately normal effect estimates and a normal random-effects model.", "However, standard methods based on this model ignore the uncertainty in estimating the between-trial heterogeneity.", "In the special setting of only two studies and in the presence of heterogeneity we investigate here alternatives such as the Hartung-Knapp-Sidik-Jonkman method (HKSJ), the modified Knapp-Hartung method (mKH, a variation of the HKSJ method) and Bayesian random-effects meta-analyses with priors covering plausible heterogeneity values.", "The properties of these methods are assessed by applying them to five examples from various rare diseases and by a simulation study.", "Whereas the standard method based on normal quantiles has poor coverage, the HKSJ and mKH generally lead to very long, and therefore inconclusive, confidence intervals.", "The Bayesian intervals on the whole show satisfying properties and offer a reasonable compromise between these two extremes." ], [ "Introduction", "Meta-analyses are used to combine evidence of treatment effects from multiple studies.", "Since treatment effects may vary across trials due to some slight differences in study characteristics including study populations, trial designs, endpoints and standardization of treatments, heterogeneous treatment effects are quite natural and must be accounted for to achieve valid statistical inferences.", "Therefore, random-effects meta-analysis has become the standard to combine treatment effects from several studies when the presence of between-trial heterogeneity is suspected which is often the case.", "The standard model for random-effects meta-analysis assumes approximately normal effect estimates and a normal random-effects model, the normal-normal hierarchical model [17].", "Based on this model, standard inference methods based on normal quantiles to construct confidence intervals for the combined effect ignore the uncertainty in the estimation of the between-study heterogeneity and they are only valid for large numbers of trials.", "However, the combination of only a few studies is quite common [4], [32].", "This is not only the case in rare diseases, but in this context it poses a particular challenge since increased levels of heterogeneity are common [13].", "For instance, in a recent systematic review by [3] six studies on acute graft rejections and three studies on steroid-restistant rejections were combined in random-effects meta-analyses to assess the efficacy and safety of Interleukin-2 receptor antibodies for immunosuppression following liver transplantation in children.", "All studies were controlled, but only two were randomised as it is often the case in paediatrics.", "Furthermore, there were some differences between the studies with respect to their control groups and other design characteristics suggesting some degree of between-trial heterogeneity.", "For random-effects meta-analyses with few studies methods based on $t$ -distributions have been suggested [12], [15], [16], [21], [29].", "Furthermore, the use of priors covering plausible between-trial standard deviations has been advocated when dealing with few studies [31], [13], [24], [28].", "Here we consider the special case of only two studies which has recently attracted some attention [14].", "Examples for meta-analyses of two studies include the summary of two pivotal studies of a clinical development programme [10], [11], [20].", "As we will see when discussing several examples below, meta-analyses of two studies are not uncommon in orphan diseases.", "For instance, two randomised controlled trials were included in the systematic review by [3] and the Cochrane Review by [23] on Riluzole in amyotrophic lateral sclerosis (ALS).", "In the presence of heterogeneity, however, the meta-analysis of only two studies may be considered an unsolved problem [14].", "Therefore, we assess here the performance of alternative approaches to real-life examples from rare diseases and by exploring their charactersistics in an extensive simulation study.", "Based on these findings we give some recommendations on how to approach the problem successfully in practice.", "The paper is organised as follows.", "In Section the statistical model is introduced and methods for frequentist and Bayesian inference are reviewed.", "Five examples in various rare diseases are presented in Section before an extensive simulation study is presented in Section .", "In Section we close with a brief discussion of the findings.", "Standard meta-analytic models assume either a common (fixed) effect or random effects across studies.", "For the latter, the normal-normal hierarchical model (NNHM) is the most popular.", "At the first level, the sampling model assumes approximately normally distributed estimates $Y_1,\\ldots ,Y_k$ for the trial-specific parameters $\\theta _1,\\ldots ,\\theta _k$ $Y_j \\vert \\theta _j \\;\\sim \\; \\mathrm {N}(\\theta _j,s_j^2), \\quad j=1,\\ldots ,k .$ Here, we will follow the standard assumption which treats the standard errors $s_j$ as known, although this could be relaxed if necessary.", "At the second level, the parameter model assumes normally distributed study effects $\\theta _j \\vert \\mu , \\tau \\;\\sim \\; \\mathrm {N}(\\mu ,\\tau ^2), \\quad j=1,\\ldots ,k.$ The between-trial standard deviation $\\tau $ determines the degree of heterogeneity across studies.", "If the parameter of interest is $\\mu $ (rather than the study effects $\\theta _j$ ), inference can be simplified by using the marginal model $Y_j \\vert \\mu ,\\tau \\;\\sim \\; \\mathrm {N}(\\mu ,s_j^2+\\tau ^2), \\quad j=1,\\ldots ,k .$ The two main approaches to infer $\\mu $ and the nuisance parameter $\\tau $ are frequentist and Bayesian.", "If $\\tau $ were known, frequentist and Bayesian (with a non-informative prior for $\\mu $ ) conclusions would be analogous.", "In fact, in the frequentist setting $ \\hat{\\mu } = \\sum _{j=1}^k w_jY_j \\bigg / \\sum _{j=1}^k w_j \\;\\sim \\;\\mathrm {N}(\\mu ,1/w_{+}),\\qquad w_j = 1 \\big / \\bigl (s_j^2+\\tau ^2\\bigr ), \\qquad j=1,\\ldots k,$ where $w_j$ are inverse-variance (precision) weights, and $w_+ =\\sum _{j=1}^k w_j$ is the total precision; the respective variance $1/w_+$ is important to construct confidence intervals for $\\mu $ , as shown in Section REF .", "The Bayesian result (posterior distribution) is $\\mu \\vert Y_1,\\ldots ,Y_k \\;\\sim \\; \\mathrm {N}\\Biggl (\\sum _{j=1}^k w_jY_j \\bigg / \\sum _{j=1}^k w_j,\\;1/w_{+}\\Biggr ) .$ For unknown $\\tau $ , this frequentist-Bayesian “equivalence” breaks down, since the two approaches handle estimation uncertainty for $\\tau $ differently." ], [ "Frequentist inference", "For unknown $\\tau $ we first consider frequentist methods to infer $\\mu $ , which comprise two steps.", "An estimate $\\hat{\\tau }$ is derived, from which estimated weights $\\hat{w}_j = 1/(s_j^2+\\hat{\\tau }^2)$ and a corresponding estimate $\\hat{\\mu }$ in (REF ) are obtained.", "Various estimators for $\\tau $ have been proposed (for an overview see [6], [27], [34]), the most prominent being the moment-estimator due to DerSimonian and Laird (DL).", "Alternatives are the maximum likelihood (ML) estimator, the restricted maximum-likelihood estimator (REML), and the Paule-Mandel estimator (PM).", "While these estimates can differ considerably, for the special case of two trials they coincide [27].", "We will refer to this common estimate $\\hat{\\tau }^2 = \\frac{(y_1-y_2)^2-s_1^2-s_2^2}{2}.$ as the DL estimate, whereby negative values are set to zero.", "A confidence interval for $\\mu $ is then derived.", "Here we will investigate three methods.", "The simplest approach, which was proposed in the seminal paper by [7], uses the following normal approximation $\\mbox{(DL)} \\qquad \\hat{\\mu } \\pm \\hat{\\sigma }_{\\mu } \\, z_{(1-\\alpha /2)}, \\qquad \\mbox{where} \\quad \\hat{\\sigma }_{\\mu }^2 = 1 \\Big / \\sum _{j=1}^k \\hat{w}_j,$ and $z_p$ is the $p$ -quantile of the standard normal distribution.", "This method is known to be problematic for small $k$ , since it ignores the uncertainty of $\\hat{\\tau }$ and will therefore give too narrow confidence intervals and inflated type-I errors.", "Various improvements using a $t$ -distribution with $k\\!-\\!1$ degrees of freedom and alternative estimators for $\\sigma _{\\mu }$ have been proposed [15], [16], [21], [29].", "The HKSJ confidence interval is given by $ \\mbox{(HKSJ)} \\qquad \\hat{\\mu } \\pm \\tilde{\\sigma }_{\\mu } \\, t_{k-1,(1-\\alpha /2)}, \\qquad \\mbox{where} \\quad \\tilde{\\sigma }_{\\mu }^2 = \\frac{1}{k-1}\\sum _{j=1}^k \\hat{w}_j(y_j-\\hat{\\mu })^2 \\bigg / \\sum _{j=1}^k \\hat{w}_j,$ and $t_{k-1,(1-\\alpha /2)}$ is the $(1\\!-\\!\\alpha /2)$ -quantile of the Student-$t$ distribution with $k\\!-\\!1$ degrees of freedom.", "It works well for any number of studies if study-specific standard errors $s_j$ are of similar magnitude.", "Otherwise, coverage probabilities can be below the nominal level.", "To address the limitations of the HKSJ method, a modified interval $\\mbox{(mKH)} \\qquad \\hat{\\mu } \\pm \\sigma _{\\mu }^{\\star } \\, t_{k-1,(1-\\alpha /2)}, \\qquad \\mbox{where} \\quad \\sigma _{\\mu }^{\\star } = \\mbox{max}\\lbrace \\hat{\\sigma }_{\\mu },\\tilde{\\sigma }_{\\mu }\\rbrace $ has been proposed [26].", "By taking the maximum of $\\hat{\\sigma }_{\\mu }$ and $\\tilde{\\sigma }_{\\mu }$ , the problems of undercoverage and occasional counterintuitive results can be resolved." ], [ "Bayesian inference", "In the Bayesian framework, uncertainty of $\\tau $ is automatically accounted for.", "Inference for $\\mu $ and $\\tau $ is captured by the joint posterior distribution of the two parameters, from which the marginal distribution of $\\mu $ is used to derive, for example, point estimates and probability intervals for $\\mu $ .", "While automatic, the approach requires sensible prior distributions for $\\mu $ and $\\tau $ .", "For the main parameter $\\mu $ , we will use a noninformative (improper) uniform prior.", "For $\\tau $ , however, the choice of prior is critical, in particular if the number of studies is small [8], [9], [33].", "For the case of two studies and in the absence of relevant external data, information about between-trial heterogeneity is clearly very small.", "Therefore, the main feature of the Bayesian approach is its ability to average over the uncertain between-trial heterogeneity.", "This requires a prior distribution for $\\tau $ that covers plausible between-trial standard deviations.", "If information about heterogeneity is weak, the 95% prior interval should capture small to large heterogeneity.", "Table: Characteristics of the two half-normal priors for log-odds-ratios.What constitues small to large heterogeneity depends on the parameter scale.", "For example, for log-odds-ratios (see examples in Section 3), values for $\\tau $ equal to 0.25, 0.5, 1, and 2 represent moderate, substantial, large, and very large heterogeneity.", "We will use two half-normal (HN) prior distributions [31] in the examples (Section 3) and the simulation study (Section 4), with scale parameters 0.5 and 1.0; for prior medians and 95%-intervals see Table REF .", "The HN(0.5) prior captures heterogeneity values typically seen in meta-analyses of heterogeneous studies, and will therefore be a sensible choice in many applications.", "If very large between-trial heterogeneity is deemed possible, the more conservative HN(1.0) prior may be advised." ], [ "Introductory remarks", "In this section we discuss five real-life examples of meta-analyses of two randomized controlled trials in various rare conditions.", "The first two are from the literature whereas the other three examples are based on US Food and Drug Administration (FDA) approvals in orphan diseases for the following drugs: Romiplostim, Mozobil, and Krystexxa.", "In neither of these approvals, a formal meta-analysis was presented in the official documents.", "All examples have a binary endpoint comparing a treatment (T) to a control (C).", "The following normal approximation on the log-odds-ratio scale $Y = \\log \\biggl (\\frac{r_\\mathrm {T}(n_\\mathrm {C}-r_\\mathrm {C})}{r_\\mathrm {C}(n_\\mathrm {T}-r_\\mathrm {T})} \\biggl ),\\qquad s^2 = \\frac{1}{r_\\mathrm {T}} + \\frac{1}{n_\\mathrm {T}-r_\\mathrm {T}} + \\frac{1}{r_\\mathrm {C}} + \\frac{1}{n_\\mathrm {C}-r_\\mathrm {C}}$ will be used, where $r$ and $n$ denote the number of responders and number of subjects, respectively.", "Figure: Forest plots for the five examplesfrom rare diseases with various combined estimates of thetreatment effect.", "The top two rows show the underlying data(numbers of cases and events in experimental and controlgroups) and illustrate the resulting estimates with their 95%confidence intervals.", "The following rows show the differentcombined estimates along with the estimated amount ofheterogeneity (posterior medians for the Bayesianapproaches)." ], [ "Systematic review of Interleukin-2 receptor antibodies in pediatric liver transplantation {{cite:7682b7f7accb8b0f3e738654ca9cc26f77ceb31e}}", "[3] conducted a systematic review of controlled trials providing evidence on the efficacy and safety of immunosuppressive therapy with Interleukin-2 receptor antibodies (IL-2RA) Basiliximab and Daclizumab following liver transplantation in children.", "Six studies were included in a meta-analysis of acute graft rejections, of which only two were randomized (Heffron 2003, Spada 2006).", "In both studies about 80 patients were randomized, with 2:1 allocation in [18] and 1:1 allocation in [30].", "For the purpose of illustration we present here a meta-analysis of the two randomised studies in Figure REF .", "Both studies yielded statistically significant results.", "However, there were some differences in the estimated odds ratios resulting in moderate to substantial estimates of the between-trial standard deviation.", "Although the two studies were statistically significant, the HKSJ and mKH methods result in confidence intervals that include the null hypothesis and are extremely wide (0–129 on the odds ratio scale).", "In contrast, the other three meta-analyses yield statistically significant results with the standard method based on normal quantiles giving the shortest confidence interval." ], [ "Cochrane review of Riluzole in ALS {{cite:068c1ef156f23c17f32fa0747df435c34c0fe6dc}}", "A Cochrane Review of Riluzole for amyotrophic lateral sclerosis (ALS) combined two randomized, placebo controlled, double-blind trials [1], [22] with information on 12-month mortality in a meta-analysis [23].", "[23] combined the three active doses of the dose-ranging study by [22] into one group for the purpose of the presented analysis.", "Whereas they used relative risks for their analyses we present here the results in terms of odds ratios (Figure REF ).", "As with the previous example both studies demonstrated statistically significant effects of the experimental drug over control (see Figure REF ).", "Whereas in the previous example the DL estimate of the between-trial heterogeneity was positive, here it is zero.", "In comparison to the [3] example, here the HKSJ and the mKH methods are more informative as they are not quite as long.", "However, they are still considerably longer than the Bayesian intervals, which appear to be conservative since they include odds ratios of 1 although the confidence intervals of the individual studies both exclude 1." ], [ "FDA approval in orphan disease: Romiplostim", "Romiplostim [2] was approved to treat Idiopathic Thrombocytopenic Purpura based on two 2:1 randomized studies.", "The two studies, 20030105 and 20030212, enrolled splenectomized and non-splenectomized patients, but were similar in their designs.", "Here we focus on patients requiring rescue medications (a secondary endpoint).", "Both studies showed statistically significant odds ratios (ORs): 0.27 (0.09, 0.80) for 20030105 and 0.13 (0.04, 0.42) for 20030212 (Figure REF ).", "The ratio of ORs is 2.12, suggesting that between-trial heterogeneity should be considered.", "However, the frequentist estimate $\\hat{\\tau }$ is zero, resulting in a narrow confidence interval for $\\mu $ .", "On the other hand, the HSKJ and mKN intervals are very wide and do not allow sensible conclusions about the treatment effect.", "The respective Bayesian intervals are much more plausible.", "Additionally, for both Bayesian analyses, the posterior medians for $\\tau $ are smaller than the respective prior medians, indicating that the two half-normal priors do not unduly favour small homogeneity." ], [ "FDA approval in orphan disease: Mozobil", "Mozobil [35] was approved for the mobilization of hematopoietic stem cells in patients with lymphoma and multiple myeloma.", "The two 1:1 randomized studies were conducted in two different indications: 3101 in Non-Hodgkin's Lymphoma, and 3102 in Multiple Myeloma.", "However, no differential treatment effect with respect to the primary endpoint was expected, which justifies a meta-analysis of the two studies.", "Both studies show statistically significant odds ratios (Figure REF ).", "The ratio of the odds ratios is 1.25, suggesting possibly small between-trial heterogeneity.", "Unsurprisingly, the frequentist estimate $\\hat{\\tau }$ is zero.", "The HKSJ and mKH methods again provide very wide (but fairly different) CIs: the HKSJ method leads to a conclusive result, whereas the more conservative mKH does not; this is clearly implausible, since both studies showed highly significant results.", "The respective Bayesian intervals are much narrower, suggesting a sensible compromise between the rather extreme (narrow and wide) frequentist counterparts." ], [ "FDA approval in orphan disease: Krystexxa", "For Krystexxa [5], two 2:2:1 randomized studies were used for approval.", "Here, we consider only one of two treatment arms (approved dose of 8mg every 2 weeks) and analyze a safety endpoint (infusion reaction).", "The two studies showed the following ORs: 6.55 (0.78, 54.60) for C405 and 7.77 (0.94, 64.72) for C406 (Figure REF ), which suggest an increase in infusion reaction.", "In this example, the HKSJ and mKH intervals, which are usually very wide, give completely different answers.", "The HKSJ interval is even narrower than the interval based on normal approximations, whereas the mKH interval is unrealistically wide.", "The overly narrow HKSJ interval is due to the similar log-odds-ratios $y_j$ (1.88 and 2.05), which lead to a very small estimate $\\tilde{\\sigma }_{\\mu }=0.089$ in equation (REF ); the classical estimate $\\hat{\\sigma }_{\\mu }=0.765$ , which is used for mKH, is dramatically larger." ], [ "Some concluding remarks on the examples", "In this section we presented five examples from a range of rare diseases.", "In each of these, two studies were combinded in meta-analyses in situations where between-study heterogeneity had to be suspected to be present.", "Still the DL estimator for the between-study heterogeneity was zero in four out of the five examples.", "Furthermore, the standard approach based on normal quantiles led to the shortest intervals in all but the Krystexxa example, in which the HKSJ interval was very narrow.", "Otherwise the HSKJ and mKH methods yielded overall long to extremely long confidence intervals not conveying useful information on the size of the treatment effect.", "This is not surprising, since the $97.5 \\%$ quantile of a $t$ -distribution with 1 degree of freedom is about 12.7.", "Although the Bayesian intervals appeared to be conservative, they led to interpretable results and a sensible compromise between the very short intervals based on normal quantiles and the often extremly long intervals based on $t$ -quantiles." ], [ "Setup", "For the simulation study of this section, we used the NNHM of Section .", "Simulations were limited to the case of two studies.", "The study sample sizes $n_1$ and $n_2$ were set to 25, 100, or 400, which leads to six different combinations of $(n_1,n_2)$ .", "In the following figures (2–4), the first rows show results for equally sized studies, while the second rows illustrate the imbalanced settings.", "Standard errors for the estimated log-odds-ratios $Y_j$ were set to $2/\\sqrt{n_1}$ and $2/\\sqrt{n_2}$ .", "Without loss of generality, $\\mu $ was set to zero.", "In terms of the “relative” amount of heterogeneity $I^2$ [19], the different settings correspond to $I^2 \\in [0.20, 0.80]$ for $\\tau =0.2$ , and to $I^2 \\in [0.86,0.99]$ for $\\tau =1.0$ .", "The number of simulations, which were performed using R, was $15\\,000$ .", "Simulation results are shown for the bias of $\\tau $ estimates, the fraction of $\\tau $ estimates equal to zero, the coverage probabilities and the interval lengths for $\\mu $ ." ], [ "Bias in estimators of the between-study heterogeneity $\\tau $", "Figure REF shows the bias of $\\tau $ estimates.", "The DL estimator tends to overestimate $\\tau $ if heterogeneity is small ($\\tau =0, 0.1$ ).", "On the other hand, $\\tau $ will be underestimated for substantial ($\\tau =0.5$ ) and large ($\\tau =1$ ) heterogeneity.", "The Bayesian estimators (posterior medians) show similar patterns, with the magnitude of bias depending on the prior.", "The overestimation under small heterogeneity is obviously more pronounced for the HN(1.0) than for the HN(0.5) prior, because the former favours larger values of $\\tau $ .", "On the other hand, underestimation of $\\tau $ only occurs (and is fairly small) if heterogeneity is substantial to large.", "It should be noted that, in contrast to the frequentist methods, the Bayesian estimates for $\\tau $ are less important, since the inference for $\\mu $ takes into account the uncertainty of $\\tau $ via the posterior distribution.", "Figure: Bias in estimating the between-study heterogeneity τ\\tau in the different simulation settings." ], [ "Fraction of zero $\\tau $ estimates", "Table REF shows that the DL estimates for $\\tau $ are often zero even if heterogeneity is substantial $(\\tau =0.5)$ or large $(\\tau =1.0)$ .", "This is a well-known problem, which, if not appropriately addressed, results in too optimistic inferences for $\\mu $ .", "Table: Fractions (in %) of heterogeneityestimates turning out as zero." ], [ "Coverage probabilities and interval lengths for $\\mu $", "As can be seen from Figure REF , if heterogeneity is small, all methods work well save for the normal approximation (DL-normal), for which the coverage can be below the nominal level even for small heterogeneity $(\\tau =0.1)$ .", "The HKSJ method is known to work well for equally sized studies, but can be problematic for unequal study sizes and considerable heterogeneity.", "As can be seen, the modified method (mKH) resolves this problem.", "The Bayesian intervals show good coverage in the range of the prior, irrespective of the study sizes.", "For example, under the more optimistic HN(0.5) prior, coverage is reasonable for $\\tau $ up to 0.5, but will drop considerably for larger values that are a-priori less likely.", "Figure: Coverage of interval estimators in the different simulation settings.Coverage is obviously linked to interval length: higher coverage generally comes at the price of longer intervals.", "Figure REF shows that this price can be very high.", "The two frequentist methods with good coverage (HKSJ, mKH) exhibit exorbitantly long and implausible 95%-intervals, for which practical relevance is unclear.", "Interestingly, the Bayesian intervals are much shorter and provide a sensible compromise between the HSJK or mKH and the DL-normal intervals.", "These findings are consistent with results of the examples in Section .", "Figure: Mean lengths of confidence / credibility intervals in the different simulation settings." ], [ "Discussion", "There is a need for random-effects meta-analyses with only two studies, in particular in rare diseases.", "To gain insights into the properties of various meta-analytic methods for two trials, in this special case we considered examples from the area of rare diseases and conducted an extensive simulation study.", "The simulations allowed us to assess the coverage probabilities and mean lengths of the confidence and credibility intervals.", "The examples led to further insights into the interpretability of results.", "We can summarize our findings as follows.", "The confidence intervals based on normal quantiles do not have the right coverage and cannot be recommended for use in the case of two studies.", "The HKSJ intervals provide good coverage if the standard errors of the treatment effects observed in the two studies are of similar size.", "In general, however, the HKSJ intervals are either so wide that they do not allow any conclusion, or are very narrow.", "The latter occurs rarely (if the two study estimates are very close, (REF )), but can lead to problematically narrow confidence intervals and unfavourable coverage.", "This can be fixed by the ad-hoc modification (mKH), which is in agreement with findings by [26].", "The mKH method yields generally coverage probabilities in excess of the nominal level, but the intervals are generally so wide that they do not allow any meangingful conclusion.", "In this sense we agree with [14] that there is currently no solution for random-effects meta-analysis in the frequentist setting.", "However, Bayesian random-effects meta-analyses with a reasonable prior yield interpretable results in our examples and showed satisfying properties in the simulations.", "Therefore, the Bayesian intervals appear to be a reasonable compromise between the extremes of the confidence intervals based on normal quantiles that suffer from poor coverage and the $t$ -distribution based intervals that tend to be so long that they are inconclusive.", "Use of a Bayesian approach of course entails the question of what constitutes sensible prior information in a given context.", "This may be argued on the basis of the endpoint in question, i.e., what is the plausible amount of heterogeneity expected e.g.", "among log-ORs, as in the motivating examples above.", "Otherwise the problem may be to determine what constitutes relevant external data, and how this information may be utilized to formulate a prior, as was done e.g.", "by [32] and [25].", "Here we investigated several meta-analytic methods for two studies, with a focus on rare diseases.", "While a definite answer to this challenging problem is under dispute, the proposed Bayesian approach works well in our examples and simulation settings.", "The current frequentist methods have severe limitations, which may be addressed with future research.", "Until these limitations are resolved, we recommend to meta-analyze two heterogeneous studies in a Bayesian way using plausible priors." ], [ "Acknowledgements", "This research has received funding from the EU's 7th Framework Programme for research, technological development and demonstration under grant agreement number FP HEALTH 2013602144 with project title (acronym) “Innovative methodology for small populations research” (InSPiRe).", "Conflict of Interest BN and SW are employees of Novartis, the manufacturer of an Interleukin-2 receptor antagonist used in the examples." ] ]	1606.04969
[ [ "Giant amplification of noise in fluctuation-induced pattern formation" ], [ "Abstract The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature.", "Here, we show that a large class of spatially-extended dynamical systems driven by intrinsic noise can exhibit giant amplification, yielding patterns whose amplitude is comparable to that of deterministic Turing instabilities.", "The giant amplification results from the interplay between noise and non-orthogonal eigenvectors of the linear stability matrix, yielding transients that grow with time, and which, when driven by the ever-present intrinsic noise, lead to persistent large amplitude patterns.", "This mechanism provides a robust basis for fluctuation-induced biological pattern formation based on the Turing mechanism, without requiring fine tuning of diffusion constants." ], [ "Giant amplification of noise in fluctuation-induced pattern formation Tommaso Biancalani Present address: Physics of Living Systems, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA Farshid Jafarpour T. Biancalani and F. Jafarpour contributed equally to this work.", "Nigel Goldenfeld Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080.", "Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1206 West Gregory Drive, Urbana, Illinois 61801.", "The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature.", "Here, we show that a large class of spatially-extended dynamical systems driven by intrinsic noise can exhibit giant amplification, yielding patterns whose amplitude is comparable to that of deterministic Turing instabilities.", "The giant amplification results from the interplay between noise and non-orthogonal eigenvectors of the linear stability matrix, yielding transients that grow with time, and which, when driven by the ever-present intrinsic noise, lead to persistent large amplitude patterns.", "This mechanism provides a robust basis for fluctuation-induced biological pattern formation based on the Turing mechanism, without requiring fine tuning of diffusion constants.", "Since the seminal paper of Turing [1], it has been recognized that pattern forming dynamical instabilities could potentially underlie various examples of biological pattern formation and development [2].", "The Turing mechanism has two major assumptions: first, that two chemical species behave as an activator-inhibitor system (but see a recent extension [3]), and secondly, that the spatial diffusion constant of the inhibitor is greater than that of the activator, typically by two orders of magnitude or more [4].", "However, this second condition is not generally present in experimental observations [5], [6].", "The widely-held conclusion is that biological patterns reflect gene expression and the interplay of developmental processes, so that the Turing mechanism itself is not generally operative [7].", "This conclusion relies upon a third assumption of Turing patterns: that they are deterministic.", "However, many biological systems exhibit strong fluctuations due to demographic stochasticity, arising from (e.g.)", "finite population size (ecology) or copy number (gene expression), and these fluctuations could potentially couple to the underlying pattern-forming instabilities.", "Detailed analysis shows that the length scale of fluctuation-induced patterns is set by the same condition as in the deterministic Turing analysis, but remarkably the pattern exists over a wide range of parameter values, even where the diffusion constants of activator and inhibitor are of similar magnitudes [8], [9], [10], [11], [12], [13].", "These fluctuation-induced or stochastic patterns arise physically because, even though the uniform unpatterned state is linearly stable, the demographic fluctuations are constantly pushing the system slightly away from its stable fixed point; if the resulting small amplitude dynamics is dominated by an eigenvalues with a non-zero wavelength, then a spatial pattern can arise.", "This mechanism suggests that the amplitude of fluctuation-induced patterns would be set by $\\Omega ^{-1/2}$ , where $\\Omega $ indicates the population size within a correlation volume of the system, ie.", "the spatial patch within which the system can be considered to be well mixed [8], [9].", "Thus in situations where $\\Omega \\gg 1$ , fluctuation-induced patterns might have a very small amplitude compared to deterministic Turing patterns, potentially diminishing their relevance for biological and ecological pattern formation.", "Figure: (Color online) Turing-like pattern with largeamplitude and comparable diffusivities.", "(right panel) Stochasticsimulations of a two-speciesmodel () with diffusivities δ U =3.9\\delta _U = 3.9, δ V =3.4δ U \\delta _V =3.4\\,\\delta _U and system size Ω=10 4 \\Omega = 10^4.", "Patterns arenoise-induced as they arise from a stable homogeneous state u * u^,i.e., the eigenvalues λ\\lambda plotted against thewavelength kk are negative (left panel).", "However, the patternamplitude results of the order of one (right bar).", "Otherparameters: a=3a = 3, b=5.8b = 5.8, c=e=1c = e = 1.The purpose of this Letter is to show that fluctuation-induced Turing patterns can readily be observed, even when the noise is very small and the ratio of diffusion constants is close to one.", "Specifically we present an analytical theory showing the presence of giant amplification, due to an interplay between a separation of time scales and non-normality of the eigenvectors in the linear stability analysis about a uniform stable steady state.", "We present a measure of non-normality for a general stochastic dynamical system near a stable fixed point, with a clear geometrical interpretation.", "We then show that giant amplification occurs in a wide class of fluctuation-induced pattern-forming systems.", "An example of our key result described below is shown in Fig.", "REF : stochastic simulations of the generic pattern-forming model of Ridolfi et al.", "[15], performed on a linear chain of $10^2$ spatial cells, each cell with a system size of $\\Omega =10^4$ .", "Patterns are noise-induced as they arise from a stable homogeneous state (left panel), but despite the factor $\\Omega ^{-1/2} =10^{-2}$ the resulting amplitude is of order unity.", "This giant amplification is due to the counterintuitive fact that the dynamics following a small displacement from a stable fixed point need not relax back to the fixed point monotonically: there can be an initial transient amplification if the linear stability matrix is non-normal: that is, it does not admit an orthogonal set of eigenvectors (Fig.", "REF ).", "Non-normality has been thoroughly investigated, at a deterministic level, in fluid dynamics [16], [17], and in ecology [18], [19], and is a common feature of pattern-forming systems [20], [15].", "Low-dimensional stochastic non-normal systems may also exhibit strong amplification of noise [21].", "The specific contribution of the present paper is to systematically analyze the role of non-normality in fluctuation-induced spatial patterns, and to show that its widespread occurrence suggests a new way in which fluctuation-induced Turing patterns may play a wider role in biological and ecological pattern formation than previously recognized.", "Non-normality in stochastic dynamics:- We begin by introducing a measure to quantify the degree of amplification in a well-mixed stochastic system.", "Consider the linear stochastic differential equation for an $m$ -component state vector $\\vec{y}$ : $ \\dot{\\vec{y}} = {A}\\, \\vec{y} + \\sigma \\, \\vec{\\eta }(t),$ where the components of $\\vec{\\eta }$ , are normalized Gaussian white noises and the model-dependent matrix ${A}$ has negative real eigenvalues, $\\lambda _i$ ($i = 1, \\dots , m$ ).", "Therefore, the fixed point $\\vec{y}_0 = 0$ is stable.", "The coefficient $\\sigma $ represents the strength of the fluctuations and scales with the system size $\\Omega ^{-1/2}$ in the case of demographic noise.", "Equation (REF ) is the prototypical linearization of stochastic dynamics near a stable fixed point, and we analyze the mean square displacement from the fixed point, $\\left\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2\\right\\rangle $ , where $\\left\\Vert \\vec{y}\\right\\Vert = \\sqrt{\\vec{y}^T\\vec{y}}$ , is the Euclidean norm.", "Since all the eigenvalues of ${A}$ are negative, under the deterministic part of Eq.", "(REF ), all the components of $\\vec{y}$ decay exponentially to zero along the eigenvectors of ${A}$ , with decay time scales $\\tau _i = \\lambda _i^{-1}$ .", "In contrast, the noise term provides stochastic agitation with a strength proportional to $\\sigma $ .", "One might intuitively expect that an upper bound for $\\left\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2\\right\\rangle $ could be found by replacing all the eigenvalues by the eigenvalues corresponding to the slowest decaying mode, $\\lambda = \\text{max}\\lbrace \\lambda _i\\rbrace $ .", "Therefore, the norm of $\\vec{y}_u$ with the dynamics $\\dot{\\vec{y}}_u =\\lambda \\, \\vec{y}_u + \\sigma \\, \\vec{\\eta }(t)$ , should provide an upper bound for $\\left\\Vert \\vec{y}\\right\\Vert $ .", "The mean square norm of $\\vec{y}_u$ is given by ($\\tau = -\\lambda ^{-1}$ ): $\\left\\langle \\left\\Vert \\vec{y}_u\\right\\Vert ^2\\right\\rangle = \\left\\langle \\left\\Vert \\int _0^{\\tau /2}\\vec{\\eta }(t)dt\\right\\Vert ^2\\right\\rangle = \\frac{m}{2}\\tau \\sigma ^2.$ However, this upper bound is only valid when the matrix ${A}$ is normal, i.e.", "it has an orthogonal set of eigenvectors (for instance, Hermitian matrices are normal) [21].", "This can be understood by analyzing the behavior of Eq.", "(REF ) in the deterministic limit ($\\sigma = 0$ ).", "Although the asymptotic decay rate of $\\left\\Vert \\vec{y}\\right\\Vert $ is set by the eigenvalues of ${A}$ , the instantaneous response is given by the eigenvalues of ${H} = ({A} +{A}^T)/2$ , the Hermitian part of ${A}$ [18].", "If ${A}$ is non-normal, then the short-time dynamics of $\\left\\Vert \\vec{y}\\right\\Vert $ cannot be predicted by the eigenvalues of ${A}$ .", "Remarkably, ${H}$ can admit positive eigenvalues even though ${A}$ possesses all negative eigenvalues, in which case $\\left\\Vert \\vec{y}\\right\\Vert $ can experience a transient growth, for suitable initial conditions, before it starts decaying (Fig.", "REF ).", "This mechanism, sometimes termed as reactivity [18], occurs because the transformation that takes $\\vec{y}$ to the eigenbasis of ${A}$ is not unitary if the eigenvectors of ${A}$ are not orthogonal, and thus does not preserve the norm of $\\vec{y}$ .", "Clearly, if the stable matrix amplifies perturbations, the bound (REF ) cannot hold.", "Figure: (Color online) Stable linear systems can amplifyperturbations .", "Dynamics of theEuclidean norm y →\\left\\Vert \\vec{y}\\right\\Vert obtained by solving y → ˙=A i y →\\dot{\\vec{y}} ={A}_i \\vec{y}.", "Reactive systems exhibit transient amplificationbefore relaxing to fixed point (blue lines), in contrast withconventional response of stable systems (yellow lines).", "MatricesA 1 {A}_1 and A 2 {A}_2 (respectively A 3 {A}_3 and A 4 {A}_4) havesame real (respectively complex conjugate) eigenvalues.In the presence of noise, this transient effect in the deterministic part of Eq.", "(REF ) has a lasting effect on the steady state amplitude of the stochastic dynamics.", "This can be demonstrated by solving the steady state probability density of $\\vec{y}$ for Eq.", "(REF ).", "The detailed derivation of what follows is presented in the supplemental material (SM).", "For every stable matrix ${A}$ , we define a matrix ${G}$ such that the Hermitian part of its inverse is the identity, and its product with ${A}$ is Hermitian, that is, $\\begin{split}&\\frac{1}{2}\\left({G}^{-1}+\\left({G}^{-1}\\right)^T\\right) = 1, \\quad \\left({G}{A}\\right)^T = {G} {A}.\\end{split}$ Note that ${G}$ is the identity matrix if ${A}$ is Hermitian.", "In terms of this matrix ${G}$ , the steady state probability density of $\\vec{y}$ is given by $P\\left(\\vec{y}\\right) = \\sqrt{\\det \\left(-\\frac{{G} {A}}{\\pi \\sigma ^2}\\right)}\\exp \\left(\\frac{\\vec{y}^{\\,T}{G}{A} \\vec{y}}{\\sigma ^2}\\right),$ hence the mean square value of $\\left\\Vert \\vec{y}\\right\\Vert $ is (tr stands for the trace function) $\\left\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2\\right\\rangle = -\\frac{\\sigma ^2}{2} \\mathcal {H}({A})\\,\\text{tr}\\left({A}^{-1}\\right),$ where we have defined the non-normality index $\\mathcal {H}$ by: $\\mathcal {H}({A}) =\\text{tr}\\left({G}^{-1} {A}^{-1}\\right)/\\text{tr}\\left( {A}^{-1}\\right).$ Note that we always have $\\mathcal {H}\\ge 1$ , and $\\mathcal {H}$ is equal to one if and only if the matrix ${A}$ is normal.", "Moreover, the further ${A}$ is from normal, the larger is the index $\\mathcal {H}$ .", "In the case of a two-dimensional matrix ${A}$ , the non-normality index $\\mathcal {H}$ simplifies to the following simple expression, where $\\cot \\theta $ is the cotangent of the angle between the two eigenvectors: $\\mathcal {H} = 1 + \\cot ^2(\\theta ) \\left( \\frac{\\lambda _1 - \\lambda _2}{\\lambda _1+\\lambda _2}\\right)^2.$ This expression gives us quantitative understanding about how transient amplification occurs (Fig.", "REF ).", "Two ingredients are necessary: non-orthogonal eigenvectors and a separation of time scales given by eigenvalues of different magnitudes.", "If the system is not subject to noise, suitable initial conditions are also required (e.g.", "the blue vector in Fig.", "REF ).", "Because of the separation of time scales, the component of $\\vec{y}$ along the eigenvector associated with the faster eigenvalue decays quickly, whereas in the slow direction the dynamics is approximately constant.", "However, because of non-orthogonality, the norm of $\\vec{y}$ instantaneously increases as $\\vec{y}$ moves along the fast eigenvector, until the slow manifold starts attracting the trajectory back to fixed point.", "Figure: (Color online) Transient amplification is caused bynon-orthogonal eigenvectors and a separation of timescales.", "Thestable fixed point is subject to the perturbation y →(0)\\vec{y}(0).Because of the separation of timescales, the deterministictrajectory (blue arrowed line) is initially parallel to the fasteigenvector before relaxing to the slow manifold.", "From AA to BB,the trajectory has magnitude greater than \|\|y → 0 \|\|\|\|\\vec{y}_0 \|\|.Non-normality in spatially-extended pattern formation:- We now analyze spatially-extended, diffusively-coupled pattern-forming systems driven by noise.", "Specifically, we consider the generic equation $\\frac{\\partial {\\vec{q}}}{\\partial {t}} = \\vec{f}(\\vec{q})+{D}\\nabla ^2\\vec{q}+\\sigma \\vec{\\xi }(\\vec{x},t),$ where $\\vec{x}$ is a space variable, the vector $\\vec{q} = (q_1, q_2)$ , the diffusion matrix ${D} = \\text{diag}(D_1, D_2)$ , and $\\xi _i$ 's, the components of $\\vec{\\xi }(\\vec{x},t)$ are normalized $\\delta $ -correlated Gaussian white noises.", "Also, we assume that $\\vec{f}(\\vec{q})$ has a stable fixed point $\\vec{q}^{\\,}$ , and all of the eigenvalues of the linear stability or Jacobian matrix ${J} = \\left.\\nabla _{\\vec{q}}f(\\vec{q})\\right\|_{\\vec{q}^{\\,}}$ have negative real part.", "Our goal is to show that in the presence of noise, system (REF ) exhibits patterns in a parameter regime where the fixed point $\\vec{q}^{\\,}$ is stable.", "The stability of $\\vec{q}^$ can be inspected by defining the deviation $\\vec{p} = \\vec{q} - \\vec{q}^{\\,}$ and linearizing near $\\vec{q}\\,^$ , yielding $\\frac{\\partial {\\vec{p}}}{\\partial {t}} = {J}\\vec{p}+{D}\\nabla ^2\\vec{p}+\\sigma \\vec{\\xi }(\\vec{x},t).$ The spatial degrees of freedom can be diagonalized by a Fourier transform ($\\vec{x} \\mapsto \\vec{k}$ ), resulting in $\\frac{\\text{d}{\\vec{p}_{\\vec{k}}}}{\\text{d}{t}} = {K}\\vec{p}_{\\vec{k}}+\\sigma \\vec{\\xi }(\\vec{k},t),\\qquad {K} = {J} - k^2{D}.$ The equations are now decoupled and are therefore tantamount to Eq.", "(REF ).", "We start by reviewing the stability of the deterministic part of Eq.", "(REF ).", "If $D_1=D_2$ , matrix ${D}$ is a multiple of the identity, and the eigenvalues of ${K}$ will be the eigenvalues of ${J}$ shifted by $-k^2D$ for each $\\vec{k}$ , resulting in a more stable operator.", "However, in the case that the diffusion rates are sufficiently different, the largest eigenvalue of ${K}$ can have a non-monotonic behavior as a function of $\\vec{k}$ , and in some cases have positive eigenvalues for a small range of $\\vec{k}$ peaked around some non-zero value $\\vec{k}_0$ .", "In this case, the modes near $\\vec{k}_0$ will grow leading to the formation of deterministic Turing patterns [1].", "Therefore, the formation of deterministic Turing patterns is dependent on a large separation of the diffusion constants [4], [5], [6].", "In contrast, consider an intermediate scenario with diffusion constants different enough so that they can cause a non-monotonic behavior for the largest eigenvalue of ${K}$ as a function of $\\vec{k}$ peaked around some value $\\vec{k}_0$ , but not enough for the largest eigenvalue to become positive at any $\\vec{k}$ (left panel of Fig.", "REF ).", "In this case, all the $\\vec{k}$ modes decay quickly to zero, but the modes with $\\vec{k} \\sim \\vec{k}_0$ decay slower than the others, causing a transient pattern.", "In the presence of the noise term $\\vec{\\xi }(\\vec{k},t)$ in Eq.", "(REF ), while the modes with smaller eigenvalues decay quickly to zero, the slow modes drift away from the fixed point under the influence of the noise.", "The drift of the $\\vec{k}$ modes near $\\vec{k}_0$ produces persistent steady-state fluctuation-induced patterns with well-defined length-scales [8], [9].", "While the stochastic Turing patterns have a less stringent requirement than the deterministic Turing patterns for the ratio of the diffusion constants, their amplitude is limited to the amplitude of the drift under the noise suppressed by the slow deterministic decay.", "As discussed in the previous section, the mean square amplitude is of order $\\lambda ^{-1}\\sigma ^2$ , unless we can show that the system is non-normal.", "We now prove that in order for a system described by Eq.", "(REF ) to produce stochastic patterns, it is necessary for the matrix ${J}$ in Eq.", "(REF ) to be non-normal.", "We show this by finding a lower bound on the difference between the largest eigenvalue of ${H} = ({J} + {J}^{T})/2$ and that of matrix ${J}$ .", "The proof relies on the fact that for the system to exhibit stochastic patterns, the real part of the largest eigenvalue, $\\lambda _1$ , of ${K}$ as a function of the wave vector $\\vec{k}$ should peak at some value $\\vec{k}_0 \\ne 0$ [22], [9], and therefore, $\\delta = \\Re (\\lambda _1({K}_0)) - \\Re (\\lambda _1({J})) >0$ , for ${K}_0 = {K}(\\vec{k}_0)$ .", "It is a well known fact that the real part of the largest eigenvalue of a matrix is less than or equal to that of its Hermitian part (e.g.", "see Ref.", "[23]), therefore, $\\Re (\\lambda _1({K}_0))\\le \\lambda _1({H} -k_0^2 {D})$ .", "Since both ${H}$ and $ -k_0^2 {D}$ are Hermitian, by Weyl inequality $\\lambda _1({H} -k_0^2 {D}) \\le \\lambda _1({H})+\\lambda _1(-k_0^2 {D}) = \\lambda _1({H}) - k_0^2 D_{min}$ .", "Adding $\\,k_0^2\\, D_{min} - \\Re (\\lambda _1({J}))$ to both sides of this inequality, we arrive at $\\lambda _1({H}) - \\Re (\\lambda _1({J})) \\ge \\delta + k_0^2 D_{min}.$ Since the non-normality of ${J}$ should be independent of the diffusion constants, this lower bound can be extended to the supremum of the right hand side of the inequality (REF ) over all the matrices ${D}$ that produce spatial patterns and their corresponding $\\vec{k}_0$ .", "In particular, if a system admits deterministic Turing patterns for some set of diffusion constants, i.e.", "$\\Re (\\lambda _1({K}_0)) >0$ , $\\delta $ would be greater than $ -\\Re (\\lambda _1({J}))$ , and therefore ${J}$ would be reactive (this special case was previously proven by Neubert et al. [20]).", "In this case, if experimentally measured values of diffusion constants do not fall within the Turing pattern regime, the system is still reactive and capable of exhibiting amplified stochastic patterns.", "Stochastic extension of model by Ridolfi et al.", ":- Finally, we apply our theory to a concrete model that is representative of a large class of systems.", "On a deterministic level, the model is given by Eq.", "(REF ) with two species $U$ and $V$ with densities $\\vec{q} = (u,\\,v)$ , and $\\vec{f}(u, v) = \\left(u (a u v -e),\\, v(b - c u^2 v)\\right)$ , with $a,b,c, e >0$ [15].", "The corresponding stochastic model is defined by considering the following individual-level processes that occur on a discretized $D$ -dimensional space with $L^D$ lattice sites, Figure: (Color online) Stochasticity allows pattern formation for similar diffusivities.", "(left) Phase diagram of model () showing that the pattern forming behavior of this model depends only on the ratios b/ab/a and D V /D U D_V/D_U (see SM for analytic expression for the boundaries).", "(right) Semi-log plot of non-normality index for the point PP as a function of a/c 2/3 a/c^{2/3}.", "Black markers are amplifications measured in simulation.$\\begin{split}&2U_i+V_i\\xrightarrow{}3U_i+V_i,\\hspace{21.3513pt}V_i\\xrightarrow{}2V_i,\\\\&U_i\\xrightarrow{}\\varnothing ,\\hspace{21.3513pt}2V_i+2U_i\\xrightarrow{}V_i+2U_i,\\\\&U_i\\xrightarrow{}U_j,\\hspace{21.3513pt}V_i\\xrightarrow{}V_j,\\hspace{21.3513pt} j\\in \\langle i\\rangle \\end{split}$ where $U_i$ and $V_i$ are the species $U$ and $V$ on the site $i$ for $i = 1\\dots L^D$ and $\\langle i\\rangle $ is the set of sites neighboring $i$ .", "The state of the system is specified by the concentration vectors $\\vec{q}_i\\equiv (u_i, v_i) \\equiv (U_i, V_i)/\\Omega $ , where $\\Omega $ is the volume of each site.", "The diffusion rates $\\delta _u$ and $\\delta _v$ are related to the diffusion constants by $(\\delta _u,\\,\\delta _v) =(D_U, D_V)/\\Omega ^{2/D}$ .", "The discrete-space version of Eqs.", "(REF ), (REF ) and (REF ) are derived by expanding in powers of $\\Omega ^{-1/2}$ the master equation corresponding to scheme (REF ) (see the SM for the derivations).", "The pattern forming behavior of the model described by (REF ) only depends on the ratio of the diffusion constants $D_V/D_U$ and the ratio of the reaction rates of the two linear reactions $b/e$ .", "The left panel of Figure REF shows the regime of parameters in which the system exhibits either stochastic or deterministic Turing patterns.", "As expected, deterministic patterns emerge only when the ratio $D_V/D_U$ of diffusion constants is very large (above the blue line in Fig.", "REF which steeply grows outside of the figure), while the requirement on this ratio for the stochastic patterns is drastically reduced (see the SM for analytic expressions for the boundaries).", "In the absence of the non-normality effect, one would expect that only stochastic patterns with parameters very close to the deterministic regime would be observed, since far from this regime, the amplitude of the patterns would be too small to detect.", "However, since for all $b/e>1$ , there is a $D_V/D_U$ above which the system exhibits deterministic Turing patterns, ${J}$ is reactive.", "Therefore, even when the system is far from the parameter regime of deterministic patterns, the amplitude of the stochastic patterns is far larger than what one would expect from the analysis of the eigenvalues from Eq.", "(REF ).", "We can see this by analyzing the amplitude of the patterns at the point $P$ in Fig.", "REF .", "This point has ratios $b/e = 5.8$ and $D_V/D_U = 3.4$ and is chosen to be very far from the deterministic Turing pattern regime.", "At this $b/e$ ratio, the ratio of the diffusion constants has to be at least ten times larger than the chosen value for the system to exhibit deterministic Turing patterns.", "The amplitude of the patterns as determined by Eq.", "(REF ) is dependent on the eigenvalues of ${K}$ (fixed by the choice of the point $P$ ) and the non-normality index $\\mathcal {H}({K})$ which can be tuned by changing the ratio $a/c^{2/3}$ without changing the point $P$ (see SM for the analytic expression).", "The right panel of Figure REF shows that the amplification of stochastic patterns for the point $P$ varies over orders of magnitude for a small range of $a/c^{2/3}$ .", "The right panel of Figure REF shows the time series of the amplified stochastic Turing patterns in the concentration of the species $U$ , in a simulation of our model in one dimension.", "The mean square amplitude of these spatial patterns is about $0.21$ , while the upper bound for the amplitude of the pattern in the absence of reactivity from Eq.", "(REF ) is $2.5\\times 10^{-3}$ .", "The non-normality index $\\mathcal {H}$ of the slowest Fourier mode $k_0 = 6$ is about 103 justifying the two order of magnitude amplification in the amplitude of the stochastic patterns (see the right panel of Fig.", "REF ).", "In conclusion, fluctuation-induced Turing patterns have larger amplitude than previously expected, even when the ratio of the diffusion coefficients is far from the requirement for deterministic Turing patterns.", "This large amplitude is due to non-normality of the type of interactions that are required for a system to produce Turing-like patterns.", "We have introduced a new measure of non-normality that is applicable to all stochastic dynamical systems and measures the amplification of the expected value of the distance that a non-equilibrium system maintains from its fixed point at steady state.", "We have used this measure to quantify the effect of non-normality on stochastic Turing patterns and explain the unexpectedly large amplitude observed in the simulations.", "By analyzing an example of an activator-inhibitor system, we have shown that the demographic stochasticity drastically expands the range of parameters in which the system exhibits Turing-like patterns, and that these patterns have amplitudes that are orders of magnitude larger that expected in all but a narrow region in parameter space.", "We conclude that fluctuation-induced Turing patterns can readily be observed, and therefore, provide a potential mechanism explaining a wide range of patterns formations observed in ecology, biology, and development This work was supported by the National Aeronautics and Space Administration Astrobiology Institute (NAI) under Cooperative Agreement No.", "NNA13AA91A issued through the Science Mission Directorate.", "T.B acknowledges partial funding from the National Science Foundation under Grant No.", "PHY-105515.", "T. B. and F. J.", "Contributed equally to this work.", "Supplemental Materials Linear response of stochastic reactive systems Linear Fokker-Planck equation and its stationary distribution In the main text, we encounter multiple times the linear stochastic differential equation (SDE) of the form $ \\frac{d \\vec{y}}{dt} = {A} \\vec{y} + \\vec{\\eta }(t),$ where ${A}$ is independent of $\\vec{y}$ and $\\vec{\\eta }$ are Gaussian white noises with zero mean and correlator $\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }^T(t^{\\prime }) \\rangle = {B} \\delta (t-t^{\\prime }).$ The noise matrix ${B}$ is symmetric (i.e ${B}^T = {B}$ ) and also supposed independent of $\\vec{y}$ .", "Equation (REF ) is tantamount to the Fokker-Planck equation for the probability density $P(\\vec{y}, t)$ [24]: $ \\frac{\\partial P(\\vec{y}, t)}{\\partial t} = - \\sum _{i,j} A_{ij} \\frac{\\partial }{\\partial y_i} (y_j P) + \\frac{1}{2} \\sum _{i,j} \\frac{\\partial ^2}{\\partial y_i \\partial y_j} (B_{ij} P).$ As shown in (e.g.)", "[25], the stationary distribution is Gaussian and takes the form $ P_s(\\vec{y}) = \\frac{1}{\\sqrt{\\det (2 \\pi {\\Xi })}} \\exp \\left( -\\frac{1}{2} \\vec{y}^T\\, {\\Xi }^{-1}\\, \\vec{y} \\right),$ where the covariance matrix ${\\Xi }$ is symmetric and given by the Sylvester's equation, $ {A} {\\Xi }+ {\\Xi }{A}^T + {B} = 0.$ In two dimensions, this equation can be solved [24] leading to an explicit formula for ${\\Xi }$ : ${\\Xi }= \\frac{\\left({A} - 1_2\\,\\text{tr} {A}\\right) {B}\\left(1_2\\,\\text{tr} {A} - {A} \\right)^T - {B}\\,\\det {A}}{2\\, \\text{tr} {A} \\,\\text{det} {A}}.$ The mean amplification factor $\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle $ We now wish to find an expression for the mean amplification factor, $\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle $ , used in the main text to quantify the linear response of a stochastic reactive system.", "The norm of $\\vec{y}$ is the Euclidean norm $\\left\\Vert \\vec{y}\\right\\Vert = \\sqrt{\\sum _i \\left\|y_i^2\\right\|}$ .", "Specifically, we want to compute the integral: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\int _{\\mathbb {R}^D} d\\vec{y}\\, P_s(\\vec{y}) \\left\\Vert \\vec{y}\\right\\Vert ^2,$ where the distribution $P_s(\\vec{y})$ is given by Eq.", "(REF ).", "Therefore, $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\frac{1}{\\sqrt{\\det (2 \\pi {\\Xi })}} \\int d\\vec{y} \\exp \\left( -\\frac{1}{2} \\vec{y}^T\\, {\\Xi }^{-1}\\, \\vec{y} \\right) \\left\\Vert \\vec{y}\\right\\Vert ^2.$ To evaluate this integral, we use the identity $\\int \\left\\Vert \\vec{p}\\right\\Vert ^2 e^{-\\vec{p}^{\\,T} {M} \\vec{p}} d\\vec{p} = \\frac{1}{2} \\text{Tr}\\left({M}^{-1}\\right)\\int e^{-\\vec{p}^{\\,T} {M} \\vec{p}} d\\vec{p},$ with ${M} = 1/2{\\Xi }^{-1}$ , which yields the compact expression: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\, \\text{Tr}\\,{\\Xi }$ In the following, we assume for convenience that the noise matrix ${B}$ is a multiple of identity identity matrix 1 (${B} = \\sigma ^2 1$ ), a choice that can be made without losing in generality.", "In fact, since ${B}$ is symmetric, it is diagonalized by an orthogonal matrix which one can use to transform the noises; the resulting diagonal matrix can then be mapped to the identity matrix simply by rescaling the variables $\\vec{y}$ .", "Now, we will write the matrix ${\\Xi }$ in terms of ${A}$ and what we call the Hermitianizer of ${A}$ , defined as $ {G} = - \\frac{1}{2}\\,\\sigma ^2\\, {\\Xi }^{-1} {A}^{-1},$ which yields a symmetrization of matrix ${A}$ : even though ${A}$ is not symmetric, ${A} \\ne {A}^T$ , the product ${G} {A} = -2^{-1}\\sigma ^2 {\\Xi }^{-1}$ is a symmetric matrix.", "Sylvester equation (REF ) written in terms of ${G}$ simplifies to $ \\frac{1}{2} ({G}^{-1} + {G}^{-T}) = 1,$ indicating that the hermitian part of ${G}^{-1}$ is identity.", "Alternatively, the Hermitianizer of ${A}$ can be defined as the unique matrix satisfying Eq.", "(REF ) whose product with ${A}$ is Hermitian.", "Now we can write the mean squared value of the norm $\\vec{y}$ in terms of ${A}$ and ${G}$ by substituting Eq.", "(REF ) in Eq.", "(REF ): $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{1}{2} \\sigma ^2\\, \\text{Tr}\\left({A}^{-1} {G}^{-1}\\right)$ When ${A}$ is a $2\\times 2$ matrix, the trace of the inverse can be written as trace over determinant: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{1}{2} \\sigma ^2\\, \\frac{\\text{Tr}\\left({G} {A}\\right)}{\\det ({G})\\det ({A})}$ $\\text{Tr}({G} {A})$ can be simplified by taking the trace of Eq.", "(REF ) $ \\text{Tr}({G}{A}) = -\\frac{1}{2}\\,\\sigma ^2\\, \\text{Tr}({\\Xi }^{-1}).$ Also, by multiplying the right-hand side of the Sylvester equation (REF ) by ${\\Xi }^{-1}$ : ${A} + {\\Xi }{A}^T {\\Xi }^{-1} = - \\sigma ^2\\,{\\Xi }^{-1}.$ and taking the trace we have (recalling that $ \\text{Tr}({\\Xi }{A}^T {\\Xi }^{-1})=\\text{Tr}({A}^T)= \\text{Tr}({A})$ ): $ \\sigma ^2\\,\\text{Tr}({\\Xi }^{-1}) = - 2\\, \\text{Tr}({A})$ From Eq.", "(REF ) and Eq.", "(REF ) it follows that $\\text{Tr}({G}{A}) = \\text{Tr}({A})$ .", "which we can use to simply Eq.", "(REF ): $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{\\,\\sigma ^2}{2\\,\\det {G}} \\frac{\\text{Tr}\\,{A}}{\\text{det}\\,{A}} = -\\frac{1}{2}\\,\\sigma ^2\\, \\det \\left({G}^{-1}\\right) \\text{Tr}\\left({A}^{-1}\\right).$ Non-normality for a $2\\times 2$ matrix ${A}$ For a $2\\times 2$ matrix ${A}$ given by its elements ${A} = \\left(\\begin{array}{cc}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{array}\\right),$ we can solve for ${\\Xi }$ from Eq.", "(REF ) and substitute in Eq.", "(REF ) to find the matrix ${G}$ in terms of matrix elements of ${A}$ : ${G} = \\left(\\begin{array}{cc}\\frac{(a_{11}+a_{22})^2}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} & -\\frac{(a_{12}-a_{21}) (a_{11}+a_{22})}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} \\\\\\frac{(a_{12}-a_{21}) (a_{11}+a_{22})}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} & \\frac{(a_{11}+a_{22})^2}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} \\\\\\end{array}\\right).$ The non-normality index $\\mathcal {H}$ is given by the inverse of the determinant of ${G}$ : $\\mathcal {H}({A}) = \\det \\left({G}^{-1}\\right) = 1 + \\frac{(a_{12}-a_{21})^2}{(a_{11}+a_{22})^2}.$ If the eigenvalues of ${A}$ are real, we can rewrite this expression in terms of the eigenvalues and the angle between the eigenvectors of ${A}$ .", "Let $\\Delta >0$ be the discriminant of the characteristic polynomial of ${A}$ : $\\Delta = (a_{11}-a_{22})^2 + 4\\, a_{12} \\, a_{21}.$ If $\\lambda _1$ and $\\lambda _2$ are the two eigenvalues of ${A}$ , and $\\vec{v}_1$ and $\\vec{v}_2$ are the two eigenvectors, we have $\\begin{split}&\\left(\\lambda _1 + \\lambda _2\\right)^2 = (a_{11}+a_{22})^2, \\qquad \\left(\\lambda _1 - \\lambda _2\\right)^2 = \\Delta , \\\\&\\cos ^2(\\theta ) = \\left(\\frac{\\vec{v}_1 \\cdot \\vec{v}_2}{\\left\\Vert \\vec{v}_1\\right\\Vert \\left\\Vert \\vec{v}_2\\right\\Vert }\\right)^2, \\qquad \\cot ^2(\\theta ) = \\frac{\\cos ^2(\\theta )}{1-\\cos ^2(\\theta )} = \\frac{ (a_{11}-a_{22})^2}{\\Delta }.\\end{split}$ Now it is clear that $\\mathcal {H}({A}) = 1 + \\cot ^2(\\theta )\\left( \\frac{ \\lambda _1 - \\lambda _2}{ \\lambda _1 + \\lambda _2}\\right)^2.$ Linear stochastic differential equations with complex variables Consider a similar set of SDEs of the the form $ \\frac{d \\vec{y}}{dt} = {A} \\vec{y} + \\vec{\\eta }(t),$ where now $\\vec{y}$ and $\\vec{\\eta }$ are vectors with complex variables, and $\\vec{\\eta }$ is a Gaussian white noise with zero mean and correlator $\\begin{split}&\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }\\,^\\dagger (t^{\\prime }) \\rangle = {B} \\delta (t-t^{\\prime }),\\\\&\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }\\,^T(t^{\\prime }) \\rangle = 0.\\end{split}$ where the $^\\dagger $ symbol represents the transpose conjugate.", "The analysis in the previous section can be generalized by evaluating the expected value of $\\vec{y}(t) \\vec{y}\\,^\\dagger (\\tau )$ and $\\vec{y}(t) \\vec{y}\\,^T(\\tau )$ at steady state for $t = \\tau $ to obtain the following relationships for the covariance and relation matrices $\\begin{split}&{A}\\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle + \\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle {A}^\\dagger + {B} = 0\\\\&{A}\\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle + \\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle {A}^T = 0\\end{split}$ The first equation is the analogue of equation of Sylvester Eq.", "(REF ) for the Hermitian covariance matrix ${\\Xi }= \\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle $ , while the second equation implies that the symmetric relation matrix ${C} = \\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle $ is equal to zero.", "Therefore, at steady state, $\\vec{y}$ obeys a circularly symmetric complex Gaussian distribution of the form $P_s(\\vec{y}) = \\frac{1}{\\det (2 \\pi {\\Xi })} \\exp \\left( -\\frac{1}{2} \\vec{y}\\,^\\dagger \\, {\\Xi }^{-1}\\, \\vec{y} \\right).$ Notice the different normalization factor compared to Eq (REF ), as it is normalized over $\\mathbb {C}^D$ instead of $\\mathbb {R}^D$ .", "To compute the mean square value of the norm of $\\vec{y}$ , we can follow similar analysis to that of section REF .", "Here, we highlight the differences.", "The mean square norm is define as $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\int _{\\mathbb {C}^D} d\\vec{y}\\, P_s(\\vec{y}) \\left\\Vert \\vec{y}\\right\\Vert ^2,$ with the norm $\\left\\Vert \\vec{y}\\right\\Vert = \\sqrt{\\vec{y}\\,^\\dagger \\vec{y}}$ .", "The complex version of Eq.", "(REF ) can be evaluated by diagonalizing the matrix ${M}$ and write the integral on a $2D$ -dimensional real space.", "The result is given by $\\int _{\\mathbb {C}^D} \\left\\Vert \\vec{p}\\right\\Vert ^2 e^{-\\vec{p}^{\\,\\dagger } {M} \\vec{p}} d\\vec{p} = \\text{Tr}\\left({M}^{-1}\\right)\\int _{\\mathbb {C}^D} e^{-\\vec{p}^{\\,\\dagger } {M} \\vec{p}} d\\vec{p}\\,,$ where the factor $1/2$ is canceled by the fact that each eigenvalue of ${M}^{-1}$ should be counted twice in the $2D$ -dimensional space, once for the real part and once for the imaginary part.", "As a result, there will be an extra factor 2 in Eq.", "(REF ), Eq.", "(REF ), and Eq.", "(REF ).", "In particular , $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = - \\sigma ^2\\, \\text{Tr}\\left({A}^{-1} {G}^{-1}\\right)$ Analysis of model by Ridolfi et al.", "From individual level model to SDEs In this section we derive a the stochastic extension of the model by Ridolfi et al.", "[15] by expanding the master equation corresponding to the individual level model defined by the following set of reactions $\\begin{split}&2U_i+V_i\\xrightarrow{}3U_i+V_i,\\hspace{21.3513pt}V_i\\xrightarrow{}2V_i,\\\\&U_i\\xrightarrow{}\\varnothing ,\\hspace{21.3513pt}2V_i+2U_i\\xrightarrow{}V_i+2U_i,\\end{split}$ where $U_i$ and $V_i$ are the species $U$ and $V$ in the site $i$ , and the diffusion reactions $U_i\\xrightarrow{}U_j,\\hspace{21.3513pt}V_i\\xrightarrow{}V_j,\\hspace{21.3513pt} j\\in \\langle i\\rangle $ where $\\langle i\\rangle $ is the set of sites neighboring $i$ , $\\delta _u=D_U/\\Omega ^{2/D}$ , $\\delta _v=D_V/\\Omega ^{2/D}$ , $D_U$ and $D_V$ are the diffusion constants, and $\\Omega $ is the volume of each site.", "The state of the system is specified by the concentration vectors $\\vec{q}_i\\equiv (u_i, v_i) \\equiv (U_i, V_i)/\\Omega $ .", "Each reaction of reaction scheme (REF ) takes the system from a state $\\lbrace \\vec{q}_i\\rbrace $ to $\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace $ with probability per unit time $T(\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace \|\\lbrace \\vec{q}_i\\rbrace )$ .", "These transition rates are given from the law of mass action: $\\begin{split}&T\\left(\\left.", "\\vec{q}_i +\\vec{s}_1\\right\|\\vec{q}_i\\right) = \\Omega a u_i^2 v_i, \\hspace{12.80948pt}T\\left(\\left.", "\\vec{q}_i + \\vec{s}_2\\right\|\\vec{q}_i\\right) = \\Omega b v_i,\\\\&T\\left(\\left.", "\\vec{q}_i - \\vec{s}_1\\right\|\\vec{q}_i\\right) = \\Omega e u_i,\\hspace{12.80948pt}T\\left(\\left.", "\\vec{q}_i - \\vec{s}_2\\right\|\\vec{q}_i\\right) = \\Omega c u_i^2 v_i^2,\\\\\\end{split}$ and for every $j\\in \\langle i\\rangle $ $\\begin{split}&T\\left(\\left.", "\\vec{q}_i -\\vec{s}_1,\\vec{q}_j + \\vec{s}_1\\right\|\\vec{q}_i,\\vec{q}_j\\right) = \\Omega \\delta _u u_i,\\\\&T\\left(\\left.", "\\vec{q}_i -\\vec{s}_2,\\vec{q}_j + \\vec{s}_2\\right\|\\vec{q}_i,\\vec{q}_j\\right) = \\Omega \\delta _v v_i,\\end{split}$ where $\\vec{s}_1 = \\Omega ^{-1} \\left(\\begin{array}{c}1\\\\0\\end{array}\\right), \\hspace{21.3513pt}\\vec{s}_2 = \\Omega ^{-1} \\left(\\begin{array}{c}0\\\\1\\end{array}\\right).$ The master equation for the time evolution of the probability of finding the system at a state $\\lbrace \\vec{q}_i\\rbrace $ , $P(\\lbrace \\vec{q}_i\\rbrace ,t)$ can be written as $\\frac{\\text{d}{P(\\lbrace \\vec{q}_i\\rbrace ,t)}}{\\text{d}{t}} = \\sum _{\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace } \\left(T(\\lbrace \\vec{q}_i\\rbrace \|\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace )-T(\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace \|\\lbrace \\vec{q}_i\\rbrace )\\right)$ Following [22], we can expand the right hand side of Eq.", "(REF ) to second order in $\\Omega ^{-1}$ obtaining a Fokker-Planck equation corresponding the following set of stochastic differential equations $\\begin{split}&\\frac{\\text{d}{u_i}}{\\text{d}{t}} = u_i(a u_i v_i-e)+\\delta _u\\sum _{j\\in \\langle i\\rangle }(u_j-u_i)+\\xi _i(t),\\\\&\\frac{\\text{d}{v_i}}{\\text{d}{t}} = v_i(b-c u_i^2 v_i)+\\delta _v\\sum _{j\\in \\langle i\\rangle }(v_j-v_i)+\\eta _i(t),\\\\\\end{split}$ where $\\xi _i$ 's and $\\eta _i$ 's are zero mean Gaussian noise with correlations $\\begin{split}\\langle \\xi _i(t)\\xi _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Bigg (\\bigg ( u_i(a u_i v_i+e)+\\delta _u\\sum _{k\\in \\langle i\\rangle }(u_i+u_k)\\bigg )\\delta _{i,j}-\\delta _u(u_i+u_j){\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Bigg )\\\\\\langle \\eta _i(t)\\eta _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Bigg (\\bigg ( v_i(b+c u_i^2 v_i)+\\delta _v\\sum _{k\\in \\langle i\\rangle }(v_i+v_k)\\bigg )\\delta _{i,j}-\\delta _v(v_i+v_j){\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Bigg )}}and the characteristic function, {\\raisebox {}{\\chi }_{\\langle i\\rangle }, of \\langle i\\rangle is defined as{\\begin{equation}{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j) = {\\left\\lbrace \\begin{array}{ll}1 & j\\in \\langle i\\rangle \\\\0 & j\\notin \\langle i\\rangle \\end{array}\\right.}\\;.", "}\\end{equation}By defining \\vec{f}(\\vec{q}) \\equiv (f,\\; g) \\equiv ( u(a u v-e), \\;v(b-c u^2 v)), \\vec{\\xi }_i \\equiv (\\xi _i,\\; \\eta _i), {\\delta }\\equiv {diag}(\\delta _u,\\;\\delta _v), and \\left(\\Delta \\vec{q}\\right)_i \\equiv \\sum _{j\\in \\langle i\\rangle }(\\vec{q}_j-\\vec{q}_i), Eq.~(\\ref {eq:slangevin}) can be written in the simple form{\\begin{equation}\\frac{\\text{d}{\\vec{q}_i}}{\\text{d}{t}} = \\vec{f}(\\vec{q}_i)+{\\delta }\\left(\\Delta \\vec{q}\\right)_i+\\vec{\\xi }_i(t).\\end{equation}}Equation~(\\ref {eq:snonlinear}) is the discrete space version of Eq.~(\\ref {eq:nonlinear}) of the main text.", "Continuous limit can be taken at any point in the following analysis to recover the continuous space stochastic partial differential equations of type analyzed in the main text.", "We continue with the discrete version where the analytic results can be more readily compared to the simulation.", "}The deterministic part of our model has a fixed point \\vec{q}\\,^ \\equiv (u^,\\; v^) = (ba/ce,\\; e^2c/a^2b), obtained by setting \\vec{f}(\\vec{q}) equal to zero.", "We can linearize Eq.~(\\ref {eq:snonlinear}) around the fixed point \\vec{q}\\,^, by defining \\vec{p}_i \\equiv \\big ((u_i-u^)/\\sqrt{2u^e},\\,(v_i-v^)/\\sqrt{2v^b}\\big ) which are the rescaled deviations of \\vec{q}_i from \\vec{q}\\,^,{\\begin{equation}\\frac{\\text{d}{\\vec{p}_i}}{\\text{d}{t}} = {J}\\vec{p}_i+{\\delta }(\\Delta \\vec{p})_i + \\vec{\\xi }_i(t),\\end{equation}}where the linear stability operator {J} is defined as the Jacobian of the transformed function f at the fixed point \\vec{p} = 0 is given by{\\begin{equation}{J} = \\left(\\begin{array}{cc}e & \\frac{b^\\frac{3}{2}a^\\frac{3}{2}}{ce}\\\\-\\frac{2e^2 c}{a^\\frac{3}{2} b^{\\frac{1}{2}}} & -b\\end{array}\\right)\\end{equation}}Evaluating Eq.~(\\ref {eq:snoise1}) at \\vec{q}\\,^{\\begin{equation}\\begin{split}\\langle \\xi _i(t)\\xi _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Big (\\big ( 1+\\delta _u n/e\\big )\\delta _{i,j}-\\delta _u{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Big ),\\\\\\langle \\eta _i(t)\\eta _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Big (\\big ( 1+\\delta _v n/b\\big )\\delta _{i,j}-\\delta _v{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Big ),}}\\end{split}where n \\equiv \\left\|\\langle i\\rangle \\right\| is the number of neighbors of each site.", "Note that for b>e, both of the eigenvalues of {J} have negative real parts, making \\vec{q}\\,^ an attractor of the dynamics in the absence of the diffusion.\\end{equation}To examine the spatial stability of \\vec{q}\\,^, we need to diagonalize the discrete Laplacian operator \\Delta , by defining the discrete Fourier transform of a sequence \\lbrace s_{\\vec{n}}\\rbrace as{\\begin{equation}\\tilde{s}_{\\vec{k}} \\equiv \\left(\\mathcal {F}[\\lbrace s_{\\vec{n}}\\rbrace ]\\right)_{\\vec{k}} \\equiv \\frac{1}{\\sqrt{N^D}}\\sum _{\\vec{n}} \\text{e}^{-2\\pi \\vec{k}.\\vec{n}/N}s_{\\vec{n}}.\\end{equation}}We drop the tildes on the Fourier variable with the convention that the variables with index k are Fourier variables.", "Equation~(\\ref {eq:slinear}) under this transformation becomes{\\begin{equation}\\frac{\\text{d}{\\vec{p}_{\\vec{k}}}}{\\text{d}{t}} ={K}\\vec{p}_{\\vec{k}} + \\vec{\\xi }_{\\vec{k}}(t), \\hspace{12.80948pt} {K} = {J}+\\Delta (\\vec{k}) {\\delta },\\end{equation}}where \\Delta (\\vec{k}) is the discrete Fourier transform of the discrete Laplacian operator given by{\\begin{equation}\\Delta (\\vec{k} ) \\equiv -2\\sum _{l = 1}^D\\big (1-\\cos (2\\pi k_l/N)\\big )\\end{equation}}and{\\begin{equation}\\begin{split}\\langle \\xi _{\\vec{k}}(t)\\xi _{\\vec{k}^{\\prime }{}}^(t^{\\prime }{})\\rangle = \\Omega ^{-1}\\left(1-e^{-1}\\delta _u\\Delta (\\vec{k})\\right)\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{}),\\\\\\langle \\eta _{\\vec{k}}(t)\\eta _{\\vec{k}^{\\prime }{}}^(t^{\\prime }{})\\rangle =\\Omega ^{-1}\\left(1-b^{-1}\\delta _v\\Delta (\\vec{k})\\right)\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{}).\\\\\\end{split}\\end{equation}}}For the regime that we observe stochastic patterns, the contribution of the diffusion process in the amplitude of the noise in Eq.~(\\ref {eq:snoise2}) is very small and will be neglected for simplicity.", "This approximation is not necessary, since there is always a change of variables that simplifies the correlation matrix to a multiple of the identity matrix (this is the reason for the rescaling in the definition of \\vec{p}).", "With this approximation{\\begin{equation}\\left\\langle \\vec{\\xi }_{\\vec{k}}(t)\\vec{\\xi }_{\\vec{k}^{\\prime }{}}^{\\,\\dagger }(t^{\\prime }{})\\right\\rangle = \\Omega ^{-1}\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{})\\,1\\end{equation}}where \\vec{\\xi }_{\\vec{k}^{\\prime }{}}^{\\,\\dagger } is the conjugate transpose of \\vec{\\xi }_{\\vec{k}^{\\prime }{}}, and 1 is the 2\\times 2 identity matrix.", "}\\end{split}$ Phase diagram of pattern formation The pattern forming behavior of the model defined by (REF ) can be understood by analyzing the eigenvalues of ${K}$ as a function of $\\vec{k}$ .", "Matrix ${K}$ can be written in elements from Eq.", "() and Eq.", "(): ${K} = \\left(\\begin{array}{cc}e + \\Delta (\\vec{k}) \\delta _u & \\frac{b^\\frac{3}{2}a^\\frac{3}{2}}{ce}\\\\-\\frac{2e^2 c}{a^\\frac{3}{2} b^{\\frac{1}{2}}} & -b + \\Delta (\\vec{k}) \\delta _v\\end{array}\\right)$ As it will become clear, most of the properties of the system depend on the following three parameters $\\rho = \\frac{b}{e},\\qquad \\nu = \\frac{e\\, c}{a^{\\frac{3}{2}} b^{\\frac{1}{2}}}, \\qquad r = \\frac{\\delta _v}{\\delta _u} = \\frac{D_V}{D_U}$ in the following analysis, we will write various expression in terms of these parameters, wherever we can.", "We start with ${K}$ ${K} = \\left(\\begin{array}{cc}e + \\Delta (\\vec{k}) \\delta _u & b/\\nu \\\\-2 e \\,\\nu & -b + \\Delta (\\vec{k}) \\delta _v\\end{array}\\right)$ The largest eigenvalue of ${K}$ is given by $\\lambda (\\vec{k}) = \\frac{1}{2} \\left(\\sqrt{b^2-2 b \\Delta (\\vec{k}) (\\delta _v-\\delta _u)-6 b e+\\left(e-\\Delta (\\vec{k}) (\\delta _v-\\delta _u)\\right)^2}-b+\\Delta (\\vec{k}) (\\delta _v+\\delta _u)-e\\right).$ Notice that the eigenvalues of ${K}$ are independent of $\\nu $ .", "For small $\\vec{k}$ , $\\Delta (\\vec{k})$ is a monotonically decreasing function of $\\vec{k}$ (proportional to $-k^2$ ).", "We define $y = -\\Delta (\\vec{k})$ .", "To determine if $\\lambda $ monotonically decays or if it has a maximum at some $\\vec{k}_0 \\ne 0$ , we can differentiate $\\lambda $ with respect to $y$ and see if it has a positive root.", "The largest root of $\\frac{\\text{d}{\\lambda }}{\\text{d}{y}}$ is given by $y_0 = -\\Delta (\\vec{k}_0) = \\frac{ (r+1) \\sqrt{2\\, b\\, e\\, r}-b\\, r-e\\, r}{\\delta _u \\,(r-1)\\, r}.$ For $y_0$ to be greater than zero we need $\\rho < \\frac{\\left(1+ r+r^2+(r+1) \\sqrt{r^2+1}\\right)}{r}.$ We can find the condition on the ratio of the diffusion constants by inverting this inequality: $r > \\frac{1-2\\, \\rho + \\rho ^2+(1+ \\rho ) \\sqrt{1 + \\rho \\, (\\rho -6)}}{4 \\,\\rho } = f_1(\\rho ).$ The condition for formation of stochastic pattern is $\\lambda (\\vec{k}_0) > \\Re (\\lambda (0))$ .", "We can find $\\lambda (\\vec{k}_0)$ and $\\lambda (0)$ by substituting $y_0 = y(\\vec{k}_0)$ from Eq.", "(REF ) and $y(0) = 0$ in Eq.", "(REF ): $\\lambda (\\vec{k}_0) = \\frac{b+e\\, r- \\sqrt{8\\,b\\, e\\, r}}{r-1}, \\qquad \\lambda (0) = \\frac{1}{2} \\left(\\sqrt{b^2-6\\, b\\, e+e^2}-b+e\\right).$ Then, $\\lambda (\\vec{k}_0) > \\Re (\\lambda (0))$ simplifies to $r > \\frac{- 1 + 14\\, \\rho - \\rho ^2+4 \\sqrt{-2\\,\\rho \\, (1 + \\rho \\, (\\rho -6)))}}{(1 + \\rho )^2} = f_2(\\rho ).$ Condition for deterministic Turing pattern is a lot simpler; we just need $\\lambda (\\vec{k}_0) > 0$ which simplifies to $r > \\left(3+2 \\sqrt{2}\\right) \\rho = f_3(\\rho ).$ When $r$ is greater than $f_1(\\rho )$ and $f_2(\\rho )$ but less than $f_3(\\rho )$ , the system exhibits stochastic patterns (blue region in Fig.", "3 of the main text), while we observe the deterministic patterns when $r$ is greater than $f_3$ (orange region of Fig.", "3 of the main text).", "Non-normality of the model The amplification of our stochastic patterns depend on the non-normality index of ${K}_0 = {K}(\\vec{k}_0)$ given by ${K}_0 = \\left(\\begin{array}{cc}e - y_0\\, \\delta _u & b/\\nu \\\\-2\\, e \\,\\nu & -b - y_0\\, \\delta _v\\end{array}\\right),$ where $y_0 = -\\Delta (\\vec{k}_0)$ .", "We use Eq.", "(REF ) to calculate the non-normality index of ${K}_0$ : $\\mathcal {H}({K}_0) = 1 + \\left(\\frac{b+2 e \\nu ^2}{\\nu (b-e+y_0\\, (\\delta _u+\\delta _v))}\\right)^2.$ We substitute $y_0$ from Eq.", "(REF ) and rewrite the resulting expression in terms of $\\rho $ , $r$ , and $\\nu $ : $\\mathcal {H}({K}_0) = 1+\\left(\\frac{2 \\nu ^2+\\rho }{\\nu \\left(\\rho - 1 +\\frac{(r+1) \\left(-\\rho r+ (r+1) \\sqrt{2\\rho \\, r}-r\\right)}{(r-1) r}\\right)}\\right)^2$ Since the eigenvalues of ${K}$ do not depend on $\\nu $ , one can change $\\mathcal {H}({K}_0)$ by changing $\\nu $ without moving the system in its phase diagram (see Fig.", "3 of the main text).", "This can be done by changing the ratio of $a/c^{2/3}$ without affecting $\\rho $ .", "Supplemental Materials In the main text, we encounter multiple times the linear stochastic differential equation (SDE) of the form $ \\frac{d \\vec{y}}{dt} = {A} \\vec{y} + \\vec{\\eta }(t),$ where ${A}$ is independent of $\\vec{y}$ and $\\vec{\\eta }$ are Gaussian white noises with zero mean and correlator $\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }^T(t^{\\prime }) \\rangle = {B} \\delta (t-t^{\\prime }).$ The noise matrix ${B}$ is symmetric (i.e ${B}^T = {B}$ ) and also supposed independent of $\\vec{y}$ .", "Equation (REF ) is tantamount to the Fokker-Planck equation for the probability density $P(\\vec{y}, t)$ [24]: $ \\frac{\\partial P(\\vec{y}, t)}{\\partial t} = - \\sum _{i,j} A_{ij} \\frac{\\partial }{\\partial y_i} (y_j P) + \\frac{1}{2} \\sum _{i,j} \\frac{\\partial ^2}{\\partial y_i \\partial y_j} (B_{ij} P).$ As shown in (e.g.)", "[25], the stationary distribution is Gaussian and takes the form $ P_s(\\vec{y}) = \\frac{1}{\\sqrt{\\det (2 \\pi {\\Xi })}} \\exp \\left( -\\frac{1}{2} \\vec{y}^T\\, {\\Xi }^{-1}\\, \\vec{y} \\right),$ where the covariance matrix ${\\Xi }$ is symmetric and given by the Sylvester's equation, $ {A} {\\Xi }+ {\\Xi }{A}^T + {B} = 0.$ In two dimensions, this equation can be solved [24] leading to an explicit formula for ${\\Xi }$ : ${\\Xi }= \\frac{\\left({A} - 1_2\\,\\text{tr} {A}\\right) {B}\\left(1_2\\,\\text{tr} {A} - {A} \\right)^T - {B}\\,\\det {A}}{2\\, \\text{tr} {A} \\,\\text{det} {A}}.$ We now wish to find an expression for the mean amplification factor, $\\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle $ , used in the main text to quantify the linear response of a stochastic reactive system.", "The norm of $\\vec{y}$ is the Euclidean norm $\\left\\Vert \\vec{y}\\right\\Vert = \\sqrt{\\sum _i \\left\|y_i^2\\right\|}$ .", "Specifically, we want to compute the integral: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\int _{\\mathbb {R}^D} d\\vec{y}\\, P_s(\\vec{y}) \\left\\Vert \\vec{y}\\right\\Vert ^2,$ where the distribution $P_s(\\vec{y})$ is given by Eq.", "(REF ).", "Therefore, $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\frac{1}{\\sqrt{\\det (2 \\pi {\\Xi })}} \\int d\\vec{y} \\exp \\left( -\\frac{1}{2} \\vec{y}^T\\, {\\Xi }^{-1}\\, \\vec{y} \\right) \\left\\Vert \\vec{y}\\right\\Vert ^2.$ To evaluate this integral, we use the identity $\\int \\left\\Vert \\vec{p}\\right\\Vert ^2 e^{-\\vec{p}^{\\,T} {M} \\vec{p}} d\\vec{p} = \\frac{1}{2} \\text{Tr}\\left({M}^{-1}\\right)\\int e^{-\\vec{p}^{\\,T} {M} \\vec{p}} d\\vec{p},$ with ${M} = 1/2{\\Xi }^{-1}$ , which yields the compact expression: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\, \\text{Tr}\\,{\\Xi }$ In the following, we assume for convenience that the noise matrix ${B}$ is a multiple of identity identity matrix 1 (${B} = \\sigma ^2 1$ ), a choice that can be made without losing in generality.", "In fact, since ${B}$ is symmetric, it is diagonalized by an orthogonal matrix which one can use to transform the noises; the resulting diagonal matrix can then be mapped to the identity matrix simply by rescaling the variables $\\vec{y}$ .", "Now, we will write the matrix ${\\Xi }$ in terms of ${A}$ and what we call the Hermitianizer of ${A}$ , defined as $ {G} = - \\frac{1}{2}\\,\\sigma ^2\\, {\\Xi }^{-1} {A}^{-1},$ which yields a symmetrization of matrix ${A}$ : even though ${A}$ is not symmetric, ${A} \\ne {A}^T$ , the product ${G} {A} = -2^{-1}\\sigma ^2 {\\Xi }^{-1}$ is a symmetric matrix.", "Sylvester equation (REF ) written in terms of ${G}$ simplifies to $ \\frac{1}{2} ({G}^{-1} + {G}^{-T}) = 1,$ indicating that the hermitian part of ${G}^{-1}$ is identity.", "Alternatively, the Hermitianizer of ${A}$ can be defined as the unique matrix satisfying Eq.", "(REF ) whose product with ${A}$ is Hermitian.", "Now we can write the mean squared value of the norm $\\vec{y}$ in terms of ${A}$ and ${G}$ by substituting Eq.", "(REF ) in Eq.", "(REF ): $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{1}{2} \\sigma ^2\\, \\text{Tr}\\left({A}^{-1} {G}^{-1}\\right)$ When ${A}$ is a $2\\times 2$ matrix, the trace of the inverse can be written as trace over determinant: $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{1}{2} \\sigma ^2\\, \\frac{\\text{Tr}\\left({G} {A}\\right)}{\\det ({G})\\det ({A})}$ $\\text{Tr}({G} {A})$ can be simplified by taking the trace of Eq.", "(REF ) $ \\text{Tr}({G}{A}) = -\\frac{1}{2}\\,\\sigma ^2\\, \\text{Tr}({\\Xi }^{-1}).$ Also, by multiplying the right-hand side of the Sylvester equation (REF ) by ${\\Xi }^{-1}$ : ${A} + {\\Xi }{A}^T {\\Xi }^{-1} = - \\sigma ^2\\,{\\Xi }^{-1}.$ and taking the trace we have (recalling that $ \\text{Tr}({\\Xi }{A}^T {\\Xi }^{-1})=\\text{Tr}({A}^T)= \\text{Tr}({A})$ ): $ \\sigma ^2\\,\\text{Tr}({\\Xi }^{-1}) = - 2\\, \\text{Tr}({A})$ From Eq.", "(REF ) and Eq.", "(REF ) it follows that $\\text{Tr}({G}{A}) = \\text{Tr}({A})$ .", "which we can use to simply Eq.", "(REF ): $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = -\\frac{\\,\\sigma ^2}{2\\,\\det {G}} \\frac{\\text{Tr}\\,{A}}{\\text{det}\\,{A}} = -\\frac{1}{2}\\,\\sigma ^2\\, \\det \\left({G}^{-1}\\right) \\text{Tr}\\left({A}^{-1}\\right).$ For a $2\\times 2$ matrix ${A}$ given by its elements ${A} = \\left(\\begin{array}{cc}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{array}\\right),$ we can solve for ${\\Xi }$ from Eq.", "(REF ) and substitute in Eq.", "(REF ) to find the matrix ${G}$ in terms of matrix elements of ${A}$ : ${G} = \\left(\\begin{array}{cc}\\frac{(a_{11}+a_{22})^2}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} & -\\frac{(a_{12}-a_{21}) (a_{11}+a_{22})}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} \\\\\\frac{(a_{12}-a_{21}) (a_{11}+a_{22})}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} & \\frac{(a_{11}+a_{22})^2}{(a_{12}-a_{21})^2+(a_{11}+a_{22})^2} \\\\\\end{array}\\right).$ The non-normality index $\\mathcal {H}$ is given by the inverse of the determinant of ${G}$ : $\\mathcal {H}({A}) = \\det \\left({G}^{-1}\\right) = 1 + \\frac{(a_{12}-a_{21})^2}{(a_{11}+a_{22})^2}.$ If the eigenvalues of ${A}$ are real, we can rewrite this expression in terms of the eigenvalues and the angle between the eigenvectors of ${A}$ .", "Let $\\Delta >0$ be the discriminant of the characteristic polynomial of ${A}$ : $\\Delta = (a_{11}-a_{22})^2 + 4\\, a_{12} \\, a_{21}.$ If $\\lambda _1$ and $\\lambda _2$ are the two eigenvalues of ${A}$ , and $\\vec{v}_1$ and $\\vec{v}_2$ are the two eigenvectors, we have $\\begin{split}&\\left(\\lambda _1 + \\lambda _2\\right)^2 = (a_{11}+a_{22})^2, \\qquad \\left(\\lambda _1 - \\lambda _2\\right)^2 = \\Delta , \\\\&\\cos ^2(\\theta ) = \\left(\\frac{\\vec{v}_1 \\cdot \\vec{v}_2}{\\left\\Vert \\vec{v}_1\\right\\Vert \\left\\Vert \\vec{v}_2\\right\\Vert }\\right)^2, \\qquad \\cot ^2(\\theta ) = \\frac{\\cos ^2(\\theta )}{1-\\cos ^2(\\theta )} = \\frac{ (a_{11}-a_{22})^2}{\\Delta }.\\end{split}$ Now it is clear that $\\mathcal {H}({A}) = 1 + \\cot ^2(\\theta )\\left( \\frac{ \\lambda _1 - \\lambda _2}{ \\lambda _1 + \\lambda _2}\\right)^2.$ Consider a similar set of SDEs of the the form $ \\frac{d \\vec{y}}{dt} = {A} \\vec{y} + \\vec{\\eta }(t),$ where now $\\vec{y}$ and $\\vec{\\eta }$ are vectors with complex variables, and $\\vec{\\eta }$ is a Gaussian white noise with zero mean and correlator $\\begin{split}&\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }\\,^\\dagger (t^{\\prime }) \\rangle = {B} \\delta (t-t^{\\prime }),\\\\&\\langle \\vec{\\eta }(t)\\, \\vec{\\eta }\\,^T(t^{\\prime }) \\rangle = 0.\\end{split}$ where the $^\\dagger $ symbol represents the transpose conjugate.", "The analysis in the previous section can be generalized by evaluating the expected value of $\\vec{y}(t) \\vec{y}\\,^\\dagger (\\tau )$ and $\\vec{y}(t) \\vec{y}\\,^T(\\tau )$ at steady state for $t = \\tau $ to obtain the following relationships for the covariance and relation matrices $\\begin{split}&{A}\\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle + \\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle {A}^\\dagger + {B} = 0\\\\&{A}\\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle + \\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle {A}^T = 0\\end{split}$ The first equation is the analogue of equation of Sylvester Eq.", "(REF ) for the Hermitian covariance matrix ${\\Xi }= \\left\\langle \\vec{y} \\vec{y}\\,^\\dagger \\right\\rangle $ , while the second equation implies that the symmetric relation matrix ${C} = \\left\\langle \\vec{y} \\vec{y}\\,^T \\right\\rangle $ is equal to zero.", "Therefore, at steady state, $\\vec{y}$ obeys a circularly symmetric complex Gaussian distribution of the form $P_s(\\vec{y}) = \\frac{1}{\\det (2 \\pi {\\Xi })} \\exp \\left( -\\frac{1}{2} \\vec{y}\\,^\\dagger \\, {\\Xi }^{-1}\\, \\vec{y} \\right).$ Notice the different normalization factor compared to Eq (REF ), as it is normalized over $\\mathbb {C}^D$ instead of $\\mathbb {R}^D$ .", "To compute the mean square value of the norm of $\\vec{y}$ , we can follow similar analysis to that of section REF .", "Here, we highlight the differences.", "The mean square norm is define as $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = \\int _{\\mathbb {C}^D} d\\vec{y}\\, P_s(\\vec{y}) \\left\\Vert \\vec{y}\\right\\Vert ^2,$ with the norm $\\left\\Vert \\vec{y}\\right\\Vert = \\sqrt{\\vec{y}\\,^\\dagger \\vec{y}}$ .", "The complex version of Eq.", "(REF ) can be evaluated by diagonalizing the matrix ${M}$ and write the integral on a $2D$ -dimensional real space.", "The result is given by $\\int _{\\mathbb {C}^D} \\left\\Vert \\vec{p}\\right\\Vert ^2 e^{-\\vec{p}^{\\,\\dagger } {M} \\vec{p}} d\\vec{p} = \\text{Tr}\\left({M}^{-1}\\right)\\int _{\\mathbb {C}^D} e^{-\\vec{p}^{\\,\\dagger } {M} \\vec{p}} d\\vec{p}\\,,$ where the factor $1/2$ is canceled by the fact that each eigenvalue of ${M}^{-1}$ should be counted twice in the $2D$ -dimensional space, once for the real part and once for the imaginary part.", "As a result, there will be an extra factor 2 in Eq.", "(REF ), Eq.", "(REF ), and Eq.", "(REF ).", "In particular , $ \\langle \\left\\Vert \\vec{y}\\right\\Vert ^2 \\rangle = - \\sigma ^2\\, \\text{Tr}\\left({A}^{-1} {G}^{-1}\\right)$ In this section we derive a the stochastic extension of the model by Ridolfi et al.", "[15] by expanding the master equation corresponding to the individual level model defined by the following set of reactions $\\begin{split}&2U_i+V_i\\xrightarrow{}3U_i+V_i,\\hspace{21.3513pt}V_i\\xrightarrow{}2V_i,\\\\&U_i\\xrightarrow{}\\varnothing ,\\hspace{21.3513pt}2V_i+2U_i\\xrightarrow{}V_i+2U_i,\\end{split}$ where $U_i$ and $V_i$ are the species $U$ and $V$ in the site $i$ , and the diffusion reactions $U_i\\xrightarrow{}U_j,\\hspace{21.3513pt}V_i\\xrightarrow{}V_j,\\hspace{21.3513pt} j\\in \\langle i\\rangle $ where $\\langle i\\rangle $ is the set of sites neighboring $i$ , $\\delta _u=D_U/\\Omega ^{2/D}$ , $\\delta _v=D_V/\\Omega ^{2/D}$ , $D_U$ and $D_V$ are the diffusion constants, and $\\Omega $ is the volume of each site.", "The state of the system is specified by the concentration vectors $\\vec{q}_i\\equiv (u_i, v_i) \\equiv (U_i, V_i)/\\Omega $ .", "Each reaction of reaction scheme (REF ) takes the system from a state $\\lbrace \\vec{q}_i\\rbrace $ to $\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace $ with probability per unit time $T(\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace \|\\lbrace \\vec{q}_i\\rbrace )$ .", "These transition rates are given from the law of mass action: $\\begin{split}&T\\left(\\left.", "\\vec{q}_i +\\vec{s}_1\\right\|\\vec{q}_i\\right) = \\Omega a u_i^2 v_i, \\hspace{12.80948pt}T\\left(\\left.", "\\vec{q}_i + \\vec{s}_2\\right\|\\vec{q}_i\\right) = \\Omega b v_i,\\\\&T\\left(\\left.", "\\vec{q}_i - \\vec{s}_1\\right\|\\vec{q}_i\\right) = \\Omega e u_i,\\hspace{12.80948pt}T\\left(\\left.", "\\vec{q}_i - \\vec{s}_2\\right\|\\vec{q}_i\\right) = \\Omega c u_i^2 v_i^2,\\\\\\end{split}$ and for every $j\\in \\langle i\\rangle $ $\\begin{split}&T\\left(\\left.", "\\vec{q}_i -\\vec{s}_1,\\vec{q}_j + \\vec{s}_1\\right\|\\vec{q}_i,\\vec{q}_j\\right) = \\Omega \\delta _u u_i,\\\\&T\\left(\\left.", "\\vec{q}_i -\\vec{s}_2,\\vec{q}_j + \\vec{s}_2\\right\|\\vec{q}_i,\\vec{q}_j\\right) = \\Omega \\delta _v v_i,\\end{split}$ where $\\vec{s}_1 = \\Omega ^{-1} \\left(\\begin{array}{c}1\\\\0\\end{array}\\right), \\hspace{21.3513pt}\\vec{s}_2 = \\Omega ^{-1} \\left(\\begin{array}{c}0\\\\1\\end{array}\\right).$ The master equation for the time evolution of the probability of finding the system at a state $\\lbrace \\vec{q}_i\\rbrace $ , $P(\\lbrace \\vec{q}_i\\rbrace ,t)$ can be written as $\\frac{\\text{d}{P(\\lbrace \\vec{q}_i\\rbrace ,t)}}{\\text{d}{t}} = \\sum _{\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace } \\left(T(\\lbrace \\vec{q}_i\\rbrace \|\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace )-T(\\lbrace \\vec{q}_i\\!^{\\prime }{}\\rbrace \|\\lbrace \\vec{q}_i\\rbrace )\\right)$ Following [22], we can expand the right hand side of Eq.", "(REF ) to second order in $\\Omega ^{-1}$ obtaining a Fokker-Planck equation corresponding the following set of stochastic differential equations $\\begin{split}&\\frac{\\text{d}{u_i}}{\\text{d}{t}} = u_i(a u_i v_i-e)+\\delta _u\\sum _{j\\in \\langle i\\rangle }(u_j-u_i)+\\xi _i(t),\\\\&\\frac{\\text{d}{v_i}}{\\text{d}{t}} = v_i(b-c u_i^2 v_i)+\\delta _v\\sum _{j\\in \\langle i\\rangle }(v_j-v_i)+\\eta _i(t),\\\\\\end{split}$ where $\\xi _i$ 's and $\\eta _i$ 's are zero mean Gaussian noise with correlations $\\begin{split}\\langle \\xi _i(t)\\xi _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Bigg (\\bigg ( u_i(a u_i v_i+e)+\\delta _u\\sum _{k\\in \\langle i\\rangle }(u_i+u_k)\\bigg )\\delta _{i,j}-\\delta _u(u_i+u_j){\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Bigg )\\\\\\langle \\eta _i(t)\\eta _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Bigg (\\bigg ( v_i(b+c u_i^2 v_i)+\\delta _v\\sum _{k\\in \\langle i\\rangle }(v_i+v_k)\\bigg )\\delta _{i,j}-\\delta _v(v_i+v_j){\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Bigg )}}and the characteristic function, {\\raisebox {}{\\chi }_{\\langle i\\rangle }, of \\langle i\\rangle is defined as{\\begin{equation}{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j) = {\\left\\lbrace \\begin{array}{ll}1 & j\\in \\langle i\\rangle \\\\0 & j\\notin \\langle i\\rangle \\end{array}\\right.}\\;.", "}\\end{equation}By defining \\vec{f}(\\vec{q}) \\equiv (f,\\; g) \\equiv ( u(a u v-e), \\;v(b-c u^2 v)), \\vec{\\xi }_i \\equiv (\\xi _i,\\; \\eta _i), {\\delta }\\equiv {diag}(\\delta _u,\\;\\delta _v), and \\left(\\Delta \\vec{q}\\right)_i \\equiv \\sum _{j\\in \\langle i\\rangle }(\\vec{q}_j-\\vec{q}_i), Eq.~(\\ref {eq:slangevin}) can be written in the simple form{\\begin{equation}\\frac{\\text{d}{\\vec{q}_i}}{\\text{d}{t}} = \\vec{f}(\\vec{q}_i)+{\\delta }\\left(\\Delta \\vec{q}\\right)_i+\\vec{\\xi }_i(t).\\end{equation}}Equation~(\\ref {eq:snonlinear}) is the discrete space version of Eq.~(\\ref {eq:nonlinear}) of the main text.", "Continuous limit can be taken at any point in the following analysis to recover the continuous space stochastic partial differential equations of type analyzed in the main text.", "We continue with the discrete version where the analytic results can be more readily compared to the simulation.", "}The deterministic part of our model has a fixed point \\vec{q}\\,^ \\equiv (u^,\\; v^) = (ba/ce,\\; e^2c/a^2b), obtained by setting \\vec{f}(\\vec{q}) equal to zero.", "We can linearize Eq.~(\\ref {eq:snonlinear}) around the fixed point \\vec{q}\\,^, by defining \\vec{p}_i \\equiv \\big ((u_i-u^)/\\sqrt{2u^e},\\,(v_i-v^)/\\sqrt{2v^b}\\big ) which are the rescaled deviations of \\vec{q}_i from \\vec{q}\\,^,{\\begin{equation}\\frac{\\text{d}{\\vec{p}_i}}{\\text{d}{t}} = {J}\\vec{p}_i+{\\delta }(\\Delta \\vec{p})_i + \\vec{\\xi }_i(t),\\end{equation}}where the linear stability operator {J} is defined as the Jacobian of the transformed function f at the fixed point \\vec{p} = 0 is given by{\\begin{equation}{J} = \\left(\\begin{array}{cc}e & \\frac{b^\\frac{3}{2}a^\\frac{3}{2}}{ce}\\\\-\\frac{2e^2 c}{a^\\frac{3}{2} b^{\\frac{1}{2}}} & -b\\end{array}\\right)\\end{equation}}Evaluating Eq.~(\\ref {eq:snoise1}) at \\vec{q}\\,^{\\begin{equation}\\begin{split}\\langle \\xi _i(t)\\xi _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Big (\\big ( 1+\\delta _u n/e\\big )\\delta _{i,j}-\\delta _u{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Big ),\\\\\\langle \\eta _i(t)\\eta _j(t^{\\prime }{})\\rangle &= \\frac{\\delta (t-t^{\\prime }{})}{\\Omega }\\Big (\\big ( 1+\\delta _v n/b\\big )\\delta _{i,j}-\\delta _v{\\raisebox {}{\\chi }_{\\langle i\\rangle }(j)\\Big ),}}\\end{split}where n \\equiv \\left\|\\langle i\\rangle \\right\| is the number of neighbors of each site.", "Note that for b>e, both of the eigenvalues of {J} have negative real parts, making \\vec{q}\\,^ an attractor of the dynamics in the absence of the diffusion.\\end{equation}To examine the spatial stability of \\vec{q}\\,^, we need to diagonalize the discrete Laplacian operator \\Delta , by defining the discrete Fourier transform of a sequence \\lbrace s_{\\vec{n}}\\rbrace as{\\begin{equation}\\tilde{s}_{\\vec{k}} \\equiv \\left(\\mathcal {F}[\\lbrace s_{\\vec{n}}\\rbrace ]\\right)_{\\vec{k}} \\equiv \\frac{1}{\\sqrt{N^D}}\\sum _{\\vec{n}} \\text{e}^{-2\\pi \\vec{k}.\\vec{n}/N}s_{\\vec{n}}.\\end{equation}}We drop the tildes on the Fourier variable with the convention that the variables with index k are Fourier variables.", "Equation~(\\ref {eq:slinear}) under this transformation becomes{\\begin{equation}\\frac{\\text{d}{\\vec{p}_{\\vec{k}}}}{\\text{d}{t}} ={K}\\vec{p}_{\\vec{k}} + \\vec{\\xi }_{\\vec{k}}(t), \\hspace{12.80948pt} {K} = {J}+\\Delta (\\vec{k}) {\\delta },\\end{equation}}where \\Delta (\\vec{k}) is the discrete Fourier transform of the discrete Laplacian operator given by{\\begin{equation}\\Delta (\\vec{k} ) \\equiv -2\\sum _{l = 1}^D\\big (1-\\cos (2\\pi k_l/N)\\big )\\end{equation}}and{\\begin{equation}\\begin{split}\\langle \\xi _{\\vec{k}}(t)\\xi _{\\vec{k}^{\\prime }{}}^(t^{\\prime }{})\\rangle = \\Omega ^{-1}\\left(1-e^{-1}\\delta _u\\Delta (\\vec{k})\\right)\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{}),\\\\\\langle \\eta _{\\vec{k}}(t)\\eta _{\\vec{k}^{\\prime }{}}^*(t^{\\prime }{})\\rangle =\\Omega ^{-1}\\left(1-b^{-1}\\delta _v\\Delta (\\vec{k})\\right)\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{}).\\\\\\end{split}\\end{equation}}}For the regime that we observe stochastic patterns, the contribution of the diffusion process in the amplitude of the noise in Eq.~(\\ref {eq:snoise2}) is very small and will be neglected for simplicity.", "This approximation is not necessary, since there is always a change of variables that simplifies the correlation matrix to a multiple of the identity matrix (this is the reason for the rescaling in the definition of \\vec{p}).", "With this approximation{\\begin{equation}\\left\\langle \\vec{\\xi }_{\\vec{k}}(t)\\vec{\\xi }_{\\vec{k}^{\\prime }{}}^{\\,\\dagger }(t^{\\prime }{})\\right\\rangle = \\Omega ^{-1}\\delta _{\\vec{k},\\vec{k}^{\\prime }{}}\\delta (t-t^{\\prime }{})\\,1\\end{equation}}where \\vec{\\xi }_{\\vec{k}^{\\prime }{}}^{\\,\\dagger } is the conjugate transpose of \\vec{\\xi }_{\\vec{k}^{\\prime }{}}, and 1 is the 2\\times 2 identity matrix.", "}\\end{split}$ The pattern forming behavior of the model defined by (REF ) can be understood by analyzing the eigenvalues of ${K}$ as a function of $\\vec{k}$ .", "Matrix ${K}$ can be written in elements from Eq.", "() and Eq.", "(): ${K} = \\left(\\begin{array}{cc}e + \\Delta (\\vec{k}) \\delta _u & \\frac{b^\\frac{3}{2}a^\\frac{3}{2}}{ce}\\\\-\\frac{2e^2 c}{a^\\frac{3}{2} b^{\\frac{1}{2}}} & -b + \\Delta (\\vec{k}) \\delta _v\\end{array}\\right)$ As it will become clear, most of the properties of the system depend on the following three parameters $\\rho = \\frac{b}{e},\\qquad \\nu = \\frac{e\\, c}{a^{\\frac{3}{2}} b^{\\frac{1}{2}}}, \\qquad r = \\frac{\\delta _v}{\\delta _u} = \\frac{D_V}{D_U}$ in the following analysis, we will write various expression in terms of these parameters, wherever we can.", "We start with ${K}$ ${K} = \\left(\\begin{array}{cc}e + \\Delta (\\vec{k}) \\delta _u & b/\\nu \\\\-2 e \\,\\nu & -b + \\Delta (\\vec{k}) \\delta _v\\end{array}\\right)$ The largest eigenvalue of ${K}$ is given by $\\lambda (\\vec{k}) = \\frac{1}{2} \\left(\\sqrt{b^2-2 b \\Delta (\\vec{k}) (\\delta _v-\\delta _u)-6 b e+\\left(e-\\Delta (\\vec{k}) (\\delta _v-\\delta _u)\\right)^2}-b+\\Delta (\\vec{k}) (\\delta _v+\\delta _u)-e\\right).$ Notice that the eigenvalues of ${K}$ are independent of $\\nu $ .", "For small $\\vec{k}$ , $\\Delta (\\vec{k})$ is a monotonically decreasing function of $\\vec{k}$ (proportional to $-k^2$ ).", "We define $y = -\\Delta (\\vec{k})$ .", "To determine if $\\lambda $ monotonically decays or if it has a maximum at some $\\vec{k}_0 \\ne 0$ , we can differentiate $\\lambda $ with respect to $y$ and see if it has a positive root.", "The largest root of $\\frac{\\text{d}{\\lambda }}{\\text{d}{y}}$ is given by $y_0 = -\\Delta (\\vec{k}_0) = \\frac{ (r+1) \\sqrt{2\\, b\\, e\\, r}-b\\, r-e\\, r}{\\delta _u \\,(r-1)\\, r}.$ For $y_0$ to be greater than zero we need $\\rho < \\frac{\\left(1+ r+r^2+(r+1) \\sqrt{r^2+1}\\right)}{r}.$ We can find the condition on the ratio of the diffusion constants by inverting this inequality: $r > \\frac{1-2\\, \\rho + \\rho ^2+(1+ \\rho ) \\sqrt{1 + \\rho \\, (\\rho -6)}}{4 \\,\\rho } = f_1(\\rho ).$ The condition for formation of stochastic pattern is $\\lambda (\\vec{k}_0) > \\Re (\\lambda (0))$ .", "We can find $\\lambda (\\vec{k}_0)$ and $\\lambda (0)$ by substituting $y_0 = y(\\vec{k}_0)$ from Eq.", "(REF ) and $y(0) = 0$ in Eq.", "(REF ): $\\lambda (\\vec{k}_0) = \\frac{b+e\\, r- \\sqrt{8\\,b\\, e\\, r}}{r-1}, \\qquad \\lambda (0) = \\frac{1}{2} \\left(\\sqrt{b^2-6\\, b\\, e+e^2}-b+e\\right).$ Then, $\\lambda (\\vec{k}_0) > \\Re (\\lambda (0))$ simplifies to $r > \\frac{- 1 + 14\\, \\rho - \\rho ^2+4 \\sqrt{-2\\,\\rho \\, (1 + \\rho \\, (\\rho -6)))}}{(1 + \\rho )^2} = f_2(\\rho ).$ Condition for deterministic Turing pattern is a lot simpler; we just need $\\lambda (\\vec{k}_0) > 0$ which simplifies to $r > \\left(3+2 \\sqrt{2}\\right) \\rho = f_3(\\rho ).$ When $r$ is greater than $f_1(\\rho )$ and $f_2(\\rho )$ but less than $f_3(\\rho )$ , the system exhibits stochastic patterns (blue region in Fig.", "3 of the main text), while we observe the deterministic patterns when $r$ is greater than $f_3$ (orange region of Fig.", "3 of the main text).", "The amplification of our stochastic patterns depend on the non-normality index of ${K}_0 = {K}(\\vec{k}_0)$ given by ${K}_0 = \\left(\\begin{array}{cc}e - y_0\\, \\delta _u & b/\\nu \\\\-2\\, e \\,\\nu & -b - y_0\\, \\delta _v\\end{array}\\right),$ where $y_0 = -\\Delta (\\vec{k}_0)$ .", "We use Eq.", "(REF ) to calculate the non-normality index of ${K}_0$ : $\\mathcal {H}({K}_0) = 1 + \\left(\\frac{b+2 e \\nu ^2}{\\nu (b-e+y_0\\, (\\delta _u+\\delta _v))}\\right)^2.$ We substitute $y_0$ from Eq.", "(REF ) and rewrite the resulting expression in terms of $\\rho $ , $r$ , and $\\nu $ : $\\mathcal {H}({K}_0) = 1+\\left(\\frac{2 \\nu ^2+\\rho }{\\nu \\left(\\rho - 1 +\\frac{(r+1) \\left(-\\rho r+ (r+1) \\sqrt{2\\rho \\, r}-r\\right)}{(r-1) r}\\right)}\\right)^2$ Since the eigenvalues of ${K}$ do not depend on $\\nu $ , one can change $\\mathcal {H}({K}_0)$ by changing $\\nu $ without moving the system in its phase diagram (see Fig.", "3 of the main text).", "This can be done by changing the ratio of $a/c^{2/3}$ without affecting $\\rho $ ." ] ]	1606.04916
[ [ "Network Valuation in Financial Systems" ], [ "Abstract We introduce a general model for the balance-sheet consistent valuation of interbank claims within an interconnected financial system.", "Our model represents an extension of clearing models of interdependent liabilities to account for the presence of uncertainty on banks' external assets.", "At the same time, it also provides a natural extension of classic structural credit risk models to the case of an interconnected system.", "We characterize the existence and uniqueness of a valuation that maximises individual and total equity values for all banks.", "We apply our model to the assessment of systemic risk, and in particular for the case of stress-testing.", "Further, we provide a fixed-point algorithm to carry out the network valuation and the conditions for its convergence." ], [ "Introduction", "Uncertainty and interdependence are two fundamental features of financial systems.", "While uncertainty over the future value of assets is traditionally very central in the financial literature [24], interdependence of financial claims' values, especially in interconnected banking systems, has been investigated only more recently [25], [19], [15], [2], [17], taking center stage mostly after the recent financial crisis [27], [16], [1], [9].", "When the two features are considered together, the valuation of interdependent claims at a given time with arbitrary maturity is, in general, a non-trivial problem [15] with crucial policy implications.", "The seminal work of [15] (EN), which has been very influential in the literature on interbank contagion and systemic risk, has developed a framework to deal with the problem of interdependence in the absence of ex-ante uncertainty and bankruptcy costs.", "Their main result is that, in the case in which contracts are interbank debt securities, mild conditions on the network topology, and a simple maturity structure, it is possible to uniquely determine the so-called clearing payment vector, i.e.", "how much each bank is required to pay to the other banks in order to maximise the total payments in the system.", "A more recent work [26] has extended the EN model by showing that, despite the fact that bankruptcy costs may imply multiple solutions, one can still uniquely determine the solution that is preferable for all banks.", "An important limitation of the EN framework is that the valuation of claims is carried out ex-post, i.e.", "at the maturity of the contracts, once the amount of external funds of each bank is known.", "One only needs to compute how losses, if any, should be redistributed among the surviving financial institutions.", "While EN emphasized the importance of moving towards an ex-ante valuation accounting for uncertainty (deriving, e.g., from possible shocks or cashflows between the valuation date and the maturity date), and despite the great interest in financial contagion spurred by the 2008 crisis, most works on systemic risk and stress-testing of interbank networks have focused on the original ex-post framework in which valuation and maturity times coincide [13], [17], [26], [22], [1].", "Therefore, a gap has emerged between this growing body of works on interconnectedness with ex-post valuation and the vast literature on the ex-ante valuation of corporate obligations.", "The latter, building on the classic Merton approach [24], deals with the problem of ex-ante valuation in the presence of uncertainty but does not encompass interdependence between claims, with a few exceptions [28], [14], [18], [23].", "The framework introduced in [28] applies also to the case in which cross-ownerships exist and [18] further generalizes it to the case of obligations with different seniorities and derivatives.", "However, such results rely on two crucial assumptions.", "First, the payments made by each firm are a continuous function of the payments made by all firms.", "The assumption of continuity rules out the presence of mechanisms such as costs of default and abrupt loss of values in assets due to fire sales.", "Second, for any given level of seniority of the cross-holdings of debt each firm must also hold external liabilities with the same seniority.", "Moreover, in the approach of [28] and [18], in order to carry out an ex-ante valuation of the claims on a given institution (under the assumption of no arbitrage opportunities and market completeness), one needs to compute all possible trajectories of the stochastic processes followed by the assets of all institutions involved.", "In other words, this approach can be thought of as global valuation mechanism, since it implicitly assumes that the computation can be carried out by an entity with full knowledge of all the parameters of the financial system including all interbank exposures.", "In contrast, as we show, the computation of the EN solution requires banks to know only local information about their counterparties at each step, although this holds at the expenses of not accounting for uncertainty.", "While such a global valuation approach is of great theoretical interest, it remains unclear whether the computation could be decentralized and therefore how feasible it would be its application as a valuation mechanism.", "Another gap in the literature has also emerged between the theoretical insights from the stress-testing exercises based on the EN approach and the experience of practitioners and policy makers.", "Indeed, according to the BIS, the largest part of losses suffered by financial institutions during the financial crisis was not due to actual counterparties' defaults, but to the mark-to-market revaluation of obligations following the deterioration of counterparties' creditworthiness.", "This approach is called Credit Valuation Adjustment (CVA)The Basel Committee on Banking Supervision states that “roughly two-thirds of losses attributed to counterparty credit risk were due to CVA losses and only about one-third were due to actual defaults.” [6]..", "This means that, while in the EN framework the default of a bank is the only event that matters for triggering losses down the chain of lending contracts, in practice also the deterioration of a bank's book matters for triggering those losses.", "Currently, this mechanism is not taken into account by most models of systemic risk [20], [21], [29], and in particular by all those based on the EN model, although its importance has been increasingly acknowledged [22].", "While the framework of DebtRank [10], [7] is one of the few models building on the idea of distress propagating even in the absence of defaults, its current formulation does not provide a consistent endogenous treatment of the recovery rate.", "In light of the above considerations, in this paper we develop a novel general framework, which we refer to as Network Valuation Model (NEVA), for the valuation of claims among institutions interconnected within a network of liabilities, with the following characteristics.", "Similarly to EN, it is possible to endogenously determine a set of consistent values for the claims, following the default of some institutions.", "Differently from EN, we account for ex-ante uncertainty on the values of external assets, arising when the valuation is carried out at some time before the maturity of the claims, as commonly done in practice.", "From this point of view, while our approach is perfectly compatible with a global valuation mechanism (as in [28], [18]), it also provides a decentralized valuation mechanism in which banks perform an ex-ante valuation of their claims in a decentralized fashion.", "In other words, the financial system as a whole performs the valuation recursively via a distributed mechanism through which each bank only needs information about the valuation of the claims of its own counterparties at each step of the calculation.", "Finally, in contrast with [28] and [18] we avoid specific assumptions about the continuity of the valuation performed by banks, therefore allowing for the possibility to account for costs of default and fire sales, e.g.", "as in [26].", "More in detail, the timing of the framework can be best conceived of in four steps.", "At time zero all contracts are set, resulting from various possible investment allocation strategies, which are not explicitly modelled here and are not relevant to our results.", "At time one there is a shock on one or more of the external assets of the banks.", "At time $t\\ge 1$ the valuation is carried out, while contracts mature at time $T \\ge t$ .", "The framework also allows for contracts with multiple maturities.", "Between $t$ and $T$ possible changes in the value of external assets are accounted for.", "Remarkably, on the one hand, we obtain the ex-post approach of Eisenberg and Noe, as well as the Rogers and Veraart extension as limit cases of the NEVA model when the time of the valuation is assumed to be the same as the time of the maturity and there is no uncertainty on the value of the external assets held by banks.", "On the other hand, the classic ex-ante valuation Merton approach can also be obtained as a limit case in which there are no interbank claims and external assets follow a geometric Brownian motion up to maturity.", "More interestingly, it is possible to extend the EN decentralized computation of consistent valuations to the case in which valuations are consistent expected values of the claims under local knowledge of the shock distribution.", "In particular, we show that the DebtRank model [10], [8], [3] is obtained as a limit case in which shocks on external assets before the maturity follow a uniform distribution.", "We characterize the existence and uniqueness of the solutions of the valuation problem under general conditions on the functional form of the valuation function, i.e.", "on how the value of each claim depends on the equity of the counterparty.", "Further, we define an algorithm to carry out the network valuation and provide sufficient conditions for convergence in finite time to the greatest solution (in the sense that the equity of each banks is greater or equal than in the other solutions) with a given precision.", "Finally, under additional assumptions, we show that the solutions of parametric families of ex-ante NEVAs models (i.e.", "before the maturity) smoothly converge to the solutions of the corresponding ex-post NEVA models (i.e.", "at the maturity)." ], [ "Framework and Definitions", "We consider a financial system consisting of $n$ institutions (for brevity “banks” in the following) engaging in credit contracts with some others.", "Our goal is to set out a general framework (NEVA) in which banks can evaluate their own interbank claims on other banks (and thus their own equity) in a network of liabilities, by taking into account simultaneously the claims of all the banks in the network.", "Credit contracts are established at time 0 and are taken as given, with $L_{ij}$ denoting the book value of the debt of bank $i$ towards bank $j$ , and $A_{ji}$ denoting the book value of the corresponding asset of bank $j$ , with $A_{ji}=L_{ij}$ .", "Banks also have assets and liabilities external to the interbank system, which we denote respectively with $A_i^{e}$ and $L_i^{e}$ .", "The external assets of banks are subject to stochastic shocks, and this is the only source of stochasticity in the model.", "We denote by $T_{ij}$ the maturity of the contract between $i$ and $j$ (with respect to a reference time zero) and by $t$ the time at which the evaluation of the financial claims is carried out.", "In the special case in which $t = T_{ij}$ , for all $i$ , $j$ , the evaluation takes place at maturity.", "This is precisely the case considered in the ex-post clearing procedure in most works based on [15].", "However, more in general, it is of practical interest the case in which the evaluation takes place before the maturity, hereafter referred to as ex-ante evaluation.", "In this case, we want to determine at $t < T$ the value of banks' liabilities knowing the underlying distribution of shocks that could affect banks external assets between $t$ and $T$ .", "We denote the book value of the equity of bank $i$ , i.e.", "the difference between its total assets and liabilities taken at their book (face) value as $M_i$ : $ M_i = A_i^e - L_i^e + \\sum _{j=1}^n A_{ij} - \\sum _{j=1}^n L_{ij} \\, .$ However, a proper valuation of equity of bank $i$ , denoted here with $E_i$ , will depend on how much bank $i$ values its own assets.", "Such valuation can markedly differ from the face value, and will certainly depend on several parameters associated with specific contracts.", "Most importantly, it will depend on other banks' equities, or more precisely on their valuation of their equities.", "For instance, one can expect the value of an asset corresponding to a loan extended from bank $i$ to bank $j$ to depend on both $E_i$ and $E_j$ and, reasonably, larger values of equities will imply larger valuations of the assets.", "Such intuition is formalised in the following definition: Definition 1 Given an integer $q \\le n$ , a function $\\mathbb {V}: \\mathbb {R}^q \\rightarrow [0, 1]$ is called feasible valuation function if and only if: it is nondecreasing: $\\mathbf {E} \\le \\mathbf {E}^\\prime $ $\\Rightarrow $ $\\mathbb {V}(\\mathbf {E}) \\le \\mathbb {V}(\\mathbf {E}^\\prime ), \\forall \\mathbf {E}, \\mathbf {E}^\\prime \\in \\mathbb {R}^q$ it is continuous from above.", "The general idea behind the definition of feasible valuation function is that one can write the value of any asset as the product of its face value times a valuation function, such that it ranges from the face value of the asset to zero.", "In general, the valuation performed by bank $i$ of its claim against bank $j$ , under the assumption that such valuation depends only on banks' equities and other parameters associated with their specific contract, is: $A_{ij} \\mathbb {V}_{ij}(\\mathbf {E}\| \\mathbf {\\alpha }) \\, ,$ where $\\mathbb {V}_{ij}: \\mathbb {R}^n \\rightarrow [0, 1]$ is a valuation function and $\\mathbf {\\alpha }$ is a set of parameters.", "Analogously, we can write the value of external assets of bank $i$ assuming that it is function of equities (and additional parameters): $ A_i^e \\mathbb {V}_i^e (E_i \| \\mathbf {\\alpha }) \\, ,$ where $\\mathbb {V}_i^e: \\mathbb {R} \\rightarrow [0, 1]$ is a valuation function.", "In the remainder of this paper we mostly focus on examples in which the valuation function of external assets has the form (REF ) and in which the valuation function of interbank assets depends on the equity of the borrower, i.e.", "$\\mathbb {V}_{ij}(\\mathbf {E} \| \\ldots ) = \\mathbb {V}_{ij}(E_j \| \\ldots )$ .", "However, we point out that all the results that we derive still hold in the more general case in which the valuation functions depend on all the equities.", "Each bank will assess the value of its equity at time $t$ as the difference between the valuation of its assets minus the value of its liabilities: $ E_i(t) = A_i^e \\mathbb {V}_i^e (\\mathbf {E}(t) \| \\ldots ) - L_i^e + \\sum _{j=1}^n A_{ij} \\mathbb {V}_{ij}(\\mathbf {E}(t) \| \\ldots ) - \\sum _{j=1}^n L_{ij}\\quad \\forall i \\, ,$ where, as customary, we consider all liabilities to be fixed at their book value.", "Since all valuation functions take values in the interval $[0, 1]$ , equities $E_i(t)$ are bounded both from below and from above: $m_i \\equiv - L_i^e - \\sum _{j=1}^n L_{ij} \\le E_i(t) \\le M_i \\quad \\forall i \\, .$ By introducing the following map: $\\Phi \\; : \\; \\varprod _{i=1}^n [m_i,M_i] \\rightarrow \\varprod _{i=1}^n [m_i,M_i]$ $ \\Phi _i(\\mathbf {E}(t)) = A_i^e \\mathbb {V}_i^e (\\mathbf {E}(t) \| \\ldots ) - L_i^e + \\sum _{j=1}^n A_{ij} \\mathbb {V}_{ij}(\\mathbf {E}(t) \| \\ldots ) - \\sum _{j=1}^n L_{ij} \\quad \\forall i \\, ,$ the set of equations (REF ) can be rewritten in compact form: $ \\mathbf {E}(t) = \\Phi (\\mathbf {E}(t)) \\, .$ The map $\\Phi $ allows each bank to compute its own equity given the equities of all the banks in the network.", "Such valuations are self-consistent only for the equity vectors $\\mathbf {E}(t)$ that satisfy relation (REF ).", "In order to implement a consistent network-based valuation of interbank claims it is essential to prove the existence of solutions of (REF ).", "For the sake of readability in the following we will drop the explicit dependence of equities on the time $t$ at which the valuation is performed." ], [ "Main results", "We now outline the most general results, which apply to generic feasible valuation functions.", "Theorem 1 (Existence) The set of solutions of (REF ) is a complete lattice.", "This implies in particular that the set of solutions is non-empty and that there exist a least $\\mathbf {E}^-$ and greatest solution $\\mathbf {E}^+$ such that for any solution $\\mathbf {E}^$ , $E^-_i\\le E^_i\\le E^+_i$ , for all $i$ .", "Within the set of solutions, the greatest solution is the most desirable outcome for all banks, as it simultaneously minimizes losses for all of them.", "Understanding how to compute such solution is therefore of paramount importance.", "Let us explicitly note that every solution $\\mathbf {E}^$ of (REF ) corresponds to a fixed point of the iterative map $ \\mathbf {E}^{(k+1)} = \\Phi (\\mathbf {E}^{(k)}) \\, ,$ and viceversa.", "Eq.", "(REF ) defines the usual Picard iteration algorithm (called “fictitious sequential default algorithm” in [15]) and in principle provides a method to compute the solutions with arbitrary precision, as we will show in the following.", "Iterating the map starting from an arbitrary $\\mathbf {E}^{(0)}$ does not guarantee that the solutions $\\mathbf {E}^+$ and $\\mathbf {E}^-$ can be attained.", "In fact different solutions of (REF ) can be found depending on the chosen starting point.", "Moreover, some solutions might be unstable, in the sense that, while still satisfying (REF ), choosing a starting point for Picard iteration algorithm arbitrary close to (but not equal to) such solutions, will result in the iterative map converging to another solution of (REF ).", "The problem of finding the least and greatest solution this problem is solved by the following theorems: Theorem 2 (Convergence to the greatest solution) If ${E}^{(0)} = {M}$ : the sequence $\\lbrace {E}^{(k)} \\rbrace $ is monotonic non-increasing: $\\forall k \\ge 0$ , ${E}^{(k+1)} \\le {E}^{(k)}$ , the sequence $\\lbrace {E}^{(k)} \\rbrace $ is convergent: $\\lim _{k\\rightarrow \\infty } {E}^{(k)} = {E}^{\\infty }$ , ${E}^{\\infty }$ is a solution of (REF ) and furthermore ${E}^{\\infty } = {E}^+$ .", "Theorem REF shows that, if the starting point of the iteration is $\\mathbf {E}^{(0)} = \\mathbf {M}$ , which corresponds to taking all assets at their face value, the iterative map (REF ) converges to the greatest solution $\\mathbf {E}^+$ .", "Theorem REF guarantees that for all $\\epsilon > 0$ , there exists $K(\\epsilon )$ such that for all $k > K(\\epsilon )$ we have that $\|\| \\mathbf {E}^{(k)} - \\mathbf {E}^+ \|\| < \\epsilon $ .", "In other words, once a precision $\\epsilon $ has been chosen, starting from the face values of equities $\\mathbf {M}$ , and after a finite number of iterations, the Picard algorithm provides equities (REF ) that are undistinguishable from the greatest solution, within precision $\\epsilon $ .", "Mutatis mutandis, it is possible to prove that: Theorem 3 (Convergence to the least solution) If ${E}^{(0)} = {m}$ and the valuation functions in $\\Phi $ are continuous from below, then: the sequence $\\lbrace {E}^{(k)} \\rbrace $ is monotonic non-decreasing: $\\forall k \\ge 0$ , ${E}^{(k+1)} \\ge {E}^{(k)}$ , the sequence $\\lbrace {E}^{(k)} \\rbrace $ is convergent: $\\lim _{k\\rightarrow \\infty } {E}^{(k)} = {E}^{\\infty }$ , ${E}^{\\infty }$ is a solution of (REF ) and furthermore ${E}^{\\infty } = {E}^-$ .", "Analogous results to the ones proved after Theorem REF also hold in this case.", "Therefore, Theorems REF and REF , provide a simple algorithmic way to check whether the solution of (REF ) is unique within numerical precision: Corollary 1 (Uniqueness) If ${E}^+ = {E}^-$ , the solution of (REF ) is unique.", "Let us now put these results in the context of the existing literature.", "In order to prove the existence of a solution, [28] and [18] exploit the Brouwer-Schauder fixed point theorem, which requires payments made by each firm to be a continuous function of the payments made by all firms.", "The assumption of continuity does not allow to account for default costs.", "However, in [28] and [18] the iterative map is not required to be monotonic, allowing to model some derivatives having a specific functional form.", "Since the Brouwer-Schauder fixed point theorem does not give any information about the structure of the solution space (e.g.", "the existence of a greatest and a least solution) it is important to have a unique solution.", "In order to prove uniqueness [28] and [18] resort to the additional hypothesis that the ownership matrix (the analogous of our matrix $A_{ij}$ ) is strictly left substochastic, meaning that for any given level of seniority of the cross-holdings of debt each firm must also hold external liabilities with the same seniority.", "Here we use instead the Knaster-Tarski fixed point theorem, which requires valuation function to be monotonic – preventing a straightforward modeling of derivatives – and not necessarily continuous.", "As a consequence, default costs and analogous mechanisms can be easily accommodated in our framework (see Sec.", ").", "Through the Knaster-Tarski fixed point theorem we prove, not only the existence of a solution, but also the existence of a greatest and a least solution.", "Remarkably, Theorem REF shows that the greatest solution is attained if the starting point of the valuation is the face value of claims, providing a clear prescription to perform the valuation even when multiple solutions exist." ], [ "Results on Directed Acyclic Graphs (DAGs)", "Proof of the existence of an algorithm that ensures the convergence to a solution in a finite time are usually based on assumptions on the form of the valuation function [23].", "In contrast, here we show that such result holds for a specific topology of the network of interbank liabilities, namely a DAG (Directed Acyclic Graph), regardless of the functional form of the interbank valuation functions.", "Proposition 1 (DAG) If the matrix defined by interbank assets $A_{ij}$ is the adjacency matrix of a DAG and $\\mathbb {V}_i^e(E) = 1$ , $\\forall i$ : the map (REF ) converges in a finite number of iterations, the solution of (REF ) is unique.", "We define source banks as those banks that do not hold interbank assets, i.e.", "$S_0=\\lbrace i :\\, A_{ij}=0, \\forall j \\rbrace $ , which is a non-empty set if the matrix of interbank exposures is a DAG.", "We then partition banks based on the maximum graph distance from the set of source banks $S_0$ , the partition being $\\lbrace S_d\\rbrace _{d=0}^{d_{\\mathrm {max}}}$ .", "Starting from the initial condition $\\mathbf {M}$ , banks in $S_0$ converge in zero iterations to their face value as their equity does not depend on the equity of any other bank (neither their own).", "Banks in $S_1$ converge in one iteration as their equity only depends on the equities of banks in $S_0$ .", "By induction, banks in $S_{d_{\\mathrm {max}}}$ converge in $d_{\\mathrm {max}}$ iterations.", "Starting from the initial condition $\\mathbf {m}$ banks in $S_0$ converge in one iteration to their face value as the Picard iteration algorithm corrects the value of their equities exactly in one iteration.", "Consequently, $\\Phi ^{(d_{\\mathrm {max}})}(\\textbf {M})=\\Phi ^{(d_{\\textrm {max}}+1)}(\\textbf {m})$ , and therefore all banks converge to $\\textbf {E}^-=\\textbf {E}^+$ in (at most) $d_{\\mathrm {max}} + 1$ iterations." ], [ "Examples", "We now highlight the generality of the NEVA outlined in Section by presenting a few relevant examples.", "More specifically, we show that four different models well known in the literature about systemic risk can be recovered as limit cases.", "Proposition 2 (Eisenberg and Noe) If: $\\mathbb {V}_i^e(E_i) = 1$ , $\\forall i$ , $\\mathbb {V}_{ij}(E_j) = \\mathbb {1}_{E_j \\ge 0} + \\left( \\frac{E_j + \\bar{p}_j}{\\bar{p}_j} \\right)^+ \\mathbb {1}_{E_j<0}$ , $\\forall i$ , $j$ , there is a one-to-one correspondance between the solutions of (REF ) and the solutions of the map $\\Phi $ introduced in [15].", "As already noted, since in EN the evaluation happens at maturity, $t = T_{ij}$ for all $i$ , $j$ .", "Under the assumptions of (i) limited liabilities, (ii) priority of debt over equity, (iii) proportional repayments, EN aims at computing a clearing payment vector $\\mathbf {p}^$ whose component $p_i^$ is the total payment made by bank $i$ to its counterparties.", "To conform to their notation, we also introduce the obligation vector $\\bar{\\mathbf {p}}$ , defined as $\\bar{p}_i = \\sum _j L_{ij}$ , which is the total interbank liability that bank $i$ needs to settle.", "[15] show that: $ p_i^ = \\min \\left[ e_i + \\sum _j L_{ji} \\frac{p_j^}{\\bar{p}_j}, \\bar{p}_i \\right] \\, ,$ where $e_i = A_i^e - L_i^e$ , and external liabilities are also due at the same maturity of interbank liabilities.", "Eq.", "(REF ) can be equivalently rewritten as: $ p_i^ = \\bar{p}_i \\mathbb {1}_{E_i \\ge 0} + \\left( E_i(\\mathbf {p}^{}) + \\bar{p}_i \\right)^+ \\mathbb {1}_{E_i<0} \\, ,$ with $ E_i(\\mathbf {p}) = A_i^e - L_i^e + \\sum _{j} A_{ij} \\frac{p_j}{\\bar{p}_j} - \\sum _j L_{ij} \\, ,$ where $\\mathbb {1}_{x>0}$ is the indicator function relative to the set defined by the condition $x>0$ and $(x)^+ = (x + \|x\|) / 2$ .", "The above equations are equivalent to (REF ) by choosing the valuation functions as in the hypotheses of the Proposition REF .", "In fact, when $E_j > 0$ , the cash inflow of bank $j$ is enough to cover its due payments, and therefore $\\bar{\\mathbf {p}} = \\mathbf {p}^$ .", "In contrast, when $E_j < 0$ , bank $j$ employs its residual assets $\\left( E_j + \\bar{p}_j \\right)^+$ to repay its creditors proportionally as much as it can.", "Proposition 3 (Rogers and Veraart) If: $\\mathbb {V}_i^e(E_i) = \\mathbb {1}_{E_i\\ge 0} + \\alpha \\mathbb {1}_{E_i<0}$ , $\\forall i$ , $\\mathbb {V}_{ij}(E_i, E_j) = \\left[ \\mathbb {1}_{E_i\\ge 0} + \\beta \\mathbb {1}_{E_i<0} \\right] \\left[ \\mathbb {1}_{E_j \\ge 0} + \\left( \\frac{E_j + \\bar{p}_j}{\\bar{p}_j} \\right)^+ \\mathbb {1}_{E_j<0} \\right]$ , $\\forall i$ , $j$ , there is a one-to-one correspondance between the solutions of (REF ) and the solutions of the map $\\Phi $ introduced in [26].", "The proof is entirely analogous to the proof of Proposition REF .", "Let us note that, if $\\alpha < 1$ ($\\beta < 1$ ), then $\\mathbb {V}_i^e$ ($\\mathbb {V}_{ij}$ ) is not a continuous function.", "In particular, a value of $\\alpha $ ($\\beta $ ) strictly smaller than one means that when a bank defaults its external (interbank) assets will suddenly experience a relative loss of $\\alpha - 1$ ($\\beta - 1$ ), due e.g.", "to the necessity to liquidate them in a fire sale.", "Proposition 4 (Furfine) If: $\\mathbb {V}_i^e(E_i) = 1$ , $\\forall i$ , $\\mathbb {V}_{ij}(E_j) = \\mathbb {1}_{E_j \\ge 0} + R\\mathbb {1}_{E_j < 0}$ , $\\forall i$ , $j$ , there is a one-to-one correspondence between the solutions of (REF ) and the solutions of the map $\\Phi $ introduced in [20].", "According to the Furfine algorithm a counterparty with non-negative equity is always able to fully repay its liabilities, while, if its equity is negative, it will only repay a fraction $R$ of them.", "This is exactly what the valuation function in Proposition REF accounts for.", "Proposition 5 (Linear DebtRank) If: $\\mathbb {V}_i^e(E_i) = 1$ , $\\forall i$ , $\\mathbb {V}_{ij}(E_j) = \\frac{E_j^+}{M_j}$ , $\\forall i$ , $j$ , there is a one-to-one correspondance between the solutions of (REF ) and the solution of recursive map (linear DebtRank) introduced in [3].", "The easiest way to prove the correspondence is to compute the incremental variation of the iterative map (REF ), which in this case reads: $E_i^{(k+1)} - E_i^{(k)} = \\sum _j A_{ij} \\frac{\\left(E_j^{(k)}\\right)^+ - \\left(E_j^{(k-1)}\\right)^+}{M_j}$ , for all $i$ .", "Starting the Picard iteration algorithm from $\\mathbf {M}$ we recover (7) in [3], in which $\\mathbf {M}$ has been denoted with $\\mathbf {E}(0)$ .", "As soon as the equity of bank $j$ becomes equal to zero in the iterative map in [3], it will not change anymore, which is consistent with the incremental variation derived above.", "DebtRank has been introduced in [10] as an effective model to propagate shocks in the interbank network.", "Subsequently, a generalization of the model (which we call linear DebtRank here) has been derived from the balance sheet identity and from simplified assumptions about the propoagation of distress in the interbank market in [3].", "The model has been further extended in [5], [4].", "In Fig.", "REF we plot several interbank valuation functions: EN (see Proposition REF ), Furfine (see Proposition REF ), Linear DebtRank (see Proposition REF ), and ex-ante EN, which will be introduced in Sec.", ".", "Figure: Interbank valuation functions as a function of the equity of the borrower.", "Parameters as follows.", "EN: p ¯=2\\bar{p} = 2, Furfine: R=1R = 1, Linear DebtRank: M=2.5M = 2.5, Ex-ante EN: A e =1A^e = 1, p ¯=2\\bar{p} = 2, β=1\\beta = 1, σ=1\\sigma = 1." ], [ "An application: Eisenberg-Noe with ex-ante valuation", "On one hand, as already remarked, EN allows to compute the payments banks have to make to their counterparties.", "From Proposition REF it is clear that to compute such payments one needs to know the values of all equities, thus implying that such computation should happen at the maturity.", "On the other hand, the evaluation of corporate debt before the maturity done by means of the Merton model [24] does not account for the recursive evaluations that are needed when creditors and debtors form a complex interconnected network.", "The aim of this section is to bridge this gap by introducing a set of valuation functions that allows to perform the ex-ante valuation of interbank claims, therefore accounting for the additional source of uncertainty deriving by the impossibility to have an unerring estimate of the assets values and equities before the maturity.", "In the context of Asset Pricing Theory (APT), [18] and [28] show that, assuming no arbitrage and market completeness, the ex-ante valuation at time $t<T$ of the random variable $E_i(T)$ at maturity can be performed by computing its conditional expectation with respect to the (unique) Equivalent Martingale Measure (EMM) $\\mathbb {Q}$ : $E_i(t) = \\mathbb {E}_{\\mathbb {Q}} [E_i(T) \| \\mathcal {F}(t)~]$ , where $\\mathcal {F}(t)$ is the filtration at time $t$ associated with $E_i$ and, for simplicity, the returns of the riskless bond have been set to zero.", "In order to compute such conditional expectation, one needs to find the fixed point of (REF ), for any realization of the underlying stochastic processes.", "Such computation can be only performed by a central authority with full knowledge of all the parameters of the system.", "Although the global ex-ante valuation can be also performed within our framework, here we focus instead on local ex-ante valuation, in the sense that, while the valuation is performed collectively by all banks, each bank only has information about its own counterparties.", "This happens precisely because, (i) the aforementioned uncertainty is entirely incorporated into the valuation functions and (ii), as it can be seen from (REF ), in order to compute an iteration of the Picard iteration algorithm (REF ), each bank only needs information about its own counterparties.", "Moreover, such set of valuation functions will naturally extend EN.", "As regards the valuation functions of external assets, we simply take them as in Proposition REF .", "For what concerns interbank assets, we note that if the valuation occurred at maturity banks could use the interbank valuation functions in Proposition REF , in which the equity would be the equity at the maturity $T$ .", "This information, however, is not available to banks before $T$ .", "We now assume that, from the time $t$ at which the valuation happens until maturity $T$ , (i) the only uncertainty that banks must consider during the evaluation process is due to external assets, and (ii) external assets follow a stochastic process ${A}^e(t)$ .", "Hence, the variation of equity between $t$ and $T$ is equal to the variation of the external assets, i.e.", "${E}(T) = {E} + \\Delta {A}^e$ .", "From a technical point of view, the difference between global and local ex-ante valuations is that in the first case one computes the expectation of the fixed point of (REF ), while in the second case one computes the fixed point of (REF ) in which the expectations of the valuation functions appear: $\\begin{split}E_i(t) = \\mathbb {E}_{\\mathbb {Q}}[E_i(T) \| \\mathcal {F}(t)] &= \\mathbb {E}_{\\mathbb {Q}}[A_i^e(T) \| \\mathcal {F}(t)] - L_i^e + \\sum _{j=1}^n A_{ij} \\mathbb {E}_{\\mathbb {Q}}[ \\mathbb {V}_{ij}^{(\\mathrm {EN})}(E_j(T)) \| \\mathcal {F}(t)] - \\sum _{j=1}^n L_{ij} \\\\&= A_i^e(t) - L_i^e + \\sum _{j=1}^n A_{ij} \\mathbb {E}_{\\mathbb {Q}}[ \\mathbb {V}_{ij}^{(\\mathrm {EN})}(E_j(T)) \| \\mathcal {F}(t)] - \\sum _{j=1}^n L_{ij} \\, ,\\end{split}$ where $\\mathbb {V}_{ij}^{(\\mathrm {EN})}$ are the valuation functions in Proposition REF .", "Once the stochastic process followed by external assets is known, we can identify the valuation functions for interbank assets with their expectation conditioned on the observation at time $t$ : $ \\begin{split}\\mathbb {V}_{ij}(E_{j}) \\equiv \\mathbb {E}_{\\mathbb {Q}}[ \\mathbb {V}_{ij}^{(\\mathrm {EN})}(E_j) \| \\mathcal {F}(t)] &= \\mathbb {E}\\left[\\mathbb {1}_{E_{j}(T)\\ge 0} + \\beta \\left(\\frac{E_{j}(T) + \\bar{p}_j}{\\bar{p}_j}\\right)^+\\mathbb {1}_{E_{j}(T)<0} \\Big \| A_j^e(t) \\right] \\\\&= \\mathbb {E}\\left[\\mathbb {1}_{E_{j}(T)\\ge 0} \| A_j^e(t) \\right] + \\beta \\mathbb {E}\\left[\\left(\\frac{E_{j}(T) + \\bar{p}_j}{\\bar{p}_j}\\right)^+\\mathbb {1}_{E_{j}(T)<0} \\Big \| A_j^e(t) \\right] \\\\&= 1 - p_j^D(E_j) + \\beta \\rho _j(E_j) \\, ,\\end{split}$ where we have defined the probability of default: $ \\begin{split}p_j^D(E_j) &= \\mathbb {E}\\left[\\mathbb {1}_{E_{j}(T)<0} \| A_j^e(t) \\right] \\\\&= \\mathbb {E}\\left[\\mathbb {1}_{\\Delta A^e_j < -E_j}\\right]\\end{split}$ and the endogenous recovery: $\\begin{split}\\rho _j(E_j) &= \\mathbb {E}\\left[\\left(\\frac{E_{j}(T) + \\bar{p}_j}{\\bar{p}_j}\\right)^+\\mathbb {1}_{E_{j}(T)<0} \\Big \| A_j^e(t) \\right] \\\\&=\\mathbb {E}\\left[\\left(\\frac{E_{j}(T) + \\bar{p}_j}{\\bar{p}_j}\\right) \\mathbb {1}_{-\\bar{p}_j -E_j \\le \\Delta A^e_j < -E_j} \\right] \\, .\\end{split}$ Strictly speaking, $\\beta $ appearing in (REF ) would be equal to one in EN.", "Here, we include it to account for additional default costs.", "Moreover, its presence will be relevant in the context of Proposition REF .", "From (REF ) we can see that the valuation function can be thought of as the expectation over a two-valued probability distribution: If the borrower $j$ does not default at maturity, bank $i$ will recover the full amount $A_{ij}$ , while if bank $j$ defaults, bank $i$ will in general recover a smaller amount.", "From this point of view, $\\beta $ can be thought of as an additional exogenous recovery rate on top of the endogenous recovery rate $\\rho _j$ .", "Finally, we note that (REF ) defines feasible valuation functions.", "Proposition 6 In the limit in which the maturity is approached, i.e.", "$t \\rightarrow T$ , the interbank valuation function (REF ) converges to the interbank valuation function of EN (Proposition REF ).", "First we notice that, as $t \\rightarrow T$ the variation in external assets goes to zero with probability approaching one, and therefore from () we have that $p_j^D(E) \\rightarrow \\mathbb {1}_{E_j<0}$ and that $\\rho _j(E) \\rightarrow \\left(\\frac{E_j + \\bar{p}_j}{\\bar{p}_j}\\right)^+\\mathbb {1}_{E_j<0}$ , from which the proposition easily follows." ], [ "Ex-ante valuation with geometric Brownian motion", "In the spirit of the Merton model we will compute the expected value of assets at maturity (here, for the sake of convenience, maturity $T$ is common to all interbank claims) given our observation of the value of external assets before the maturity (at time $t$ ), assuming that external assets follow independent geometric brownian motions: $ dA_i^e(s) = \\sigma A_i^e(s) dW_i(s) \\qquad \\forall i \\, s \\, ,$ where, for simplicity, we consider the drift to be equal to zero.", "The PDF of $\\Delta A_i^e$ is: $ p(\\Delta A_i^e) = \\frac{1}{\\sqrt{2\\pi (T-t)}\\sigma (\\Delta A_i^e + A_i^e)}e^{\\frac{-\\left[\\log \\left(1+\\frac{\\Delta A_i^e}{A_i^e}\\right)+\\frac{1}{2}\\sigma ^2(T-t)\\right]^2}{2\\sigma ^2(T-t)}} \\, .$ From () we then have: $p_{j}^D(E) = \\frac{1}{2} \\left[ 1 + \\mathrm {erf} \\left[\\frac{\\log (1-E/A_j^e) + \\sigma ^2 (T-t)/2}{\\sqrt{2(T-t)} \\sigma } \\right] \\right] \\mathbb {1}_{E<A_j^{e}}$ $\\rho _{j}(E) = \\left( 1+\\frac{E}{\\bar{p}} \\right) \\left(p_j^D(E) - p_j^D(E + \\bar{p})\\right) + \\frac{1}{2 \\bar{p}}c_{j}(E)$ with $c_{j}(E) =& -\\text{erf}\\left[{\\frac{\\sigma ^2(T-t)/2 - \\log \\left(1-E/A_j^e\\right)}{\\sqrt{2(T-t)}\\sigma }}\\right] \\mathbb {1}_{E<A_j^e} \\\\&-\\text{erf}\\left[{\\frac{\\sigma ^2(T-t)/2 + \\log \\left(1-E/A_j^e\\right)}{\\sqrt{2(T-t)}\\sigma }}\\right] \\mathbb {1}_{E<A_j^e} \\\\&+\\text{erf}\\left[{\\frac{\\sigma ^2(T-t)/2 + \\log \\left(1-(E+\\bar{p})/A_j^e\\right)}{\\sqrt{2(T-t)}\\sigma }}\\right] \\mathbb {1}_{E<A_j^e-\\bar{p}} \\\\&+\\text{erf}\\left[{\\frac{\\sigma ^2(T-t)/2 - \\log \\left(1-(E+\\bar{p})/A_j^e\\right)}{\\sqrt{2(T-t)}\\sigma }}\\right] \\mathbb {1}_{E<A_j^e-\\bar{p}} \\, .$ Theorems REF and REF ensure that there exists a greatest solution (therefore optimal for all banks) and that such solution can be computed with arbitrary precision using the Picard iteration algorithm (REF )." ], [ "Comparison between valuation functions", "For illustrative purposes here we perform a stress test on a small financial system composed by three banks, $A$ , $B$ , $C$ .", "We choose a simple ring topology, $A \\rightarrow B \\rightarrow C \\rightarrow A$ with the following parameters: $A^e = \\begin{pmatrix}10 \\\\5 \\\\3\\end{pmatrix}\\qquad L^e = \\begin{pmatrix}9 \\\\4 \\\\3\\end{pmatrix}\\qquad A = \\begin{pmatrix}0 & 0.5 & 0 \\\\0 & 0 & 0.5 \\\\0.5 & 0 & 0\\end{pmatrix} \\, ,$ so that all three banks have a book value of their equity equal to one.", "Total leverages, defined as the ratio between total assets and book values of equity, range from 10.5 to 3.5.", "Our stress test consists in applying an exogenous shock to the external assets of all banks, resulting in a relative devaluation $\\alpha $ , i.e.", "$A^e_i \\rightarrow (1-\\alpha ) A^e_i$ .", "The variation in external assets of bank $i$ , measured as the difference between its external assets before the shock and its external assets after the shock is $\\Delta A^e_i = \\alpha A^e_i$ .", "Using (REF ) we can readily compute the corresponding variation in equity, again measured as the difference between the equity before the shock (i.e.", "its book value) and the equity after the shock: $\\Delta E_i = \\alpha A^e_i + \\sum _j A_{ij}(1 - \\mathbb {V}_{ij}(E_j^))$ .", "Network effects can be quantified as the total losses in the system minus the losses directly caused by the exogenous shock: $\\sum _i \\Delta E_i - \\Delta A^e_i = \\sum _{ij} A_{ij}(1 - \\mathbb {V}_{ij}(E_j^))$ , which can be conveniently normalised by its maximum, $\\sum _{ij} A_{ij}$ : $ \\frac{\\sum _i \\Delta E_i - \\Delta A^e_i}{\\sum _{ij} A_{ij}} = \\frac{\\sum _{ij} A_{ij}\\left[1 - \\mathbb {V}_{ij}(E_j^)\\right]}{\\sum _{ij} A_{ij}} \\, .$ In the left panel of Fig.", "REF we show the behaviour of the quantity (REF ) as a function of the exogenous shock on external assets, for several valuation functions.", "For Furfine we use $R = 0$ , while for ex-ante EN (NEVA in the legend) we use $\\beta = 1$ and (REF ) with $\\sigma = 0.1$ , for all banks.", "Interestingly, we can see that for smaller values of the exogenous shock network effects are larger for the ex-ante EN than for EN at maturity, while the situation is reversed for larger values of the exogenous shock.", "This is consistent with the fact that uncertainty deriving from being before the maturity can both lead to lower and higher valuations of interbank claims.", "Lower valuations correspond to potential sizeable losses that can happen even in the presence of smaller shocks, while higher valuations correspond to positive fluctuations in the value of external assets that can lead to a recovery in the presence of larger shocks.", "Another way to assess the extent of network effects is the following.", "Let us imagine that each bank wants to valuate the interbank assets of its counterparty using the standard Merton approach.", "This amounts to using the valuation function (REF ) and evaluating it in the book value of the equity of the counterparty.", "Hence, the lender $i$ discounts its interbank assets $A_{ij}$ towards the borrower $j$ by a factor $\\mathbb {V}_{ij}(M_j)$ .", "If the same valuation is performed using NEVA, more specifically using ex-ante EN, the discount factor equals to $\\mathbb {V}_{ij}(E_j^)$ .", "In the right panel of Fig.", "REF we show the difference between such discount factors, i.e.", "between the discount factor of the valuation of an interbank claim performed with the standard Merton approach and the valuation of an interbank claim performed with ex-ante EN (NEVA in the legend) valuation functions with $\\beta = 1$ and (REF ) $\\sigma = 0.1$ , for all banks.", "The difference is maximal for intermediate values of the exogenous shock.", "In fact, for small values of the shock network effects are small, while for large values of the shock the valuation of interbank claims becomes less and less important, as most losses will be direct losses due to the exogenous shock.", "Figure: Stress test consisting in applying an exogenous shock to external assets of all banks and by re-evaluating interbank claims.", "Left panel: network contribution (normalised to its maximum value) as a function of the exogenous shock, for several valuation functions.", "Right panel: difference between the discount factor of the valuation of an interbank claim performed with the standard Merton approach and discount factor of the valuation of an interbank claim performed with ex-ante EN (NEVA)." ], [ "Limit behavior of solutions", "We now introduce a sequence $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ of valuation functions.", "For example, different values of $l$ could correspond to different values of a parameter in the NEVA.", "For each value of $l$ we have a different equation (REF ) of the form $\\mathbf {E} = \\Phi _l(\\mathbf {E})$ .", "We will denote the $k$ -th iteration of the corresponding map in (REF ) with $\\Phi ^{(k)}_l$ .", "In order to clarify the rationale behind the introduction of sequences of valuation functions, we will consider the following example.", "Let $l$ be an index associated with the distance to maturity, so that maturity is approached as $l$ increases.", "For a given $l$ we can compute the solution ${E}_{l}$ corresponding to the given maturity.", "The question now arises: in the limit $l \\rightarrow \\infty $ (that is, as maturity approaches) will the solutions of the ex-ante valuations corresponding to the equations ${E} = \\Phi _l({E})$ approach the solution of the ex-post valuation (that is, at maturity) corresponding to the equation ${E} = \\Phi _\\infty ({E})$ ?", "Solving this problem essentially boils down to identifying sufficient conditions under which an interchange of the two limits (one on the model parameters and the other on the iterations of the Picard algorithm) is legit.", "A positive answer to this question will allow us to relate ex-ante and ex-post valuation models, correctly identifying ex-ante models as genuine generalizations of ex-post ones to the case of arbitrary maturity.", "The following theorem provides an affirmative answer for non-decreasing valuation functions.", "Theorem 4 If: the sequences $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ are monotonic non-decreasing: $\\mathbb {V}^l_{ij}(E) \\le \\mathbb {V}^{l+1}_{ij}(E)$ , $\\forall E$ , $i$ , $j$ , $l$ , the sequences $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ are pointwise convergent: $\\lim _{l\\rightarrow \\infty } \\mathbb {V}^l_{ij}(E) = \\mathbb {V}^\\infty _{ij}(E)$ , $\\forall E$ , $i$ , $j$ , there exists a unique solution for ${E^} = \\Phi _\\infty ({E}^)$ , where $\\Phi _\\infty $ is the map corresponding to the valuation functions $\\mathbb {V}^\\infty _{ij}(E)$ , $\\mathbb {V}^{\\infty }_{ij}(E)$ is a continuous function, $\\forall i$ , $j$ .", "then $\\lim _{l\\rightarrow \\infty } {E}_{l} = {E^}$ , where ${E}_{l} = \\lim _{k\\rightarrow \\infty } \\Phi _l^{(k)}({M})$ .", "For non-increasing valuation functions the requirement that $\\Phi _\\infty $ has a unique solution can be relaxed.", "Theorem 5 If: the sequences $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ are monotonic non-increasing: $\\mathbb {V}^l_{ij}(E) \\ge \\mathbb {V}^{l+1}_{ij}(E)$ , $\\forall E$ , $i$ , $j$ , $l$ , the sequences $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ are pointwise convergent: $\\lim _{l\\rightarrow \\infty } \\mathbb {V}^l_{ij}(E) = \\mathbb {V}^\\infty _{ij}(E)$ , $\\forall E$ , $i$ , $j$ , then $\\lim _{l\\rightarrow \\infty } {E}_{l} = {E^}$ , where ${E}_{l} = \\lim _{k\\rightarrow \\infty } \\Phi _l^{(k)}({M})$ ." ], [ "Limit to linear DebtRank", "As a first application of the limit theorems we now consider the following sequence of valuation functions: $ \\mathbb {V}_{ij}^l(E_{j}) = 1 - p_j^D(E_j) + \\beta _l \\rho _j(E_j) \\qquad l = 1, 2, \\ldots \\, ,$ where $\\lbrace \\beta _l\\rbrace $ is a monotone non-increasing sequence of real parameters such that $\\lim _{l \\rightarrow \\infty } \\beta _l = 0$ and $\\Delta A^e_j$ has the uniform distribution in the interval $[-M_j,\\, 0]$ , $\\forall j$ .", "Then: $p_j^D(E) = 1 - \\frac{(E)^+}{M_j}$ $\\rho _j(E_j) = \\left[ \\frac{E_j + \\bar{p_j}}{\\bar{p}_j M_j} (b-a) + \\frac{b^2-a^2}{2\\bar{p}_jM_j} \\right] \\mathbb {1}_{b > a} \\, ,$ where $b(E) = -(E)^+$ and $a(E) = \\max (-\\bar{p}_j-E, -M_j)$ .", "The valuation functions $\\mathbb {V}_{ij}^l$ are monotonic non-increasing in $l$ and converge to the valuation functions, $\\mathbb {V}_{ij}^{\\infty }(E_j) = (E_j)^+/M_j$ .", "Proposition 7 If: the sequence $\\lbrace \\beta _l\\rbrace $ is such that $\\lim _{l \\rightarrow \\infty } \\beta _l = 0$ , the sequences $\\lbrace \\mathbb {V}^l_{ij} \\rbrace $ are chosen as in (REF ), $\\forall i$ , $j$ , the probability of default and the endogenous recovery () are computed with $\\Delta A^e_j$ having a uniform distribution in the interval $[-M_j,\\, 0]$ , $\\forall j$ , then the solution of the corresponding equation (REF ) converges to the solution ${E}^$ of the linear DebtRank.", "The proof follows immediately from Theorem REF .", "By using the previous proposition one can effectively re-interpret linear DebtRank as an ex-ante EN model in which shocks are negative and the exogeneous recovery rate $\\beta $ is equal to zero." ], [ "The case of geometric Brownian motion", "We have already discussed the case in which external assets follow a geometric brownian motion in Section REF .", "Here we consider a sequence $\\lbrace \\mathbb {V}_{ij}^l\\rbrace $ of valuation functions with corresponding maturities $\\lbrace T_l\\rbrace $ , with $\\lim _{l \\rightarrow \\infty } T_l = t$ , i.e.", "the limit $l \\rightarrow \\infty $ corresponds to the limit in which the distance to maturity goes to zero.", "In this limit the valuation functions $\\mathbb {V}_{ij}^l$ converge pointwisely to the valuation functions of EN (see Proposition REF ).", "If we could apply either of Theorems REF and REF , the solutions ${E}_l$ of NEVA with interbank valuation functions in (REF ) (the ex-ante EN) would converge to the solution ${E}^$ of the (ex-post) EN.", "Unfortunately, $\\mathbb {V}_{ij}^l$ are neither non-increasing nor non-decreasing and the theorems cannot be applied.", "To see why this is the case, let us note that the interbank valuation functions of EN are equal to zero for $E<-\\bar{p}$ and equal to one for $E>0$ (see Proposition REF ).", "From (REF ) we can see that there is always a non-zero probability that the variation of external assets $\\Delta A^{e}$ is either positive or negative.", "Hence, the interbank valuation functions obtained by plugging (REF ) into (REF ) take values in the open interval $(0,1)$ .", "Therefore, for $E<-\\bar{p}$ ($E > 0$ ) they are larger (smaller) than interbank valuation functions of EN.", "Nevertheless, in the Appendix we present three numerical examples in which the convergence of the solutions of the NEVA with interbank valuation functions in (REF ) (the ex-ante EN) with external assets following a geometric Brownian motion to the solutions of EN holds.", "This provides a sound background to conjecture that the convergence might be proven to hold under more general hypotheses." ], [ "Conclusions", "In this paper, we introduce a general framework that allows financial institutions to perform an ex-ante network-adjusted valuation of interbank claims in a decentralized fashion.", "On the one hand, our framework encompasses some of the most widely used models of financial contagion [15], [20], [26], [3], in the precise sense that the model is equivalent to those models for specific choices of the valuation functions and the parameters.", "On the other hand, our framework relates also to the stream of literature [18], [28] carrying out the valuation of claims à la Merton when cross-holdings of debt exist between different firms.", "An important contribution of our approach is that the valuation is decentralized, meaning that it does not assume the existence of an entity with perfect information on the parameters of the financial system.", "Our main result is that, under mild assumptions about valuation functions, the valuation problem admits a greatest solution, i.e.", "a solution in which the losses of all banks are minimal.", "Moreover, we provide a simple iterative algorithm to compute such solution.", "Furthermore, we derive a set of conditions under which the solution of the valuation problem at the maturity time T is equal to the limit of the sequence of solutions obtained for the valuation problems at $t<T$ as the maturity is approached (i.e.", "$t\\rightarrow T$ ).", "In other words, the solution of the problem at the maturity coincides with the limit for the valuation time approaching the maturity of the solutions of problems at a given valuation time.", "A natural application of our framework is in devising stress-tests to assess losses on banks' portfolios in a network of liabilities, conditional to shocks on their external assets in order to determine capital requirements and value at risk.", "Indeed, to any given shock on the external assets of the banks it corresponds a different valuation of banks' equities.", "Therefore, by assuming a known distribution of shocks, one can derive a corresponding distribution of equity losses.", "Finally, such distribution can be taken as the input of any axiomatic risk measure [12], [11]." ], [ "Acknowledgments", "PB, MB, MDE, GV, SB, and GC acknowledge support from: FET Project SIMPOL nr.", "610704, FET Project DOLFINS nr.", "640772, and FET IP Project MULTIPLEX nr.", "317532.", "FC acknowledges support of the Economic and Social Research Council (ESRC) in funding the Systemic Risk Centre (ES/K002309/1).", "SB acknowledges the Swiss National Fund Professorship grant nr.", "PP00P1-144689.", "GC acknowledges also EU projects SoBigData nr.", "654024 and CoeGSS nr.", "676547.", "To prove it we just need to show that: (a) the function $\\Phi $ maps a complete lattice into itself, $\\Phi : T \\rightarrow T$ , (b) the function $\\Phi $ is an order-preserving function.", "To prove (a) we notice that if valuation functions are feasible then: $\\forall {E} \\in \\mathbb {R}^n \\quad m_i = -L^{e}_i - \\sum _{j}L_{ij}\\le \\Phi _i({E}) \\le A^{e}_i-L^{e}_i + \\sum _{j}A_{ij} - \\sum _{j}L_{ij} = M_i$ and consequently $T=\\varprod _{i=1}^n [m_i,M_i]$ is a complete lattice such that $\\Phi : T \\rightarrow T$ , that proves (a).", "Since $\\Phi $ is a linear combination of monotonic non-decreasing functions in ${E}$ , then $\\forall {E},{E^{\\prime }}$ if ${E}<{E}^{\\prime }$ , follows $\\Phi ({E}) \\le \\Phi ({E^{\\prime }})$ , where the partial ordering relation in $T$ is component-wise, i.e.", "${x}\\le {y}$ iff $\\forall i$ $x_i \\le y_i$ .", "So both conditions (a) and (b) hold and the Knaster-Tarski theorem applies.", "The set of solutions $S$ of (REF ) is then a complete lattice, therefore it is non-empty (the empty set cannot contain its own supremum) and, more importantly, it admits a supremum solution, ${E}^+$ , and an infimum solution, ${E}^-$ , such that $\\forall {E}^{} \\in S$ , ${E}^- \\le {E}^{} \\le {E}^+$ .", "Convergence will be proved by induction.", "For $n = 0$ we have $ {E}^{(1)} = \\Phi ({E}^{(0)}) \\le {M} = {E}^{(0)} $ Assume now that the claim is true for all $0 \\le m \\le n$ , then $ {E}^{(n+1)} = \\Phi ({E}^{(n)}) \\le \\Phi ({E}^{(n - 1)}) = {E}^{(n)}$ where we have used the fact that $\\Phi $ is monotonic non-decreasing and ${E}^{(n)} \\le {E}^{(n - 1)}$ by hypothesis, We know that $\\lbrace {E}^{(n)}\\rbrace $ is bounded below and monotonic non-increasing, by the Monotone Convergence Theorem we have that ${E}^{} = \\lim _{n\\rightarrow \\infty } {E}^{(n)} = \\inf _n\\lbrace {E}^{(n)}\\rbrace $ exists and is finite.", "By hypothesis $\\Phi $ is continuous from above (because under assumptions of Theorem (REF ) we know that the valuation functions are feasible), hence $ \\Phi ({E}^{}) = \\Phi (\\lim _{n} {E}^{(n)}) = \\lim _n \\Phi ({E}^{(n)}) = \\lim _n {E}^{(n+1)} = {E}^{}$ So that ${E}^{} \\in S$ .", "We will now prove it must be that ${E}^{} = {E}^{+}$ .", "First we need to establish a preliminary result, namely that ${E}^{(n)} \\ge {E}^+, \\forall n$ .", "Reasoning by induction, it is trivially true for the initial point that ${E}^{(0)}\\ge {E}^+$ .", "Suppose now that it is true up to a given $\\bar{n}$ , ${E}^{(\\bar{n})} \\ge {E}^+$ then, since $\\Phi $ is order-preserving, $ {E}^{(\\bar{n}+1)} = \\Phi ({E}^{(\\bar{n})}) \\ge \\Phi ({E}^+) = {E}^+$ Now, knowing that ${E}^{(n)} \\ge {E}^{+}, \\forall n$ we have that ${E}^{} = \\inf _n\\lbrace {E}^{(n)}\\rbrace \\ge {E}^+$ .", "But ${E}^{} \\in S$ , hence ${E}^{} = {E}^+$ .", "Convergence will be proved by induction.", "For $n = 0$ we have $ {E}^{(1)} = \\Phi ({E}^{(0)}) \\ge {m} = {E}^{(0)} $ Assume now that the claim is true for all $0 \\le m \\le n$ , then $ {E}^{(n+1)} = \\Phi ({E}^{(n)}) \\ge \\Phi ({E}^{(n - 1)}) = {E}^{(n)}$ where we have used the fact that $\\Phi $ is monotonic non-decreasing and ${E}^{(n)} \\ge {E}^{(n - 1)}$ by hypothesis.", "We know that $\\lbrace {E}^{(n)}\\rbrace $ is bounded above and monotonic non-decreasing, by the Monotone Convergence Theorem we have that ${E}^{} = \\lim _{n} {E}^{(n)} = \\sup _n\\lbrace {E}^{(n)}\\rbrace $ exists and is finite.", "By hypothesis $\\Phi $ is continuous from below, hence $ \\Phi ({E}^{}) = \\Phi (\\lim _{n} {E}^{(n)}) = \\lim _{n\\rightarrow \\infty } \\Phi ({E}^{(n)}) = \\lim _{n\\rightarrow \\infty } {E}^{(n+1)} = {E}^{}$ So that ${E}^{} \\in S$ .", "We will now prove it must be that ${E}^{} = {E}^{-}$ .", "First we need to establish a preliminary result, namely that ${E}^{(n)} \\le {E}^-, \\forall n$ .", "Reasoning by induction, it is trivially true for the initial point that ${E}^{(0)}\\le {E}^-$ .", "Suppose now that it is true up to a given $\\bar{n}$ , ${E}^{(\\bar{n})} \\le {E}^-$ then, since $\\Phi $ is order-preserving, $ {E}^{(\\bar{n}+1)} = \\Phi ({E}^{(\\bar{n})}) \\le \\Phi ({E}^-) = {E}^-$ Now, knowing that ${E}^{(n)} \\le {E}^-, \\forall n$ we have that ${E}^{} = \\sup _n\\lbrace {E}^{(n)}\\rbrace \\le {E}^{-}$ .", "But ${E}^{} \\in S$ , hence ${E}^{} = {E}^{-}$ .", "Let us consider the sequences $\\Phi _l^n \\equiv \\Phi _{l}^{(n)}({M})$ , where the index $n$ denotes composition of $\\Phi $ with itself $n$ times.", "Since the valuation functions are monotonically non-decreasing in $l$ then $\\forall {E}\\, \\Phi _{l}({E})\\le \\Phi _{l+1}({E})$ that implies $\\Phi _l^n \\le \\Phi _{l+1}^n$ .", "Since $\\mathbb {V}^l_{ij}(E)$ are all feasible valuation functions we also have that $\\Phi _l^n \\ge \\Phi _{l}^{n+1}$ .", "From this follows, and boundedness of the sequences in both indices, it follows that $\\lim _l\\lim _n\\Phi _l^n = {\\tilde{E}}$ exists.", "Monotonicity, punctual convergence and continuity of the limit valuation function imply, by Dini's Theorem, uniform convergence of $\\Phi _l({E})$ to $\\Phi _\\infty ({E})$ .", "Uniform convergence and continuity of $\\Phi _{\\infty }$ imply that ${E^} = \\Phi _{\\infty }({E^})$ .", "Since, by assumption, the solution is unique then we must have that ${\\tilde{E}} = {E^} $ .", "Thus $\\lim _l\\lim _n\\Phi _l^n = \\lim _l\\lim _n\\Phi _l^n$ that is equivalent to the thesis.", "Let us consider the sequences $\\Phi _l^n \\equiv \\Phi _{l}^{(n)}({M})$ .", "Since the valuation function is monotonic non-increasing in $l$ then $\\forall {E}\\, \\Phi _{l}({E})\\ge \\Phi _{l+1}({E})$ that implies $\\Phi _l^n \\ge \\Phi _{l+1}^n$ .", "Since $\\mathbb {V}^l_{ij}(E)$ are all feasible valuation functions we also have that $\\Phi _l^n \\ge \\Phi _{l}^{n+1}$ .", "Hence, the sequences $\\Phi _l^n$ are all non-increasing in both indices and bounded from below, from this follows that $\\lim _l\\lim _n\\Phi _l^n= \\lim _l\\lim _n\\Phi _l^n$ that is equivalent to the thesis.", "In our examples, the financial system is composed of three banks that use the interbank valuation functions (REF ) and in which external assets follow the geometric brownian motion (REF ).", "For each network topology we vary the time to maturity $T_{l}$ and we compute equities via (REF ).", "We also compute equities for the (ex-post) EN, also via (REF ).", "We set $\\beta = 1.0$ , i.e.", "we do not include an additional exogeneous recovery.", "External liabilities are equal to zero for all banks in all cases.", "$A\\rightarrow B\\rightarrow C$ $A^e = \\begin{pmatrix}1 \\\\1 \\\\1\\end{pmatrix}\\qquad A = \\begin{pmatrix}0 & 1.2 & 0 \\\\0 & 0 & 1.2 \\\\0 & 0 & 0\\end{pmatrix}$ Figure: Open chain.", "Solid blue: equities E j,l E_{j, l}^* as a function of the distance to maturity T l T_l for the NEVA with interbank valuation functions in () (ex-ante Eisenberg and Noe model).", "Dashed red: equities E j * E_{j}^* for the (ex-post) Eisenberg and Noe model.", "Panels from the top to the bottom refer to banks A, B, and C.$B\\leftarrow A\\rightarrow C$ $A^e = \\begin{pmatrix}1 \\\\0.1 \\\\1\\end{pmatrix}\\qquad A = \\begin{pmatrix}0 & 1 & 1 \\\\0 & 0 & 0 \\\\0 & 0 & 0\\end{pmatrix}$ Figure: Analogous of Fig.", "for a tree.$A\\rightarrow B\\rightarrow C\\rightarrow A$ $A^e = \\begin{pmatrix}1 \\\\1 \\\\1\\end{pmatrix}\\qquad A = \\begin{pmatrix}0 & 1.1 & 0 \\\\0 & 0 & 1.2 \\\\1.5 & 0 & 0\\end{pmatrix}$ Figure: Analogous of Fig.", "for a closed chain." ] ]	1606.05164
[ [ "Sentiment Aggregate Functions for Political Opinion Polling using\n Microblog Streams" ], [ "Abstract The automatic content analysis of mass media in the social sciences has become necessary and possible with the raise of social media and computational power.", "One particularly promising avenue of research concerns the use of sentiment analysis in microblog streams.", "However, one of the main challenges consists in aggregating sentiment polarity in a timely fashion that can be fed to the prediction method.", "We investigated a large set of sentiment aggregate functions and performed a regression analysis using political opinion polls as gold standard.", "Our dataset contains nearly 233 000 tweets, classified according to their polarity (positive, negative or neutral), regarding the five main Portuguese political leaders during the Portuguese bailout (2011-2014).", "Results show that different sentiment aggregate functions exhibit different feature importance over time while the error keeps almost unchanged." ], [ "Introduction", "Surveys and polls using the telephone are widely used to provide information of what people think about parties or political personalities[1].", "Surveys randomly select the electorate sample, avoiding selection bias, and are designed to collect the perception of a population regarding some subject, such as in politics or marketing.", "However this method is expensive and time consuming [2], [1].", "Furthermore, over the years it is becoming more difficult to contact people and persuade them to participate in these surveys [3].", "On the other hand, online publication of news articles is a standard behavior of news outlets and the raise of social media, namely Twitter and Facebook, has changed the way people interact with news [4], [5].", "This way, people are able to react and comment any news in real time.", "One challenge that several research works have been trying to solve is to understand how opinions expressed on social media, and their sentiment, can be a leading indicator of public opinion.", "However, at the same time there might exist simultaneously positive, negative and neutral opinions regarding the same subject.", "Thus, we need to obtain a value that reflects the general image of each political target in social media, for a given time period.", "To that end, we use sentiment aggregate functions.", "In summary, a sentiment aggregate function calculates a global value based on the number of positive, negative, and neutral mentions of each political target, in a given period.", "We conducted an exhaustive study and collected and implemented several sentiment aggregate functions from the state of the art [4], [6], [7], [8], [9], [10], [11], [12], [13].", "Thus, the main objective of our work is to study and define a methodology capable of successfully estimating the polls results, based on opinions expressed on social media, represented by sentiment aggregators.", "We applied this problem to the Portuguese bailout case study, using Tweets from a sample of the Portuguese Tweetosphere and Portuguese polls as gold standard.", "Given the monthly periodicity of polls, we needed to monthly aggregate data.", "This approach allows each aggregator value to represent the monthly sentiment for each political party.", "Due to the absence of a general sentiment aggregate function suitable for different case studies, we decided to include all aggregate functions as features of the regression model.", "Therefore the learning algorithm is able to adapt to the most informative aggregate functions through time.", "In the next Section we review related work.", "In Section we present the methodology we implemented.", "We describe data in Section followed by the experimental setup in Section .", "In Section we present and discuss the results we obtained, while Section is reserved for some conclusions taken from our study, and for future work." ], [ "Related Work", "Content analysis of mass media has an established tradition in the social sciences, particularly in the study of effects of media messages, encompassing topics as diverse as those addressed in seminal studies of newspaper editorials [14], media agenda-setting [15], or the uses of political rhetoric [16], among many others.", "By 1997, Riffe and Freitag [17], reported an increase in the use of content analysis in communication research and suggested that digital text and computerized means for its extraction and analysis would reinforce such trend.", "Their expectation has been fulfilled: the use of automated content analysis has by now surpassed the use of hand coding [18].", "The increase in the digital sources of text, on the one hand, and current advances in computation power and design, on the other, are making this development both necessary and possible, while also raising awareness about the inferential pitfalls involved [19], [20].", "One avenue of research that has been explored in recent years concerns the use of social media to predict present and future political events, namely electoral results [4], [6], [7], [8], [9], [10], [11], [12], [13].", "Although there is no consensus about methods and their consistency [21], [22].", "Gayo-Avello [23] summarizes the differences between studies conducted so far by stating that they vary about period and method of data collection, data cleansing and pre-processing techniques, prediction approach and performance evaluation.", "One particular challenge when using sentiment is how to aggregate opinions in a timely fashion that can be fed to the prediction method.", "Two main strategies have been used to predict elections: buzz, i.e., number of tweets mentioning a given candidate or party and the use of sentiment polarity.", "Different computational approaches have been explored to process sentiment in text, namely machine learning and linguistic based methods [24], [25], [26].", "In practice, algorithms often combine both strategies.", "Johnson et al.", "[1] concluded that more than predicting elections, social media can be used to gauge sentiment about specific events, such as political news or speeches.", "Defending the same idea, Diakopoulos el al.", "[27] studied the global sentiment variation based on Twitter messages of an Obama vs McCain political TV debate while it was still happening.", "Tumasjan et al.", "[10] used Twitter data to predict the 2009 Federal Election in Germany.", "They stated that “the mere number of party mentions accurately reflects the election result”.", "Bermingham et al.", "[4] correctly predicted the 2011 Irish General Elections also using Twitter data.", "Gayo-Avello et al.", "[22] also tested the share of volume as predictor in the 2010 US Senate special election in Massachusetts.", "On the other hand, several other studies use sentiment as a polls result indicator.", "Connor et al.", "[12] used a sentiment aggregate function to study the relationship between the sentiment extracted from Twitter messages and polls results.", "They defined the sentiment aggregate function as the ratio between the positive and negative messages referring an specific political target.", "They used the sentiment aggregate function as predictive feature in the regression model, achieving a correlation of 0.80 between the results and the poll results, capturing the important large-scale trends.", "Bermingham et al.", "[4] also included in their regression model sentiment features.", "Bermingham et al.", "introduced two novel sentiment aggregate functions.", "For inter-party sentiment, they modified the share of volume function to represent the share of positive and negative volume.", "For intra-party sentiment , they used a log ratio between the number of positive and negative mentions of a given party.", "Moreover, they concluded that the inclusion of sentiment features augmented the effectiveness of their model.", "Gayo-Avello et al.", "[22] introduced a different aggregate function.", "In a two-party race, all negative messages on party $c2$ are interpreted as positive on party $c1$ , and vice-versa." ], [ "Methodology", "Figure REF depicts an overview of the data mining process pipeline applied in this work.", "To collect and process raw Twitter data, we use an online reputation monitoring platform [28] which can be used by researchers interested in tracking political opinion on the web.", "It collects tweets from a pre-defined sample of users, applies named entity disambiguation [29] and generates indicators of both frequency of mention and polarity (positivity/negativity) of mentions of entities across time.", "In our case, tweets are collected from the stream of 100 thousand different users, representing a sample of the Portuguese community on Twitter.", "The platform automatically classifies each tweet according to its sentiment polarity.", "If a message expresses a positive, negative or neutral opinion regarding an entity (e.g.", "politicians), it is classified as positive, negative or neutral mention, respectively.", "The sentiment classifier uses a corpus of 1500 annotated tweets as training set and it is reported an accuracy over 80% using 10-fold cross validation.", "These 1500 tweets were manually annotated by 3 political science students.", "Mentions of entities and respective polarity are aggregated by counting positive, negative, neutral and total mentions for each entity in a given period.", "Sentiment aggregate functions use these cumulative numbers as input to generate a new value for each specific time period.", "Since we want to use sentiment aggregate functions as features of a regression model to produce an estimate of the political opinion, we decided to use traditional poll results as gold standard." ], [ "Sentiment Aggregate Functions", "The following list presents the sentiment aggregate functions applied to the aggregated data between polls: - $entity\\_buzz$ : the monthly sum of the number of mentions (buzz) of a given entity (political party leader) between consecutive polls.", "- $entity\\_positives$ : the monthly sum of the positively classified mentions of a given entity (political party leader) between consecutive polls.", "- $entity\\_neutrals$ : the monthly sum of the neutral classified mentions of a given entity (political party leader) between consecutive polls.", "- $entity\\_negatives$ : the monthly sum of the negatively classified mentions of a given entity (political party leader) between consecutive polls.", "- $bermingham$ [4]: $\\log _{10}{\\frac{entity\\_posistives +1}{entitty\\_negatives +1}}$ - $berminghamsovn$ [4]: $\\frac{entity\\_negatives}{total\\_negatives} $ , $total\\_negatives$ corresponds to the sum of the negative mentions of all entities between polls.", "- $berminghamsovp$ [4]: $\\frac{entity\\_positives}{total\\_positives}$ , $total\\_positives$ corresponds to the sum of the positives mentions of all entities between polls.", "- $connor$ [2]: $\\frac{entity\\_positives}{entity\\_negatives}$ - $gayo$ [30]: $\\frac{entity\\_positives + others\\_negatives}{total\\_positives + total\\_negatives}$ - $polarity$ : $entity\\_positives - entity\\_negatives$ - $polarityONeutral$ : $\\frac{entity\\_positives - entity\\_negatives}{entity\\_neutrals}$ - $polarityOTotal$ : $\\frac{entity\\_positives - entity\\_negatives}{entity\\_buzz}$ - $subjOTotal$ : $\\frac{entity\\_positives + entity\\_negatives}{entity}$ - $subjNeuv$ : $\\frac{entity\\_positives + entity\\_negatives}{entity\\_neutrals}$ - $subjSoV$ : $\\frac{entity\\_positives + entity\\_negatives}{total\\_positives + total\\_negatives }$ - $subjVol$ : $entity\\_positives + entity\\_negatives$ - $share$ [4]: $\\frac{entity\\_buzz}{total\\-buzz}$ - $shareOfNegDistribution$ : $\\frac{ \\frac{entity\\_negatives}{entity\\_buzz}}{\\sum _{i=0}^n \\frac{entity\\_negatives_i}{entity\\_buzz_i}}$ , where $n$ is the number of political entities in the poll - $normalized\\_positive$ : $\\frac{entity\\_positivesi}{entity\\_buzz}$ - $normalized\\_negative$ : $\\frac{entity\\_positivesi}{entity\\_buzz}$ - $normalized\\_neutral$ : $\\frac{entity\\_positivesi}{entity\\_buzz}$ - $normalized\\_bermingham$ : $\\log _{10}{\\frac{normalized\\_positives +1}{normalized\\_negatives +1}} $ - $normalized\\_connor$ : $\\frac{normalized\\_positives}{normalized\\_negatives}$ - $normalized\\_gayo$ : $\\frac{normalized\\_positives + normalized\\_others\\_negatives}{normalized\\_total\\_positives + normalized\\_total\\_negatives}$ - $normalized\\_polarity$ : $normalized\\_positives - normalized\\_negatives$ The sentiment aggregate functions are used as features in the regression models." ], [ "Data", "The data used in this work consists of tweets mentioning Portuguese political party leaders and polls from August 2011 to December 2013.", "This period corresponds to the Portuguese bailout when several austerity measures were adopted by the incumbent right wing governmental coalition of PSD and CDS." ], [ "Twitter", "The Twitter data set contains 232 979 classified messages, collected from a network of 100 thousand different users classified as Portuguese.", "Table REF presents the distribution of positive, negative, and neutral mentions of the political leaders of the 5 most voted political parties in Portugal (PSD, PS, CDS, PCP and BE).", "The negative mentions represent the majority of the total mentions, except for CDU where the number of negative mentions is smaller than the neutral ones.", "The positive mentions represent less than 1% of the total mentions of each party, except for BE where they represent 2% of the total mentions.", "The most mentioned parties are PS, PSD and CDS.", "The total mentions of these three parties represent 90% of the data sample total mentions.", "Figure REF depicts the time series of the $berminghamsovn$ (negatives share) sentiment aggregate function.", "The higher the value of the function the higher is the percentage of negative tweets mention a given political entity in comparison with the other entities.", "As expected, Pedro Passos Coelho (PSD) as prime-minister is the leader with the higher score through the all time period under study.", "Paulo Portas (CDS) leader of the other party of the coalition, and also member of the government is the second most negatively mentioned in the period, while António José Seguro (PS) is in some periods the second higher.", "PSD and CDS are the incumbent parties while PS is the main opposition party in the time frame under study.", "PSD and CDS as government parties were raising taxes and cutting salaries.", "PS was the incumbent government during the years that led to the bailout and a fraction of the population considered responsible for the financial crisis.", "The bailout and the consequent austerity measures could explain the overwhelming percentage of negative mentions although we verified that in other time periods the high percentage of negatives mentions remains.", "We can say that Twitter users of this sample when mentioning political leaders on their tweets tend to criticize them.", "Figure: Negatives share (berminghamsovnberminghamsovn) of political leaders in Twitter." ], [ "Political Opinion Polls", "The polling was performed by Eurosondagem, a Portuguese private company which collects public opinion.", "This data set contains the monthly polls results of the five main Portuguese parties, from June 2011 to December 2013.", "Figure REF represents the evolution of Portuguese polls results.", "We can see two main party groups: The first group, where both PSD and PS are included, has a higher value of vote intention (above 23%).", "PSD despite starting as the preferred party in vote intention, has a downtrend along the time, losing the leadership for PS in September 2012.", "On the other hand, PS has in general an uptrend.", "The second group, composed by CDS, PCP and BE, has a vote intention range from 5% to 15%.", "While CDS has a downtrend in public opinion, PCP has an ascendent one.", "Although the constant tendencies (up- and downtrends), we noticed that the maximum variation observed between two consecutive months is 3%.", "In June 2013 there was political crises in the government when CDS threaten to leave the government coalition due to the austerity measures being implemented and corresponds to the moment when PS takes the lead in the polls.", "Figure: Representation of the monthly poll results of each political candidate" ], [ "Experimental Setup", "We defined the period of 2011 to December 2012 as training set and the all year of 2013 as test set.", "We applied a sliding window setting in which we start to predict the poll results of January 2013 using the previous 16 months as training set.", "The second poll to estimate is February 2013 and we train a new model using the previous 16 months to the target month under prediction.", "Training set – containing the monthly values of the aggregators (both sentiment and buzz aggregator) for 16 months prior the month intended to be predicted.", "Testing set - containing the values of the aggregators (both sentiment and buzz aggregator) of the month intended to be predicted.", "We select the values of the aggregators of the 16 months prior January 2013 (September 2011 to December 2012).", "We use that data to train our regression model.", "Then we input the aggregators' values of January 2013 - the first record of the testing set - in the the trained model, to obtain the poll results prediction.", "We select the next month of the testing set and repeat the process until all months are predicted.", "The models are created using two regression algorithms: a linear regression algorithm (Ordinary Least Squares - OLS) and a non-linear regression algorithm (Random Forests - RF).", "We also run an experiment using the derivative of the polls time series as gold standard, i.e., poll results variations from poll to poll.", "Thus, we also calculate the variations of the aggregate functions from month to month as features.", "Furthermore, we repeat each experiment including and excluding the lagged self of the polls, i.e., the last result of the poll for a given candidate ($y_{t-1}$ ) or the last polls result variation ($\\Delta y_{t-1}$ ) when predicting polls variations.", "We use Mean Absolute Error (MAE) [31] as evaluation measure, to determine the absolute error of each prediction.", "Then, we calculate the average of the twelve MAE's so we could know the global prediction error of our model.", "$MAE = \\frac{\\sum _{i=1}^{n}\|f_{i} - y_{i}\|}{n}$ $n$ is the number of forecasts, $f_{i}$ is the model's forecast and $y_{i}$ the real outcome." ], [ "Results and Discussion", "In this Section we explain in detail the experiments and its results.", "We perform two different experiments: (1) using absolute values and (2) using monthly variations." ], [ "Predicting Polls Results", "In this experiment, the sentiment aggregators take absolute values in order to predict the absolute values of polls results.", "Mathematically speaking, this experiment can be seen as: $y \\leftarrow $ {$y_{t-1}$ , $buzzAggregators$ , $sentimentAggregators$ }.", "In figure REF we see the global errors we obtained.", "Figure: Error predictions for polls results.The results shows that we obtain an Mean Absolute Error for the 5 parties poll results over 12 months of 6.55 % using Ordinary Least Squares and 3.1 % using Random Forests.", "The lagged self of the polls, i.e., assuming the last known poll result as prediction results in a MAE of 0.61 % which was expectable since the polls exhibit slight changes from month to month.", "This experiment shows that the inclusion of the lagged self ($y_{t-1}$ ) produces average errors similar to the lagged self." ], [ "Predicting Polls Results Variation", "According to our exploratory data analysis, the polls results have a small variation between two consecutive months.", "Thus, instead of predicting the absolute value of poll results, we tried to predict the variation, $\\Delta y \\leftarrow $ {$\\Delta (y_{t-1})$ , $\\Delta buzzAggregators$ , $\\Delta sentimentAggregators$ } Figure: Error predictions for polls results variation.In this particular experiment, the inclusion of the $\\Delta y_{t-1}$ as feature in the regression model has not a determinant role (figure REF ).", "Including that feature we could not obtain lower MAE than excluding it.", "It means that the real monthly poll variation is not constant over the year.", "In general, using a non-linear regression algorithm we obtain lower MAE.", "The results show that when leading with polls results with slight changes from poll to poll it makes sense to transform the dataset by derivation." ], [ "Buzz and Sentiment", "Several studies state that the buzz has predictive power and reflects correctly the public opinion on social media.", "Following that premise, we trained our models with buzz and sentiment aggregators separately to predict polls variations: $\\Delta y \\leftarrow $ {$\\Delta (y_{t-1})$ , $\\Delta buzzAggregators$ } $\\Delta y \\leftarrow $ {$\\Delta (y_{t-1})$ , $\\Delta sentimentAggregators$ } This experiment allowed us to compare the behavior of buzz and sentiment aggregators.", "Figure: Mean absolute error buzz vs sentiment.According to figure REF , buzz and sentiment aggregators have similar results.", "Although the OLS algorithm combined only with buzz aggregators has a slightly lower error than the other models, it is not a significant improvement.", "These results also show that Random Forests algorithm performs the best when combined only with sentiment aggregators." ], [ "Feature Selection", "One of the main goals of our work is to understand which aggregator (or group of aggregators) better suits our case study.", "According to the previous experiments, we can achieve lower prediction errors when training our model with buzz and sentiment aggregators separately.", "However, when training our model with these two kinds of aggregators separately, we are implicitly performing feature selection.", "We only have two buzz features ($share$ and $total\\_mentions$ ).", "Due to that small amount of features, it was not necessary to perform any feature selection technique within buzz features.", "Thus, we decided to apply a feature selection technique to the sentiment aggregators, in order to select the most informative ones to predict the monthly polls results variation.", "We use univariate feature selection, selecting 10% of the sentiment features (total of 3 features).", "Using this technique, the Random Forests' global error raise from 0.65 to 0.73.", "However, OLS presents an MAE drop from 0.72 to 0.67.", "Another important fact to notice is that if we perform univariate feature selection to all aggregators (buzz and sentiment), we will achieve the same MAE value that when applied only to sentiment aggregators.", "We try a different approach and perform a recursive feature elimination technique.", "In this technique, features are eliminated recursively according to a initial score given by the external estimator.", "This method allow us to determine the number of features to select.", "Thus, also selecting 3 features, the OLS' MAE drop to 0.63.", "Once again, none of the buzz features were selected.", "Furthermore, both feature selection techniques select different features for each monthly prediction.", "Figure: Aggregate functions importance in the Random Forests models." ], [ "Feature Importance", "We select the Random Forest model of monthly variations to study the features importance as depicted in figure REF .", "The higher, the more important the feature.", "The importance of a feature is computed as the (normalized) total reduction of the criterion brought by that feature.", "It is also known as the Gini importance.", "Values correspond to the average of the Gini importance over the different models trained in the experiments.", "The single most important feature is the $bermingham$ aggregate function, followed by $neutrals$ .", "It is important to notice that when combining all the aggregate functions as features in a single regression model, the $buzz$ does not comprises a high Gini importance, even if when used as a single feature produces similar results with the sentiment aggregate functions.", "In general, the standard deviation of the GIni importance is relatively high.", "This has to due with our experimental setup, as the values depicted in the bar chart correspond to the average of the Gini importance over 12 different models (12 months of testing set).", "Therefore, feature importances vary over time while the MAE tends to remain unchanged.", "We can say that different features have different informative value over time and consequently it is useful to combine all the sentiment aggregation functions as features of the regression models over time." ], [ "Conclusions", "We studied a large set of sentiment aggregate functions to use as features in a regression model to predict political opinion poll results.", "The results show that we can estimate the polls results with low prediction error, using sentiment and buzz aggregators based on the opinions expressed on social media.", "We introduced a strong baseline for comparison, the lagged self of the polls.", "In our study, we built a model where we achieve the lowest MAE using the linear algorithm (OLS), combined only with buzz aggregators, using monthly variations.", "The model has an MAE of 0.63%.", "We performed two feature selection techniques: (1) Univariate feature selection and (2) recursive feature elimination.", "Applying the recursive technique to the sentiment features, we can achieve an MAE of 0.63, equating our best model.", "Furthermore, the chosen features are not the same in every prediction.", "Regarding feature importance analysis our experiments showed that $bermingham$ aggregate function represents the higher Gini importance in the Random Forests model.", "The next immediate step is to implement a methodology using time series analysis.", "Furthermore, it is desirable to test this methodology with difference data sources, such as Facebook messages, blogs or news.", "Other alternative approach we intend to implement and evaluate is to interpret this problem as a classification problem - predict only the changing direction of opinion poll result (i.e., up or down)." ] ]	1606.05242
[ [ "Mid-Infrared intersubband polaritons in dispersive metal-insulator-metal\n resonators" ], [ "Abstract We demonstrate room-temperature strong-coupling between a mid-infrared ($\\lambda$=9.9 $\\mu$m) intersubband transition and the fundamental cavity mode of a metal-insulator-metal resonator.", "Patterning of the resonator surface enables surface-coupling of the radiation and introduces an energy dispersion which can be probed with angle-resolved reflectivity.", "In particular, the polaritonic dispersion presents an accessible energy minimum at k=0 where - potentially - polaritons can accumulate.", "We also show that it is possible to maximize the coupling of photons into the polaritonic states and - simultaneously - to engineer the position of the minimum Rabi splitting at a desired value of the in-plane wavevector.", "This can be precisely accomplished via a simple post-processing technique.", "The results are confirmed using the temporal coupled mode theory formalism and their significance in the context of the concept of strong critical coupling is highlighted." ], [ "Mid-Infrared intersubband polaritons in dispersive metal-insulator-metal resonators J-M. Manceau$^{1,a)}$ , S. Zanotto$^{2,c)}$ , T. Ongarello$^{1}$ , L. Sorba$^{2}$ , A. Tredicucci$^{2,3}$ , G. Biasiol$^{4}$ , R. Colombelli$^{1,b)}$ $^1$ Institut d'Electronique Fondamentale, Univ.", "Paris Sud, UMR8622 CNRS, 91405 Orsay, France $^2$ NEST, Istituto Nanoscienze, CNR and Scuola Normale Superiore, Piazza San Silvestro 12, Pisa, Italy $^3$ Dipartimento di Fisica, Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy $^4$ Laboratorio TASC, CNR-IOM, Area Science Park, Trieste I-34149, Italy We demonstrate room-temperature strong-coupling between a mid-infrared ($\\lambda $ =9.9 $\\mu $ m) intersubband transition and the fundamental cavity mode of a metal-insulator-metal resonator.", "Patterning of the resonator surface enables surface-coupling of the radiation and introduces an energy dispersion which can be probed with angle-resolved reflectivity.", "In particular, the polaritonic dispersion presents an accessible energy minimum at k=0 where – potentially – polaritons can accumulate.", "We also show that it is possible to maximize the coupling of photons into the polaritonic states and - simultaneously - to engineer the position of the minimum Rabi splitting at a desired value of the in-plane wavevector.", "This can be precisely accomplished via a simple post-processing technique.", "The results are confirmed using the temporal coupled mode theory formalism and their significance in the context of the concept of strong critical coupling is highlighted.", "42.79.Gn, 78.67.De, 85.30.-z Light emitting devices based on microcavity polaritons have experienced a tremendous development in the last two decades with the successful demonstration of electroluminescent diodes and optically pumped \"bosonic lasers\" operating at near-infrared wavelengths [1], [2].", "The extension of such devices to mid-infrared (mid-IR) and Terahertz (THz) wavelengths ($\\lambda >10\\mu $ m) has been recently explored, taking advantage of the design flexibility offered by intersubband (ISB) transitions in semiconductor quantum wells.", "The strong coupling between an ISB transition (or – more precisely – an ISB plasmon [3]) and a microcavity photonic mode was first demonstrated in the mid-IR [4] and then in the THz range [5].", "Devices based on microcavity ISB polaritons hold great potential since in the strong-coupling regime a periodic energy exchange between the light and matter degrees of freedom takes place on an ultrafast time scale (the Rabi oscillation time).", "On one hand, ISB polaritons can in principle exhibit radiatiave decay times faster than a bare ISB transition.", "This effect could yield more efficient electroluminescent devices at such wavelengths [6], [7].", "On the other hand, due to their bosonic nature, ISB polaritons are subject to final state stimulation, as it is also the case for their excitonic counterparts, and they can potentially lead to the demonstration of bosonic mid-IR or THz lasers [8], [9], [10], which would not rely on population inversion.", "Quantum cascade structures embedded in microcavities have been used to demonstrate electrically pumped light emitting polaritonic devices in the mid-IR [11].", "Furthermore, phonon-assisted polariton scattering processes have been observed [12].", "This constitutes an encouraging step towards the development of efficient electroluminescent polaritonic devices, since it is possible to rely on a proven scattering process.", "However, in the polaritonic light emitting devices (LED) demonstrated to date a key parameter is missing.", "It is not possible with a total internal reflection cavity geometry to obtain an energy minimum at very low in-plane wavevector ($k_{\\parallel }$ ) values, where the density of states could favor a high bosonic population and – in principle – final state stimulation [10], [13].", "One dimensional surface plasmon photonic crystal membranes have been successfully used to obtain ISB polaritons at low $k_{\\parallel }$ , but this geometry is unfortunately incompatible with electrical injection [14].", "In this letter, we demonstrate an optical resonator which offers an energy minimum at $ k_{\\parallel }$ =0 in the polaritonic dispersion.", "In essence, we manage to mimic the polaritonic dispersion of exciton-polariton systems based on Fabry-Perot cavities, which has been a crucial tool behind the demonstration of exciton-polariton lasers [2], [15].", "This photonic resonator is compatible with electrical injection.", "Furthermore, it can be post-processed to maximize the coupling of photons into the polaritonic states and to engineer the position of the minimum energy splitting between upper and lower polaritons at a specific position in k-space.", "The device relies on a metal-insulator-metal geometry as depicted in Figure 1a, obtained using standard gold thermo-compression waferbonding technique as detailed, for instance, in Ref.", "[16].", "The bottom mirror is a planar gold layer, 1-$\\mu $ m-thick.", "The active region consists of a multiple quantum well (QW) system: 35 repetitions of 8.3-nm-wide GaAs QWs separated by 20-nm-thick Al$_{0.33}$ Ga$_{0.67}$ As barriers.", "Figure: (a)Schematic of the device and experimental probing conditions.", "(b)Quantum well ISB absorption recorded at the Brewster angle and at room temperature.", "It is centered at 125 meV (E 12 _{12}) with a FWHM (Γ 12 \\Gamma _{12}) of 12.5 meV.Two of these structures have been grown by molecular beam epitaxy: one with a uniform Si-doping (n$_{2d}$ = $7\\times 10^{11} cm^{-2}$ ) within the wells (sample $\\#$ HM3703), and a second undoped serving as reference (sample $\\#$ HM3705).", "The bare ISB transition of the doped sample has been measured in transmission at Brewster angle.", "A clear absorption peak is detected at an energy of 125 meV ($\\lambda $ =9.9 $\\mu $ m, Figure 1b showing a Q-factor of 10.", "The Q-factor is defined as the ratio E$_{12}$ /$\\Gamma _{12}$ where $\\Gamma _{12}$ is the measured full width at half maximum (FWHM).", "The surface of the resonator is lithographically patterned with a top 1D metallic grating (Ti/Au, 5/65 nm and a thin layer of Cr, 30nm), with period $\\Lambda $ and filling factor ${ff}$ , to enable coupling of the system to the external world.", "We probe the reflectivity R($\\omega $ ,$\\theta $ ) over a large bandwidth (200 to 2000 $cm^{-1}$ ), and over a wide angular range ($13^\\circ \\le $ $\\theta $ $\\le 73^\\circ $ ) using a Fourier transform infrared spectrometer equipped with a Globar thermal source.", "The incoming radiation is P-polarized (electric field in the plane of incidence) using a wire-grid polarizer and the reflected signal is detected with a deuterated triglycine sulfate detector.", "The absolute reflectivity is obtained by normalizing the sample spectrum against a reference obtained on a planar gold surface (see supplementary material [17] for reflectance spectra at all the explored angles of incidence).", "The photonic dispersion (energy vs in-plane wavevector) R(E,$ k_{\\parallel }$ ) is then readily inferred from R($\\omega $ ,$\\theta $ ) using the relationship $k_{\\parallel }=\\frac{\\omega }{c}sin(\\theta )$ and E=$\\hbar \\omega $ .", "Figure: (a) Experimental band-diagram of the undoped sample (the color-bar provides the scale for the reflectivity).", "The grating parameters are Λ\\Lambda =3.81μ\\mu m and ff=83%.", "(b) Experimental band-diagram of the doped sample (n 2d _{2d} = 7×10 11 cm -2 7\\times 10^{11} cm^{-2}) with the appearance of the two polariton branches.", "(c) Dots are the points of minimum reflectivity extracted from the experimental data.", "Solid lines are the polaritonic dispersions simulated with the RCWA code.", "(d) Hopfield coefficients deduced from experimental data (dots) and simulated with RCWA (solid lines).", "The evenly mixed polariton quasi-particles lie at k ∥ k_{\\parallel }=0.43.The dispersion of the undoped device ($\\Lambda $ =3.81$\\mu $ m and ff=83%) measured at room temperature is shown in Figure 2a and provides information on the resonator.", "The two dispersive branches of the transverse magnetic (TM) mode folded in the first Brillouin zone are clearly observable.", "At the $\\Gamma $ -point (2nd order Bragg scattering) they lie at approximately 110 meV (upper branch) and 100 meV (lower branch), respectively.", "The upper branch, which is of interest to us since it shows a positive, quadratic dispersion, exhibits a Q-factor of 22 at the energy of the ISB transition (125 meV).", "We have then probed a doped sample featuring an identical top 1D grating.", "The sole difference with respect to the previous sample is the presence of the ISB plasmon, which is now active thanks to the presence of electrons from the Si donors.", "Figure 2b reveals that the upper dispersive branch of the TM mode is now split into two dispersive branches which correspond to the new eigenstates of the system, respectively named lower (LP) and upper (UP) polariton modes.", "A clear anti-crossing with a minimum splitting of 19 meV has been measured at an incidence angle of 37 degrees.", "The minimum Rabi splitting, which is to be gauged instead in the (E,$k_{\\parallel }$ ) space, is 17.8 meV and it is measured in this structure at a normalized $k_{\\parallel }$ = 0.43.", "The experimental results can be predicted using rigorous coupled wave analysis simulations (RCWA).", "The numerical details can be found in Refs.", "[18], [19].", "The ISB plasmon couples only to the z-component of the electric field (the component along the growth direction) because of the dipolar selection rule.", "The multiple QW structure can therefore be modelled as an anisotropic dispersive medium.", "The in-plane components of the effective permittivity tensor take into account the optical phonons and can be found in Ref.", "[20].The ISB transition contribution is included in the z-component of the tensor using the Zaluzny-Nalewajko approach[21]: $\\epsilon _{z}(\\omega )=\\epsilon _{\\infty }\\left(1-f_{0}\\frac{\\epsilon _{\\infty }^{2}}{\\epsilon ^{2}_{w}}\\frac{\\omega ^{2}_{p}}{\\omega ^{2}_{12}-\\omega ^{2}-i\\omega \\Gamma _{12}}\\right)^{-1}$ where $\\Gamma _{12}$ is the FWHM of the ISB transition, $\\epsilon _{w}$ is the dielectric constant in the well material and $f_{0}$ is the oscillator strength approximated to one for this two level system.", "The plasma frequency is expressed as follows: $\\omega _{p}= \\sqrt{\\frac{\\pi e^2 n_{2d}}{\\epsilon _{\\infty } m^* (L_{b}+L_{w})}}$ where $L_{b}$ and $L_{w}$ are respectively the barrier and well thicknesses, and $n_{2d}$ stands for the dopants’ concentration.", "Finally the dielectric function of gold is defined according to Ref.[22].", "Figure 2c shows the simulated polariton dispersion (continuous lines) superimposed onto the reflectivity minima as extracted from the experimental data (dots).", "Theory and experiment are in excellent agreement and - most importantly – the dispersion presents an energy minimum at $k_{\\parallel }$ =0.", "The eigenstates of the coupled system are a linear superposition of the uncoupled eigenstates $\|\\psi _{i,k}\\rangle $ (with $\|\\psi _{1,k}\\rangle $ the fundamental state and $\|\\psi _{2,k}\\rangle $ the excited state of the ISB plasmon) and $\|n\\rangle $ the state with n photons in the cavity, as follows: $•\|UP\\rangle =\\alpha _{UP}\|\\psi _{1,k},1\\rangle +\\beta _{UP}\|\\psi _{2,k},0\\rangle ,$ $•\|LP\\rangle =\\alpha _{LP}\|\\psi _{1,k},1\\rangle +\\beta _{LP}\|\\psi _{2,k},0\\rangle $ where UP is the higher energy eigenstate and LP is the lower energy eigenstate.", "The coefficients $\\alpha _{LP,UP}$ , $\\beta _{LP,UP}$ are called the Hopfield coefficients [23] and they permit to gauge the weight of the photonic/material components within the polariton branches.", "Using analytical formulas (See supplementary material [17] for details on the extraction of the Hopfield coefficients) and the set of experimental and simulated data, we have inferred the fractional contribution of each component, as reported in Fig.", "2d.", "An even mixing (50-50%) is observed at a normalized $k_{\\parallel }$ value of 0.43, which corresponds to the minimum Rabi splitting.", "Further away from the minimum splitting, the quasi-particles lose their mixed nature.", "Note however that even at $k_{\\parallel }$ =0 the LP still maintains more than 10% of photonic component.", "In the perspective of developing polaritonic light emitting devices and especially lasers, a few aspects need to be taken into account.", "First, the control of the Hopfield coefficients is of crucial importance, since it permits to tailor / maximise the polaritonic lifetime.", "For instance, the possibility to engineer the minimum energy splitting at a desired position in the $k_{\\parallel }$ -space would be an important asset.", "The second key point is the need to efficiently populate the polaritonic states, via optical or electrical pumping.", "The first point, and also the second in the case of optical pumping, can be addressed with a post-processing approach that we have recently demonstrated on a similar resonator geometry [24].", "In the general framework of temporal coupled mode theory (TCMT) [25], [26], the device employing the undoped active region can be described as a one port system.", "Its electromagnetic coupling to the external world is described by a radiative damping rate $\\gamma _{r}$ , while the resonator losses are lumped into a non-radiative rate $\\gamma _{nr}$ .", "When the two damping rates are matched ($\\gamma _{r}=\\gamma _{nr}$ ), all the incoming photons are coupled into the resonant photonic mode, a situation known as critical coupling [24], [27].", "Interestingly, this general concept also holds for the regime of strong coupling, as developed in Ref.", "[28].", "To this scope, it is necessary to add an additional oscillator to model the ISB plasmon, and introduce the light-matter coupling constant [29].", "The TCMT equations become: $\\frac{db}{dt}=(i\\omega _{12}-\\gamma _{12})b+i\\Omega a$ $\\frac{da}{dt}=(i\\omega _{c}-\\gamma _{c})b+i\\Omega b+ks^+$ $s^-=cs^+ +ad$ where $\|s^\\pm \|^2$ is the energy flux per unit time ingoing (outgoing) into (out of) the system; $\\vert a\\vert ^2$ is the total electromagnetic energy stored in the cavity; $\\vert b\\vert ^2$ is the total energy stored in the matter resonator; $\\gamma _{12}$ is the ISB plasmon damping rate and $\\omega _{12}$ its frequency; $\\gamma _{c}= \\gamma _{r}+\\gamma _{nr}$ is the total cavity damping rate; and $\\Omega $ is the light-matter coupling constant (i.e.", "the Rabi frequency).", "If $2\\Omega \\gg \\mid \\gamma _{r}+ \\gamma _{nr}-\\gamma _{12}\\mid ^2$ the system is in the strong coupling regime [32].", "It is possible to show that perfect energy feeding into the polaritonic states takes place under a novel strong critical coupling (SCC) condition which is $\\gamma _{r}=\\gamma _{nr} +\\gamma _{12}$ .", "The TCMT description of the system is of crucial importance as it allows one to naturally describe on the same footing the coupling between the resonator and the material excitations – which gives rise to polaritons – simultaneously with the coupling of the system to the external world.", "Note: the damping rates used in the frame of TCMT are twice smaller than the experimental ones since the time-dependent amplitude is now considered.", "Figure: (a) Angularly-resolved experimental reflectance at the minimum energy splitting for different etching depths.", "The angle of measurement is mentioned next to the curve.", "(b) Simulated reflectance with the CMT.", "The radiative damping rate (γ r \\gamma _{r} in meV) is the tuning parameter.", "The round points represent the regime of strong critical coupling.To experimentally demonstrate the existence of this regime, we iteratively etched the semiconductor material between the grating metallic fingers with an inductively coupled plasma reactor and using the metallic grating as mask.", "Etching has a direct impact on $\\gamma _{r}$ while it essentially does not affect $\\gamma _{nr}$ as demonstrated in Ref.", "[28].", "After each etching step, we measure the polaritonic dispersion with our experimental set-up.", "Figure 3a shows the reflectivity of the sample at an incidence angle corresponding to the minimum splitting.", "The energy scale (x-axis) is renormalized by the frequency of the ISB transition.", "As the sample is etched, more photons get coupled into the polaritonic states.", "From an initial absorption of $\\sim 50\\%$ , we reach $80\\%$ at an etch depth of 190 nm.", "Further etching brings the minimum splitting point outside the experimentally accessible light cone and makes the measurement impossible.", "This behaviour is well reproduced by the TCMT, as shown by the simulations in Fig.", "3b, which report $\|r(\\omega )\|^2=\|{s^-}/{s^+}\|^2$ as obtained from the equations above.", "The sole fitting parameter is $\\gamma _{r}$ , which reveals that the resonator is under-coupled: the etching procedure increases $\\gamma _{r}$ thus driving it towards the critical coupling condition.", "Note: the slight reduction of the splitting, as inferred from the reflectance minima in Fig.", "3a, further confirms that we are affecting the damping rates of the system.", "We can now explore how to engineer the position of the minimum Rabi splitting with respect to the in-plane wave-vector $k_{\\parallel }$ .", "As the sample gets etched, the effective refractive index contrast increases as we are carving air-holes into a 1D photonic crystal.", "This gradual change of index contrast broadens the gap at $k_{\\parallel }$ =0 and permits to spectrally blue-shift the upper photonic branch of the resonator to a desired value.", "Figure 4a presents the Hopfield coefficients extracted from the experimental results on the non-etched sample (black line) and after a 290-nm-deep etch (dotted line).", "The Hopfield coefficients of this last etched configuration have been deduced with the RCWA code (See supplementary material [17] for details on the extraction of the Hopfield coefficients).", "In this case, we have been able to engineer the regime of evenly mixed polaritonic states (i.e.", "equal Hopfield coefficients) from $k_{\\parallel }$ =0.43 to $k_{\\parallel }$ =0.", "This post-processing approach which allows one to tune the position of the minimum splitting is even more remarkable as it occurs simultaneously with the onset of the strong critical coupling regime.", "Figure: (a) Calculated Hopfield coefficients for the initial device and after 290 nm of etching (dots).", "The evenly mixed polaritons occur now at the Γ\\Gamma point of the polaritonic dispersion.", "(b) Experimental band-diagram (the color-bar provides the scale for the reflectivity) of the device final configuration with the injection angle for optical pumping experiment at 55 ∘ ^\\circ (dotted line) and the representation of a longitudinal optical phonon energy separation (E LO _{LO}) as a potential scattering mechanism.", "The crosses represent the reflectance minima.The final resonator configuration (Fig.", "4b) is of particular relevance in the context of polariton scattering experiments under optical or electrical pumping.", "Figure 4b presents the polaritonic dispersion at the final etch depth of 290 nm: the minimum splitting occurs at $k_{\\parallel }$ =0 and the light coupling is greatly improved over the whole dispersion diagram.", "In particular, it is possible to envision an optical pumping experiment where an incident beam at $\\theta =63^\\circ $ efficiently generates ISB polaritons, since the absorption is larger than $80\\%$ .", "Efficient population of the $k_{\\parallel }$ =0 state in turn relies on polariton-LO phonon scattering [17].", "On one hand, this is a promising architecture for polaritonic LEDs.", "On the other hand, with the appropriate lifetime engineering, the occupation number of the ground state could be improved and even reach the value of 1, necessary for the final state stimulation process [16].", "In conclusion, we have demonstrated room temperature strong coupling between a mid-infrared intersubband plasmon and the fundamental cavity mode of a metal-insulator-metal resonator, with a polaritonic dispersion presenting an energy minimum at $k_{\\parallel }$ =0.", "We have also shown the possibility to maximize the coupling of photons into the polaritonic states (close to critical coupling) and simultaneously to engineer the position of the minimum Rabi splitting at a desired value of the in-plane wavevector.", "These results constitute a building block for future developments of long wavelength optically or electrically pumped polaritonic light emitting devices.", "We thank N. Isac for help with the wafer-bonding process, and we acknowledge financial support from the Triangle de la Physique (Project 'INTENSE'), and from the French National Research Agency (ANR-09-NANO-017 'HI-TEQ').", "This work was partly supported by the French RENATECH network.", "R.C.", "acknowledges partial support from the ERC 'GEM' grant (Grant agreement $\\#$ 306661), and A.T. from the ERC Advanced Grant 'SoulMan'." ] ]	1606.05090
[ [ "PECOK: a convex optimization approach to variable clustering" ], [ "Abstract The problem of variable clustering is that of grouping similar components of a $p$-dimensional vector $X=(X_{1},\\ldots,X_{p})$, and estimating these groups from $n$ independent copies of $X$.", "When cluster similarity is defined via $G$-latent models, in which groups of $X$-variables have a common latent generator, and groups are relative to a partition $G$ of the index set $\\{1, \\ldots, p\\}$, the most natural clustering strategy is $K$-means.", "We explain why this strategy cannot lead to perfect cluster recovery and offer a correction, based on semi-definite programing, that can be viewed as a penalized convex relaxation of $K$-means (PECOK).", "We introduce a cluster separation measure tailored to $G$-latent models, and derive its minimax lower bound for perfect cluster recovery.", "The clusters estimated by PECOK are shown to recover $G$ at a near minimax optimal cluster separation rate, a result that holds true even if $K$, the number of clusters, is estimated adaptively from the data.", "We compare PECOK with appropriate corrections of spectral clustering-type procedures, and show that the former outperforms the latter for perfect cluster recovery of minimally separated clusters." ], [ "Introduction", "The problem of variable clustering is that of grouping similar components of a $p$ -dimensional vector $X=(X_{1},\\ldots ,X_{p})$ .", "These groups are referred to as clusters.", "In this work we investigate the problem of cluster recovery from a sample of $n$ independent copies of $X$ .", "Variable clustering has had a long history in a variety of fields, with important examples stemming from gene expression data [22], [8], [10] or protein profile data [3].", "The solutions to this problem are typically algorithmic and entirely data based.", "They include applications of $K$ -means, spectral clustering, or versions of them.", "The statistical properties of these procedures have received a very limited amount of investigation.", "It is not currently known what probabilistic cluster model on $X$ can be estimated by these popular techniques, or by their modifications.", "Our work offers an answer to this question.", "We study variable clustering in $G$ -models, introduced in Bunea et al.", "[4], which is a class of models that offer a probabilistic framework for cluster similarity.", "For a given partition $G=\\lbrace G_{k} \\rbrace _{k=1,\\ldots ,K}$ of $\\lbrace 1,\\ldots ,p\\rbrace $ , the most general of these models, called the $G$ -block covariance model, makes the mild assumption that permuting two variables with indices in the same group $G_k$ of the partition does not affect the covariance $\\Sigma $ of the vector $X=(X_{1},\\ldots ,X_{p})$ .", "Hence, the off-diagonal entries of the covariance $\\Sigma $ depend only on the group membership of the entries, enforcing a block-structure on $\\Sigma $ .", "This block-structure relative to a partition $G$ can be summarized by the matrix decomposition $\\Sigma =A C A^t +\\Gamma ,$ where the $p \\times K$ matrix $A$ with entries $A_{ak} : = 1_{\\lbrace a\\in G_k\\rbrace }$ assigns the index of a variable $X_a$ to a group $G_{k}$ , the matrix $C$ is symmetric, and $\\Gamma $ is a diagonal matrix with $\\Gamma _{aa} = \\gamma _{k}$ , for all $a \\in G_k$ .", "We focus on an important sub-class of the $G$ -block covariance models, the class of $G$ -latent models, also discussed in detail in Bunea et al.", "[4].", "We say that a zero mean vector $X$ has a latent decomposition with respect to a generic partition $G$ , if $X_{a}=Z_{k}+E_{a},\\ \\textrm {for all}\\ a\\in G_{k},\\ \\textrm {and all}\\ k=1,\\ldots , K,$ with $Z=(Z_{1},\\ldots ,Z_{K})$ a $K$ -dimensional zero-mean latent vector assumed to be independent of the zero-mean error vector $E=(E_{1},\\ldots ,E_{p})$ , which itself has independent entries, and the error variances are equal within a group.", "It is immediate to see that if (REF ) holds, then the covariance matrix $\\Sigma $ of $X$ has a $G$ -block structure (REF ), with $C = Cov(Z)$ and $\\Gamma = Cov(E)$ .", "Therefore, the latent $G$ -models are indeed a sub-class of the $G$ -covariance models, and the models are not necessarily equivalent, since, for instance, the matrix $C$ in (REF ) can be negative definite.", "Intuitively, it is clear that if the clusters of $X$ are well separated, they will be easy to estimate accurately, irrespective of the model used to define them.", "This motivates the need for introducing metrics for cluster separation that are tailored to $G$ -models, and for investigating the quality of cluster estimation methods relative to the size of cluster separation.", "In what follows, we refer to the $X$ -variables with indices in the same group $G_k$ of a partition $G$ given by either (REF ) or (REF ) as a cluster.", "When the $G$ -latent model (REF ) holds, two clusters are separated if they have different generators.", "Therefore, separation between clusters can be measured in terms of the canonical \"within-between group\" covariance gap $\\Delta (C):= \\min _{j<k}\\left(C_{kk}+C_{jj}-2C_{jk} \\right)=\\min _{j<k}\\mathbf {E}\\left[(Z_{j}-Z_{k})^2\\right],$ since $\\Delta (C) = 0$ implies $Z_j = Z_k$ a.s.", "When the $G$ -block covariance model (REF ) holds, the canonical separation metric between clusters has already been considered in [4] and is given by $\\textsc {MCord}(\\Sigma ):=\\min _{a\\stackrel{G}{\\nsim } b}\\max _{c\\ne a,b}\|\\Sigma _{ac}-\\Sigma _{bc}\|.", "$ The two metrics are connected via the following chain of inequalities, valid as soon as the size of the smallest cluster is larger than one: $ \\Delta (C) \\le 2\\textsc {MCord}(\\Sigma ) \\le 2 \\sqrt{\\Delta (C)}\\ \\max _{k=1,\\ldots ,K} \\sqrt{C_{kk}}.", "$ The variable clustering algorithm Cord introduced in [4] was shown to recover clusters given by (REF ), and in particular by (REF ), as soon as $ \\textsc {MCord}(\\Sigma ) \\gtrsim \\sqrt{\\log (p)\\over n}.$ Moreover, the rate of $\\sqrt{\\log (p)/n}$ was shown in [4] to be the minimax optimal cluster separation size for correct cluster recovery, with respect to the $\\textsc {MCord}$ metric, in both $G$ -block covariance matrix and $G$ -latent models.", "The first inequality in (REF ) shows that if we are interested in the class of $G$ -latent models together with their induced canonical cluster separation metric $\\Delta (C)$ , the Cord algorithm of [4] also guarantees correct cluster recovery as soon as $\\Delta (C)\\gtrsim \\sqrt{\\log (p)/n}$ .", "However, the second inequality in (REF ) suggests that $\\Delta (C)$ can be, in order, as small as $ [\\textsc {MCord}(\\Sigma )]^2$ , implying that, with respect to this metric, we could recover clusters that are closer together.", "This motivates a full investigation of variable clustering in $G$ -latent models (REF ), relative to the $\\Delta (C)$ cluster separation metric, as outlined below." ], [ "Our contribution", "We assume that the data consist in i.i.d.", "observations $X^{(1)},\\ldots ,X^{(n)}$ of a random vector $X$ with mean 0 and covariance matrix $\\Sigma $ , for which (REF ) holds relative to a partition $G$ .", "Our work is devoted to the development of a computationally feasible method that yields an estimate $\\widehat{G}$ of $G$ , such that $\\widehat{G} = G$ , with high probability, when $\\Delta (C)$ is as small as possible, and to the characterization of the minimal value of $\\Delta (C)$ , from a minimax perspective.", "We begin by highlighting our main results.", "For simplicity, we discuss here the case where the $K$ clusters have a similar size, so that the size of the smaller cluster is $m\\approx p/K$ .", "We refer to Section for the general case.", "When $X$ is Gaussian, Theorem REF below shows that no algorithm can estimate $G$ correctly, with high probability, when the latent model (REF ) holds with $ C$ fulfilling $\\Delta (C) \\lesssim \|\\Gamma \|_{\\infty } \\left(\\sqrt{\\log (p)\\over nm}\\bigvee {\\log (p)\\over n}\\right).", "$ This result shows that the minimax optimal value for exact variable clustering according to model (REF ), and with respect to the metric $\\Delta (C)$ , can be much smaller than the above-mentioned $\\sqrt{{\\log (p)}/{n}}$ and that the threshold for $\\Delta (C)$ is sensitive to the size of $m$ : As $m$ increases the clustering problem becomes easier, in that a smaller degree of cluster separation is needed for exact recovery.", "This property is in contrast with the fact that the $\\textsc {MCord}$ metric is not affected by the size of $m$ .", "Our main result, Theorem REF of Section , shows that perfect cluster recovery is possible, with high probability, via the polynomial-time PECOK algorithm outlined below, when $\\Delta (C) \\gtrsim \|\\Gamma \|_{\\infty } \\left(\\sqrt{\\log (p)\\vee K\\over nm}\\bigvee {\\log (p)\\vee K\\over n}\\right).$ PECOK is therefore minimax optimal as long as the number $K$ of clusters is bounded from above by $\\log (p)$ , and nearly minimax optimal otherwise.", "To describe our procedure, we begin by defining the block matrix $B$ with entries $B_{ab}= {\\left\\lbrace \\begin{array}{ll} {1\\over \|G_{k}\|} & \\textrm {if a and b are in the same group G_{k},}\\\\0 & \\textrm {if a and b are in a different group.}\\end{array}\\right.", "}$ The groups in a partition $G$ are in a one-to-one correspondence with the non-zero blocks of $B$ .", "Our PECOK algorithm has three steps, and the main step 2 produces an estimator $\\widehat{B}$ of $B$ from which we derive the estimated partition $\\widehat{G}$ .", "The three steps of PECOK are: Compute an estimator $\\widehat{\\Gamma }$ of the matrix $\\Gamma $ .", "Solve the semi-definite program (SDP) $\\widehat{B}=\\operatornamewithlimits{argmax}_{B \\in \\mathcal {C}}\\langle \\widehat{\\Sigma } - \\widehat{\\Gamma }, B\\rangle , $ where $\\widehat{\\Sigma }$ the empirical covariance matrix and $\\mathcal {C}:=\\left\\lbrace B \\in \\mathbb {R}^{p\\times p}:\\begin{array}{l}\\bullet \\ B \\succcurlyeq 0 \\ \\ \\text{(symmetric and positive semidefinite)} \\\\\\bullet \\ \\sum _a B_{ab} = 1,\\ \\forall b\\\\\\bullet \\ B_{ab}\\ge 0,\\ \\forall a,b\\\\\\bullet \\ \\operatorname{tr}(B) = K\\end{array}\\right\\rbrace .$ Compute $\\widehat{G}$ by applying a clustering algorithm on the rows (or equivalently columns) of $\\widehat{B}$ .", "The construction of an accurate estimator $\\widehat{\\Gamma }$ of $\\Gamma $ is a crucial step for guaranteeing the statistical optimality of the PECOK estimator.", "Estimating $\\Gamma $ before estimating the partition itself is a non-trivial task, and needs to be done with extreme care.", "We devote Section REF below to the construction of an estimator $\\widehat{\\Gamma }$ for which the results of Theorem REF hold.", "Sections REF and REF are devoted to the second step.", "In Section REF we motivate the SDP (REF ) for estimating $B$ , and present its analysis in Section REF , proving that $\\widehat{B}=B$ when (REF ) holds.", "The required inputs for Step 2 of our algorithm are: (i) $\\widehat{\\Sigma }$ , the sample covariance matrix; (ii) $\\widehat{\\Gamma }$ , the estimator produced at Step 1; and (iii) $K$ , the number of groups.", "When $K$ is not known, we offer a procedure for selecting it in a data adaptive fashion in Section below.", "Theorem REF of this section shows that the resulting estimator enjoys the same properties as $\\widehat{B}$ given by (REF ), under the same conditions.", "The construction of $\\widehat{B}$ , when $K$ is either known, or estimated from the data, requires solving the SDP (REF ) or a variant of it, over the convex domain $\\mathcal {C}$ .", "Finally, in Step 3, we recover the estimated partition $\\widehat{G}$ from $\\widehat{B}$ by using any clustering method that employs the rows or, equivalently, columns, of $\\widehat{B}$ as input.", "This step is done at no additional accuracy cost, as shown in Corollary REF , Section REF below.", "We summarize our three-fold contributions below.", "(i) Minimax lower bounds.", "In Theorem REF we establish minimax limits on the size of the $\\Delta (C)$ -cluster separation metric, for exact partition recovery, over the class of identifiable $G$ -latent variable models.", "The bounds indicate that the difficulty of the problem decreases not only when the sample size $n$ is large, but also, as expected, when $m$ , the size of the smallest cluster, is large.", "To the best of our knowledge, this is the first result of this nature for clustering via $G$ -latent variable models.", "Our results can be contrasted with the minimax cluster separation rates in [4] with respect to the $\\textsc {MCord}$ metric.", "They can also be contrasted with results regarding clustering of $p$ variables from data of a different nature, network data, based on a different model, the Stochastic Block Model (SBM).", "We elaborate on this point in the remark below.", "(ii) Near minimax-optimal procedure.", "We introduce and analyze PECOK, a new variable clustering procedure based on Semi-Definite Programing (SDP) for variable clustering, that can be used either when the number of clusters, $K$ , is known, or when it is unknown, in which case we estimate it.", "We prove that, in either case, the resulting partition estimator $\\widehat{G}$ recovers $G$ , with high probability, at a near-optimal $\\Delta (C)$ -cluster separation rate.", "(iii) Perfect partition recovery requires corrections of $K$ -means or spectral clustering.", "We view PECOK as a correction of existing clustering strategies, the correction being tailored to $G$ -models.", "We use the connection between $K$ -means clustering and SDP outlined in Peng and Lei [20] to explain why $K$ -means cannot perfectly recover $G$ in latent models, and present the details in Section REF .", "To the best of our knowledge, this is the first work that addresses the statistical properties of corrections of $K$ -means for variable clustering.", "Moreover, we connect PECOK with another popular algorithm, spectral clustering, and explain why the latter cannot be directly used, in general, for perfect clustering in $G$ -models.", "The details are presented in Section ." ], [ "Remark.", "Variable clustering from network data via Stochastic Block Models (SBM) has received a large amount of attention in the past years.", "We contrast our contribution with that made in the SBM literature, for instance in [9], [15], [6], [16], [1], [19], [14].", "Although general strategies such as SDP and spectral clustering methods are also employed for cluster estimation from network data, and share some similarities with those analyzed in this manuscript, there are important differences.", "The most important difference stems from the nature of the data: the data analyzed via SBM is a $p\\times p$ binary matrix, called the adjancency matrix, with entries assumed to have been generated as independent Bernoulli random variables.", "In contrast, the data matrix $\\mathbf {X}$ generated from a G-latent model is a $n\\times p$ matrix with real entries, and rows viewed as i.i.d copies of a $p$ -dimensional vector with dependent entries.", "In the SBM literature, SDP-type and spectral clustering procedures are directly applied to the adjacency matrix, whereas we need to apply them to the empirical covariance matrix $\\widehat{\\Sigma }:=\\mathbf {X}^t \\mathbf {X}/n$ , not directly on the observed ${\\bf X}$ .", "This has important repercussions on the statistical analysis of the cluster estimates.", "Contrary to the SBM framework, $\\widehat{\\Sigma }$ does not simply decompose as the sum between the clustering signal “$A C A^t$ ” (REF ) and a noise component, but its decomposition also contains cross-product terms.", "The analysis of these additional terms is non-standard, and needs to be done with care, as illustrated by the proof of our Theorem REF .", "Moreover, in contrast to procedures tailored to the SBM underlying model, SDP and spectral methods for $G$ -latent models need to be corrected in a non-trivial fashion, as mentioned in (iii) above." ], [ "Organization of the paper.", "In Section we give conditions for partition identifiability in $G$ -latent models.", "We also introduce the notation used throughout the paper.", "In Section REF we present the connections between $K$ -means and PECOK.", "In Section REF we prove that one can construct an estimator of $\\Gamma $ , with $\\sqrt{\\log p/n}$ accuracy in supremum norm, before estimating the partition $G$ .", "In Section REF , Theorem REF and Corollary REF show that PECOK can recover $G$ exactly, with high probability, as long as $\\Delta (C)$ is sufficiently large and the number of groups $K$ is known.", "In Section , Theorem REF and Corollary REF show that the same results hold, under the same conditions, when Step 2 of the PECOK algorithm is modified to include a step that allows for the data dependent selection of $K$ .", "Theorem REF of Section gives the minimax lower bound for the $\\Delta (C)$ cluster separation for perfect recovery.", "In Section we give the connections between PECOK and spectral clustering.", "Theorem REF gives sufficient conditions under which spectral clustering recovers partially the target partition in $G$ -models, as a function of a given misclassification proportion.", "We present extensions in Section .", "All our proofs are collected in Section and the Appendix." ], [ "Model and Identifiability", "We begin by observing that as soon as the latent decomposition (REF ) holds for $G$ it also holds for a sub-partition of $G$ .", "It is natural therefore to seek the smallest of such partitions, that is the partition $G^$ with the least number of groups for which (REF ) holds.", "The smallest partition is defined with respect to a partial order on sets, and one can have multiple minimal partitions.", "If a $G$ -model holds with respect to a unique minimal partition $G^$ , we call the partition identifiable.", "We present below sufficient conditions for identifiability that do not require any distributional assumption for $X$ .", "We assume that $X$ is a centered random variable with covariance matrix $\\Sigma $ .", "If $X$ follows the latent decomposition (REF ) with respect to $G^$ , we recall that $\\Sigma =A C^ A^t +\\Gamma .$ with $C^=cov(Z)$ , a semi-positive definite matrix, and $\\Gamma $ a diagonal matrix.", "We also assume that the size $m$ of the smallest cluster is larger than 1.", "Since $C^$ is positive semi-definite, we always have $\\Delta (C^)=\\min _{j<k} (e_{j}-e_{k})^tC^(e_{j}-e_{k})\\ge 2 \\lambda _{K}(C^)\\ge 0.$ Lemma REF below shows that requiring $\\Delta (C^)\\ne 0$ ensures that the latent decomposition (REF ) holds with respect to a unique partition $G^$ .", "Lemma 1 If the latent decomposition (REF ) holds with $m>1$ and $\\Delta (C^)>0$ , then the partition $G^$ is identifiable.", "We remark that when $m = 1$ , the partition may not be identifiable, and we refer to [4] for a counterexample.", "We also remark that when $\\Delta (C^)=0$ and $\\Gamma =\\gamma I$ , the partition $G^$ is not identifiable.", "Therefore, the sufficient conditions for identifiability given by Lemma REF are almost necessary, and we refer to Section for further discussion.", "In the remaining of the paper, we assume that we observe $n$ i.i.d.", "realizations $X^{(1)},\\ldots ,X^{(n)}$ of a vector $X$ following the latent decomposition (REF ) with $m>1$ and $\\Delta (C^)>0$ , so that the partition $G^$ is identifiable.", "We will also assume that $X \\sim N(0, \\Sigma )$ .", "We refer to Section for the sub-Gaussian case." ], [ "Notation", "In the sequel, $\\mathbf {X}$ , $\\mathbf {E}$ , and $\\mathbf {Z}$ respectively refer to the $n\\times p$ (or $n\\times K$ for $\\mathbf {Z}$ ) matrices obtained by stacking in rows the realizations $X^{(i)}$ , $E^{(i)}$ and $Z^{(i)}$ , for $i=1,\\ldots ,n$ .", "We use $M_{:a} $ , $M_{b:}$ , to denote the $a$ -th column or, respectively, $b$ -th row of a generic matrix $M$ .", "The sample covariance matrix $\\widehat{\\Sigma }$ is defined by $ \\widehat{\\Sigma } = \\frac{1}{n} {\\bf X}^{t}{\\bf X}={1\\over n}\\sum _{i=1}^nX^{(i)}(X^{(i)})^t.", "$ Given a vector $v$ and $p\\ge 1$ , $\|v\|_{p}$ stands for its $\\ell _p$ norm.", "Similarly $\|A\|_p$ refers to the entry-wise $\\ell _p$ norm.", "Given a matrix $A$ , $\\Vert A\\Vert _{op}$ is its operator norm and $\\Vert A\\Vert _F$ refers to the Frobenius norm.", "The bracket $\\langle .,.\\rangle $ refers to the Frobenius scalar product.", "Given a matrix $A$ , denote $\\mathrm {supp}(A)$ its support, that is the set of indices $(i,j)$ such that $A_{ij}\\ne 0$ .", "We use $[p]$ to denote the set $\\lbrace 1, \\ldots , p\\rbrace $ and $I$ to denote the identity matrix.", "We define the variation semi-norm of a diagonal matrix $D$ as $\|D\|_{V}:=\\max _{a} D_{aa}-\\min _{a} D_{aa}$ .", "We use $B \\succcurlyeq 0$ to denote a symmetric and positive semidefinite matrix.", "We use the notation $a\\stackrel{G}{\\sim }b$ whenever $a, b \\in G_{k}$ , for the same $k$ .", "Also, $m=\\min _k \|G_k\|$ stands for the size of the smallest group.", "The notation $\\gtrsim $ and $\\lesssim $ is used for whenever the inequalities hold up to multiplicative numerical constants." ], [ " A convex relaxation of penalized $K$ -means ", "For the remaining of the paper we will discuss the estimation of the identifiable partition $G^$ discussed above.", "Knowing $G^$ is equivalent with knowing whether “$a$ and $b$ are in the same group” or “$a$ and $b$ are in different groups\", which is encoded by the normalized partnership matrix $B^$ given by (REF ).", "In what follows, we provide a constructive representation of $B^$ , that holds when $\\Delta (C^) > 0$ , and that can be used as the basis of an estimation procedure.", "To motivate this representation, we begin by noting that the most natural variable clustering strategy, when the $G$ -latent model (REF ) holds, would be $K$ -means [17], when $K$ is known.", "The estimator offered by the standard $K$ -means algorithm is $ \\widehat{G} \\in \\mathop {\\textrm {argmin}}_{G}\\text{crit}({\\bf X},G)\\quad \\textrm {with}\\quad \\text{crit}({\\bf X},G)=\\sum _{a=1}^p\\min _{k=1,\\ldots ,K}\\Vert {\\bf X}_{:a}-\\bar{\\bf X}_{G_{k}}\\Vert ^2,$ and $\\bar{\\bf X}_{G_{k}}=\|G_{k}\|^{-1} \\sum _{a\\in G_{k}} {\\bf X}_{:a}$ .", "Theorem 2.2 in Peng and Wei [20] shows that solving the $K$ -means problem is equivalent to finding the global maximum $\\widehat{B}=\\operatornamewithlimits{argmax}_{B \\in \\mathcal {D}} \\langle \\widehat{\\Sigma },B\\rangle $ for $\\mathcal {D}$ given by $\\mathcal {D}:=\\left\\lbrace B \\in \\mathbb {R}^{p\\times p}:\\begin{array}{ll}\\bullet \\ B \\succcurlyeq 0 \\\\\\bullet \\ \\sum _a B_{ab} = 1,\\ \\forall b\\\\\\bullet \\ B_{ab}\\ge 0,\\ \\forall a,b\\\\\\bullet \\ \\operatorname{tr}(B) = K \\\\\\bullet \\ B^2 = B\\end{array}\\right\\rbrace ,$ and then recovering $\\widehat{G}$ from $\\widehat{B}$ .", "However, we show below that we cannot expect the $K$ -means estimator $\\widehat{B}$ given by (REF ) to equal $B^$ , with high probability, unless additional conditions are met.", "This stems from the fact that $B^$ does not equal $\\underset{B \\in \\mathcal {D}}{\\operatornamewithlimits{argmax}} \\ \\langle {\\Sigma } , B \\rangle $ under the identifiability condition $\\Delta (C^) > 0$ , but rather under the stronger condition (REF ) below, which is shown in Proposition REF to be sufficient.", "Proposition 1 Assume model (REF ) holds.", "If $\\Delta (C^)> \\frac{2}{m}\|\\Gamma \|_{V} ,$ then $ B^* = \\underset{B \\in \\mathcal {D}}{\\operatornamewithlimits{argmax}} \\ \\langle {\\Sigma } , B \\rangle .$ Proposition REF shows that, moreover, Condition (REF ) is needed.", "Proposition 2 Consider the model (REF ) with $C^=\\left[{\\scriptsize \\begin{array}{ccc}\\alpha & 0 & 0\\\\0 & \\beta & \\beta -\\tau \\\\0 & \\beta -\\tau & \\beta \\end{array}}\\right]\\ , \\quad \\quad \\Gamma =\\left[{\\scriptsize \\begin{array}{ccc}\\gamma _+&0&0\\\\ 0& \\gamma _-&0 \\\\ 0& 0& \\gamma _-\\end{array}}\\right],\\quad \\text{ and \\ }\|G_1\|=\|G_1\|=\|G_3\|=m\\ .$ The population maximizer $B_{\\Sigma }=\\operatornamewithlimits{argmax}_{B\\in \\mathcal {D}} \\langle \\Sigma , B\\rangle $ is not equal to $B^$ as soon as $2\\tau =\\Delta (C^)< \\frac{2}{m}\|\\Gamma \|_{V}\\ .$ Corollary 1 Assume model (REF ) holds with $\\Delta (C^) > 0$ .", "Then $ B^= \\underset{B \\in \\mathcal {D}}{\\operatornamewithlimits{argmax}} \\ \\langle \\Sigma -\\Gamma , B \\rangle .$ Propositions REF and REF therefore show that the $K$ -means algorithm does not have the capability of estimating $B^$ , unless $\\Gamma = \\gamma I$ , whereas Corollary REF suggests that a correction of the type $\\widetilde{B} = \\underset{B \\in \\mathcal {D}}{\\operatornamewithlimits{argmax}} \\ \\left\\lbrace \\langle \\widehat{ \\Sigma }, B \\rangle - \\langle \\widehat{\\Gamma }, B \\rangle \\right\\rbrace .$ might be successful.", "In light of (REF ), we can view this correction as a penalization of the $K$ -means criterion.", "There are two difficulties with this estimation strategy.", "The first one regards the construction of the estimator $\\widehat{\\Gamma }$ of $\\Gamma $ .", "Although, superficially, this may appear to be a simple problem, recall that we do not know the partition $G^$ , in which case the problem would, indeed, be trivial.", "Instead, we need to estimate $\\Gamma $ exactly for the purpose of estimating $G^$ .", "We show how this vicious circle can be broken in a satisfactory manner in Section REF below.", "The second difficulty regards the optimization problem (REF ): although the objective function is linear, $\\mathcal {D}$ is not convex.", "Following Peng and Wei [20], we consider its convex relaxation $\\mathcal {C}$ given in (REF ) above, in which we drop the constraint $B^2=B$ .", "This leads to our proposed estimator announced in (REF ), the PEnalized COnvex relaxation of K-means (PECOK) summarized below: PECOK algorithm $& \\text{Step 1.", "Estimate} \\ \\Gamma \\ \\text{by} \\ \\widehat{\\Gamma }.", "\\nonumber \\\\& \\text{Step 2.", "Estimate} \\ B^* \\, \\text{ by} \\ \\widehat{B}=\\operatornamewithlimits{argmax}_{B \\in \\mathcal {C}}\\langle \\widehat{\\Sigma } - \\widehat{\\Gamma }, B\\rangle .", "\\nonumber \\\\& \\text{Step 3.", "Estimate} \\ G^* \\, \\text{ by applying a clustering algorithm to the columns of }\\, \\widehat{B}.", "\\nonumber $ Our only requirement on the clustering algorithm applied in Step 3 is that it succeeds to recover the partition $G^$ when applied to $B^$ .", "The standard $K$ -means algorithm [17] seeded with $K$ distinct centroids, kmeans++ [2], or any approximate $K$ -means as defined in (REF ) in Section , fulfill this property.", "We also have: Proposition 3 Assume model (REF ) holds.", "$\\text{If} \\ \\ \\ \\ \\Delta (C^)> \\frac{2}{m}\|\\Gamma \|_{V} , \\ \\text{ then} \\ B^ = \\underset{B \\in \\mathcal {C}}{\\operatornamewithlimits{argmax}} \\ \\langle {\\Sigma } , B \\rangle .\\ $ In particular, when $\\Delta (C^) > 0 $ , $ B^= \\underset{B \\in \\mathcal {C}}{\\operatornamewithlimits{argmax}} \\ \\langle \\Sigma -\\Gamma , B \\rangle .$ Proposition REF shows that Proposition REF and Corollary REF continue to hold when the non-convex set $\\mathcal {D}$ is replaced by the convex set $\\mathcal {C}$ , and we notice that the counterexample of Proposition REF also continues to be valid.", "On the basis Proposition REF , we expect the PECOK estimator $\\widehat{B}$ to recover $B^$ , with high probability, and we show that this is indeed the case in Section REF below, for the estimator $\\widehat{\\Gamma }$ given in the following section." ], [ "Estimation of $\\Gamma $", "If the groups $G_{k}$ of the partition $G^$ were known, we could immediately estimate $\\widehat{\\Gamma }$ by a method of moments, and obtain an estimator with $\\sqrt{\\log (p)/(nm)}$ rate with respect to the $\\ell ^\\infty $ norm.", "If the groups are not known, estimation of $\\Gamma $ is still possible, but one needs to pay a price in terms of precision.", "Fortunately, as explained in (REF ) below, we only need to estimate $\\Gamma $ at a $\\sqrt{\\log (p)/n}$ rate.", "Specifically, in this section, our goal is to build an estimator $\\widehat{\\Gamma }$ of $\\Gamma $ which fulfills $ \|\\widehat{\\Gamma }- \\Gamma \|_{\\infty }\\lesssim \|\\Gamma \|_\\infty \\sqrt{\\log (p)/ n}$ , with high probability.", "In Appendix we show the construction of an intuitive estimator of $\\Gamma $ , for which $ \|\\widehat{\\Gamma }- \\Gamma \|_{\\infty }\\lesssim \|\\Sigma \|_\\infty ^{1/2}\|\\Gamma \|_{\\infty }^{1/2}\\sqrt{\\log (p)/ n}$ , which is not fully satisfactory, as it would suggest that the precision of an estimate of $\\Gamma $ depends on the size of parameters not in $\\Gamma $ .", "In order to correct this and achieve our goal we need the somewhat subtler estimator $\\widehat{\\Gamma }$ constructed below.", "For any $a,b\\in [p]$ , define $V(a,b):= \\max _{c,d \\in [p]\\setminus \\lbrace a,b\\rbrace }\\big \|\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:b}, \\frac{\\mathbf {X}_{:c}-\\mathbf {X}_{:d}}{\|\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\|_2}\\rangle \\big \| \\ ,$ with the convention $0/0=0$ .", "Guided by the block structure of $\\Sigma $ , we define $ne_1(a):= \\operatornamewithlimits{argmin}_{b\\in [p]\\setminus \\lbrace a\\rbrace }V(a,b)\\quad \\text{ and }\\quad ne_2(a):= \\operatornamewithlimits{argmin}_{b\\in [p]\\setminus \\lbrace a,ne_1(a)\\rbrace }V(a,b) ,$ to be two “neighbors” of $a$ , that is two indices $b_1 = ne_1(a)$ and $b_2 = ne_2(a)$ such that the covariance $\\langle \\mathbf {X}_{:b_i} ,\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\\rangle $ , $i =1,2$ , is most similar to $\\langle \\mathbf {X}_{:a} ,\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\\rangle $ , for all variables $c$ and $d$ not equal to $a$ or $b_{i}$ , $i = 1,2$ .", "It is expected that $ne_1(a)$ and $ne_2(a)$ belong to the same group as $a$ , or at least that $(\\mathbf {Z}A^t)_{:a}-(\\mathbf {Z}A^t)_{:ne_i(a)}$ is small.", "Then, the estimator $\\widehat{\\Gamma }$ , which is a diagonal matrix, is defined by $\\widehat{\\Gamma }_{aa}= {1 \\over n}\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne_{1}(a)}, \\mathbf {X}_{:a}-\\mathbf {X}_{:ne_{2}(a)}\\rangle ,\\quad \\text{ for $a=1,\\ldots , p$.", "}$ The population version of the quantity above is of order $\\Gamma _{aa}+ C^_{k(a)k(a)}+C^_{k(ne_{1}(a))k(ne_{2}(a))}-C^_{k(a)k(ne_{1}(a))}-C^_{k(a)k(ne_{2}(a))}\\ ,$ where $k(b)$ stands for the group of $b$ .", "It should therefore be of order $\\Gamma _{aa}$ if the above intuition holds.", "Proposition REF below shows that this is indeed the case.", "Proposition 4 There exist three numerical constants $c_1$ –$c_3$ such that the following holds.", "Assume that $m\\ge 3$ and that $\\log (p)\\le c_1 n$ .", "With probability larger than $1-c_{3}/p$ , the estimator $\\widehat{\\Gamma }$ defined by (REF ) satisfies $\|\\widehat{\\Gamma }- \\Gamma \|_{V}\\le 2 \|\\widehat{\\Gamma }- \\Gamma \|_{\\infty }\\le c_2 \|\\Gamma \|_\\infty \\sqrt{\\frac{\\log (p)}{n}}\\ .$ We remark on the fact that, even though the above proposition does not make any separation assumption between the clusters, we are still able to estimate the diagonal entries $\\Gamma _{aa}$ at rate $\\sqrt{\\log (p)/n}$ in $\\ell ^\\infty $ norm." ], [ " Perfect clustering with PECOK ", "Whereas Lemma REF above guarantees that $B^$ is identifiable when $\\Delta (C^) > 0$ , a larger cluster separation level is needed for estimating $B^$ consistently from noisy observations.", "Theorem REF below shows that $\\widehat{B} = B^$ with high probability whenever $\\Delta (C^)$ is larger than the sum between $ \|\\widehat{\\Gamma }-\\Gamma \|_{V}/m$ and terms accounting for the variability of $\\widehat{\\Sigma }$ , which are the dominant terms if $\\widehat{\\Gamma }$ is constructed as in (REF ) above.", "Theorem 1 There exist $c_1,\\ldots , c_3$ three positive constants such that the following holds.", "Let $\\widehat{\\Gamma }$ be any estimator of $\\Gamma $ , such that $\|\\widehat{\\Gamma }-\\Gamma \|_{V}\\le \\delta _{n,p}$ with probability $1-c_{3}/(2p)$ .", "Then, assuming that $\\log (p)\\le c_1 n$ , and that $\\Delta (C^) \\ge c_2 \\left[\|\\Gamma \|_{\\infty }\\left\\lbrace \\sqrt{ \\frac{\\log p}{mn} }+ \\sqrt{\\frac{p}{nm^2}} + \\frac{\\log (p)}{n}+ \\frac{p}{nm}\\right\\rbrace + \\frac{\\delta _{n,p}}{m} \\right]\\ ,$ we have $\\widehat{B} = B^$ , with probability higher than $1 - c_3/p$ .", "We stress once again the fact that exact partition recovery is crucially dependent on the quality of the estimation of $\\Gamma $ .", "To make this fact as transparent as possible, we discuss below condition (REF ) in a simplified setting.", "The same reasoning applies in general.", "When all groups $G_k$ have equal size, so that $p=mK$ (or more generally when $p\\approx mK$ ), and when the number $K$ of groups is smaller than $\\log (p)$ , Condition (REF ) simplifies to $\\Delta (C^) \\gtrsim \|\\Gamma \|_{\\infty }\\left[\\sqrt{ \\frac{\\log p}{mn} }+ \\frac{\\log (p)}{n}\\right]+ \\frac{\\delta _{n,p}}{m} \\ .$ The first term in the right-hand side of (REF ) is of order $\\sqrt{ \\tfrac{\\log p}{mn} }+ \\tfrac{\\log (p)}{n}$ .", "It is shown to be minimax optimal in Theorem REF of Section below.", "The order of magnitude of the second term, ${\\delta _{np}}/{m}$ , depends on the size of $\|\\widehat{\\Gamma }-\\Gamma \|_{V}$ and can become the dominant term for poor estimates of $\\Gamma $ .", "We showcase below two cases of interest.", "Suboptimal cluster recovery with an uncorrected convex relaxation of $K$ -means: $\\widehat{\\Gamma }=0$ .", "Theorem REF shows that if we took $\\widehat{\\Gamma }=0$ in the definition of our estimator (REF ), we could only guarantee recovery of clusters with a relatively large separation.", "Specifically, when $\\widehat{\\Gamma }=0$ , then $\\delta _{n,p} \\approx \|\\Gamma \|_{V}/m$ and (REF ) becomes $\\Delta (C^)\\gtrsim \|\\Gamma \|_{V}/m$ , when $m < n$ , which would be strongly sub-optimal relative to the minimax optimal separation rate of Theorem REF below.", "We note that the corresponding unpenalized estimator $ \\widehat{B}_1=\\operatornamewithlimits{argmax}_{B \\in \\mathcal {C}}\\langle \\widehat{\\Sigma }, B\\rangle $ is a convex relaxation of $K$ -means, and would still be computationally feasible, but not statistically optimal for variable clustering in $G$ -models.", "Optimal cluster recovery with a penalized convex relaxation of $K$ -means, when $\|\\widehat{\\Gamma }-\\Gamma \|_{\\infty }\\lesssim \|\\Gamma \|_{\\infty } \\sqrt{\\log (p)/n}$ .", "The issues raised above can be addressed by using the corrected estimator PECOK corresponding to an estimator $\\widehat{\\Gamma }$ for which $\|\\widehat{\\Gamma }-\\Gamma \|_{\\infty }\\lesssim \|\\Gamma \|_{\\infty } \\sqrt{\\log (p)/n}$ , such as the estimator given in (REF ) above.", "Then, as desired, the second term of (REF ) becomes small relative to the first term of (REF ): ${\\delta _{np}\\over m} \\lesssim \|\\Gamma \|_{\\infty } \\sqrt{\\log (p)\\over nm^2} \\le {\|\\Gamma \|_{\\infty }\\over \\sqrt{m}}\\left[\\sqrt{ \\frac{\\log p}{mn} }+ \\frac{\\log (p)}{n}\\right],$ since $\|D\|_{V}\\le 2 \|D\|_{\\infty }$ .", "With these ingredients, we can then show that the PECOK estimator corresponding to $\\widehat{\\Gamma }$ defined by (REF ) recovers the true partition, at a near-minimax optimal separation rate.", "Corollary 2 There exist $c_1, c_2, c_3$ three positive constants such that the following holds.", "Assuming that $\\widehat{\\Gamma }$ is defined by (REF ), $\\log (p)\\le c_1 n$ , and that $\\Delta (C^) \\ge c_2 \|\\Gamma \|_{\\infty }\\left\\lbrace \\sqrt{ \\frac{\\log p}{mn} }+ \\sqrt{\\frac{p}{nm^2}} + \\frac{\\log (p)}{n}+ \\frac{p}{nm}\\right\\rbrace \\ ,$ then $\\widehat{B} = B^$ , with probability higher than $1 - c_3/p$ .", "Moreover, $\\widehat{G} = G^$ , for $\\widehat{G}$ given by Step 3 of the PECOK algorithm, with probability higher than $1 - c_3/p$ ." ], [ "Adaptation to the number of groups", "In the previous section, we assumed that the number $K$ of groups is known in advance.", "In many situations, however, $K$ is not known, and we address this situation here.", "The information on $K$ was used to build our estimator $\\widehat{B}$ , via the constraint $\\operatorname{tr}(B)=K$ present in the definition of $\\mathcal {C}$ given in (REF ).", "When $K$ is not known, we drop this constraint from the definition of $\\mathcal {C}$ , and instead penalize the scalar product $\\langle \\widehat{\\Sigma }- \\widehat{\\Gamma } , B\\rangle $ by the trace of $B$ .", "Specifically, we define the adaptive estimator $\\widehat{B}_{adapt}$ given by $\\widehat{B}_{adapt} := \\underset{B \\in \\mathcal {C}_0}{\\operatornamewithlimits{argmax}} \\ \\langle \\widehat{\\Sigma }- \\widehat{\\Gamma } , B \\rangle - \\widehat{\\kappa } \\operatorname{tr}(B) \\ .$ where $\\mathcal {C}_0:=\\left\\lbrace B \\in \\mathbb {R}^{p\\times p}:\\begin{array}{ll}\\bullet \\ B \\text{ is in } \\mathcal {S}^+ = \\lbrace \\text{symmetric and positive semidefinite}\\rbrace \\\\\\bullet \\ \\sum _a B_{ab} = 1,\\ \\forall b\\\\\\bullet \\ B_{ab}\\ge 0,\\ \\forall a,b\\\\\\end{array}\\right\\rbrace ,$ and $\\widehat{\\kappa }$ is a data-driven tuning parameter.", "The following theorem gives conditions on $\\widehat{\\kappa }$ , $\\widehat{\\Gamma }$ and $\\Delta (C^)$ which ensure exact recovery of $B^$ .", "Theorem 2 There exist $c_1, c_2, c_3$ three positive constants such that the following holds.", "Let $\\widehat{\\Gamma }$ be any estimator of $\\Gamma $ , such that $\|\\widehat{\\Gamma }-\\Gamma \|_{V}\\le \\delta _{n,p}$ with probability $1-c_{3}/(3p)$ .", "Then, assuming that $\\log (p)\\le c_1 n$ , and that $\\Delta (C^) \\ge c_2 \\left[\|\\Gamma \|_{\\infty }\\left\\lbrace \\sqrt{ \\frac{\\log p}{mn} }+ \\sqrt{\\frac{p}{nm^2}} + \\frac{\\log (p)}{n}+ \\frac{p}{nm}\\right\\rbrace + \\frac{\\delta _{n,p}}{m} \\right]$ and that, with probability larger than $1-c_3/(3p)$ $ 4\|\\Gamma \|_{\\infty }\\left(\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right) + \\delta _{n,p} < \\widehat{\\kappa } < \\frac{m}{8}\\Delta (C^)$ then $\\widehat{B}_{adapt} = B^$ , with probability higher than $1 - c_3/p$ .", "Condition (REF ) in the above theorem encourages us to consider the following data dependent value of $\\widehat{\\kappa }$ : $\\widehat{\\kappa }= : 5\|\\widehat{\\Gamma }\|_{\\infty }\\left(\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right), \\quad \\quad \\text{ where $\\widehat{\\Gamma }$ is defined in (\\ref {eq:estim:gamma2})}\\ .$ We note that the constant 5 may not be optimal, but further analysis of this constant is beyond the scope of this paper.", "Equipped with the estimator $\\widehat{\\Gamma }$ defined in (REF ) and $\\widehat{\\kappa }$ defined in (REF ), the adaptive estimator (REF ) then fulfills the following recovering property.", "Corollary 3 There exist $c_1,\\ldots , c_3$ three positive constants such that the following holds.", "Assuming that $\\widehat{\\Gamma }$ and $\\widehat{\\kappa }$ are defined by (REF ) and (REF ), $\\log (p)\\le c_1 n$ , and that $\\Delta (C^) \\ge c_2 \|\\Gamma \|_{\\infty }\\left\\lbrace \\sqrt{ \\frac{\\log p}{mn} }+ \\sqrt{\\frac{p}{nm^2}} + \\frac{\\log (p)}{n}+ \\frac{p}{nm}\\right\\rbrace \\ ,$ then we have $\\widehat{B}_{adapt} = B^$ , with probability higher than $1 - c_3/p$ .", "We observe that the Condition (REF ) that ensures perfect recovery of $B^$ when $K$ is unknown is the same as the condition (REF ) employed in Corollary REF when $K$ was assumed to be known.", "This condition is shown to be near-minimax optimal in the next section." ], [ "Minimax lower bound", "To ease the presentation, we restrict ourselves in this section to the toy model with $C^* = \\tau I_K$ and $\\Gamma =I_{p}$ , so that, given a partition $G$ , the covariance matrix decomposes as $\\Sigma _G = A_G \\big (\\tau I_K \\big ) A_G^t + I_p\\ ,$ where $A_G$ is the assignment matrix associated to the partition $G$ .", "Note that, in this case $\\Delta (C^) = 2\\tau $ .", "Define $\\mathcal {G}$ the class of all partitions of $\\lbrace 1,\\ldots , p\\rbrace $ into $K$ groups of identical size $m$ , therefore $p=mK$ .", "In the sequel, $\\operatorname{\\mathbb {P}}_{\\Sigma _G}$ refers to the normal distribution with covariance $\\Sigma _G$ .", "The minimax optimal error probability for partition recovery is defined as: $\\overline{\\mathbf {R}}^[\\tau ,n,m,p] := \\inf _{\\hat{G}} \\sup _{G\\in \\mathcal {G}} \\mathbb {P}_{\\Sigma _G} \\big [ \\hat{G}\\ne G\\big ] .$ Theorem 3 There exists a numerical constant $c>0$ such that the following holds.", "The optimal error probability of recovery $\\overline{\\mathbf {R}}^[\\tau ,n,m,p]$ is larger than $1/7$ as soon as $\\Delta (C^) = 2 \\tau \\le c \\left[\\sqrt{\\frac{\\log (p)}{n(m-1)}}\\bigvee \\frac{\\log (p)}{n} \\right]\\ .$ In view of Corollary REF (see also Corollary REF ), a sufficient condition on the size of $\\tau $ under which one obtains perfect partition recovery is that $\\tau $ be of order $\\sqrt{\\frac{\\log (p)}{nm}} + \\frac{\\log (p)}{n}$ , when the ratio between the number $K$ of groups and $\\log (p)$ is bounded from above.", "However, the necessary conditions (REF ) and sufficient conditions (REF ) scale differently with $K$ , when $K$ is large.", "This discrepancy between minimax lower bounds and the performance of estimators obtained via convex optimization algorithms has also been pinpointed in network clustering via the stochastic block model [6].", "It has been conjectured that, for large $K$ , there is a gap between the statistical boundary, i.e.", "the minimal cluster separation for which a statistical method achieves perfect clustering with high probability, and the polynomial boundary, i.e.", "the minimal cluster separation for which there exists a polynomial-time algorithm that achieves perfect clustering.", "Further investigation of this gap is beyond the scope of this paper and we refer to [6] for more details." ], [ "A comparison between PECOK and Spectral Clustering", "In this section we discuss connections between the clustering methods introduced above and spectral clustering, a method that has become popular in network clustering.", "When used for variable clustering, uncorrected spectral clustering consists in applying a clustering algorithm, such as $K$ -means, on the rows of the $p\\times K$ -matrix obtained by retaining the $K$ leading eigenvectors of $\\widehat{\\Sigma }$ .", "Similarly to Section , we propose below a correction of this algorithm.", "First, we recall the premise of spectral clustering, adapted to our context.", "For $G^$ -block covariance models as (REF ), we have $\\Sigma - \\Gamma = AC^A^{t}$ .", "Let $U$ be the $p \\times K$ matrix collecting the $K$ leading eigenvectors of $\\Sigma - \\Gamma $ .", "It has been shown, see e.g.", "Lemma 2.1 in Lei and Rinaldo [15], that $a$ and $b$ belong to the same cluster if and only if $U_{a:} = U_{b:}$ if and only if $[UU^{t}]_{a:} = [UU^{t}]_{b:} $ .", "Therefore, the partition $G^$ could be recovered from $\\Sigma - \\Gamma $ via any clustering algorithm applied to the rows of $U$ or $UU^{t}$ , for instance by a $K$ -means.", "It is natural therefore to consider the possibility of estimating $G^$ by clustering the rows of $\\widehat{U}$ , the matrix of the $K$ leading eigenvectors of $\\widetilde{\\Sigma } := \\widehat{\\Sigma } - \\widehat{\\Gamma }$ .", "Since $\\widehat{U}$ is an orthogonal matrix, when the clustering algorithm is rotation invariant, it is equivalent to cluster the rows of $\\widehat{U}\\widehat{U}^{t}$ .", "We refer to this algorithm as Corrected Spectral Clustering (CSC), as it is relative to $\\widehat{\\Sigma } - \\widehat{\\Gamma }$ , not $\\widehat{\\Sigma }$ .", "The two steps of CSC are then: CSC algorithm Compute $\\widehat{U}$ , the matrix of the $K$ leading eigenvectors of $\\widetilde{\\Sigma } := \\widehat{\\Sigma } - \\widehat{\\Gamma }$ Estimate $G^$ by clustering the rows of $\\widehat{U}$ , via an $\\eta $ -approximation of $K$ -means, defined in (REF ).", "An $\\eta $ -approximation of $K$ -means is defined as follows.", "Let $\\eta >1$ be a given positive number.", "Denote $\\mathcal {A}_{p,K}$ the collection of membership matrices, that is $p\\times K$ binary matrices whose rows contain exactly one non-zero entry.", "Note that a membership matrix $A\\in \\mathcal {A}_{p,K}$ defines a partition $G$ .", "Given a $p\\times K$ matrix $\\widehat{U}$ , the membership matrix $\\widehat{A}$ is said to be an $\\eta $ -approximation $K$ -means problem on $\\widehat{U}$ if there exists a $K\\times K$ matrix $\\widehat{Q}$ such that $\\Vert \\widehat{U}- \\widehat{A} \\widehat{Q}\\Vert _F^2 \\le \\eta \\min _{A\\in \\mathcal {A}{p,k}}\\min _Q \\Vert \\widehat{U} - AQ\\Vert _F^2\\ .$ Note then that $G^$ will be estimated by $\\widehat{G}$ , the partition corresponding to $\\widehat{A}$ .", "An example of polynomial time approximate $K$ -means algorithm is given in Kumar et al. [12].", "We show below how CSC relates to our proposed PECOK estimator.", "Lemma 2 When the clustering algorithm applied at the second step of Corrected Spectral Clustering (CSC) is rotation invariant, then CSC is equivalent to the following algorithm: Step 1.", "Find $\\overline{B}=\\operatornamewithlimits{argmax}\\lbrace \\langle \\widetilde{\\Sigma },B\\rangle \\ : \\ tr(B)=K,\\ I \\succcurlyeq B \\succcurlyeq 0\\rbrace .$ Step 2.", "Estimate $G^$ by clustering the rows of $\\overline{B}$ , via an $\\eta $ -approximation of $K$ -means, defined in (REF ).", "The connection between PECOK and spectral clustering now becomes clear.", "The PECOK estimator involves the calculation of $\\widehat{B}=\\operatornamewithlimits{argmax}_{B}\\lbrace \\langle \\widetilde{\\Sigma },B\\rangle \\ : \\ B1=1,\\ B_{ab}\\ge 0,\\ tr(B)=K,\\ B \\succcurlyeq 0\\rbrace .$ Since the matrices $B$ involved in (REF ) are doubly stochastic, their eigenvalues are smaller than 1 and hence (REF ) is equivalent to $\\widehat{B}=\\operatornamewithlimits{argmax}_{B}\\lbrace \\langle \\widetilde{\\Sigma },B\\rangle \\ : \\ B1=1,\\ B_{ab}\\ge 0,\\ tr(B)=K,\\ I \\succcurlyeq B \\succcurlyeq 0\\rbrace .", "$ Note then that $\\overline{B}$ can be viewed as a less constrained version of $\\widehat{B}$ , in which $\\mathcal {C}$ is replaced by $ \\overline{\\mathcal {C}} = \\lbrace B: \\ tr(B)=K,\\ I \\succcurlyeq B \\succcurlyeq 0\\rbrace , $ where we have dropped the $p(p+1)/2$ constraints given by $B1=1$ , and $B_{ab}\\ge 0$ .", "We show in what follows that the possible computational gains resulting from such a strategy may result in severe losses in the theoretical guarantees for exact partition recovery.", "In addition, the proof of Lemma REF shows that $\\overline{B}=\\widehat{U}\\widehat{U}^{t}$ so, contrary to $\\widehat{B}$ , the estimator $\\overline{B}$ is (almost surely) never equal to $B^$ .", "To simplify the presentation, we assume in the following that all the groups have the same size $\|G^_1\|=\\ldots =\|G^_K\|=m=p/K$ .", "We emphasize that this information is not required by either PECOK or CSC, or in the proof of Theorem REF below.", "We only use it here to illustrate the issues associated with CSC in a way that is not cluttered by unnecessary notation.", "We denote by $\\mathcal {S}_{K}$ the set of permutations on $\\lbrace 1,\\ldots ,K\\rbrace $ and we denote by $\\overline{L}(\\widehat{G},G^)= \\min _{\\sigma \\in \\mathcal {S}_{K}}\\sum _{k=1}^K{\|G^_{k}\\setminus \\widehat{G}_{\\sigma (k)}\|\\over m}$ the sum of the ratios of miss-assigned variables with indices in $G^_k$ .", "In the previous sections, we studied perfect recovery of $G^$ , which would correspond to $\\overline{L}(\\widehat{G},G^) = 0$ , with high probability.", "We give below conditions under which $\\overline{L}(\\widehat{G},G^) \\le \\rho $ , for an appropriate quantity $\\rho < 1$ , and we show that very small values of $\\rho $ require large cluster separation, possibly much larger than the minimax optimal rate.", "We begin with a general theorem pertaining to partial partition recovery by CSC, under restrictions on the smallest eigenvalue $\\lambda _{K}(C^)$ of $C^$ .", "Theorem 4 We let $Re(\\Sigma )=tr(\\Sigma )/\\Vert \\Sigma \\Vert _{op}$ denote the effective rank of $\\Sigma $ .", "There exist $c_{\\eta }>0$ and $c^{\\prime }_{\\eta }>0$ only depending on $\\eta $ and numerical constants $c_1$ and $c_2$ such that the two following bounds hold.", "For any $0<\\rho < 1$ , if $\\lambda _{K}(C^)\\ge {c^{\\prime }_{\\eta }\\sqrt{K} \\Vert \\Sigma \\Vert _{op}\\over m\\sqrt{\\rho }}\\sqrt{\\frac{Re(\\Sigma )\\vee \\log (p)}{n}},$ then $\\overline{L}(\\widehat{G},G^)\\le \\rho $ , with probability larger than $1-c_2/p$ .", "The proof extends the arguments of [15], initially developped for clustering procedures in stochastic block models, to our context.", "Specifically, we relate the error $\\overline{L}(\\widehat{G},G^)$ to the noise level, quantified in this problem by $\\Vert \\widetilde{\\Sigma }-AC^A^t\\Vert _{op}$ .", "We then employ the results of [11] and [5] to show that this operator norm can be controlled, with high probability, which leads to the conclusion of the theorem.", "We observe that $\\Delta (C^)\\ge 2 \\lambda _{K}(C^)$ , so the lower bound (REF ) on $ \\lambda _{K}(C^)$ enforces the same lower-bound on $\\Delta (C^)$ .", "To further facilitate the comparison with the performances of PECOK, we discuss both the conditions and the conclusion of this theorem in the simple setting where $C^=\\tau I$ and $\\Gamma =I$ .", "Then, the cluster separation measures coincide up to a factor 2, $\\Delta (C^) = 2\\lambda _K(C^) = 2\\tau $ .", "Corollary 4 (Illustrative example: $C^=\\tau I$ and $\\Gamma =I$ ) There exist three positive numerical constants $c_{1,\\eta }$ , $c_{2,\\eta }$ and $c_3$ such that the following holds.", "For any $0<\\rho <1$ , if $\\rho \\ge c_{1,\\eta }\\Big [ \\frac{K^2}{n}+ \\frac{K\\log (p)}{n}\\Big ]\\quad \\quad \\text{ and }\\quad \\quad \\tau \\ge c_{2,\\eta }\\Big [ {K^2\\over \\rho n} \\vee \\frac{K}{\\sqrt{\\rho nm}}\\Big ]\\ ,$ then $\\overline{L}(\\widehat{G},G^)\\le \\rho $ , with probability larger than $1-c_3/p$ .", "Recall that, as a benchmark, Corollary REF above states that, when $\\widehat{G}$ is obtained via the PECOK algorithm, and if $\\tau \\gtrsim \\sqrt{ \\frac{K\\vee \\log p }{mn} }+ \\frac{\\log (p)\\vee K }{n}\\ ,$ then $\\overline{L}(\\widehat{G},G^) = 0$ , or equivalently, $\\widehat{G} = G^$ , with high probability.", "We can therefore provide the following summary, for the simple case $C^= \\tau I$ .", "Summary: PECOK vs CSC when $C^=\\tau I$ .", "1.", "$\\rho $ is a user specified small value, independent of $n$ or $p$ , and the number of groups $K$ is either a constant or grows at most as $\\log p$ .", "In this case, the size of the cluster separation given by either Condition (REF ) and (REF ) are essentially the same, up to unavoidable $\\log p$ factors.", "The difference is that, in this regime, CSC guarantees recovery up to a fixed, small, fraction of mistakes, whereas PECOK guarantees exact recovery.", "2.", "$\\rho \\rightarrow 0 $ .", "Although perfect recovery, with high probability, cannot be guaranteed for CSC, we could be close to it by requiring $\\rho $ to be close to zero.", "In this case, the distinctions between Conditions (REF ) and (REF ) become much more pronounced.", "Notice that whereas the latter condition is independent of $\\rho $ , in the former there is a trade-off between the precision $\\rho $ and the size of the cluster separation.", "Condition (REF ) is the near-optimal separation condition that guarantees that $\\rho = 0$ when PECOK is used.", "However, if in (REF ) we took, for instance, $\\rho $ to be proportional to $K^2/n$ , whenever the latter is small, the cluster separation requirement for CSC would become $ \\tau \\gtrsim 1, $ which is unfortunately very far from the optimal minimax rate.", "The phenomena summarized above have already been observed in the analysis of spectral clustering algorithms for network clustering via the Stochastic Block Model (SBM), for instance in [15].", "When we move away from the case $C^=\\tau I$ discussed above, the sufficient condition (REF ) of the general Theorem REF for CSC compares unfavorably with condition (REF ) of Corollary REF for PECOK even when $\\rho $ is a fixed value.", "For instance, consider $C^=\\tau I+\\alpha J$ , with $J$ being the matrix with all entries equal to one, and $\\Gamma =I$ .", "Notice that in this case we continue to have $\\Delta (C^)= 2\\lambda _{K}(C^)= 2\\tau $ .", "Then, for a given, fixed, value of $\\rho $ and $K$ fixed, condition (REF ) of the general Theorem REF guarantees $\\rho $ -approximately correct clustering via CSC for the cluster separation $\\tau \\gtrsim \\frac{\\alpha \\sqrt{\\log (p)}}{\\sqrt{n\\rho }}\\ ,$ which is independent of $m$ , unlike the minimax cluster separation rate that we established in Theorem REF above.", "Although we only compare sufficient conditions for partition estimation, this phenomenon further supports the merits the PECOK method proposed in this work." ], [ "Extensions", "In this section we discuss briefly immediate generalizations of the framework presented above.", "First we note that we only assumed that $X$ is Gaussian in order to keep the notation as simple as possible.", "All our arguments continue to hold if $X$ is sub-Gaussian, in which case all the concentration inequalities used in our proofs can be obtained via appropriate applications of Hanson-Wright inequality [21].", "Moreover, the bounds in operator norm between covariance matrices and their estimates continue to hold, at the price of an additional $\\log (p)$ factor, under an extra assumption on the moments of $X$ , as explained in section 2.1 of [5].", "We can also generalize slightly the modeling framework.", "If (REF ) holds for $X$ , we can alternatively assume that the error variances within a block are not equal.", "The implication is that in the decomposition of the corresponding $\\Sigma $ the diagonal matrix $\\Gamma $ will have arbitrary non-zero entries.", "The characterization (REF ) of $B^$ that motivates PECOK is unchanged, and so is the rest of the paper, with the added bonus that in Lemma REF the sufficient identifiability condition $\\Delta (C^) > 0$ also becomes necessary.", "We have preferred the set-up in which $\\Gamma $ has equal diagonal entries per cluster only to facilitate direct comparison with [4], where this assumption is made." ], [ "Proof of Lemma ", "When (REF ) holds, then we have $\\Sigma =A C^ A^t+\\Gamma ^$ , with $A_{ak}=1_{a \\in G^_{k}}$ .", "In particular, writing $k^(a)$ for the integer such that $a\\in G^_{k^(a)}$ , we have $\\Sigma _{ab}=C^_{k^(a)k^(b)}$ for any $a\\ne b$ .", "Let $a$ be any integer between 1 and $p$ and set $V(a)=\\lbrace a^{\\prime }:a^{\\prime }\\ne a,\\ Cord(a,a^{\\prime })=0\\rbrace ,\\quad \\text{where}\\quad Cord(a,a^{\\prime })=\\max _{c\\ne a,a^{\\prime }}\|\\widehat{\\Sigma }_{ac}-\\widehat{\\Sigma }_{a^{\\prime }c}\|.$ We prove below that $V(a)=G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ , and hence the partition $G^$ is identifiable from $\\Sigma $ .", "First, if $a^{\\prime }\\in G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ , then $k^(a^{\\prime })=k^(a)$ , so $\\Sigma _{ac}-\\Sigma _{a^{\\prime }c}=C^_{k^(a)k^(c)}-C^_{k^(a)k^(c)}=0$ for any $c\\ne a,a^{\\prime }$ .", "So $a^{\\prime }\\in V(a)$ and hence $G^_{k^(a)}\\setminus \\lbrace a\\rbrace \\subset V(a)$ .", "Conversely, let us prove that $V(a) \\subset G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ .", "Assume that it is not the case, hence there exists $a^{\\prime }\\in V(a)$ such that $k^(a^{\\prime })\\ne k^(a)$ .", "Since $m>1$ and $a^{\\prime }\\notin G^_{k^(a)}$ , we can find $b\\in G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ and $b^{\\prime }\\in G^_{k^(a^{\\prime })}\\setminus \\lbrace a^{\\prime }\\rbrace $ .", "Since $a^{\\prime }\\in V(a)$ , we have $Cord(a,a^{\\prime })=0$ so $0&=\\Sigma _{ab}-\\Sigma _{a^{\\prime }b}=C_{k^(a)k^(a)}^-C_{k^(a)k^(a^{\\prime })}^\\\\0&=\\Sigma _{ab^{\\prime }}-\\Sigma _{a^{\\prime }b^{\\prime }}=C^_{k^(a)k^(a^{\\prime })}-C^_{k^(a^{\\prime })k^(a^{\\prime })}.$ In particular, we have $\\Delta (C^)\\le C^_{k^(a)k^(a)}+C^_{k^(a^{\\prime })k^(a^{\\prime })}-2C^_{k^(a)k^(a^{\\prime })}=0,$ which is in contradiction with $\\Delta (C^)>0$ .", "So it cannot hold that $k^(a^{\\prime })\\ne k^(a)$ , which means that any $a^{\\prime }\\in V(a)$ belongs to $G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ , i.e.", "$V(a) \\subset G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ .", "This conclude the proof of the equality $V(a)=G^_{k^(a)}\\setminus \\lbrace a\\rbrace $ and the proof of the Lemma is complete.", "In order to avoid notational clutter in the remainder of the paper, we re-denote $C^$ by $C$ and $G^$ by $G$ ." ], [ " Proofs of Proposition ", "Since ${\\cal D}\\subset {\\cal C}$ , and $B^* \\in \\mathcal {D}$ , the proofs of Proposition REF and Corollary REF follow from the proof of Proposition REF , given below.", "The basis of the proof of Proposition REF is the following Lemma.", "Lemma 3 The collection $\\mathcal {C}$ contains only one matrix whose support is included in $\\mathrm {supp}(B^)$ , that is $\\mathcal {C}\\cap \\big \\lbrace B,\\ \\mathrm {supp}(B)\\subset \\mathrm {supp}(B^)\\big \\rbrace = \\lbrace B^\\rbrace \\ .$ Consider any matrix $B\\in \\mathcal {C}$ whose support is included in $\\mathrm {supp}(B^)$ .", "Since $B1=1$ , it follows that each submatrix $B_{G_kG_k}$ is symmetric doubly stochastic.", "Since $B_{G_kG_k}$ is also positive semidefinite, we have $tr(B_{G_kG_k})\\ge \\Vert B_{G_kG_k}\\Vert _{op}\\ge 1^tB_{G_kG_k}1/\|G_k\|=1\\ .", "$ As $B\\in \\mathcal {C}$ , we have $\\operatorname{tr}(B)= K$ , so all the submatrices $B_{G_kG_k}$ have a unit trace.", "Since $\\Vert B_{G_kG_k}\\Vert _{op}\\ge 1$ , this also enforces that $B_{G_kG_k}$ contains only one non-zero eigenvalue and that a corresponding eigenvector is the constant vector 1.", "As a consequence, $B_{G_kG_k}=11^t/\|G_k\|$ for all $k=1,\\ldots ,K$ and $B=B^$ .", "As a consequence of Lemma REF , we only need to prove that, $ \\langle {\\Sigma }, B^-B \\rangle > 0, \\ \\mbox{ for all} \\ B \\in \\mathcal {C}\\text{ such that } \\ \\mathrm {supp}(B)\\nsubseteq \\mathrm {supp}(B^).$ We have $ \\langle {\\Sigma }, B^-B \\rangle =\\langle ACA^t,B^-B\\rangle + \\langle \\Gamma , B^-B \\rangle .$ Define the $p$ -dimensional vector $v$ by $v=diag(ACA^t)$ .", "Since $B1=1$ for all $B\\in \\cal {C}$ , we have $\\langle v1^t+1v^t,B^-B\\rangle =0$ .", "Hence, we have $\\langle ACA^t,B^-B\\rangle &=\\langle ACA^t-{1\\over 2}(v1^t+1v^t),B^-B\\rangle \\nonumber \\\\&= \\sum _{j,k}\\sum _{a\\in G_{j},\\,b\\in G_{k}} \\left(C_{jk}-{C_{jj}+C_{kk}\\over 2}\\right)(B^_{ab}-B_{ab})\\nonumber \\\\&= \\sum _{j\\ne k}\\sum _{a\\in G_{j},\\,b\\in G_{k}} \\left({C_{jj}+C_{kk}\\over 2}-C_{jk}\\right)B_{ab}\\nonumber \\\\&= \\sum _{j\\ne k} \\left({C_{jj}+C_{kk}\\over 2}-C_{jk}\\right) \|B_{G_jG_k}\|_1,$ where $B_{G_jG_k}=[B_{ab}]_{a\\in G_{j},\\, b\\in G_{k}}$ .", "Next lemma lower bounds $\\langle \\Gamma , B^-B \\rangle $ .", "Lemma 4 $\\langle \\Gamma , B^-B \\rangle \\ge - \\frac{\\max _k \\gamma _k -\\min _k \\gamma _k}{m}\\sum _{k\\ne j}\|B_{G_jG_k}\|_1\\ .$ [Proof of Lemma REF ] By definition of $B^$ and since $\\operatorname{tr}(B)=\\operatorname{tr}(B^)=K$ , we have $\\langle \\Gamma , B^-B \\rangle &=\\langle \\Gamma -\|\\Gamma \|_{\\infty }I, B^-B \\rangle \\nonumber \\\\&= \\sum _{k=1}^K (\\gamma _{k}-\|\\Gamma \|_{\\infty })\\Big [1- \\operatorname{tr}(B_{G_kG_k})\\Big ] \\nonumber \\\\& \\ge - (\\max _k \\gamma _k -\\min _k \\gamma _k )\\sum _{k=1}^K \\Big [1- \\operatorname{tr}(B_{G_kG_k})\\Big ]_{+} $ Since $B_{G_kG_k}$ is positive semidefinite, we have $0\\le 1^t B_{G_kG_k} 1/\|G_k\|\\le \\Vert B_{G_kG_k}\\Vert _{op}\\le \\operatorname{tr}(B_{G_kG_k})\\ .$ As a consequence, $1-\\operatorname{tr}(B_{G_kG_k})\\le 1 - 1^t B_{G_kG_k} 1/\|G_k\|$ .", "Since $B1=1$ , we conclude that $\\big [1-\\operatorname{tr}(B_{G_kG_k})\\big ]_{+}\\le 1 - {1^t B_{G_kG_k} 1\\over \|G_k\|}=\\frac{1}{\|G_k\|}\\sum _{j:j\\ne k}\|B_{G_jG_k}\|_1.$ Coming back to (REF ), this gives us $\\langle \\Gamma , B^-B \\rangle \\ge - \\frac{\\max _k \\gamma _k -\\min _k \\gamma _k}{m}\\sum _{k\\ne j}\|B_{G_jG_k}\|_1\\ .$ Hence, combining (REF ) and Lemma REF , we obtain $ \\langle {\\Sigma }, B^-B \\rangle \\ge \\sum _{j\\ne k} \\left({C_{jj}+C_{kk}\\over 2}-C_{jk}- \\frac{\\max _k \\gamma _k -\\min _k \\gamma _k}{m}\\right) \|B_{G_jG_k}\|_1.$ The condition (REF ) enforces that if $\\mathrm {supp}(B)\\nsubseteq \\mathrm {supp}(B^)$ then $\\langle {\\Sigma }, B^-B \\rangle > 0$ .", "This concludes the proof of (REF ) and (REF ).", "The proof of Proposition REF is complete." ], [ "Proof of Proposition ", "By symmetry, we can assume that the true partition matrix $B^$ is diagonal block constant.", "Define the partition matrix $B_1:= \\left[{\\scriptsize \\begin{array}{ccc} 2/m & 0 & 0 \\\\0& 2/m & 0 \\\\0& 0 & 1/(2m)\\end{array}}\\right]$ where the first two blocks are of size $m/2$ and the the last block has size $2m$ .", "The construction of the matrix $B_1$ amounts to merging groups $G_2$ and $G_3$ , and to splitting $G_1$ into two groups of equal size.", "Then, $\\langle \\Sigma , B^\\rangle = \\gamma _{+}+2\\gamma _{-}+ m tr(C)\\ , \\quad \\quad \\langle \\Sigma , B_1\\rangle = 2\\gamma _+ + \\gamma _-+ m tr(C)-m\\tau \\ .$ As a consequence, $\\langle \\Sigma , B_1\\rangle < \\langle \\Sigma , B^\\rangle $ if and only if $\\tau > \\frac{\\gamma _+- \\gamma _-}{m}$ ." ], [ "Proof of Theorem ", "As a consequence of Lemma REF page REF , when $\\widehat{B}$ is given by (REF ), we only need to prove that $ \\langle \\widehat{\\Sigma }-\\widehat{\\Gamma }, B^-B \\rangle > 0, \\ \\mbox{ for all} \\ B \\in \\mathcal {C}\\text{ such that } \\ \\mathrm {supp}(B)\\nsubseteq \\mathrm {supp}(B^), $ with high probability.", "We begin by recalling the notation: ${\\bf X}$ denotes the $n \\times p$ matrix of observations.", "Similarly, ${\\bf Z}$ stands for the $n \\times K$ matrix corresponding to the un-observed latent variables and ${\\bf E}$ denotes the $n \\times p$ error matrix defined by $\\mathbf {X}=\\mathbf {Z}A^t+\\mathbf {E}$ .", "Our first goal is to decompose $ \\widehat{\\Sigma }-\\widehat{\\Gamma }$ in such a way that the distance $\|\\mathbf {Z}_{:k}-\\mathbf {Z}_{:j}\|_2^2$ becomes evident, as this is the empirical counter-part of the key quantity $Var(Z_j - Z_k) =[C_{jj}+ C_{kk}-2C_{jk}]$ which informs cluster separation.", "To this end, recall that $n\\widehat{\\Sigma }= \\mathbf {X}^t \\mathbf {X}$ and let $\\widetilde{\\Gamma }= \\frac{1}{n}{\\bf E^t E}$ .", "Using the latent model representation, we further have $ n\\widehat{\\Sigma } = A {\\mathbf {Z}}^t \\mathbf {Z}A^t + n\\widetilde{\\Gamma }+ A({\\bf Z^t E}) + ({\\bf E^t Z})A^t.", "$ Using the fact that for any vectors $v_1$ and $ v_2$ we have $\|v_1 - v_2\|^2_2 = \|v_1\|_2^2 + \|v_2\|^2 - 2<v_1, v_2>$ , we can write $ [ A {\\mathbf {Z}}^t \\mathbf {Z}A^t ]_{ab} = \\frac{1}{2} \|[A{\\bf Z}^t]_{a:}\|_2^2 + \\frac{1}{2} \|[A{\\bf Z}^t]_{b:}\|_2^2 -\\frac{1}{2} \|[A{\\bf Z}^t]_{a:}- [A{\\bf Z}^t]_{b:}\|_2^2,$ for any $1\\le a,b\\le p$ .", "We also observe that $ [ A({\\bf Z^t E}) + ({\\bf E^t Z})A^t] _{ab} = [\\mathbf {E}^t_{b:}- \\mathbf {E}^t_{a:}][( A\\mathbf {Z}^t)_{a:}- (A\\mathbf {Z}^t )_{b:}] + [A\\mathbf {Z}^t\\mathbf {E}]_{aa} +[A\\mathbf {Z}^t\\mathbf {E}]_{bb} .", "$ Define the $p\\times p$ matrix $W$ by $W_{ab}:= n(\\widehat{\\Sigma }_{ab} -\\widehat{\\Gamma }_{ab})- \\frac{1}{2} \|[A{\\bf Z}^t]_{a:}\|_2^2 - \\frac{1}{2} \|[A{\\bf Z}^t]_{b:}\|_2^2 - [A{\\bf Z^t E}]_{aa}- [A{\\bf Z^t E}]_{bb}.$ Combining the four displays above we have $ W=W_1+W_2+ n(\\widetilde{\\Gamma }-\\widehat{\\Gamma }), $ with $(W_{1})_{ab}:= -\\frac{1}{2} \|[A{\\bf Z}^t]_{a:}- [A{\\bf Z}^t]_{b:}\|_2^2, \\ \\ \\ \\ (W_2)_{ab}:= [\\mathbf {E}^t_{b:}- \\mathbf {E}^t_{a:}][(A\\mathbf {Z}^t )_{a:}- (A\\mathbf {Z}^t)_{b:}],$ for any $1\\le a,b\\le p$ .", "Observe that $W-n(\\widehat{\\Sigma }-\\widehat{\\Gamma })$ is a sum of four matrices, two of which are of the type $1v_1^t$ , and two of the type $v_21^t$ , for some vectors $v_1, v_2 \\in \\mathcal {R}^p$ .", "Since for any two matrices $B_1$ and $B_2$ in $\\mathcal {C}$ , we have $B_1 1= B_21=1$ , it follows that $\\langle W-n(\\widehat{\\Sigma }-\\widehat{\\Gamma }), B_1-B_2\\rangle = 0 \\ .$ As a consequence and using the decomposition (REF ), proving (REF ) reduces to proving $\\langle W_1 + W_2 + n(\\widetilde{\\Gamma }-\\widehat{\\Gamma }), B^-B \\rangle > 0, \\ \\mbox{ for all} \\ B \\in \\mathcal {C}\\text{ such that } \\ \\mathrm {supp}(B)\\nsubseteq \\mathrm {supp}(B^).$ We will analyze the inner product between $B^ - B$ with each of the three matrices in (REF ) separately.", "The matrix $W_{1}$ contains the information about the clusters, as we explain below.", "Note that for two variables $a$ and $b$ belonging to the same group, $(W_{1})_{ab}=0$ and for two variables $a$ and $b$ belonging to different groups $G_j$ and $G_k$ , $(W_{1})_{ab}=-\|\\mathbf {Z}_{:i}- \\mathbf {Z}_{:k}\|_2^2/2$ .", "As a consequence, $\\langle W_1, B^\\rangle =0$ and $\\langle W_1, B^-B \\rangle = {1\\over 2} \\sum _{j\\ne k}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2^2 \\sum _{a\\in G_j,\\ b\\in G_k}B_{ab}.", "$ In the sequel, we denote by $B_{G_j,G_k}$ the submatrix $(B_{ab})_{a\\in G_j,\\,b\\in G_k}$ .", "Since all the entries of $B$ are nonnegative, $\\langle W_1, B^-B \\rangle = {1\\over 2} \\sum _{j\\ne k}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2^2 \|B_{G_jG_k}\|_1\\ .$ We will analyze below the two remaining cross products.", "As we shall control the same quantities $\\langle W_2,B^-B\\rangle $ and $\\langle \\widetilde{\\Gamma }-\\widehat{\\Gamma }, B^-B\\rangle $ for $B$ in the larger class $\\mathcal {C}_0$ given by (REF ) in the proof of Theorem REF , we state the two following lemmas for $ B \\in \\mathcal {C}_0$ .", "Their proofs are given after the proof of this theorem.", "Lemma 5 With probability larger than $1-c^{\\prime }_0/p$ , it holds that $\|\\langle W_2, B^-B \\rangle \|\\le c_1 \\sqrt{\\log (p)}\\sum _{j\\ne k}\|\\Gamma \|_{\\infty }^{1/2} \|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2 \|B_{G_jG_k}\|_1\\ ,$ simultaneously over all matrices $B\\in \\mathcal {C}_0$ .", "It remains to control the term corresponding to the empirical covariance matrix of the noise $\\mathbf {E}$ .", "This is the main technical difficulty in this proof.", "Lemma 6 With probability larger than $1-c_0/p$ , it holds that $n\|\\langle \\widetilde{\\Gamma }-\\widehat{\\Gamma }, B^-B\\rangle \|&\\le &c_2 \\left[\| \\Gamma \|_{\\infty } \\left( \\sqrt{ n\\frac{\\log p}{m} } \\vee \\sqrt{\\frac{np}{m^2}} \\vee \\frac{p}{m}\\right) + {n \|\\widehat{\\Gamma }-\\Gamma \|_{V}\\over m} \\right] \\sum _{j\\ne k }\|B_{G_jG_k}\|_1\\\\ &&+ 4\|\\Gamma \|_{\\infty }\\left[\\sqrt{\\frac{p}{n}}\\vee \\frac{p}{n}\\right](\\operatorname{tr}(B)-K)+ [\\operatorname{tr}(B)-K]_{+}\| \\Gamma -\\widehat{\\Gamma }\|_{V}, \\nonumber $ simultaneously over all matrices $B\\in \\mathcal {C}_0$ .", "For all matrices $B\\in \\mathcal {C}$ we have $\\operatorname{tr}(B)-K=0$ .", "Therefore, the second line of (REF ) is zero for the purpose of this proof.", "Combining (REF ), (REF ), and (REF ) we obtain that, with probability larger than $1-c/p$ , $\\langle W, B^-B \\rangle \\ge \\sum _{j\\ne k} \\Bigg [{1\\over 2}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2 - c_1 \\sqrt{\\log (p)} \|\\Gamma \|_{\\infty }^{1/2}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2 \\\\ -c_2 \\frac{n\|\\widehat{\\Gamma }-\\Gamma \|_{V}}{m} - c_{2}\|\\Gamma \|_{\\infty }\\left( \\sqrt{ \\frac{n\\log p}{m} } \\vee \\sqrt{\\frac{np}{m^2}} \\vee \\frac{p}{m}\\right)\\Bigg ] \|B_{G_jG_k}\|_1\\ , \\nonumber $ simultaneously for all $B\\in \\mathcal {C}$ .", "Therefore, if each term in the bracket of (REF ) is positive, with high probability, (REF ) will follow, since any matrix $B\\in \\mathcal {C}$ whose support is not included in $\\mathrm {supp}(B^)$ satisfies $ \|B_{G_jG_k}\|_1>0$ for some $j\\ne k$ , Since for any $j \\ne k$ the differences $Z_{ij} - Z_{ik}$ , for $1 \\le i \\le n$ are i.i.d.", "Gaussian random variables with mean zero and variance $C_{jj}+ C_{kk}-2C_{jk}$ , we can apply Lemma REF in Appendix with $A = -I$ and $t = \\log p$ .", "Then, if $\\log (p)<n/32$ , $\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2\\ge n[C_{jj}+ C_{kk}-2C_{jk}]/2 \\ ,$ simultaneously for all $j\\ne k$ , with probability larger than $1-1/p$ .", "Then, on the event for which (REF ) holds intersected with the event $\|\\Gamma -\\widehat{\\Gamma }\|_{V}\\le \\delta _{n,p}$ , Condition (REF ) enforces that, for all $j\\ne k$ , $ \\frac{1}{4}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2 \\ge c_1 \\log (p) \|\\Gamma \|_{\\infty }, $ and also $ \\frac{1}{4}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2 &\\ge & c_2 \\frac{n\|\\widehat{\\Gamma }-\\Gamma \|_{V}}{m} \\\\ &+& c_{2}\|\\Gamma \|_{\\infty }\\left( \\sqrt{ \\frac{n\\log p}{m} } \\vee \\sqrt{\\frac{np}{m^2}} \\vee \\frac{p}{m}\\right)$ with probability larger than $1-c^{\\prime }/p$ .", "Therefore, $ \\langle W, B^-B \\rangle > 0$ , for all $B\\in \\mathcal {C}$ , which concludes the proof of this theorem .", "$\\blacksquare $ [Proof of Lemma REF ] Consider any $a$ and $b$ in $[p]$ and let $j$ and $k$ be such that $a\\in G_j$ and $b\\in G_k$ .", "If $j=k$ , $(W_{2})_{ab}=0$ .", "If $j\\ne k$ , then $(W_{2})_{ab}$ follows, conditionally to $\\mathbf {Z}$ , a normal distribution with variance $\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2^2[\\gamma _{j}+ \\gamma _{k}]$ .", "Applying the Gaussian concentration inequality together with the union bound, we conclude that with probability larger than $1-1/p$ , $\|(W_2)_{ab}\|\\le c_{1} \|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2\\sqrt{\\log (p)}(\\gamma _{j}^{1/2}\\vee \\gamma _{k}^{1/2} )\\ ,$ simultaneously for all $a,\\ b\\in [p]$ .", "It then follows that $\|\\langle W_2, B^-B \\rangle \|\\le c_1 \\sqrt{\\log (p)}\|\\Gamma \|_{\\infty }^{1/2}\\sum _{j\\ne k}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2 \|B_{G_jG_k}\|_1\\ ,$ becauce $B^_{ab}=0$ and $B_{a,b}\\ge 0$ for all $a\\in G_j$ and $b\\in G_k$ , with $j\\ne k$ .", "[Proof of Lemma REF ] We split the scalar product $\\langle \\widetilde{\\Gamma }- \\widehat{\\Gamma }, B^* - B \\rangle $ into two terms $\\langle \\widetilde{\\Gamma }- \\widehat{\\Gamma }, B^* - B \\rangle =\\langle \\widetilde{\\Gamma }- \\Gamma , B^* - B \\rangle +\\langle \\Gamma - \\widehat{\\Gamma }, B^* - B \\rangle $ .", "(a) Control of $\\langle \\widetilde{\\Gamma }- \\Gamma , B^* - B \\rangle $ .", "Observe first that $B^$ is a projection matrix that induces the following decomposition of $\\widetilde{\\Gamma }- \\Gamma $ .", "$\\widetilde{\\Gamma }-\\Gamma &=& B^(\\widetilde{\\Gamma }-\\Gamma ) + (\\widetilde{\\Gamma }-\\Gamma ) B^* - B^* (\\widetilde{\\Gamma }-\\Gamma )B^* + (I - B^)(\\widetilde{\\Gamma }-\\Gamma )(I - B^).$ By the definition of the inner product, followed by the triangle inequality, and since $(I-B^)B^ = 0$ , we further have: $\\nonumber \|\\langle \\widetilde{\\Gamma }- \\Gamma , B^* - B \\rangle \| & \\le & 3 \|B^(\\widetilde{\\Gamma }-\\Gamma )\|_{\\infty }\|B^(B^* - B)\|_1 + \|\\langle (I - B^)(\\widetilde{\\Gamma }-\\Gamma )(I - B^), B^* - B\\rangle \| \\\\&=& 3 \|B^(\\widetilde{\\Gamma }-\\Gamma )\|_{\\infty }\|B^(B^* - B)\|_1 + \|\\langle \\widetilde{\\Gamma }-\\Gamma , (I-B^)B(I-B^)\\rangle \|.", "$ By the duality of the nuclear $\\Vert \\ \\Vert _$ and operator $\\Vert \\ \\Vert _{op}$ norms, we have $\|\\langle \\widetilde{\\Gamma }-\\Gamma , (I-B^)B(I-B^)\\rangle \|&\\le & \\Vert \\widetilde{\\Gamma }-\\Gamma \\Vert _{op} \\Vert (I-B^)B(I-B^)\\Vert _.$ We begin by bounding the nuclear norm $\\Vert (I-B^)B(I-B^)\\Vert _$ .", "Since $(I-B^)B(I-B^) \\in \\mathcal {S}^+$ , we have $\\Vert (I-B^)B(I-B^)\\Vert _ = \\operatorname{tr}((I-B^)B(I-B^)) = \\langle I-B^,B(I-B^) \\rangle = \\langle I-B^,B \\rangle .$ Using the fact that the sum of each row of $B$ is 1, we have $\\Vert (I-B^)B(I-B^)\\Vert _= \\langle I-B^,B \\rangle &= \\operatorname{tr}(B) - \\sum _{k=1}^{K}\\sum _{a,b\\in G_k}\\frac{B_{ab}}{\|G_k\|}\\nonumber \\\\&= \\operatorname{tr}(B)-K + \\sum _{k\\ne j}\\sum _{a\\in G_k,\\ b\\in G_j}\\frac{B_{ab}}{\|G_k\|} \\nonumber \\\\& \\le \\operatorname{tr}(B)-K + \\frac{1}{m}\\sum _{k\\ne j}\|B_{G_jG_k}\|_1 \\, .", "$ Next, we simplify the expression of $\|B^(B^* - B)\|_1=\|B^(I-B)\|_{1}$ .", "$\|B^(I - B)\|_1&=&\\sum _{j\\ne k}\\sum _{a\\in G_j,\\ b\\in G_k}\|(B^B)_{ab} \|+ \\sum _{k=1}^K\\sum _{a,b\\in G_k}\|[B^(I - B)]_{ab}\|\\\\&= &\\sum _{j\\ne k}\\sum _{a\\in G_j,\\ b\\in G_k} \\frac{1}{\|G_j\|}\\sum _{c\\in G_j}B_{cb}+ \\sum _{k=1}^K\\sum _{a,b\\in G_k}\\frac{1}{\|G_k\|}\\Big \|1 -\\sum _{c\\in G_k}B_{cb}\\Big \|\\\\& = & 2 \\sum _{j\\ne k }\|B_{G_jG_k}\|_1\\ ,$ where we used again $B1=1$ and that the entries of $B$ are nonnegative.", "Gathering the above bounds together with (REF ) yields: $\|\\langle \\widetilde{\\Gamma }- \\Gamma , B^-B \\rangle \| \\le 2\\Bigg [\\sum _{j\\ne k }\|B_{G_jG_k}\|_1\\Bigg ]\\left( {\\Vert \\widetilde{\\Gamma }-\\Gamma \\Vert _{op}\\over 2m}+ 3\|B^(\\widetilde{\\Gamma }-\\Gamma )\|_{\\infty }\\right) + \\big [\\operatorname{tr}(B)-K\\big ]\\Vert \\widetilde{\\Gamma }-\\Gamma \\Vert _{op}.$ We bound below the two terms in the parenthesis of (REF ).", "Since $n \\Gamma ^{-1/2}\\widetilde{\\Gamma }\\Gamma ^{-1/2}$ follows a Wishart distribution with $(n,p)$ parameters, we obtain by [7] that $\\Vert \\widetilde{\\Gamma }-\\Gamma \\Vert _{op}\\le 4\\Vert \\Gamma \\Vert _{op}\\left[\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right]=4\|\\Gamma \|_{\\infty }\\left[\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right]\\ ,$ with probability larger than $1-1/p$ .", "We now turn to $\|B^(\\widetilde{\\Gamma }-\\Gamma )\|_{\\infty }$ .", "For any $a,b$ in $[p]$ , let $k$ be such that $a\\in G_k$ .", "We have $[B^(\\widetilde{\\Gamma }-\\Gamma )]_{ab} = \\frac{1}{\|G_k\|} \\sum _{l\\in G_k} \\left(\\widetilde{\\Gamma }_{lb}- \\Gamma _{lb}\\right) = \\frac{1}{n\|G_k\|} \\sum _{l\\in G_{k}}\\sum _{i=1}^{n} [ \\epsilon _{li}\\epsilon _{bi} - \\operatorname{\\mathbb {E}}(\\epsilon _l\\epsilon _b)] \\ .$ The sum $\\sum _{l \\sim a}\\sum _{i=1}^{n} [ \\epsilon _{li}\\epsilon _{bi} - \\operatorname{\\mathbb {E}}(\\epsilon _l\\epsilon _b)]$ is a centered quadratic form of $n(\|G_k\|+1)$ (or $n\|G_k\|$ if $b\\in G_k$ ) independent normal variables whose variances belong to $\\lbrace \\gamma _l,\\ l=b\\text{ or }l\\sim a\\rbrace $ .", "Applying Lemma REF together with the assumption $\\log (p)\\le c_{1}n$ , we derive that, with probability larger than $1-c^{\\prime }/p$ , $\\sum _{l \\sim a}\\sum _{i=1}^{n} [ \\epsilon _{li}\\epsilon _{bi} - \\operatorname{\\mathbb {E}}(\\epsilon _l\\epsilon _b)]\\le c \|\\Gamma \|_{\\infty }\\left(\\sqrt{n\|G_k\|\\log (p)}+ \\log (p)\\right) \\le c^{\\prime } \|\\Gamma \|_{\\infty }\\sqrt{n\|G_k\|\\log (p)}$ simultaneously for all $l$ and all $b$ .", "In the second inequality, we used the assumption $\\log (p)\\le c_1 n$ .", "This yields $ \\mathbb {P}\\left[\|B^(\\widetilde{\\Gamma }-\\Gamma )\|_{\\infty } \\le c^{\\prime \\prime }\|\\Gamma \|_{\\infty } \\sqrt{\\frac{\\log p}{mn}} \\right]\\ge 1-c^{\\prime }/p\\ .$ Plugging into (REF ) the bounds derived in (REF ) and (REF ) above, and noticing from (REF ) that ${1\\over m} \\sum _{j\\ne k }\|B_{G_jG_k}\|_1+\\operatorname{tr}(B)-K\\ge 0,$ we obtain that $\|\\langle \\widetilde{\\Gamma }-\\Gamma , B^-B \\rangle \| \\le c \|\\Gamma \|_{\\infty }\\left( \\sqrt{ \\frac{\\log p}{mn} } \\vee \\sqrt{\\frac{p}{m^2n}} \\vee \\frac{p}{nm}\\right) \\sum _{j\\ne k }\|B_{G_jG_k}\|_1+ 4\|\\Gamma \|_{\\infty }\\left[\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right](\\operatorname{tr}(B)-K)\\ ,$ with probability larger than $ 1 - c^{\\prime }/p$ .", "(b) Control of $\\langle \\Gamma - \\widehat{\\Gamma }, B^* - B \\rangle $ .", "We follow the same approach as for $\\widetilde{\\Gamma }-\\Gamma $ .", "The additional ingredient is that $\\langle \\Gamma - \\widehat{\\Gamma }, B^-B \\rangle =\\langle \\Gamma - \\widehat{\\Gamma }-\\alpha I_{p}, B^-B \\rangle + \\alpha [K-\\operatorname{tr}(B)]$ for any $\\alpha \\in \\mathbb {R}$ , since $\\operatorname{tr}(B^)=K$ .", "Analogously to (REF ), for any $\\alpha \\in \\mathbb {R}$ , the following holds: $\|\\langle \\Gamma - \\widehat{\\Gamma }, B^-B \\rangle \| &\\le \|\\langle \\Gamma - \\widehat{\\Gamma }-\\alpha I_{p}, B^-B \\rangle \| + \|\\alpha [K-\\operatorname{tr}(B)]\| \\\\&\\le 2\\Bigg [\\sum _{j\\ne k }\|B_{G_jG_k}\|_1\\Bigg ]\\left( {\\Vert \\Gamma -\\widehat{\\Gamma }-\\alpha I_{p}\\Vert _{op}\\over 2m}+ 3\|B^(\\Gamma -\\widehat{\\Gamma })-\\alpha I_{p}\|_{\\infty }\\right)\\\\ &\\quad + \|\\alpha [K-\\operatorname{tr}(B)]\| +\\big [\\operatorname{tr}(B)-K\\big ]\\Vert \\widetilde{\\Gamma }-\\Gamma -\\alpha I\\Vert _{op}\\ .$ We fix $\\alpha = \| \\Gamma -\\widehat{\\Gamma }\|_{V}/2$ so that $\| \\Gamma -\\widehat{\\Gamma }-\\alpha I_{p}\|_{\\infty }=\| \\Gamma -\\widehat{\\Gamma }\|_{V}/2$ .", "Since $\\alpha I_{p}$ , $\\Gamma $ and $\\widehat{\\Gamma }$ are diagonal matrices, the above inequality simplifies to $\|\\langle \\Gamma - \\widehat{\\Gamma }, B^-B \\rangle \| \\le \\frac{ 7\|\\Gamma -\\widehat{\\Gamma }\|_{V}}{2m}\\sum _{j\\ne k }\|B_{G_jG_k}\|_1+ [\\operatorname{tr}(B)-K]_{+}\| \\Gamma -\\widehat{\\Gamma }\|_{V} \\ .$ The proof of Lemma REF is complete." ], [ "Poof of Corollary ", "At step 3 of PECOK, we have chosen a clustering algorithm which returns the partition $G^$ when applied to the true partnership matrix $B^$ .", "Hence, PECOK returns $G^$ as soon as $\\widehat{B}=B^$ .", "The Corollary REF then follows by combining Theorem REF and Proposition REF ." ], [ "Proof of Proposition ", "Let $k$ , $l_1$ and $l_2$ be such that $a\\in G_k$ and $ne_1(a)\\in G_{l_1}$ and $ne_2(a)\\in G_{l_2}$ .", "Starting from the identity $\\mathbf {X}_{:a}=\\mathbf {Z}_{:k}+\\mathbf {E}_{:a}$ , we developp $\\widehat{\\Gamma }_{aa}$ $\\widehat{\\Gamma }_{aa}&=& \\frac{\|\\mathbf {E}_{:a}\|_2^2}{n} + \\frac{1}{n} \\langle \\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_1}, \\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_2}\\rangle \\\\& &+ \\frac{1}{n}\\left[ \\langle \\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_1}, \\mathbf {E}_{:a}-\\mathbf {E}_{:ne_2(a)}\\rangle + \\langle \\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_2}, \\mathbf {E}_{:a}-\\mathbf {E}_{:ne_1(a)}\\rangle \\right]\\\\&& + \\frac{1}{n}\\left[ \\langle \\mathbf {E}_{:ne_1(a)}, \\mathbf {E}_{:ne_2(a)}\\rangle - \\langle \\mathbf {E}_{:a}, \\mathbf {E}_{:ne_1(a)}+\\mathbf {E}_{:ne_2(a)}\\rangle \\right]$ Since $2xy\\le x^2+y^2$ , the above expression decomposes as $\\big \|\\widehat{\\Gamma }_{aa}- \\Gamma _{aa}\\big \|&\\le & \\big \|\\frac{\|\\mathbf {E}_{:a}\|_2^2}{n}- \\Gamma _{aa} \\big \|+ U_1 + 2U_2+3U_3\\\\U_1 &:=& \\frac{1}{n} \|\\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_1}\|_2^2+ \\frac{1}{n}\|\\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_2}\|_2^2\\nonumber \\\\U_2&:= & \\frac{1}{n}\\sup _{k,j\\in [K]}\\sup _{b\\in [p]} \\langle \\frac{\\mathbf {Z}_{:k}- \\mathbf {Z}_{:j}}{\|\\mathbf {Z}_{:k}- \\mathbf {Z}_{:j}\|_2}, \\mathbf {E}_{:b}\\rangle ^2 \\ , \\quad U_3 := \\frac{1}{n}\\sup _{b\\ne c }\\langle \\mathbf {E}_{:b},\\mathbf {E}_{:c}\\rangle \\ .\\nonumber $ Recall that all the columns of $\\mathbf {E}$ are independent and that $\\mathbf {E}$ is independent from $\\mathbf {Z}$ .", "The terms $\|E_{:a}\|_2^2/n- \\Gamma _{aa}$ , $U_2$ and $U_3$ in (REF ) are quite straightforward to control as they either involve quadratic functions of Gaussian variables, suprema of Gaussian variables or suprema of centered quadratic functions of Gaussian variables.", "Applying the Gaussian tail bound and Lemma REF together with an union bound, and $\\log (p)\\le c_{1} n$ , we obtain $\\Big \|\\frac{\|E_{:a}\|_2^2}{n}- \\Gamma _{aa} \\big \|+2U_2+3U_3\\le c\|\\Gamma \|_{\\infty }\\sqrt{\\frac{\\log (p)}{n}} , $ with probability higher than $1-1/p^2$ .", "The main hurdle in this proof is to control the bias terms $\|\\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_i}\|_2^2$ for $i=1,2$ .", "Since $m\\ge 3$ , there exists two indices $b_1$ and $b_2$ other than $a$ belonging to the group $G_{k}$ .", "As a consequence, $\\mathbf {X}_{:a}-\\mathbf {X}_{:b_i}= \\mathbf {E}_{:a}-\\mathbf {E}_{:b_{i}}$ is independent from $\\mathbf {Z}$ and from all the other columns of $\\mathbf {E}$ .", "Hence, $\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:b_{i}}, \\tfrac{\\mathbf {X}_{:c}-\\mathbf {X}_{:d}}{\|\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\|_2}\\rangle $ is normally distributed with variance $2\\Gamma _{aa}$ and it follows that, with probability larger than $1-p^{-2}$ , $V(a,b_1)\\vee V(a,b_2) \\le c\|\\Gamma \|^{1/2}_{\\infty }\\sqrt{\\log (p)}\\ .$ The definition of $ne_1(a)$ and $ne_2(a)$ enforces that $V(a,ne_1(a))$ and $V(a,ne_2(a))$ satisfy the same bound.", "When $k=l_{1}$ then $\|\\mathbf {Z}_{:k}- \\mathbf {Z}_{:l_1}\|_2^2=0$ , so we only need to consider the case where $k\\ne l_{1}$ .", "Let $c\\in G_k\\setminus \\lbrace a\\rbrace $ and $d\\in G_{l_{1}}\\setminus \\lbrace ne_1(a)\\rbrace $ , which exists since $m\\ge 3$ .", "The above inequality for $V(a,ne_1(a))$ implies $\\big \|\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne_1(a)},\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\\rangle \\big \| \\le c\|\\Gamma \|^{1/2}_{\\infty }\\sqrt{\\log (p)}\|\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\|_2.$ This inequality is the key to control the norm of $t= \\mathbf {Z}_{:k}-\\mathbf {Z}_{:l_1}$ .", "Actually, since $a,c\\in G_{k}$ and $ne_{1}(a),d\\in G_{l_{1}}$ , we have $\\big \|\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne_1(a)},&\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\\rangle \\big \|= \\big \| \|t\|_{2}^2+\\langle t,E_{a}-E_{ne_{1}(a)}+E_{c}-E_{d}\\rangle +\\langle \\mathbf {E}_{:a}-\\mathbf {E}_{:ne_1(a)},\\mathbf {E}_{:c}-\\mathbf {E}_{:d}\\rangle \\big \|\\\\&\\ge \\frac{\|t\|_2^2}{2}-\\frac{1}{2}\\big \| \\langle \\frac{t}{\|t\|_2}, \\mathbf {E}_{:a}-\\mathbf {E}_{:ne_1(a)}+\\mathbf {E}_{:c}- \\mathbf {E}_{:d}\\rangle \\big \|^2 - \\big \|\\langle \\mathbf {E}_{:a}-\\mathbf {E}_{:ne_1(a)},\\mathbf {E}_{:c}-\\mathbf {E}_{:d}\\rangle \\big \|.$ Applying again a Gaussian deviation inequality and Lemma REF simultaneously for all $a,b,c,d\\in [p]$ and $k,l\\in [K]$ , we derive that with probability larger than $1-p^{-2}$ , $\\big \|\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne_1(a)},\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\\rangle \\big \|\\ge {1\\over 2}\|t\|_2^2 - c\|\\Gamma \|_{\\infty }\\sqrt{n\\log (p)}\\ ,$ since $\\log (p)\\le c_{1} n$ .", "Turning to the rhs of (REF ), we have $\|\\mathbf {X}_{:c}-\\mathbf {X}_{:d}\|_2\\le \|t\| + \|\\mathbf {E}_{:c}-\\mathbf {E}_{:d}\|_2$ .", "Taking an union bound over all possible $c$ and $d$ , we have $\|\\mathbf {E}_{:c}-\\mathbf {E}_{:d}\|_2\\le c\|\\Gamma \|_{\\infty }^{1/2}\\sqrt{\\log (p)}\\le c^{\\prime }\|\\Gamma \|_{\\infty }^{1/2}n^{1/2}$ with probability larger than $1-p^{-2}$ .", "Plugging these results in (REF ), we arrive at $\|t\|_2^2 - c_1 \|t\|_2\|\\Gamma \|_{\\infty }^{1/2}\\sqrt{\\log (p)}\\le c_2 \|\\Gamma \|_{\\infty }\\sqrt{n\\log (p)}\\ .$ This last inequality together with $\\log (p)\\le c_{1} n$ enforce that $\|t\|_{2}^2= \|\\mathbf {Z}_{:k}-\\mathbf {Z}_{:l_1}\|_2^2\\le c \|\\Gamma \|_{\\infty }\\sqrt{n\\log (p)}.$ Analogously, the same bound holds for $\|\\mathbf {Z}_{:k}-\\mathbf {Z}_{:l_2}\|_2^2$ .", "Together with (REF ) and (REF ), we have proved that $\\big \|\\widehat{\\Gamma }_{aa}- \\Gamma _{aa}\\big \|\\le c \|\\Gamma \|_{\\infty }\\sqrt{\\frac{\\log (p)}{n}}\\ ,$ with probability larger than $1-p^{-2}$ .", "The result follows." ], [ "Proof of Theorem ", "We shall follow the same approach as in the proof of Theorem REF .", "We need to prove that $ \\langle \\widehat{\\Sigma }, B^-B \\rangle + \\widehat{\\kappa } [\\operatorname{tr}(B)-K] > 0, \\ \\mbox{ for all} \\ B \\in \\mathcal {C}_0\\setminus \\lbrace B^\\rbrace .$ As in that previous proof, we introduce the matrix $W$ , so that it suffices to prove that $R(B):=\\langle W, B^-B \\rangle + n\\widehat{\\kappa } [\\operatorname{tr}(B)-K] > 0, \\ \\mbox{ for all} \\ B \\in \\mathcal {C}_0\\setminus \\lbrace B^\\rbrace \\ .$ We use the same decomposition as previously.", "Applying (REF ) together with Lemmas REF and REF , we derive that, with probability larger than $1-c/p$ , ${\\langle W, B^-B \\rangle \\ge \\sum _{j\\ne k} \\Bigg [{1\\over 2}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2 - c_1 \\sqrt{\\log (p)} \|\\Gamma \|_{\\infty }^{1/2}\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|_2} \\\\&& -c_2 \\frac{n\|\\widehat{\\Gamma }-\\Gamma \|_{V}}{m} - c_{2}\|\\Gamma \|_{\\infty }\\left( \\sqrt{ \\frac{n\\log p}{m} } \\vee \\sqrt{\\frac{np}{m^2}} \\vee \\frac{p}{m}\\right)\\Bigg ] \|B_{G_jG_k}\|_1\\ , \\nonumber \\\\&& - 4n\|\\Gamma \|_{\\infty }\\left(\\sqrt{\\frac{p}{n}}+\\frac{p}{n}\\right)(\\operatorname{tr}(B)-K)- n[\\operatorname{tr}(B)-K]_{+}\| \\Gamma -\\widehat{\\Gamma }\|_{V} \\ .\\nonumber $ As in (REF ), we use that with high probability $\|\\mathbf {Z}_{:j}-\\mathbf {Z}_{:k}\|^2_2$ is larger than $n\\Delta (C)/2$ .", "Condition (REF ) then enforces, that with high probability, the term inside the square brackets in (REF ) is larger than $n\\Delta (C)/8$ .", "As a consequence, we have $R(B)\\ge {n\\over 8}\\Delta (C) \\sum _{j\\ne k} \|B_{G_jG_k}\|_1+ n(\\operatorname{tr}(B)-K)\\left[ \\widehat{\\kappa } - 4\|\\Gamma \|_{\\infty }\\left(\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right) - \\mathbf {1}_{\\operatorname{tr}(B)\\ge K}\| \\Gamma -\\widehat{\\Gamma }\|_{V}\\right] \\ , $ uniformly over all $B\\in \\mathcal {C}_0$ .", "To finish the proof, we divide the analysis intro three cases depending on the values of $\\operatorname{tr}(B)$ .", "1.", "If $\\operatorname{tr}(B)>K$ , we apply Condition (REF ) to get $\\widehat{\\kappa } - 4\|\\Gamma \|_{\\infty }\\left(\\sqrt{\\frac{p}{n}}+ \\frac{p}{n}\\right) - \| \\Gamma -\\widehat{\\Gamma }\|_{V}>0$ , which implies that the right-hand side in (REF ) is positive.", "2.", "If $\\operatorname{tr}(B)=K$ , the right-hand side in (REF ) is positive except if $\\sum _{j\\ne k} \|B_{G_jG_k}\|_1=0$ , which implies $B=B^$ by Lemma REF .", "3.", "Turning to the case $\\operatorname{tr}(B)<K$ , (REF ) implies that $R(B) \\ge {n\\over 8}\\Delta (C) \\sum _{j\\ne k} \|B_{G_jG_k}\|_1 - \\widehat{\\kappa } n(K-\\operatorname{tr}(B)) \\ .", "$ It turns out, that when $\\operatorname{tr}(B)<K$ , the support of $B$ cannot be included in the one of $B^$ so that $ \\sum _{j\\ne k} \|B_{G_jG_k}\|_1$ is positive.", "Actually, (REF ) ensures that $\\sum _{j\\ne k }\|B_{G_jG_k}\|_1 \\ge [K-\\operatorname{tr}(B)]m$ , so together with (REF ), this gives us $R(B) \\ge n[K-\\operatorname{tr}(B)] \\left[ \\frac{m}{8}\\Delta (C) - \\widehat{\\kappa } \\right]\\ ,$ which is positive by condition (REF )." ], [ "Proof of Theorem ", "The proof is based on a careful application of Fano's lemma.", "Before doing this, we need to reformulate the clustering objective into a discrete estimation problem." ], [ "Construction of the covariance matrices $\\Sigma ^{(j)}$ .", "Let $A^{(0)}$ be the assignment matrix such that the $m$ first variables belong to the first group, the next $m$ belong to the second group and so on.", "In other words, $A^{(0)} = \\begin{bmatrix}1&0&0&\\hdots &&0\\\\\\vdots \\\\1&0&0&\\hdots &&0\\\\0&1&0&\\hdots &&0\\\\&\\vdots &\\ddots \\\\&&&&0&1\\\\&&&&&\\vdots \\\\&&&&&1\\\\\\end{bmatrix}\\text{ so that }\\Sigma ^{(0)} = \\begin{bmatrix}1+\\tau &\\tau &\\tau &0&\\hdots \\\\\\tau & \\ddots & \\tau \\\\\\tau &\\tau &1+\\tau \\\\\\end{bmatrix}0&&&1+\\tau &\\tau &\\tau &0&\\hdots \\\\\\vdots &&&\\tau &\\ddots &\\tau \\\\&&&\\tau &\\tau &1+\\tau \\\\&&&0&&&\\ddots \\\\&&&\\vdots &&&&\\\\$ , $where $ (0)= A(0) IK A(0)t + Ip$.", "Note that the associated partition for $ G(0)$ is $ { {1...m}...{p-m+1...p}}$.", "For any $ a=m+1,..., p$, denote $ a$ the transposition between $ 1$ and $ a$ in $ {1...p}$.", "Then, for any $ a=m+1,...,p$, define theassignement matrix $ A(a)$ and $ (a)$ by$$A^{(a)}_{ij}= A^{(0)}_{\\mu _{a}(i),j}\\ ,\\quad \\Sigma ^{(a)}_{ij}= \\Sigma ^{(0)}_{\\mu _{a}(i),\\mu _a(j)}\\ .$$In other words, the corresponding partition $ G(a)$ is obtained from $ G(0)$ by exchanging the role of the first and the $ a$-th node.\\medskip $ Also define the subset $M:=\\lbrace 0, m+1, m+2,\\ldots , p\\rbrace $ .", "Equipped with these notations, we observe that the minimax error probability of perfect recovery is lower bounded by $\\overline{\\mathbf {R}}^[\\tau ,n,m,p]\\ge \\inf _{\\hat{G}} \\max _{j\\in M } \\operatorname{\\mathbb {P}}_{\\Sigma ^{(j)}} \\big ( \\hat{G} \\ne G_j \\big )$ .", "According to Birgé's version of Fano's Lemma (see e.g.", "[18]), $\\inf _{\\hat{G}} \\max _{j\\in M } \\operatorname{\\mathbb {P}}_j \\big ( \\hat{G} \\ne G_j \\big )\\ge \\frac{1}{2e+1}\\bigwedge \\left(1-\\frac{\\sum _{j\\in M\\setminus \\lbrace 0\\rbrace }\\mathrm {KL}(\\operatorname{\\mathbb {P}}^{\\otimes n}_{\\Sigma ^{(j)}},\\operatorname{\\mathbb {P}}^{\\otimes n}_{\\Sigma ^{(0)}})}{(\|M\|-1)\\log (\|M\|)}\\right).$ By symmetry, all the Kullback divergences are equal.", "Since $2e/(2e+1)\\ge 0.8$ and $1/(2e+1)\\ge 1/7$ , we arrive at $\\overline{\\mathbf {R}}^[\\tau ,n,m,p] \\ge 1/7\\ , \\quad \\text{ if }\\quad n\\mathrm {KL}(\\operatorname{\\mathbb {P}}_{\\Sigma ^{(m+1)}},\\operatorname{\\mathbb {P}}_{\\Sigma ^{(0)}})\\le 0.8 \\log (p-m+1)\\ .$ As the derivation of the Kullback-Leibler discrepancy is involved, we state it here and postpone its proof to the end of the section.", "Lemma 7 For any $\\tau >1$ and any integers $p$ and $m$ , we have $\\mathrm {KL}(\\operatorname{\\mathbb {P}}_{\\Sigma ^{(m+1)}},\\operatorname{\\mathbb {P}}_{\\Sigma ^{(0)}})= \\frac{2 (m-1)\\tau ^2}{1+m\\tau }$ As a consequence, the minimax error probability of perfect recovery $\\overline{\\mathbf {R}}^[\\tau ,n,m,p]$ is larger than $1/7$ as soon as $\\frac{2n (m-1)\\tau ^2}{1+m\\tau }\\le 0.8 \\log (p-m+1)\\ .$ This last condition is satisfied as soon as $\\tau \\le c \\left[\\sqrt{\\frac{\\log (p)}{n(m-1)}}\\bigvee \\frac{\\log (p)}{n} \\right]\\ ,$ for some numerical constant $c>0$ ." ], [ "Proof of Lemma ", "The Kullback-Leibler divergence between two centered normal distributions writes as $\\mathrm {KL}(\\operatorname{\\mathbb {P}}_{\\Sigma ^{(m+1)}}, \\operatorname{\\mathbb {P}}_{\\Sigma ^{(0)}}) = \\frac{1}{2} \\big [ - \\log \\mathrm {det} \\big ((\\Sigma ^{(0)})^{-1}\\Sigma ^{(m+1)}\\big ) + \\text{trace} \\big ((\\Sigma ^{(0)})^{-1}\\Sigma ^{(m+1)} - I_p\\big )\\big ] \\ ,$ so that we only have to compute the determinant and the trace of $A:= (\\Sigma ^{(1)})^{-1}\\Sigma ^{(m+1)}$ .", "We shall see that see that $A$ is a rank 2 perturbation of the identity matrix, so that we will only need to compute its two eigenvalues different from zero.", "Observe that for $i=0,m+1$ , the matrices $A^{(i)}{A^{(i)}}^t$ admit exactly $K$ non-zero eigenvalues that are all equal to $m$ .", "As a consequence, we can decompose $A^{(i)}{A^{(i)}}^t = m \\sum _{k=1}^{K}u_k^{(i)}(u_k^{(i)})^t$ where $u_k^{(i)}$ is a unit vector whose non zero components are all equal to $1/m$ and correspond to the $k$ -th group in $G^{(i)}$ .", "Note that $u_k^{(0)}=u_{k}^{(m+1)}$ for $k=3,\\ldots , K$ as $A^{(0)}$ and $A^{(m+1)}$ only differ by rows 1 and $m+1$ .", "The orthogonal projector $P_i= \\sum _{k=1}^{K}u_k^{(i)}(u_k^{(i)})^t$ satisfies $\\Sigma ^{(i)}= m \\tau P_i + I_p = (1+m \\tau )P_i + (I_p-P_i)\\ .$ Since $P_i$ and $I_p-P_i$ are orthogonal, $(\\Sigma ^{(i)})^{-1} = (1+m\\tau )^{-1}P_i + (I_p-P_i)= I_p-\\frac{m\\tau }{1+m\\tau }P_i$ As a consequence of the above observations, we have $A &=& I_p + (\\Sigma ^{(0)})^{-1}\\Big [\\Sigma ^{(m+1)}-\\Sigma ^{(0)}\\Big ]\\\\& = & I_p + m\\tau \\big (P_{m+1}- P_{0}\\big ) - \\frac{m^2\\tau ^2}{1+m\\tau }P_{0}(P_{m+1}-P_{0}) =: I_p + B$ The matrices $P_{0}$ and $P_{m+1}$ are $k-1$ block diagonal with a first block of size $2m\\times 2m$ .", "Besides, $P_{0}$ and $P_{m+1}$ take the same values on all the $K-2$ remaining blocks.", "To compute the non-zero eigenvalues of $B$ , we only to consider the restrictions $\\overline{P}_{0}$ and $\\overline{P}_{m+1}$ of $P_{0}$ and $P_{m+1}$ to the first $2m\\times 2m$ entries.", "Also observe that the matrices $\\overline{P}_{0}$ and $\\overline{P}_{m+1}$ are $4\\times 4$ block-constant, with block size ${\\scriptsize \\begin{bmatrix}1\\times 1&1\\times (m-1) &1\\times 1& 1\\times (m-1)\\\\(m-1) \\times 1 & (m-1) \\times (m-1) & (m-1) \\times 1 & (m-1) \\times (m-1) \\\\1\\times 1&1\\times (m-1) &1\\times 1& 1\\times (m-1)\\\\(m-1) \\times 1 & (m-1) \\times (m-1) &(m-1) \\times 1&(m-1) \\times (m-1)\\end{bmatrix}}$ and the entries are ${\\scriptsize m\\overline{P}_{0} = \\begin{bmatrix}1&1&0&0\\\\1& 1 & 0 & 0\\\\0&0&1&1\\\\0&0&1&1\\end{bmatrix}\\text{ and }m \\overline{P}_{m+1} = \\begin{bmatrix}1&0&0&1\\\\0& 1 & 1 & 0\\\\0&1&1&0\\\\1&0&0&1 \\ .\\end{bmatrix}}$ As a consequence, the non zero eigenvalues of $B$ are the same as those of $C:= m\\tau \\big (\\underline{P}_{m+1}- \\underline{P}_{0}\\big ) - \\frac{m^2\\tau ^2}{1+m\\tau }\\underline{P}_{0}(\\underline{P}_{m+1}-\\underline{P}_{0}) $ where $\\underline{P}_{m+1}$ and $\\underline{P}_{0}$ are two $4\\times 4$ matrices ${\\scriptsize m\\underline{P}_{0} = \\begin{bmatrix}1&(m-1)&0&0\\\\1& (m-1) & 0 & 0\\\\0&0&1&(m-1)\\\\0&0&1&(m-1)\\end{bmatrix}\\text{ and }m \\underline{P}_{m+1} = \\begin{bmatrix}1&0&0&(m-1)\\\\0& (m-1) & 1 & 0\\\\0&(m-1)&1&0\\\\1&0&0&(m-1) \\ .\\end{bmatrix}}$ Working out the product of matrices, we get ${\\scriptsize C = - \\tau \\begin{bmatrix}0&(m-1)&0&-(m-1)\\\\1& 0 & -1 & 0\\\\0&-(m-1)&0&(m-1)\\\\-1& 0 & 1 & 0\\\\\\end{bmatrix} + \\frac{(m-1)\\tau ^2}{1+m\\tau }\\begin{bmatrix}1&1&-1&-1\\\\1& 1 & -1 & -1\\\\-1&-1&1& 1\\\\-1&-1&1&1\\end{bmatrix}}$ We observe that these two matrices have their first (resp.", "second) and third (resp.", "fourth) lines and columns opposite to each other.", "As a consequence, the two non-zero eigenvalues of $C$ are the same as those of ${\\scriptsize D}&: =& {\\scriptsize -2\\tau \\begin{bmatrix}0&(m-1)\\\\1& 0\\end{bmatrix} + \\frac{2(m-1)\\tau ^2}{1+m\\tau }\\begin{bmatrix}1&1\\\\1&1\\end{bmatrix} }\\\\&=& {\\scriptsize \\frac{2\\tau }{1+m\\tau }\\begin{bmatrix}(m-1)\\tau & -(m-1)[1+(m-1)\\tau ]\\\\-(1+\\tau )& (m-1)\\tau \\end{bmatrix}}\\ .$ Straightforward computations then lead to $tr(D) = \\frac{4(m-1)\\tau ^2}{1+m\\tau } \\ , \\quad \\mathrm {det}(D)= -tr(D) $ Coming back to (REF ), we have $2\\mathrm {KL}(\\operatorname{\\mathbb {P}}_{\\Sigma ^{(m+1)}}, \\operatorname{\\mathbb {P}}_{\\Sigma ^{(0)}}) &=& - \\log \\mathrm {det} \\big (A \\big ) + \\text{trace} \\big (A- I_p\\big )\\big ]\\\\&=& - \\log \\mathrm {det}(I+D)+ tr(D)\\\\& =& tr(D) - \\log \\big [1+tr(D)+ det(D)\\big ]\\\\& =&\\frac{4(m-1)\\tau ^2}{1+m\\tau } \\ .$" ], [ "Proof of Theorem ", "The proof is based on the following Lemma by Lei and Rinaldo [15].", "Lemma 8 Let $M$ be any matrix of the form $M=AQ$ where $A\\in \\mathcal {A}_{p,K}$ is a membership matrix and $Q\\in \\mathbb {R}^{K\\times q}$ , and denote by $\\delta $ the minimal distance between two rows of $Q$ .", "Then, there exists a constant $c_{\\eta }$ , such that, for any matrix $M^{\\prime }$ fulfilling $\\Vert M-M^{\\prime }\\Vert _{F}^2< m\\delta ^2/c_{\\eta }$ , the classification of the rows of $M^{\\prime }$ by an $\\eta $ -approximate $K$ -means provides a clustering $\\widehat{G}$ fulfilling $\\bar{L}(\\hat{G},G)\\le c_{\\eta } {\\Vert M-M^{\\prime }\\Vert _{F}^2\\over m\\delta ^2}.$ We start with the following observation.", "Since $\\Vert \\Sigma \\Vert _{op}\\ge m\\Vert C\\Vert _{op}\\ge m \\lambda _K(C)$ , Condition (REF ) enforces that $\\frac{Re(\\Sigma )\\vee \\log (p)}{n}\\le 1/c^{2}_{\\eta }\\ .$ Let $U$ be a $K\\times p$ matrix which gathers the eigenvectors of $A CA^t$ associated to the $K$ leading eigenvalues.", "The associated eigenvectors are block constant.", "Therefore $U_{0}=AQ_{0}$ , and since $A^tA=mI$ , the matrix $\\sqrt{m}Q_{0}$ is orthogonal.", "We apply Lemma REF with $M^{\\prime }=\\widehat{U}$ and $M=U_{0}\\widehat{O}$ , where $\\widehat{O}$ is a $K\\times K$ orthogonal matrix to be chosen.", "We have $M=AQ$ with $\\sqrt{m}Q=\\sqrt{m}Q_{0}\\widehat{O}$ orthogonal.", "In particular, the minimal distance between two rows of $Q$ is $\\delta =\\sqrt{2/m}$ .", "Lemma REF ensures that $\\bar{L}(\\hat{G}_{S},G) \\le c_{\\eta } {\\Vert \\widehat{U}-U_{0}\\widehat{O}\\Vert _F^2 \\over 2},$ whenever the right-hand side is smaller than 1.", "By Davis-Kahan inequality (e.g.", "[15]), there exists an orthogonal matrix $\\widehat{O}$ such that $\\Vert \\widehat{U}-U_{0}\\widehat{O}\\Vert _F^2\\le {8K \\Vert \\widetilde{\\Sigma }-ACA^t\\Vert ^2_{op} \\over m^2 \\lambda ^2_{K}(C)}\\ .$ We can upper-bound the operator norm of $\\widetilde{\\Sigma }-A CA^t$ by $\\Vert \\widetilde{\\Sigma }-A CA^t\\Vert _{op}\\le \\Vert \\widehat{\\Sigma }-\\Sigma \\Vert _{op}+ \\Vert \\widehat{\\Gamma }-\\Gamma \\Vert _{op}\\ .$ According to Theorem 1 in [11] (see also [5]), there exists a constant $c>0$ such that, with probability at least $1-1/p$ $\\nonumber \\Vert \\widehat{\\Sigma }-\\Sigma \\Vert _{op}&\\le & c \\Vert \\Sigma \\Vert _{op} \\left(\\sqrt{Re(\\Sigma )\\over n}\\bigvee {Re(\\Sigma )\\over n}\\bigvee \\sqrt{\\log (p)\\over n} \\bigvee \\frac{\\log (p)}{n} \\right)\\\\&\\le & c \\Vert \\Sigma \\Vert _{op} \\left(\\sqrt{Re(\\Sigma )\\over n} \\bigvee \\sqrt{\\log (p)\\over n} \\right) ,$ where we used (REF ) in the second line.", "Then, using that $\\Vert \\widehat{\\Gamma }-\\Gamma \\Vert _{op}=\|\\widehat{\\Gamma }-\\Gamma \|_{\\infty }$ and Proposition REF together with $\|\\Gamma \|_{\\infty }\\le \\Vert \\Sigma \\Vert _{op}$ , we obtain the inequality $\\Vert \\widetilde{\\Sigma }-A CA^t\\Vert _{op} \\le c \\Vert \\Sigma \\Vert _{op}\\left(\\sqrt{Re(\\Sigma )\\over n} \\bigvee \\sqrt{\\log (p)\\over n} \\right)\\ ,$ with probability at least $1-c/p$ .", "So combining (REF ), with (REF ) and (REF ) we obtain the existence of $c^{\\prime }_{\\eta }>0$ such that we have $\\bar{L}(\\hat{G}_S,G) \\le {c^{\\prime }_{\\eta }K \\Vert \\Sigma \\Vert _{op}^2\\over m^2\\lambda _{K}(C)^2}\\left(\\sqrt{\\frac{Re(\\Sigma )}{n}}\\bigvee \\sqrt{\\log (p)\\over n}\\right)^2,$ with probability at least $1-c/p$ , whenever the right-hand side is smaller than 1.", "The proof of Theorem REF follows." ], [ "Proof of Lemma ", "We recall that $\\widehat{U}$ is the $p\\times K$ matrix stacking the $K$ leading eigenvectors of $\\widetilde{\\Sigma } = \\widehat{\\Sigma } - \\widehat{\\Gamma }$ .", "We first prove that the matrix $\\widehat{U}\\widehat{U}^t$ is solution of (REF ).", "Let us write $\\widetilde{\\Sigma }=\\widetilde{U} \\widetilde{D} \\widetilde{U}^t$ for a diagonalisation of $\\widetilde{\\Sigma }$ with $\\widetilde{U}$ orthogonal and $\\widetilde{D}_{11}\\ge \\ldots \\ge \\widetilde{D}_{pp}\\ge 0$ .", "We observe that $\\langle \\widetilde{\\Sigma },B\\rangle = \\langle \\widetilde{D},\\widetilde{U}^t B \\widetilde{U}\\rangle $ , and that $B\\in \\overline{\\mathcal {C}}$ iff $\\widetilde{U}^t B \\widetilde{U} \\in \\overline{\\mathcal {C}}$ since the matrix $\\widetilde{B}=\\widetilde{U}^t B \\widetilde{U}$ has the same eigenvalues as $B$ .", "We observe also that $\\widehat{U}\\widehat{U}^t=\\widetilde{U}\\Pi _{K}\\widetilde{U}^t$ , where $\\Pi _{K}$ is the diagonal matrix, with 1 on the first $K$ diagonal elements and 0 on the $p-K$ remaining ones.", "So proving that $\\overline{B}=\\widehat{U}\\widehat{U}^t$ is solution of (REF ) is equivalent to proving that $\\Pi _{K}=\\operatornamewithlimits{argmax}_{\\widetilde{B}\\in \\overline{\\mathcal {C}}} \\langle \\widetilde{D},\\widetilde{B}\\rangle .$ Let us prove this result.", "To start with, we notice that $ \\sum _{k=1}^K\\widetilde{D}_{kk}=\\max _{0\\le \\widetilde{B}_{kk}\\le 1;\\ \\sum _{k}\\widetilde{B}_{kk}=K} \\langle \\widetilde{D},\\widetilde{B}\\rangle .$ Since the condition $I \\succcurlyeq \\widetilde{B}\\succcurlyeq 0$ enforces $0\\le \\widetilde{B}_{kk}\\le 1$ , we have $\\overline{\\mathcal {C}}\\subset \\lbrace B: 0\\le \\widetilde{B}_{kk}\\le 1;\\ \\sum _{k}\\widetilde{B}_{kk}=K\\rbrace $ and then $\\max _{\\widetilde{B}\\in \\overline{\\mathcal {C}}} \\langle \\widetilde{D},\\widetilde{B}\\rangle \\le \\sum _{k=1}^K\\widetilde{D}_{kk}=\\langle \\widetilde{D},\\Pi _{K}\\rangle .$ Hence $\\Pi _{K}$ is solution to the above maximisation problem and $\\overline{B}=\\widetilde{U} \\Pi _{K} \\widetilde{U}^t=\\widehat{U} \\widehat{U}^t$ .", "To conclude the proof, we notice that $\\widehat{U}_{a:}\\widehat{U}^t$ is an orthogonal transformation of $\\widehat{U}_{a:}$ , so we obtain the same results when applying a rotationally invariant clustering algorithm to the rows of $\\widehat{U}$ and to the rows of $\\widehat{U} \\widehat{U}^t$ ." ], [ "An alternative estimator of $\\Gamma $", "We propose here a more simple estimator of $\\Gamma $ .", "It has the nice feature to have a smaller computational complexity than (REF ), but the drawback to have fluctuations possibly proportional to $\|\\Sigma \|_{\\infty }^{1/2}$ .", "For any $a\\in [p]$ , define $ne(a):= \\operatornamewithlimits{argmin}_{b\\in [p]\\setminus \\lbrace a\\rbrace }\\, \\max _{c\\ne a,b}\\big \|\\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:b}, \\frac{\\mathbf {X}_{:c}}{\|\\mathbf {X}_{:c}\|_2}\\rangle \\big \| \\ ,$ the “neighbor” of $a$ , that is the variable $\\mathbf {X}_{:b}$ such that the covariance $\\langle \\mathbf {X}_{:b} ,\\mathbf {X}_{:c}\\rangle $ is most similar to $\\langle \\mathbf {X}_{:a} ,\\mathbf {X}_{:c}\\rangle $ , this for all variables $c$ .", "It is expected that $ne(a)$ belongs to the same group of $a$ , or if it is not the case that the difference $C^_{kk}-C^_{kj}$ , where $a\\in G^_k$ and $ne(a)\\in G^_j$ , is small.", "Then, the diagonal matrix $\\widehat{\\Gamma }$ is defined by $\\widehat{\\Gamma }_{aa}= {1\\over n}\\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {X}_{:a}\\rangle ,\\ \\textrm {for}\\ a=1,\\ldots ,p.$ In population version, this quantity is of order $\\Gamma _{aa}+ C^_{k(a)k(a)}-C^_{k(a)k(ne(a))}$ ($k(a)$ and $k(ne(a))$ respectively stand for the group of $a$ and $ne(a)$ ) and should therefore be of order $\\Gamma _{aa}$ if the last intuition is true.", "As shown by the following proposition, the above discussion can be made rigorous.", "Proposition 5 There exist three numerical constants $c_1$ –$c_3$ such that the following holds.", "Assume that $m\\ge 2$ and that $\\log (p)\\le c_1 n$ .", "With probability larger than $1-c_{3}/p$ , the estimator $\\widehat{\\Gamma }$ defined by (REF ) satisfies $\|\\widehat{\\Gamma }- \\Gamma \|_{V}\\le 2 \|\\widehat{\\Gamma }- \\Gamma \|_{\\infty }\\le c_2 \|\\Gamma \|^{1/2}_\\infty \|\\Sigma \|^{1/2}_{\\infty }\\sqrt{\\frac{\\log (p)}{n}}\\ .$ The PECOK estimator with $\\widehat{\\Gamma }$ defined by (REF ) then fulfills the following recovering property.", "Corollary 5 There exist $c_1,\\ldots , c_3$ three positive constants such that the following holds.", "Assuming that $\\widehat{\\Gamma }$ is defined by (REF ), $\\log (p)\\le c_1 n$ , and that $\\Delta (C^) \\ge c_2 \\left[\|\\Gamma \|_{\\infty }\\left\\lbrace \\sqrt{ \\frac{\\log p}{mn} }+ \\sqrt{\\frac{p}{nm^2}} + \\frac{\\log (p)}{n}+ \\frac{p}{nm}\\right\\rbrace + \|\\Gamma \|_{\\infty }^{1/2}\|C^\|^{1/2}_{\\infty } \\sqrt{\\frac{\\log (p)}{nm^2}} \\right]\\ ,$ then we have $\\widehat{B} = B^$ and $\\widehat{G}=G^$ , with probability higher than $1 - c_3/p$ .", "The additional term $\|\\Gamma \|_{\\infty }^{1/2}\|C^\|^{1/2}_{\\infty }\\sqrt{\\frac{\\log (p)}{nm^2}}$ term is smaller than $\|\\Gamma \|_{\\infty }\\sqrt{\\frac{\\log (p)}{nm}}$ when $\|C^\|_{\\infty }\\le m \|\\Gamma \|_{\\infty }$ , which is likely to occur when $m$ is large." ], [ "Proof of Proposition ", "Consider any $a\\in [p]$ , let $k$ be the group such that $a\\in G^_k$ .", "We now divide the analysis into two cases: (i) $ne(a)\\in G^_k$ ; (ii) $ne(a)\\notin G^_k$ .", "In case (i), we directly control the difference: $\|\\widehat{\\Gamma }_{aa} - \\Gamma _{aa}\|&=&\\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {X}_{:a}\\rangle /n - \\Gamma _{aa}\\big \|\\\\&\\le & \\big \|\\Gamma _{aa}- \|\\mathbf {E}_{:a}\|^2_2/n\\big \| + \\big \|\\langle \\mathbf {E}_{:ne(a)}, \\mathbf {E}_{:a}\\rangle /n\\big \|+ \\big \|\\langle Â~\\mathbf {E}_{:a}-\\mathbf {E}_{:ne(a)} , \\mathbf {Z}_{:k} \\rangle /n\\big \|\\\\&\\le &\\big \|\\Gamma _{aa}- \|\\mathbf {E}_{:a}\|^2_2/n\\big \| + \\big \|\\langle \\mathbf {E}_{:ne(a)}, \\mathbf {E}_{:a}\\rangle /n\\big \|+ \\big \|\\langle Â~\\mathbf {E}_{:a}-\\mathbf {E}_{:ne(a)} , \\mathbf {Z}_{:k} \\rangle /n\\big \|\\\\&\\le & \\big \|\\Gamma _{aa}- \|\\mathbf {E}_{:a}\|^2_2/n\\big \|+ \\sup _{c\\in G^_k\\setminus \\lbrace a\\rbrace }\\left(\\big \|\\langle \\mathbf {E}_{:c}, \\mathbf {E}_{:a}\\rangle /n\\big \|+ \\big \|\\langle \\mathbf {E}_{:a}-\\mathbf {E}_{:c} , \\mathbf {Z}_{:k} \\rangle /n\\big \|\\right)$ The random variable $\|\\mathbf {E}_{:a}\|^2_2/\\Gamma _{a,a}$ follows a $\\chi ^2$ distribution with $n$ degrees of freedom whereas the remaining variables are centered quadratic form of independent Gaussian variables.", "Applying the deviation inequality for Gaussian quadratic forms (REF ) together with an union bound, we arrive at $\|\\widehat{\\Gamma }_{aa} - \\Gamma _{aa}\|\\le c \|\\Gamma \|^{1/2}_{\\infty }\|\\Sigma \|^{1/2}_{\\infty } \\sqrt{\\frac{\\log (p)}{n}} ,$ with probability larger than $1-1/p^{2}$ .", "Let us turn to case (ii): $ne(a)\\notin G^_k$ .", "Let $b\\in G^_k$ , with $b\\ne a$ .", "We have $\|\\widehat{\\Gamma }_{aa} - \\Gamma _{aa}\|&=&\\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {X}_{:a}\\rangle /n - \\Gamma _{aa}\\big \|\\\\&\\le & \\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {E}_{:a}\\rangle /n - \\Gamma _{aa}\\big \| + \\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {Z}_{:k}\\rangle /n \\big \|\\\\&\\le & \\big \|\|\\mathbf {E}_{:a}\|_{2}^2/n - \\Gamma _{aa}\\big \|+ \\big \|\\langle \\mathbf {Z}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {E}_{:a}\\rangle /n + \\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {X}_{:b}\\rangle /n \\big \|\\\\&&+ \\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {E}_{:b}\\rangle /n \\big \|\\ ,$ where we used $\\mathbf {X}_{:b}= \\mathbf {E}_{:b}+ \\mathbf {Z}_{:k}$ since $b\\in G^_k$ .", "In the above inequality, the difference $\|\|\\mathbf {E}_{:a}\|_{2}^2/n - \\Gamma _{aa}\\big \|$ is handled as in the previous case.", "The second term $ \|\\langle \\mathbf {Z}_{:k}- \\mathbf {X}_{:ne(a)},\\mathbf {E}_{:a}\\rangle /n\|\\le \\sup _{c \\notin G^_k}\|\\langle \\mathbf {Z}_{:k}- \\mathbf {X}_{:c},\\mathbf {E}_{:a}\\rangle /n\|$ is bounded by a supremum of centered quadratic forms of Gaussian variables and is therefore smaller than $c\\sqrt{\\log (p)/n}\|\\Gamma \|_{\\infty }\|^{1/2}\\Sigma \|_{\\infty }^{1/2}$ with probability larger than $1-p^{-2}$ .", "The fourth term $\\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {E}_{:b}\\rangle /n \\big \|$ is handled analogously.", "To control the last quantity $\\big \| \\langle \\mathbf {X}_{:a}- \\mathbf {X}_{:ne(a)},\\mathbf {X}_{:b}\\rangle /n \\big \|$ , we use the definition (REF ) $\\big \|\\big \\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne(a)}, \\frac{\\mathbf {X}_{:b}}{\|\\mathbf {X}_{:b}\|_2}\\big \\rangle \\big \|\\le \\max _{c\\ne a, ne(a)}\\big \|\\big \\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne(a)}, \\frac{\\mathbf {X}_{:c}}{\|\\mathbf {X}_{:c}\|_2}\\big \\rangle \\big \|\\le \\max _{c\\ne a,b}\\big \|\\big \\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:b}, \\frac{\\mathbf {X}_{:c}}{\|\\mathbf {X}_{:c}\|_2}\\big \\rangle \\big \|.$ Since $b\\in G^_k$ , the random variable $\\mathbf {X}_{:a}-\\mathbf {X}_{:b}= \\mathbf {E}_{:a}-\\mathbf {E}_{:b}$ is independent from all $\\mathbf {X}_{:c}$ , with $c\\ne a,b$ .", "Since $\\operatorname{Var}\\left(E_{a}-E_{b}\\right)\\le 2\|\\Gamma \|_{\\infty }$ , we use the Gaussian concentration inequality together with an union bound to get $\\max _{c\\ne a,b} \\big \|\\big \\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:b}, \\frac{\\mathbf {X}_{:c}}{\|\\mathbf {X}_{:c}\|_2}\\big \\rangle \\big \|\\le \|\\Gamma \|^{1/2}_{\\infty }\\sqrt{12\\log (p)}$ with probability larger than $1-p^{-2}$ .", "As a consequence, $\\big \|\\big \\langle \\mathbf {X}_{:a}-\\mathbf {X}_{:ne(a)}, \\frac{\\mathbf {X}_{:b}}{\|\\mathbf {X}_{:b}\|_2}\\big \\rangle \\big \|\\le \|\\Gamma \|^{1/2}_{\\infty }\\sqrt{12\\log (p)},$ with probability larger than $1-p^{-2}$ .", "Since $\|\\mathbf {X}_{:b}\|_2^2/\\Sigma _{bb}$ follows a $\\chi ^2$ distribution with $n$ degrees of freedom, we have $\|\\mathbf {X}_{:b}\|_2\\le c n^{1/2}\|\\Sigma \|_{\\infty }^{1/2}$ with probability larger than $1-p^{-2}$ .", "Putting everything together, we have shown that $\|\\widehat{\\Gamma }_{aa} - \\Gamma _{aa}\|\\le c \|\\Gamma \|^{1/2}_{\\infty }\|\\Sigma \|_{\\infty }^{1/2} \\sqrt{\\frac{\\log (p)}{n}}\\ ,$ with probability larger than $1-c^{\\prime }/p^2$ .", "Taking an union bound over all $a\\in [p]$ concludes the proof." ], [ "Deviation inequalities", "Lemma 9 (Quadratic forms of Gaussian variables [13]) Let $Y$ stands for a standard Gaussian vector of size $k$ and let $A$ be a symmetric matrix of size $k$ .", "For any $t>0$ , $\\mathbb {P}\\left[Y^t A Y \\ge tr(A)+ 2\\Vert A\\Vert _F\\sqrt{t}+2 \\Vert A\\Vert _{op}t\\right]\\le e^{-t}\\ .$ Laurent and Massart [13] have only stated a specific version of Lemma REF for positive matrices $A$ , but their argument straightforwardly extend to general symmetric matrices $A$ ." ] ]	1606.05100
[ [ "Partial redistribution in 3D non-LTE radiative transfer in solar\n atmosphere models" ], [ "Abstract Resonance spectral lines such as H I Ly {\\alpha}, Mg II h&k, and Ca II H&K that form in the solar chromosphere are influenced by the effects of 3D radiative transfer as well as partial redistribution (PRD).", "So far no one has modeled these lines including both effects simultaneously owing to the high computing demands of existing algorithms.", "Such modeling is however indispensable for accurate diagnostics of the chromosphere.", "We present a computationally tractable method to treat PRD scattering in 3D model atmospheres using a 3D non-LTE radiative transfer code.", "To make the method memory-friendly, we use the hybrid approximation of Leenaarts et al.", "(2012) for the redistribution integral.", "To make it fast, we use linear interpolation on equidistant frequency grids.", "We verify our algorithm against computations with the RH code and analyze it for stability, convergence, and usefulness of acceleration using model atoms of Mg II with the h&k lines and H I with the Ly {\\alpha} line treated in PRD.", "A typical 3D PRD solution can be obtained in a model atmosphere with $252 \\times 252 \\times 496$ coordinate points in 50 000--200 000 CPU hours, which is a factor ten slower than computations assuming complete redistribution.", "We illustrate the importance of the joint action of PRD and 3D effects for the Mg II h&k lines for disk-center intensities as well as the center-to-limb variation.", "The proposed method allows simulating PRD lines in time series of radiation-MHD models in order to interpret observations of chromospheric lines at high spatial resolution." ], [ "Introduction", "At the beginning of the development of radiative transfer theory, astrophysicists assumed that scattering in spectral lines is coherent.", "This assumption is however unable to reproduce observed intensity profiles and their center-to-limb variation in detail.", "In the 1940s, a better approximation called complete frequency redistribution (CRD) was introduced.", "[13] validated it using systematic observations of various line profiles in the solar spectrum.", "The CRD approximation means that during line scattering there is no correlation between absorbed and emitted photons.", "CRD requires atomic levels of the line transition to be strongly perturbed by elastic collisions with other atoms.", "This is true in dense layers of stellar atmospheres such as the photosphere of the Sun.", "With a few exceptions, all lines visible in the solar spectrum are formed in CRD.", "The CRD approximation makes the line source function constant, which strongly simplifies analytical and numerical solutions of the radiative transfer problem in such lines.", "In the 1970s, as reviewed by [27] and [33], it became clear that strong chromospheric lines actually demonstrate partially-coherent scattering (nowadays more commonly referred to as partial frequency redistribution, PRD).", "Although the theory of PRD was developed earlier, only then it became possible to model PRD lines due to high computational demands for the redistribution functions.", "Contrary to CRD, PRD includes the fact that photon scattering is not arbitrary and could even be fully correlated in the most extreme case of coherent scattering.", "Line scattering becomes partially or fully coherent only if several conditions are met together [16].", "The upper atomic level of the line transition must be weakly perturbed by elastic collisions to be able to retain radiatively-excited sublevels during the finite radiative lifetime of the level.", "This is true in low-density layers of stellar atmospheres such as the chromosphere of the Sun.", "The chemical element of the line has to be abundant and exist in a dominant ionization stage so that the line extinction dominate over the continuum extinction by several orders of magnitude.", "The line has to be a resonance transition, or a transition whose lower level is metastable Only a small number of lines in the solar spectrum are affected by PRD but they are indispensable as diagnostics of the outer atmosphere of the Sun.", "Among them are: the strongest lines of HI Lyman series such as the Ly $\\beta $ 102.6 nm and the Ly $\\alpha $ 121.6 nm lines [30], [15]; the strongest chromospheric MgII k 279.6 nm and h 280.4 nm [31] as well as the CaII K 393.4 nm and H 396.8 nm lines [48], [43]; strong resonance UV lines of abundant neutrals such as the MgI 285.2 nm line [4] or the OI resonance triplet at 130 nm [32]; and resonance lines of other alkali and alkaline-earth metals such as the NaI D 589 nm doublet or the BaII 455.4 nm line, which are mostly formed in the photosphere but show PRD effects towards the extreme limb [46], [2], [39].", "In the chromosphere, where PRD lines are formed, the 3D spatial transport of radiation becomes essential [21], [26], [22], [47], [18].", "So far no one has modeled PRD lines including the effects of 3D non-LTE radiative transfer due to the large computational effort that is required.", "In this paper, we present a method to perform 3D non-LTE radiative transfer with PRD effects, which was implemented in the Multi3D code [20], and investigate whether such modeling is significant.", "We are motivated to make up for a lack of accurate physical models needed to interpret observations in the MgII h&k lines from the IRIS satellite [9], measurements in the HI Ly $\\alpha $ line obtained by the CLASP rocket experiment [19], and observations in the CaII H&K lines using the new imaging spectrometer CHROMIS at the Swedish 1-m Solar Telescope.", "The structure of the paper is as follows.", "Section briefly explains the method to solve the 3D non-LTE radiative transfer problem in PRD lines.", "Section describes the computational setup: model atmospheres, model atoms, and the code.", "Section lays out the most important computational specifics.", "Section presents the results for our calculations, which we verify, analyze for stability, convergence speed, and applicability of acceleration.", "Using intensities computed in the MgII h&k lines, we illustrate the importance of 3D non-LTE radiative transfer with PRD effects.", "In Section , we give some conclusions.", "Technical details on the algorithm are given in Appendix ." ], [ "Method", "The iterative solution algorithm of Multi3D employs preconditioning of the rate equations as formulated by [40], [41].", "This method was extended by [45] to include the effects of PRD and implemented in the RH codehttp://www4.nso.edu/staff/uitenbr/rh.html, the de facto standard for non-LTE radiative transfer calculations in plane-parallel geometry.", "The RH code has been extended to allow for parallel computation of a large number of plane-parallel atmospheres (1.5D approximation) by [36].", "Below we briefly review the algorithm and the extension by [23], who devised a method to compute a fast approximate solution of the full angle-dependent PRD problem in moving atmospheres." ], [ "The PRD algorithm in the Rybicki-Hummer framework", "The statistical equilibrium non-LTE radiative transfer problem for an atom with $ N_\\mathrm {L} $ levels consists of solving the rate equations for the atomic level populations: $ \\dfrac{n_i}{t} =\\sum _{ j,\\,j \\ne i }^{ N_\\mathrm {L} }n_{\\!j}P_{\\!ji}- n_i\\sum _{ j,\\,j \\ne i }^{ N_\\mathrm {L} }P_{i\\!j}= 0,$ with $ n $ the atomic level populations, and $ P_{i\\!j} = C_{i\\!j} + R_{i\\!j} $ the total rates consisting of collisional rates $ C_{i\\!j} $ and radiative rates $ R_{i\\!j} $ .", "The radiative rates depend on the specific intensity $ I( \\vec{n}, \\nu ) $ , which in turn depends on the direction $ \\vec{n} $ and the frequency $\\nu $ , and is computed from the transfer equation: $\\frac{ I( \\vec{n}, \\nu ) }{ s }=j( \\vec{n}, \\nu ) - \\alpha ( \\vec{n}, \\nu )\\, I( \\vec{n}, \\nu ).$ The transfer equation couples different locations in the atmosphere and makes the problem non-linear and non-local.", "The assumption of complete redistribution (CRD) is that the frequency and direction of the absorbed and emitted photon in a scattering event are uncorrelated, so that the normalized emission profile $ \\psi _{\\!ji}(\\vec{n}, \\nu ) $ and absorption profile $ \\varphi _{\\!ji}(\\vec{n}, \\nu ) $ in a transition $i {\\rightarrow } j$ are equal: $ \\psi _{\\!ji}( \\vec{n}, \\nu ) =\\varphi _{\\!ji}( \\vec{n}, \\nu ),$ for all frequencies $ \\nu $ and directions $ \\vec{n} $ .", "If PRD is important then this equality is no longer valid, and following [44] we define: $\\rho _{i\\!j}( \\vec{n}, \\nu )\\equiv \\frac{ \\psi _{\\!ji}( \\vec{n}, \\nu ) }{ \\varphi _{i\\!j}( \\vec{n}, \\nu ) }.$ The profile ratio $ \\rho _{i\\!j}(\\vec{n}, \\nu ) $ describes for the transition $ i {\\rightarrow } j $ how strongly the line emissivity correlates with the line opacity for the direction $ \\vec{n} $ and the frequency $ \\nu $ .", "It depends on the local particle densities, temperature and the local radiation field $ I( \\vec{n}, \\nu ) $ .", "It is discussed in more detail in Section REF .", "The contributions into the opacity and emissivity from the PRD transition $ i {\\rightarrow } j $ are ij (n, ) = h 4 Bij ij( n, ) [ ni - nj gi gj ij( n, ) ] and jij (n, ) = h 4 Aji ij( n, ) [ nj ij( n, ) ], with $ n $ and $ g $ the corresponding level populations and statistical weights, and $ A_{\\!ji} $ , $ B_{\\!ji} $ , and $ B_{i\\!j} $ the Einstein coefficients.", "This means that the line source function $ S_{\\!i\\!j} (\\vec{n}, \\nu ) =\\frac{ n_{\\!j} A_{\\!ji} \\rho _{i\\!j}( \\vec{n}, \\nu ) }{ n_i B_{i\\!j} - n_{\\!j} B_{\\!ji} \\rho _{i\\!j}( \\vec{n}, \\nu ) } =\\frac{ 2h\\nu ^3 }{ c^2 }\\biggl [\\frac{ 1 }{ \\rho _{i\\!j}( \\vec{n}, \\nu ) }\\frac{ n_i }{ n_{\\!j} }\\frac{ g_{\\!j} }{ g_i }- 1\\biggr ]^{-1}$ depends on frequency and direction, in contrast to CRD where it is constant.", "The radiative rates in the PRD transition $ i {\\rightarrow } j $ are Rij = Bij Jij and Rji = Aji + Bji Jji, where the mean angle-averaged intensities integrated with the absorption and emission profiles are denoted by Jij -11mu I( n, ) ij( n, ) 4 and Jji -11mu I( n, ) ij( n, ) 4.", "The Rybicki-Hummer method of solving the non-LTE problem in PRD consists of the following conceptual steps: The level populations are initialized with a guess solution.", "Popular choices are LTE populations, or populations based on assuming a zero radiation field, or a previously computed solution assuming CRD.", "The profile ratios $ \\rho _{i\\!j}( \\vec{n}, \\nu ) $ are initalized to unity for each PRD transition $ i {\\rightarrow } j $ .", "The formal solution of the radiative transfer equation is performed for all directions and frequencies to obtain intensities.", "Using the resulting intensities a preconditioned set of rate equations is formulated.", "The solution of the preconditioned equations gives an improved estimate of the true solution for the level populations.", "A number of extra PRD sub-iterations are performed, where the level populations are kept fixed and only intensity is redistributed: The formal solution is computed again only for the frequencies in the PRD transitions.", "Using the new intensities, the redistribution integral is computed in each PRD transition to update the profile ratio $ \\rho _{i\\!j}( \\vec{n}, \\nu ) $ .", "Using the new profile ratio, line opacities and emissivities are updated.", "Steps (a)–(c) are repeated until the changes to the radiation field are smaller than a desired value.", "From $ \\rho _{i\\!j}( \\vec{n}, \\nu ) $ updated opacities and emissivities are computed.", "Steps 3-7 are repeated until convergence.", "For more details we refer the reader to [45] and [40], [41]." ], [ "The profile ratio $ \\rho _{i\\!j}( \\vec{n}, \\nu ) $", "The core of the PRD scheme is the calculation of the redistribution integrals to compute the profile ratio $ \\rho _{i\\!j}( \\vec{n}, \\nu ) $ .", "Following [44], we consider the PRD transition $ i {\\rightarrow } j $ with all subordinate transitions $ k {\\rightarrow } j\\colon k < j$ , which share the same broadened upper level $ j $ .", "The redistribution integrals, computed in each subordinate transition $ k {\\rightarrow } j $ including the transition $ i {\\rightarrow } j $ itself then contribute into the profile ratio: $ \\rho _{i\\!j}( \\vec{n}, \\nu )=1 + \\sum _{k < j} \\dfrac{ n_k B_{k\\!j} }{n_{\\!j} P_{\\!j}}\\\\{\\times }\\oint {-11mu}\\int \\!\\!I\\bigl ( \\vec{n}^\\prime \\!, \\nu ^\\prime \\bigr )\\Biggl [\\frac{ R_{k\\!ji}\\bigl ( \\vec{n}^\\prime \\!, \\nu ^\\prime \\!", ";\\vec{n}, \\nu \\bigr ) }{ \\varphi _{i\\!j}( \\vec{n}, \\nu ) }-\\varphi _{k\\!j}\\bigl ( \\vec{n}^\\prime \\!, \\nu ^\\prime \\bigr )\\Biggr ]\\nu ^\\prime \\frac{\\Omega ^\\prime \\!", "}{4\\pi },$ with $ R_{k\\!ji} $ the inertial-frame redistribution function, $ P_{\\!j} = \\sum _{k \\ne j } P_{\\!jk} $ the total depopulation rate out of the upper level $ j $ , where each double redistribution integral is taken over photons with frequencies $ \\nu ^\\prime \\!", "$ coming along directions $ \\vec{n}^\\prime $ within solid angles $ \\Omega ^\\prime $ that are absorbed in the subordinate lines $ k {\\rightarrow } j $ including the resonance transition $ i {\\rightarrow } j $ .", "As the inertial frame redistribution function $ R_{k\\!ji} $ we use a function for transitions with a sharp lower level and a broadened upper level, which is generalised for two possible cascades of a normal redistribution in the $ i {\\rightarrow } j {\\rightarrow } i $ transition itself and resonance Raman scattering (also called cross-redistribution) in $ k {\\rightarrow } j {\\rightarrow } i\\colon k \\ne i $ subordinate transitions sharing the same upper level $ j $ .", "The function $ R_{k\\!ji} $ consists of two weighted components [14]: $ R_{k\\!ji}\\bigl (\\vec{n}^\\prime \\!, \\nu ^\\prime \\!", ";\\vec{n}, \\nu \\bigr )=\\gamma R_{k\\!ji}^\\mathrm {II}\\bigl (\\vec{n}^\\prime \\!, \\nu ^\\prime \\!", ";\\vec{n}, \\nu \\bigr )+(1 - \\gamma )\\varphi _{k\\!j}\\bigl ( \\vec{n}^\\prime \\!, \\nu ^\\prime \\bigr )\\varphi _{i\\!j}\\bigl ( \\vec{n}, \\nu \\bigr ).$ Here $ R_{k\\!ji}^\\mathrm {II} $ is the inertial frame correlated redistribution function for a transition with a sharp lower level and a broadened upper level, representing coherent scattering.", "The product $ \\varphi _{k\\!j} \\varphi _{i\\!j} $ approximates the inertial frame non-correlated redistribution function $ R_{k\\!ji}^\\mathrm {III} $ , representing complete redistribution.", "The quantity $ \\gamma \\equiv P_{\\!j} / \\bigl ( P_{\\!j} + Q_{\\!j}^\\mathrm {E} \\bigr ) $ is the coherence fraction, i.e., the ratio of total rate $ P_{\\!j} $ out of level $ j $ to its sum with the rate of elastic collisions $ Q_{\\!j}^\\mathrm {E} $ .", "Entering (REF ) into (REF ) we finally obtain $ \\rho _{i\\!j}(\\vec{n}, \\nu )=\\\\1 + \\gamma \\!\\sum _{k < j}\\frac{ n_k B_{k\\!j} }{ n_{\\!j} P_{\\!j} }\\left[\\oint {-11mu}\\int \\!\\!I\\bigl ( \\vec{n}^\\prime \\!, \\nu ^\\prime \\bigr )\\frac{R_{k\\!ji}^\\mathrm {II}\\bigl (\\vec{n}^\\prime \\!, \\nu ^\\prime \\!", ";\\vec{n}, \\nu \\bigr )}{\\phi _{i\\!j}( \\vec{n}, \\nu )}\\nu ^\\prime \\frac{\\Omega ^\\prime \\!", "}{4\\pi }-\\bar{J}_{\\!k\\!j}^\\varphi \\right]$" ], [ "The hybrid approximation", "Computing the profile ratio using Eq.", "(REF ) is computationally expensive, as it involves evaluating the angle-dependent redistribution function $ R_{k\\!ji}^\\mathrm {II} / \\varphi _{i\\!j} $ and computing the double integral along each absorption frequency $ \\nu ^\\prime $ and direction $ \\vec{n}^\\prime $ for each emission frequency $ \\nu $ and direction $ \\vec{n}$ .", "[23] demonstrated in plane-parallel computations using RH, that evaluating Eq.", "(REF ) takes about a factor 100 more time than computing the formal solution for all angles and frequencies.", "A common additional assumption to speed up the computations is to assume that the radiation field $ I( \\vec{n}, \\nu ) $ is isotropic in the inertial frameHere and below we distinguish between two reference frames.", "1) The inertial frame, also called the laboratory frame or the observer's frame.", "2) The comoving frame, where gas is locally at rest, in other texts also called the gas frame, the fluid frame, or the rest frame..", "In that case the angle integral in Eq.", "(REF ) can be performed analytically to obtain $ \\rho _{i\\!j}( \\nu ) =1 + \\gamma \\!\\sum _{k < j}\\frac{ n_k B_{k\\!j} }{ n_{\\!j} P_{\\!j} }\\Biggl [\\int \\!\\!J\\bigl ( \\nu ^\\prime \\bigr )\\,_{k\\!ji}^\\mathrm {II}\\bigl ( \\nu ^\\prime \\!, \\nu \\bigr )\\,\\nu ^\\prime \\!-\\bar{J}_{k\\!j}^\\varphi \\Biggr ].$ with an angle-averaged redistribution function $ _{k\\!ji}^\\mathrm {II} $ [11], [44].", "The assumption of isotropy is not valid if the velocities in the model atmosphere are larger than the Doppler width of the absorption profile, a common situation in radiation-MHD simulations of the solar atmosphere [28], [29].", "Therefore [23] proposed what they called the hybrid approximation, which is a fast and sufficiently accurate representation of Eq.", "(REF ) based on Eq.", "(REF ).", "The idea behind the hybrid approximation is to take the redistribution integral in the comoving frame assuming the radiation field to be isotropic there.", "The angle-averaged intensity in the comoving frame $ J^{\\star \\!", "}\\bigl ( \\nu ^{\\star \\!}", "\\bigr ) $ is computed from the intensity in the inertial frame $ I(\\vec{n}, \\nu ) $ , taking into account Doppler shifts due to the local velocity $ \\vec{} $ : $ J^{\\star \\!", "}\\bigl ( \\nu ^{\\star \\!}", "\\bigr )=\\oint I\\biggl (\\vec{n},\\nu ^{\\star \\!", "}\\biggl [1 + \\frac{ \\vec{n}\\cdot \\vec{} }{ c }\\biggr ]\\biggr )\\frac{\\Omega }{4\\pi }.$ The assumption of an isotropic specific intensity in the comoving frame $ I^{\\star \\!", "}\\bigl ( \\vec{n}, \\nu ^{\\star \\!}", "\\bigr ) =J^{\\star \\!", "}\\bigl ( \\nu ^{\\star \\!}", "\\bigr ) $ implies that we can use Eq.", "(REF ) with the angle-averaged redistribution function in the comoving frame to compute the comoving-frame profile ratio $ \\rho _{i\\!j}^\\star $ that depends only on frequency: $ \\rho _{i\\!j}^{\\star \\!", "}\\bigl ( \\nu ^{\\star \\!}", "\\bigr ) =1 + \\gamma \\!\\sum _{k < j}\\frac{ n_k B_{k\\!j} }{ n_{\\!j} P_{\\!j} }\\Biggl [\\int \\!\\!J^{\\star \\!", "}\\bigl ( \\nu ^\\prime \\bigr )\\,_{k\\!ji}^\\mathrm {II}\\bigl ( \\nu ^\\prime \\!, \\nu ^{\\star \\!}", "\\bigr )\\nu ^\\prime \\!-\\bar{J}_{k\\!j}^\\varphi \\Biggr ],$ where the integration is taken over the absorption frequency in the comoving frame.", "Finally, the comoving profile ratio $ \\rho _{i\\!j}^{\\star \\!}", "$ is then transformed into the inertial profile ratio $ \\rho _{i\\!j} $ using an inverse Doppler transform so that $ \\rho _{i\\!j} $ again becomes dependent on both frequency and angle: $ \\rho _{i\\!j}( \\vec{n}, \\nu )=\\rho _{i\\!j}^{\\star \\!", "}\\biggl (\\nu \\biggl [1 - \\frac{ \\vec{n}\\cdot \\vec{} }{ c }\\biggr ]\\biggr ).$ Note that Eqs.", "(8)–(9) in [23] contain two mistakes.", "First, both signs before $ \\vec{n}\\cdot \\vec{u}/c $ have to be inverted.", "Second, the subscript `r' of $ \\nu _\\mathrm {r} $ in Eq.", "(8) has to be dropped.", "We give the correct transforms here in Eqs.", "(REF ) and (REF ).", "There are several different possibilities for numerically computing the interpolations in Eqs.", "(REF ) and (REF ).", "For Eq.", "(REF ), storing $ I( \\vec{n}, \\nu ) $ and then performing the interpolation and integration in one go is the most straightforward and rather fast.", "However, storing $ I( \\vec{n}, \\nu ) $ takes typically 1–4 GiB of memory per subdomain, which is too much on current-generation supercomputers, which normally have 2 GiB of memory per core (see the discussion in Sect. ).", "Another method is not to store the intensity, but instead to interpolate and incrementally add it to $ J^{\\star \\!}", "$ during the formal solution.", "This method does not require as much storage, and we implemented various variants in Multi3D: General interpolation – interpolation indices and weights are computed on the fly.", "This method is slow but requires no additional storage.", "Precomputed interpolation – interpolation indices and weights are computed once, stored and then re-used.", "This method is fast, but requires a large memory storage.", "Interpolation on an equidistant frequency grid – this method is moderately fast, and requires only modest extra storage because the interpolation indices and weights depends only on the direction $ \\vec{n} $ and not on frequency $ \\nu $ .", "We describe each of these algorithms in detail in Appendix ." ], [ "Model atmospheres", "We tested our PRD algorithm using two different 1D model atmospheres with different velocity distributions.", "The standard FAL-C model atmosphere [10] shown in Fig.", "REF (top) was modified to contain constant vertical velocities of ${-}10,\\,0,\\,{+}10$ km s$^{-1}$ , two monotonically increasing from ${-}10$ to ${+}10$ km s$^{-1}$ and decreasing from ${+}10$ to ${-}10$ km s$^{-1}$ velocity gradients, two discontinuous velocity jumps from ${+}10$ to ${-}10$ km s$^{-1}$ and back representing toy shock waves, one smooth velocity composition of two waves with 10 and 6 km s$^{-1}$ amplitudes, and one random normal velocity distribution with a zero mean and a standard deviation of 10 km s$^{-1}$ .", "Small insets in Fig.", "REF illustrate some of these configurations.", "For timing, convergence stability, convergence acceleration, and other tests we employed a 1D column extracted from a radiation-MHD simulation performed with the Bifrost code (details given below).", "This column corresponds to coordinates $ X = 0 $ and $ Y = 0 $ of the original snapshot and is resampled into a vertical grid with 188 points in the Z-direction.", "Physical properties of this model atmosphere are illustrated in Fig.", "REF (bottom).", "In this paper we simply refer to it as the 1D Bifrost model atmosphere.", "To investigate how the algorithm performs in realistic situations, we used a snapshot from the 3D radiation-MHD simulation by [7] corresponding to run time $ t = 3850 $ s performed with the Bifrost code [12].", "This snapshot contains a bipolar region of enhanced magnetic flux with unsigned field strength of 50 G in the photosphere.", "The simulation box spans from the bottom of the photosphere up to the corona having physical sizes of $ 24 \\times 24 \\times 16.8 $ Mm.", "To save computational time, the horizontal resolution of the snapshot was halved to 96 km so that a new coarser grid with $ 252 \\times 252 \\times 496 $ points is used.", "This snapshot has, amongst other studies, been used previously to study the formation of the hydrogen H$\\alpha $ line [22], the MgII h&k lines [25], and the CII lines [38]." ], [ "Model atoms", "As a first test model atom we used a minimally sufficient model of MgII with four levels and the continuum of MgIII, where first two resonance transitions represent the k 279.64 nm and h 280.35 nm doublet.", "[24] provides more details on this $ 4{+}1 $ level model atom.", "As a second test model atom we used a model of HI with five levels and continuum of HII, where the first two resonance transitions represent the Ly $\\alpha $ 121.6 nm and the Ly $\\beta $ 102.6 nm lines.", "For the timing experiments summarized in Table REF , we initialized all corresponding model atom transitions on a frequency grid of $ N_\\nu ^\\mathrm {total} $ points with $ N_\\nu ^\\mathrm {PRD} $ of them covered by PRD lines.", "PRD frequencies cover the two overlapping h&k lines for the MgII model atom and by the Ly $\\alpha $ line for the HI model atom." ], [ "Numerical code", "We perform numerical simulations using an extensively updated version of the Multi3D code [20].", "The code solves the non-LTE radiative transfer problem for a given model atom in a given 3D model atmosphere using the method of short characteristics to integrate the formal solution of the transfer equation and the method of multilevel accelerated $\\Lambda $ -iterations (M-ALI) with pre-conditioned rates in the system of statistical equilibrium equations following [40], [41].", "The current version of the code allows MPI parallelization both for frequency and 3D Cartesian domains.", "A typical geometrical subdomain size is $ 32 \\times 32 \\times 32 $ grid points.", "The code also employs non-overshooting 3rd order Hermite interpolation for the source function and the optical depth scale [1], [17].", "We use the 24-angle quadrature (A4 set) from [5].", "To treat resonance lines in PRD, we implemented non-coherent scattering into the code using the hybrid approximation with the three interpolation schemes mentioned in Sects.", "REF –REF .", "We validate our Multi3D implementation with RH computations.", "In the RH code, the hybrid approximation for the PRD redistribution was implemented by [23] for plane-parallel model atmospheres, and shown to give results consistent with full angle-dependent PRD." ], [ "Computational expenses of interpolation methods", "We compared the time efficiency of each of the three interpolation methods mentioned in Sect.", "REF using system clock routines in the computation of Eq.", "(REF ) (which we call forward transform) and Eq.", "(REF ) (which we call backward transform).", "Given a model atmosphere of $ N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} $ spatial grid points, using $ N_\\mu $ angles, and a frequency grid with $ N_\\nu $ knots, we computed the mean time spent per spatial point per ray direction per frequency for the forward transform ($t_\\blacktriangleright $ ) and the backward transform ($t_\\blacktriangleleft $ ).", "We also estimated how much memory each algorithm requires to store interpolation indices and weights.", "The numerical details of each interpolation method are given in Appendix .", "An advantage of the general interpolation algorithm is that it is applicable to any type of frequency grid with arbitrary spacing between knots and requires no memory storage for interpolation indices and weights.", "Drawbacks are that it is very time-consuming and it does not scale linearly with the number of grid points.", "To perform the forward transform, for each knot in the inertial frame two full scans are done along knots in the comoving frame, resulting in $ 2N_\\nu ^2$ operations for the whole line profile.", "The mean time $ t_\\blacktriangleright $ spent per knot therefore scales as $ O(N_\\nu )$ .", "To perform the backward transform, for each knot in the inertial frame a bisection search is done on the whole knot range in the comoving frame.", "The search can be correlatedUsually this is called hunting in textbooks on numerical methods.", "if the interpolation is performed along the frequency direction but this is not the case in Multi3D.", "The total number of operations is therefore around $2 N_\\nu \\log _2 N_\\nu $ and $t_\\blacktriangleleft $ scales as $O(\\log _2 N_\\nu )$ ." ], [ "Precomputed interpolation", "The precomputed interpolation algorithm is applicable to any frequency grid, just like the general interpolation method.", "It is both very fast and scales well because interpolation indices and weights are computed only once based on the atmosphere, the chosen angles, and the frequency grid.", "The drawback is that the algorithm is memory consuming.", "For each knot in the inertial frame two operations are performed for the forward transform, and thus $t_\\blacktriangleright $ scales as $O(1)$ .", "In the backward transform, only one operation is done per inertial-frame knot, resulting in $t_\\blacktriangleleft = O(1)$ .", "If we have a single non-overlapping PRD line (such as Ly $\\alpha $ ), the forward transform requires storage of $3 N_\\nu N_\\mu N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} = 3 N_\\Sigma $ numbers for interpolation indices and weights.", "For the backward transform, we need only $2 N_\\Sigma $ numbers (see Appendix for details).", "For single-precision (4-byte) integers and floating point numbers this results in $20 N_\\Sigma $ bytes.", "For a typical run using the 3D model atmosphere we have $ N_\\nu \\approx 200 $ , $ N_\\mu = 24 $ , and a spatial subdomain size of $32^3$ grid points, so that storing the interpolation indices and weighs takes 2.9 GiB of memory per CPU.", "The data can be stored more efficiently by using less precise floating point numbers, so that the storage requirement is lowered to $14 N_\\Sigma $ bytes, corresponding to 2 GiB for our typical use case." ], [ "Precomputed interpolation on equidistant grid", "The general interpolation algorithm is slow, and precomputed interpolation is fast but requires a large amount of memory.", "A solution to the memory consumption of the latter is to use an equidistant frequency grid, so that the interpolation weights and indices only depend on the ray direction and spatial grid point, but not on the frequencyThis idea was suggested by J. de la Cruz Rodríguez..", "The requirement to properly sample to absorption profile sets the grid spacing to ${\\sim }1$ km s$^{-1}$ .", "Because PRD lines typically have wide wings, such sampling leads to thousands of knots.", "This is undesirable because of the corresponding increase in computing time.", "It turns out that one can keep the advantages of an equidistant grid but not have the disadvantage of the low speed by performing the formal solution only for selected knots (which we call real).", "The other knots (which we call virtual) are ignored.", "We keep all knots real close to nominal line center in order to resolve the Doppler core, and have mainly virtual knots in the line wing where a coarse frequency resolution is acceptable.", "We can then still use frequency-independent interpolations coefficients, at the price of a small computational and memory overhead to track and store real and virtual knot locations and certain relations between them.", "Usually, $\\sim $ 90 % of the real knots are in the line core.", "The number of operations required to perform the forward transform depends on whether the real frequency knot is in the line core or in the line wings.", "The number of operations to perform the forward transform on all $N_\\nu $ knots is around $2 C_\\blacktriangleright N_\\nu $ , where $1 < C_\\blacktriangleright 10$ is a factor that depends on $N_\\nu $ and on the fine grid resolution.", "The $t_\\blacktriangleright $ scales as $O(1)$ but the number of operations is higher than for precomputed interpolation.", "In the backward transform, the algorithm is essentially the same.", "The total number of operations for all frequency knots is $2 C_\\blacktriangleleft N_\\nu $ , where $1 < C_\\blacktriangleleft < C_\\blacktriangleright 10$ so that $t_\\blacktriangleleft $ scales as $O(1)$ .", "The interpolation indices and weights are the same for both the forward and backward transforms.", "To store them we need $ 2 N_\\mu N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} $ four-byte numbers resulting in 6 MiB plus some auxiliary variables amounting to another $0.3$ MiB." ], [ "Timing experiments", "We measured the performance of the three algorithms in the 1D Bifrost model atmosphere using the 5-level MgII model atom with the two overlapping h&k lines treated in PRD.", "The number of frequency points in each line profile was varied from from 15 to 1 900.", "We used a ten-point Gauss-Legendre quadrature for the angle integrals.", "The calculations were run on a computer with Intel® Xeon® CPU E5-2697 v3 2.60 GHz cores.", "Figure REF shows our measurements for both transforms using the three interpolation algorithms.", "In the general algorithm, the total computing time scales as $O(N_\\nu ^2)$ , which makes the algorithm unsuitable for wide chromospheric lines with many frequency points.", "Contrary to that, the total time for the precomputed and the precomputed equidistant algorithms scales linearly as $O(N_\\nu )$ .", "The equidistant algorithm is slower than the precomputed one, but of the same order.", "The extra book-keeping required for the equidistant interpolation causes a small performance penalty compared to the precomputed case.", "This is however a small price for the large saving in memory consumption compared to the precomputed algorithm.", "We preferably use the precomputed algorithm in 1D and small 3D model atmospheres, while for big 3D model atmospheres we use the equidistant one." ], [ "Truncated frequency grid", "Only the core and inner wings of strong chromospheric lines are formed in the chromosphere, which is the region where observers of such lines are mainly interested in.", "It is therefore interesting to see whether ignoring the extended wings has an influence on the intensity in the central part of the line.", "If this is not the case we can lower the number of frequency points in the line profile and so speed up the computations.", "Therefore we tested whether a truncation of the profile wings makes a difference.", "For the MgII h&k lines, we performed a computation using equidistant interpolation where we resolve the core on a fine frequency grid up to $\\pm 70$ km s$^{-1}$ around the line center, and use the coarse grid out to $\\pm 160$ km s$^{-1}$ , so that we only cover the core of the line profiles from just outside the h$_1$ and k$_1$ minima.", "In Fig.", "REF we show the differences between the truncated and the fully-winged profiles of the MgII h&k lines in CRD and PRD.", "In CRD, there are small differences around the k$_2$ emission peaks while in PRD differences are negligible.", "We thus conclude that we can use a truncated grid.", "The main reason why truncated grids work for chromospheric lines is that the density in the chromosphere is so low that the damping wings are weak, so that the extinction profile is well-approximated by a Gaussian.", "The averages over the extinction profile of the form $\\oint {-11mu}\\int \\cdots \\varphi ( \\vec{n}, \\nu )\\,\\nu \\frac{\\Omega }{4\\pi }$ that enter into the non-LTE problem are thus dominated by a range of few Doppler widths around the nominal line center in the local comoving frame.", "As long as the frequency grid covers this range, the radiative transfer in the chromosphere is accurate.", "In CRD there is a larger inaccuracy because photons absorbed in the line wings tend to be re-emitted in the line core due to Doppler diffusion [16].", "In PRD, this effect is weak, because the complete redistribution component of the redistribution function from Eq.", "(REF ) is strongly reduced because the coherency fraction $ \\gamma $ is nearly equal to one, so that the wings and the core are effectively decoupled and do not significantly influence each other.", "This is a rare instance where the inclusion of PRD effects actually makes computations easer instead of harder." ], [ "Comparison to RH", "To validate our method and implementation, we simulated the MgII h&k lines in CRD and PRD using both the Multi3D and the RH code, and compared the resulting intensities.", "The PRD calculations with RH were done using both the angle-dependent and the hybrid PRD algorithms, while Multi3D computations were done only using the hybrid algorithm.", "Intensity profiles of both MgII h&k lines were calculated in the FAL-C model atmosphere with modified vertical velocity distributions given in the insets in Fig.", "REF .", "The simplest cases are those with constant velocities, the most extreme is that with the random normal distribution.", "Note that the case of a random velocity distribution only illustrates that the method is stable in extreme situations.", "Both codes produce CRD intensity profiles in a good agreement.", "Comparison of the RH computations using the hybrid and angle-dependent algorithms shows that the hybrid algorithm is accurate for all cases except the strong toy-shock wave.", "In such strong gradients the assumption of isotropy of the radiation in the comoving frame is not accurate.", "Both codes yield the same hybrid PRD intensities except for some small differences around the inner wings close to the line core (see PRD intensity values at 279.58–279.61 nm and 279.65–279.68 nm in Fig.", "REF ).", "This effect is caused by slight differences in frequency grids used by the codes and differences in line broadening recipes that lead to differences in coherency fraction $\\gamma $ .", "We performed the same computations for the 1D Bifrost column.", "Corresponding profiles are given in Fig.", "REF .", "Also here both codes and algorithms yield excellent agreement." ], [ "Convergence properties", "Numerical PRD operations are computationally expensive.", "In order to minimize the computing time we investigated how many PRD subiterations are needed to lead to a converged solution, how the number of subiterations influences the convergence speed of the ALI iterations, and whether convergence acceleration using the Ng method [34] can be applied." ], [ "Number of PRD subiterations", "We tested the convergence of accelerated $\\Lambda $ -iterations (ALI) for the MgII and the HI model atoms with PRD lines in the FAL-C and the 1D Bifrost model atmospheres for different numbers of PRD subiterations.", "Similar but simpler tests were done when solving the non-LTE PRD problem in the 3D Bifrost model.", "We considered the solution to be converged if $\\max \|\\delta n/n\| \\le 10^{-4}$ .", "The atomic level populations are initialized either using the LTE or the zero radiation approximationPopulations are obtained by solving the system of statistical equilibrium equations with the radiation field set to zero throughout the atmosphere [6].", "B. Lites introduced this idea in 1983.. We show examples of such tests in Figs.", "REF –REF and Table REF .", "Figure REF shows the behavior of the maximum absolute relative change in angle-averaged intensity, $\\max \|\\delta J/J\|$ , for the MgII model atom in the FAL-C model atmosphere, as a function of the number of ALI iterations.", "With one PRD subiteration the solution diverges.", "With two or more subiterations the solution converges.", "For iterations starting with LTE populations there is a maximum in $\\max \|\\delta J/J\|$ after about 25 iterations, while $\\max \|\\delta J/J\|$ decreases monotonically when the populations are initialized using the zero radiation approximation.", "When the correction to the atomic level populations in non-LTE PRD become monotonically decreasing, the major correction of the redistributed intensity is performed during the first two PRD subiterations.", "If the first PRD subiteration corrects the redistributed intensity by some amount, then, on average, the second PRD subiteration produces a slightly smaller correction but with an opposite sign.", "Each next PRD subiteration decreases this correction by almost an order of magnitude while keeping the same sign.", "Therefore, the major adjustment to make the intensities consistent with the level populations occurs during the first two or three PRD subiterations.", "In Table REF we summarize our experiments that investigated the dependence of the number of iterations required to reach a converged solution on the number of PRD sub-iterations and the initial populations.", "A converged solution is reached faster for the zero-radiation inital condition than for LTE.", "The solution converges in practically the same number of ALI iterations independently of the number of PRD subiterations.", "As long as the number of subiterations is sufficient to lead to a converged solution, then the number of PRD subiterations has no influence on the convergence rate.", "We also investigated the convergence properties as a function of the number of PRD subiterations in the 3D Bifrost model atmosphere, for both the MgII and HI model atoms.", "This atmosphere has large velocity, density, and temperature gradients in the chromosphere, and represents a much tougher test for the algorithm than the 1D atmospheres.", "We found that for MgII we obtain a converged solution using two PRD subiterations, but for HI we needed three subiterations.", "In Fig.", "REF we illustrate the difference between two and three subiterations for HI.", "It shows the corrections in atomic level populations and the radiation field, starting with initial conditions obtained from the previous incomplete run and using three PRD subiterations.", "The top row shows the behavior after we switch to two subiterations.", "Two subiterations are not enough to make the radiation field consistent with the level populations, and from iteration 5 the corrections to the radiation field become larger and larger.", "At iteration 16 the corrections to the level populations begin to increase as well, and the solution starts to diverge.", "In the bottom row, we show the computation with three PRD subiterations, where the corrections to both the level populations and the radiation field decrease steadily.", "In summary, we found the following empirical rules: One PRD subiteration is never enough to converge for any model atom in any model atmosphere.", "Two PRD subiterations are enough for convergence in the MgII lines.", "Three PRD subiterations must be done for convergence in the HI lines.", "The number of accelerated $\\Lambda $ -iterations needed to achieve a converged solution does not depend on the number of PRD subiterations." ], [ "Convergence speed", "We measured how many iterations and how much time are needed to obtain a converged CRD or PRD non-LTE solution for the MgII and the HI model atoms in the 1D and the 3D model atmospheres.", "We estimated how many iterations $N_\\mathrm {dex}$ are needed to decrease $\\max \|\\delta n/n\|$ by an order of magnitude, how much time $t_\\mathrm {iter}$ is spent per one full iteration including PRD subiterations, and how much time $t_\\mathrm {dex}$ is spent per $N_\\mathrm {dex}$ iterations.", "Table REF summarizes our results for the 3D Bifrost snapshot.", "From our measurements we found that In the 3D model atmosphere, $N_\\mathrm {dex}$ is a few times larger in PRD than in CRD.", "The computational time per full iteration increases in PRD mostly due to the extra PRD subiterations.", "It is about 2 times bigger for magnesium (2 PRD subiterations) and 2.5–5 times bigger for hydrogen (3 PRD subiterations).", "For a typical $32^3$ subdomain, computational time per decade change in $\\max \|\\delta n/n\|$ is of the order of half a day to two days if PRD effects are computed using the hybrid approximation with equidistant grid interpolation.", "For a typical PRD run in a 3D model atmosphere, iteration until $\\max \|\\delta n/n\| \\le 10^{-4}$ is often required to get intensities accurate better than 1%.", "For our 3D model atmosphere with $252 \\times 252 \\times 496$ grid points, this means that one reaches a converged solution using 1024 CPUs in ${\\sim }2$ days for MgII and ${\\sim }8$ days for HI, corresponding to roughly 50 000 and 200 000 CPU hours.", "These numbers are sufficiently small to make it possible to model PRD lines on modern supercomputers either in a single model or a time-series of snapshots from large 3D radiation-MHD simulations.", "This is not possible if full angle-dependent PRD is used.", "Table 1 in [23], shows that the full angle-dependent algorithm is almost two orders of magnitude slower and much more memory consuming than the hybrid algorithm, requiring millions of CPU hours for a single snapshot." ], [ "Convergence acceleration", "We tested whether convergence acceleration using the algorithm by [34] can be applied to 3D PRD calculations.", "We used 2nd order Ng acceleration with 5 iterations between acceleration steps.", "We found that Ng acceleration does not work if it is applied too soon after starting a new computation.", "It only works if it is started when both $ \\max \|\\delta n/n\| $ and $ \\max \|\\delta J/J\| $ are steadily decreasing, such as in the bottom panels of Fig.", "REF .", "The steeper the decrease rate, the better the acceleration will work.", "If the acceleration is started too soon, then the convergence rate may be slower than without applying acceleration, or the computation will diverge.", "We observed that acceleration yields better results if the atomic level populations are initialized with the zero radiation approximation than if they are initialized with LTE populations.", "Initializing the populations with a previously converged CRD solution also yields good acceleration results, especially in 3D model atmospheres.", "Finally, we noticed that acceleration generally works reliably in 1D model atmospheres and for our MgII atom in the 3D model atmosphere.", "For the HI atom in the 3D atmosphere, we find that using Ng acceleration does not significantly improve the convergence rate.", "To illustrate the joint action of 3D and PRD effects on emerging line profiles we calculated intensity profiles of the MgII h&k lines in the 3D Bifrost snapshot in CRD and PRD using both 1D and 3D formal solvers.", "We compared results from the 3D PRD case (which is the most realistic) to results from the two approximations of 3D CRD and 1D PRD.", "In 3D CRD, line intensities are computed neglecting effects of partial redistribution but including effects of the horizontal radiative transfer.", "In 1D PRD, only effects of partial redistribution are present, while the radiative transfer is performed treating each vertical column as a plane-parallel atmosphere.", "For all the pixels in each computation we determined the wavelength position and intensities of the k$_\\mathrm {2v}$ , k$_3$ , and k$_\\mathrm {2r}$ spectral features using the algorithm described in [35].", "The intensities are mostly used to diagnose temperatures in the chromosphere, while the wavelength positions and separations between the features are used to measure either velocities or velocity gradients [25], [35].", "Here we verify two statements made by [24].", "These authors stated that the central depression features (h$_3$ and k$_3$ ) of the MgII h&k lines are only weakly affected by PRD and are mostly influenced by effects of the horizontal radiative transfer.", "They also stated that the emission peaks (h$_\\mathrm {2v}$ , k$_\\mathrm {2v}$ , h$_\\mathrm {2r}$ , and k$_\\mathrm {2r}$ ) are mostly controlled by PRD effects while effects of the 3D radiative transfer play a minor role.", "They thus argued that modeling of the emission peaks could be done assuming 1D PRD while modeling of the central depressions could be done assuming 3D CRD.", "They gave arguments based on 1D PRD calculations and 3D CRD to support these claims because they could not perform 3D PRD calculations." ], [ "Imaging", "Figure REF shows images of the brightness temperature in k$_\\mathrm {2r}$ , k$_3$ , and k$_\\mathrm {3r}$ for the 1D PRD, 3D PRD, and 3D CRD computations.", "These features are not always present and sometimes the automated fitting routine misidentifies them.", "These locations are indicated with fuchsia-colored pixels.", "We see that the k$_3$ images computed in 3D CRD and 3D PRD (middle row of Fig.", "REF ) indeed look similar.", "As was already shown by [24], the 1D PRD image appears very different than the images computed in 3D.", "For the emission peak images (top and bottom rows) we see substantial differences between the 1D PRD and 3D PRD computations.", "The 1D PRD images have substantially higher contrast, and bright structures have much sharper edges than in the 3D PRD computations.", "Interestingly there is also a significant structural similarity between 3D PRD and 3D CRD.", "Figure REF illustrates the same points by showing joint probability density functions (JPDFs) between brightness temperatures computed in 3D PRD and those computed in 1D PRD for the emission peaks and 3D CRD for the central depression.", "The k$_3$ brightness temperature in 3D PRD is typically slightly higher than predicted by the 3D CRD computation.", "For k$_\\mathrm {2v}$ and k$_\\mathrm {2r}$ we see that the contrast for the 3D PRD computation is lower than for the 1D PRD case.", "The 1D PRD computations underestimate the brightness where $T_\\mathrm {b} < 5$ kK, and overestimate it where $T_\\mathrm {b} > 5$ kK." ], [ "Average line profiles and center-to-limb variation", "Figures REF and REF display the center-to-limb variation of the spatially-averaged intensity profiles of the MgII k line.", "First, we discuss how PRD and 3D effects change the wavelength positions of the k$_\\mathrm {2v}$ , k$_3$ , and k$_\\mathrm {2r}$ features.", "The 1D PRD approximation correctly predicts the wavelength positions of all the features, while the 3D CRD approximation does this only for the k$_3$ feature.", "Going towards the limb, we see similar blueshifts of the k$_3$ feature in 3D CRD, 1D PRD, and 3D PRD.", "We see the same effect in the k$_\\mathrm {2v}$ peak in both PRD cases but a much stronger shift in 3D CRD.", "Going towards the limb, we see a slight redshift of the k$_\\mathrm {2r}$ peak in both PRD cases and a strong redshift in 3D CRD.", "The peak separation $ \\lambda (\\mathrm {k_{2r}}) - \\lambda (\\mathrm {k_{2v}}) $ slightly increases towards the limb in 3D PRD and it behaves similar in 1D PRD.", "However, in 3D CRD it strongly increases.", "In short, one can use 1D PRD as a valid approximation for the wavelength positions of the k$_\\mathrm {2v}$ and the k$_\\mathrm {2r}$ emission peaks as well as for the k$_3$ absorption feature.", "The 3D CRD approximation is accurate only for the wavelength of the k$_3$ feature.", "Second, we show how PRD and 3D effects change the intensities of the line profiles including the features.", "As expected, the 3D CRD approximation is not valid for the intensities in the inner wings and in the emission peaks.", "It also underestimates the intensities of the line core at disk center, but this difference gets smaller towards the limb.", "The 1D PRD approximation is very good for the inner wings, but it overestimates the intensities of all the features and this effect increases towards the limb.", "We see asymmetries in the line profiles in all three cases.", "In all three cases we see that the intensities of the k$_\\mathrm {2v}$ peak is larger than the intensities of k$_\\mathrm {2r}$ .", "This is caused by asymmetries in vertical velocity field in the simulation, for example by upward-propagating shocks [8].", "There is an interesting asymmetry in the flanks of the emission peaks in the 3D PRD case that is not present in the 1D PRD case (compare the red and green curves Fig.", "REF between 279.60 nm and 279.63 nm).", "A similar asymmetry is present in 3D CRD, except that it happens further away from the line center because of the wider emission peaks.", "This asymmetry in the 3D computations must be caused by interaction between the velocity field in the simulation and the horizontal transport of radiation, and should be investigated in more detail.", "In short, the intensities of the inner wings in the average line profile are well reproduced by the 1D PRD approximation.", "The line core is however not modeled accurately by 1D PRD.", "The 3D CRD approximation reproduces only the line core intensity towards the limb.", "In summary, we have shown for the MgII k line that its profile, as well as the profile features, are influenced by both PRD and 3D effects.", "The 1D PRD approximation reproduces the wavelength position of the features, and can be used to accurately compute intensities in the inner wings.", "The intensity and the wavelength position of k$_3$ can be computed using 3D CRD at disk center, but not towards the limb.", "Accurate quantitative modeling of the whole line profile and its center-to-limb variation requires 3D PRD." ], [ "Discussion and Conclusions", "In this paper we presented an algorithm for 3D non-LTE radiative transfer including PRD effects in atmospheres with velocity fields.", "It improves on the algorithm described by [23] by discretizing PRD line profiles on a frequency grid with intervals being integer multiples of a chosen fine resolution.", "Such a grid allows using fast linear interpolation with pre-computed indices and weights stored with a small memory footprint.", "This permits a relatively fast solution of the 3D non-LTE problem, costing 50 000–200 000 CPU hours to reach a converged solution in a model atmosphere with $252 \\times 252 \\times 496$ grid points.", "We investigated the numerical properties of the algorithm in realistic use case scenario's and found that it is stable, applicable to strongly scattering lines like Ly $\\alpha $ , and sufficiently fast to be practically useful for the computation of time-series of snapshots from radiation-MHD simulations.", "We presented an application to the formation of the MgII h&k lines in the solar chromosphere.", "We found that the conclusion of [24] that the k$_\\mathrm {2v}$ and k$_\\mathrm {2r}$ emission peaks for disk-center intensities can be accurately modeled using 1D PRD computations to be incorrect.", "The emission peaks are affected by both PRD and 3D effects, and should be modeled using both 3D and PRD effects simultaneously.", "We also showed that the center-to-limb variation in 1D PRD, 3D CRD, and 3D PRD is markedly different.", "The method presented in this paper will allow comparison of numerical models and observations of PRD-sensitive lines not only for disk-center observations but also for observations towards the limb.", "We are planning to perform a detailed comparison of the center-to-limb variation of the MgII h&k lines as predicted by numerical simulations to IRIS observations, such as those by [42].", "Computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre at Linköping University, at the High Performance Computing Center North at Umeå University, and at PDC Centre for High Performance Computing (PDC-HPC) at the Royal Institute of Technology in Stockholm.", "This study has been supported by a grant from the Knut and Alice Wallenberg Foundation (CHROMOBS)." ], [ "Definitions and notations", "The notations adopted in this Appendix are as follows.", "Parentheses ( and ) denote functional dependence, e.g., $ \\rho ( \\vec{n}, q) $ , while brackets [ and ] denote array indexing, e.g., $ q[i] $ .", "Both notations should not be mistaken with intervals of real numbers, e.g., $ x \\in {[ x_\\mathrm {A}, x_\\mathrm {B} )} $ means $ x_\\mathrm {A} \\le x < x_\\mathrm {B} $ .", "A different notation is used for ordered ranges of integers, e.g., $ l = m, \\cdots , n $ .", "Index tuples, which subscript arrays, are always named explicitly not to confuse them with open intervals, e.g., index tuple $ (x, y, z, \\mu , i) $ .", "Sections of 1D-arrays are denoted by their initial and terminal subscripts separated by a colon, e.g., $ q[i_{:}i_] $ .", "Names of auxiliary arrays are given in typewriter font, e.g., $ \\mathtt {map}[i] $ , while other quantities, stored in arrays, are given in roman font, e.g., $ \\vec{}[x, y, z] $ .", "The symbol $ \\mapsto $ means interpolation, the symbol $ \\leftarrow $ is an assignment, and the symbol $ \\overset{+}{\\leftarrow } $ indicates an increment.", "As mentioned in Sect.", "REF , we deal with the inertial frame and the comoving frame.", "Frequency-dependent quantities from both frames are Doppler-shifted in frequency with respect to each other due to the velocity field $ \\vec{}[x, y, z] $ given in the model atmosphere.", "The forward transform converts the specific intensity $ I $ from the inertial frame to the comoving frame by taking into account Doppler shifts of frequencies, while the backward transform does the same in the opposite direction for the profile ratio $ \\rho $ .", "All quantities in the comoving frame are marked with a superscript $ {\\!", "}^{\\star \\!}", "$ : specific intensity $ I^{\\star \\!}", "$ , angle-averaged intensity $ J^{\\star \\!}", "$ , profile ratio $ \\rho ^{\\star \\!}", "$ , and frequency $ q^{\\star \\!", "}$ .", "The same quantities in the inertial frame have no superscript.", "The only exception is for line frequencies, which always appear in indexed array form as either $ q[i] $ or $ q[j] $ , where different subscripts are used to tell to which of the frames $ q $ belongs.", "For brevity, we say `inertial/comoving $ X $ ' instead of `quantity $ X $ in the inertial/comoving frame'.", "Computations are performed in a Cartesian subdomain of $ N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} $ grid points with indices $ x = x_\\mathrm {min}, \\cdots , x_\\mathrm {max} $ , $ y = y_\\mathrm {min}, \\cdots , y_\\mathrm {max} $ , and $ z = z_\\mathrm {min}, \\cdots , z_\\mathrm {max} $ .", "Angle-dependent quantities are represented on an angle quadrature with $ N_\\mu $ rays having directions $ \\vec{n}[\\mu ]\\colon \\mu = 1, \\cdots , N_\\mu $ .", "Frequency-dependent quantities are discretized on a grid of $ N_\\nu $ points having frequencies $ q[i]\\colon i = i_, \\cdots , i_$ (see Sect.", "REF for the definition of $ q $ ).", "Subscript $ i $ denotes frequency points in the inertial grid, while subscript $ j $ denotes them in the comoving grid, such that their ranges and frequency grid values are identical: $ i_= j_$ , $ i_= j_$ , and $ q[i] = q[j] $ if $ i = j $ .", "For the equidistant interpolation (see Sect.", "REF ), we use two extra grids with two different subscripts: $ m $ for the fine inertial grid, and $ n $ for the fine comoving grid.", "We use three short-hands `knot', `lie', and `match' with the following meanings.", "Following textbooks on interpolation methods, we refer to frequency grid points as knots, e.g., saying `comoving knot $i$ ' we mean `frequency $ q[i] $ in the comoving frame'.", "Saying `inertial knot $ i $ lies on the comoving interval $ {[ j, j+1 )} $ ', we mean that the frequency $ q[i] $ in the inertial frame is between two frequencies $ q^{\\star \\!", "}[j] $ and $ q^{\\star \\!", "}[j + 1] $ in the comoving frame, i.e., $ q^{\\star \\!", "}[j] \\le q[i] < q^{\\star \\!", "}[j + 1] $ .", "Saying `inertial knot $ i $ matches the comoving knot $ j $ ', we mean that $ q[i] = q^{\\star \\!", "}[j] $ .", "In the listings of algorithms, we implicitly use an advantage of the iteration loop control in Fortran do-statements to avoid if-statements: if the increment is positive/negative and the starting index is larger/smaller than the ending index, then no iterations are performed.", "The same applies to array sections: if the increment is positive/negative and the lower bound is larger/smaller than the upper bound, then no array elements are indexed.", "We also use the Fortran feature that array subscripts can start at any integer number.", "Usually, velocities in the upper atmosphere of the Sun do not exceed 150 km s$^{-1}$ so that $ \\beta \\equiv / c < 5\\cdot 10^{-4} $ and we can adopt a non-relativistic approximation for the Doppler transform in both directions neglecting $ O(\\beta ^2) $ terms: = ( 1 - n c ), = ( 1 + n c ).", "We also neglect aberration of light so that $ \\vec{n}^{\\star \\!}", "= \\vec{n} $ .", "For convenience, we express frequency $ \\nu $ as dimensionless frequency displacement $ q $ given in Doppler units $ \\Delta \\nu _\\mathrm {D} = \\nu _0_\\mathrm {B} / c $ : $ q \\equiv \\dfrac{ \\nu - \\nu _0 }{ \\Delta \\nu _\\mathrm {D} }=\\Bigl ( \\dfrac{ \\nu }{ \\nu _0 } - 1 \\Bigr )\\dfrac{ c }{ _\\mathrm {B} },$ with $ \\nu _0 $ the line center frequency, and $ _\\mathrm {B} $ the characteristic broadening velocity, which we typically set to a value of a few km s$^{-1}\\!$ .", "Using frequencies $ q $ , we can linearize the Doppler transform in Eqs.", "(REF )–(REF ): q = q - q, q = q + q, with the Doppler shift along direction $ \\vec{n}[\\mu ]$ given by a scalar product $q_\\mu =\\dfrac{ \\vec{n}[\\mu ]\\cdot \\vec{}[x, y, z] }{ _\\mathrm {B} }.$ The linear approximation in Eqs.", "(REF )–(REF ) is valid if $ \|q\|\\,_\\mathrm {B} \\ll c $ , which is true even for the most extreme situation of the HI Ly $\\alpha $ line." ], [ "Piecewise linear interpolation", "Since we are restricted by available computing time, we employ the fastest and the easiest method: piecewise linear interpolation and constant extrapolationIf more accurate method is needed, the algorithms given in Sect.", "REF for equidistant linear interpolation can be easily adapted to shifted linear interpolation [3].. We especially benefit from it when we use equidistant frequency grids.", "Given data samples $ y_\\mathrm {L} $ and $ y_\\mathrm {R} $ on two knots L and R with frequencies $ q_\\mathrm {L} $ and $ q_\\mathrm {R} $ , the interpolant value $ y $ at frequency $ q \\in {[ q_\\mathrm {L}, q_\\mathrm {R} )} $ is a linear combination of both samples $y = \\,y_\\mathrm {L} + (1 - )\\,y_\\mathrm {R},$ where knot L contributes with the left-hand-side (l.h.s.)", "weight $= \\frac{ q_\\mathrm {R} - q\\phantom{_\\mathrm {R}} }{ q_\\mathrm {R} - q_\\mathrm {L} }$ and knot R contributes with the right-hand-side (r.h.s.)", "weight $1 - = \\frac{ q\\phantom{_\\mathrm {R}} - q_\\mathrm {L} }{ q_\\mathrm {R} - q_\\mathrm {L} }.$ An advantage of linear interpolation over other schemes is that only one weight $ $ has to be known, the other one always complements $ $ to one.", "Constant extrapolation is used if $ q $ is beyond the interpolation range $ {[ q_\\mathrm {A}, q_\\mathrm {Z} )} $ set by the outermost knots A and Z.", "If $ q < q_\\mathrm {A} $ , then $ y = y_\\mathrm {A} $ , and if $ q_\\mathrm {Z} \\le q $ , then $ y = y_\\mathrm {Z} $ .", "When necessary, we call a standard function $ Y \\leftarrow \\mathrm {interpolate}\\big ( X, \\vec{x}, \\vec{y} \\big ),$ which evaluates an interpolant $ Y $ at a given coordinate $ X $ for data values $ \\vec{y} $ sampled on knots with coordinates $ \\vec{x} $ .", "It uses a correlated bisection search [37]: $ l \\leftarrow \\mathrm {locate}\\big ( X, \\vec{x}, l^\\prime \\big ),$ which returns the index $l$ of the array $ \\vec{x} $ such that $ \\vec{x}[l] \\le X < \\vec{x}[l + 1] $ .", "The input parameter $ l^\\prime $ is used to initialize the search." ], [ "Interval coverage", "Given a monotonically ordered set of knots $ \\lbrace i\\colon i = i_, \\cdots , i_\\rbrace $ representing a line frequency grid $ q[i] $ from the leftmost knot $ i_$ to the rightmost knot $ i_$ , we complement it by two infinite points $ {-}\\infty $ and $ {+}\\infty $ for convenience.", "We perform interpolation and extrapolation on right-open intervals to avoid using values at knots twice on adjacent intervals.", "That is to say, linear interpolation is carried out on $ {[ i, i + 1 )}\\colon i = i_, \\cdots , i_- 1 $ intervals, and constant extrapolation is performed on the $ ( {-}\\infty , i_) $ and $ {[ i_, {+}\\infty )} $ intervals as shown in Fig.", "REF ." ], [ "Forward transform", "After the formal solution of the transfer equation is done at coordinate $ (x, y, z) $ for direction $ \\vec{n}[\\mu ] $ at frequency $ q[i] $ , we have the inertial specific intensity $ I\\bigl ( \\vec{n}[\\mu ], q[i] \\bigr ) = I[x, y, z, \\mu , i] $ .", "Owing to the local nature of the intensity redistribution and the Doppler transform, we drop dependencies on the spatial coordinates in the formulas given below.", "The forward transform consists of two parts (see orange notations in Fig.", "REF ).", "First, the inertial specific intensity $ I\\bigl ( \\vec{n}[\\mu ], q[i] \\bigr ) $ is interpolated to get the comoving specific intensity $ I^{\\star \\!", "}\\bigl ( \\vec{n}[\\mu ], q^{\\star \\!}", "\\bigr ) $ at comoving frequency $ q^{\\star \\!}", "= q[i] - q_\\mu $ : $ I\\bigl ( \\vec{n}[\\mu ], q[i] \\bigr )\\mapsto I^{\\star \\!", "}\\bigl ( \\vec{n}[\\mu ], q^{\\star \\!}", "\\bigr ),$ This part is also called forward interpolation.", "Second, the interpolated intensity $ I^{\\star \\!", "}\\bigl ( \\vec{n}[\\mu ], q^{\\star \\!}", "\\bigr ) $ contributes into (increments) the comoving angle-averaged intensity $ J^{\\star \\!", "}\\bigl ( q^{\\star \\!}", "\\bigr ) $ : $ J^{\\star \\!", "}\\bigl ( q^{\\star \\!}", "\\bigr )\\overset{+}{\\leftarrow }\\dfrac{ \\omega _\\mu }{ 4\\pi }I^{\\star \\!", "}\\bigl ( \\vec{n}[\\mu ], q^{\\star \\!}", "\\bigr ),$ with $ \\omega _\\mu / 4\\pi $ the quadrature weight for direction $ \\vec{n}[\\mu ] $ .", "If we combine Eq.", "(REF ) with Eq.", "(REF ) for all $ \\mu $ , we obtain the angle-discretized form of Eq.", "(REF ): $ J^{\\star \\!", "}\\bigl ( q^{\\star \\!}", "\\bigr ) =\\sum _{ \\mu = 1 }^{ N_\\mu }\\dfrac{ \\omega _\\mu }{ 4\\pi }I\\bigl ( \\vec{n}[\\mu ], q^{\\star \\!}", "+ q_\\mu \\bigr ).$ This equation implies first a transform of the intensity from the inertial to the comoving frame, which numerically corresponds to an interpolation.", "The problem is that the intensity $ I $ is not known for all frequencies simultaneously, due to the way how the formal solution is performed numerically in the Multi3D code.", "Instead, we compute Eqs.", "(REF )–(REF ) incrementally from each $ I\\bigl ( \\vec{n}[\\mu ], q[i] \\bigr ) $ , without ever having the intensity for other inertial frequencies and angles in memory at the same time.", "Forward interpolation $ I\\bigl [ \\mu , i \\bigr ] \\mapsto I^{\\star \\!", "}\\bigl [ \\mu , j \\bigr ] $ contributes the intensity $ I $ at the inertial knot $ i $ to intensities $ I^{\\star \\!}", "$ at related comoving knots $ j $ .", "The related knots $ j $ lie on the left and right neighboring intervals on both sides of $ i $ .", "We adopt the following notation for a series of the comoving knots lying on the inertial interval $ {[ i, i + 1 )} $ : $ j_i^{\\,(\\alpha )} $ denote the related knots, and $ _i^{(\\alpha )} $ denote the corresponding l.h.s.", "interpolation weights, where the subscript $ i $ is the starting index of the inertial interval and the counter $ \\alpha = 1, \\cdots , \\alpha _\\mathrm {max} $ numbers all the knots in the series.", "There are zero or more $ (\\alpha _\\mathrm {max} \\ge 1) $ comoving knots, whose indices $ j_i^{\\,(\\alpha )} $ and weights $ _i^{(\\alpha )} $ depend on the positions and relative distances between their comoving frequencies $ q^{\\star \\!}", "= q[j] + q_\\mu $ and the inertial frequency $ q[i] $ .", "This is illustrated in Fig.", "REF .", "The knot $ i $ always has two neighbors $ i - 1 $ and $ i + 1 $ so that related knots $ j $ are interpolated on the left $ {[ i - 1, i )} $ and right $ {[ i, i + 1 )} $ intervals.", "The inertial intensity $ I $ at the knot $ i $ is interpolated to the comoving intensity $ I^\\star $ at knots $ j_{i - 1}^{\\,(\\alpha )} $ using r.h.s.", "weights $ 1 - _{i - 1}^{(\\alpha )} $ on the left interval, and at knots $ j_{i}^{\\,(\\alpha )} $ using l.h.s.", "weights $ _{i}^{(\\alpha )} $ on the right interval.", "At the ends of the frequency grid, weights on the infinite intervals are always equal to unity to force constant extrapolation: $ 1 - _{i - 1}^{(\\alpha )} = 1 - _{{-}\\infty }^{(\\alpha )} = 1 $ for $ i = i_$ , and $ _{i}^{(\\alpha )} = _{i_}^{(\\alpha )} = 1 $ for $ i = i_$ .", "Plugging expressions for $ j $ and $ $ into the discretized Eqs.", "(REF )–(REF ) we get 2 J[ ji-1 () ] + (1 - i-1() ) 4 I[ , i - 1 ] on [ i-1, i ), J[ ji () ] + 43mui() 4 I[ , i ] on [ i, i+1 )." ], [ "Backward transform", "Once $J^{\\star }$ has been fully accumulated, we compute the comoving profile ratio $ \\rho ^{\\star \\!", "}\\bigl ( q[j] \\bigr ) = \\rho ^{\\star \\!", "}[ x, y, z, j ] $ and keep it in memory for all frequency knots $ j = j_, \\cdots , j_$ simultaneously.", "This makes the backward transform much easier than the forward one, because we can employ a standard interpolation.", "The backward transform can be directly evaluated because it is just an interpolation from the comoving frame to the inertial frame (see green notations in Fig.", "REF ): $ \\rho ^{\\star \\!", "}\\bigl ( q[i] - q_\\mu \\bigr )\\mapsto \\rho \\bigl ( \\vec{n}[\\mu ], q[i] \\bigr ).$ As follows from Fig.", "REF , each $ \\rho \\bigl ( \\vec{n}[\\mu ], q[i] \\bigr ) $ depends on either one or two comoving knots $ j $ .", "It depends on two knots, if the inertial knot $ i $ lies on the comoving interval $ {[j, j + 1)} $ , where both $ j $ and $ j + 1 $ are internal knots ($ j_\\le j < j + 1 \\le j_$ ), i.e., $ q[j] + q_\\mu \\le q[i] < q[j + 1] + q_\\mu $ .", "In this case we do linear interpolation: $ \\rho [\\mu , i]=_i\\, \\rho ^{\\star \\!", "}[j]+(1 - _i)\\, \\rho ^{\\star \\!", "}[j + 1].$ For the backward transform, knots and weights are always denoted by $ j $ and $ _i $ with no superscript $ (\\alpha ) $ to tell them from knots $ j_i^{\\,(\\alpha )} $ and weights $ _i^{(\\alpha )} $ for the forward transform.", "It depends on one knot, if the inertial knot $ i $ matches the comoving knot $ j $ , i.e., $ q[i] = q[j] + q_\\mu $ .", "It also depends on one knot, if the inertial knot $ i $ lies on intervals before $ j_$ or after $ j_$ , which means $ q[i] < q[j_] + q_\\mu $ or $ q[j_] + q_\\mu \\le q[i] $ , because we use constant extrapolation.", "Therefore, in those three cases: 2 [, i] = [j], or [, i] = [j], or [, i] = [j]." ], [ "General interpolation", "One can perform the searches and interpolations described in Sects.", "REF –REF on the fly.", "This does not require any additional storage, and can be used for a frequency grid of arbitrary spacing, but the algorithm is slow.", "We call this algorithm general interpolation." ], [ "General forward transform", "[!t] 0.5em0.5em textsl Specific intensity $ I[x, y, z, \\mu , i] $ at frequency $ q[i]$ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "Incremented angle-averaged intensity $ J^{\\star \\!", "}[x, y, z, j] $ at corresponding frequencies $ q[j] $ in the comoving frame.", "$ q_\\mu \\leftarrow \\bigl ( \\vec{n}[\\mu ] \\cdot \\vec{}[x, y, z] \\bigr ) / _\\mathrm {B} $ $ q^\\star _\\mathrm {L} \\leftarrow q[{ {\\bigl [ \\min (i + 1, i_) \\bigr ]} }][l]{\\bigl [ \\max (i - 1, i_) \\bigr ]} - q_\\mu $ $ q^\\star _\\mathrm {C} \\leftarrow q[{ {\\bigl [ \\min (i + 1, i_) \\bigr ]} }][l]{\\bigl [i\\bigr ]} - q_\\mu $ $ q^\\star _\\mathrm {R} \\leftarrow q\\bigl [ \\min ( i_, i + 1 ) \\bigr ] - q_\\mu $ ([f]Linear interpolation on $ \\bigl ( q^\\star _\\mathrm {L}, q^\\star _\\mathrm {C} \\bigr ) $ :) $ q^\\star _\\mathrm {L} \\ne q^\\star _\\mathrm {C} $ $ \\Delta \\leftarrow q^\\star _\\mathrm {C} - q^\\star _\\mathrm {L} $ $ j \\leftarrow j_$ $ j_$ $ q^\\star _\\mathrm {L} < q[j] < q^\\star _\\mathrm {C} $ $ \\xi ^\\prime \\leftarrow \\big ( q[j] - q^\\star _\\mathrm {L} \\big ) / \\Delta $ [r]The r.h.s.", "weight $ 1 - _{i - 1} $ .", "$ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow }\\xi ^\\prime \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ ($ q^\\star _\\mathrm {L} = q^\\star _\\mathrm {C} $ [f]Constant extrapolation on $ \\bigl ( {-}\\infty , q^\\star _\\mathrm {C} \\bigr ) $ :) $ j \\leftarrow j_$ $ j_$ $ q[j] < q^\\star _\\mathrm {C} $ $ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow } \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, i] $ ([f]Linear interpolation on $ {\\bigl [ q^\\star _\\mathrm {C}, q^\\star _\\mathrm {R} \\bigr )} $ :) $ q^\\star _\\mathrm {C} \\ne q^\\star _\\mathrm {R} $ $ \\Delta \\leftarrow q^\\star _\\mathrm {R} - q^\\star _\\mathrm {C} $ $ j \\leftarrow j_$ $ j_$ $ q^\\star _\\mathrm {C} \\le q[j] < q^\\star _\\mathrm {R} $ $ \\xi \\leftarrow \\big ( q^\\star _\\mathrm {R} - q[j] \\big ) / \\Delta $ [r]The l.h.s.", "weight $ _i $ .", "$ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow } \\xi \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, i] $ ($ q^\\star _\\mathrm {C} = q^\\star _\\mathrm {R} $ [f]Constant extrapolation on $ {\\bigl [ q^\\star _\\mathrm {C}, {+}\\infty \\bigr )} $ :) $ j \\leftarrow j_$ $ j_$ $ q^\\star _\\mathrm {C} \\le q[j] $ $ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow }\\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, i] $ General forward transform.", "Comments in this algorithm and those listed below are given in slanted text after `//'.", "We search for all comoving knots $ j $ that lie on the intervals $ {[i - 1, i)} $ and $ {[i, i + 1)} $ , i.e., whose comoving frequencies $ q^{\\star \\!}", "= q[j] + q_\\mu $ are either between $ q[i - 1] $ and $ q[i] $ , or between $ q[i] $ and $ q[i + 1] $ .", "We take the current inertial knot $ i $ and its left neighbor $ \\max ( i - 1, i_) $ and right neighbor $ \\min ( i_, i + 1 ) $ .", "We use $ \\max () $ and $ \\min () $ to keep the neighbors within the frequency grid range $ i = i_, \\cdots , i_$ .", "For these knots we compute their frequencies in the comoving frame, which we denote by 2 qL = q[ (i - 1, i) ] - q, qC = q[ i ] - q, qR = q[ (i, i + 1) ] - q.", "Therefore, the comoving knots $ j $ related to the inertial knot $ i $ are those, whose frequencies $ q[j] $ belong to the $ {\\bigl [ q^\\star _\\mathrm {L}, q^\\star _\\mathrm {C} \\bigr )} $ or $ {\\bigl [ q^\\star _\\mathrm {C}, q^\\star _\\mathrm {R} \\bigr )} $ intervals.", "The knots $ j $ lie either on the left or on the right side of $ q^\\star _\\mathrm {C} $ , and they might fall outside the frequency coverage of the line.", "We thus have four different cases to test for and apply Eqs.", "(REF )–(REF ).", "Algorithm REF shows the forward transform in detail." ], [ "General backward transform", "[!t] 0.5em0.5em textsl Profile ratio $ \\rho ^{\\star \\!", "}[x, y, z, j] $ for all frequencies $ q[j]\\colon j = j_, \\cdots , j_$ in the comoving frame.", "Profile ratio $ \\rho [x, y, z, \\mu , i] $ for given frequency $ q[i] $ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "$ q_\\mu \\leftarrow \\big ( \\vec{n}[\\mu ]\\cdot \\vec{}[x, y, z] \\big ) / _\\mathrm {B} $ $ \\rho [x, y, z, \\mu , i] \\leftarrow \\mathrm {interpolate}\\big (\\, X = q[i] - q_\\mu , $ $ \\phantom{ \\rho [x, y, z, \\mu , i] \\leftarrow \\mathrm {interpolate}\\big (\\, } $$ \\vec{x} = q[j_{:}j_], $ $ \\phantom{ \\rho [x, y, z, \\mu , i] \\leftarrow \\mathrm {interpolate}\\big (\\, } $$ \\vec{y} = \\rho ^{\\star \\!", "}[x, y, z, j_{:}j_] \\,\\big ) $ General backward transform.", "Algorithm REF gives the backward transform, which is just an interpolation.", "It is performed by using Eq.", "(REF ) with $ Y = \\rho [x, y, z, \\mu , i] $ , $ X = q[i] - q_\\mu $ , $ \\vec{x} = q[j_{:}j_] $ , and $ \\vec{y} = \\rho ^{\\star \\!", "}[x, y, z, j_{:}j_] $ ." ], [ "Precomputed interpolation", "The general algorithm given in Sect.", "REF is slow because for each $ x $ , $ y $ , $ z $ , $ \\mu $ , and $ i $ we perform two linear searches in the forward transform and a correlated bisection search in the backward transform along the full frequency range $ j = j_, \\cdots , j_$ .", "To speed up the computations, one can precompute and store for each index tuple $ (x, y, z, \\mu , i) $ the comoving knots $ j_i^{\\,(\\alpha )} $ and the l.h.s weights $ _i^{(\\alpha )} $ used in the forward transform and the nearest comoving knot $ j_i $ and its l.h.s.", "weight $ _i $ used in the backward transform." ], [ "Precomputation for the forward interpolation", "[] 0.5em0.5em textsl Line frequencies $ q[i_{:}i_] $ , directions $ \\vec{n}[1{:}N_\\mu ] $ , velocities $ \\vec{}[x_\\mathrm {min}{:}x_\\mathrm {max},y_\\mathrm {min}{:}y_\\mathrm {max},z_\\mathrm {min}{:}z_\\mathrm {max}] $ .", "Arrays address, forward_j, and forward_w.", "$ \\mathtt {address}[] \\leftarrow 0 $ [r]Initialize array address $ c \\leftarrow 0 $ [r]and its counter.", "$ x \\leftarrow x_\\mathrm {min} $ $ x_\\mathrm {max} $ $ y \\leftarrow y_\\mathrm {min} $ $ y_\\mathrm {max} $ $ z \\leftarrow z_\\mathrm {min} $ $ z_\\mathrm {max} $ $ \\mu \\leftarrow 1 $ $ N_\\mu $ $ q_\\mu \\leftarrow \\big ( \\vec{n}[\\mu ]\\cdot \\vec{}[x, y, z] \\big ) /_\\mathrm {B} $ $ i \\leftarrow i_- 1 $ [r]Start searching from the leftmost [r]knot $ ({-}\\infty ) $ in the inertial frame.", "([f]For each comoving knot $j$ ,) $ j \\leftarrow j_$ $ j_$ [r]find which inertial knot $ i $ [r]contributes into it.", "$ q \\leftarrow q[j] + q_\\mu $ [r]Inertial frequency.", "$i \\leftarrow \\mathrm {locate}\\big ( q, q[i_{:}i_], i \\big )$ [r]$ q[i] \\le q < q[i + 1] $ ([f]Linear interpolation) $i_\\le i < i_$ $ \\leftarrow \\dfrac{ q[i + 1] - q{16mu} }{ q[i + 1] - q[i] } $ [r]on $ {[ i, i + 1 )} $ .", "([f]Constant extrapolation) $ i = i_- 1 $ $ \\leftarrow 0 $ [r]on $ {( {-}\\infty , i_)} $ .", "($ i = i_$ [f]Constant extrapolation) $ \\leftarrow 1 $ [r]on $ {[ i_, {+}\\infty )} $ .", "$ o \\leftarrow o(x, y, z, \\mu , i) $ [r]Memory-offset by Eq.", "(REF ).", "[r]Fill address between $ c $ and $ o $ [r]with the last stored address: $ \\mathtt {address}[c + 1{:}o - 1] \\leftarrow \\mathtt {address}[c] $ [r]Update the current address: $ \\mathtt {address}[o] \\leftarrow \\mathtt {address}[c] + 1 $ $ c \\leftarrow o $ [r]Set counter to the current address.", "$ \\mathtt {forward\\_j}\\left[\\mathtt {address}[c]\\right] \\leftarrow j $ [r]Save comoving $ \\mathtt {forward\\_w}\\left[\\mathtt {address}[c]\\right] \\leftarrow $ [r]index & weight.", "Initialization of the precomputed forward interpolation.", "In the forward transform, each inertial knot $ i $ contributes to zero, one or more comoving knots $ j $ .", "Since the indices and the weights depend on $ x $ , $ y $ , $ z $ , $ \\mu $ , $ i $ , and $ \\alpha $ , they require jagged 6D-arrays with the last variable-length dimension for index $ \\alpha $ .", "Neither Fortran nor C allows such a data structure in their latest language standards.", "A typical solution is to use 5D-arrays of pointers or linked lists.", "Both solutions are inconvenient and memory-inefficient because they require at least one 8-byte pointer (on the default 64-bit architecture) for each $ \\alpha $ -series of $ j_i^{\\,(\\alpha )} $ and $ _i^{(\\alpha )} $ .", "It is more efficient to flatten these jagged arrays into two continous 1D-arrays, one for $ j_i^{\\,(\\alpha )} $ (forward_j) and one for $ _i^{\\,(\\alpha )} $ (forward_w), which use the same indexing subscripts.", "Each of them consecutively stores $ N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} N_\\mu N_\\nu $ values.", "In that case, for each index tuple $ (x, y, z, \\mu , i) $ we need to keep track of the related knots $ j_i^{\\,(\\alpha )} $ and weights $ _i^{(\\alpha )} $ .", "We use the 1D-array address to do so.", "For each $ i = i_- 1, \\dots , i_$ , this array keeps the address of the last comoving knot $ j_i^{\\,(\\alpha _\\mathrm {max})} $ that lies on the inertial interval $ {[i, i + 1)} $ .", "The extra knot $ i_- 1 $ corresponds to the comoving frequency $ {-}\\infty $ and is needed to treat extrapolation beyond the left end of the frequency grid.", "For convenience, one more element is added to the address array at the beginning so that the array subscript $ o $ starts from $-1$ .", "Hence, address has $ N_\\mathrm {X} N_\\mathrm {Y} N_\\mathrm {Z} N_\\mu (N_\\nu + 1) + 1 $ elements.", "The values stored in address start from 0 and are non-decreasing.", "For each $ x $ , $ y $ , $ z $ , and $ \\mu $ , we scan comoving knots $ j = j_, \\cdots , j_$ to find the inertial nearest-left neighbor knot $ i $ such that $ q[i] \\le q[j] + q_\\mu < q[i + 1] $ .", "Then we compute the l.h.s.", "weight $ $ , using interpolation for internal knots, and constant extrapolation for external knots.", "We set the subscript $ o $ to the memory-offset given by the index tuple $ (x, y, z, \\mu , i) $ : o( x, y, z, , i ) = i - (i- 1) + (N+ 1) ( - 1 + N ( z - zmin + NZ ( y - ymin + NY ( x - xmin ) ) ) ) ).", "It is important that the order of indices in the tuple follows the order of the nested for-loops in Algs.", "REF –REF so that address is filled continuously.", "Frequency is the fastest (continuous) dimension in memory.", "At this point, counter $ c $ subscripts the last modified element of address such that address[$ c $ ] points to the last comoving knot saved in forward_j.", "Then we point address[$ o $ ] to the current comoving knot $ j $ : since $ j $ is stored contiguously in index_j, we use an incremented address of the last comoving knot: $\\mathtt {address}[o] = \\mathtt {address}[c] + 1.$ If $ o = c $ , then both the last and the current comoving knots lie on the same interval $ {[i, i+1)} $ and address[$o = c$ ] is just incremented.", "If $ o > c + 1 $ , then there are inertial knots between the last and the current comoving knots, which do not contribute to any comoving knots.", "In this case, the elements of address between $ c $ and $ o $ are filled with address[$ c $ ]: $\\mathtt {address}[c + 1{:}o - 1] = \\mathtt {address}[c],$ so that they point to the last comoving knot.", "Then we adjust the counter $ c $ to the current memory-offset $ o $ $c = o$ and store the current comoving knot and its weight: forward_j[address[c]] = j, forward_w[address[c]] = .", "Now we proceed to the next comoving knot.", "We summarize this complicated procedure in Alg.", "REF and illustrate it in Fig.", "REF ." ], [ "Precomputation for the backward interpolation", "[!t] 0.5em0.5em textsl Line frequencies $ q[i_{:}i_] $ , directions $ \\vec{n}[1{:}N_\\mu ] $ , velocities $ \\vec{}[x_\\mathrm {min}{:}x_\\mathrm {max},y_\\mathrm {min}{:}y_\\mathrm {max},z_\\mathrm {min}{:}z_\\mathrm {max}] $ .", "Arrays backward_j and backward_w.", "$ x \\leftarrow x_\\mathrm {min} $ $ x_\\mathrm {max} $ $ y \\leftarrow y_\\mathrm {min} $ $ y_\\mathrm {max} $ $ z \\leftarrow z_\\mathrm {min} $ $ z_\\mathrm {max} $ $ \\mu \\leftarrow 1 $ $ N_\\mu $ $ q_\\mu \\leftarrow \\big ( \\vec{n}[\\mu ]\\cdot \\vec{}[x, y, z] \\big ) /_\\mathrm {B} $ $ j \\leftarrow j_- 1 $ [r]Start searching from the leftmost [r]knot $ ( {-}\\infty ) $ in the comoving frame.", "([f]For each inertial knot $i$ , find) $ i \\leftarrow i_$ $ i_$ [r]which comoving knot $j$ [r]contributes into it.", "$ q^{\\star \\!}", "\\leftarrow q[i] - q_\\mu $ [r]Comoving frequency.", "$ j \\leftarrow \\mathrm {locate}\\big ( q^{\\star \\!", "}, q[j_{:}j_], j \\big ) $ [r]$ q[j]\\,{\\le }\\,q^\\star {<}\\,q[j{+}1] $ ()$ j = j_- 1 $ $ [{}][l]{j^\\prime } \\leftarrow j_$ [r]Constant extrapolation $ \\leftarrow 1 $ [r]on $ {( {-}\\infty , j_)} $ .", "$ j = j_$ $ [{}][l]{j^\\prime } \\leftarrow j_$ [r]Constant extrapolation $ \\leftarrow 1 $ [r]on $ {[ j_, {+}\\infty )} $ .", "$ q^\\star = q[j] $ $ [{}][l]{j^\\prime } \\leftarrow j $ [r]Exact match of $j$ to $i$ .", "$ \\leftarrow 1$ ($ j_\\le j < j_$ ) $ [{}][l]{j^\\prime } \\leftarrow j $ [r]Linear interpolation $ \\leftarrow \\dfrac{ q[j + 1] - q^\\star \\;\\: }{ q[j + 1] - q[j] } $ [r]on $ {[ j, j + 1 )} $ .", "$ \\mathtt {backward\\_j}[x, y, z, \\mu , i] \\leftarrow j^\\prime $ [l]Store comoving $ \\mathtt {backward\\_w}[x, y, z, \\mu , i] \\leftarrow $ [l]index and weight.", "Initialization of the precomputed backward interpolation.", "Precomputation for the backward transform is similar to the one for the forward interpolation and is explained in Alg.", "REF , but there are two differences.", "First, each inertial knot $ i $ has only one related comoving knot $ j $ , as illustrated in Fig.", "REF .", "This strict correspondence allows us to use 5D-arrays to store the indices and weights, no address array is needed.", "Second, we scan the inertial knots to find the related comoving knots, instead of the other way around." ], [ "Precomputed forward transform", "[!t] 0.5em0.5em textsl Specific intensity $ I[x, y, z, \\mu , i] $ at frequency $ q[i] $ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "Incremented angle-averaged intensity $ J^{\\star \\!", "}[x, y, z, j] $ at corresponding frequencies $ q[j] $ in the comoving frame.", "$ o \\leftarrow o(x, y, z, \\mu , i) $ [r]Memory-offset by Eq.", "(REF ) $ [{a_{i - 2}}][l]{a_i} \\leftarrow \\mathtt {address}[{14mu}o{15mu}] $ [r]Addresses of comoving knots $ [{a_{i - 2}}][l]{a_{i - 1}} \\leftarrow \\mathtt {address}[o - 1] $ [r]for current $ i $ , previous $ i - 1 $ , and $ [{a_{i - 2}}][l]{a_{i - 2}} \\leftarrow \\mathtt {address}[o - 2] $ [r]pre-previous $ i - 2 $ inertial knots.", "([f]Interpolate on $ {[ i - 1, i )} $ :) $ a \\leftarrow a_{i - 2} + 1 $ $ a_{i - 1} $ $ [{}][l]{j} \\leftarrow \\mathtt {forward\\_j}[a] $ $ \\leftarrow \\mathtt {forward\\_w}[a] $ $ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow }(1 - ) \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ [r]Use r.h.s.", "weights.", "([f]Interpolate on $ {[i, i + 1)} $ :) $ a \\leftarrow a_{i - 1} + 1 $ $ a_i $ $ [{}][l]{j} \\leftarrow \\mathtt {forward\\_j}[a] $ $ \\leftarrow \\mathtt {forward\\_w}[a] $ $ J^{\\star \\!", "}[x, y, z, j] \\overset{+}{\\leftarrow }\\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i]$ [r]Use l.h.s.", "weights.", "Precomputed forward transform.", "With precomputed indices and weights, the forward transform can be done very quickly following Alg.", "REF .", "At grid point $ (x, y, z) $ , angle direction $ \\vec{n}[\\mu ] $ , and inertial frequency $ q[i] $ , we compute the memory-offset $ o $ by using Eq.", "(REF ).", "Now $ a_i = \\mathtt {address}[o] $ is the address of the last comoving knot $ j_i^{\\,(\\max \\alpha )} $ that lies on the inertial interval $ {[ i, i + 1 )} $ .", "The previous knot $ i - 1 $ has memory-offset $ o - 1 $ so that $ a_{i - 1} = \\mathtt {address}[o - 1] $ is the address of the last comoving knot $ j_{i - 1}^{\\,(\\max \\alpha )} $ that lies on the inertial interval $ {[i - 1, i)} $ .", "If $ a_{i - 1} < a_i $ then $ a_i - a_{i - 1} $ comoving knots lie on the interval $ {[ i, i + 1 )} $ , and their addresses are $ a_{i - 1} + 1, \\cdots , a_i $ .", "Using these addresses, we extract indices the $ j_i^{\\,(\\alpha )} $ from $ \\mathtt {forward\\_j} $ and the l.h.s.", "weights $ _i^{(\\alpha )} $ from $ \\mathtt {forward\\_w} $ and perform the transform by using Eq.", "(REF ).", "If $ a_{i - 1} = a_i $ , then there are no comoving knots that lie on the inertial interval $ {[ i, i + 1 )} $ and no interpolation is done.", "Applying the same procedure to the previous $ i - 1 $ and pre-previous $ i - 2 $ knots, we obtain the corresponding comoving indices $ j_{i - 1}^{\\,(\\alpha )} $ and r.h.s.", "weights $ 1 - _{i - 1}^{(\\alpha )} $ for the inertial interval $ {[ i - 1, i )} $ and perform the transform by using Eq.", "(REF ).", "It is necessary to use three knots (the current $ i $ , the previous $ i - 1 $ , and the pre-previous $ i - 2 $ ) to compute the related indices and weights for both the $ {[ i - 1, i )} $ and $ {[i, i + 1 )} $ intervals because we store only the l.h.s.", "weights to save memory.", "This is why the subscript of address starts at $-1$ , so that the algorithm handles the case $ i = i_$ without using an if-statement." ], [ "Precomputed backward transform", "[!t] 0.5em0.5em textsl Profile ratio $ \\rho ^{\\star \\!", "}[x, y, z, j] $ for all frequencies $ q[j]\\colon j = j_, \\cdots , j_$ in the comoving frame.", "Profile ratio $ \\rho [x, y, z, \\mu , i] $ at given frequency $ q[i] $ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "$ [{}][l]{j} \\leftarrow \\mathtt {backward\\_j}[x, y, z, \\mu , i] $ [r]Comoving index $ \\leftarrow \\mathtt {backward\\_w}[x, y, z, \\mu , i] $ [r]and weight.", "([f]Linear interpolation on ${[j, j + 1)}$ :)$0 \\le < 1$ $ \\rho [x, y, z, \\mu , i] \\leftarrow \\,\\rho ^{\\star \\!", "}[x, y, z, j] +\\left( 1 - \\right) \\rho ^{\\star \\!", "}[x, y, z, j + 1] $ ($ = 1 $ [f]Either constant extrapolation if $j \\notin {[ j_, j_)}$ ,) $ \\rho [x, y, z, \\mu , i] \\leftarrow \\rho ^{\\star \\!", "}[x, y, z, \\mu , j] $ [r]or $j$ exactly matches $i$ .", "Precomputed backward transform.", "With precomputed indices and weights the backward interpolation is performed as in Alg.", "REF .", "We select the proper type of interpolation from Eqs.", "(REF )–(REF ) depending on the value of the r.h.s.", "weight $ $ .", "The address array needs to store big numbers (typically, $ o < 10^9 $ ) and therefore must be of 4-byte integer type, while values of forward_j and backward_j usually fit in 2-byte integer type.", "Values of forward_w and backward_w can be packed into 1-byte integer type because interpolation weights are real numbers on the interval $ {[0, 1]} $ .", "The limited precision of a one-byte representation is acceptable because linear interpolation itself is not very accurate.", "Hence, we need 10 bytes per spatial grid point, direction, and frequency." ], [ "Equidistant interpolation", "To get rid of frequency dependence in the arrays that store precomputed indices and weights, we modify the frequency grid so that it is equidistant.", "For a given velocity resolution $ \\delta $ (typically, 1–2 km s$^{-1}$ ), the new frequency grid resolution is $\\delta q = \\frac{ \\delta }{ _\\mathrm {B} }.$ From $ q[i_] $ to $ q[i_] $ we have $m_\\mathrm {max} = 1 + \\frac{ q[i_] - q[i_] }{ \\delta q }$ knots separated by equal intervals of fine resolution $ \\delta q $ and numbered from 1 to $ m_\\mathrm {max} $ having frequencies $q^\\prime [m] = q[i_] + ( m - 1 )\\,\\delta q.$ so that $ q^\\prime [1] = q[i_] $ and $ q^\\prime [m_\\mathrm {max}] = q[i_] $ .", "Note that indices $ m $ and $ i $ do not coincide.", "Fine resolution is needed only in the line core but not in the line wings, so we solve the radiative transfer equation only in selected knots called real and do not do this in the remaining knots called virtual.", "We call the ordered set of real knots the real grid and we number them in the same way as before using index $ i = i_, \\cdots , i_$ for the inertial real grid and using index $ j = j_, \\cdots , j_$ for the comoving real grid.", "We call the ordered set of both real and virtual knots the fine grid and we number them using index $ m = 1, \\cdots , m_\\mathrm {max} $ for the inertial fine grid and using index $ n = 1, \\cdots , n_\\mathrm {max} $ for the comoving fine grid, with $ m_\\mathrm {max} = n_\\mathrm {max}$ .", "We make interpolation on the real grid fast by using extra maps, that track positions of real and virtual knots and specify relations between them.", "We store these maps as 1D arrays.", "Example contents of these arrays are illustrated in Fig.", "REF .", "The array map with size $ N_\\nu $ is indexed from $ i_$ to $ i_$ and for the real grid index $ i $ , map[$ i $ ] is the related fine grid index $ m $ .", "The array unmap with size $ m_\\mathrm {max} $ is indexed from 1 to $ m_\\mathrm {max} $ , and for the fine grid index $ m $ , unmap[$ m $ ] either is the related real grid index $ i $ if $ m $ is a real knot, otherwise it is 0.", "In essence, map matches the real grid with the fine grid, while unmap does the opposite.", "The logical array real_knot with size $ m_\\mathrm {max} $ is indexed from 1 to $ m_\\mathrm {max} $ and specifies whether fine grid knots are real or virtual.", "The two arrays prior_real and next_real with size $ m_\\mathrm {max} $ are indexed from 1 to $ m_\\mathrm {max} $ .", "They specify for each $ m $ in the fine grid, the nearest-left real or nearest-right real knot." ], [ "Precomputation for equidistant interpolation", "[!t] 0.5em0.5em textsl Line frequencies $ q[i_{:}i_] $ , directions $ \\vec{n}[1{:}N_\\mu ] $ , velocities $ \\vec{}[x_\\mathrm {min}{:}x_\\mathrm {max},y_\\mathrm {min}{:}y_\\mathrm {max},z_\\mathrm {min}{:}z_\\mathrm {max}] $ .", "Arrays shift and weight.", "$ z \\leftarrow z_\\mathrm {min} $ $ z_\\mathrm {max} $ $ y \\leftarrow y_\\mathrm {min} $ $ y_\\mathrm {max} $ $ x \\leftarrow x_\\mathrm {min} $ $ x_\\mathrm {max} $ $ \\mu \\leftarrow 1 $ $ N_\\mu $ $ q_\\mu \\leftarrow \\bigl ( \\vec{n}[\\mu ]\\cdot \\vec{}[x, y, z] \\bigr ) /_\\mathrm {B} $ $ s \\leftarrow \\bigg \\lfloor \\dfrac{q_\\mu }{\\delta q} \\bigg \\rfloor $ [r]Integer shift between the two grids.", "$ r \\leftarrow q_\\mu - s \\cdot \\delta q $ [r]Fractional remainder.", "$ \\leftarrow 1 - \\dfrac{r}{\\delta q} $ [r]The l.h.s.", "weight $ [{\\mathtt {weight}}][l]{\\mathtt {shift}}[x, y, z, \\mu ] \\leftarrow s $ [r]Store shift and weight.", "$ \\mathtt {weight}[x, y, z, \\mu ] \\leftarrow $ Precomputation for equidistant interpolation.", "At each grid point $ (x, y, z) $ , for each direction $ \\vec{n}[\\mu ] $ , we compute the Doppler shift $ q_\\mu $ .", "The comoving knots $ n $ are Doppler-shifted along the inertial knots $ m $ by $ s $ intervals of $ \\delta q $ length and a fractional remainder $ r $ such that $q_\\mu = s\\cdot \\delta q + r.$ If inertial knot $ m $ has a comoving nearest-right neighbor knot $ n $ , then $ m - n = s $ .", "The frequency displacement between these knots is $ q_\\mu + q[n] - q[m] = r $ .", "The remainder $ r $ determines the interpolation weights for the comoving knot $ n $ on the inertial interval $ {[m, m + 1)} $ .", "The l.h.s.", "weight is $= \\frac{ \\delta q - r }{ \\delta q } = 1 - \\frac{ r }{ \\delta q }.$ There are three advantages of using equidistant frequency grids.", "First, for a given index tuple $ (x, y, z, \\mu ) $ , the values of $ s $ , $ r $ , and $ $ are independent of frequency as illustrated in Fig.", "REF .", "Second, during the transforms, we can limit the searching range for comoving knots by using the $ m = n + s $ relation.", "Third, both the forward and backward transforms use the same values of $ s $ and $ $ .", "Therefore, we precompute and store $ s $ and $ $ in two 4D-arrays (shift and weight) following Alg.", "REF ." ], [ "Equidistant forward transform", "[!t] 0.5em0.5em textsl Specific intensity $ I[x, y, z, \\mu , i] $ at frequency $ q[i] $ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "Incremented angle-averaged intensity $ J^{\\star \\!", "}[x, y, z, j] $ at corresponding frequencies $ q[j] $ in the comoving frame.", "$ n_\\mathrm {max} \\leftarrow \\mathrm {size}( \\mathtt {map} ) $ [f]Fine grid length.", "$ [{}][l]{s} \\leftarrow [{\\mathtt {weight}}][l]{\\mathtt {shift}}[x, y, z, \\mu ] $ $ \\leftarrow \\mathtt {weight}[x, y, z, \\mu ] $ $ n_\\mathrm {L} \\leftarrow \\mathtt {map}[{[\\min (i_, i + 1)]}][l]{\\bigl [\\max (i - 1, i_)\\bigr ]} - s $ [h]Nearest-right neighbors of $ n_\\mathrm {C} \\leftarrow \\mathtt {map}[{[\\min (i_, i + 1)]}][l]{\\bigl [i\\bigr ]} - s $ [h]$i-1$ , $i$ , and $i+1$ on the fine $ n_\\mathrm {R} \\leftarrow \\mathtt {map}\\bigl [\\min (i_, i + 1)\\bigr ] - s $ [h]grid in the comoving frame.", "([f]Linear interpolation on $ {[i - 1, i)} $ :)$ n_\\mathrm {L} \\ne n_\\mathrm {C} $ $ n \\leftarrow \\min \\bigl ( n_\\mathrm {C} - 1, n_\\mathrm {max} \\bigr ) $ ()$ 1 \\le n $ ()not $ \\mathtt {real\\_knot}[n] $$ n \\leftarrow \\mathtt {prior\\_real}[n] $ ()$ \\max \\bigl ( 1, n_\\mathrm {L} \\bigr ) \\le n $ $ \\xi ^\\prime \\leftarrow \\dfrac{ n - n_\\mathrm {L} + 1 - }{ n_\\mathrm {C} - n_\\mathrm {L} } $ [f]The r.h.s.", "weight.", "$ J^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n]\\bigr ] \\overset{+}{\\leftarrow }\\xi ^\\prime \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ $ n \\leftarrow \\mathtt {prior\\_real}[n] $ ($ n_\\mathrm {L} = n_\\mathrm {C} $ [f]Constant extrapolation on $ {({-}\\infty , i_)} $ :) $ n \\leftarrow \\min \\bigl ( n_\\mathrm {C} - 1, n_\\mathrm {max} \\bigr ) $ ()$ 1 \\le n $ ()not $ \\mathtt {real\\_knot}[n] $$ n \\leftarrow \\mathtt {prior\\_real}[n] $ ()$ 1 \\le n $ $ J^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n]\\bigr ] \\overset{+}{\\leftarrow }\\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ $ n \\leftarrow \\mathtt {prior\\_real}[n] $ ([f]Linear interpolation on $ {[i, i+1)} $ :)$ n_\\mathrm {C} \\ne n_\\mathrm {R} $ $ n \\leftarrow \\max \\bigl ( 1, n_\\mathrm {C} \\bigr ) $ ()$ n \\le n_\\mathrm {max} $ ()not $ \\mathtt {real\\_knot}[n] $$ n \\leftarrow \\mathtt {next\\_real}[n] $ ()$ n \\le \\min \\bigl ( n_\\mathrm {R} - 1, n_\\mathrm {max} \\bigr ) $ $ \\xi \\leftarrow \\dfrac{ n_\\mathrm {R} - n - (1 - ) }{ n_\\mathrm {R} - n_\\mathrm {C} } $ [f]The l.h.s.", "weight.", "$ J^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n]\\bigr ] \\overset{+}{\\leftarrow }\\xi \\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ $ n \\leftarrow \\mathtt {next\\_real}[n] $ ($ n_\\mathrm {C} = n_\\mathrm {R} $ [f]Constant extrapolation on $ {[i_, {+}\\infty )} $ :) $ n \\leftarrow \\max \\bigl (1, n_\\mathrm {C} \\bigr ) $ ()$ n \\le n_\\mathrm {max} $ ()not $ \\mathtt {real\\_knot}[n] $$ n \\leftarrow \\mathtt {next\\_real}[n] $ ()$ n \\le n_\\mathrm {max} $ $ J^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n]\\bigr ] \\overset{+}{\\leftarrow }\\dfrac{\\omega _\\mu }{4\\pi } I[x, y, z, \\mu , i] $ $ n \\leftarrow \\mathtt {next\\_real}[n] $ Equidistant forward transform.", "The forward transform for the equidistant grid is given by Alg.", "REF .", "Figure REF provides an illustration.", "Interpolation is performed on the fine grid.", "We project the current knot $ i $ and its left $ i - 1 $ and right $ i + 1 $ neighbors from the inertial real grid onto the inertial fine grid: mL = map[(i - 1, i) ], mC = map[i ], mR = map[(i, i + 1)].", "We use $ \\max (i - 1, i_) $ and $ \\min (i_, i + 1) $ to avoid going beyond the profile ends $ i = i_$ and $ i = i_$ .", "Indices of the related nearest-right neighbors in the comoving fine grid are computed from the shift $ s $ : 2 nL = map[(i - 1, i) ] - s, nC = map[i ] - s, nR = map[(i, i + 1)] - s. To interpolate on the $ {[i, i + 1)} $ interval, we scan the comoving fine grid knots $ n_\\mathrm {C}, \\cdots , n_\\mathrm {R} - 1 $ that lie on the $ {[ m_\\mathrm {C}, m_\\mathrm {R} ]} $ interval.", "Moving from $ n = n_\\mathrm {C} $ to $ n_\\mathrm {R} - 1$ , we look for real knots.", "If $ n $ is a virtual knot, we jump to the next real knot using the next_real array.", "If $ n $ is a real knot, we find the corresponding index $ j $ in the comoving real grid using the unmap array.", "Then the desired l.h.s.", "weight $ \\xi $ is computed from $ n $ , $ n_\\mathrm {C} $ , $ n_\\mathrm {R} $ , and the fine grid weight $ $ : $\\xi = \\dfrac{ n_\\mathrm {R} - n - (1 - ) }{ n_\\mathrm {R} - n_\\mathrm {C} }.$ To interpolate on the $ {[i - 1, i)} $ interval, we scan the comoving fine grid knots $ n_\\mathrm {L}, \\cdots , n_\\mathrm {C} - 1 $ that lie on the $ {[ m_\\mathrm {L}, m_\\mathrm {C} ]} $ interval.", "Moving from $ n = n_\\mathrm {C} - 1 $ to $ n_\\mathrm {L} $ we look for real knots.", "If $ n $ is a virtual knot, we jump to the prior real knot using the prior_real array.", "If $ n $ is a real knot, we find the corresponding index $ j $ in the comoving real grid using the unmap array.", "The desired r.h.s.", "weight $ 1 - \\xi $ is then computed as $\\xi ^\\prime \\equiv 1 - \\xi = \\dfrac{ n - n_\\mathrm {L} + 1 - }{ n_\\mathrm {C} - n_\\mathrm {L} }.$ Extrapolation at the $ i = i_$ or $ i = i_$ ends is done likewise.", "In all four cases, we truncate the running index $ n $ by taking $ \\max (1, n) $ or $ \\min (n, n_\\mathrm {max}) $ to keep it within the fine grid range $ 1, \\cdots , n_\\mathrm {max} $ ." ], [ "Equidistant backward transform", "[!t] 0.5em0.5em textsl Profile ratio $ \\rho ^{\\star \\!", "}[x, y, z, j] $ for all frequencies $ q[j]\\colon j = j_, \\cdots , j_$ in the comoving frame.", "Profile ratio $ \\rho [x, y, z, \\mu , i] $ at given frequency $ q[i] $ and direction $ \\vec{n}[\\mu ] $ in the inertial frame.", "$ n_\\mathrm {max} \\leftarrow \\mathrm {size}( \\mathtt {map} ) $ [f]Fine grid length.", "$ [{}][l]{s} \\leftarrow [{\\mathtt {weight}}][l]{\\mathtt {shift}}[x, y, z, \\mu ] $ $ \\leftarrow \\mathtt {weight}[x, y, z, \\mu ] $ $ n_\\mathrm {C} \\leftarrow \\mathtt {map}[i] - s $ [r]Nearest-right neighbor of $i$ on the [r]fine grid in the comoving frame.", "([f]Linear interpolation on $ {[j, j+1)} $ :)$ 1 < n_\\mathrm {C} \\le n_\\mathrm {max} $ $ n_\\mathrm {L} \\leftarrow n_\\mathrm {C} - 1 $ [r]Nearest-left and right neighbors $ n_\\mathrm {R} \\leftarrow n_\\mathrm {C} $ [r]in the comoving fine grid.", "()not $ \\mathtt {real\\_knot}[n_\\mathrm {L}] $ $ n_\\mathrm {L} \\leftarrow \\mathtt {prior\\_real}[n_\\mathrm {L}] $ ()not $ \\mathtt {real\\_knot}[n_\\mathrm {R}] $ $ n_\\mathrm {R} \\leftarrow \\mathtt {next\\_real\\ \\,}[n_\\mathrm {R}] $ $ \\xi \\leftarrow \\dfrac{ n_\\mathrm {R} - n_\\mathrm {C} + 1 - }{ n_\\mathrm {R} - n_\\mathrm {L} } $ [f]The l.h.s.", "weight.", "[r]Interpolate on the comoving grid: $ \\rho [x, y, z, \\mu , i] \\leftarrow \\xi \\, \\rho ^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n_\\mathrm {L}]\\bigr ] $ $ {57mu} {} + (1 - \\xi )\\,\\rho ^{\\star \\!", "}\\bigl [x, y, z, \\mathtt {unmap}[n_\\mathrm {R}]\\bigr ] $ ([f]Constant extrapolation on $ {[ j_, {+}\\infty )} $ :)$ n_\\mathrm {max} < n_\\mathrm {C} $ $ \\rho [x, y, z, \\mu , i] \\leftarrow \\rho ^{\\star \\!", "}[x, y, z, j_] $ ($ n_\\mathrm {C} \\le 1 $ *[f]Constant extrapolation on $ {( {-}\\infty , j_)} $ :) $ \\rho [x, y, z, \\mu , i] \\leftarrow \\rho ^{\\star \\!", "}[x, y, z, j_] $ Equidistant backward transform.", "The backward interpolation employs the same ideas as the equidistant forward interpolation.", "It is given in Alg.", "REF and illustrated in Fig.", "REF .", "The current knot $ i $ is projected from the inertial real onto the inertial fine grid using the map array to get $ m_\\mathrm {C} $ .", "Then we compute its nearest-right neighbor $ n_\\mathrm {C} = m_\\mathrm {C} - s $ in the comoving grid.", "We test whether $ n_\\mathrm {C} $ is beyond the fine grid ends 1 or $ n_\\mathrm {max} $ .", "If so, then we extrapolate from the comoving real grid ends $ j_$ or $ j_$ .", "If not, then knot $ i $ lies between knots $ n_\\mathrm {C} - 1 $ and $ n_\\mathrm {C} $ .", "We set $ n_\\mathrm {L} = n_\\mathrm {C} - 1 $ and $ n_\\mathrm {R} = n_\\mathrm {C} $ .", "If $ n_\\mathrm {L} $ or $ n_\\mathrm {R} $ are virtual, we use prior_real or next_real to set them to the nearest real knots.", "Then we project $ n_\\mathrm {L} $ and $ n_\\mathrm {R} $ onto the comoving real grid using the unmap array to get the comoving real knots $ j $ and $ j + 1 $ .", "The desired l.h.s.", "weight is computed using $ n_\\mathrm {L} $ , $ n_\\mathrm {C} $ , $ n_\\mathrm {R} $ , and $ $ : $\\xi = \\dfrac{ n_\\mathrm {R} - n_\\mathrm {C} + 1 - }{ n_\\mathrm {R} - n_\\mathrm {L} }.$" ] ]	1606.05180
[ [ "Insights into the location and dynamics of the coolest X-ray emitting\n gas in clusters of galaxies" ], [ "Abstract We extend our previous study of the cool gas responsible for the emission of OVII X-ray lines in the cores of clusters and groups of galaxies.", "This is the coolest X-ray emitting phase and connects the 10,000 K H {\\alpha} emitting gas to the million degree phase, providing a useful tool to understand cooling in these objects.", "We study the location of the O VII gas and its connection to the intermediate Fe XVII and hotter O VIII phases.", "We use high-resolution X-ray grating spectra of elliptical galaxies with strong Fe XVII line emission and detect O VII in 11 of 24 objects.", "Comparing the O VII detection level and resonant scattering, which is sensitive to turbulence and temperature, suggests that OVII is preferably found in cooler objects, where the FeXVII resonant line is suppressed due to resonant scattering, indicating subsonic turbulence.", "Although a larger sample of sources and further observations is needed to distinguish between effects from temperature and turbulence, our results are consistent with cooling being suppressed at high turbulence as predicted by models of AGN feedback, gas sloshing and galactic mergers.", "In some objects the OVII resonant-to-forbidden line ratio is decreased by either resonant scattering or charge-exchange boosting the forbidden line, as we show for NGC 4636.", "Charge-exchange indicates interaction between neutral and ionized gas phases.", "The Perseus cluster also shows a high Fe XVII forbidden-to- resonance line ratio, which can be explained with resonant scattering by low-turbulence cool gas in the line-of-sight." ], [ "Introduction", "The intracluster medium (ICM) embedded in the deep gravitational well of clusters of galaxies has a complex multi-temperature structure with different cospatial phases ranging from $\\sim 10^{6}$ to above $10^{8}$ K. It is thought to contain most of the baryonic mass of the clusters and its density strongly increases in their cores where the radiative cooling time is less than 1 Gyr and therefore shorter than their age.", "In the absence of heating, this would imply the cooling of hundreds of solar masses of gas per year below $10^{6}$ K [6].", "The gas is expected to produce prominent emission lines from O VI in UV, peaking at $T\\sim 3\\times 10^{5}$ K, as well as O VII ($T\\sim 2\\times 10^{6}$ K) and Fe XVII ($T\\sim 6\\times 10^{6}$ K) in X-rays, suggesting that spectroscopy is a key to understand the cooling processes in clusters of galaxies.", "Evidence of weak O VI UV lines was found by [3], [2] at levels of $30\\,M_{\\odot }$ yr$^{-1}$ or lower, significantly less than the predicted $100\\,M_{\\odot }$ yr$^{-1}$ .", "Fe XVII emission lines have been discovered, but with luminosities much lower than expected from cooling-flow models (see e.g.", "[19]).", "O VII lines were detected for the first time in a stacked spectrum of a sample of cool objects by [24] and more recently in individual elliptical galaxies by our group [20], but in most cases their fluxes are lower than those predicted by cooling flow models.", "There is an overal deficit of cool ($\\lesssim 0.5$ keV or $\\lesssim 6\\times 10^{6}$ K) gas in the cores of clusters of galaxies and nearby elliptical galaxies.", "Several energetic phenomena are occurring in the cores and in the outskirts of clusters of galaxies and isolated galaxies such as feedback from active galactic nuclei (AGN, see e.g.", "[4], [18], [7]).", "Briefly, energetic AGN outflows drive turbulence in the surrounding ICM, which then dissipates and heats the ICM balancing the cooling [30].", "AGN can also heat the surrounding gas via dissipation of sound waves (see e.g.", "[9], [8]).", "The phenomenology can be more complex because galactic mergers and sloshing of gas within the gravitational potential also produce high turbulence (see e.g.", "[1]; [15]).", "In this work we study the coolest X-ray emitting gas in clusters and groups of galaxies and in elliptical galaxies, which is crucial to understand the ICM cooling from $10^8$ K down to $10^4$ K. We use high quality archival data and new observations taken with the high-resolution Reflection Grating Spectrometer (RGS) aboard XMM-Newton.", "We search for a relationship between the cool O VII gas and the turbulence, evidence of resonant scattering and charge exchange in the ICM where neutral gas is observed.", "We present the data in Sect.", "and the spectral modeling in Sect. .", "We discuss the results in Sect.", "and give our conclusions in Sect. .", "Table: XMM-Newton/RGS observations used in this paper, extraction regions and O VII detection." ], [ "The data", "The observations used in this paper are listed in Table REF .", "Most objects were already included in our recent work [21], but here we only focus on those which exhibit cool gas producing Fe XVII emission lines.", "The original catalog, also known as the CHEERS sample, consists of 44 nearby, bright clusters and groups of galaxies and elliptical galaxies with a $\\gtrsim 5\\sigma $ detection of the O VIII 1s–2p line at 19 Å and with a well-represented variety of strong, weak, and non cool-core objects.", "In addition to the CHEERS sources exhibiting Fe XVII emission, here we include two cool objects: NGC 1332 and IC 1459.", "In NGC 1332, O VIII was detected just below $5\\sigma $ , however its Fe XVII emission lines are much stronger.", "IC 1459 data were enriched by $\\sim 120$ ks new data awarded during the AO-14.", "In total we have 24 sources.", "The XMM-Newton satellite is provided with two main X-ray instruments: RGS and EPIC (European Photon Imaging Camera).", "We have used RGS data for the spectral analysis and EPIC (MOS 1 detector) data for imaging.", "The RGS spectrometers are slitless and the spectral lines are broadened by the source extent.", "We correct for spatial broadening through the use of EPIC/MOS 1 surface brightness profiles.", "We repeat the data reduction as previously done in [21], but with newer calibration files and software versions (available by January, 2016).", "All the observations have been reduced with the XMM-Newton Science Analysis System (SAS) v14.0.0.", "We correct for contamination from soft-proton flares with the standard procedure.", "The sources in our sample span a large range of distances (Table REF ).", "Therefore, we tried to extract spectra in slices with widths of the same physical size.", "Before chosing an absolute scale, we have tested several extraction regions.", "For the nearby objects, such as NGC 4636 which is the nearest X-ray bright giant elliptical galaxy, we adopted a width of about 0.8' ($\\sim 4$ kpc) because it provides a good coverage of the inner Fe XVII bright core, strengthens the Fe XVII lines with respect to those produced by the hotter gas phase, and maximizes the detection of the O VII emission lines.", "The spectra of all objects were then extracted in regions centered on the Fe XVII emission peak with widths scaled by the ratio between the distance of the objects and that of NGC 4636.", "For NGC 507, 533, 3411 and 4325 we had to adopt a slightly larger width because it was the minimum to provide enough statistics in the Fe XVII lines.", "The spectra extracted in these regions of approximately equal physical size have been used to measure the Fe XVII line ratios.", "Finally, we have also extracted spectra in different regions, with widths up to 3.4' which is the RGS sensitive field of view, to improve the O VII detection.", "We subtracted the model background spectrum, which is created by the standard RGS pipeline and is a template background file based on the count rate in CCD 9.", "The spectra were converted to SPEXwww.sron.nl/spex format through the SPEX task trafo.", "We produced MOS 1 images in the $8-27$ Å wavelength band and extracted surface brightness profiles to model the RGS line spatial broadening with the following equation: $\\Delta \\lambda = 0.138 \\, \\Delta \\theta \\, {\\mbox{Å}}$ (see the XMM-Newton Users Handbook)." ], [ "Baseline model", "Our analysis focuses on the $8-27$ Å first and second order RGS spectra.", "We perform the spectral analysis with SPEX version 3.00.00.", "We scale elemental abundances to the proto-Solar abundances of [17], which are the default in SPEX, use C-statistics and adopt $1\\,\\sigma $ errors.", "We have described the ICM emission with an isothermal plasma model of collisional ionization equilibrium (cie).", "The basis for this model is given by the mekal model, but several updates have been included (see the SPEX manual).", "Free parameters in the fits are the emission measure $Y=n_{\\rm e}\\,n_{\\rm H}\\,dV$ , the temperature $T$ , and the abundances (N, O, Ne, Mg, and Fe).", "Nickel abundance was coupled to iron.", "Most objects required two cie components (see Table REF ).", "Here, we coupled the abundances of the two cie components and assumed that the gas phases have the same abundances because the spectra do not allow us to measure them separately.", "The cie emission models were corrected for redshift, Galactic absorption, see Table REF , and line-spatial-broadening through the multiplicative lpro component that receives as input the MOS 1 surface brightness profile (see Sect. ).", "We do not explicitly model the cosmic X-ray background in the RGS spectra because any diffuse emission feature would be smeared out into a broad continuum-like component.", "For several objects, including the Perseus and Virgo clusters, we have added a further power-law emission component to account for any emission from the central AGN (see [22] and references therein).", "This is not convolved with the spatial profile because it is produced by a point source.", "For each source, we have simultaneously fitted the spectra of individual observations by adopting the same model, apart from the emission measures of the cie components which were uncoupled to account for the different roll angles of the observations.", "We have successfully applied this multi-temperature model to the RGS spectra.", "However, as previously shown in [21], the model underestimates the 17 Å Fe XVII line peaks and overestimates its broadening for some sources, e.g.", "Fornax, M 49, M 86, NGC 4636, and NGC 5813.", "This is due to the different spatial distribution of the gas responsible for the cool Fe XVII emission lines and that producing most of the high-ionization Fe-L and O VIII lines.", "The Fe XVII gas is indeed to be found predominantly in the cores showing a profile more peaked than that of the hotter gas.", "The spatial profiles estimated with MOS 1 images strongly depend on the emission of the hotter gas due to its higher emission measure and therefore they overestimate the spatial broadening of the 15–17 Å lines.", "It is difficult to extract a spatial profile for these lines because MOS 1 has a limited spectral resolution and the images extracted in such a narrow band will lack the necessary statistics (see e.g.", "[25]).", "The 17 Å / 15 Å line ratio is also affected by resonant scattering (see e.g.", "[11], [26]), which requires a different approach.", "In Sect.", "REF and REF we account for the different location of the different phases and the Fe XVII (and O VII) resonant scattering." ], [ "Search for O ", "Following [20], we have removed the O VII ion from the model and fitted two delta lines fixed at 21.6 Å and 22.1 Å, which reproduce the O VII resonance and forbidden lines, respectively.", "The intercombination line at 21.8 Å is generally weak or insignificant and blends with the resonance line.", "These lines are corrected by the redshift, the Galactic absorption, and the spatial line broadening as done for the cie models.", "If the resonant line was comparable or stronger than the forbidden lines, we have determined the O VII total significance by fixing the resonance-to-forbidden line flux ratio to $(r/f) = 1.3$ as predicted by the thermal model.", "Otherwise the O VII total significance was calculated as the squared-sum of the significance of each line.", "The latter refers to Perseus, M 89, and NGC 4636 and 5813.", "We applied this technique to spectra extracted in regions of different widths in order to search for that one maximizing the O VII detection.", "We adopt as threshold for the O VII detection the 99% confidence level because the objects are distributed in two subsamples with detection levels $<2.0\\sigma $ and $>2.6\\sigma $ showing a gap in between.", "The results are reported in Table REF and discussed in Sect.", "." ], [ "The location of the cool gas", "It is possible to probe the extent of the cool (O VII $-$ Fe XVII) gas by comparing its linewidth to that of the hot (O VIII $-$ Fe XVIII+) gas.", "The dominant line broadening effect in grating spectra is indeed produced by the spatial extent of the source (normally a few 1000 km s$^{-1}$ ), which is almost an order of magnitude larger than the thermal + turbulent broadening (few 100s km s$^{-1}$ , see e.g.", "[21] and references therein).", "The turbulence and thermal broadening are not expected to be significantly different between the two phases (see e.g.", "[21]).", "We therefore did the same exercise for the Fe XVII emission lines by removing the Fe XVII ion from the model and fitting four delta lines fixed at 15.01 Å, 15.26 Å, 16.78 Å, and 17.08 Å, which are the main Fe XVII transitions.", "We do not tabulate the significance of the Fe XVII lines because they are typically much larger than $5\\sigma $ .", "The lpro model in SPEX that corrects for the line broadening has an additional scale parameter s, which allows to fit the width of the spatial broadening by a factor free to vary (see the SPEX manual).", "We therefore use one lpro model to account for the spatial broadening in the cie components that produce the high-temperature lines and another lpro model to fit the spatial broadening of the low-temperature O VII and Fe XVII lines.", "Averaging between all objects in our sample, we find that the lpro scale parameter of the cool gas is half of that measured for the hot gas.", "In Fig.", "REF we show the RGS spectra of three interesting sources, the Centaurus cluster, M 84 and M 89 (from top to bottom).", "Overlaid on the data there are three spectral models: the baseline cie model (thick black line), the delta line model for O VII and Fe XVII lines adopting the same spatial broadening as the cie models (solid green line), and finally the O VII and Fe XVII lines with the spatial scale parameter s free to vary (dashed red line).", "In order to better visualize the effect on the fit from spatial broadening, we calculate the ratios from the best-fit models obtained with the delta lines and the best-fitting cie model and display them in the bottom panel of each figure.", "The color is coded similarly to the top panel: the green line is the ratio between the O VII–Fe XVII delta model and the cie components (with the same spatial broadening); the red line shows the same ratio but with a different spatial broadening.", "The Fe XVII lines appear clearly narrower then the hot–gas lines in the Centaurus cluster (A 3526), even taking into account the slightly different thermal broadening, but there is no significant wavelength shift (in agreement with [26]).", "This suggests that the Fe XVII cool ($\\sim 6\\times 10^6$ K) gas peaks in the central regions and has a smaller extent than that of the hot gas responsible for the O VIII at 19 Å and the higher-ionization Fe XX+ ($\\gtrsim 10^7$ K) lines between $11-13$ Å.", "A similar trend is observed in Fornax, M 49, M 84, NGC 4636–5044–5813, and Perseus.", "The M 84 and M 89 elliptical galaxies, whose spectra are dominated by the Fe XVII lines, show an O VII excess with respect to the two-phase cie model.", "Interestingly, in M 84 (and NGC 5846) the O VII resonant line at 21.6 Å is in excess, while in M 89 (and NGC 4636) the excess is shown by the forbidden line at 22.1 Å.", "The quality of the spectra of the other objects is not good enough to detected O VII in excess to that already produced by the two-cie model.", "A stronger forbidden line may indicate resonant scattering for both O VII and Fe XVII lines as we previously suggested in [20].", "Figure: From top to bottom: RGS spectra of the Centaurus cluster,M 84, and M 89.", "Three spectral models are overlaid:2-cie model (thick black line), delta-line Fe XVII model(thick green line) and different spatial broadening (dashed red line).The bottom panels show the ratios between the Fe XVII line modelsand the 2-cie model.", "The blue dotted lines show the 1σ1\\sigma uncertainties.Figure: Fe XVII forbidden-to-resonance line ratio versusaverage temperature.", "O VII detections are reported with red points.The dashed green line shows the best fit in the log-log space for the objects below 1 keV.The theoretical predictions from SPEX and Atomdb v3.0.2 are also shown.The objects above 1 keV are grey-shaded because there is little Fe XVII at thoseaverage temperatures and most of it should be produced by a cooler phase.The small grey box shows the average Fe XVII ratio (2.00±0.292.00\\pm 0.29)of the elliptical galaxies below 1 keV with O VII detected abovethe 99% confidence level." ], [ "O ", "When turbulence is low the resonant line can be optically thick; it is therefore absorbed and re-emitted in a random direction with the line being suppressed towards the bright core and enhanced outside.", "This does not occur at high turbulence due to the energy shift of the transitions (see e.g [29] and [5]).", "The forbidden lines have a smaller oscillator strength and are much less affected.", "It is then interesting to measure the Fe XVII line resonant scattering of the sources in our sample, which is an indicator of (low-) turbulence, and compare it to the O VII detection, in a certain temperature range.", "We have used the Fe XVII line fluxes measured in Sect.", "REF to calculate the Fe XVII (f/r) line ratios for the models with a different spatial broadening between these lines and the hot gas, and quote the results in Table REF .", "In order to estimate an average temperature for each source, we fit again the RGS spectra with only one single cie component.", "The average temperatures estimated through these models are quoted in Table REF .", "We plot the Fe XVII (f/r) line ratios versus the temperature in Fig.", "REF with the red points showing the sources with O VII detection above the 99% confidence level.", "The point size scales with the average S/N ratio of the RGS spectra at 17 Å.", "We also show the Fe XVII line ratios as predicted by a thermal model without resonant scattering according to the Atomdb v3.0.2 and SPEX to visualize the strength of the resonant scattering in each source and the systematic uncertainties in the atomic data.", "Points below the theoretical predictions would be unphysical because it is difficult to strengthen only the resonant line in the line-of-sight towards the center of the galaxies (although charge-exchange can slightly enhance the Fe XVII resonant line).", "All our Fe XVII (f/r) line ratios agree with the theoretical predictions or are above the theoretical curves, indicating moderate resonant scattering and therefore low-to-mild (subsonic) turbulence.", "We fit a straight-line in the log-log space for the objects with $T<1$ keV, and found a significant anti-correlation between the Fe XVII (f/r) line ratio and the average temperature (well above the $3\\sigma $ confidence level, $p-$ value$=0.00026$ , with slope $-0.79\\pm 0.18$ , see the green dashed line in Fig.", "REF ).", "This may indicate either a decrease in optical depth or an increase in turbulence or both.", "We caution against the comparison with the brightest cluster galaxies (BCG), i.e.", "those in A 262, Centaurus, Fornax, Perseus, and Virgo clusters, because above 1 keV the ICM becomes optically thin to the Fe XVII emitted by the cooler gas phases ($kT<0.9$ keV) and therefore resonance scattering becomes insensitive to turbulence in the cores of these systems.", "All results are discussed in Sect.", "." ], [ "Systematic effects", "There are several systematics that may affect our results and their interpretation such as the background subtraction, the line blending and the uncertainties in the atomic database.", "The model background spectra used throughout this work adopt long exposures of blank fields.", "This is a safe approach since any background contribution to the weak O VII or the strong Fe XVII lines would be smeared out in a continuum like feature.", "For some bright and compact objects such as NGC 1316–1404 M 89 we could extract a background spectrum in the outer regions of the RGS detector and match it with the model background spectrum.", "The spectra were comparable and no significant difference in the line ratios were found.", "We also tested a different continuum with a local (14.5–18.0 Å) fit using a power-law and a few delta lines obtaining larger statistical uncertainties and consistent results with the previous Fe XVII measurements.", "We tested a power-law continuum for the 19.5–22.5 Å range obtaining similar O VII detections.", "We have also checked the effects of blending with other lines.", "The O VII resonant and forbidden lines are located in a rather clean spectral range apart from O VI and O VII intercombination lines.", "As mentioned in Sect.", ", there are only small amounts of O VI in these objects as clearly shown in far-UV spectra.", "The O VI stronger line at 22.0 Å is also expected to be resolved by RGS due to the smaller extent, and therefore line broadening, of the O VI-VII cool phases.", "The O VII intercombination line is 5.5 weaker than the resonant line and also not expected to significantly affect our results.", "The Fe XVII resonant and forbidden lines are in a crowded spectral region, but they are much stronger than the neighbor lines.", "We have artificially doubled the flux of the brightest neighbor lines, which is more than the statistical uncertainties.", "The Fe XVII (f/r) line ratio was consistent with the standard measurements.", "The uncertainties in the atomic database do not affect our measurements of line ratios, but of course the interpretation of resonant scattering.", "There is a significant ($>20$ %) difference between the Fe XVII (f/r) line ratio as predicted by AtomDB and SPEX.", "This means that we do not know the absolute value of resonant scattering in our sources, which is crucial to estimate the absolute scale of turbulence, but the relative differences between line ratios measured in different objects should not be highly affected." ], [ "Discussion", "In Sect.", "REF we have searched for O VII ($\\sim 2 \\times 10^6$ K) gas in a sample of 24 objects, including clusters and groups of galaxies and elliptical galaxies, with strong ($>5\\sigma $ ) Fe XVII line emission.", "We have detected O VII above the 99% confidence level in 11 sources and shown that O VII is preferably found in the cores of the sources, possibly following the distribution of the Fe XVII ($> 5 \\times 10^6$ K) gas.", "Exceptions are IC 1459 and M 89 where the lower count rate requires to integrate photons over a larger region.", "For M 86, NGC 4636 and NGC 5813 the O VII is also better detected in the wider slit most likely due to their more extended cool cores.", "In order to search for a link between cooling and turbulence, we have plotted the Fe XVII forbidden-to-resonant line ratio with the temperature and the O VII significant detections in Fig.", "REF .", "The high quality data points show some evidence for O VII to be mainly detected in sources with significant resonant scattering, which indicates the low level of turbulence.", "Although our sample is incomplete and the resonant scattering is more sensitive at lower temperatures, our results are consistent with a picture where turbulence is heating the gas and preventing it to cool below $\\sim 0.45$ keV, where O VII line emission begins to be important." ], [ "O ", "At temperatures of 0.2-to-0.6 keV the O VII resonance-to-forbidden line ratio is predicted to be between 1.25–1.35.", "We found (r/f) line ratios lower than 1.25 in the RGS spectra of NGC 4636, M 89, and NGC 1404 as already shown in [20].", "Our values could be due to either suppression of the resonant line via resonant scattering or enhancement of the forbidden line by photoionization or charge exchange.", "The O VII resonance line at 21.6 Å may be subject to resonant scattering.", "At the temperature of $\\sim 0.5$ keV, where the Fe XVII ionic concentration peaks, the O VII is optically thin and no longer self-absorbed along the line-of-sight.", "However, it is possible that the gas is distributed in various non-volume filling phases at different temperatures.", "We multiplied the two cie emission components for a collisionally-ionized absorbing model (hot model in SPEX) and re-fit the RGS spectrum of NGC 4636.", "We obtained a column density of $1.24\\pm 0.30\\times 10^{20}\\,{\\rm cm}^{-2}$ with a temperature of $0.23\\pm 0.03$ keV, which is lower than the $0.43\\pm 0.07$ keV value measured for the cie component responsible for the O VII emission.", "The presence of such cool gas is suggested by the detection of a large amount of H$\\alpha $ emission in the core of NGC 4636 [28].", "It is suspicious, however, that the absorbing gas is cooler than the emitting gas despite the need to be located (on average) in outer regions where higher temperatures are expected unless the cool gas is clumpy.", "The astrophysical processes that strengthen the O VII forbidden line emission are photoionization and charge exchange.", "We can rule out photoionization because no bright AGN is observed in NGC 4636.", "Charge exchange (CX hereafter) occurs when ions interact with neutral atoms or molecules; one or more electrons are transferred to the ion into an excited state, which decays and emits a cascade of photons increasing the forbidden-to-resonance ratios of triplet transitions.", "This process is often observed in supernova remnants (e.g.", "Puppis A, [14]), starburst galaxies (e.g.", "M 82, [16]) and colliding stellar winds (e.g.", "Solar Wind, [27]).", "The CX plasma code recently provided by [12] is implemented in SPEX v3.00.00 (cx model).", "[13] first used this code to successfully describe the unidentified 3.5 keV feature in the lower resolution CCD spectrum of the Perseus cluster.", "We re-fit the NGC 4636 spectrum with a new cie (driven by the Fe XVII-XVIII lines) + cx (mainly, O VII-VIII, Ne X, and Mg XI) model corrected by redshift and Galactic absorption and obtain results comparable to the resonant scattering (hot) model described above.", "In the fit we exclude the $13.8-15.5$ Å spectral range because it contains several Fe XVII lines suppressed by resonant scattering which would lead to a wrong estimate of the temperature.", "In Fig.", "REF we show the best fit with the contribution from the cie and cx components.", "Charge exchange provides a reasonable description of the O VII lines and produces significant O VIII, Ne X, and Mg XI emission and accounts for $\\sim 10\\%$ of the flux in the 0.3–2.0 keV energy band.", "Figure: NGC 4636 RGS first order spectrum with hybrid modelconsisting of isothermal and charge-exchange components.The ionic temperature of the cx component was coupled to the $\\sim 0.7$ keV temperature of the cie component.", "If left free to vary, a better fit provides $T_{\\rm ion}=0.40\\pm 0.05$ keV, in agreement with the 2-cie model, which may suggest that the charge exchange is occurring between neutrals and the cooler O VII phase rather than the hotter gas phase associated with the Fe XVII lines.", "This may indicate that the cool O VII gas is a better tracer of the cold neutral phase and that they could be somewhat cospatial, both distributed in clumps.", "The CX code calculates velocity-dependent rates with which we measure a collision velocity lower than 50 km s$^{-1}$ (at 68% level), in agreement with the low turbulence found in NGC 4636 [29].", "This is the first time that a charge exchange model is successfully applied to a high-resolution X-ray spectrum of a giant elliptical galaxy." ], [ "Resonant scattering in Perseus?", "In Fig.", "REF we have shown that the Perseus cluster has an unexpected, high ($4\\pm 2$ ), Fe XVII (f/r) line ratio.", "The spectrum extracted within a larger region of width $\\sim 0.8^{\\prime }$ (see Fig.", "REF ) holds much smaller error bars and constrains Fe XVII (f/r) $\\ge 4$ .", "This value is higher than that measured in any other object and remarkable if compared to the other clusters (A 262, Centaurus, Fornax and Virgo).", "The inner core of the Perseus cluster is dominated by a hot $\\sim 3$ keV plasma, but it has been clearly shown to be multiphase with the inner arcminute ($\\sim 20$ kpc) having significant emission from 0.5–4 keV [23].", "Below 1 keV and in a low-turbulence regime the 15 Å resonance line is optically thick and it may therefore be subject to resonant scattering in the line-of-sight.", "We have therefore re-fitted the Perseus spectrum multiplying the two thermal components by a collisionally-ionized absorption model (hot model) to test the suppression of the Fe XVII resonant line (as previously done for the O VII lines in NGC 4636 in Sect.", "REF ).", "We have ignored the first order spectra between 10 and 14 Å due to high pile up and use the second order RGS 1 and 2 spectra because they are not significantly affected by pileup and their statistics peak in this wavelength range.", "This model reasonably describes the 15–17 Å Fe XVII lines (see Fig.", "REF ) and provides a column density of $\\sim 2\\times 10^{20}\\,{\\rm cm}^{-2}$ and a temperature of $\\sim 0.6$ keV.", "[10] suggested that high-resolution X-ray spectra enable to search for evidence of ICM absorption onto the AGN continuum in NGC 1275, the brightest cluster galaxy in Perseus, with a focus on the hard X-ray band where Fe K lines dominate.", "We have tested the same approach in the soft RGS band by applying the hot absorption model only to the nucleus, which was fitted with a power law; the two cie emission line components are only absorbed by the Galactic neutral ISM.", "This AGN-only absorption model is statistically indistinguishable to the previous one, with $\\Delta \\chi ^2$ and $\\Delta $ C-stat of 6 for 1948 degrees of freedom, but a column density of $\\sim 1.5\\times 10^{21}\\,{\\rm cm}^{-2}$ is required, in good agreement with the predictions of [10].", "If the suppression of the 15 Å Fe XVII resonant line and the detection of absorption are interpreted as resonant scattering, which is a very likely scenario, then this means that the cool gas in Perseus is characterized by low turbulence.", "Figure: Perseus RGS first order spectrum with multiphase thermal emission modelabsorbed by isothermal gas at 0.6 keV." ], [ "Conclusions", "In this work we have confirmed and extended our previous discovery of O VII emission lines in spectra of elliptical galaxies as well as groups and clusters of galaxies.", "This is the coolest X-ray emitting intracluster gas and seems to be connected to the mild Fe XVII gas, being located preferably at small (1-10 kpc) scales.", "The O VII is often detected in objects with strong resonant scattering of photons in the Fe XVII lines, indicating subsonic turbulence.", "This would be consistent with a scenario where cooling is suppressed by turbulence in agreement with models of AGN feedback, gas sloshing and galactic mergers.", "We note that a larger sample of sources and consequently more observations are needed to better disentangle resonant scattering effects due to temperature and turbulence; the current sample is incomplete.", "In some objects the O VII resonant line is weaker than the forbidden line either due to resonant scattering or to charge-exchange processes occurring in the gas as we have shown for NGC 4636.", "The Perseus cluster shows an anomalous, high, Fe XVII forbidden-to-resonance line ratio, which can be explained with resonant scattering by cool gas in the line-of-sight under a regime of low turbulence.", "In two forthcoming papers (Ogorzalek et al., Pinto et al.)", "we will compare the measurements of Fe XVII line ratios with those predicted by theoretical models of resonant scattering that take into account thermodynamic properties of these objects in order to estimate the turbulence in both their cores and outskirts.", "This will provide further insights onto the link between cooling, turbulence, and the phenomena of AGN feedback, sloshing, and mergers occurring in clusters and groups of galaxies." ], [ "Acknowledgments", "This work is based on observations obtained with XMM-Newton, an ESA science mission funded by ESA Member States and USA (NASA).", "We also acknowledge support from ERC Advanced Grant Feedback 340442 and new data from the awarded XMM-Newton proposal ID 0760870101.", "Y.Y.Z.", "acknowledges support by the German BMWi through the Verbundforschung under grant 50OR1506." ] ]	1606.04954
[ [ "Singular inflation from Born-Infeld-f(R) gravity" ], [ "Abstract Accelerating dynamics from Born-Infeld-$f(R)$ gravity are studied in a simplified conformal approach without matter.", "Explicit unification of inflation with late-time acceleration is realized within this singular inflation approach, which is similar to Odintsov-Oikonomou singular $f(R)$ inflation.", "Our model turns out to be consistent with the latest release of Planck data." ], [ "Introduction", "Various cosmological observations support the current accelerated expansion of our universe.", "To explain this phenomenon it is necessary to assume either the existence of dark energy, which has a negative pressure, or the fact that gravity must be modified.", "Indeed, a quite natural approach to the universe evolution is the description where both the early-time as well as the late-time universe acceleration is achieved by a modification of the standard theory of General Relativity [1]-[6].", "Work in this direction has shown that, indeed, a unified description of the early-time inflation with the late-time dark energy appears rather naturally in modified gravity, as was clearly shown by Nojiri-Odintsov [7] (for further unified models of this sort see Refs. [10]-[14]).", "However, the transition from a decelerating phase to the dark-energy universe is not yet well understood (possibly because there is not yet a clear understanding of what dark energy itself actually is).", "In the recent literature, increasing interest has appeared for theories of gravity formulated in the Palatini scheme, in particular the so-called Born-Infeld theories [8], [9].", "The Palatini formulation brings about a number of restrictions and additional constraints to the metrics under consideration.", "As a consequence, it turns out to be quite difficult to get consistent generalizations of the original Born-Infeld model.", "However, it was recently demonstrated that a non-perturbative and consistent generalization of Born-Infeld gravity is possible, under the form of a new Born-Infeld-$f(R)$ theory, which was introduced in Ref. [15].", "It was shown there that Born-Infeld-$F(R)$ gravity without matter can be easily reconstructed in the conformal approach.", "In this way, eventually any kind of dark energy cosmology could in principle be derived from the above theory.", "In the present letter we will apply the techniques of [16] in order to explicitly show that Born-Infeld-$f(R)$ gravity is able to give rise to a very realistic singular inflation theory accurately matching the most recent Planck data.", "This provides a natural possibility for the unification of singular inflation with dark energy within the theory under discussion here." ], [ "Born-Infeld-$f(R)$ theory in a conformal ansatz", "In what follows we are going to consider the Born-Infeld-$f(R)$ theory in a conformal ansatz (see Refs.", "[15], [16] for details), which can be used with the purpose to discuss a number of relevant situations.", "In particular, we can explicitly work out in detail the unification of the inflation epoch with the late-time acceleration stage by using metrics proposed in [18].", "As already advanced above, in order to enhance the capabilities of the theory, we here propose a modified action of Born-Infeld type but containing an arbitrary function $f(R)$ , where $R=g^{\\mu \\nu } R_{\\mu \\nu }(\\Gamma )$ [15], namely $S=\\frac{2}{\\kappa }\\int d^4x\\left[\\sqrt{\|\\det {\\left(g_{\\mu \\nu }+\\kappa R_{\\mu \\nu }(\\Gamma )\\right)}\|}-\\lambda \\sqrt{\|g\|}\\right]+\\int d^4x\\sqrt{\|g\|}f(R).$ As noted above, matter will be absent from our model, the purpose being to retain the full power of the conformal approach (the massive case, with its particularities, will be the subject of a future investigation).", "Under the conformal approach we understand the situation where the metric $g_{\\mu \\nu }$ and the auxiliary metric (on which the Christoffel symbols are built, both metrics, as is known, being independent in the Palatini formalism) are connected by a transformation having the form of a conformal one ($g_{\\mu \\nu }=\\Omega \\, u_{\\mu \\nu }$ ).", "Varying action (REF ) with respect to the connection, the following equation results $\\nabla _\\alpha \\left[\\sqrt{q}\\left( q^{-1}\\right)^{\\mu \\nu }+\\sqrt{g} g^{\\mu \\nu }f_R\\right]=0 \\ ,$ where $q_{\\mu \\nu }=g_{\\mu \\nu }+\\kappa R_{\\mu \\nu }(\\Gamma )$ and being $f_R\\equiv df/dR$ .", "The corresponding equation which follows by variation over the metric has the form $\\sqrt{q}\\left(q^{-1}\\right)^{\\mu \\nu }-\\lambda \\sqrt{g}g^{\\mu \\nu }+\\frac{\\kappa }{2}\\sqrt{g}g^{\\mu \\nu } f(R)-\\kappa \\sqrt{g}f_R R^{\\mu \\nu }=0.$ Since we work in the conformal approach, it is just sufficient to require the fulfillment of the following condition $q_{\\mu \\nu }=k(t) g_{\\mu \\nu }.$ In this case we have an auxiliary metric, $u_{\\mu \\nu }$ , which defines the covariant derivative and, hence, the Christoffel symbols, as $\\Gamma ^\\alpha _{\\mu \\nu }=\\frac{1}{2} u^{\\alpha \\beta }\\left(\\partial _\\mu u_{\\nu \\beta }+\\partial _\\nu u_{\\mu \\beta }-\\partial _\\beta u_{\\mu \\nu }\\right),$ where $u_{\\mu \\nu }=(k(t)+f_R)g_{\\mu \\nu }.$ From the condition (REF ), together with the definition $q_{\\mu \\nu }$ , it is clear that the Ricci tensor must be also proportional to the metric $g_{\\mu \\nu }$ .", "One can write the relationship between the Ricci tensor and the metric, as $R_{\\mu \\nu }=\\frac{1}{\\kappa }[k(t)-1]g_{\\mu \\nu }.$ Let us now consider the spatially-flat FRW universe, with metric $ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2})\\ .$ The auxiliary metric will be given by the expression (REF ).", "We denote the function connecting the main and the auxiliary metrics by $u(t)=k(t)+f_R$ .", "Assume now that $R_{\\mu \\nu }=r(t) g_{\\mu \\nu }$ , where $r(t)$ is easy to find from Eq.", "(REF ).", "Finally, after all these considerations are taken into account, the equations acquire the following form $r(t)&=&3H^2+\\frac{3H\\dot{u}}{u(t)}+\\frac{3\\dot{u}(t)^2}{4u(t)^2},\\\\2\\dot{H}&=&H\\frac{\\dot{u}(t)}{u(t)}+\\frac{3\\dot{u}(t)^2}{2u(t)^2}-\\frac{\\ddot{u}(t)^2}{u(t)},$ $H$ being the Hubble rate, $H=\\frac{\\dot{a}}{a}$ .", "From these expressions, it follows that $u(t)=c \\,\\,r(t)$ where $c$ is a constant.", "The remaining equations lead to $H=\\pm \\sqrt{\\frac{u}{3c}}-\\frac{\\dot{u}}{2u}.$ From this result (see [15]), the form of the function $f(R)$ can be found explicitly to be given by $f(R)=\\frac{2}{\\kappa }(\\lambda -1)-R+\\frac{c-\\kappa }{8}R^2.$" ], [ "Our model", "In what follows we will go one step further and consider a particular universe with scale factor of the following type [20], [21] $a(t)=\\mathrm {e}^{-[f_0 (-t+t_s)^{1+\\alpha }]/(1+\\alpha )}\\, ,$ where $f_0$ , $t_s$ and $\\alpha $ are constants.", "Interest in this type of metrics has arisen in the literature when considering the so-called singular inflationary cosmologies [19], [20], [21].", "The Hubble parameter takes the form $H=f_0 (-t+t_s)^{\\alpha }$ In order to look now for the relation between the metrics $g_{\\mu \\nu }$ and $u_{\\mu \\nu }$ , it is necessary to find solutions to Eq.", "(REF ), as $u(t)=\\frac{3 (1+\\alpha )^2 c e^{\\frac{2 f_0 (-t+t_s)^{1+\\alpha }}{1+\\alpha }}\\left(-\\frac{f_0(-t+t_s)^{1+\\alpha }}{1+\\alpha }\\right)^{\\frac{2}{1+\\alpha }}}{\\left\\lbrace \\alpha (1+\\alpha )f_0 \\left(-\\frac{f_0 (-t+t_s)^{1+\\alpha }}{1+\\alpha }\\right)^{\\frac{1}{1+\\alpha }}C\\pm (t-t_s){\\Gamma }\\left[\\frac{1}{1+\\alpha },-\\frac{f_0(-t+t_s)^{1+\\alpha }}{1+\\alpha }\\right]\\right\\rbrace ^2}.$ Here $C$ is a constant and $\\Gamma [a,z]$ is the incomplete gamma function.", "The scale factor $a(t)$ (REF ) for the spatially-flat FRW universe (REF ) is a solution of equation (REF ) (obtained by varying the action (REF ) with respect to the connection) provided the conditions (REF ) and (REF ) are imposed.", "In fact, taking into account (REF ) and (REF ), Eq.", "(REF ) takes the form: $&&-1+\\Lambda -u(t)+\\frac{1}{8 (t-t_s) u(t)^2}3 (c-\\kappa ) \\left(4 f_0 (-t+t_s)^{\\alpha } \\left(\\alpha -2 f_0 (-t+t_s)^{1+\\alpha }\\right) u(t)^2+\\right.\\nonumber \\\\&&\\left.+ (-t+t_s) u^{\\prime }(t)^2+2 (t-t_s) u(t) \\left(3 f_0 (-t+t_s)^{\\alpha } u^{\\prime }(t)+u^{\\prime \\prime }(t)\\right)\\right)+\\nonumber \\\\&&+\\left(u(t)-\\frac{1}{8 (t-t_s) u(t)^2}3 (c-\\kappa ) \\left(4 f_0 (-t+t_s)^{\\alpha } \\left(\\alpha -2 f_0 (-t+t_s)^{1+\\alpha }\\right) u(t)^2+\\right.\\right.", "\\nonumber \\\\&&\\left.\\left.+(-t+t_s) u^{\\prime }(t)^2+2 (t-t_s) u(t) \\left(3 f_0 (-t+t_s)^{\\alpha } u^{\\prime }(t)+u^{\\prime \\prime }(t)\\right)\\right)\\right)\\times \\nonumber \\\\&&\\times \\left(-1+\\frac{1}{8 (t-t_s) u(t)^2}3 (c-\\kappa ) \\left(4 f_0 (-t+t_s)^{\\alpha } \\left(\\alpha -2 f_0 (-t+t_s)^{1+\\alpha }\\right) u(t)^2+\\right.\\right.\\nonumber \\\\&&\\left.\\left.+(-t+t_s) u^{\\prime }(t)^2+2 (t-t_s) u(t) \\left(3 f_0 (-t+t_s)^{\\alpha } u^{\\prime }(t)+u^{\\prime \\prime }(t)\\right)\\right)\\right)-\\nonumber \\\\&&-\\frac{1}{2} \\kappa \\left(\\frac{2-2 \\Lambda }{\\kappa }-\\frac{1}{2 (t-t_s) u(t)^2}3 \\left(4 f_0 (-t+t_s)^{\\alpha } \\left(\\alpha -2 f_0 (-t+t_s)^{1+\\alpha }\\right) u(t)^2+\\right.\\right.\\nonumber \\\\&&\\left.+(-t+t_s) u^{\\prime }(t)^2+2 (t-t_s) u(t) \\left(3 f_0 (-t+t_s)^{\\alpha } u^{\\prime }(t)+u^{\\prime \\prime }(t)\\right)\\right)+\\nonumber \\\\&&+\\frac{1}{32 (t-t_s)^2 u(t)^4}9 (c-\\kappa ) \\left(4 f_0 (-t+t_s)^{\\alpha } \\left(\\alpha -2 f_0 (-t+t_s)^{1+\\alpha }\\right) u(t)^2+\\right.\\nonumber \\\\&&\\left.\\left.+(-t+t_s) u^{\\prime }(t)^2+2 (t-t_s) u(t) \\left(3 f_0 (-t+t_s)^{\\alpha } u^{\\prime }(t)+u^{\\prime \\prime }(t)\\right)\\right)^2\\right)$ And if we now substitute in this equation the corresponding expression for the function $u(t)$ (REF ) then it is not difficult to see that we obtain an identity.", "We will now discuss several specific cases in more detail.", "First, consider the situation when $f_0<0$ ($f_0=-1$ ).", "In this case, the evolution of the metric, of the Hubble rate, and of the equation of state (EoS) parameter are given in Figs.", "1-3, respectively.", "Figure: The effective EoS parameter for f 0 =-1f_0=-1, t s =2t_s=2, α=1\\alpha =1For this choice of constants it is not difficult to construct the corresponding interaction function (REF ), Figs. 4-5.", "Figure: u(t) for f 0 =-1f_0=-1, t s =2t_s=2, α=1\\alpha =1, c=1c=1, C=1C=1One can also select alternative values for the parameter $\\alpha $ .", "Thus, for $\\alpha =3$ one gets a different picture, as is clear from Figs. 6-7.", "Figure: u(t) for f 0 =-1f_0=-1, t s =2t_s=2, α=3\\alpha =3, c=1c=1, C=1C=1For even values of the parameter $\\alpha $ we are bound to have half of the solutions only.", "Thus, for $\\alpha =2$ , we end up with Figs. 8-9.", "Figure: u(t) for f 0 =-1f_0=-1, t s =2t_s=2, α=2\\alpha =2, c=1c=1, C=1C=1The opposite situation occurs for positive values of the parameter $f_0$ .", "For example, Figs.", "10-13 depict the the behavior of the metric, the Hubble rate and the EoS parameter, respectively, for $f_0=1$ , $t_s=2$ , and $\\alpha =2$ .", "Figure: The effective EoS parameter for f 0 =1f_0=1, t s =2t_s=2, α=2\\alpha =2In this case, the function connecting the metrics (REF ) only exists for $t>t_s$ or for $t<t_s$ , either.", "Figure: u(t) for f 0 =3f_0=3, t s =2t_s=2, α=2\\alpha =2, c=1c=1, C=0.0001C=0.0001To summarize, we have here demonstrated that for $\\alpha $ bigger than 1, when the so-called Type IV singularity [22] occurs, we get singular inflation of the sort observed in Ref. 10.", "Then, precisely in the same way as in Ref.", "10 one can demonstrate that if a Type IV singularity occurs at the end of inflation it may naturally induce a graceful exit from it (see [23])" ], [ "Conclusions", "After reviewing the non-perturbative, consistent generalization of Born-Infeld gravity, under the form of the so-called Born-Infeld-$f(R)$ theory, introduced recently [15], we have here continued this development by applying the techniques of [16] to explicitly show, with specific examples, that Born-Infeld-$f(R)$ gravity is able to give rise to very interesting universe models and to realize singular inflation.", "For a metric chosen in the form (REF ) and with the same constants of the theory as in Ref.", "[24], we have obtained for the inflationary parameters, calculated in a similar way, the following values: $n_s = 0.96491,\\;\\;\\; r = 0.1403,\\;\\;\\;a_s =-0.000307$ .", "These values of the inflationary parameters are in agreement with the observational bounds for them obtained from the very latest release of Planck data.", "To finish, here the Born-Infeld-$f(R)$ theory has been discussed in a conformal ansatz, without matter, with the purpose to retain the full power of the conformal approach.", "The massive case, with its particularities, will be the subject of a future investigation.", "Acknowledgements.", "E.E.", "was supported in part by MINECO (Spain), Project FIS2013-44881-P, by the CSIC I-LINK1019 Project, and by the CPAN Consolider Ingenio Project.", "A.N.M.", "was supported by a grant of the Russian Ministry of Education and Science." ] ]	1606.05211
[ [ "Vacuum Energy Sequestering and Graviton Loops" ], [ "Abstract We recently formulated a local mechanism of vacuum energy sequester.", "This mechanism automatically removes all matter loop contributions to vacuum energy from the stress energy tensor which sources the curvature.", "Here we adapt the local vacuum energy sequestering mechanism to also cancel all the vacuum energy loops involving virtual gravitons, in addition to the vacuum energy generated by matter fields alone." ], [ " June 2016 Vacuum Energy Sequestering and Graviton Loops Nemanja Kaloper$^{a, }[email protected] and Antonio Padilla$^{b, }[email protected] $^a$ Department of Physics, University of California, Davis, CA 95616, USA $^b$ School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK ABSTRACT We recently formulated a local mechanism of vacuum energy sequester.", "This mechanism automatically removes all matter loop contributions to vacuum energy from the stress energy tensor which sources the curvature.", "Here we adapt the local vacuum energy sequestering mechanism to also cancel all the vacuum energy loops involving virtual gravitons, in addition to the vacuum energy generated by matter fields alone.", "The cosmological constant problem [1], [2], [3], [4], [5], [6] follows from two simple statements: in a local Quantum Field Theory (QFT), off-shell dynamics facilitated by virtual particles renormalizes the Lagrangian, and in General Relativity (GR), the universality of gravitational couplings ensures that all energy gravitates in the same way.", "The vacuum energy induced by virtual particles, scaling as $({\\rm cutoff})^4$ , behaves just like a cosmological constant, curving the geometry of space-time even in vacuum.", "Cosmological observations constrain the vacuum energy density to be less than $(\\textrm {meV})^4$ .", "This is at least 60 orders of magnitude below a naive theoretical estimate based on the possible value of the cutoff of the low energy QFT and the absence of a dynamical cancellation mechanism below that cutoff.", "In QFT coupled to GR, the real cosmological constant problem is more subtle.", "Consider a low energy effective field theory (EFT), defined up to a cutoff $M$ .", "In the absence of symmetries that can enforce cancellationsFor example, supersymmetry and/or exact scale invariance, its vacuum energy loops generically scale as $M^4$ [3], [7], [8].", "This means that the observable that is corrected by the vacuum energy loops – the curvature of the empty space – is UV sensitive, and must be renormalized.", "Its physical value is the sum total of the quantum vacuum contributions and the bare counterterm.", "This is a finite quantity which cannot be predicted, but must be measured.", "When the cutoff-dependent pieces are subtracted, the renormalized cosmological constant is a function of the finite contributions coming from all the physical scales $m$ below the cutoff and the arbitrary subtraction scale $\\bar{M}$ , where it is measured.", "The problem is that the renormalized cosmological constant depends on the powers of the physical scales below the cutoff, implying that it can be greatly affected by any existing heavy field beyond the threshold of local measurements at sub-TeV scales.", "This sensitivity to the scales that govern unknown physics is the problem.", "To reproduce the measured value at any level of perturbation theory we must choose the bare counterterm with great precision in the units of the cutoff.", "This counterterm serves as an avatar for the unknown high scales, and its choice is not robust against modifications of the UV sector of the theory, even though the physics which sets the value of the renormalized cosmological constant is in the far infra-red.", "If we view the bare counterterm as a cosmological initial condition, this shows a great deal of sensitivity to the cosmological initial conditions in the vacuum energy sector.", "These difficulties with the cosmological constant should be contrasted with the mass of the electron in QED.", "The renormalized electron mass depends logarithmically on the cutoff due to the restoration of the electron chiral symmetry in the massless limit [9], [10].", "This means that the electron mass cannot be calculated in QFT, but must also be measured.", "The important point, however, is that the UV sensitivity is only logarithmic, too mild to significantly affect the cancellation between the quantum corrections and the counterterm.", "The quantum corrections are proportional to the symmetry-breaking parameters – the electron mass – so they are never much larger than the renormalized value at any loop order.", "This is how supersymmetry and/or scaling symmetry could stabilize the vacuum energy in principle.", "The problem is that if these symmetries exist, the relevant numerical scales controlling their breaking are much larger than the values required by observation.", "Nevertheless, this argument has been the raison d'être behind the quest for the `dynamical solution' of the cosmological constant's radiative instability, obstructed by Weinberg's venerable no-go theorem [3].", "Following early work [11], [12], [13], [14], we have recently proposed a mechanism for eliminating the contributions to vacuum energy from matter loops, dubbed vacuum energy sequestering [15], [16], [17].", "The procedure is a very conservative, minimalistic modification of the gravitational sector of the theory, deviating from the on-shell behavior of GR only at the global level, in the infinite wavelength limit.", "Exploiting the fact that vacuum energy is the only source of curvature that is spacetime filling (and locally constant), we introduce constraints that operate at the largest wavelengths, along with new gauge symmetries that prohibit any additional local degrees of freedom.", "The constraints ensure that counterterms always cancel the power-sensitive contributions to vacuum energy from matter loops, so that they do not gravitate.", "Gravity is sourced by a renormalized vacuum energy that is radiatively stable, albeit with a value that is incalculable and should be set by measurement (just like the electron mass).", "For additional recent explorations see [18], [19].", "Here we pursue a modification of the mechanism that addresses vacuum energy loops that include virtual gravitons.", "We show that higher dimensional operators can be used as conjugate variables to construct constraints that sequester vacuum energy loops with graviton lines from the gravitational field equations.", "The mechanism can be embedded in the formulation of gravity as a quantum EFT defined up to some scale $M \\lesssim M_{Pl}$ , below which the EFT is unitary and weakly coupled, following the “classic lore\" of [20], [21], along with more recent ideas developed by Donoghue [22].", "To set the stage, we review the local matter vacuum energy sequestering, given by [16] $S = \\int d^4 x \\left\\lbrace \\sqrt{g} \\left[ \\frac{\\kappa ^{2}(x)}{2} R - \\Lambda (x) - {\\cal L}_m( g^{\\mu \\nu } , \\Phi ) \\right] + \\frac{\\epsilon ^{\\mu \\nu \\lambda \\sigma }}{4!}", "\\left[\\sigma \\left(\\frac{ \\Lambda }{ \\mu ^4}\\right){F_{\\mu \\nu \\lambda \\sigma } } +\\hat{\\sigma }\\left(\\frac{ \\kappa ^{2}}{ M_{Pl}^2}\\right){\\hat{F}_{\\mu \\nu \\lambda \\sigma }} \\right] \\right\\rbrace \\, .$ In addition to the metric and the matter fields $\\Phi $ we include a pair of 4-forms $F_{\\mu \\nu \\lambda \\sigma } = 4\\partial _{[\\mu }A_{\\nu \\lambda \\sigma ]}$ and $\\hat{F}_{\\mu \\nu \\lambda \\sigma } = 4\\partial _{[\\mu }\\hat{A}_{\\nu \\lambda \\sigma ]}$ and a pair of scalar fields $\\kappa (x)$ and $\\Lambda (x)$ .", "The gauge symmetries of the 4-forms enforce that these scalars have no fluctuating modes [23].", "The arguments of the smooth functions $\\sigma $ and $\\hat{\\sigma }$ are normalized to the two high energy scales $\\mu $ , $M_{Pl}$ which are close to the EFT cutoff, $M \\lesssim \\mu , M_{Pl}$ .", "The properties of $\\sigma ,\\hat{\\sigma }$ are discussed in [17]; we stress that they cannot be linear functions to avoid any hidden fine tunings in the vacuum energy cancellations.", "The matter sector ${\\cal L}_m$ couples to the metric minimally.", "The last two terms of (REF ) are a non-gravitating, topological sector by virtue of the absence of the metric.", "To get the dynamics, we could now vary (REF ), obtain the local field equations, and then concentrate on the global, infinite wavelength sector to address the curvature (in)sensitivity to vacuum energy loops [16].", "That yields very similar equations to the original global constraints of [15], with the main difference being that the global constraints appear as integrals of the local field equations, zooming in on the vacuum energy [14], [16].", "In particular, the cancellation works in spacetimes with infinite 4-volume and finite field theory scales.", "Since we are mainly interested in the mechanics of cancellation of vacuum energy loops, we can shortcut our analysis by working “in the action\" and focusing on the global sector from the start.", "To this end, we integrate out the 3-forms from (REF ), bearing in mind that they impose the `rigidity' of the scalars $\\kappa $ and $\\Lambda $ , forcing them to be spacetime constants.", "The effective dynamics of the global sector of the theory is now controlled by $S= \\int d^4 x \\sqrt{g} \\left[ \\frac{\\kappa ^{2}}{2} R - \\Lambda - {\\cal L}_m( g^{\\mu \\nu } , \\Phi ) \\right] +\\sigma \\left(\\frac{ \\Lambda }{ \\mu ^4}\\right)c+\\hat{\\sigma }\\left(\\frac{ \\kappa ^{2}}{ M_{Pl}^2}\\right)\\hat{c} \\, .$ Here $c$ and $\\hat{c}$ respectively describe the flux of 3-forms, $A$ and $\\hat{A}$ , with the additional constraints obtained by varying with respect to the rigid scalars $\\kappa ^2$ and $\\Lambda $ .", "The field equations are $\\kappa ^{2} G^\\mu {}_\\nu = T^\\mu {}_\\nu -\\Lambda \\delta ^\\mu {}_\\nu \\, , ~~~~~~\\frac{\\sigma ^{\\prime }}{\\mu ^4} c = \\int \\sqrt{g} d^4 x, ~~~~~~~\\frac{\\hat{\\sigma }^{\\prime }}{M_{Pl}^2}\\hat{c} =-\\frac{1}{2} \\int R \\sqrt{g} d^4 x \\, .", "$ Tracing the gravity equation and averaging over spacetime fixes $\\Lambda $ in terms of $\\langle T^\\alpha {}_\\alpha \\rangle $ and $\\langle R \\rangle $ , where $T^\\alpha {}_\\alpha = g^{\\mu \\nu } T_{\\mu \\nu }$ is the regularized trace of the matter stress energy tensor at a given loop order and angled brackets denote the spacetime average.", "This gives $\\Lambda = \\langle T^\\alpha {}_\\alpha \\rangle /4 + \\Delta \\Lambda $ where $\\Delta \\Lambda = \\kappa ^{2} \\langle R \\rangle /4=-\\frac{\\mu ^4}{2} \\frac{\\kappa ^{2} \\hat{\\sigma }^{\\prime }}{M_{Pl}^2 \\sigma ^{\\prime }}\\frac{\\hat{c}}{c} $ , the last equality following from the ratio of the global equations in (REF ).", "Inserting this expression into the gravity equation yields $\\kappa ^{2} G^\\mu {}_\\nu = T^\\mu {}_\\nu - \\frac{1}{4} \\delta ^\\mu {}_\\nu \\langle T^\\alpha {}_\\alpha \\rangle - \\Delta \\Lambda \\delta ^\\mu {}_\\nu \\, .$ This equation shows that vacuum energy $ \\langle vac \|T^\\mu {}_\\nu \| vac \\rangle = -\\delta ^\\mu {}_\\nu V_{vac}$ completely drops out of the gravitational dynamics.", "Matter radiative corrections do affect the finite renormalized cosmological constant $\\Delta \\Lambda $ in (REF ) since $\\Lambda \\rightarrow \\Lambda +{\\cal O}( M^4)$ , and $\\kappa ^2 \\rightarrow \\kappa ^2+{\\cal O}(M^2)$ [24].", "However, when $\\sigma $ and $\\hat{\\sigma }$ are smooth (ie $\\sigma ({\\cal O}(1) z ) \\sim {\\cal O}(1)\\sigma (z)$ , etc) and non-degenerate functions [16], [17], the renormalized cosmological constant is radiatively stable since $M \\lesssim \\mu , M_{Pl}$ .", "The vacuum energy cancellation described above is enforced by two approximate symmetries of the theory [15].", "The first is the shift symmetry ${\\cal L}_m \\rightarrow {\\cal L}_m+\\nu ^4$ , $\\Lambda \\rightarrow \\Lambda -\\nu ^4$ , where $\\nu $ is a constant, which is broken by the topological terms, yet restored in the limit $c/\\mu ^4 \\rightarrow 0$ .", "The second approximate symmetry is the scaling symmetry of [15], which in terms of the variables we employ here dwells in the $\\kappa ^2$ sector.", "To see it consider metric and $\\kappa ^2$ fluctuations about a flat background in the limit of vanishing cosmological constant.", "Setting $\\kappa ^2=M_{Pl}^2(1+ {\\phi }/{M_{Pl}})$ and $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+{h_{\\mu \\nu }}/{M_{Pl}}$ , the theory is invariant under $\\phi \\rightarrow \\phi +\\hat{\\nu }$ in the limit $\\hat{c}/M_{Pl} \\rightarrow 0$ , $M_{Pl} \\rightarrow \\infty $ .", "Clearly this symmetry is broken at finite $M_{Pl}$ .", "While the breaking is weak for the matter sector loops, it implies that graviton loops will not be cancelled in the theory (REF ).", "To see this explicitly consider the vacuum energy renormalization of the 1PI effective action by loops that involve both matter and gravitons.", "For simplicity, we compute them in a locally Lorentzian frame, treating the background geometry as flat.", "This correctly captures all the UV contributions.", "Expanding in the gravitational coupling, the result is $-\\left[ a_0 M^4 +a_1 \\frac{M^6}{\\kappa ^2} +a_2\\frac{ M^8}{\\kappa ^4}+\\ldots \\right] \\int \\sqrt{g} d^4 x \\, ,$ where $a_i \\sim {\\cal O}(1)$ .", "The terms $\\sim M^4$ are the contributions from matter vacuum energy loops.", "Pure gravity loop diagrams also contribute in the same way.", "They are automatically sequestered away from sourcing curvature in the theory (REF ).", "The terms that go as powers of $1/\\kappa ^2$ contain graviton interactions.", "The $\\sim M^6/\\kappa ^2$ terms can arise from diagramsHere we also include the pure gravity loops, although eg.", "they vanish around flat space in dimensional regularization since they are given by scaleless integrals [20], [7], [8].", "However, in principle they are still sensitive to the details of the UV and IR regulator, as is seen from mixed matter-gravity loops involving heavy fields and/or in curved backgrounds.", "such as $\\begin{fmffile}{grav2}{\\begin{array}{c}\\begin{fmfgraph}(75,75){i}{o}{phantom,tension=5}{i,v1}{phantom,tension=5}{v2,o}{plain,left,tension=0.4}{v1,v2,v1}{photon}{v1,v2}\\end{fmfgraph}\\end{array}}\\end{fmffile}\\textrm {and}\\begin{fmffile}{grav4}{\\begin{array}{c}\\begin{fmfgraph}(75,75){i}{o}{phantom,tension=5}{i,v1}{phantom,tension=5}{v2,o}{photon,left,tension=0.4}{v1,v2,v1}{photon}{v1,v2}\\end{fmfgraph}\\end{array}}\\end{fmffile}\\nonumber $ where the solid lines denote matter propagators, and the wiggly lines are gravitons.", "For $\\kappa \\sim M_{Pl}$ , and a cutoff as low as $M \\sim $ TeV, the numerical value of their regularized contributions is already thirty orders of magnitude above the dark energy scale.", "The contributions $\\sim M^8/\\kappa ^4$ come from diagrams with either more, or higher order, graviton interactions, such as $\\begin{fmffile}{grav3}{\\begin{array}{c}\\begin{fmfgraph}(75,75){i}{o}{phantom,tension=10}{i,v1}{phantom,tension=10}{v2,o}{plain,left,tension=0.4}{v1,v2,v1}{photon,left=0.5}{v1,v2}{photon,right=0.5}{v1,v2}\\end{fmfgraph}\\end{array}}\\end{fmffile}\\textrm {and}\\begin{fmffile}{grav5}{\\begin{array}{c}\\begin{fmfgraph}(75,75){i}{o}{phantom,tension=10}{i,v1}{phantom,tension=10}{v2,o}{photon,left,tension=0.4}{v1,v2,v1}{photon,left=0.5}{v1,v2}{photon,right=0.5}{v1,v2}\\end{fmfgraph}\\end{array}}\\end{fmffile}\\nonumber $ Curiously, if the cutoff is as low as TeV their regularized values would be at the dark energy scale or below.", "However, for higher cutoffs they are dangerously large.", "In any case, the $\\kappa ^2$ -dependent corrections to vacuum energy will not be sequestered in the theory (REF ).", "They scale differently with the cutoff, spoiling the cancellation implied by (the trace of) Einstein equations and the geometric constraint found by varying (REF ) with respect to the rigid scalar $\\kappa $ .", "Indeed, calculating the vacuum field equations we find that $&&~~~~~~~~~~~~~~~~ \\kappa ^{2} G^\\mu {}_\\nu = -\\left(\\Lambda + a_0 M^4 +a_1 \\frac{M^6}{\\kappa ^2} +a_2\\frac{ M^8}{\\kappa ^4}+\\ldots \\right)\\delta ^\\mu {}_\\nu \\, , \\nonumber \\\\&&\\frac{\\sigma ^{\\prime }}{\\mu ^4} c = \\int \\sqrt{g} d^4 x, ~~~~~~~~~~\\frac{\\hat{\\sigma }^{\\prime }}{M_{Pl}^2}\\hat{c} =-\\frac{1}{2} \\int \\left(R +2a_1 \\frac{M^6}{\\kappa ^4} +4a_2\\frac{ M^8}{\\kappa ^6}+\\ldots \\right) \\sqrt{g} d^4 x \\, , $ Tracing, averaging over the spacetime, and eliminating constraints yields $\\kappa ^{2} G^\\mu {}_\\nu = - \\left[\\Delta \\Lambda -a_1 \\frac{M^6}{2\\kappa ^2} -a_2\\frac{ M^8}{\\kappa ^4}+\\ldots \\right] \\delta ^\\mu {}_\\nu \\, .$ with $\\Delta \\Lambda = \\kappa ^{2} \\langle R \\rangle /4=-\\frac{\\mu ^4}{2} \\frac{\\kappa ^{2} \\hat{\\sigma }^{\\prime }}{M_{Pl}^2 \\sigma ^{\\prime }}\\frac{\\hat{c}}{c} $ as before.", "The regularized vacuum energy terms independent of $\\kappa $ cancel out, and the only residual dependence remains through the radiatively stable finite term $\\Delta \\Lambda $ .", "In contrast, the $\\kappa $ dependent pieces do not cancel.", "Clearly, the largest contributions come from terms $\\sim \\frac{M^6}{2\\kappa ^2}$ , but others are in principle dangerous too, requiring some additional mechanism to keep them under control.", "The main purpose of this Letter is to point out that such a mechanism can be obtained by a straightforward modification of the local sequestering theory (REF ).", "The logic behind local sequestering was threefold: (i) promote the gravitational parameters $\\kappa ^2, \\Lambda $ into local fields; (ii) project out their local fluctuations by the gauge symmetries of the 4-forms; (iii) retain variational equations with respect to the rigid fields $\\kappa ^2,\\Lambda $ because they fix the counterterms and divert radiative instabilities away from the metric and into the physically unobservable sector of local 4-form fluctuations.", "The key condition arises from the variation with respect to $\\kappa $ , whose global limit is the condition that the spacetime average of the scalar curvature is fixed by a radiatively stable quantity $\\Delta \\Lambda $ , controlled by the ratio of the 4-form fluxes.", "By Einstein's equations, the cutoff-dominated terms in $\\Lambda - \\langle T^\\alpha {}_\\alpha \\rangle /4$ automatically cancel.", "One can immediately verify that a qualitatively similar condition would follow from vanishing of the spacetime average of any generic curvature invariant not constructed purely out of the Weyl tensor and the traceless part of the Ricci tensor.", "Any such invariant would not be scale invariant and would therefore involve the Ricci scalar.", "By the vacuum Einstein's equation, it follows that this curvature invariant would be polynomial in the difference $\\Lambda - \\langle T^\\alpha {}_\\alpha \\rangle /4$ .", "Fixing it on shell by a variational principle to a radiatively stable quantity would yield a behavior similar to that which follows from (REF ).", "Furthermore, if the variational constraint is not directly dependent on the Planck scale, the cancellation of Planck-mass dependent vacuum energy contributions – namely, those involving graviton virtual lines – will also cancel from the residual stress energy tensor to leading order.", "Although we can construct many suitable curvature invariants in four dimensions, one immediately arises as a most minimal candidate: the Gauss-Bonnet invariant, $R_{GB}=R_{\\mu \\nu \\alpha \\beta }^2-4R_{\\mu \\nu }^2+R^2$ .", "Because it is a total derivative, adding it to the action merely changes the topological sector, and does not affect any local phenomena at finite wavelength.", "Since it involves the Ricci scalar, and so is not scale invariant, it yields a desired constraint that picks the correct counterterms to sequester all large contributions from loops from the source of the Einstein's equations.", "Let us demonstrate this explicitly.", "We start with the action $S= \\int d^4 x \\left\\lbrace \\sqrt{g} \\left[ \\frac{M_{Pl}^2}{2} R+\\theta (x)R_{GB} - \\Lambda (x) - {\\cal L}_m \\right] +\\frac{\\epsilon ^{\\mu \\nu \\lambda \\sigma }}{4!}", "\\left[\\sigma \\left(\\frac{ \\Lambda }{ \\mu ^4}\\right){F_{\\mu \\nu \\lambda \\sigma } } +\\hat{\\sigma }\\left(\\theta \\right){\\hat{F}_{\\mu \\nu \\lambda \\sigma }} \\right] \\right\\rbrace \\, .$ where we now vary over the auxiliary scalar $\\theta (x)$ controlling the Gauss-Bonnet coupling.", "Again for simplicity, we focus only on the global limit of the theory, and integrate out the 3-forms as before, yielding an effective action $S= \\int d^4 x \\sqrt{g} \\left[ \\frac{M_{Pl}^2}{2} R+\\theta R_{GB} - \\Lambda - {\\cal L}_m( g^{\\mu \\nu } , \\Phi ) \\right] +\\sigma \\left(\\frac{ \\Lambda }{ \\mu ^4}\\right)c+\\hat{\\sigma }\\left(\\theta \\right)\\hat{c} \\, .$ The two scalars are now rigid, with no local variations off-shell, and $c$ and $\\hat{c}$ are the fluxes of the 3-forms through the boundary.", "The resulting field equations are $M_{Pl}^{2} G^\\mu {}_\\nu = T^\\mu {}_\\nu -\\Lambda \\delta ^\\mu {}_\\nu \\, , ~~~~~~\\frac{\\sigma ^{\\prime }}{\\mu ^4} c = \\int \\sqrt{g} d^4 x, ~~~~~~\\hat{\\sigma }^{\\prime } \\hat{c} =- \\int R_{GB} \\sqrt{g} d^4 x \\, .", "$ We can write $R_{GB}=W_{\\mu \\nu \\alpha \\beta }^2-2\\left(R_{\\mu \\nu }-\\frac{1}{4} R g_{\\mu \\nu }\\right)^2+\\frac{1}{6} R^2$ , where $W_{\\mu \\nu \\alpha \\beta }$ is the Weyl tensor, whose contribution drops out in the vacuum.", "Again, taking traces, averages, and eliminating constraints, we find $M_{Pl}^{2} G^\\mu {}_\\nu = T^\\mu {}_\\nu - \\frac{1}{4} \\delta ^\\mu {}_\\nu \\langle T^\\alpha {}_\\alpha \\rangle - \\Delta \\Lambda \\delta ^\\mu {}_\\nu \\, ,$ where now $\\Delta \\Lambda $ satisfies $\\Delta \\Lambda ^2=\\frac{3 M_{Pl}^4 }{8} \\left[\\langle R_{GB} \\rangle -\\langle W_{\\mu \\nu \\alpha \\beta }^2 \\rangle +\\frac{2}{M_{Pl}^4} \\langle (T_{\\mu \\nu }-\\frac{1}{4} T g_{\\mu \\nu })^2\\rangle \\right.\\left.", "-\\frac{1}{6 M_{Pl}^4}\\left(\\langle T^2 \\rangle -\\langle T\\rangle ^2 \\right)\\right] \\, .$ The spacetime average of the Gauss-Bonnet invariant is constrained by the ratio of the global equations, such that $\\langle R_{GB} \\rangle =-\\mu ^4\\frac{ \\hat{\\sigma }^{\\prime }}{ \\sigma ^{\\prime }}\\frac{\\hat{c}}{c} \\, .$ Of course, we stress that the full system of equations from (REF ) is more complicated.", "However the global limit is faithfully reproduced by the equations obtained from (REF ).", "As before, the regularized vacuum energy, $\\langle vac \|T^\\mu {}_\\nu \| vac \\rangle = -\\delta ^\\mu {}_\\nu V_{vac}$ , completely drops out of the first two terms in equation (REF ).", "By its scale invariance, the Weyl tensor contribution vanishes, and so radiative corrections can only affect $\\Delta \\Lambda $ through its dependence on $\\langle R_{GB} \\rangle $ .", "These corrections yield $\\Lambda \\rightarrow \\Lambda +{\\cal O}(M^4)$ , and $\\theta \\rightarrow \\theta +{\\cal O} (1)\\ln ({M}/{m})$ , where $m$ is a typical mass scale in the EFT [24].", "Therefore, by the same line of reasoning given after Eq.", "(REF ), we see that the corresponding source of curvature is radiatively stable.", "This cancellation now occurs for both the matter vacuum energy contributions, and for the vacuum energy loops involving virtual gravitons.", "The fact that Gauss-Bonnet is a topological invariant ensures that graviton loops cannot introduce any additional $\\theta $ dependence in the off-shell effective action.", "One might worry about generating extra dependence on the rigid scalars from background curvature effects and the IR corrections they induce, which have so far been neglected.", "However these corrections are suppressed by the background curvature scale, and one expects them to be harmless.", "Other radiatively induced curvature corrections, obtained by renormalizing (REF ) will likewise remain subleading below the cutoff $M \\lesssim M_{Pl}$ .", "The reason behind the improved behavior of the global sector (REF ) of the theory (REF ) over the global sector of the theory (REF ) is that the second approximate shift symmetry now involves $\\theta \\rightarrow \\theta + \\alpha $ , which is only broken by the topological terms in the theory.", "The bulk terms remain invariant even at finite $M_{Pl}$ .", "This means that as $\\hat{c} \\rightarrow 0$ , but keeping $M_{Pl}$ finite, the symmetry is completely restored.", "This improved approximate shift symmetry, which was absent in our previous set-up at finite $M_{Pl}$ , yields the cancellation of the cutoff dominated vacuum energy contributions that include virtual gravitons.", "The fact that the symmetry is broken only by the topological terms prevents the generation of potential terms in $\\theta $ that would otherwise spoil the vacuum energy sequester.", "To summarize, in this Letter we have adapted the vacuum energy sequestering mechanism to yield a cancellation of all cutoff-dominated vacuum energy contributions, computed in the loop expansion using arbitrary bubble diagrams.", "We include all such diagrams, with or without virtual gravitons, treating gravity as an effective field theory with a cutoff below the Planck scale.", "The generalization utilizes the Gauss-Bonnet topological invariant, yielding a better approximate shift symmetry that remains unbroken in the bulk even at finite $M_{Pl}$ .", "This generalization should be viewed as a particular example of what may well be a considerably broader class of models.", "The guideline may be that in formulating the low energy effects of quantum gravity, one may need to promote all the UV sensitive `couplings' in the theory to independent fields, which are “stiffened\" by their mixing with the non-gravitating 4-form sectors.", "The gauge symmetries of the 4-forms render the local fluctuations of the new fields unphysical.", "In turn the rigid fields divert the vacuum energy contributions dominated by the cutoff into the local part of the 4-form fields, which do not gravitate.", "In our original set-up this logic applied to the bare cosmological constant and the Planck mass.", "Here we used the bare cosmological constant and the Gauss-Bonnet coupling.", "We can imagine combining the two frameworks into one, by writing the action $S &=& \\int d^4 x \\sqrt{g} \\left[ \\frac{\\kappa ^{2}(x)}{2} R - \\Lambda (x) - {\\cal L}_m( g^{\\mu \\nu } , \\Phi ) +\\theta (x) R_{GB} +\\ldots \\right] \\nonumber \\\\&& +\\int \\frac{dx^{\\mu }dx^{\\nu } \\ldots }{4!}", "\\left[\\sigma _1\\left(\\frac{ \\Lambda }{ \\mu ^4}\\right){F^{(1)}_{\\mu \\nu \\lambda \\sigma } }+\\sigma _2\\left(\\frac{ \\kappa ^{2}}{ M_{Pl}^2}\\right){{F}^{(2)}_{\\mu \\nu \\lambda \\sigma }} +\\sigma _3\\left(\\theta \\right){F^{(3)}_{\\mu \\nu \\lambda \\sigma } } +\\ldots ~~~ \\right] \\,$ Here instead of two constraint equations we would have three of them.", "For non-degenerate functions $\\sigma _i$ there would be no fine tunings in the theory, since the three constraints would fix three independent quantities $\\kappa ^2, \\Lambda , \\theta $ in terms of the three 4-form fluxes.", "Vacuum loops of both matter and gravitons should still be sequestered in this theory, as long as all the $\\sigma $ 's satisfy the generic smoothness condition, $\\sigma \\left({\\cal O}(1)z\\right)\\sim {\\cal O}(1)\\sigma (z)$ .", "The reason is that loop corrections are still guaranteed to be independent of $\\theta $ , preserving the effectiveness of the geometric constraint arising from $\\theta $ variation.", "From a symmetry perspective, we see that in the limit of vanishing 4-form flux, the action is invariant under constant shifts in $\\theta $ , even at finite values of the other couplings.", "This symmetry enhancement protects the observable part of the cosmological constant from large corrections.", "We note that the theory (REF ) shares some features with the frameworks introduced in the past for a phenomenologically motivated attempt to resolve cosmological singularities [25].", "We believe that it would be interesting to explore cosmological and phenomenological properties of this and similar theories.", "Acknowledgments: We would like to thank T. Hamill, D. Stefanyszyn and G. Zahariade for useful discussions.", "N.K.", "is supported in part by the DOE Grant DE-SC0009999.", "AP was funded by a Royal Society URF." ] ]	1606.04958
[ [ "Leadership Network and Team Performance in Interactive Contests" ], [ "Abstract Over the years, the concept of leadership has experienced a paradigm shift - from solitary leader (centralized leadership) to de-centralized leadership or distributed leadership.", "This paper explores the idea that centralized leadership, as earlier suggested, negatively impacts team performance.", "I applied the hypothesis to cricket, a sport in which leaders play an important role in team's success.", "I generated batting partnership network and evaluated the central-most player in the team, applying tools of social network analysis.", "Analyzing 3420 matches in one day international cricket and 1979 Test matches involving 10 teams, I examined the impact of centralized leadership in outcome of a contest.", "I observed that the odds for winning a one day international match under centralized leadership is 30% higher than the odds for winning under de-centralized leadership.", "In both forms of cricket (Test and one day international ), I failed to find evidence that distributed leadership is associated with higher team performance.", "These results suggest important implications for cricket administrators in development and management of working teams." ], [ "Introduction", "There exists a corpus of work about the benefits of working in teams, a trend which is gaining importance.", "In academia, it has been shown that works with highest scientific impact have been produced by teams[1], [2].", "Team coordination is also prized in sports [3], [4], [5], [6] and military [7], [8], where team members coordinate with each other for a common objective of being more successful than the opponent.", "A recent survey conducted on high-level managers concluded that teams are central to organizational success[9].", "The effect of leadership on team performance has been a topic of interest for a long time.", "Previous works on leadership have dealt with role of leadership in coaching related activities[10], [11] or managing events in context of teams[12].", "Some works have also focussed on how leadership is shared in teams[13], [14], [15].", "However, earlier body of work on effect of leadership on team performance was conducted at the level of survey analysis and narrow set of leadership activities[16], [17].", "One of the major drawback of such studies is that team performances were assessed in a subjective manner in which team leaders rated the performance of their own teams.", "An earlier work has shown that team leaders tend to over-rate team performance, since a team's performance reflects the ability of the leader [18].", "The decisive role of leaders in team's performance has been a long debated topic[19], [20].", "Prior works focussed on the paradigm of leader-centeredness, in which the leadership is viewed as a top-down process between the leader and the followers[21].", "Recent works have also focussed on the idea of shared leadership or distributed leadership in which other team members emerge as leaders [22].", "An earlier meta-analysis of 37 studies of teams in natural contexts discus how the network position of team leaders influences team performance[23].", "It was observed that teams with stronger interpersonal ties are more successful and teams with leaders who are central in the intra-group networks display better performance [23].", "One of the main limitations of the earlier studies is that they are restricted to cross-sectional data, primarily due to the limited availability of longitudinal data.", "To overcome the limitations of previous works, I employed the treasury of data available in sports [24], [25], [26], [27] and objectively investigate the association between leadership structure and team performance in interactive contests.", "I applied the social network analysis approach to diagnose the role and qualities of a leader effectively.", "Leadership involving team activities is a relational construct.", "Again, social network analysis emphasizes on the relationship of social actors and subsequently elucidates the patterns and theories of such relationships [28].", "Network analysis has been applied to explore the significance of structure of various relationship in organizations [29], [30].", "Social network approach to leadership demonstrated how would-be leaders perfectly perceives the relationship among team members in various organizations [31].", "Social network analysis provides an understanding of the dynamics of centralized leadership and distributed leadership[22], [32].", "Here, I quantified the extent to which leadership potentials are associated with games won across all teams in the history of cricket.", "Even though cricket is the second most popular game in the world after soccer, compared to other professional sports it has been relatively understudied by academics, although there is no dearth of match statistics.", "Cricket is chosen for the following reasons.", "First, cricket is a game in which an outcome depends a lot on the leadership.", "Compared with other sports the role of a captain is elevated in cricket.", "A cricket captain's direct involvement in the proceedings of a game can be viewed as team-leadership in the corporate world, leadership in politics, social capital [33] or organizational communication tactics [34].", "The captain chooses the batting order, sets up fielding positions and shoulders the responsibility of on-field decision-making and is also responsible at all times for ensuring that play is conducted within the Spirit of the Game as well as within the Laws.", "However, a coach in soccer or manager in baseball takes decisions off the field, which includes player substitution or deciding batting line-up.", "In cricket, the role of a captain is not restricted to off-the-field decisions but also to deliver winning performance for the team while playing [35].", "It is to be noted that in cricket, there is no substitution unlike Soccer or Basketball, where a player is substituted by the coach.", "To quote Sir Don Bradman “A captain must make every decision before he knows what its effect will be, and he must carry the full responsibility, not whether his decision will be right or wrong, but whether it brings success\" [36] .", "In cricket, the captains are appointed based on their performance and position in the team (often the role is given to batsmen).", "One of the key role performed by the captain is leading by example [35], a quality that is gaining importance in business domains [19], [20].", "The captain is expected to win the match for his team, commonly referred by fans and commentators as `captain's knock'.", "Legendary players like Sir Don Bradman, Richie Benaud or Sir Gary Sobers, were great performers and inspired their team through their own performance $-$ example of centralized leadership.", "Even though in cricket there are always formally appointed captains, the emergence of leaders has been seen in many games.", "These emergent leaders were responsible for leading their team to victories.", "While captains like Mike Brearley or Ray Illingworth were not the best players in their side but were known to extract maximum performance from their players.", "Again, Sir Gary Sobers and Sachin Tendulkar were best players in their sides, they were not successful captains.", "In an earlier study it was shown that Steve Waugh was the most successful captain in the history of Test cricket ($1877-2010$ )[37].", "Again, presence of legendary performers like Adam Gilchrist, Shane Warne, Glenn McGrath and Ricky Ponting in Steve Waugh's Australian team, leads to the well debated topic whether distributed leadership is more successful than centralized leadership.", "Secondly, in a team game like cricket, one can objectively assess the role of leader-position in the network and team performance.", "Motivated by the above observations I set to explore the role of leaders in a team game like cricket and the impact of leadership structure on the outcome of a match.", "I analyzed the data of batting partnership (publicly available in cricinfo website [40]) in Test cricket between 1877 and 2013 and also one day international cricket between 1971 and 2013.", "Cricinfo has recorded the information for all 3420 one day international matches played between 1971 and 2013 and all 1979 Test matches played between 1877 and 2013.", "For every match I recorded and analyzed the score-cards which contain the information of match outcome, amount of runs scored by a pair of batsmen and run-rate of each team after the game is over.", "In order to control for team talent, I also collected the information about the International cricket Council (ICC) points awarded to every player each year as well as the batting average of every player (including the captain) in a year.", "Data are available upon request The author will share the data in an online repository post publication of the manuscript.", "To articulate the social network analysis approach of studying the pattern of leadership in cricket, I first outline the methodology of identifying the leadership style between two competing teams.", "Next I discuss the nature of leadership networks and finally discuss the effect of centralized and distributed leadership on the outcome of a game.", "In cricket two batsmen always bat in partnership, although only one is on strike at any time.", "The partnership of two batsmen comes to an end when one of them is dismissed or at the end of an innings.", "Fig REF demonstrates the formation of batting partnership network.", "Two opening batsmen $a$ and $b$ start the innings for their team.", "In network terminology, this can be visualized as a network with two nodes $a$ and $b$ , the link representing the partnership between the two players.", "Weight of the link reflects the amount of runs scored in partnership.", "Now, if batsman $a$ is dismissed by a bowler, then a new batsman $c$ arrives to form a new partnership with batsman $b$ .", "Thus a new node $c$ gets linked with node $b$ .", "Subsequently one can generate an entire network of batting-partnership till the end of an innings.", "The innings comes to an end when 10 players are dismissed or when the duration of play comes to an end[38].", "The score of a team is the sum of all the runs scored during a batting partnership.", "The outcome of a match depends on the batting partnerships between batsmen.", "Long lasting partnerships not only add runs on the team's score, it may also serve to exhaust the tactics of the fielding team.", "Again, the concept of partnerships become vital if only one recognized batsman remains.", "It is therefore important to identify the key players in a team by constructing network of batting partners.", "Two batsmen are connected if they formed a batting partnership in the match.", "An undirected and weighted batting partnership network is generated for each team and for every match played through 2013.", "I examined two network metrics which captures the position of a captain in the batting partnership network in cricket.", "A similar approach using network metrics to capture team performance and strategies has been used in basketball [3] and soccer [39]." ], [ " Centralized leadership and de-centralized leadership", "To quantify the centrality of a captain, I evaluated the betweenness centrality of players in the batting partnership network.", "The betweenness centrality is defined as $C^{w}_{B}(i) = \\frac{g^{w}_{jl}(i)}{g^{w}_{jl}}$ Where $w$ is the weight of the link between two nodes $j$ and $l$ , $g_{jl}$ is the number of shortest paths between two nodes and $g_{jl}(i)$ is the number of shortest paths that pass through node $i$ [41], [42].", "Betweenness centrality measures the extent to which one batsman is between two other batsmen who are not connected to each other.", "In other words, betweenness centrality measures how the run scoring by a player during a batting partnership depends on another player.", "Batsmen with high betweenness centrality are crucial for the team for scoring runs without losing his wicket.", "These batsmen are important because their dismissal has a huge impact on the structure of the network[43].", "So a single player with a high betweenness centrality is also a weakness, since the entire team is vulnerable to the loss of his wicket.", "In an ideal case, every captain would seek a combination of players where betweenness scores are uniformly distributed among players.", "Hence betweenness centrality is a measure of dependence on other team members[43].", "Centralized leadership refers to the post-match situation when captain is the player with highest betweenness centrality, else it is an example of emergent de-centralized leadership." ], [ " Distributed leadership", "To predict a continuos measure of leadership structure I measure the network de-centralization proposed by Mayo et al [28].", "The variance of centrality is given by the equation $\\omega = \\frac{\\sum _{i=1}^N (k_{max} - k(i))}{(N-1)(N-2)}$ Where $N$ is the number of players in the batting partnership network and $\\omega $ is the variance of centrality of the network and $k$ is the degree of the node, with $k_{max}$ being the maximum degree.", "Here degree of a node refers to the number of batting partners of a player.", "Variance of centrality ($\\omega $ ) captures the aspects of batting performance of players under the leadership of different captains.", "As mentioned earlier the captain takes the bulk of the decision in forming the batting line-up.", "A good leader will always allot effective batting positions such that the team is benefited the most.", "In this sense the $\\omega $ highlights the extent to which the leadership is distributed.", "The index $\\omega $ varies from 0 to 1.", "A value of 0 indicates that the leadership is distributed equally among the individuals and a value of 1 indicates that the team is centralized around an individual, not necessarily the captain." ], [ "Normal approximation method of the Binomial confidence interval", "The equation for Normal approximation method[44] to evaluate 95% Binomial confidence intervals is given as, $CI = p~ \\pm ~1.96~\\sqrt{\\frac{}{}}{p(1-p)}{M}$ Where $p$ is the proportion of interest and $M$ is the number of matches played.", "For example if out of $M$ matches, $m$ matches results in win under centralized leadership, then the 95% Binomial confidence intervals fall between $\\frac{m}{M} - 1.96~\\sqrt{\\frac{}{}}{\\frac{m}{M}(1-\\frac{m}{M})}{M}$ and $\\frac{m}{M} + 1.96~\\sqrt{\\frac{}{}}{\\frac{m}{M}(1-\\frac{m}{M})}{M}$ ." ], [ " Regression Methods and Control variables", "Below I report detailed results for predicting the probability of win using logistic regression analysis.", "The dependent variable is a binary indicator for win$-$ loss.", "The explanatory variable of interest is an indicator variable of centrality of the captain, as defined above.", "The control variables are as follows:" ], [ " Batting average of captain", "Batting average of a player is defined as the amount of runs scored divided by the number of times the player is dismissed.", "Batting average of a captain serves as an indicator of his ability and skills as a batsman irrespective of external factors like match situation or strength of opposition.", "In the logistic regression analysis, I include a dummy variable which takes the value 1 if the captain's batting average is higher than the median batting average of the team; 0 otherwise." ], [ " Talent of captain", "While the batting average of a player serves as a good metric of a batsman's ability, it is independent of match situations, quality of bowling attacks, or strength of opponents.", "The ICC player rankings provides a sophisticated ranking of batsmen based on amount of runs scored, quality of opposition, winning performance for the team.", "Players are rated on a point scale of 0 to 1000, more points being granted if the opponent is stronger or the player's performance results in team's win.", "The ICC points of the captain is an indicator of his batting talent.", "In the logistic regression analysis, I introduce a dummy variable which takes the value 1 if the captain's ICC points is higher than the median ICC points of the team; 0 otherwise." ], [ " Fixed effects", "The logistic regression includes a full set of team fixed-effects, fixed effects of each year the match was played and fixed effects of batting position of the captain during the match.", "The logistic regression takes the form, $logit(W_{i}) = \\beta _{1}~C(i) + \\beta _{2}~S_{b}(i) + \\beta _{3}~S_{p}(i) + {\\sum _{t=1}^m \\gamma _{t}}~Team_{ti} + {\\sum _{y} \\gamma _{y}}~Year_{yi} + {\\sum _{p} \\gamma _{p}}~Pos_{pi}\\,$ Where $C(i)$ is an indicator variable for centrality of captain, $W(i)$ is an indicator variable of win-loss by a captain.", "The indicator variable $S_{b}$ ($S_{p}$ ) takes a value 1 if the batting average (ICC points) of the captain is above the median batting average (median ICC points) of the team; 0 otherwise.", "The logistic regression includes a full set of fixed effects for each of teams, where the indicator variables $[(Team)]_{ti}$ $\\in $ {0,1} are equal to 1 if the match $i$ involves team $t$ .", "I also controlled for fixed effect of the year the match was played - indicator variables $[(Year)]_{yi}$ $\\in $ {0,1} are equal to 1 if the match $i$ was played in the year $y$ and fixed effect of the batting position of the captain - $[(Pos)]_{pi}$ $\\in $ {0,1} are equal to 1 if the batting position is $p$ in match $i$." ], [ "Distributed leadership model", "The dependent variable is the difference of run-rates of a team, defined as the ratio of number of runs scored by a team to the total number of overs played by the team $-$ if a team scored 140 runs in 20 overs, run-rate is 7.", "The explanatory variable is the difference of variance of centrality of two teams.", "The control variables include difference in batting average of captains as well as difference in team talent of competing teams.", "Team talent is measured as the coefficient of variation of ICC points of every player in the team.", "The linear model takes the form, $\\delta ~r_{12}(i) = A_0 + A_1~\\delta ~\\omega _{12}(i) + A_2~\\delta ~C^{v}_{12}(i) + A_3~\\delta ~B^{Avg}_{12}(i) + A_4~{\\sum _{g=1}^m \\gamma _{g}}~Ground_{gi} + A_5~{\\sum _{y} \\gamma _{y}}~Year_{yi} \\,$ Where the dependent variable $\\delta ~r_{12}$ is the difference of run-rate of the two competing teams ($r_{1} - r_{2}$ ), $\\delta ~\\omega _{12}$ is the difference of degree centralities of the two teams ($\\omega _{1} - \\omega _{2}$ ), $\\delta ~C^{v}_{12}$ is the difference in coefficient of variation of team talent (ICC points) of the teams and $\\delta ~B^{Avg}_{12}$ is the difference in batting average of the captains leading the two teams.", "The subscripts 1 and 2 indicates the first and second batting innings.", "The regression includes a full set of fixed effects for each games, where the indicator variables $[(Ground)]_{gi}$ $\\in $ ${(0,1)}$ are equal to 1 if the game $i$ was played in ground $g$ ." ], [ " Centralized leadership and team performance", "I empirically investigated whether team performance is negatively associated with centralized leadership.", "If the captain emerge as the player with highest betweenness centrality, then the team is under centralized leadership.", "On the other hand, in many cases, other players may emerge as most central player $-$ an example of emerging de-centralized leadership.", "In Fig REF I compared the batting partnership network of two competing teams $-$ England and Australia during the first Test match in Ashes $2013/2014$ .", "The Australian captain M. J. Clarke is not the central-most player, whereas for England, captain A. N. Cook emerged as the player with highest betweenness centrality.", "The first Test match resulted in a defeat for England $-$ an example where centralized leadership is negatively associated with team performance.", "I analyzed whether the first Test match in Ashes $2013/2014$ is a special case and whether leadership structure has any relationship with team performance in cricket matches.", "For each game the betweenness centrality of captains in the batting partnership network is evaluated.", "An indicator variable $C_{i}$ is introduced for centrality of captain $i$ , which takes a value 1 if the captain emerges as the player with highest betweenness centrality and 0 otherwise.", "In Test cricket, there are three possible outcomes $-$ win, loss or draw.", "The success or failure of a team is directly related to the success or failure of the team's captain.", "In order to assess a team's success, I assigned a score of 0 if the team loses, a score of 1 if the match is drawn (indecisive) and score of 2 if team wins the match.", "In the entire history of Test cricket there has been only two games which resulted in tie.", "Here I assigned the score of 1 for tied matches.", "In one day international cricket there are three possible outcomes $-$ win, loss or tie.", "As mentioned earlier a score of 0 was assigned if the team loses, a score of 1 if the match was tied and score of 2 if the team won the match.", "The average of all the scores for each team is an indicator of the performance of the captain.", "Fig REF shows that in Test cricket, the leadership structure doesn't have a significant effect on team performance.", "Interestingly, in one day international cricket a significant advantage is seen for centralized leadership ($p < 0.001$ ).", "Team performance can be assessed if the captain is the most central player ($C(i)=1$ ) or captain is not the most central player ($C(i)=0$ ) at the end of a game.", "I hypothesize that the centrality of the captain has a stronger effect on the outcome of a game compared to the batting average and talent of the captain.", "To assess the robustness of the association between leadership structure and the team performance, the relationship is quantified with a logistic regression of the form $logit(W_{i}) = \\beta _{1}~C(i) + \\beta _{2}~S_{b}(i) + \\beta _{3}~S_{p}(i) + {\\sum _{t=1}^m \\gamma _{t}}~Team_{ti} + {\\sum _{y} \\gamma _{y}}~Year_{yi}$ (Materials and Methods).", "As summarized in Table REF , I observe that in one day international cricket, the probability of winning depends positively and significantly on centrality $C$ of the captain ($p<1\\times 10^{-16}$ ), negatively depends on the batting average as well as captain's talent.", "For the latter variable, no significant relationship is observed with the outcome of a match.", "The odds of winning a one day international match for centralized leadership ($C=1$ ) over the odds of winning for de-centralized leadership ($C=0$ ) is $\\exp ~(0.262)$ $=1.299$ .", "In other words, the odds for centralized leadership is 30% higher than the odds for de-centralized leadership.", "The same logistic analysis reveals that in Test cricket, captain's centrality, captain's batting average, and captain's talent are statistically unrelated to the performance of the team ($p > 0.05$ )." ], [ " Binomial confidence interval", "In an alternate approach, I implemented a non-parametric test to evaluate the confidence of the results.", "I estimated the number of wins when the captain is the player with highest centrality as well as number of wins when the captain is not the player with highest centrality and evaluate the Binomial Confidence Interval (BCI) using the Normal Approximation Method[44].", "In one day international cricket, out of 891 matches, 459 resulted in win when the captain is the player with highest centrality, with the $95\\%$ BCI falling between $48.2\\%$ and $54.8\\%$ .", "Considering the one day international matches where the captain is not the player with highest centrality, I observe that the $95\\%$ BCI falls between $43.8\\%$ and $46.6\\%$ (2293 wins out of 5067 matches), strengthening the hypothesis that centralized leadership is more successful than de-centralized leadership.", "However, in Test cricket I don't observe any significant difference between success under centralized leadership or de-centralized leadership - the $95\\%$ BCI falling between $61.5\\%$ and $69.8\\%$ for centralized leadership (338 wins and draws out of 514 matches) and between $65.6\\%$ and $68.7\\%$ for de-centralized leadership (2360 wins and draws out of 3512 matches)." ], [ " Distributed leadership and team performance", "So far I described centralized leadership and de-centralized leadership based on the betweenness centrality of the captain in batting partnership network.", "In emergent de-centralized leadership, a player other than the formally appointed leader emerge as the most central player and takes the responsibility for winning performance for the team.", "In such situations, team members other than the captain also score high as well.", "There also exists situations where different team members rotate to take leadership responsibilities - an example of distributed leadership, a team level construction [45].", "As defined in Equation 2, I utilized the variance of centrality as an indicator for distributed leadership.", "Next, I investigated the impact of leadership structure on team performance.", "The team performance is judged by the run-rate of the team.", "A higher run-rate indicates a superior performance of the team.", "In a team game, performance of the team depends on the team talent [50].", "As considered earlier, the ICC points for every player in a team serves as a quantifier for overall team talent (Materials and Methods).", "I collected the ICC points for every player in a team per year and evaluate the coefficient of variation of the ICC points.", "If I hypothesize that distributed leadership positively affect the outcome of a game, then it would follow that the difference of variance of centrality of two teams is negatively associated with the difference of run-rate of teams.", "Controlling for the team talent and batting average of the captain, the relationship between difference in team run-rate and difference in variance of centrality is quantified by a linear model mentioned in Equation 5 (Materials and Methods).", "The linear model indicates that in one day international cricket team talent ($\\delta ~C^{v}_{12}$ ) and captain batting average ($B^{Avg}_{12}$ ) have no significant explanatory power for $\\delta ~r_{12}$ .", "Consistent with the conjecture, there exists a positive and significant relationship between $\\delta ~r_{12}$ and variance of centrality ($\\delta ~\\omega _{12}$ ) in one day international cricket (Table REF ).", "Together, they account for about $10\\%$ of the observed variance in the data.", "No significant association was observed between $\\delta ~r_{12}$ and $\\delta ~\\omega _{12}$ in Test cricket ($p=0.883$ ), although team talent plays a significant effect on the result ($p<1\\times 10^{-16}$ ).", "In Test cricket, variance of centrality, team talent and captain batting average account for only $4\\%$ of the observed variance in the data.", "Figure REF shows the standardized coefficients for one day international and Test cricket and compares the relative effects of the variables $-$ $\\delta ~\\omega _{12}$ , $\\delta ~B^{Avg}_{12}$ and $\\delta ~C^{v}_{12}$ .", "In one day international matches the relative effect of the difference of variance of centrality is significantly higher than the difference of batting average of captain and difference of coefficient of variation of ICC points of team $-$ a one standard deviation increase in $\\delta ~\\omega _{12}$ yields a $0.269$ standard deviation increase in the predicted difference of team run-rate.", "In Test cricket, difference in coefficient of variation of ICC points (team talent) has significantly higher effect than leadership structure or difference in batting average of captains of two teams $-$ one standard deviation decrease in $\\delta ~C^{v}_{12}$ results in $0.097$ standard deviation increase in $\\delta ~r_{12}$ ." ], [ "Discussion", "Contribution of this paper is of practical importance in research involving leadership perception in teams.", "While effect of leadership on team performance has long been analyzed under the premises of survey analysis, an extensive empirical evidence was lacking.", "Contrary to the example discussed in Fig REF, at least in the context of Test cricket, my results extensively demonstrates that there is no evidence to suggest distributed leadership as well as de-centralized leadership is associated with better team performance in competitions involving small teams in cricket matches.", "The magnitude of the coefficients enable us to infer that, in one day international cricket, centralized leadership shows positive effect on team performance, which can be justified by the fact that one day international cricket is more competitive and result oriented than Test cricket, where there are situations where teams attempt to draw a match and the outcome remains indecisive.", "These findings perhaps indicate that depending on the level of competitiveness, centralized leadership is positively related with team performance.", "While this positive correlation cannot establish a causal relationship, it nonetheless suggests strongly the positive relation between centralized leadership and team performance.", "Contrary to my findings, an earlier study on 28 field-based sales teams which showed that certain types of distributed leadership (distributed-fragmented leadership) are positively related to team performance [22].", "My results confirm the earlier findings of link between centralized leadership and greater team performance, as observed in the meta-analysis of study of 37 teams in natural contexts [23].", "Beyond cricket this approach could be extended to serve as template for analyzing other small team collaborations.", "It would be interesting to conduct similar research on other professional domains like basketball and soccer in which the most central player is identified by the ball passing networks among players.", "One of the potential limitation of the current work involves the process of captain selection which is an endogenous process.", "A captain is assigned by a selection committee to maximize the chances of winning.", "Currently, I am unable to deal with this crucial endogeneity due to lack of available information about the selection process.", "Nevertheless, these findings leave a lot of potential for future research.", "For example, one of the key aspects of leadership is experience.", "It has been shown in earlier works that on average basketball teams with coaches early in their careers benefit relatively more from timeouts than teams with high-experienced coaches[46].", "Previous research has also shown that in mathematics, mentors early in their careers can have a stronger positive impact on protégés than later in their careers [47].", "It remains to be seen if there exists a link between experience and leadership in sports teams, academia or business.", "While the effect of centralized or distributed leadership in academia or business cannot be demonstrated, preliminary results provide insights into the structure of leadership and performance of cricket teams.", "It remains an open challenge to extrapolate these findings to domains of scientific and managerial impact.", "Finally there are few more aspects which deserve closer scrutiny.", "The relationship between gender diversity and decision making hasn't been explored effectively.", "While an earlier survey of executives observed that there exists a direct connection between gender diversity and business success[48], the connections between success, leadership structure and gender diversity still remains an unexplored challenge.", "Performance in academia or sports and leadership abilities depending on gender remains an open idea for further research.", "Again, teams are prone to conflict between team members.", "A recent study has empirically attempted to predict future conflict in team-members[49].", "Leadership will play a critical role in diagnosing and resolving potential conflicts and confrontations among team members." ], [ "Limitations", "This paper is limited to the situation where captains led by example, specifically identifying whether the most central player in the batting partnership network is captain of the team.", "Batting partnership network provides a visual summary of how the batting is centralized around an individual batsman or distributed around multiple batsmen.", "Since the captain is responsible in selecting an effective batting order, batting partnership network reflects a limited set of roles performed by the captain.", "There are various other roles which are performed by the captain $-$ inspiring and motivating the players they lead, communicating effectively with the coach and selectors, remaining positive in match situations and devising strategies accordingly.", "The work is also limited by the fact that I have considered static network of batting partnerships, when the innings come to an end.", "This limitation is due to the available data in Cricinfo.", "A more robust analysis is possible by considering the dynamic version of the batting partnership network similar to the idea of “dynamic exchange networks\" [51], in which the players change as players exchange their batting-striking position.", "Traditionally, 90% of the captains have been specialist batsmen.", "The present study deals with captains who are part of batting partnership network.", "However, captains like Courtney Walsh of West Indies, who are not specialist batsmen, would not end up being the most central player solely due to his low batting order and poor batting abilities.", "Future analysis involves new techniques to analyze the effect of captains who are specialist bowlers.", "Also, the work is limited to the structural approach to leadership and doesn't explore concepts like work environment under leadership structure.", "Whether members in centralized-leadership network experience higher levels of conflict than members of distributed-leadership network is a matter worthy of future investigations." ] ]	1606.05248
[ [ "Colorful vortex intersections in SU(2) lattice gauge theory and their\n influencs on chiral properties" ], [ "Abstract We introduce topological non-trivial colorful regions around intersection points of two perpendicular vortex pairs and investigate their influence on topological charge density and eigenmodes of the Dirac operator.", "With increasing distance between the vortices the eigenvalues of the lowest modes decrease.", "We show that the maxima and minima of the chiral densities of the low modes follow mainly the distributions of the topological charge densities.", "The topological non-trivial color structures lead in some low modes to distinct peaks in the chiral densities.", "The other low modes reflect the topological charge densities of the intersection points." ], [ "Introduction", "Non-perturbative QCD is dominated by the phenomena of color confinement and spontaneous chiral symmetry breaking ($\\chi $ SB).", "The non-perturbative nature of these phenomena is caused by the non-triviality of the QCD vacuum.", "There is still intensive discussion about the nature of the quantum fluctuations responsible for this non-triviality.", "The idea that the QCD vacuum is dominated by center vortices [1], [2], [3], [4], [5], [6], by quantized magnetic flux tubes, is strongly supported by lattice simulations [7], [8], [9], [10], [11], [12] and infrared models [13], [14], [15], [16], [17], [18], [19].", "Via the Stokes theorem vortices piercing Wilson loops modify the trace of these loops by center elements.", "A non-vanishing density of center vortices causes an exponential decay of Wilson loops proportional to the minimal area of the loops and a confinement potential getting linear in the infrared.", "Vortex removal destroys confinement and restores chiral symmetry [20], [21].", "This observation encourages the conclusion that vortices are responsible for $\\chi $ SB, but it does not yet explain how vortices may lead to this phenomenon.", "According to the Banks-Casher relation [22], the density of near-zero modes of the Dirac operator is proportional to the value of the chiral condensate, the order parameter of $\\chi $ SB.", "The Atiyah-Singer index theorem [23], [24], [25] relates the difference in the numbers of zero-modes of positive and negative chirality to the topological charge of gauge field configurations.", "A well established theory of $\\chi $ SB relies on instantons [26], [27], [28], [29], which are localized in space-time and carry a topological charge of modulus one.", "A zero mode of the Dirac operator arises, which is concentrated at the instanton core.", "The instanton liquid model [30], [31], [32] provides a mechanism how overlapping would-be zero modes split into low-lying nonzero modes which create a chiral condensate.", "But instantons are minima of the action and therefore only of indirect importance in the ensemble of gauge field configurations as far as they increase the number of local minima of the action and all lumps of topological charge can finally be deformed via cooling or smoothing procedures to instantons.", "Since smooth deformations of field configurations do not change the homotopy class, these observations indicate that all types of contributions to the topological charge may influence the density of near-zero modes via their interaction.", "Lattice simulations have shown that center vortices contribute to the topological charge via writhing, vortex intersections [33], [34], [35], [36], [37], [38], [39], [40] and their color structure [41], [42], [43], [44].", "Vortices lead also to spontaneous $\\chi $ SB [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56].", "As Engelhardt and Reinhardt [45], [33] have indicated in SU(2) gauge theory in addition to the location of the vortex surfaces, one needs their orientation to determine contributions of vortices to the topological charge.", "This relates to the common vortex identification method on the lattice, resulting in P-vortices [7], thin connected surfaces consisting of plaquettes projected to center elements.", "Regions of different orientations are separated by lines, attributed to the worldlines of magnetic monopoles in maximal abelian gauge [57].", "In a Monte-Carlo ensemble of field configuration vortices are neither thin nor do they have a given orientation, they have a certain average thickness and colors of the full gauge group.", "At present the determination of the color structure of vortices is impossible.", "Therefore, the only possibility to study the influence of the color structure of thick vortices on the topological charge density and low lying eigenmodes of the Dirac operator is by studies of artificial gauge configurations.", "In recent articles the importance of colorful vortices has been underlined [44], [39].", "The contributions of intersections with such colorful vortices has not been investigated yet and is at the focus of this article.", "We restrict our investigation to SU(2) lattice gauge theory.", "On periodic lattices plane vortices have to appear in parallel or anti-parallel pairs.", "We arrange two such pairs appropriately to get intersections in four points, as discussed in Ref.", "[52] where the intersections of uni-color vortices, of vortices in a U(1)-subgroup are investigated.", "In this paper we modify such a configuration and follow the suggestion of Ref.", "[39] and make one of the plane vortices colorful, this means the links of this colorful vortex are distributed over the full SU(2) gauge group.", "After a gauge transformation it gets obvious that a colorful vortex is a vacuum to vacuum transition along a direction perpendicular to the vortex.", "In Sect.", "we describe the gluon field configurations investigated in this article: two perpendicular plain anti-parallel vortex pairs intersecting in four points, where one of the vortices is colorful around one of the intersections.", "We investigate the influence of the position of the colorful region on the topological charge $Q$ in Sect. .", "There we check also the details of the topological charge density and give an interpretation of its behavior.", "In Sect.", "we analyze the low lying modes of the overlap Dirac operator [58], [59], [60], [61] in the background of the considered vortex configurations.", "We study the influence of the distances of the vortex pairs on the lowest eigenvalues of the overlap Dirac operator.", "We determine the chiral densities of zero modes and near-zero modes and compare them with excited modes.", "We find pronounced peaks and regions with oscillating chiral density and try to find correlations of chiral and topological charge densities." ], [ "Uni-color and Colorful SU(2) plane vortices ", "The configurations which we want to investigate in SU(2) lattice gauge theory are thick plane vortices [41], [37] extending along two coordinate axes, thickness in a third coordinate direction and formulated with non-trivial links in the forth direction.", "One of these vortices will get a special color structure and will be smoothed in the forth direction.", "On periodic lattices due to their quantization with the non-trivial center element, plane vortices have to occur in pairs.", "We use two different arrangements of vortex sheets, xy-vortices formulated with t-links in a given t-slice $t_\\perp $ changing in $z$ -direction and zt-vortices with non-trivial y-links at the y-slice $y_\\perp $ varying in $x$ -direction.", "For uni-color vortices the nontrivial links are elements in a U(1) subgroup of SU(2), usually the $\\sigma _3$ -subgroup with links of the form $U_\\mu (x)=\\exp \\lbrace \\mathrm {i}\\alpha (x)\\sigma _3\\rbrace .$ To such U(1)-vortices we can assign an orientation given by the gradient of the angle $\\alpha $ .", "We treat pairs of anti-parallel plane vortices where in a region of $2d$ given by the vortex thickness the angle $\\alpha $ increases in the first vortex linearly from 0 to $\\pi $ and decreases in the next vortex linearly from $\\pi $ to 0.", "We define the profile function $\\alpha (z)$ for a pair of xy-vortices as $\\alpha (z)={\\left\\lbrace \\begin{array}{ll}0&0<z\\le z_1-d,\\\\\\frac{\\pi }{2d}[z-(z_1-d)]&z_1-d< z\\le z_1+d,\\\\\\pi &z_1+d<z\\le z_2-d,\\\\\\pi \\left[1-\\frac{z-(z_2-d)}{2d}\\right]&z_2-d<z\\le z_2+d,\\\\0&z_2+d<z\\le N_z.\\end{array}\\right.", "}$ A diagram for such a profile function is shown in Fig.", "REF a, see also ref [37].", "An analog profile we use for $\\alpha (x)$ for a pair of anti-parallel zt-vortices centered around $x_1$ and $x_2$ .", "The two vortex pairs intersect in the $y_\\perp ,t_\\perp $ -plane at four points with the coordinates $x_1,x_2$ and $z_1,z_2$ .", "The intersection of such vortex pairs is displayed in Fig.", "REF b.", "There is a two-dimensional manifold of U(1)-subgroups of SU(2).", "These subgroups can be characterized by a unit vector $\\vec{n}$ in the SU(2) group element $\\exp \\lbrace \\mathrm {i}\\alpha \\vec{n}\\vec{\\sigma }\\rbrace $ defining a unit sphere S$^2$ in $\\mathbb {R}^3$ .", "Mapping this S$^2$ to a two-dimensional vortex plane leads to a colorful vortex as introduced in Ref. [39].", "We map this S$^2$ to the time-like links of a circular region with radius $R$ around $x_0,y_0$ of the xy-vortex at $z_1,t_\\perp $ by $\\begin{aligned}&\\vec{n}\\vec{\\sigma }=\\sigma _1\\,\\sin \\theta (\\rho )\\cos \\phi (x,y)+\\sigma _2\\,\\sin \\theta (\\rho )\\sin \\phi (x,y)+\\sigma _3\\,\\cos \\theta (\\rho ),\\\\&\\rho =\\sqrt{(x-x_0)^2+(y-y_0)^2},\\\\&\\theta (\\rho )=\\pi (1-\\frac{\\rho }{R})H(R-\\rho )\\;\\in \\,[0,\\pi ],\\quad \\phi =\\arctan _2\\frac{y-y_0}{x-x_0}\\;\\in \\,[0,2\\pi ),\\end{aligned}$ where $H$ is the Heaviside step function.", "The color structure of such a vortex is displayed in Fig.", "REF c. As discussed in Refs.", "[44], [39] colorful vortices defined by links in one time-slice of the lattice do not contribute to $\\vec{E}_a\\vec{B}_a$ and have vanishing gluonic topological charge.", "The sum over the index $a$ runs over the 3 directions $\\sigma _a$ of the SU(2) color algebra.", "The vanishing of this contribution is a lattice artifact due to the singularity of the vortex in time direction.", "By a gauge transformation rotating the non-trivial time-like links to unit matrices it gets obvious that a colorful plane vortex defines a transition between vacua of different winding number.", "After this gauge transformation it is possible to smooth the vortex in time direction without creating a singularity of the gauge field [44], [39].", "Increasing the smoothing region $\\Delta t$ of the colorful vortex described in Eq.", "(REF ) the gluonic topological charge approaches $Q=-1$ , as will be also described in Sect. .", "Figure: a) The profile function α(z)\\alpha (z) of an anti-parallel xyxy vortex pair with t-links varying in z-direction.", "The arrows indicate t-links rotating in zz direction.", "The centers of the thick vortices (dashed lines) are located at z 1 z_1 and z 2 z_2.", "In the shaded areas the links have positive, otherwise negative trace.", "b) Three-dimensional detail of two pairs of intersecting vortices in 4D.", "The horizontal planes represent xy-vortices and the vertical lines a t-slice of zt-vortices.", "The two pairs intersect in four points of a yt-plane.", "c) Diagram of a colorful xy-vortex.", "The color direction n →\\vec{n} of t-links is displayed in the xy-plane for R=1R=1 and x 0 =y 0 =0x_0=y_0=0 by maps to RGB-colors, ±i ^↦\\pm \\hat{i}\\mapsto green, ±j ^↦\\pm \\hat{j}\\mapsto red and ±k ^↦\\pm \\hat{k}\\mapsto blue.After defining uni-color and colorful vortices we are ready to compare the topological charge contributions of colorful intersections with those of uni-color intersections." ], [ "Topological Charges and their Densities", "According to the definition $Q=-\\frac{1}{32\\pi ^2}\\int d^4x\\,\\epsilon _{\\mu \\nu \\rho \\sigma }\\,\\mbox{tr}[{\\mathcal {F}}_{\\mu \\nu }{\\mathcal {F}}_{\\rho \\sigma }]=\\frac{1}{4\\pi ^2}\\int d^4x\\,\\vec{E}_a\\cdot \\vec{B}_a$ only regions with common presence of electric and magnetic fields of same spatial directions and colors $a$ contribute to the topological charge $Q$ .", "The configurations introduced in Sect.", ", two perpendicular plain anti-parallel vortex pairs, contribute at vortex intersections and in colorful regions.", "We intersect an anti-parallel $xy$ - with an anti-parallel zt-vortex pair, as shown in Fig.", "REF b.", "For uni-color vortices each intersection point gives rise to a lump of topological charge $Q=\\pm 1/2$ [62].", "For anti-parallel vortex pairs two of the intersection points carry topological charge $Q=+1/2$ while the other two intersection points have $Q=-1/2$ [37].", "They sum up to total topological charge $Q=0$ .", "It is interesting to investigate the modification of these contributions when one of the uni-color vortices is substituted by a colorful vortex, with the colorful region centered at one of the intersection points.", "The continuum action $S$ for a colorful region with radius $R$ and smoothing region $\\Delta t$ is calculated as [39] $\\frac{S(\\Delta t)}{S_\\mathrm {Inst}}=\\frac{0.51\\,\\Delta t}{R}+\\frac{1.37\\,R}{\\Delta t}$ where the instanton action $S_\\mathrm {Inst}=8\\pi ^2/g^2$ .", "Its minimum value is reached around $R=\\Delta t$ with 1.68 $S_\\textrm {Inst}$ .", "Figure: a) The total topological charge of the vortex configurations corresponding to Fig.", "a and b.", "In the left (right) diagram the colorful region in the xy-plane is centered at x 0 =x 1 (x 0 =x 2 )x_0=x_1\\;(x_0=x_2) and y 0 =y ⊥ y_0=y_\\perp .", "Increasing the radius RR of the colorful region and increasing the lattice size the total topological charge converges to Q=-2Q=-2 for the configuration of Fig.", "a and to Q=0Q=0 for Fig.", "b.The total topological charge of the configuration with the colorful region around $(x_1,z_1)$ is shown in Fig.", "REF a and with the colorful region around $(x_2,z_1)$ in Fig.", "REF b for $\\Delta t=R$ for various values of $R$ and increasing lattice sizes.", "The values of the topological charge approach the values $Q=$ -2 and 0.", "To explain these asymptotic values we display in Fig.", "REF schematic diagrams for the intersection planes of these configurations.", "Figure: The geometry, field strength and the contribution to the topological charge of the intersection regions in the intersection plane of two anti-parallel vortex pairs.", "In diagram (a) the red line at the intersection point (x 1 ,z 1 )(x_1,z_1) indicates that the uni-color vortex is in this region substituted by a colorful region.", "In the right diagram (b) the region around (x 2 ,z 1 )(x_2,z_1) is substituted.These diagrams show that by the insertion of the colorful regions the topological charge contributions at the intersection points change sign leading to sums of their four contributions of $\\mp 1$ .", "Such a sign change corresponds to the change of orientation of the corresponding uni-color vortex within a circle of radius $R$ , by the insertion of a circular monopole line around the intersection point.", "But in the case of a colorful vortex it is a monopole line which changes its color along the circle in a non-trivial way, such that this line contributes itself with a value of -1 to the total topological charge $Q=\\mp 1-1$ of the two investigated configurations.", "To check the details of the contribution to the topological charge we show in Figs.", "REF a and REF b characteristic charge densities of an anti-parallel xy-vortex pair at ($z_1=6,z_2=13$ ) with an anti-parallel $zt$ -pair at ($x_1=6,x_2=13$ ) at $t_\\perp =y_\\perp =6$ with thickness $d=3$ and $\\Delta t=R=4$ on a $16^4$ -lattice.", "In both diagrams, the center of the colorful region with radius $R = 4$ in the $xy$ plane is located at $x_0=x_1=6,\\;y_0=y_\\perp =6$ .", "In Fig.", "REF a the topological charge density of the vortex configuration is plotted in the intersection plane, the xz-plane with $y=y_\\perp ,t=t_\\perp $ , where we can identify the positive and negative contributions indicated in Fig.", "REF a.", "In Fig.", "REF b we show the perpendicular yt-plane at the colorful intersection with the coordinates $x_1=z_1=6$ .", "The broad shallower structure of the colorful vortex extends in y-direction with radius $R$ and in t-direction with $\\pm \\Delta t/2$ around $t_\\perp $ .", "There is a further contribution to the topological charge density from the intersection region of the xy-vortex with the zt-vortex.", "It is narrow in y since the zt-vortex is constructed from non-trivial y-links in one y-slice at $y_\\perp $ only.", "Figure: The topological charge density in two characteristic planes for two intersecting anti-parallel xy- and zt-vortex pairs with (z 1 =6,z 2 =13z_1=6,z_2=13) and (x 1 =6,x 2 =13x_1=6,x_2=13) at t ⊥ =y ⊥ =6t_\\perp =y_\\perp =6 with thickness d=3d=3 and Δt=R=4\\Delta t=R=4 on a 16 4 16^4-lattice is displayed in the diagrams a) and b) for the situation of Fig.", "a, where the center of the colorful region of the xy-vortex surrounds the intersection at lower xx and lower zz, x 0 =x 1 x_0=x_1 and z 0 =z 1 z_0=z_1.", "Diagram a) shows the topological charge density in the intersection plane y ⊥ =t ⊥ =6y_\\perp =t_\\perp =6 with the four intersection regions.", "The area around the intersection point x 1 =z 1 =6x_1=z_1=6 is influenced by the surrounding color structure shown in the perpendicular yt-plane in diagram b) as a broad, shallow depression.", "The intersection region at x 1 =z 1 =6x_1=z_1=6 forms a narrow, deep structure since the zt-vortex is defined by y-links in one y-slice only.", "For the situation of Fig.", "b the two contributions at the intersection point x 0 =x 2 x_0=x_2 and z 0 =z 1 z_0=z_1, a broad structure from the colorful region and a narrow one from the intersection, can be better distinguished in diagram c), due the difference in their signs.", "Diagram c) displays the topological charge density in the yt-plane at the intersection point (x 2 ,z 1 )(x_2,z_1) for the situation of Fig.", "b, where now this other intersection is surrounded by the colorful region.When the colorful region is shifted from $x_1$ to $x_2$ as schematically depicted in Fig.", "REF b the total topological charge gets $Q=0$ .", "Since the orientation of the xy-vortex is now flipped at the intersection point $(x_2,z_1)$ , the contribution of this intersection changes sign, $-\\frac{1}{2}\\rightarrow +\\frac{1}{2}$ , as indicated in Fig.", "REF b.", "Since the colorful region contributes again with $-1$ this results in the total topological charge $Q=0$ .", "It is not difficult to imagine the analog of Fig.", "REF a after this shift of the colorful region.", "Therefore, it is not shown by a diagram.", "But it may be instructive to see the analog of Fig.", "REF b.", "It is displayed in Fig.", "REF c. The shallower negative contribution reflects the contribution of the colorful vortex and the ridge in t-direction originates in the contribution of the vortex intersection which in this case is positive.", "Eqs.", "(REF ) and (REF ) can also be used to insert a colorful region with topological charge contribution of $+1$ instead of $-1$ which was discussed above.", "From Eq.", "(REF ) we can read that the gradient of $\\alpha $ has opposite $z$ -direction at $z_1$ and $z_2$ .", "Thus, a shift of the center coordinate $z_0$ from $z_1$ to $z_2$ flips the sign of this gradient, but it leaves the $xy$ -structure (REF ) untouched and leads therefore to a sign change of the topological charge of the colorful region.", "Combined with the contributions of the intersections, one of them modified by a surrounding colorful region, we get total topological charges $Q=0$ for $(x_0=x_1,z_0=z_2)$ and $Q=2$ for $(x_0=x_2,z_0=z_2)$ .", "By symmetry considerations we can easily imply the consequences on the topological charge densities and also on eigenvalues of the Dirac operator and chiral densities which are discussed in the next section." ], [ "Dirac Eigenmodes and Chiral Densities", "In the previous section, we defined two colorful configurations which are combinations of two anti-parallel plane vortex pairs.", "Now, we investigate the effect of these configurations on fermions $\\psi $ by determining the low-lying eigenvectors and eigenvalues $\|\\lambda \| \\in [0,2/a]$ of the overlap Dirac operator [58], [59], [60], [61] $D_{ov}=\\frac{1}{a}\\left[1+ \\gamma _5 \\frac{H}{\|H\|}\\right]\\textrm { with }H=\\gamma _5 A,\\;A=a D_\\mathrm {W}-m,$ where $m$ describes one species of single massless Dirac fermions and has to be in the range $(0,2)$ and the massless Wilson Dirac operator $D_W$ [63], [64] on a lattice with lattice constant $a$ reads $D_\\mathrm {W}(x,y)=\\frac{4}{a}\\delta _{x,y}-\\frac{1}{2a}\\sum _{\\mu =\\mp 1}^{\\pm 4}(1-\\gamma _\\mu )\\;U_\\mu (x)\\;\\delta _{x+\\hat{\\mu },y}\\textrm { with }\\gamma _{-\\mu }=-\\gamma _\\mu ,\\;U_{-\\mu }(x)=U_\\mu ^\\dagger (x-\\hat{\\mu }).$ The coordinates and the results for eigenvalues and densities we give in units of the lattice constant $a$ , i.e we put further on $a=1$ .", "The vectors $\\hat{\\mu }$ connect nearest neighbors $x$ and $y$ in $x_\\mu $ -direction.", "$U_\\mu (x)\\in SU(2)$ are the parallel transporters from $x$ to $x+\\hat{\\mu }$ .", "The mass parameter $m$ is chosen with $m=+1.5$ .", "The eigenvalues of the overlap Dirac operator as a Ginsparg-Wilson operator are restricted to a circle in the complex plane.", "The absolute value $\|\\lambda \|$ of the two complex conjugate eigenvalues of $D_{ov}$ is simply written as $\\lambda $ .", "According to the Atiyah–Singer index theorem [23] a configuration with non-vanishing topological charge has to be related to zero modes of the Dirac operator.", "If lumps with topological charge are parts of a larger configuration they could localize the fermionic modes, interact and contribute to a finite density of near-zero modes.", "According to the Banks-Casher relation [22] a finite density of near-zero modes leads to non-zero chiral condensate and spontaneous $\\chi $ SB.", "As mentioned above, the topological charge of two intersecting anti-parallel vortex pairs where one of the vortices is colorful, negatively charged, leads to a total topological charge $Q=-2$ or $Q=0$ depending on the location of the colorful region.", "In Fig.", "REF we study the lowest eigenvalues of the overlap Dirac operator in the background of these configurations and compare them with those of the free overlap Dirac operator.", "For the fermions we use anti-periodic boundary conditions in temporal direction and periodic boundary conditions in spatial directions on a $16^4$ -lattice.", "In diagram a) the parameters of the configurations are the same as those in the previous section, in diagram b) these data are compared with analogous vortex configurations where the distances between the vortices in each pair are reduced from 7 to 3 and the vortex thickness parameters d of Eq.", "(REF ) are correspondingly decreased from 3 to 1.5.", "Figure: a) The lowest overlap eigenvalues for the vortex configurations schematically displayed in Fig.", "with Q=-2Q=-2 (Fig.", "a) and Q=0Q=0 (Fig. b).", "These values are compared with those of the free Dirac operator on a 16 4 16^4 lattice.", "b) The influence of the distance between the vortex sheets is investigated for the Q=-2Q=-2 configuration.", "Increasing this distance the interaction decreases and the lowest lying non-zero modes move from the lowest Matsubara frequency towards zero.The topological charge of these configurations agrees with the analytical index, $\\mathrm {ind}D[A]=n_--n_+=Q$ [23], [24], [25].", "For a single configuration, one never finds zero modes of both chiralities and at least one of the numbers $n_-$ or $n_+$ vanishes.", "As shown in Fig.", "REF a, we find two zero modes of positive chirality for the $Q=-2$ configuration and no zero-mode for $Q=0$ .", "Note, for better comparison we indicated the two zero-modes with mode numbers $\\#(-1)$ and $\\#0$ .", "By the influence of the vortices the eigenvalues occupy the space between the Matsubara frequencies, visible in the spectrum of the trivial configuration.", "For both non-trivial configurations four of the modes move from the first Matsubara frequency down towards zero eigenvalues.", "Some of them let expect to contribute in the infinite volume limit to the density of near-zero modes.", "In Fig.", "REF b we compare the lowest eigenvalues of configurations with decreased distances, as mentioned above, with those of the original, larger distances.", "With increasing distance between the lumps of topological charge we expect decreasing interaction and observe that the four lowest non-zero modes shift down from the first Matsubara frequency towards zero for both topological charges.", "Figure: The chiral densities χ n (x)\\chi _n(x) of the first three modes of the Q=-2Q=-2 configuration in the x-z-planes through the points with the maximal (absolute) values of the chiral density.", "The plot titles indicate the plane positions, the chirality (chi=0,±10,\\pm 1), the numbers n=#n=\\# of plotted modes and the maximal (minimal) density in the plotted area, \"max=...\" (\"min=...\").", "\"n=-1n=-1\" means we plot the density χ -1 (x)\\chi _{-1}(x) for the mode with number #(-1)\\#(-1).", "Note, to ease the comparison of the various modes in Fig.", "we use the mode numbers #(-1)\\#(-1) and #0\\#0 for the two zero-modes and start counting positive mode numbers for non-zero modes.The two gauge field configurations with $Q=-2$ and $Q=0$ give the nice opportunity to study the properties of zero modes and near-zero modes and to compare them with excited modes.", "For the first one, $Q=-2$ , we compare in Fig.", "REF the chiral densities of the first three modes in the x-z-planes through the points with the maximal values of the chiral density.", "The first two modes, $\\#(-1)$ and $\\#0$ , are zero-modes with positive chirality.", "The corresponding diagrams, Figs.", "REF a and b show clear maxima, located in the intersection plane close to the center of the colorful region with topological charge $Q=-1$ , slightly shifted against each other by $\\Delta x_\\mu =(2,1,0,1)$ lattice units.", "The maximum of the first mode is more pronounced than that of the second mode.", "The same clear peak structure we can find also in all other cross-section through the maxima.", "This means the peaks are nearly spherical in four dimensions.", "Due to the missing space we do not show the corresponding cross-sections.", "The next modes are non-zero modes, see Fig.", "REF a.", "The third mode, depicted in Fig.", "REF c looks completely different.", "Oscillations extend through the whole intersection plane, but they have large amplitudes only in a region of a few lattice constant in the perpendicular y- and t-directions.", "The corresponding densities are again not shown.", "Comparing Fig.", "REF c with Fig.", "REF a we see that the maxima and minima of $\\chi _1(x)$ reflect the position of the topological charge density.", "Figure: Maximal and minimal values of the chiral densities of the 20 lowest modes for the above described configuration with Q=-2Q=-2 (left) and Q=0Q=0 (right).The lumps of topological charge at the intersection points of the vortex pairs seem to contribute jointly to the shape of this mode.", "Lumps with positive topological charge density are correlated with the components with negative chiral density and vice versa.", "The chiral densities of the next modes we can only discuss in words.", "$\\chi _2(x)$ behaves similar to $\\chi _1(x)$ , $\\chi _3(x)$ and $\\chi _4(x)$ have again positive peaks similar to $\\chi _{-1}(x)$ and $\\chi _0(x)$ but only around 1/3 of their height.", "This indicates that the maxima and minima of the topological density keep their influence on some of the next modes.", "In this respect it may be interesting to compare the values of maxima and minima of the chiral densities of the lowest modes.", "Fig.", "REF a depicts the values of maxima and minima of the chiral densities for the 20 lowest modes for $Q=-2$ configuration.", "Large maxima of the density indicate localized peaks influenced by the colorful vortex region centered at $x_\\mu =(6,6,6,6)$ .", "Besides the two zero-modes such maxima can be found for the mode numbers $\\#3,\\#4,\\#11,\\#12$ and $\\#13$ .", "Inspections of these modes shows nice peaks with a height given in Fig.", "REF and a shallow negative see compensating the peak in order to reach the integrated chiral density $\\chi =0$ .", "In regions far away from the peaks there appears a wavy character due to the neighborhood of these modes to the corresponding Matsubara frequency.", "For points in Fig.", "REF a with nearly equal sizes of maxima and minima we can imply a wavy character due to the four vortex intersections and the increasing momenta of higher Matsubara frequencies.", "Fig.", "REF b depicts maximal and minimal values of the chiral densities of the 20 lowest modes for the $Q=0$ configuration.", "The modes $\\#1,\\#5,\\#13$ are striking for their large maxima and point to their peaky structure at the position of the colorful vortex around $x_\\mu =(13,6,6,6)$ similar to Figs.", "REF a and b, but with lower height.", "These three modes are interestingly some type of band-heads in the eigenvalue spectrum of Fig.", "REF a.", "The other modes have wavy character, e.g.", "mode $\\#2$ behaves similar to mode $\\#1$ of the $Q=-2$ configuration displayed in Fig.", "REF c." ], [ "Conclusion", "In four dimensions vortices are two-dimensional surfaces with some thickness.", "On a periodic lattice plain vortices can only be defined in parallel or anti-parallel pairs.", "In the past mainly uni-color vortices were investigated.", "Intersections of plain uni-color vortices contribute to the topological charge with $\\pm 1/2$ .", "The four intersections of two vortex pairs result therefore in topological charges of $Q=0,\\,\\pm 1$ or $\\pm 2$ .", "In this article we considered pairs of anti-parallel vortices yielding $Q=0$ .", "It was shown recently that vortices can also have a color structure with non-vanishing topological charge.", "To get more insight into the effect of such color structures on vortices we investigated the influence of a circular colorful region with topological charge $Q=-1$ around one of the four intersection points of two intersecting anti-parallel vortex pairs.", "We studied the consequences of this insertion on topological charge density, zero-modes and near-zero modes.", "To uni-color vortices one can attribute an orientation.", "In this picture of unicolor vortices the above colorful region introduces a monopole line on its vortex surface, a line surrounding the intersection point and changes the surface orientation inside this circular region.", "This leads to a sign change of the topological charge contribution at the intersection.", "The contributions of the intersections aggregate thus to $Q=\\pm 1$ .", "The monopole line itself has a non-trivial topology and leads to further contribution to the topological charge, in the considered cases with a contribution of $-1$ .", "For our configurations we can therefore distinguish the two cases $Q=0$ and $Q=-2$ .", "In both configurations we have identified the regions with non-vanishing contributions to the topological charge density.", "Further, we have analyzed the low-lying modes of the Dirac operator.", "The number of zero-modes agrees with the expectations from the Atiyah-Singer index theorem.", "In both configurations we found four low lying modes which are shifted from the first Matsubara frequency down towards zero eigenvalues.", "We increased the distance between the lumps of topological charge in the expectation of decreasing interaction and observe a decrease of the eigenvalues of these four modes.", "We found that the lumps of topological charge influence strongly the spatial distribution of the low-lying modes of the Dirac operator.", "The colorful region with topological charge $Q=-1$ leads in some of the lowest modes to distinct positive peaks of the chiral density.", "The other modes reflect the positions of the intersections and their contributions to the topological charge density.", "It turned out that a good indication for the chiral properties of the eigenmodes is the relation between maxima and minima of the chiral densities.", "We would like to thank Roman Höllwieser and Urs M. Heller for their support in the preparation of the programs.", "SMHN thanks Sedigheh Deldar for her support and is grateful to the Iran National Science Foundation (INSF) for supporting this study." ] ]	1606.04887
[ [ "Deep Learning for Music" ], [ "Abstract Our goal is to be able to build a generative model from a deep neural network architecture to try to create music that has both harmony and melody and is passable as music composed by humans.", "Previous work in music generation has mainly been focused on creating a single melody.", "More recent work on polyphonic music modeling, centered around time series probability density estimation, has met some partial success.", "In particular, there has been a lot of work based off of Recurrent Neural Networks combined with Restricted Boltzmann Machines (RNN-RBM) and other similar recurrent energy based models.", "Our approach, however, is to perform end-to-end learning and generation with deep neural nets alone." ], [ "confposter block titlefg=DarkRed,bg=white block bodyfg=black,bg=white block alerted titlefg=white,bg=dblue!70 block alerted bodyfg=black,bg=dblue!10 Deep Learning for Music Allen Huang, Raymond Wu Stanford University block end block alerted end headline [remember picture,overlay] shift=(-10 cm,-5cm)] at (current page.north east) Figure: NO_CAPTION 2ex t] block titlefg=white,bg=CardinalRed block bodyfg=black,bg=white Deep Learning for Music [remember picture,overlay] shift=(-10 cm,-6.5cm)] at (current page.north east) Figure: NO_CAPTION Allen Huang, Raymond Wu Stanford University block titlefg=CardinalRed,bg=white block bodyfg=black,bg=white [t] Introduction and Background Our goal is to be able to build a generative model from a deep neural network architecture to try to create music that has both harmony and melody and is passable as music composed by humans.", "Previous work in polyphonic music modeling has centered around time series probability density estimation.", "In particular, there has been a lot of work based off of Recurrent Neural Networks combined with Restricted Boltzmann Machines (RNN-RBM).", "Recurrent Temporal Restricted Boltzmann Machines (RTRBM) have also been successful.", "Our approach, however, is to perform end-to-end learning and generation with deep neural nets alone.", "Midi Data Midi files are structured as a series of tracks, each containing a list of meta messages and messages.", "We extract the messages pertaining to the notes and their duration and encode the entire message as a unique token.", "We flatten the tracks so that the tokens of the separate tracks of a piece would be concatenated end-to-end.", "Table: NO_CAPTION Muse Piano Roll Data We represent the midi files as a series of time steps.", "Each time step is a list of note ids that are playing.", "The MuseData dataset had 524 pieces for a total of 245,202 time steps.", "We encode each time step by concatenating the note ids together to form a token.", "For example, we encode a C-Major chord as \"60-64-67.\"", "Furthermore, in order to reduce the number of unique tokens, we randomly chose 3 notes if the polyphony exceeded 4.", "Table: NO_CAPTION [t,totalwidth=] Bach-Midi Experiment We use a 2-layered Long Short Term Memory (LSTM) recurrent neural network (RNN) architecture on the \"Bach Only\" dataset.", "The output of the LSTM is fed into softmax layer with a corresponding cross-entropy objective function.", "50 epochs took around 4 hours to train on a AWS g2.2xlarge instance.", "Table: NO_CAPTION Figure: Music generated from 'Bach Only' dataset.Classical-Midi Experiment We use the same architecture as in the Bach-Midi experiment on the \"Truncated Classical\" dataset due to time constraints.", "15 epochs took 22 hours on a AWS g2.2xlarge instance.", "Furthermore, due to limitations on device memory on AWS's g2.2xlarge, we were forced to reduce the batch size and the sequence length.", "Table: NO_CAPTION Figure: Music generated from the 'Truncated Classical' dataset.", "Muse-Piano-Roll Experiments We ran this experiment with the same parameters as the \"Bach-Midi Experiment.\"", "We ran it with for 800 epochs, which took 7 hours on a AWS g2.2xlarge instance.", "We also ran the same configuration on the truncated dataset for 100 epochs, which took 7 hours on a CPU.", "Figure: Music generated from the Muse piano roll data.", "The top 4 lines are from the 'Muse-All' dataset and the last two lines are from the 'Muse-Truncated' dataset.Piano-Roll Sigmoid Model Instead of encoding a list of notes for each time step as one token, we first represent each note with its own embedding vector.", "For each time step the input vector would be a sum of these vectors.", "The output of the LSTM is projected back into the input space and fed into a sigmoid layer.", "The objective function is standard cross-entropy.", "Figure: Music generated from our sigmoid model.Results Figure: t-SNE visualization of single note embedding vectors from the Classical-Midi experiment.", "The circles denote ON messages while the x's represent OFF messages.", "Note that there are clear clusters between the on and off messages for the medium frequency notes (the notes that are played most often), while the blue and red notes corresponding to the low and high notes are clumped together in an indistinct cloud in the center.Figure: t-SNE visualization of single note embedding vectors from the Muse-Piano-Roll experiment.", "Note that vectors are able to disambiguate among the low, medium, and high notes cleanly.Figure: t-SNE visualization of single note embedding vectors from the Piano-Roll Sigmoid Model experiment.", "It looks like our sigmoid model was not able to learn effectively, which is coincidentally reflected in the music generated.Conclusion Early analysis of our embedding vectors suggest that we have moderate success in producing a viable generative model.", "That being said, we plan on crowd-sourcing the evaluation of our model against the state-of-the-art results in a blind experiment.", "Please volunteer!" ] ]	1606.04930
[ [ "Contravariance through enrichment" ], [ "Abstract We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution.", "For this purpose we introduce \"2-categories with contravariance\", a sort of enhanced 2-category with a basic notion of \"contravariant morphism\", which can be regarded either as generalized multicategories or as enriched categories.", "This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem." ], [ "Introduction", "One of the more mysterious bits of structure possessed by the 2-category $\\mathcal {C}\\mathit {at}$ is its duality involution $ (-)^{\\mathrm {op}}: \\mathcal {C}\\mathit {at} ^{\\mathrm {co}}\\rightarrow \\mathcal {C}\\mathit {at}.", "$ (As usual, the notation $(-)^{\\mathrm {co}}$ denotes reversal of 2-cells but not 1-cells.)", "Many familiar 2-categories possess similar involutions, such as 2-categories of enriched or internal categories, the 2-category of monoidal categories and strong monoidal functors, or $[\\underline{A},\\mathcal {C}\\mathit {at} ]$ whenever $\\underline{A}$ is a locally groupoidal 2-category; and they are an essential part of much standard category theory.", "However, there does not yet exist a complete abstract theory of such “duality involutions”.", "A big step forward was the observation by Day and Street [7] that $A^{\\mathrm {op}}$ is a monoidal dual of $A$ in the monoidal bicategory of profunctors.", "As important and useful as this fact is, it does not exhaust the properties of $(-)^{\\mathrm {op}}$ ; indeed, it does not even determine $A^{\\mathrm {op}}$ up to equivalence!", "In this paper we study duality involutions like $(-)^{\\mathrm {op}}$ acting on 2-categories like $\\mathcal {C}\\mathit {at}$ , rather than bicategories like $\\mathcal {P}\\mathit {rof}$ .", "(We leave it for future work to combine the two, perhaps with a theory of “duality involutions on proarrow equipments”.", "One step in that direction was taken by [36], in the case where profunctors are represented by discrete two-sided fibrations.)", "Note that in most of the examples cited above, $(-)^{\\mathrm {op}}$ is a 2-functor that is a strict involution, in that we have $(A^{\\mathrm {op}})^{\\mathrm {op}}= A$ on the nose.", "On the other hand, from a higher-categorical perspective it would be more natural to ask only for a weak duality involution, where $(-)^{\\mathrm {op}}$ is a pseudofunctor that is self-inverse up to coherent pseudo-natural equivalence.", "For instance, strict duality involutions are not preserved by passage to a biequivalent bicategory, but weak ones are.", "The main result of this paper is that there is no loss of generality in considering only strict involutions.", "More precisely, we prove the following coherence theorem.", "Theorem 1.1 Every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution, by a biequivalence which respects the involutions up to coherent equivalence.", "Let me now say a few words about the proof of label@name@thm:intro REF , which I regard as more interesting than its statement.", "Often, when proving a coherence theorem for categorical structure at the level of objects, it is helpful to consider first an additional structure at the level of morphisms, whose presence enables the object-level structure to be characterized by a universal property.", "For instance, instead of pseudofunctors $A^{\\mathrm {op}}\\rightarrow \\mathcal {C}\\mathit {at} $ , we may consider categories over $A$ , among which those underlying some pseudofunctor (the fibrations) are characterized by the existence of cartesian arrows, which have a universal property.", "Similarly, instead of monoidal categories, we may consider multicategories, among which those underlying some monoidal category are characterized by the existence of representing objects, which also have a universal property.", "An abstract framework for this procedure is the theory of generalized multicategories; see [15], [5] and the numerous other references in [5].", "In general, for a suitably nice 2-monad $T$ , in addition to the usual notions of strict and pseudo $T$ -algebra, there is a notion of virtual $T$ -algebra, which contains additional kinds of morphisms whose domain “ought to be an object given by a $T$ -action if such existed”.", "For example, if $T$ is the 2-monad for strict monoidal categories, then a virtual $T$ -algebra is an ordinary multicategory, in which there are “multimorphisms” whose domains are finite lists of objects that “ought to be tensor products if we had a monoidal category”.", "In our case, it is easy to write down a 2-monad whose strict algebras are 2-categories with a strict duality involution: it is $T\\mathcal {A} = \\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}$ .", "A virtual algebra for this 2-monad is, roughly speaking, a 2-category equipped with a basic notion of “contravariant morphism”.", "That is, for each pair of objects $x$ and $y$ , there are two hom-categories $\\underline{A} ^+(x,y)$ and $\\underline{A} ^-(x,y)$ , whose objects we call covariant and contravariant morphisms respectively.", "Composition is defined in the obvious way: the composite of two morphisms of the same variance is covariant, while the composite of two morphisms of different variances is contravariant.", "In addition, postcomposing with a contravariant morphism is contravariant on 2-cells.", "We call such a gadget a 2-category with contravariance.", "As with any sort of generalized multicategory, we can characterize the virtual $T$ -algebras that are pseudo $T$ -algebras by a notion of representability.", "This means that for each object $x$ , we have an object $x^\\circ $ and isomorphisms $\\underline{A} ^-(x,y) \\cong \\underline{A} ^+(x^\\circ ,y)$ and $\\underline{A} ^+(x,y) \\cong \\underline{A} ^-(x^\\circ ,y)$ , jointly natural in $y$ .", "We call an object $x^\\circ $ with this property a (strict) opposite of $x$ .", "The corresponding pseudo $T$ -algebra structure describes this operation $(-)^\\circ $ as a strong duality involution on the underlying 2-category $\\underline{A} ^+$ , meaning a strict 2-functor $(\\underline{A} ^+)^{\\mathrm {co}}\\rightarrow \\underline{A} ^+$ that is self-inverse up to coherent strict 2-natural isomorphism.", "Now, it turns out that 2-categories with contravariance are not just generalized multicategories: they are also enriched categories.Representation of additional structure on a category as enrichment occurs in many other places; see for instance [10], [21], [8], [11], [9].", "Namely, there is a (non-symmetric) monoidal category, denoted (for Variance), such that -enriched categories are the same as 2-categories with contravariance.", "(As a category, is just $\\mathbf {Cat} \\times \\mathbf {Cat} $ , but its monoidal structure is not the usual one.)", "From this perspective, we can alternatively describe strict opposites as weighted colimits: $x^\\circ $ is the copower (or “tensor”) of $x$ by a particular object ${1} ^{-}$ of , called the dual unit.", "Since ${1} ^{-}$ is dualizable in , opposites are an absolute or Cauchy colimit in the sense of [35]: they are preserved by all -enriched functors.", "It follows that any 2-category-with-contravariance has a “completion” with respect to opposites, and this operation is idempotent.", "We have now moved into a context having a straightforward bicategorical version.", "We simply observe that can be made into a monoidal 2-category, and consider -enriched bicategories; we call these bicategories with contravariance.", "In such a bicategory we can consider “weak opposites”, asking only for pseudonatural equivalences $\\underline{A} ^-(x,y) \\simeq \\underline{A} ^+(x^\\circ ,y)$ and $\\underline{A} ^+(x,y) \\simeq \\underline{A} ^-(x^\\circ ,y)$ ; these are “absolute weighted bicolimits” in the sense of [13].", "Since any isomorphism of categories is an equivalence, any strict opposite is also a weak one.", "(More abstractly, strict opposites should be flexible colimits [2] in a suitable sense, but we will not make this precise.)", "Now, it is straightforward to generalize the coherence theorem for bicategories to a coherence theorem for enriched bicategories.", "Therefore, any bicategory with contravariance is biequivalent to a 2-category with contravariance.", "This suggests that the process by which we arrived at -enriched categories could be duplicated on the bicategorical side, yielding the following “ladder” strategy for proving label@name@thm:intro REF : $@C=4pc{\\text{\\parbox {1.5in}{\\centering -enriched bicategories\\\\with weak opposites}}[r]^{\\text{coherence theorem}}_{\\text{for bicategories}} &\\text{\\parbox {1.5in}{\\centering -enriched categories\\\\with strict opposites}}[d]\\\\\\text{\\parbox {2in}{\\centering representable\\\\$T$-multi-bicategories}}[u] &\\text{\\parbox {2in}{\\centering representable\\\\$T$-multicategories\\\\(virtual $T$-algebras)}}[d]\\\\\\text{\\parbox {2in}{\\centering bicategories with\\\\weak duality involution}}[u] &\\text{\\parbox {2in}{\\centering 2-categories with\\\\strong duality involution\\\\(pseudo $T$-algebras)}}[d]\\\\& \\text{\\parbox {2in}{\\centering 2-categories with\\\\strict duality involution\\\\(strict $T$-algebras)}}}$ There are three problems with this idea, two minor and one major.", "The first is that it (apparently) produces only a strong duality involution rather than a strict one, necessitating an extra step at the bottom-right of the ladder, as shown.", "However, the strictification of pseudo-algebras for 2-monads is fairly well-understood, so we can apply a general coherence result [24], [19].", "The second problem is that a priori, the coherence theorem for -enriched bicategories does not also strictify the weak opposites into strict opposites.", "However, this is also easy to remedy: since the strictification of a -bicategory with weak opposites will still have weak opposites, and any strict opposite is also a weak one, it will be biequivalent to its free cocompletion under strict opposites.", "The third, and more major, problem with this strategy is that there is no extant theory of “generalized multi-bicategories”.", "We could develop such a theory, but it would take us rather far afield.", "Thus, instead we will “hop over” that rung of the ladder by constructing a $$ -enriched bicategory with weak opposites directly from a bicategory with a weak duality involution, by a “beta-reduced” and weakened version of the analogous operation on the other side.", "Since this direct construction also includes the strict case, we could, formally speaking, dispense with the multicategories on the other side as well.", "Indeed, the entire proof can be beta-reduced into a more compact form: if we prove the coherence theorem for enriched bicategories using a Yoneda embedding, the strictification and cocompletion processes could be combined into one and tweaked slightly to give a strict duality involution directly.", "In fact, there are not many applications of label@name@thm:intro REF anyway.", "First of all, it is not all that easy to think of naturally occurring duality involutions that are not already strict.", "But here are a few: The 2-category of fibrations over some base category has a “fiberwise” duality involution, but since its action on non-vertical arrows has to be constructed in a more complicated way than simply turning them around, it is not strict.", "If is a compact closed bicategory [7], [32], then its bicategory $\\mathcal {M}\\mathit {ap} ()$ of maps (left adjoints) has a duality involution that is not generally strict.", "If $\\mathcal {A}$ is a bicategory with a duality involution, and is a class of morphisms in $\\mathcal {A}$ admitting a calculus of fractions [26] and closed under the duality involution, then the bicategory of fractions $\\mathcal {A} [^{-1}]$ inherits a duality involution that is not strict (even if the one on $\\mathcal {A}$ was strict).", "However, even in these cases label@name@thm:intro REF is not as important as it might be, because Lack's coherence theorem (“naturally occurring bicategories are biequivalent to naturally occurring 2-categories”) applies very strongly to duality involutions: nearly all naturally occurring bicategories with duality involutions are biequivalent to some naturally occurring strict 2-category with a strict duality involution.", "For the examples above, we have: The 2-category of fibrations over is biequivalent to the 2-category of -indexed categories, which has a strict duality involution inherited from $\\mathcal {C}\\mathit {at}$ .)", "For the standard examples of compact closed bicategories such as $\\mathcal {P}\\mathit {rof}$ or $\\mathcal {S}\\mathit {pan}$ , the bicategory of maps is biequivalent to a well-known strict 2-category with a strict duality involution, such as $\\mathcal {C}\\mathit {at} _{\\mathrm {cc}}$ (Cauchy-complete categories) or $\\mathbf {Set}$ .", "Many naturally occurring examples of bicategories of fractions are also biequivalent to well-known 2-categories with strict duality involutions, such as some 2-category of stacks.", "Thus, if label@name@thm:intro REF were the main point of this paper, it would be somewhat disappointing.", "However, I regard the method of proof, and the entire ladder it gives rise to, as more important than the result itself.", "Representating contravariance using generalized multicategories and enrichment seems a promising avenue for future study of further properties of duality involutions.", "From this perspective, the paper is primarily a contribution to enhanced 2-category theory in the sense of [21], which just happens to prove a coherence theorem to illustrate the ideas.", "Furthermore, our abstract approach also generalizes to other types of contravariance.", "The right-hand side of the ladder, at least, works in the generality of any group action on any monoidal category .", "The motivating case of duality involutions on 2-categories is the case when $/2$ acts on $\\mathbf {Cat}$ by $(-)^{\\mathrm {op}}$ ; but other actions representing other kinds of contravariance include the following.", "[leftmargin=2em] $/2\\times /2$ acts on $\\mathbf {2Cat}$ by $(-)^{\\mathrm {op}}$ and $(-)^{\\mathrm {co}}$ .", "When $\\mathbf {2Cat}$ is given the Gray monoidal structure, this yields a theory of duality involutions on Gray-categories.", "$(/2)^n$ acts on strict $n$ -categories (including the case $n=\\omega $ ), yielding duality involutions for strict $(n+1)$ -categories.", "not as interesting as weak ones, but their theory can point the way towards a weak version.", "$/2$ acts on the category $\\mathbf {sSet}$ of simplicial sets by reversing the directions of all the simplices.", "With simplicial sets modeling $(\\infty ,1)$ -categories as quasicategories, this yields a theory of duality involutions on a particular model for $(\\infty ,2)$ -categories (see for instance [29]).", "Combining the ideas of the last two examples, $(/2)^n$ acts on the category of $\\Theta _n$ -spaces by reversing direction at all dimensions, leading to duality involutions on an enriched-category model for $(\\infty ,n+1)$ -categories [3].", "We will not develop any of these examples further here, but the perspective of describing contravariance through enrichment may be useful for all of them as well.", "We begin in label@name@sec:strong-duality by defining weak, strong, and strict duality involutions.", "Then we proceed up the ladder from the bottom right.", "In label@name@sec:2-monadic-approach we express strong and strict duality involutions as algebra structures for a 2-monad, and deduce that strong ones can be strictified.", "In label@name@sec:genmulti we express strong duality involutions using generalized multicategories, and in sections – we reexpress them using enrichment.", "In label@name@sec:bicat-contra we jump over to the other side of the ladder, showing that weak duality involutions on bicategories can be expressed using bicategorical enrichment.", "Then finally in label@name@sec:coherence we cross the top of the ladder with a coherence theorem for enriched bicategories." ], [ "Duality involutions", "In this section we define strict, strong, and weak duality involutions, allowing us to state label@name@thm:intro REF precisely.", "Definition 1 A weak duality involution on a bicategory $\\mathcal {A}$ consists of: [leftmargin=2em] A pseudofunctor $(-)^\\circ : \\mathcal {A} ^{\\mathrm {co}}\\overset{}{\\longrightarrow }\\mathcal {A} $ .", "A pseudonatural adjoint equivalence $ {\\mathcal {A} [dr]_{((-)^\\circ )^{\\mathrm {co}}} @{=}[rr] && \\mathcal {A}.\\\\& \\mathcal {A} ^{\\mathrm {co}}[ur]_{(-)^\\circ } @{}[u]\|(.6){\\Downarrow }}$ An invertible modification $ {\\mathcal {A} ^{\\mathrm {co}}[r]^{(-)^\\circ } & \\mathcal {A} [dr]_{((-)^\\circ )^{\\mathrm {co}}} @{=}[rr] && \\mathcal {A} \\\\&& \\mathcal {A} ^{\\mathrm {co}}[ur]_{(-)^\\circ } @{}[u]\|(.6){\\Downarrow }}$ $whose components are therefore 2-cells $ x: x $\\scriptstyle \\sim $(x)$.\\item For any $ x$\\mathcal {A}$ $, we have$${x [r]^{_x} &x^\\circ {}^\\circ @(ur,ul)[rr]^{_{x^\\circ {}^\\circ }}@(dr,dl)[rr]_{(_{x^\\circ })^\\circ }@{}[rr]\|{\\Downarrow \\zeta _{x^\\circ }} &&x^\\circ {}^\\circ {}^\\circ {}^\\circ }$ = $(the unnamed isomorphism is a pseudonaturality constraint for ).$ If $\\mathcal {A}$ is a strict 2-category, a strong duality involution on $\\mathcal {A}$ is a weak duality involution for which [leftmargin=2em] $(-)^\\circ $ is a strict 2-functor, $$ is a strict 2-natural isomorphism, and $\\zeta $ is an identity.", "If moreover $$ is an identity, we call it a strict duality involution.", "In particular, and $\\zeta $ in a weak duality involution exhibit $(-)^\\circ $ and $((-)^\\circ )^{\\mathrm {co}}$ as a biadjoint biequivalence between $\\mathcal {A}$ and $\\mathcal {A} ^{\\mathrm {co}}$ , in the sense of [14].", "Similarly, in a strong duality involution, exhibits $(-)^\\circ $ and $((-)^\\circ )^{\\mathrm {co}}$ as a 2-adjoint 2-equivalence between $\\mathcal {A}$ and $\\mathcal {A} ^{\\mathrm {co}}$ .", "And, of course, in a strict duality involution, $(-)^\\circ $ and $((-)^\\circ )^{\\mathrm {co}}$ are inverse isomorphisms of 2-categories.", "Definition 2 If $\\mathcal {A}$ and are bicategories equipped with weak duality involutions, a duality pseudofunctor $F:\\mathcal {A} \\rightarrow $ is a pseudofunctor equipped with [leftmargin=2em] A pseudonatural adjoint equivalence ${\\mathcal {A} ^{\\mathrm {co}}[d]_{(-)^\\circ }[r]^-{F^{\\mathrm {co}}} {\\mathfrak {i}} &^{\\mathrm {co}}[d]^{(-)^\\circ }\\\\\\mathcal {A} [r]_-F &.", "}$ $\\item An invertible modification$ $whose components are therefore 2-cells in of the following shape:$ $\\item For any $ $\\mathcal {A}$ $, we have\\begin{multline}@R=3pc@C=1.5pc{(Fx)^\\circ {}^\\circ {}^\\circ [rr]^{(\\mathfrak {i}_x)^\\circ {}^\\circ } {{(\\theta _x)^\\circ }} &&(F(x^\\circ ))^\\circ {}^\\circ [d]^{(\\mathfrak {i}_{x^\\circ })^\\circ }\\\\(Fx)^\\circ [rr]_{(F_x)^\\circ } [d]_{\\mathfrak {i}_x} {\\cong }^{{_{(F x)^\\circ }}}_{{(_{F x})^\\circ }}{\\zeta _{F x}} &&(F(x^\\circ {}^\\circ ))^\\circ [d]^{\\mathfrak {i}_{x^\\circ {}^\\circ }}\\\\F(x^\\circ ) [rr]_{F((_x)^\\circ )} && F(x^\\circ {}^\\circ {}^\\circ )} \\end{multline}=@R=4pc@C=1.5pc{(Fx)^\\circ {}^\\circ {}^\\circ [r]^{(\\mathfrak {i}_x)^\\circ {}^\\circ } &(F(x^\\circ ))^\\circ {}^\\circ [rr]^{(\\mathfrak {i}_{x^\\circ })^\\circ } {{\\theta _{x^\\circ }}} &&(F(x^\\circ {}^\\circ ))^\\circ [d]^{\\mathfrak {i}_{x^\\circ {}^\\circ }}\\\\(Fx)^\\circ [u]^{_{(Fx)^\\circ }} [r]_{\\mathfrak {i}_x} {\\cong } &F(x^\\circ ) [u]\|{_{F(x^\\circ )}} ^{F(_{x^\\circ })}_{F((_x)^\\circ )}{{F(\\zeta _x)}} &&F(x^\\circ {}^\\circ {}^\\circ )}$ (the unnamed isomorphisms are pseudonaturality constraints for $\\mathfrak {i}$ and $$ ).", "If $\\mathcal {A}$ and are strict 2-categories with strong duality involutions, then a (strong) duality 2-functor $F:\\mathcal {A} \\rightarrow $ is a duality pseudofunctor such that [leftmargin=2em] $F$ is a strict 2-functor, $\\mathfrak {i}$ is a strict 2-natural isomorphism, and $\\theta $ is an identity.", "If $\\mathfrak {i}$ is also an identity, we call it a strict duality 2-functor.", "Note that if the duality involutions of $\\mathcal {A}$ and are strict, then the identity $\\theta $ says that $(\\mathfrak {i}_x)^\\circ = (\\mathfrak {i}_{x^\\circ })^{-1}$ .", "On the other hand, if $\\mathcal {A} $ is a strict 2-category with two strong duality involutions $(-)^\\circ $ and $(-)^\\circ {}^{\\prime }$ , to make the identity 2-functor into a duality 2-functor is to give a natural isomorphism $A^\\circ \\cong A^\\circ {}^{\\prime }$ that commutes with the isomorphisms $$ and $^{\\prime }$ .", "Now label@name@thm:intro REF can be stated more precisely as: Theorem 2.1 If $\\mathcal {A} $ is a bicategory with a weak duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality pseudofunctor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a biequivalence.", "We could make this more algebraic by defining a whole tricategory of bicategories with weak duality involution and showing that our biequivalence lifts to an internal biequivalence therein, but we leave that to the interested reader.", "In fact, the correct definitions of transformations and modifications can be extracted from our characterization via enrichment.", "(It does turn out that there is no obvious way to define non-invertible modifications.)", "The definition of duality involution may seem a little ad hoc.", "In label@name@sec:bicat-contra we will rephrase it as a special case of a “twisted group action”, which may make it seem more natural.", "We end this section with some examples.", "With nearly any reasonable set-theoretic definition of “category” and “opposite”, the 2-category $\\mathcal {C}\\mathit {at} $ of categories and functors has a strict duality involution.", "The same is true for the 2-category of categories enriched over any symmetric monoidal category, or the 2-category of categories internal to some category with pullbacks.", "If $\\mathcal {A}$ is a bicategory with a weak duality involution and is a locally groupoidal bicategory, then the bicategory $[,\\mathcal {A} ]$ of pseudofunctors, pseudonatural transformations, and modifications inherits a weak duality involution by applying the duality involution of $\\mathcal {A}$ pointwise.", "Local groupoidalness of ensures that $\\cong ^{\\mathrm {co}}$ , so that we can define the dual of a pseudofunctor $F:\\rightarrow \\mathcal {A} $ to be $F^\\circ : \\cong ^{\\mathrm {co}}\\xrightarrow{} \\mathcal {A} ^{\\mathrm {co}}\\xrightarrow{} \\mathcal {A}.$ The rest of the structure follows by whiskering.", "If $\\mathcal {A}$ is a 2-category and its involution is strong or strict, the same is true for $[,\\mathcal {A} ]$ .", "If $\\mathcal {A} $ is a bicategory with a weak duality involution and $F:\\mathcal {A} \\rightarrow $ is a biequivalence, then $$ can be given a weak duality involution making $F$ a duality pseudofunctor.", "We first have to enhance $F$ to a biadjoint biequivalence as in [14]; then we define all the structure by composing with $F$ and its inverse.", "The 2-category of fibrations over a base category has a strong duality involution constructed as follows.", "Given a fibration $P:\\rightarrow $ , in its dual $P^\\circ :^\\circ \\rightarrow $ the objects of $^\\circ $ are those of , while the morphisms from $x$ to $y$ over a morphism $f:a\\rightarrow b$ in are the morphisms $f^y \\rightarrow x$ over $a$ in $$ .", "Here $f^y$ denotes the pullback of $y$ along $f$ obtained from some cartesian lifting; the resulting “set of morphisms from $x$ to $y$ ” in $^\\circ $ is independent, up to canonical isomorphism, of the choice of cartesian lift.", "However, there is no obvious way to define it such that $^\\circ {}^\\circ $ is equal to $$ , rather than merely canonically isomorphic.", "Of course, the 2-category of fibrations over is biequivalent to the 2-category of -indexed categories, which has a strict duality involution induced from its codomain $\\mathcal {C}\\mathit {at}$ .", "Let $\\mathcal {A}$ be a bicategory with a duality involution, let be a class of morphisms of $\\mathcal {A}$ admitting a calculus of right fractions in the sense of [26], and suppose moreover that if $v\\in $ then $v^\\circ \\in $ .", "Then the bicategory of fractions $\\mathcal {A} [^{-1}]$ also admits a duality involution, constructed using its universal property [26] as follows.", "Let $\\ell : \\mathcal {A} \\rightarrow \\mathcal {A} [^{-1}] $ be the localization functor.", "By assumption, the composite $\\mathcal {A} \\xrightarrow{}\\mathcal {A} ^{\\mathrm {co}}\\xrightarrow{} \\mathcal {A} [^{-1}] ^{\\mathrm {co}}$ takes morphisms in to equivalences.", "Thus, it factors through $\\ell $ , up to equivalence, by a functor that we denote $((-)^{\\lozenge })^{\\mathrm {co}}: \\mathcal {A} [^{-1}] \\rightarrow \\mathcal {A} [^{-1}] ^{\\mathrm {co}}$ (that is, a functor whose 2-cell dual we denote $(-)^{\\lozenge }$ ).", "Now the pasting composite composite ${& && \\mathcal {A} [dr]^\\ell \\\\\\mathcal {A} [dr]_{\\ell } [rr]_{((-)^\\circ )^{\\mathrm {co}}} @{=}@/^5mm/[urrr] @{}[drrr]\|{\\Downarrow \\simeq } @{}[urrr]\|{\\Downarrow } &&\\mathcal {A} ^{\\mathrm {co}}[dr]_{\\ell ^{\\mathrm {co}}} [ur]_{(-)^\\circ } @{}[rr]\|{\\Downarrow \\simeq } && \\mathcal {A} [^{-1}] \\\\& \\mathcal {A} [^{-1}] [rr]_{((-)^{\\lozenge })^{\\mathrm {co}}} && \\mathcal {A} [^{-1}] ^{\\mathrm {co}}[ur]_{(-)^{\\lozenge }}}$ is a pseudonatural equivalence from $\\ell $ to $(-)^{\\lozenge }\\circ ((-)^{\\lozenge })^{\\mathrm {co}}\\circ \\ell $ .", "Hence, by the universal property of $\\ell $ , it is isomorphic to the whiskering by $\\ell $ of some pseudonatural equivalence $ {\\mathcal {A} [^{-1}] [dr]_{((-)^{\\lozenge })^{\\mathrm {co}}} @{=}[rr] && \\mathcal {A} [^{-1}].\\\\& \\mathcal {A} [^{-1}] ^{\\mathrm {co}}[ur]_{(-)^{\\lozenge }} @{}[u]\|(.6){\\Downarrow ^{\\prime }}}$ Similar whiskering arguments produce the modification $\\zeta ^{\\prime }$ and verify its axiom.", "Note that this induced duality involution on $\\mathcal {A} [^{-1}]$ will not generally be strict, even if the one on $\\mathcal {A}$ is.", "Specifically, with careful choices we can make $(-)^{\\lozenge }$ strictly involutory on objects, 1-cells, and 2-cells, but there is no obvious way to make it a strict 2-functor.", "(On the other hand, as remarked in label@name@sec:introduction , often $\\mathcal {A} [^{-1}]$ is biequivalent to some naturally-occurring 2-category having a strict duality involution, such as the examples of étendues and stacks considered in [26].)", "A related special case is that if we work in an ambient set theory not assuming the axiom of choice, then we might take $\\mathcal {A} =\\mathcal {C}\\mathit {at} $ and the class of fully faithful and essentially surjective functors.", "In this case $\\mathcal {A} [^{-1}]$ is equivalent to the bicategory of categories and anafunctors [23], [28], which therefore inherits a weak duality involution.", "Let be a compact closed bicategory (also called symmetric autonomous) as in [7], [32].", "Thus means that is symmetric monoidal, and moreover each object $x$ has a dual $x^\\circ $ with respect to the monoidal structure, with morphisms $\\eta :{1} \\rightarrow x\\otimes x^\\circ $ and $\\varepsilon : x^\\circ \\otimes x \\rightarrow {1} $ satisfying the triangle identities up to isomorphism.", "If we choose such a dual for each object, then $(-)^\\circ $ can be made into a biequivalence $^{\\mathrm {op}}\\rightarrow $ , sending a morphism $g:y\\rightarrow x$ to the composite ${ x^\\circ [r]^-{\\eta _y} & x^\\circ \\otimes y\\otimes y^\\circ [r]^-{g} & x^\\circ \\otimes x \\otimes y^\\circ [r]^-{\\varepsilon _x} & y^\\circ },{}$ with $\\eta $ and $\\varepsilon $ becoming pseudonatural transformations.", "Moreover, this functor $^{\\mathrm {op}}\\rightarrow $ looks exactly like a duality involution except that $(-)^{\\mathrm {co}}$ has been replaced by $(-)^{\\mathrm {op}}$ : we have a pseudonatural adjoint equivalence $ {[dr]_{((-)^\\circ )^{\\mathrm {op}}} @{=}[rr] && .\\\\& ^{\\mathrm {op}}[ur]_{(-)^\\circ } @{}[u]\|(.6){\\Downarrow }}$ and an invertible modification $ {^{\\mathrm {op}}[r]^{(-)^\\circ } & [dr]_{((-)^\\circ )^{\\mathrm {op}}} @{=}[rr] && \\\\&& ^{\\mathrm {op}}[ur]_{(-)^\\circ } @{}[u]\|(.6){\\Downarrow }}$ $satisfying the same axiom as in \\ref {label@name@defn:duality-involution}~\\ref {defn:duality-involution}.Explicitly, $ is the composite $ x \\xrightarrow{} x\\otimes x^\\circ \\otimes x^\\circ {}^\\circ \\xrightarrow{}x^\\circ \\otimes x\\otimes x^\\circ {}^\\circ \\xrightarrow{} x^\\circ {}^\\circ $ and $\\zeta $ is obtained as a pasting composite ${ & x^\\circ x^\\circ {}^\\circ x^\\circ {}^\\circ {}^\\circ [r]^{\\sim } [dd] & x^\\circ {}^\\circ x^\\circ x^\\circ {}^\\circ {}^\\circ [dr]^{\\varepsilon _{x^\\circ } x^\\circ {}^\\circ {}^\\circ } [dd] \\\\x^\\circ [ur]^{x^\\circ \\eta _{x^\\circ {}^\\circ }} [dr]_{(x \\eta _{x^\\circ })^\\circ }@{}[r]\|{\\Downarrow \\cong } & @{}[r]\|{\\Downarrow \\cong }& @{}[r]\|{\\Downarrow \\cong } &x^\\circ {}^\\circ {}^\\circ \\\\&(x x^\\circ x^\\circ {}^\\circ )^\\circ [r]_\\sim & (x^\\circ x x^\\circ {}^\\circ )^\\circ [ur]_{(\\varepsilon _x x^\\circ {}^\\circ )^\\circ }}$ using the fact that if $x^\\circ $ is a dual of $x$ , then by symmetry of , $x$ is a dual of $x^\\circ $ .", "Now let $\\mathcal {A}$ be the locally full sub-bicategory of maps (left adjoints) in .", "Since passing from left to right adjoints reverses the direction of 2-cells, we have a “take the right adjoint” functor $\\mathcal {A} ^{\\mathrm {coop}}\\rightarrow $ , or equivalently $\\mathcal {A} ^{\\mathrm {co}}\\rightarrow ^{\\mathrm {op}}$ .", "Composing with the above “duality” functor $^{\\mathrm {op}}\\rightarrow $ , we have a functor $\\mathcal {A} ^{\\mathrm {co}}\\rightarrow $ , and since right adjoints in $$ are left adjoints in $^{\\mathrm {op}}$ , this functor lands in $\\mathcal {A}$ , giving $(-)^\\circ :\\mathcal {A} ^{\\mathrm {co}}\\rightarrow \\mathcal {A} $ .", "Of course, equivalences are maps, so the above $$ and $\\zeta $ lie in $\\mathcal {A}$ , and therefore equip $\\mathcal {A} $ with a duality involution.", "This duality involution on $\\mathcal {A}$ is not generally strict or even strong.", "However, as remarked in label@name@sec:introduction , in many naturally-ocurring examples $\\mathcal {A}$ is equivalent to some naturally-ocurring 2-category with a strict duality involution.", "For instance, if $=\\mathcal {P}\\mathit {rof} $ then $\\mathcal {A} \\simeq \\mathcal {C}\\mathit {at} _{\\mathrm {cc}}$ , the 2-category of Cauchy-complete categories; while if $=\\mathcal {S}\\mathit {pan} $ then $\\mathcal {A} \\simeq \\mathbf {Set} $ , and similarly for internal and enriched versions.", "A 2-monadic approach Let $T(\\mathcal {A}) = \\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}$ , an endo-2-functor of the 2-category 2-$\\mathcal {C}\\mathit {at}$ of 2-categories, strict 2-functors, and strict 2-natural transformations.", "We have an obvious strict 2-natural transformation $\\eta :\\mathsf {Id}\\rightarrow T$ , and we define $\\mu :TT\\rightarrow T$ by $ (\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}) + (\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}})^{\\mathrm {co}}\\xrightarrow{}\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}+ \\mathcal {A} ^{\\mathrm {co}}+ \\mathcal {A} \\xrightarrow{} \\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}$ where $\\nabla $ is the obvious “fold” map.", "Theorem 3.1 $T$ is a strict 2-monad, and: Normal pseudo $T$ -algebras are 2-categories with strong duality involutions; Pseudo $T$ -morphisms are duality 2-functors; and Strict $T$ -algebras are 2-categories with strict duality involutions.", "The 2-monad laws for $T$ are straightforward to check.", "By a normal pseudo algebra we mean one for which the unit constraint identifying $\\mathcal {A} \\rightarrow T\\mathcal {A} \\rightarrow \\mathcal {A} $ with the identity map is itself an identity.", "Thus, when $T\\mathcal {A} = \\mathcal {A} +\\mathcal {A} ^{\\mathrm {co}}$ , this means the action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ contains no data beyond a 2-functor $(-)^\\circ :\\mathcal {A} ^{\\mathrm {co}}\\rightarrow \\mathcal {A} $ .", "The remaining data is a 2-natural isomorphism ${TT\\mathcal {A} [r]^{Ta}[d]_{\\mu } {\\cong } &T\\mathcal {A} [d]^{a}\\\\T\\mathcal {A} [r]_{a} &\\mathcal {A}}$ that is satisfying three axioms that can be found, for instance, in [19].", "The right-hand square commutes strictly on the first three summands in its domain, and the second and third of the coherence axioms say exactly that the given isomorphism in these cases is an identity.", "Thus, what remains is the component of the isomorphism on the fourth summand, which has precisely the form of $$ in label@name@defn:duality-involution REF , and it is easy to check that the first coherence axiom reduces in this case to the identity $\\zeta $ .", "This proves REF , and REF follows immediately.", "Similarly, for REF , a pseudo $T$ -morphism is a 2-functor $F:\\mathcal {A} \\rightarrow $ together with a 2-natural isomorphism ${T\\mathcal {A} [r]^{T F}[d] {\\cong } &T[d]\\\\\\mathcal {A} [r]_F &}$ that is satisfying two coherence axioms also listed in [19].", "This square commutes strictly on the first summand of its domain, and the second coherence axiom ensures that the isomorphism is the identity there.", "So the remaining data is the isomorphism on the second summand, which has precisely the form of $\\mathfrak {i}$ in label@name@defn:duality-functor REF , and the first coherence axiom reduces to the identity $\\theta $ .", "In particular, we obtain an automatic definition of a “duality 2-natural transformation”: a $T$ -2-cell between pseudo $T$ -morphisms.", "This also gives us another source of examples.", "The 2-category $T\\text{-}\\mathcal {A}\\mathit {lg} _s$ of strict $T$ -algebras and strict $T$ -morphisms is complete with limits created in $2\\text{-}\\mathcal {C}\\mathit {at} $ , including in particular Eilenberg–Moore objects [33].", "Thus, for any monad $M$ in this 2-category — which is to say, a 2-monad that is a strict duality 2-functor and whose unit and multiplication are duality 2-natural transformations — the 2-category $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ of strict $M$ -algebras and strict $M$ -morphisms is again a strict $T$ -algebra, i.e.", "has a strict duality involution.", "Similarly, by [1] the 2-category $T\\text{-}\\mathcal {A}\\mathit {lg} $ of strict $T$ -algebras and pseudo $T$ -morphisms has PIE-limits, including EM-objects.", "Thus, we can reach the same conclusion even if $M$ is only a strong duality 2-functor.", "And since 2-$\\mathcal {C}\\mathit {at}$ is locally presentable and $T$ has a rank, there is another 2-monad $T^{\\prime }$ whose strict algebras are the pseudo $T$ -algebras; thus we can argue similarly in the 2-category $T\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ of pseudo $T$ -algebras and pseudo $T$ -morphisms, so that strong duality involutions also lift to $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ .", "Usually, of course, we are more interested in the 2-category $M\\text{-}\\mathcal {A}\\mathit {lg} $ of strict $M$ -algebras and pseudo $M$ -morphisms.", "It might be possible to enhance the above abstract argument to apply to this case using techniques such as [18], [25], but it is easy enough to check directly that if $M$ lies in $T\\text{-}\\mathcal {A}\\mathit {lg} $ or $T\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ , then so does $M\\text{-}\\mathcal {A}\\mathit {lg} $ .", "If $(A,a)$ is an $M$ -algebra, then the induced $M$ -algebra structure on $A^\\circ $ is the composite $ M(A^\\circ ) \\xrightarrow{} (MA)^\\circ \\xrightarrow{} A^\\circ $ and if $(f,):(A,a) \\rightarrow (B,b)$ is a pseudo $M$ -morphism (where $: a\\circ Mf \\xrightarrow{}f \\circ b$ ), then $f^\\circ $ becomes a pseudo $M$ -morphism with the following structure 2-cell: ${M(B^\\circ )[r]^{\\mathfrak {i}} &(MB)^\\circ [rr]^{b^\\circ } &&B^\\circ \\\\M(A^\\circ )[r]_{\\mathfrak {i}}[u]^{M(f^\\circ )} {\\cong } &(MA)^\\circ [rr]_{a^\\circ }[u]\|{(Mf)^\\circ } {{(^{-1})^\\circ }} &&A^\\circ [u]_{f^\\circ }}$ The axiom $\\theta $ of $\\mathfrak {i}$ (which is an equality since $M$ is a strong duality 2-functor) ensures that $$ lifts to $M\\text{-}\\mathcal {A}\\mathit {lg} $ (indeed, to $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ ), and its own $\\theta $ axiom is automatic.", "A similar argument applies to $M\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ .", "Thus, 2-categories of algebraically structured categories such as monoidal categories, braided or symmetric monoidal categories, and so on, admit strict duality involutions, even when their morphisms are of the pseudo sort.", "(Of course, this is impossible for lax or colax morphisms, since dualizing the categories involved would switch lax with colax.)", "In theory, this could be another source of weak duality involutions that are not strong: if for a 2-monad $M$ the transformation $\\mathfrak {i}$ were not a strictly 2-natural isomorphism or its axiom $\\theta $ were not strict, then $M\\text{-}\\mathcal {A}\\mathit {lg} $ would only inherit a weak duality involution, even if the duality involution on the original 2-category were strict.", "However, I do not know any examples of 2-monads that behave this way.", "We end this section with the strong-to-strict coherence theorem.", "Theorem 3.2 If $\\mathcal {A}$ is a 2-category with a strong duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality 2-functor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a 2-equivalence.", "The 2-category $2\\text{-}\\mathcal {C}\\mathit {at} $ admits a factorization system $(,)$ in which $$ consists of the 2-functors that are bijective on objects and $$ of the 2-functors that are 2-fully-faithful, i.e.", "an isomorphism on hom-categories.", "Moreover, this factorization system satisfies the assumptions of [19], and we have $T\\subseteq $ .", "Thus, [19] (which is an abstract version of [24]), together with the characterizations of label@name@thm:2monad REF , implies the desired result.", "Inspecting the proof of the general coherence theorem, we obtain a concrete construction of $\\mathcal {A}^{\\prime }$ : it is the result of factoring the pseudo-action map $T\\mathcal {A} \\rightarrow \\mathcal {A} $ as a bijective-on-objects 2-functor followed by a 2-fully-faithful one.", "In other words, the objects of $\\mathcal {A}^{\\prime }$ are two copies of the objects of $\\mathcal {A} $ , one copy representing each object and one its opposite, with the duality interchanging them.", "The morphisms and 2-cells are then easy to determine.", "It remains, therefore, to pass from a weak duality involution on a bicategory to a strong one on a 2-category.", "We proceed up the right-hand side of the ladder from label@name@sec:introduction .", "Contravariance through virtualization As mentioned in label@name@sec:introduction , for much of the paper we will work in the extra generality of “twisted group actions”.", "Specifically, let be a complete and cocomplete closed symmetric monoidal category, and let $G$ be a group that acts on by strong symmetric monoidal functors.", "We write the action of $g\\in G$ on $W\\in $ as ${W}^{g}$ .", "For simplicity, we suppose that the action is strict, i.e.", "${({W}^{g})}^{h} = {W}^{gh}$ and ${W}^{1} = W$ strictly (and symmetric-monoidal-functorially).", "The case we are most interested in, which will yield our theorems about duality involutions on 2-categories, is when $=\\mathbf {Cat} $ with $G$ the 2-element group $\\lbrace +,-\\rbrace $ with $+$ the identity element (a copy of $/2$ ), and ${A}^{-}=A^{\\mathrm {op}}$ .", "However, there are other examples as well.", "Here are a few, also mentioned in label@name@sec:introduction , that yield “duality involutions” with a similar flavor.", "Let $=\\mathbf {2}\\text{-}\\mathbf {Cat}$ , with $G$ as the 4-element group $\\lbrace ++,-+,+-,--\\rbrace $ (a copy of $/2\\times /2$ ), and ${A}^{-+}=A^{\\mathrm {op}}$ , ${A}^{+-}=A^{\\mathrm {co}}$ , and hence ${A}^{--}=A^{\\mathrm {coop}}$ .", "If we give $$ the Gray monoidal structure as in [12], this example leads to a theory of duality involutions on Gray-categories.", "Let $$ be the category of strict $n$ -categories, with $G = (/2)^n$ acting by reversal of $k$ -morphisms at all levels.", "Since a category enriched in strict $n$ -categories is exactly a strict $(n+1)$ -category, we obtain a theory of duality involutions on strict $(n+1)$ -categories.", "Let $=\\mathbf {sSet}$ , the category of simplicial sets, with $G=\\lbrace +,-\\rbrace $ , and ${A}^{-}$ obtained by reversing the directions of all simplices in $A$ .", "This leads to a theory of duality involutions on simplicially enriched categories that is appropriate when the simplicial sets are regarded as modeling $(\\infty ,1)$ -categories as quasicategories [16], [22], so that simplicially enriched categories are a model for $(\\infty ,2)$ -categories.", "For example, such simplicially enriched categories are used in [29] to define a notion of “$\\infty $ -cosmos” analogous to the “fibrational cosmoi” of [34], so such duality involutions could be a first step towards an $\\infty $ -version of [36].", "Combining the ideas of the last two examples, if $$ is the category of $\\Theta _n$ -spaces as in [27], then $(/2)^n$ acts on it by reversing directions at all dimensions.", "Thus, we obtain a theory of duality involutions on categories enriched in $\\Theta _n$ -spaces, which in [3] were shown to be a model of $(\\infty ,n+1)$ -categories.", "Note that we do not assume the action of $G$ on is by -enriched functors, since in most of the above examples this is not the case.", "In particular, $(-)^{\\mathrm {op}}$ is not a 2-functor.", "We also note that most or all of the theory would probably be the same if $G$ were a 2-group rather than just a group, but we do not need this extra generality.", "Since the action of $G$ on is symmetric monoidal, it extends to an action on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ applied homwise, which we also write ${\\mathcal {A}}^{g}$ , i.e.", "${\\mathcal {A}}^{g}(x,y) = {(\\mathcal {A} (x,y))}^{g}$ .", "In our motivating example we have ${\\mathcal {A}}^{-}= \\mathcal {A} ^{\\mathrm {co}}$ for a 2-category $\\mathcal {A}$ .", "We now define a 2-monad $T$ on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ by $ T\\mathcal {A} = \\sum _{g\\in G} {\\mathcal {A}}^{g}.", "$ The unit $\\mathcal {A} \\rightarrow T\\mathcal {A} $ includes the summand indexed by $1\\in G$ , and the multiplication uses the fact that each action, being an equivalence of categories (indeed, an isomorphism of categories), is cocontinuous: $TT\\mathcal {A} = \\sum _{g\\in G} {(T\\mathcal {A})}^{g}= \\sum _{g\\in G} {\\left(\\sum _{h\\in G} {\\mathcal {A}}^{h}\\right)}^{g}\\cong \\sum _{g\\in G} \\sum _{h\\in G} {({\\mathcal {A}}^{h})}^{g}\\cong \\sum _{g\\in G} \\sum _{h\\in G} {\\mathcal {A}}^{hg}$ which we can map into $T\\mathcal {A} $ by sending the $(g,h)$ summand to the $hg$ -summand.", "We will refer to a normal pseudo $T$ -algebra structure as a twisted $G$ -action; it equips a -category $\\mathcal {A}$ with actions ${(-)}^{g} : {\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ that are suitably associative up to coherent isomorphism (with ${(x)}^{1}=x$ strictly).", "In our motivating example of $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ , the monad $T$ agrees with the one we constructed in label@name@sec:2-monadic-approach ; thus twisted $G$ -actions are strong duality involutions (and likewise for their morphisms and 2-cells).", "If we write $[x,y]$ for the internal-hom of , then we have maps ${[x,y]}^{g} \\rightarrow [{x}^{g}, {y}^{g}]$ obtained by adjunction from the composite $ {[x,y]}^{g} \\otimes {x}^{g} \\xrightarrow{}{([x,y]\\otimes x)}^{g}\\rightarrow {y}^{g}$ Since the $[x,y]$ are the hom-objects of the -category , these actions assemble into a -functor ${(-)}^{g}:{}^{g}\\rightarrow $ , and as $g$ varies they give itself a twisted $G$ -action.", "(Thus, among the three different actions we are denoting by ${(-)}^{g}$ — the given one on , the induced one on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ , and an arbitrary twisted $G$ -action — the first is a special case of the third.)", "In particular, we obtain in this way the canonical strong (in fact, strict) duality involution on $\\mathbf {Cat}$ .", "Now we note that $T$ extends to a normal monad in the sense of [5] on the proarrow equipment $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ , as follows.", "As in [30], [5], we view equipments as pseudo double categories satisfying with a “fibrancy” condition saying that horizontal arrows (the “proarrow” direction, for us) can be pulled back universally along vertical ones (the “functor” direction).", "In $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ the objects are -categories, a horizontal arrow $\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}$ is a profunctor (i.e.", "a -functor $^{\\mathrm {op}}\\otimes \\mathcal {A} \\rightarrow $ ), a vertical arrow $\\mathcal {A} \\rightarrow $ is a -functor, and a square ${\\mathcal {A} [r]\|{\|}^M[d]_F @{}[dr]\|{\\Downarrow } &[d]^G\\\\[r]\|{\|}_N &}$ is a -natural transformation $M(b,a)\\rightarrow N(G(b),F(a))$ .", "A monad on an equipment is strictly functorial in the vertical direction, laxly functorial in the horizontal direction, and its multiplication and unit transformations consist of vertical arrows and squares.", "In our case, we already have the action of $T$ on -categories and -functors.", "A -profunctor $M:\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}$ induces another one ${M}^{g} : {\\mathcal {A}}^{g}0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}{}^{g}$ by applying the $G$ -action objectwise, and by summing up over $g$ we have $T M : T \\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T $ .", "This is in fact pseudofunctorial on profunctors.", "Finally, the unit and multiplication are already defined as vertical arrows, and extend to squares in an evident way: ${\\mathcal {A} [r]\|{\|}^M[d]_\\eta @{}[dr]\|{\\Downarrow } &[d]^\\eta \\\\T\\mathcal {A} [r]\|{\|}_{T M} &T}$ Since we have a monad on an equipment, we can define “$T$ -multicategories” in $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ , which following [5] we call virtual $T$ -algebras.", "For our specific monad $T$ , we will refer to virtual $T$ -algebras as $G$ -variant -categories.", "Such a gadget is a -category $\\mathcal {A}$ together with a profunctor $\\underline{A}:\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T\\mathcal {A} $ , a unit isomorphism $\\mathcal {A} (x,y) \\xrightarrow{}\\underline{A} (\\eta (x),y)$ , and a composition ${\\mathcal {A} [r]\|{\|}^\\underline{A} @{=}[d] @{}[drr]\|{\\Downarrow } &T\\mathcal {A} [r]\|{\|}^{T\\underline{A}} &TT\\mathcal {A} [d]^\\mu \\\\\\mathcal {A} [rr]\|{\|}_\\underline{A} & &T\\mathcal {A}}$ satisfying associativity and unit axioms.", "If we unravel this explicitly, we see that a $G$ -variant -category has a set of objects along with, for each pair of objects $x,y$ and each $g\\in G$ , a hom-object $\\underline{A} ^g(x,y)\\in $ , plus units ${1} \\rightarrow \\underline{A} ^1(x,x)$ and compositions $\\underline{A} ^g(y,z) \\otimes {(\\underline{A} ^h(x,y))}^{g} \\rightarrow \\underline{A} ^{hg}(x,z)$ satisfying the expected axioms.", "(Technically, in addition to the hom-objects $\\underline{A} ^1(x,y)$ it has the hom-objects $\\mathcal {A} (x,y)$ that are isomorphic to them, but we may ignore this duplication of data.)", "We may refer to the elements of $\\underline{A} ^g(x,y)$ as $g$ -variant morphisms.", "The rule for the variance of composites is easier to remember when written in diagrammatic order: if we denote $\\alpha \\in \\underline{A} ^g(x,y)$ by $\\alpha :x\\xrightarrow[g]{}y$ , then the composite of $x\\xrightarrow[g]{} y \\xrightarrow[h]{} z$ is $x\\xrightarrow[gh]{} z$ .", "(Of course, in our motivating example $G$ is commutative, so the order makes no difference.)", "In the specific case of $G=\\lbrace +,-\\rbrace $ acting on $\\mathbf {Cat} $ , we can unravel the definition more explicitly as follows.", "Definition 3 A 2-category with contravariance is a $G$ -variant -category for $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ .", "Thus it consists of [leftmargin=2em] A collection $\\operatorname{ob}\\underline{A} $ of objects; For each $x,y\\in \\operatorname{ob}\\underline{A} $ , a pair of categories $\\underline{A} ^+(x,y)$ and $\\underline{A} ^-(x,y)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , an object $1_x \\in \\underline{A} ^+(x,x)$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ , composition functors $\\underline{A} ^+(y,z) \\times \\underline{A} ^+(x,y) &\\overset{}{\\longrightarrow }\\underline{A} ^+(x,z)\\\\\\underline{A} ^-(y,z) \\times \\underline{A} ^-(x,y)^{\\mathrm {op}}&\\overset{}{\\longrightarrow }\\underline{A} ^+(x,z)\\\\\\underline{A} ^+(y,z) \\times \\underline{A} ^-(x,y) &\\overset{}{\\longrightarrow }\\underline{A} ^-(x,z)\\\\\\underline{A} ^-(y,z) \\times \\underline{A} ^+(x,y)^{\\mathrm {op}}&\\overset{}{\\longrightarrow }\\underline{A} ^-(x,z);$ such that [leftmargin=2em] Four ($2\\cdot 2^1$ ) unitality diagrams commute; and Eight ($2^3$ ) associativity diagrams commute.", "Like any kind of generalized multicategory, $G$ -variant -categories form a 2-category.", "We leave it to the reader to write out explicitly what the morphisms and 2-cells in this 2-category look like; in our example of interest we will call them 2-functors preserving contravariance and 2-natural transformations respecting contravariance.", "Now, according to [5], any twisted $G$ -action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ gives rise to a $G$ -variant -category with $\\underline{A} = \\mathcal {A} (a,1)$ , which in our situation means $\\underline{A} ^g(x,y) = \\mathcal {A} ({x}^{g},y)$ (where ${x}^{g}$ refers, as before, to the $g$ -component of the action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ ).", "In particular, any 2-category with a strong duality involution can be regarded as a 2-category with contravariance, where $\\underline{A} ^+(x,y) = \\mathcal {A} (x,y)$ and $\\underline{A} ^-(x,y) = \\mathcal {A} (x^\\circ ,y)$ .", "Moreover, by [5], a $G$ -variant -category $\\underline{A}$ arises from a twisted $G$ -action exactly when The profunctor $\\underline{A}: \\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T\\mathcal {A} $ is representable by some $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ , and The induced 2-cell $\\overline{a} : a\\circ \\mu \\rightarrow a \\circ Ta$ is an isomorphism.", "Condition REF means that for every $x\\in \\underline{A} $ and every $g\\in G$ , there is an object “${x}^{g}$ ” and an isomorphism $\\underline{A} ^g(x,y) \\cong \\underline{A} ^1({x}^{g},y)$ , natural in $y$ .", "The Yoneda lemma implies this isomorphism is mediated by a “universal $g$ -variant morphism” $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ .", "Condition REF then means that for any $x\\in \\underline{A} $ and $g,h\\in G$ , the induced map $\\psi _{h,g,x}:{x}^{gh} \\rightarrow {({x}^{g})}^{h}$ is an isomorphism.", "(This map arises by composing $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ with $\\chi _{h,{x}^{g}} \\in \\underline{A} ^h({x}^{g},{({x}^{g})}^{h})$ to obtain a map in $\\underline{A} ^{gh}(x,{({x}^{g})}^{h})$ , then applying the defining isomorphism of ${x}^{gh}$ .)", "As usual for generalized multicategories, this is equivalent to requiring a stronger universal property of ${x}^{g}$ : that precomposing with $\\chi _{g,x}$ induces isomorphisms $\\underline{A} ^h({x}^{g},y) \\xrightarrow{}\\underline{A} ^{gh}(x,y){}$ for all $h\\in G$ .", "(This again is more mnemonic in diagrammatic notation: any arrow $x \\xrightarrow[gh]{} y$ factors uniquely through $\\chi _{g,x}$ by a morphism ${x}^{g} \\xrightarrow[h]{} y$ , i.e.", "variances on the arrow can be moved into the action on the domain, preserving order.)", "This is because the following diagram commutes by definition of $\\psi _{h,g,x}$ , and the vertical maps are isomorphisms by definition of $\\chi $ : $@C=4pc{\\underline{A} ^{gh}(x,y)@{<-}[r]^{-\\circ \\chi _{g,x}}@{<-}[d]_{-\\circ \\chi _{gh,x}} &\\underline{A} ^h({x}^{g},y)@{<-}[d]^{-\\circ \\chi _{h,{x}^{g}}}\\\\\\underline{A} ^1({x}^{gh},y)@{<-}[r]_{-\\circ \\psi _{h,g,x}} &\\underline{A} ^1({({x}^{g})}^{h},y)}$ If ${x}^{g}$ is an object equipped with a morphism $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ satisfying this stronger universal property (), we will call it a $g$ -variator of $x$ .", "In our motivating example $=\\mathbf {Cat} $ with $g=-$ , we call a $-$ -variator an opposite.", "Explicitly, this means the following.", "Definition 4 In a 2-category with contravariance $\\underline{A}$ , a (strict) opposite of an object $x$ is an object $x^\\circ $ equipped with a contravariant morphism $\\chi _x\\in \\underline{A} ^-(x,x^\\circ )$ such that precomposing with $\\chi _x$ induces isomorphisms of hom-categories for all $y$ : $\\underline{A} ^+(x^\\circ ,y) &\\xrightarrow{}\\underline{A} ^-(x,y)\\\\\\underline{A} ^-(x^\\circ ,y) &\\xrightarrow{}\\underline{A} ^+(x,y).$ In fact, $g$ -variators can also be characterized more explicitly.", "The second universal property of $\\chi _{g,x}\\in {\\underline{A}}^{g}(x,{x}^{g})$ means in particular that the identity $1_x\\in {\\underline{A}}^{1}(x,x)$ can be written as $\\xi _{g,x} \\circ \\chi _{g,x}$ for a unique $\\xi _{g,x} \\in \\underline{A} ^{g^{-1}}({x}^{g},x)$ .", "(This is the first place where we have used the fact that $G$ is a group rather than just a monoid.)", "Moreover, since $(\\chi _{g,x}\\circ \\xi _{g,x}) \\circ \\chi _{g,x} = \\chi _{g,x} \\circ (\\xi _{g,x}\\circ \\chi _{g,x}) = \\chi _{g,x} $ it follows by the first universal property of $\\chi _{g,x}$ that $\\chi _{g,x}\\circ \\xi _{g,x} = 1_{{x}^{g}}$ as well.", "Thus, $\\chi _{g,x}$ and $\\xi _{g,x}$ form a “$g$ -variant isomorphism” between $x$ and ${x}^{g}$ .", "On the other hand, it is easy to check that any such $g$ -variant isomorphism between $x$ and an object $y$ makes $y$ into a $g$ -variator of $x$ .", "Thus, we have: Proposition 1 Any $g$ -variant -functor $F:\\underline{A} \\rightarrow $ preserves $g$ -variators.", "In particulary, any 2-functor preserving contravariance also preserves opposites.", "It obviously preserves “$g$ -variant isomorphisms”.", "Thus we have: Theorem 4.1 The 2-category of 2-categories with strong duality involutions, duality 2-functors, and duality 2-natural transformations is 2-equivalent to the 2-category of 2-categories with contravariance in which every object has a strict opposite, 2-functors preserving contravariance, and 2-natural transformations respecting contravariance.", "By [5] and the remarks preceding label@name@defn:opposite REF , the latter 2-category is equivalent to the 2-category of pseudo $T$ -algebras, lax $T$ -morphisms, and $T$ -2-cells.", "However, label@name@thm:gm-abs REF implies that in fact every lax $T$ -morphism is a pseudo $T$ -morphism.", "Finally, every pseudo $T$ -algebra is isomorphic to a normal pseudo one obtained by re-choosing ${(-)}^{1}$ to be the identity (which it is assumed to be isomorphic to).", "Contravariance through enrichment We continue with our setup from label@name@sec:genmulti , with a complete and cocomplete closed monoidal category and a group $G$ acting on .", "We start by noticing that the monad $T$ on $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ constructed in label@name@sec:genmulti can actually be obtained in a standard way from a simpler monad.", "Recall that there is another equipment $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ whose objects are sets, whose vertical arrows are functions, and whose horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}Y$ are “-valued matrices”, which are just functions $Y\\times X \\rightarrow $ ; we call them matrices because we compose them by “matrix multiplication”.", "The equipment $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ is obtained from $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ by applying a functor $\\mathbb {M}\\mathsf {od}$ that constructs monoids (monads) and modules in the horizontal directions (see [30], [5]).", "We now observe that our monad $T$ , like many monads on equipments of profunctors, is also in the image of $\\mathbb {M}\\mathsf {od}$ .", "Let $S$ be the following monad on $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ .", "On objects and vertical arrows, it acts by $S(X)=X\\times G$ .", "On a -matrix $M:Y\\times X \\rightarrow $ it acts by $SM((y,h),(x,g)) ={\\left\\lbrace \\begin{array}{ll}{(M(y,x))}^{g} &\\quad g=h\\\\\\emptyset &\\quad g\\ne h\\end{array}\\right.", "}$ We may write this schematically using a Kronecker delta as $SM((y,h),(x,g)) = \\delta _{g,h}\\cdot {(M(y,x))}^{g}.$ On a composite of matrices $X 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{M} Y 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{N} Z$ we have $(SM \\odot SN)((z,k),(x,g)) &= \\sum _{(y,h)} (\\delta _{g,h}\\cdot {M(y,x)}^{g}) \\otimes (\\delta _{h,k}\\cdot {N(z,y)}^{h})\\\\&\\cong \\delta _{k,g}\\sum _{y} {M(y,x)}^{g} \\otimes {N(z,y)}^{g}\\\\&\\cong \\delta _{k,g} \\Big (\\sum _{y} \\big (M(y,x) \\otimes N(z,y)\\big )\\Big )^{g}\\\\&= \\delta _{k,g}\\cdot {(M\\odot N)(z,x)}^{g}\\\\&= S(M\\odot N)((z,k),(x,g))$ making $S$ a pseudofunctor.", "The monad multiplication and unit are induced from the multiplication and unit of $G$ ; the squares ${X[r]\|{\|}^M[d]_\\eta @{}[dr]\|{\\Downarrow } &Y[d]^\\eta \\\\SX[r]\|{\|}_{S M} &SY}$ map the components $M(y,x)$ and ${({M(y,x)}^{g})}^{h}$ isomorphically to ${M(y,x)}^{1}$ and ${M(y,x)}^{gh}$ respectively.", "Now, recalling that $T\\mathcal {A} = \\sum _{g\\in G} {\\mathcal {A}}^{g}$ , we see that $\\operatorname{ob}(T\\mathcal {A}) = \\operatorname{ob}(\\mathcal {A}) \\times G$ and $T\\mathcal {A} ((y,h),(x,g)) = \\delta _{h,g}\\cdot {(\\mathcal {A} (y,x))}^{g},$ and so in fact $T \\cong \\mathbb {M}\\mathsf {od} (S)$ .", "Thus, by [5], virtual $T$ -algebras can be identified with “$S$ -monoids”; these are defined like virtual $S$ -algebras, with sets and matrices of course replacing categories and profunctors, and omitting the requirement that the unit be an isomorphism.", "Thus, an $S$ -monoid consists of a set $X$ of objects, a function $\\underline{A}:S(X)\\times X = X\\times G\\times X \\rightarrow $ , unit maps $1_x:I\\rightarrow \\underline{A} ^1(x,x)$ , and composition maps that turn out to look like $\\underline{A} ^g(y,z) \\otimes {(\\underline{A} ^h(x,y))}^{g} \\rightarrow \\underline{A} ^{hg}(x,z)$ .", "Note that this is exactly what we obtain from a virtual $T$ -algebra by omitting the redundant data of the hom-objects $\\mathcal {A} (x,y)$ and their isomorphisms to $\\underline{A} ^1(x,y)$ ; this is essentially the content of [5] in our case.", "In [5], the construction of $S$ -monoids is factored into two: first we build a new equipment $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ whose objects and vertical arrows are the same as $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ but whose horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}Y$ are the horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SY$ in $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ , and then we consider horizontal monoids in $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ .", "In fact, $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ is in general only a “virtual equipment” (i.e.", "we cannot compose its horizontal arrows, though we can “map out of composites” like in a multicategory), but in our case it is an ordinary equipment because $S$ is “horizontally strong” [5].", "This means that $S$ is a strong functor (which we have already observed) and that the induced maps of matrices $(\\eta ,1)_!M &\\rightarrow (1,\\eta )^S M \\\\(\\mu ,1)_!SSM &\\rightarrow (1,\\mu )^\\odot SSM$ are isomorphisms, where $f^$ and $f_!$ denote the pullback and its left adjoint pushforward of matrices along functions.", "Indeed, we have $(\\eta ,1)_!M((y,h),x) &= \\delta _{h,1} \\cdot M(y,x)\\hspace{56.9055pt}\\text{while}\\\\(1,\\eta )^S M((y,h),x) &= SM((y,h),(x,1))\\\\&= \\delta _{h,1} \\cdot {(M(y,x))}^{1}\\\\&= \\delta _{h,1} \\cdot M(y,x)$ and likewise $(\\mu ,1)_!SSM((y,h),((x,g_1),g_2)) &= \\textstyle \\sum _{h_2 h_1 = h} SSM(((y,h_1),h_2),((x,g_1),g_2))\\\\&= \\textstyle \\sum _{h_1 h_2 = h} \\delta _{h_2,g_2} \\cdot \\Big (\\delta _{h_1,g_1} \\cdot {M(y,x)}^{g_1}\\Big )^{g_2}\\\\&= \\textstyle \\sum _{h_1 h_2 = h} \\delta _{h_2,g_2}\\delta _{h_1,g_1} \\cdot {M(y,x)}^{g_1 g_2}\\\\&= \\delta _{h,g_1 g_2} \\cdot {M(y,x)}^{g_1 g_2}\\\\\\multicolumn{2}{l}{\\text{while}}\\\\(1,\\mu )^\\odot SSM((y,h),((x,g_1),g_2)) &= SSM((y,h),(x,g_1 g_2))\\\\&= \\delta _{h,g_1 g_2} \\cdot {M(y,x)}^{g_1 g_2}.$ Inspecting the definition of composition in [5], we see that the composite of $M:X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SY$ and $N:Y0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SZ$ is $ (M\\odot _S N)((z,h),x) = \\sum _y \\sum _{g_1g_2 = h} M((y,g_1),x) \\odot {N((z,g_2),y)}^{g_1}$ Note that what comes after the $\\sum _y$ depends only on $M((y,-),x)$ and $N((z,-),y)$ , which are objects of $^G$ .", "Thus, if we write $\\textstyle \\int _G\\!$ for the category $^G$ with the following monoidal structure: $ (M \\otimes N)(h) = \\sum _{g_1 g_2 = h} M(g_1) \\odot {N(g_2)}^{g_1}$ then we have $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S) \\cong (\\textstyle \\int _G\\!", ")\\text{-}\\mathbb {M}\\mathsf {at} $ .", "It follows that $S$ -monoids (that is, $G$ -variant -categories) can equivalently be regarded as ordinary monoids in the equipment $(\\textstyle \\int _G\\!", ")\\text{-}\\mathbb {M}\\mathsf {at} $ .", "But since monoids in an equipment of matrices are simply enriched categories, we can identify $G$ -variant -categories with $\\textstyle \\int _G\\!", "$ -enriched categories.", "Note that this monoidal structure on $\\textstyle \\int _G\\!$ is not symmetric.", "It is a version of Day convolution [6] that is “twisted” by the action of $G$ on (see [20] for further discussion).", "Like an ordinary Day convolution monoidal structure, it is also closed on both sides (as long as is); that is, we have left and right hom-functors $\\!\\!\\,$ and $$ with natural isomorphisms ${}(\\textstyle \\int _G\\!", ")(A\\otimes B, C) \\cong (\\textstyle \\int _G\\!", ")(A, B\\mathrel {} C) \\cong (\\textstyle \\int _G\\!", ")(B, C \\mathrel {\\!\\!}", "A).$ Inspecting the definition of the tensor product in $\\textstyle \\int _G\\!$ , it suffices to define $(B\\mathrel {} C)(g) &\\prod _{h} \\big ({B(h)}^{g}\\mathrel {} C(gh)\\big )\\\\(C \\mathrel {\\!\\!}", "A)(g) &\\prod _{h} \\left({C(hg)}^{h^{-1}} \\mathrel {\\!\\!}", "{A(h)}^{h^{-1}}\\right)$ (This is another place where we use the fact that $G$ is a group rather than a monoid.)", "As usual, it follows that $\\textstyle \\int _G\\!$ can be regarded as a $\\textstyle \\int _G\\!$ -category (that is, as a $g$ -variant -category), with hom-objects $\\underline{\\textstyle \\int _G\\!", "}(A,B) (A\\mathrel {} B)$ .", "(The fact that a closed monoidal category becomes self-enriched is often described only for closed symmetric monoidal categories, but it works just as well for closed non-symmetric ones, as long as we use the right hom.)", "Bringing things back down to each a bit, in our specific case with $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ , let us write $= \\int _{\\lbrace +,-\\rbrace }\\!\\mathbf {Cat} $ .", "The underlying category of is just $\\mathbf {Cat} \\times \\mathbf {Cat} $ , but we denote its objects as $A = (A^+,A^-)$ , with $A^+$ the covariant part and $A^-$ the contravariant part.", "The monoidal structure on is the following nonstandard one: $(A\\otimes B)^+&\\big (A^+\\times B^+\\big ) \\amalg \\big (A^-\\times (B^-)^{\\mathrm {op}}\\big )\\\\(A\\otimes B)^-&\\big (A^+\\times B^-\\big ) \\amalg \\big (A^-\\times (B^+)^{\\mathrm {op}}\\big )$ The unit object is $ {1} (1,0) $ where 1 denotes the terminal category and 0 the initial (empty) one.", "The conclusion of our equipment-theoretic digression above is then the following: Theorem 5.1 The 2-category of 2-categories with contravariance, 2-functors preserving contravariance, and 2-natural transformations respecting contravariance is 2-equivalent to the 2-category of -enriched categories.", "This theorem is easy to prove explicitly as well, of course.", "A -category has, for each pair of objects $x,y$ , a pair of hom-categories $(\\underline{A} ^+(x,y),\\underline{A} ^-(x,y))$ , together with composition functors that end up looking just like those in label@name@defn:2cat-contra REF , and so on.", "But I hope that the digression makes this theorem seem less accidental; it also makes it clear how to generalize it to other examples.", "The underlying ordinary category $\\underline{A} _{\\,o}$ of a 2-category $\\underline{A}$ with contravariance, in the usual sense of enriched category theory, consists of its objects and its covariant 1-morphisms (the objects of the categories $\\underline{A} ^+(x,y)$ ).", "It also has an underlying ordinary 2-category, induced by the lax monoidal forgetful functor $(-)^+: \\rightarrow \\mathbf {Cat} $ , whose hom-categories are the categories $\\underline{A} ^+(x,y)$ ; we denote this 2-category by $\\underline{A} ^+$ .", "Of course, there is no 2-category to denote by “$\\underline{A} ^-$ ”, but we could say for instance that $\\underline{A} ^-$ is a profunctor from $\\underline{A} ^+$ to itself.", "Opposites through enrichment For most of this section, we let $(,\\otimes ,{1})$ be an arbitrary biclosed monoidal category, not assumed symmetric.", "We are, of course, thinking of our from the last section, or more generally $\\textstyle \\int _G\\!$ .", "Suppose $\\underline{A}$ is a -category, that $x\\in \\operatorname{ob}\\underline{A} $ , and $\\omega \\in \\operatorname{ob}$ .", "A copower (or tensor) of $x$ by $\\omega $ is an object $\\omega \\odot x$ of $\\underline{A} $ together with isomorphisms in : ${}\\underline{A} (\\omega \\odot x,y) \\cong \\omega \\mathrel {} \\underline{A} (x,y)$ for all $y\\in \\operatorname{ob}\\underline{A} $ , which are -natural in the sense that for any $y,z\\in \\operatorname{ob}\\underline{A} $ , the following diagram commutes: ${\\underline{A} (y,z) \\otimes \\underline{A} (\\omega \\odot x,y)[r]^-\\cong [d] &\\underline{A} (y,z) \\otimes (\\omega \\mathrel {} \\underline{A} (x,y)) [r] &\\omega \\mathrel {} (\\underline{A} (y,z)\\otimes \\underline{A} (x,y)) [d]\\\\\\underline{A} (\\omega \\odot x, z)[rr]_\\cong &&\\omega \\mathrel {} \\underline{A} (x,z)}$ $Taking $ =x$ in~(\\ref {eq:copower}), we obtain from$ 1x$ a canonical map $$\\underline{A}$ (x,x)$, which bythe Yoneda lemma determines~(\\ref {eq:copower}) uniquely.", "Of course,this is just the usual definition of copowers in enriched categories,specialized to enrichment over .", "We have spelled it out explicitlyto emphasize that it makes perfect sense even though is notsymmetric, as long as we choose the correct hom $$ and not$ $ (see~\\cite {street:absolute}, which treats the even moregeneral case of enrichment over a \\emph {bicategory}).$ Note that if $\\underline{A} =$ (the category regarded as a -category), then the tensor product $\\omega \\otimes x$ is a copower $\\omega \\odot x$ .", "Moreover, for general $\\underline{A}$ , if $\\omega ,\\varpi \\in $ and the copowers $\\omega \\odot x$ and $\\varpi \\odot (\\omega \\odot x)$ exist, we have $\\underline{A} (\\varpi \\odot (\\omega \\odot x), y)&\\cong \\varpi \\mathrel {} \\underline{A} (\\omega \\odot x,y)\\\\&\\cong \\varpi \\mathrel {} (\\omega \\mathrel {} \\underline{A} (x,y))\\\\&\\cong (\\varpi \\otimes \\omega )\\mathrel {} \\underline{A} (x,y)$ so that $\\varpi \\odot (\\omega \\odot x)$ is a copower $(\\varpi \\otimes \\omega ) \\odot x$ .", "In particular, these observations mandate writing the copower as $\\omega \\odot x$ rather than $x\\odot \\omega $ .", "Frequently one defines a power in a -category $\\underline{A}$ to be a copower in $\\underline{A} ^{\\mathrm {op}}$ , but since our is not symmetric, -categories do not have opposites.", "Thus, we must define directly a power of $x$ by $\\omega $ to be an object $x \\mathrel {\\oslash } \\omega \\in \\operatorname{ob}\\underline{A} $ together with isomorphisms $\\underline{A} (y,x \\mathrel {\\oslash } \\omega ) \\cong \\underline{A} (y,x) \\mathrel {\\!\\!}", "\\omega $ for all $y\\in \\operatorname{ob}\\underline{A} $ , which are -natural in that the following diagram commutes: ${\\underline{A} (y,x \\mathrel {\\oslash } \\omega ) \\otimes \\underline{A} (z,y)[r]^-\\cong [d] &(\\underline{A} (y,x) \\mathrel {\\!\\!}", "\\omega ) \\otimes \\underline{A} (z,y)[r] &(\\underline{A} (y,x) \\otimes \\underline{A} (z,y)) \\mathrel {\\!\\!}", "\\omega [d]\\\\\\underline{A} (z,x \\mathrel {\\oslash } \\omega ) [rr]_-\\cong & &\\underline{A} (z,x) \\mathrel {\\!\\!}", "\\omega }$ $Analogous arguments to those for copowers show that when $ A =$,then $ x $ is a power $ x $, and that in general we have$ (x ) x ()$.", "If boththe copower $ x$ and the power $ x $ exist, then we have{\\begin{@align}{1}{-1}\\underline{A} _{\\,o}(\\omega \\odot x,y)&\\cong ({1},{\\underline{A} (\\omega \\odot x,y)})\\\\&\\cong ({1}, \\omega \\mathrel {} \\underline{A} (x,y))\\\\&\\cong (\\omega ,{\\underline{A} (x,y)})\\\\&\\cong ({1},\\underline{A} (x,y) \\mathrel {\\!\\!}", "\\omega )\\\\&\\cong ({1},\\underline{A} (x,y \\mathrel {\\oslash } \\omega ))\\\\&\\cong \\underline{A} _{\\,o}(x,y \\mathrel {\\oslash } \\omega ).\\end{@align}}so that the endofunctors $ (-)$ and $ (- )$ on theunderlying 1-category $$\\underline{A}$ o$ are adjoint.", "They are \\emph {not}adjoint -functors, even when $$\\underline{A}$ =$: in our motivating example,the isomorphisms~(\\ref {eq:biclosed}) do not even lift from the1-category $ o=$ to the 2-category $ +$.$ Now suppose that $\\omega $ is right dualizable in , i.e.", "that we have an object $\\omega ^\\in $ and morphisms $\\omega ^ \\otimes \\omega \\rightarrow {1} $ and ${1} \\rightarrow \\omega \\otimes \\omega ^$ satisfying the triangle identities.", "Then $(-\\otimes \\omega ^)$ is right adjoint to $(-\\otimes \\omega )$ , hence isomorphic to $(\\omega \\mathrel {} -)$ ; and dually we have $(\\omega \\otimes -) \\cong (- \\mathrel {\\!\\!}", "\\omega ^)$ .", "Thus, a copower $\\omega \\odot x$ in a -category $\\underline{A}$ is equivalently characterized by an isomorphism $\\underline{A} (\\omega \\odot x,-) \\cong \\underline{A} (x,-) \\otimes \\omega ^,{}$ while a power $x \\mathrel {\\oslash } \\omega ^$ is characterized by an isomorphism $\\underline{A} (-,x \\mathrel {\\oslash } \\omega ^) \\cong \\omega \\otimes \\underline{A} (-,x).", "{}$ However, for fixed $x$ , the right-hand sides of () and () are adjoint in the bicategory of -modules.", "Since $\\underline{A} (\\omega \\odot x,-)$ always has an adjoint $\\underline{A} (-,\\omega \\odot x)$ , and likewise $\\underline{A} (-,x \\mathrel {\\oslash } \\omega ^)$ always has an adjoint $\\underline{A} (x \\mathrel {\\oslash } \\omega ^,-)$ , it follows that giving a copower $\\omega \\odot x$ is equivalent to giving a power $x \\mathrel {\\oslash } \\omega ^$ .", "Now let us specialize to the case of $\\textstyle \\int _G\\!", "$ .", "Then for any $g\\in G$ , we have a twisted unit ${1} ^{g} \\in \\textstyle \\int _G\\!", "$ , defined by ${1} ^{g}(h) = \\delta _{g,h} \\cdot {1} $ .", "By definition of $$ and $\\!\\!\\,$ , we have $({1} ^{h}\\mathrel {} \\underline{A} (x,y))(g) \\;&\\cong \\; \\underline{A} ^{gh}(x,y) \\qquad \\text{and}\\\\(\\underline{A} (x,y) \\mathrel {\\!\\!}", "{1} ^{h})(g) \\;&\\cong \\; {(\\underline{A} ^{hg}(x,y))}^{h^{-1}}.", "$ Thus, ${1} ^{h}\\odot x$ , if it exists, is characterized by isomorphisms $\\underline{A} ^g({1} ^{h}\\odot x,y) \\cong \\underline{A} ^{gh}(x,y)$ that are suitably and jointly natural in $y$ .", "In other words, a copower ${1} ^{h}\\odot x$ is precisely an $h$ -variator of $x$ as defined in label@name@sec:genmulti .", "And in our particular case of $=\\mathbf {Cat} $ , a copower ${1} ^{-}\\odot x$ is precisely an opposite of $x$ as defined in label@name@defn:opposite REF .", "Thus we have: Theorem 6.1 A 2-category with contravariance has opposites, as in label@name@defn:opposite REF , exactly if when regarded as a -category it has all copowers by ${1} ^{-}$ .", "Note that since ${1} ^{h}\\otimes {1} ^{h^{-1}} \\cong {1} $ , in particular ${1} ^{h}$ is dualizable.", "Thus, copowers by ${1} ^{h}$ are equivalent to powers by ${1} ^{h^{-1}}$ .", "In particular, since $-\\in \\lbrace +,-\\rbrace $ is its own inverse, it follows that ${1} ^{-}$ is self-dual, and opposites are also characterized by isomorphisms $\\underline{A} ^+(y,x^\\circ ) \\cong \\underline{A} ^-(y,x)^{\\mathrm {op}}\\qquad \\text{and}\\qquad \\underline{A} ^-(y,x^\\circ ) \\cong \\underline{A} ^+(y,x)^{\\mathrm {op}}.$ This gives another reason why a 2-functor preserving contravariance must preserve opposites: copowers by a dualizable object are absolute colimits [35].", "Bicategories with contravariance We have now reached the top of the right-hand side of the ladder from label@name@sec:introduction .", "It remains to move across to the other side and head down, starting with a bicategorical version of -categories for our $=\\int _{\\lbrace +,-\\rbrace }\\mathbf {Cat} $ .", "In fact, it will be convenient to stay in a more general setting.", "Thus, suppose that our monoidal category is actually a 2-category , and that our group $G$ acts on it by 2-functors.", "In this case, the construction of $\\textstyle \\int _G\\!$ can all be done with 2-categories, obtaining a monoidal 2-category $\\textstyle \\int _G\\!$ (in the strict sense of a monoidal $\\mathbf {Cat}$ -enriched category).", "Since a monoidal 2-category is a fortiori a monoidal bicategory, we can consider $\\textstyle \\int _G\\!$ -enriched bicategories, which we call $g$ -variant -bicategories.", "The most comprehensive extant reference on enriched bicategories seems to be [13], though the basic definition dates back at least to [4], [17].", "The definition of an enriched bicategory is quite simple: we just write out the definition of bicategory and replace all hom-categories by objects of $\\textstyle \\int _G\\!$ , cartesian products of categories by $\\otimes $ , and functors and natural transformations by morphisms and 2-cells in $\\textstyle \\int _G\\!$ .", "If we write this out explicitly, it consists of the following.", "[leftmargin=2em] A collection $\\operatorname{ob}\\underline{A} $ of objects; For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a category $\\underline{A} ^g(x,y)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , a unit morphism $1_x : {1} \\rightarrow \\underline{A} ^1(x,x)$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , composition morphisms $ \\underline{A} ^h(y,z) \\otimes {(\\underline{A} ^g(x,y))}^{h} \\rightarrow \\underline{A} ^{gh}(x,z) $ For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , two natural unitality isomorphisms; For each $x,y,z,w\\in \\operatorname{ob}\\underline{A} $ and $g,h,k\\in G$ , an associativity isomorphism; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , a unitality axiom holds; and For each $x,y,z,w,u\\in \\operatorname{ob}\\underline{A} $ and $g,h,k,\\ell \\in G$ , an associativity pentagon holds.", "Enriched bicategories, of course, come naturally with a notion of enriched functor.", "(In fact, as described in [13] we have a whole tricategory of enriched bicategories, but we will not need the higher structure.)", "Explicitly, a $\\textstyle \\int _G\\!$ -enriched functor $F:\\underline{A} \\rightarrow $ consists of [leftmargin=2em] A function $F:\\operatorname{ob}\\underline{A} \\rightarrow \\operatorname{ob}$ ; and For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a morphism $F:\\underline{A} ^g(x,y)\\rightarrow ^g(Fx,Fy)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , an isomorphism $F(1_x) \\cong 1_{F x}$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , a natural functoriality isomorphism of the form $(Fg)(Ff) \\cong F(gf)$ ; For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a unit coherence diagram commmutes; For each $x,y,z,w\\in \\operatorname{ob}\\underline{A} $ and $g,h,k\\in G$ , an associativity coherence diagram commutes.", "In the case of interest, we have $=\\mathbf {Cat} $ , which is of course enhances to the 2-category $\\mathcal {C}\\mathit {at}$ .", "However, we cannot take $=\\mathcal {C}\\mathit {at} $ , because as we have remarked, $(-)^{\\mathrm {op}}$ is not a 2-functor on $\\mathcal {C}\\mathit {at}$ , so $\\lbrace +,-\\rbrace $ does not act on $\\mathcal {C}\\mathit {at}$ through 2-functors.", "However, $(-)^{\\mathrm {op}}$ is a 2-functor on $\\mathcal {C}\\mathit {at} _g$ , the 2-category of categories, functors, and natural isomorphisms; so this is what we take as our $$ .", "We denote the resulting monoidal 2-category $\\textstyle \\int _G\\!$ by , and make the obvious definition: Definition 5 A bicategory with contravariance is a -enriched bicategory, and a pseudofunctor preserving contravariance is a -enriched functor.", "If we write this out explicitly in terms of covariant and contravariant parts, we see that a bicategory with contravariance has four kinds of composition functors, eight kinds of associativity isomorphisms, and sixteen coherence pentagons.", "Working with an abstract and $G$ thus allows us to avoid tedious case-analyses.", "We now generalize the enriched notion of $g$ -variator (and hence of “opposite”) from label@name@sec:opposites to the bicategorical case.", "For any $\\omega \\in \\textstyle \\int _G\\!", "$ , any $\\textstyle \\int _G\\!$ -bicategory $\\underline{A}$ , and any $x\\in \\underline{A} $ , a copower of $x$ by $\\omega $ is an object $\\omega \\odot x$ together with a map $\\omega \\rightarrow \\underline{A} (x,\\omega \\odot x)$ such that for any $y$ the induced map $\\underline{A} (\\omega \\odot x,y) \\rightarrow \\omega \\mathrel {} \\underline{A} (x,y)$ is an equivalence (not necessarily an isomorphism).", "(This is essentially the special case of [13] when is the unit $\\textstyle \\int _G\\!$ -bicategory.)", "As in label@name@sec:opposites , we are mainly interested in the case when $\\omega $ is one of the twisted units ${1} ^{g}$ .", "In this case we again write ${x}^{g}$ for ${1} ^{g}\\odot x$ , and the map ${1} ^{g} \\rightarrow \\underline{A} (x,{x}^{g})$ is just a $g$ -variant morphism $\\chi _{g,x}\\in \\underline{A} ^g(x,{x}^{g})$ .", "Its universal property says that any $gh$ -variant morphism $x\\xrightarrow[gh]{} y$ factors essentially uniquely through $\\chi _{g,x}$ via an $h$ -variant morphism ${x}^{g} \\xrightarrow[h]{} y$ (and similarly for 2-cells); that is, we have equivalences $ \\underline{A} ^h({x}^{g},y) \\xrightarrow{}\\underline{A} ^{gh}(x,y)$ As before, by Yoneda arguments this is equivalent to having a $g$ -variant morphism $x \\xrightarrow[g]{} {x}^{g}$ and a $g^{-1}$ -variant morphism ${x}^{g} \\xrightarrow[g^{-1}]{} x$ whose composites in both directions are isomorphic to identities; that is, a “$g$ -variant equivalence”.", "In the specific example of $=\\mathcal {C}\\mathit {at} _g$ and $G=\\lbrace +,-\\rbrace $ , we of course call ${x}^{-}$ a (weak) opposite of $x$ , written $x^\\circ $ .", "Our goal now is to show that any weak duality involution on a bicategory $\\mathcal {A}$ gives it the structure of a bicategory with contravariance having weak opposites; but to minimize case analyses, we will work in the generality of and $G$ .", "Thus, we first define a (weak, strictly unital) twisted $G$ -action on a -category $\\mathcal {A}$ to consist of: [leftmargin=2em] For each $g\\in G$ , a -functor ${(-)}^{g}:{\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ .", "(Note that here ${\\mathcal {A}}^{g}$ denotes the hom-wise action, ${\\mathcal {A}}^{g}(x,y) = {\\mathcal {A} (x,y)}^{g}$ .)", "When $g=1$ is the unit element of $G$ , we ask that ${(-)}^{1}$ be exactly equal to the identity functor.", "For each $g,h\\in G$ , a -pseudonatural adjoint equivalence $ {{(\\mathcal {A})}^{gh} [dr]_{{({(-)}^{g})}^{h}} [rr]^{{(-)}^{gh}} && \\mathcal {A}.\\\\& {\\mathcal {A}}^{h}[ur]_{{(-)}^{h}} @{}[u]\|(.6){\\Downarrow }}$ (Note that ${{({(-)}^{g})}^{h}}$ means the homwise endofunctor ${(-)}^{h}$ of $$ -bicategories applied to the action functor ${(-)}^{g}:{\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ .)", "When $g$ or $h$ is $1\\in G$ , we ask that $$ be exactly the identity transformation.", "For each $g,h,k\\in G$ , an invertible -modification $ {&{\\mathcal {A}}^{hk} [dr] [dd] \\\\{\\mathcal {A}}^{ghk} [ur] [dr] @{}[r]\|(.6){{_{g,h}}^{k}} &@{}[r]\|(.4){_{h,k}}& \\mathcal {A} \\\\& {\\mathcal {A}}^{k} [ur] }$ $As before, when $ $, $ h$, or $ k$ is $ 1G$, we ask that $$ beexactly the identity.\\item For each $ g,h,k,G$, a 4-simplex diagram of instances of$$ commutes.$ In our motivating example of $=\\mathcal {C}\\mathit {at} _g$ and $G=\\lbrace +,-\\rbrace $ , the strict identity requirements mean that: [leftmargin=2em] The only nontrivial action is ${(-)}^{-}$ , which we write as $(-)^\\circ $ .", "The only nontrivial $$ is $_{-,-}$ , which has the same type as the $$ in label@name@defn:duality-involution REF .", "The only nontrivial $\\zeta $ is $\\zeta _{-,-,-}$ , which has an equivalent type to the $\\zeta $ in label@name@defn:duality-involution REF (since $--=+$ is the identity, $_{-,--}$ and $_{--,-}$ are identities, so the type of $\\zeta $ displayed above has moved one copy of $$ from the codomain to the domain).", "The only nontrivial axiom likewise reduces to the one given in label@name@defn:duality-involution REF .", "Thus, this really does generalize our notion of duality involution.", "Now we will show: Theorem 7.1 Let $\\mathcal {A}$ be a -bicategory with a twisted $G$ -action, and for $x,y\\in \\mathcal {A} $ and $g\\in G$ define $\\underline{A} ^g(x,y) = \\mathcal {A} ({x}^{g},y)$ .", "Then $\\underline{A} $ is a $\\textstyle \\int _G\\!$ -bicategory with copowers by all the twisted units.", "We define the composition morphisms as follows: $\\begin{array}{rcl}\\underline{A} ^h(y,z) \\otimes {(\\underline{A} ^g(x,y))}^{h}&=& \\mathcal {A} ({y}^{h},z) \\otimes {(\\mathcal {A} ({x}^{g},y))}^{h}\\\\&=& \\mathcal {A} ({y}^{h},z) \\otimes {\\mathcal {A}}^{h}({x}^{g},y)\\\\&\\xrightarrow{}& \\mathcal {A} ({y}^{h},z) \\otimes \\mathcal {A} ({({x}^{g})}^{h},{y}^{h})\\\\&\\xrightarrow{}& \\mathcal {A} ({({x}^{g})}^{h}, z)\\\\&\\xrightarrow{}& \\mathcal {A} ({x}^{gh},z)\\\\&=&\\underline{A} ^{gh}(x,z)\\end{array}$ Informally (or, formally, in an appropriate internal “linear type theory” of ), we can say that the composite of $\\beta \\in \\underline{A} ^h(y,z)$ and $\\alpha \\in {(\\underline{A} ^g(x,y))}^{h}$ is $\\beta \\circ {\\alpha }^{h}\\circ _{g,h}$ where $\\circ $ denotes composition in $\\mathcal {A}$ .", "Expressed in the same way, the associator for $\\alpha \\in {(\\underline{A} ^g(x,y))}^{hk}$ , $\\beta \\in {(\\underline{A} ^h(y,z))}^{k}$ , and $\\gamma \\in \\underline{A} ^k(z,w)$ is $(\\gamma \\circ {\\beta }^{k}\\circ _{h,k}) \\circ {\\alpha }^{hk}\\circ _{g,hk}&\\cong \\gamma \\circ {\\beta }^{k}\\circ {({\\alpha }^{h})}^{k} \\circ _{h,k} \\circ _{g,hk}\\\\&\\cong \\gamma \\circ {\\beta }^{k}\\circ {({\\alpha }^{h})}^{k} \\circ {_{g,h}}^{k} \\circ _{gh,k}\\\\&\\cong \\gamma \\circ {(\\beta \\circ {\\alpha }^{h}\\circ _{g,h})}^{k} \\circ _{gh,k}$ using the naturality of , the modification $\\zeta $ , and the functoriality of ${(-)}^{k}$ (and omitting the associativity isomorphisms of $\\mathcal {A}$ , by coherence for bicategories).", "For the unit, since $\\underline{A} ^1(x,y)= \\mathcal {A} (x,y)$ , the unit map ${1} \\rightarrow \\underline{A} ^1(x,y)$ is just the unit of $\\mathcal {A}$ .", "One unit isomorphism is just that of $\\mathcal {A}$ , while the other is that of $\\mathcal {A}$ together with the unit isomorphism of the pseudofunctor ${(-)}^{g}$ .", "And the associator appearing in the unit axiom has $g=k=1$ , so all the $$ 's collapse and it is essentially trivial, and the unit axiom follows immediately from that of $\\mathcal {A}$ .", "To show that $\\underline{A}$ is a $\\textstyle \\int _G\\!$ -bicategory, it remains to consider the pentagon axiom.", "Omitting $\\circ $ from now on, the pentagon axiom is an equality of two morphisms $ \\delta {\\gamma }^{\\ell }_{k,\\ell } {\\beta }^{k\\ell }_{h,k\\ell } {\\alpha }^{hk\\ell }_{g,hk\\ell }\\longrightarrow \\delta {(\\gamma {(\\beta {\\alpha }^{h}_{g,h})}^{k} _{gh,k})}^{\\ell } _{ghk,\\ell }$ By naturality of the functoriality isomorphisms for the actions ${(-)}^{g}$ , we can certainly push all applications of them to the end where they will be equal; thus it suffices to compare the morphisms $ \\delta {\\gamma }^{\\ell }_{k,\\ell } {\\beta }^{k\\ell }_{h,k\\ell } {\\alpha }^{hk\\ell }_{g,hk\\ell }\\longrightarrow \\delta {\\gamma }^{\\ell }{({\\beta }^{k})}^{\\ell } {({({\\alpha }^{h})}^{k})}^{\\ell } {({_{g,h}}^{k})}^{\\ell } {_{gh,k}}^{\\ell } _{ghk,\\ell }$ This is done in Figure REF , where most of the regions are naturality, except for the one at the bottom left which is the 4-simplex axiom for $\\zeta $ .", "Figure: The pentagon axiomNow we must show that $\\underline{A}$ has copowers by the twisted units; of course we will use ${x}^{g}$ as the copower ${1} ^{g} \\odot x$ .", "Since $\\underline{A} ^g(x,{x}^{g}) = \\mathcal {A} ({x}^{g},{x}^{g})$ by definition, for $\\chi _{g,x}$ we can take the identity map of ${x}^{g}$ in $\\mathcal {A}$ .", "By definition of composition in $\\underline{A}$ , the induced precomposition map $\\underline{A} ^h({x}^{g},y) \\rightarrow \\underline{A} ^{gh}(x,y)$ is essentially just precomposition with $$ : $ \\mathcal {A} ({({x}^{g})}^{h},y) \\rightarrow \\mathcal {A} ({x}^{gh},y) $ and hence is an equivalence.", "Thus, $\\underline{A}$ has copowers by the twisted units.", "Inspecting the construction, we also conclude: Scholium 1 If a 2-category $\\mathcal {A}$ has a twisted $G$ -action in the sense of label@name@sec:genmulti , and we regard this as a weak twisted $G$ -action in the sense defined above with the actions being strict functors, $$ strictly natural, and $\\zeta $ an identity, then the $\\textstyle \\int _G\\!$ -bicategory constructed in label@name@thm:bicat REF is actually a strict $\\textstyle \\int _G\\!$ -category, and this construction agrees with the one in §–.", "In particular, if $\\mathcal {A}$ is a 2-category with a strong duality involution, and we regard it as a bicategory with a weak duality involution to construct a bicategory with contravariance $\\underline{A}$ , the result is the 2-category with contravariance we already obtained from it in label@name@sec:genmulti .", "With some more work we could enhance label@name@thm:bicat REF to a whole equivalence of tricategories.", "However, all we will need for our coherence theorem, in addition to label@name@thm:bicat REF and label@name@thm:bicat-2cat REF , is to go backwards on biequivalences.", "Before stating such a theorem, we have to define what we want to get out of it.", "Suppose $\\mathcal {A}$ and are -bicategories with twisted $G$ -action; by a twisted $G$ -functor $F:\\mathcal {A} \\rightarrow $ we mean a functor of -bicategories together with: [leftmargin=2em] For each $g\\in G$ , a -pseudonatural adjoint equivalence ${{\\mathcal {A}}^{g}[d]_{{(-)}^{g}}[r]^-{{F}^{g}} {\\mathfrak {i}} &{}^{g}[d]^{{(-)}^{g}}\\\\\\mathcal {A} [r]_-F &.", "}$ $\\item For each $ ,hG$, an invertible -modification$$@R=3pc@C=3pc{&& {{\\mathcal {A}}^{gh}} [dl]_{{({(-)}^{g})}^{h}} [r]^{{F}^{gh}} {\\mathfrak {i}}& {}^{gh}[dl]\|{{({(-)}^{g})}^{h}} ^{{{(-)}^{gh}}}{}\\\\&{\\mathcal {A}}^{h}[d]_{{(-)}^{h}}[r]\|-{{F}^{h}} {\\mathfrak {i}} &{}^{h}[d]\|{{(-)}^{h}}\\\\&\\mathcal {A} [r]_-F &}$ $(As before, $ ((-)g)h$ denotes the functorial action ofthe homwise endofunctor $ (-)h$ of -bicategories on the givenaction functor $ (-)g: $\\mathcal {A}$ h$\\mathcal {A}$ $.)", "This can bewritten formally as$$ \\mathfrak {i}_h \\circ {\\mathfrak {i}_g}^{h} \\circ ^_{g,h} \\cong ^\\mathcal {A} _{g,h} \\circ \\mathfrak {i}_{gh} $$\\item For all $ g,h,kG$, an axiom holds that can be writtenformally as the commutative diagram shown in Figure~\\ref {fig:theta-ax}.\\begin{figure}\\centering {\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} {({\\mathfrak {i}_g}^{h})}^{k} ^_{h,k} ^_{g,hk} [r] [d]_\\zeta &\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} ^_{h,k} {\\mathfrak {i}_g}^{hk} ^_{g,hk} [r]^\\theta &^\\mathcal {A} _{h,k} \\mathfrak {i}_{hk} {\\mathfrak {i}_g}^{hk} ^_{g,hk} [d]^\\theta \\\\\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} {({\\mathfrak {i}_g}^{h})}^{k} {^_{g,h}}^{k} ^_{gh,k} [d] && ^\\mathcal {A} _{h,k} ^\\mathcal {A} _{g,hk} \\mathfrak {i}_{ghk} [d]^\\zeta \\\\\\mathfrak {i}_k {(\\mathfrak {i}_h {{\\mathfrak {i}_g}^{h}} {^_{g,h}})}^{k} ^_{gh,k} [d]_\\theta & &{(^\\mathcal {A} _{g,h})}^{k} ^\\mathcal {A} _{gh,k} \\mathfrak {i}_{ghk}\\\\\\mathfrak {i}_k {( {^\\mathcal {A} _{g,h}} \\mathfrak {i}_{gh})}^{k} ^_{gh,k} [r] &\\mathfrak {i}_k {(^\\mathcal {A} _{g,h})}^{k} {(\\mathfrak {i}_{gh})}^{k} ^_{gh,k} [r] &{(^\\mathcal {A} _{g,h})}^{k} \\mathfrak {i}_k {(\\mathfrak {i}_{gh})}^{k} ^_{gh,k} [u]_\\theta & }\\caption {The axiom for \\theta }{}\\end{figure}$ Theorem 7.2 Suppose $\\mathcal {A}$ and are -bicategories with twisted $G$ -action, with resulting $\\textstyle \\int _G\\!$ -bicategories $\\underline{A}$ and .", "If $\\underline{A}$ and are biequivalent as $\\textstyle \\int _G\\!$ -bicategories, then $\\mathcal {A}$ and are biequivalent by a twisted $G$ -functor.", "In particular, if two bicategories $\\mathcal {A}$ and with duality involution give rise to biequivalent bicategories-with-contravariance, then $\\mathcal {A}$ and are biequivalent by a duality pseudofunctor.", "Let $F:\\underline{A} \\rightarrow $ be a $\\textstyle \\int _G\\!$ -biequivalence.", "In particular, then, it is a biequivalence on the 1-parts, hence a biequivalence $\\mathcal {A} \\simeq $ .", "Now by label@name@thm:bicat REF , for any $x\\in \\mathcal {A} $ we have a “$g$ -variant equivalence” $x\\xrightarrow[g]{} {x}^{g}$ with inverse ${x}^{g} \\xrightarrow[g^{-1}]{} x$ .", "This structure is preserved by $F$ , so we have a $g$ -variant equivalence between $Fx$ and $F({x}^{g})$ .", "But we also have a $g$ -variant equivalence between $Fx$ and ${(Fx)}^{g}$ , and composing them we obtain an ordinary (1-variant) isomorphism ${(Fx)}^{g}\\cong F({x}^{g})$ .", "These supply the components of $\\mathfrak {i}$ ; their pseudonaturality is straightforward.", "Now, by construction of the copowers by twisted units, it follows that $_{g,h} : {x}^{gh} \\rightarrow {({x}^{g})}^{h}$ is isomorphic to the composite of the variant equivalences $ {x}^{gh} \\xrightarrow[(gh)^{-1}]{} x \\xrightarrow[g]{} {x}^{g} \\xrightarrow[h]{} {({x}^{g})}^{h}$ while $\\zeta $ is obtained by canceling and uncanceling some of these equivalences.", "In particular, when $$ is composed with $\\mathfrak {i}$ , we can simply cancel some inverse variant equivalences to obtain the components of $\\theta $ .", "As for Figure , its source is ${(F(x))}^{ghk}\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[g]{} {(F(x))}^{g}\\xrightarrow[hk]{} {({(F(x))}^{g})}^{hk}\\\\\\xrightarrow[(hk)^{-1}]{} {(F(x))}^{g}\\xrightarrow[h]{} {({(F(x))}^{g})}^{h}\\xrightarrow[k]{} {({({(F(x))}^{g})}^{h})}^{k}\\\\\\xrightarrow[g^{-1}]{} {({(F(x))}^{h})}^{k}\\xrightarrow[g]{} {({(F({{x}^{g}}))}^{h})}^{k}\\\\\\xrightarrow[h^{-1}]{} {(F({{x}^{g}}))}^{k}\\xrightarrow[h]{} {(F({({x}^{g})}^{h}))}^{k}\\xrightarrow[k^{-1}]{} {F({({x}^{g})}^{h})}\\xrightarrow[k]{} F({({({x}^{g})}^{h})}^{k})$ while its target is ${(F(x))}^{ghk}\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[ghk]{} F({x}^{ghk})\\\\\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[gh]{} F({x}^{gh})\\xrightarrow[k]{} F({({x}^{gh})}^{k})\\\\\\xrightarrow[(gh)^{-1}]{} F({x}^{k})\\xrightarrow[g]{} F({({x}^{g})}^{k})\\xrightarrow[h]{} F({({({x}^{g})}^{h})}^{k})$ Here we have applied functors such as ${(-)}^{k}$ to variant morphisms; we can define this by simply “conjugating” with the variant equivalences $x\\xrightarrow[k]{} {x}^{k}$ .", "We leave it to the reader to apply naturality and cancel all the redundancy in these composites, reducing them both to $ {(F(x))}^{ghk} \\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[g]{} F({{x}^{g}})\\xrightarrow[h]{} F({({x}^{g})}^{h})\\xrightarrow[k]{} F({({({x}^{g})}^{h})}^{k}) $ so that they are equal.", "Therefore, to strictify a bicategory with duality involution, it will suffice to strictify its corresponding bicategory with contravariance.", "This is the task of the next, and final, section.", "Coherence for enriched bicategories We could continue in the generality of $G$ and , but there seems little to be gained by it any more.", "Theorem 8.1 Any bicategory with contravariance is biequivalent to a 2-category with contravariance.", "Just as there are two ways to prove the coherence theorem for ordinary bicategories, there are two ways to prove this coherence theorem.", "The first is an algebraic one, involving formally adding strings of composable arrows that hence compose strictly associatively.", "This can be expressed abstractly using the same general coherence theorem for pseudo-algebras over a 2-monad that we used in label@name@sec:2-monadic-approach .", "As sketched at the end of [31], this theorem (or a slight generalization of it) applies as soon as we observe that our 2-category is closed monoidal and cocomplete.", "The other method is by a Yoneda embedding.", "To generalize this to the enriched (and non-symmetric) case, first note that for any -bicategory $\\underline{A}$ , by [13] we have a -bicategory $\\underline{A} $ of moderate $\\underline{A}$ -modules, and a Yoneda embedding $\\underline{A} \\rightarrow \\underline{A} $ that is fully faithful.", "Thus, $\\underline{A}$ is biequivalent to its image in $\\underline{A} $ .", "However, since is a strict 2-category that is closed and complete, $\\underline{A} $ is actually a strict -category, and hence so is any full subcategory of it.", "Explicitly, an $\\underline{A}$ -module consists of categories $F^+(x)$ and $F^-(x)$ for each $x\\in \\underline{A} $ together with actions $F^+(y) \\times \\underline{A} ^+(x,y) &\\rightarrow F^+(x)\\\\F^+(y) \\times \\underline{A} ^-(x,y) &\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^+(x,y)^{\\mathrm {op}}&\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^-(x,y)^{\\mathrm {op}}&\\rightarrow F^+(x)$ and coherent associativity and unitality isomorphisms.", "A covariant $\\underline{A}$ -module morphism consists of functors $F^+(x) \\rightarrow G^+(x)$ and $F^-(x)\\rightarrow G^-(x)$ that commute up to coherent natural isomorphism with the actions, while a contravariant one consists similarly of functors $F^+(x)^{\\mathrm {op}}\\rightarrow G^-(x)$ and $F^-(x)^{\\mathrm {op}}\\rightarrow G^+(x)$ .", "Since $\\mathcal {C}\\mathit {at}$ is a strict 2-category, the bicategory-with-contravariance of modules is in fact a strict 2-category with contravariance.", "The Yoneda embedding, of course, sends each $z\\in \\underline{A} $ to the representable $Y_z$ defined by $Y_z^+(x) \\underline{A} ^+(x,z)$ and $Y_z^-(x)\\underline{A} ^-(x,z)$ .", "We have almost completed our trip over the ladder; it remains to make the following observation and then put all the pieces together.", "Theorem 8.2 If $\\underline{A}$ is a 2-category with contravariance that has weak opposites, then it is biequivalent to a 2-category with contravariance having strict opposites.", "Let $\\underline{A}^{\\prime }$ be the free cocompletion of $\\underline{A}$ , as a strict -category, under strict opposites (a strict -weighted colimit).", "It is easy to see that this can be done in one step, by considering the collection of all opposites of representables in the presheaf -category of $\\underline{A}$ .", "Thus, the embedding $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is -fully-faithful, and every object of $\\underline{A}^{\\prime }$ is the strict opposite of something in the image.", "However, $\\underline{A}$ has weak opposites, which are preserved by any -functor, and any strict opposite is a weak opposite.", "Thus, every object of $\\underline{A}^{\\prime }$ is equivalent to something in the image of $\\underline{A}$ , since they are both a weak opposite of the same object.", "Hence $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is bicategorically essentially surjective, and thus a biequivalence.", "Finally, we can prove label@name@thm:main REF .", "Theorem 8.3 If $\\mathcal {A} $ is a bicategory with a weak duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality pseudofunctor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a biequivalence.", "By label@name@thm:bicat REF , we can regard $\\mathcal {A}$ as a bicategory with contravariance $\\underline{A}$ having weak opposites.", "By label@name@thm:bicat-coherence REF , it is therefore biequivalent to a 2-category with contravariance and weak opposites, and therefore by label@name@thm:opposites-weaktostrict REF also biequivalent to a 2-category with contravariance and strict opposites.", "Now by label@name@thm:gm-2cat REF and label@name@thm:contrav-enriched REF , the latter is equivalently a 2-category with a strong duality involution.", "Thus, by label@name@thm:2monad-coherence REF it is equivalent to a 2-category with a strict duality involution, say $\\mathcal {A}^{\\prime }$ .", "So we have a biequivalence $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a pseudofunctor preserving contravariance, and by label@name@thm:biequiv REF , we can also regard it as a duality pseudofunctor.", "As mentioned in label@name@sec:introduction , we could actually dispense with the right-hand side of the ladder as follows.", "Let $\\mathcal {A}^{\\prime }$ be the full sub–bicategory of $\\mathcal {A} $ , as in label@name@thm:bicat-coherence REF , consisting of the modules that are either of the form $Y_z$ or of the form $Y_z^\\circ $ , where $Y_z^\\circ $ is defined by $(Y_z^\\circ )^+(x) \\underline{A} ^-(x,z)^{\\mathrm {op}}$ and $(Y_z^\\circ )^-(x) \\underline{A} ^+(x,z)^{\\mathrm {op}}$ .", "This $\\mathcal {A}^{\\prime }$ is a 2-category with a strict duality involution that interchanges $Y_z$ and $Y_z^\\circ $ , and the Yoneda embedding is a biequivalence for the same reasons.", "However, this quicker argument still depends on the description of weak duality involutions using bicategorical enrichment 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Math., Vol.", "420.", "Springer, Berlin, 1974.", "Ross Street.", "Absolute colimits in enriched categories.", "Cahiers Topologie Géom.", "Différentielle, 24(4):377–379, 1983.", "Mark Weber.", "Yoneda structures from 2-toposes.", "Appl.", "Categ.", "Structures, 15(3):259–323, 2007." ], [ "A 2-monadic approach", "Let $T(\\mathcal {A}) = \\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}$ , an endo-2-functor of the 2-category 2-$\\mathcal {C}\\mathit {at}$ of 2-categories, strict 2-functors, and strict 2-natural transformations.", "We have an obvious strict 2-natural transformation $\\eta :\\mathsf {Id}\\rightarrow T$ , and we define $\\mu :TT\\rightarrow T$ by $ (\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}) + (\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}})^{\\mathrm {co}}\\xrightarrow{}\\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}+ \\mathcal {A} ^{\\mathrm {co}}+ \\mathcal {A} \\xrightarrow{} \\mathcal {A} + \\mathcal {A} ^{\\mathrm {co}}$ where $\\nabla $ is the obvious “fold” map.", "Theorem 3.1 $T$ is a strict 2-monad, and: Normal pseudo $T$ -algebras are 2-categories with strong duality involutions; Pseudo $T$ -morphisms are duality 2-functors; and Strict $T$ -algebras are 2-categories with strict duality involutions.", "The 2-monad laws for $T$ are straightforward to check.", "By a normal pseudo algebra we mean one for which the unit constraint identifying $\\mathcal {A} \\rightarrow T\\mathcal {A} \\rightarrow \\mathcal {A} $ with the identity map is itself an identity.", "Thus, when $T\\mathcal {A} = \\mathcal {A} +\\mathcal {A} ^{\\mathrm {co}}$ , this means the action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ contains no data beyond a 2-functor $(-)^\\circ :\\mathcal {A} ^{\\mathrm {co}}\\rightarrow \\mathcal {A} $ .", "The remaining data is a 2-natural isomorphism ${TT\\mathcal {A} [r]^{Ta}[d]_{\\mu } {\\cong } &T\\mathcal {A} [d]^{a}\\\\T\\mathcal {A} [r]_{a} &\\mathcal {A}}$ that is satisfying three axioms that can be found, for instance, in [19].", "The right-hand square commutes strictly on the first three summands in its domain, and the second and third of the coherence axioms say exactly that the given isomorphism in these cases is an identity.", "Thus, what remains is the component of the isomorphism on the fourth summand, which has precisely the form of $$ in label@name@defn:duality-involution REF , and it is easy to check that the first coherence axiom reduces in this case to the identity $\\zeta $ .", "This proves REF , and REF follows immediately.", "Similarly, for REF , a pseudo $T$ -morphism is a 2-functor $F:\\mathcal {A} \\rightarrow $ together with a 2-natural isomorphism ${T\\mathcal {A} [r]^{T F}[d] {\\cong } &T[d]\\\\\\mathcal {A} [r]_F &}$ that is satisfying two coherence axioms also listed in [19].", "This square commutes strictly on the first summand of its domain, and the second coherence axiom ensures that the isomorphism is the identity there.", "So the remaining data is the isomorphism on the second summand, which has precisely the form of $\\mathfrak {i}$ in label@name@defn:duality-functor REF , and the first coherence axiom reduces to the identity $\\theta $ .", "In particular, we obtain an automatic definition of a “duality 2-natural transformation”: a $T$ -2-cell between pseudo $T$ -morphisms.", "This also gives us another source of examples.", "The 2-category $T\\text{-}\\mathcal {A}\\mathit {lg} _s$ of strict $T$ -algebras and strict $T$ -morphisms is complete with limits created in $2\\text{-}\\mathcal {C}\\mathit {at} $ , including in particular Eilenberg–Moore objects [33].", "Thus, for any monad $M$ in this 2-category — which is to say, a 2-monad that is a strict duality 2-functor and whose unit and multiplication are duality 2-natural transformations — the 2-category $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ of strict $M$ -algebras and strict $M$ -morphisms is again a strict $T$ -algebra, i.e.", "has a strict duality involution.", "Similarly, by [1] the 2-category $T\\text{-}\\mathcal {A}\\mathit {lg} $ of strict $T$ -algebras and pseudo $T$ -morphisms has PIE-limits, including EM-objects.", "Thus, we can reach the same conclusion even if $M$ is only a strong duality 2-functor.", "And since 2-$\\mathcal {C}\\mathit {at}$ is locally presentable and $T$ has a rank, there is another 2-monad $T^{\\prime }$ whose strict algebras are the pseudo $T$ -algebras; thus we can argue similarly in the 2-category $T\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ of pseudo $T$ -algebras and pseudo $T$ -morphisms, so that strong duality involutions also lift to $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ .", "Usually, of course, we are more interested in the 2-category $M\\text{-}\\mathcal {A}\\mathit {lg} $ of strict $M$ -algebras and pseudo $M$ -morphisms.", "It might be possible to enhance the above abstract argument to apply to this case using techniques such as [18], [25], but it is easy enough to check directly that if $M$ lies in $T\\text{-}\\mathcal {A}\\mathit {lg} $ or $T\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ , then so does $M\\text{-}\\mathcal {A}\\mathit {lg} $ .", "If $(A,a)$ is an $M$ -algebra, then the induced $M$ -algebra structure on $A^\\circ $ is the composite $ M(A^\\circ ) \\xrightarrow{} (MA)^\\circ \\xrightarrow{} A^\\circ $ and if $(f,):(A,a) \\rightarrow (B,b)$ is a pseudo $M$ -morphism (where $: a\\circ Mf \\xrightarrow{}f \\circ b$ ), then $f^\\circ $ becomes a pseudo $M$ -morphism with the following structure 2-cell: ${M(B^\\circ )[r]^{\\mathfrak {i}} &(MB)^\\circ [rr]^{b^\\circ } &&B^\\circ \\\\M(A^\\circ )[r]_{\\mathfrak {i}}[u]^{M(f^\\circ )} {\\cong } &(MA)^\\circ [rr]_{a^\\circ }[u]\|{(Mf)^\\circ } {{(^{-1})^\\circ }} &&A^\\circ [u]_{f^\\circ }}$ The axiom $\\theta $ of $\\mathfrak {i}$ (which is an equality since $M$ is a strong duality 2-functor) ensures that $$ lifts to $M\\text{-}\\mathcal {A}\\mathit {lg} $ (indeed, to $M\\text{-}\\mathcal {A}\\mathit {lg} _s$ ), and its own $\\theta $ axiom is automatic.", "A similar argument applies to $M\\text{-}\\mathcal {P}\\mathit {s} \\mathcal {A}\\mathit {lg} $ .", "Thus, 2-categories of algebraically structured categories such as monoidal categories, braided or symmetric monoidal categories, and so on, admit strict duality involutions, even when their morphisms are of the pseudo sort.", "(Of course, this is impossible for lax or colax morphisms, since dualizing the categories involved would switch lax with colax.)", "In theory, this could be another source of weak duality involutions that are not strong: if for a 2-monad $M$ the transformation $\\mathfrak {i}$ were not a strictly 2-natural isomorphism or its axiom $\\theta $ were not strict, then $M\\text{-}\\mathcal {A}\\mathit {lg} $ would only inherit a weak duality involution, even if the duality involution on the original 2-category were strict.", "However, I do not know any examples of 2-monads that behave this way.", "We end this section with the strong-to-strict coherence theorem.", "Theorem 3.2 If $\\mathcal {A}$ is a 2-category with a strong duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality 2-functor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a 2-equivalence.", "The 2-category $2\\text{-}\\mathcal {C}\\mathit {at} $ admits a factorization system $(,)$ in which $$ consists of the 2-functors that are bijective on objects and $$ of the 2-functors that are 2-fully-faithful, i.e.", "an isomorphism on hom-categories.", "Moreover, this factorization system satisfies the assumptions of [19], and we have $T\\subseteq $ .", "Thus, [19] (which is an abstract version of [24]), together with the characterizations of label@name@thm:2monad REF , implies the desired result.", "Inspecting the proof of the general coherence theorem, we obtain a concrete construction of $\\mathcal {A}^{\\prime }$ : it is the result of factoring the pseudo-action map $T\\mathcal {A} \\rightarrow \\mathcal {A} $ as a bijective-on-objects 2-functor followed by a 2-fully-faithful one.", "In other words, the objects of $\\mathcal {A}^{\\prime }$ are two copies of the objects of $\\mathcal {A} $ , one copy representing each object and one its opposite, with the duality interchanging them.", "The morphisms and 2-cells are then easy to determine.", "It remains, therefore, to pass from a weak duality involution on a bicategory to a strong one on a 2-category.", "We proceed up the right-hand side of the ladder from label@name@sec:introduction ." ], [ "Contravariance through virtualization", "As mentioned in label@name@sec:introduction , for much of the paper we will work in the extra generality of “twisted group actions”.", "Specifically, let be a complete and cocomplete closed symmetric monoidal category, and let $G$ be a group that acts on by strong symmetric monoidal functors.", "We write the action of $g\\in G$ on $W\\in $ as ${W}^{g}$ .", "For simplicity, we suppose that the action is strict, i.e.", "${({W}^{g})}^{h} = {W}^{gh}$ and ${W}^{1} = W$ strictly (and symmetric-monoidal-functorially).", "The case we are most interested in, which will yield our theorems about duality involutions on 2-categories, is when $=\\mathbf {Cat} $ with $G$ the 2-element group $\\lbrace +,-\\rbrace $ with $+$ the identity element (a copy of $/2$ ), and ${A}^{-}=A^{\\mathrm {op}}$ .", "However, there are other examples as well.", "Here are a few, also mentioned in label@name@sec:introduction , that yield “duality involutions” with a similar flavor.", "Let $=\\mathbf {2}\\text{-}\\mathbf {Cat}$ , with $G$ as the 4-element group $\\lbrace ++,-+,+-,--\\rbrace $ (a copy of $/2\\times /2$ ), and ${A}^{-+}=A^{\\mathrm {op}}$ , ${A}^{+-}=A^{\\mathrm {co}}$ , and hence ${A}^{--}=A^{\\mathrm {coop}}$ .", "If we give $$ the Gray monoidal structure as in [12], this example leads to a theory of duality involutions on Gray-categories.", "Let $$ be the category of strict $n$ -categories, with $G = (/2)^n$ acting by reversal of $k$ -morphisms at all levels.", "Since a category enriched in strict $n$ -categories is exactly a strict $(n+1)$ -category, we obtain a theory of duality involutions on strict $(n+1)$ -categories.", "Let $=\\mathbf {sSet}$ , the category of simplicial sets, with $G=\\lbrace +,-\\rbrace $ , and ${A}^{-}$ obtained by reversing the directions of all simplices in $A$ .", "This leads to a theory of duality involutions on simplicially enriched categories that is appropriate when the simplicial sets are regarded as modeling $(\\infty ,1)$ -categories as quasicategories [16], [22], so that simplicially enriched categories are a model for $(\\infty ,2)$ -categories.", "For example, such simplicially enriched categories are used in [29] to define a notion of “$\\infty $ -cosmos” analogous to the “fibrational cosmoi” of [34], so such duality involutions could be a first step towards an $\\infty $ -version of [36].", "Combining the ideas of the last two examples, if $$ is the category of $\\Theta _n$ -spaces as in [27], then $(/2)^n$ acts on it by reversing directions at all dimensions.", "Thus, we obtain a theory of duality involutions on categories enriched in $\\Theta _n$ -spaces, which in [3] were shown to be a model of $(\\infty ,n+1)$ -categories.", "Note that we do not assume the action of $G$ on is by -enriched functors, since in most of the above examples this is not the case.", "In particular, $(-)^{\\mathrm {op}}$ is not a 2-functor.", "We also note that most or all of the theory would probably be the same if $G$ were a 2-group rather than just a group, but we do not need this extra generality.", "Since the action of $G$ on is symmetric monoidal, it extends to an action on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ applied homwise, which we also write ${\\mathcal {A}}^{g}$ , i.e.", "${\\mathcal {A}}^{g}(x,y) = {(\\mathcal {A} (x,y))}^{g}$ .", "In our motivating example we have ${\\mathcal {A}}^{-}= \\mathcal {A} ^{\\mathrm {co}}$ for a 2-category $\\mathcal {A}$ .", "We now define a 2-monad $T$ on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ by $ T\\mathcal {A} = \\sum _{g\\in G} {\\mathcal {A}}^{g}.", "$ The unit $\\mathcal {A} \\rightarrow T\\mathcal {A} $ includes the summand indexed by $1\\in G$ , and the multiplication uses the fact that each action, being an equivalence of categories (indeed, an isomorphism of categories), is cocontinuous: $TT\\mathcal {A} = \\sum _{g\\in G} {(T\\mathcal {A})}^{g}= \\sum _{g\\in G} {\\left(\\sum _{h\\in G} {\\mathcal {A}}^{h}\\right)}^{g}\\cong \\sum _{g\\in G} \\sum _{h\\in G} {({\\mathcal {A}}^{h})}^{g}\\cong \\sum _{g\\in G} \\sum _{h\\in G} {\\mathcal {A}}^{hg}$ which we can map into $T\\mathcal {A} $ by sending the $(g,h)$ summand to the $hg$ -summand.", "We will refer to a normal pseudo $T$ -algebra structure as a twisted $G$ -action; it equips a -category $\\mathcal {A}$ with actions ${(-)}^{g} : {\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ that are suitably associative up to coherent isomorphism (with ${(x)}^{1}=x$ strictly).", "In our motivating example of $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ , the monad $T$ agrees with the one we constructed in label@name@sec:2-monadic-approach ; thus twisted $G$ -actions are strong duality involutions (and likewise for their morphisms and 2-cells).", "If we write $[x,y]$ for the internal-hom of , then we have maps ${[x,y]}^{g} \\rightarrow [{x}^{g}, {y}^{g}]$ obtained by adjunction from the composite $ {[x,y]}^{g} \\otimes {x}^{g} \\xrightarrow{}{([x,y]\\otimes x)}^{g}\\rightarrow {y}^{g}$ Since the $[x,y]$ are the hom-objects of the -category , these actions assemble into a -functor ${(-)}^{g}:{}^{g}\\rightarrow $ , and as $g$ varies they give itself a twisted $G$ -action.", "(Thus, among the three different actions we are denoting by ${(-)}^{g}$ — the given one on , the induced one on $\\mathbf {W}\\text{-}\\mathcal {C}\\mathit {at} $ , and an arbitrary twisted $G$ -action — the first is a special case of the third.)", "In particular, we obtain in this way the canonical strong (in fact, strict) duality involution on $\\mathbf {Cat}$ .", "Now we note that $T$ extends to a normal monad in the sense of [5] on the proarrow equipment $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ , as follows.", "As in [30], [5], we view equipments as pseudo double categories satisfying with a “fibrancy” condition saying that horizontal arrows (the “proarrow” direction, for us) can be pulled back universally along vertical ones (the “functor” direction).", "In $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ the objects are -categories, a horizontal arrow $\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}$ is a profunctor (i.e.", "a -functor $^{\\mathrm {op}}\\otimes \\mathcal {A} \\rightarrow $ ), a vertical arrow $\\mathcal {A} \\rightarrow $ is a -functor, and a square ${\\mathcal {A} [r]\|{\|}^M[d]_F @{}[dr]\|{\\Downarrow } &[d]^G\\\\[r]\|{\|}_N &}$ is a -natural transformation $M(b,a)\\rightarrow N(G(b),F(a))$ .", "A monad on an equipment is strictly functorial in the vertical direction, laxly functorial in the horizontal direction, and its multiplication and unit transformations consist of vertical arrows and squares.", "In our case, we already have the action of $T$ on -categories and -functors.", "A -profunctor $M:\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}$ induces another one ${M}^{g} : {\\mathcal {A}}^{g}0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}{}^{g}$ by applying the $G$ -action objectwise, and by summing up over $g$ we have $T M : T \\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T $ .", "This is in fact pseudofunctorial on profunctors.", "Finally, the unit and multiplication are already defined as vertical arrows, and extend to squares in an evident way: ${\\mathcal {A} [r]\|{\|}^M[d]_\\eta @{}[dr]\|{\\Downarrow } &[d]^\\eta \\\\T\\mathcal {A} [r]\|{\|}_{T M} &T}$ Since we have a monad on an equipment, we can define “$T$ -multicategories” in $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ , which following [5] we call virtual $T$ -algebras.", "For our specific monad $T$ , we will refer to virtual $T$ -algebras as $G$ -variant -categories.", "Such a gadget is a -category $\\mathcal {A}$ together with a profunctor $\\underline{A}:\\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T\\mathcal {A} $ , a unit isomorphism $\\mathcal {A} (x,y) \\xrightarrow{}\\underline{A} (\\eta (x),y)$ , and a composition ${\\mathcal {A} [r]\|{\|}^\\underline{A} @{=}[d] @{}[drr]\|{\\Downarrow } &T\\mathcal {A} [r]\|{\|}^{T\\underline{A}} &TT\\mathcal {A} [d]^\\mu \\\\\\mathcal {A} [rr]\|{\|}_\\underline{A} & &T\\mathcal {A}}$ satisfying associativity and unit axioms.", "If we unravel this explicitly, we see that a $G$ -variant -category has a set of objects along with, for each pair of objects $x,y$ and each $g\\in G$ , a hom-object $\\underline{A} ^g(x,y)\\in $ , plus units ${1} \\rightarrow \\underline{A} ^1(x,x)$ and compositions $\\underline{A} ^g(y,z) \\otimes {(\\underline{A} ^h(x,y))}^{g} \\rightarrow \\underline{A} ^{hg}(x,z)$ satisfying the expected axioms.", "(Technically, in addition to the hom-objects $\\underline{A} ^1(x,y)$ it has the hom-objects $\\mathcal {A} (x,y)$ that are isomorphic to them, but we may ignore this duplication of data.)", "We may refer to the elements of $\\underline{A} ^g(x,y)$ as $g$ -variant morphisms.", "The rule for the variance of composites is easier to remember when written in diagrammatic order: if we denote $\\alpha \\in \\underline{A} ^g(x,y)$ by $\\alpha :x\\xrightarrow[g]{}y$ , then the composite of $x\\xrightarrow[g]{} y \\xrightarrow[h]{} z$ is $x\\xrightarrow[gh]{} z$ .", "(Of course, in our motivating example $G$ is commutative, so the order makes no difference.)", "In the specific case of $G=\\lbrace +,-\\rbrace $ acting on $\\mathbf {Cat} $ , we can unravel the definition more explicitly as follows.", "Definition 3 A 2-category with contravariance is a $G$ -variant -category for $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ .", "Thus it consists of [leftmargin=2em] A collection $\\operatorname{ob}\\underline{A} $ of objects; For each $x,y\\in \\operatorname{ob}\\underline{A} $ , a pair of categories $\\underline{A} ^+(x,y)$ and $\\underline{A} ^-(x,y)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , an object $1_x \\in \\underline{A} ^+(x,x)$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ , composition functors $\\underline{A} ^+(y,z) \\times \\underline{A} ^+(x,y) &\\overset{}{\\longrightarrow }\\underline{A} ^+(x,z)\\\\\\underline{A} ^-(y,z) \\times \\underline{A} ^-(x,y)^{\\mathrm {op}}&\\overset{}{\\longrightarrow }\\underline{A} ^+(x,z)\\\\\\underline{A} ^+(y,z) \\times \\underline{A} ^-(x,y) &\\overset{}{\\longrightarrow }\\underline{A} ^-(x,z)\\\\\\underline{A} ^-(y,z) \\times \\underline{A} ^+(x,y)^{\\mathrm {op}}&\\overset{}{\\longrightarrow }\\underline{A} ^-(x,z);$ such that [leftmargin=2em] Four ($2\\cdot 2^1$ ) unitality diagrams commute; and Eight ($2^3$ ) associativity diagrams commute.", "Like any kind of generalized multicategory, $G$ -variant -categories form a 2-category.", "We leave it to the reader to write out explicitly what the morphisms and 2-cells in this 2-category look like; in our example of interest we will call them 2-functors preserving contravariance and 2-natural transformations respecting contravariance.", "Now, according to [5], any twisted $G$ -action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ gives rise to a $G$ -variant -category with $\\underline{A} = \\mathcal {A} (a,1)$ , which in our situation means $\\underline{A} ^g(x,y) = \\mathcal {A} ({x}^{g},y)$ (where ${x}^{g}$ refers, as before, to the $g$ -component of the action $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ ).", "In particular, any 2-category with a strong duality involution can be regarded as a 2-category with contravariance, where $\\underline{A} ^+(x,y) = \\mathcal {A} (x,y)$ and $\\underline{A} ^-(x,y) = \\mathcal {A} (x^\\circ ,y)$ .", "Moreover, by [5], a $G$ -variant -category $\\underline{A}$ arises from a twisted $G$ -action exactly when The profunctor $\\underline{A}: \\mathcal {A} 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}T\\mathcal {A} $ is representable by some $a:T\\mathcal {A} \\rightarrow \\mathcal {A} $ , and The induced 2-cell $\\overline{a} : a\\circ \\mu \\rightarrow a \\circ Ta$ is an isomorphism.", "Condition REF means that for every $x\\in \\underline{A} $ and every $g\\in G$ , there is an object “${x}^{g}$ ” and an isomorphism $\\underline{A} ^g(x,y) \\cong \\underline{A} ^1({x}^{g},y)$ , natural in $y$ .", "The Yoneda lemma implies this isomorphism is mediated by a “universal $g$ -variant morphism” $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ .", "Condition REF then means that for any $x\\in \\underline{A} $ and $g,h\\in G$ , the induced map $\\psi _{h,g,x}:{x}^{gh} \\rightarrow {({x}^{g})}^{h}$ is an isomorphism.", "(This map arises by composing $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ with $\\chi _{h,{x}^{g}} \\in \\underline{A} ^h({x}^{g},{({x}^{g})}^{h})$ to obtain a map in $\\underline{A} ^{gh}(x,{({x}^{g})}^{h})$ , then applying the defining isomorphism of ${x}^{gh}$ .)", "As usual for generalized multicategories, this is equivalent to requiring a stronger universal property of ${x}^{g}$ : that precomposing with $\\chi _{g,x}$ induces isomorphisms $\\underline{A} ^h({x}^{g},y) \\xrightarrow{}\\underline{A} ^{gh}(x,y){}$ for all $h\\in G$ .", "(This again is more mnemonic in diagrammatic notation: any arrow $x \\xrightarrow[gh]{} y$ factors uniquely through $\\chi _{g,x}$ by a morphism ${x}^{g} \\xrightarrow[h]{} y$ , i.e.", "variances on the arrow can be moved into the action on the domain, preserving order.)", "This is because the following diagram commutes by definition of $\\psi _{h,g,x}$ , and the vertical maps are isomorphisms by definition of $\\chi $ : $@C=4pc{\\underline{A} ^{gh}(x,y)@{<-}[r]^{-\\circ \\chi _{g,x}}@{<-}[d]_{-\\circ \\chi _{gh,x}} &\\underline{A} ^h({x}^{g},y)@{<-}[d]^{-\\circ \\chi _{h,{x}^{g}}}\\\\\\underline{A} ^1({x}^{gh},y)@{<-}[r]_{-\\circ \\psi _{h,g,x}} &\\underline{A} ^1({({x}^{g})}^{h},y)}$ If ${x}^{g}$ is an object equipped with a morphism $\\chi _{g,x} \\in \\underline{A} ^g(x,{x}^{g})$ satisfying this stronger universal property (), we will call it a $g$ -variator of $x$ .", "In our motivating example $=\\mathbf {Cat} $ with $g=-$ , we call a $-$ -variator an opposite.", "Explicitly, this means the following.", "Definition 4 In a 2-category with contravariance $\\underline{A}$ , a (strict) opposite of an object $x$ is an object $x^\\circ $ equipped with a contravariant morphism $\\chi _x\\in \\underline{A} ^-(x,x^\\circ )$ such that precomposing with $\\chi _x$ induces isomorphisms of hom-categories for all $y$ : $\\underline{A} ^+(x^\\circ ,y) &\\xrightarrow{}\\underline{A} ^-(x,y)\\\\\\underline{A} ^-(x^\\circ ,y) &\\xrightarrow{}\\underline{A} ^+(x,y).$ In fact, $g$ -variators can also be characterized more explicitly.", "The second universal property of $\\chi _{g,x}\\in {\\underline{A}}^{g}(x,{x}^{g})$ means in particular that the identity $1_x\\in {\\underline{A}}^{1}(x,x)$ can be written as $\\xi _{g,x} \\circ \\chi _{g,x}$ for a unique $\\xi _{g,x} \\in \\underline{A} ^{g^{-1}}({x}^{g},x)$ .", "(This is the first place where we have used the fact that $G$ is a group rather than just a monoid.)", "Moreover, since $(\\chi _{g,x}\\circ \\xi _{g,x}) \\circ \\chi _{g,x} = \\chi _{g,x} \\circ (\\xi _{g,x}\\circ \\chi _{g,x}) = \\chi _{g,x} $ it follows by the first universal property of $\\chi _{g,x}$ that $\\chi _{g,x}\\circ \\xi _{g,x} = 1_{{x}^{g}}$ as well.", "Thus, $\\chi _{g,x}$ and $\\xi _{g,x}$ form a “$g$ -variant isomorphism” between $x$ and ${x}^{g}$ .", "On the other hand, it is easy to check that any such $g$ -variant isomorphism between $x$ and an object $y$ makes $y$ into a $g$ -variator of $x$ .", "Thus, we have: Proposition 1 Any $g$ -variant -functor $F:\\underline{A} \\rightarrow $ preserves $g$ -variators.", "In particulary, any 2-functor preserving contravariance also preserves opposites.", "It obviously preserves “$g$ -variant isomorphisms”.", "Thus we have: Theorem 4.1 The 2-category of 2-categories with strong duality involutions, duality 2-functors, and duality 2-natural transformations is 2-equivalent to the 2-category of 2-categories with contravariance in which every object has a strict opposite, 2-functors preserving contravariance, and 2-natural transformations respecting contravariance.", "By [5] and the remarks preceding label@name@defn:opposite REF , the latter 2-category is equivalent to the 2-category of pseudo $T$ -algebras, lax $T$ -morphisms, and $T$ -2-cells.", "However, label@name@thm:gm-abs REF implies that in fact every lax $T$ -morphism is a pseudo $T$ -morphism.", "Finally, every pseudo $T$ -algebra is isomorphic to a normal pseudo one obtained by re-choosing ${(-)}^{1}$ to be the identity (which it is assumed to be isomorphic to)." ], [ "Contravariance through enrichment", "We continue with our setup from label@name@sec:genmulti , with a complete and cocomplete closed monoidal category and a group $G$ acting on .", "We start by noticing that the monad $T$ on $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ constructed in label@name@sec:genmulti can actually be obtained in a standard way from a simpler monad.", "Recall that there is another equipment $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ whose objects are sets, whose vertical arrows are functions, and whose horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}Y$ are “-valued matrices”, which are just functions $Y\\times X \\rightarrow $ ; we call them matrices because we compose them by “matrix multiplication”.", "The equipment $\\mathbf {W}\\text{-}\\mathbb {P}\\mathsf {rof} $ is obtained from $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ by applying a functor $\\mathbb {M}\\mathsf {od}$ that constructs monoids (monads) and modules in the horizontal directions (see [30], [5]).", "We now observe that our monad $T$ , like many monads on equipments of profunctors, is also in the image of $\\mathbb {M}\\mathsf {od}$ .", "Let $S$ be the following monad on $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ .", "On objects and vertical arrows, it acts by $S(X)=X\\times G$ .", "On a -matrix $M:Y\\times X \\rightarrow $ it acts by $SM((y,h),(x,g)) ={\\left\\lbrace \\begin{array}{ll}{(M(y,x))}^{g} &\\quad g=h\\\\\\emptyset &\\quad g\\ne h\\end{array}\\right.", "}$ We may write this schematically using a Kronecker delta as $SM((y,h),(x,g)) = \\delta _{g,h}\\cdot {(M(y,x))}^{g}.$ On a composite of matrices $X 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{M} Y 0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{N} Z$ we have $(SM \\odot SN)((z,k),(x,g)) &= \\sum _{(y,h)} (\\delta _{g,h}\\cdot {M(y,x)}^{g}) \\otimes (\\delta _{h,k}\\cdot {N(z,y)}^{h})\\\\&\\cong \\delta _{k,g}\\sum _{y} {M(y,x)}^{g} \\otimes {N(z,y)}^{g}\\\\&\\cong \\delta _{k,g} \\Big (\\sum _{y} \\big (M(y,x) \\otimes N(z,y)\\big )\\Big )^{g}\\\\&= \\delta _{k,g}\\cdot {(M\\odot N)(z,x)}^{g}\\\\&= S(M\\odot N)((z,k),(x,g))$ making $S$ a pseudofunctor.", "The monad multiplication and unit are induced from the multiplication and unit of $G$ ; the squares ${X[r]\|{\|}^M[d]_\\eta @{}[dr]\|{\\Downarrow } &Y[d]^\\eta \\\\SX[r]\|{\|}_{S M} &SY}$ map the components $M(y,x)$ and ${({M(y,x)}^{g})}^{h}$ isomorphically to ${M(y,x)}^{1}$ and ${M(y,x)}^{gh}$ respectively.", "Now, recalling that $T\\mathcal {A} = \\sum _{g\\in G} {\\mathcal {A}}^{g}$ , we see that $\\operatorname{ob}(T\\mathcal {A}) = \\operatorname{ob}(\\mathcal {A}) \\times G$ and $T\\mathcal {A} ((y,h),(x,g)) = \\delta _{h,g}\\cdot {(\\mathcal {A} (y,x))}^{g},$ and so in fact $T \\cong \\mathbb {M}\\mathsf {od} (S)$ .", "Thus, by [5], virtual $T$ -algebras can be identified with “$S$ -monoids”; these are defined like virtual $S$ -algebras, with sets and matrices of course replacing categories and profunctors, and omitting the requirement that the unit be an isomorphism.", "Thus, an $S$ -monoid consists of a set $X$ of objects, a function $\\underline{A}:S(X)\\times X = X\\times G\\times X \\rightarrow $ , unit maps $1_x:I\\rightarrow \\underline{A} ^1(x,x)$ , and composition maps that turn out to look like $\\underline{A} ^g(y,z) \\otimes {(\\underline{A} ^h(x,y))}^{g} \\rightarrow \\underline{A} ^{hg}(x,z)$ .", "Note that this is exactly what we obtain from a virtual $T$ -algebra by omitting the redundant data of the hom-objects $\\mathcal {A} (x,y)$ and their isomorphisms to $\\underline{A} ^1(x,y)$ ; this is essentially the content of [5] in our case.", "In [5], the construction of $S$ -monoids is factored into two: first we build a new equipment $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ whose objects and vertical arrows are the same as $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ but whose horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}Y$ are the horizontal arrows $X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SY$ in $\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} $ , and then we consider horizontal monoids in $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ .", "In fact, $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S)$ is in general only a “virtual equipment” (i.e.", "we cannot compose its horizontal arrows, though we can “map out of composites” like in a multicategory), but in our case it is an ordinary equipment because $S$ is “horizontally strong” [5].", "This means that $S$ is a strong functor (which we have already observed) and that the induced maps of matrices $(\\eta ,1)_!M &\\rightarrow (1,\\eta )^S M \\\\(\\mu ,1)_!SSM &\\rightarrow (1,\\mu )^\\odot SSM$ are isomorphisms, where $f^$ and $f_!$ denote the pullback and its left adjoint pushforward of matrices along functions.", "Indeed, we have $(\\eta ,1)_!M((y,h),x) &= \\delta _{h,1} \\cdot M(y,x)\\hspace{56.9055pt}\\text{while}\\\\(1,\\eta )^S M((y,h),x) &= SM((y,h),(x,1))\\\\&= \\delta _{h,1} \\cdot {(M(y,x))}^{1}\\\\&= \\delta _{h,1} \\cdot M(y,x)$ and likewise $(\\mu ,1)_!SSM((y,h),((x,g_1),g_2)) &= \\textstyle \\sum _{h_2 h_1 = h} SSM(((y,h_1),h_2),((x,g_1),g_2))\\\\&= \\textstyle \\sum _{h_1 h_2 = h} \\delta _{h_2,g_2} \\cdot \\Big (\\delta _{h_1,g_1} \\cdot {M(y,x)}^{g_1}\\Big )^{g_2}\\\\&= \\textstyle \\sum _{h_1 h_2 = h} \\delta _{h_2,g_2}\\delta _{h_1,g_1} \\cdot {M(y,x)}^{g_1 g_2}\\\\&= \\delta _{h,g_1 g_2} \\cdot {M(y,x)}^{g_1 g_2}\\\\\\multicolumn{2}{l}{\\text{while}}\\\\(1,\\mu )^\\odot SSM((y,h),((x,g_1),g_2)) &= SSM((y,h),(x,g_1 g_2))\\\\&= \\delta _{h,g_1 g_2} \\cdot {M(y,x)}^{g_1 g_2}.$ Inspecting the definition of composition in [5], we see that the composite of $M:X0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SY$ and $N:Y0055{\\m@th \\hspace{-3.88885pt}\\hspace{-1.111pt}\\hspace{-1.111pt}}\\hfill {}{}$$$${}{}SZ$ is $ (M\\odot _S N)((z,h),x) = \\sum _y \\sum _{g_1g_2 = h} M((y,g_1),x) \\odot {N((z,g_2),y)}^{g_1}$ Note that what comes after the $\\sum _y$ depends only on $M((y,-),x)$ and $N((z,-),y)$ , which are objects of $^G$ .", "Thus, if we write $\\textstyle \\int _G\\!$ for the category $^G$ with the following monoidal structure: $ (M \\otimes N)(h) = \\sum _{g_1 g_2 = h} M(g_1) \\odot {N(g_2)}^{g_1}$ then we have $\\mathbb {H}\\text{-}\\mathsf {Kl}(\\mathbf {W}\\text{-}\\mathbb {M}\\mathsf {at} ,S) \\cong (\\textstyle \\int _G\\!", ")\\text{-}\\mathbb {M}\\mathsf {at} $ .", "It follows that $S$ -monoids (that is, $G$ -variant -categories) can equivalently be regarded as ordinary monoids in the equipment $(\\textstyle \\int _G\\!", ")\\text{-}\\mathbb {M}\\mathsf {at} $ .", "But since monoids in an equipment of matrices are simply enriched categories, we can identify $G$ -variant -categories with $\\textstyle \\int _G\\!", "$ -enriched categories.", "Note that this monoidal structure on $\\textstyle \\int _G\\!$ is not symmetric.", "It is a version of Day convolution [6] that is “twisted” by the action of $G$ on (see [20] for further discussion).", "Like an ordinary Day convolution monoidal structure, it is also closed on both sides (as long as is); that is, we have left and right hom-functors $\\!\\!\\,$ and $$ with natural isomorphisms ${}(\\textstyle \\int _G\\!", ")(A\\otimes B, C) \\cong (\\textstyle \\int _G\\!", ")(A, B\\mathrel {} C) \\cong (\\textstyle \\int _G\\!", ")(B, C \\mathrel {\\!\\!}", "A).$ Inspecting the definition of the tensor product in $\\textstyle \\int _G\\!$ , it suffices to define $(B\\mathrel {} C)(g) &\\prod _{h} \\big ({B(h)}^{g}\\mathrel {} C(gh)\\big )\\\\(C \\mathrel {\\!\\!}", "A)(g) &\\prod _{h} \\left({C(hg)}^{h^{-1}} \\mathrel {\\!\\!}", "{A(h)}^{h^{-1}}\\right)$ (This is another place where we use the fact that $G$ is a group rather than a monoid.)", "As usual, it follows that $\\textstyle \\int _G\\!$ can be regarded as a $\\textstyle \\int _G\\!$ -category (that is, as a $g$ -variant -category), with hom-objects $\\underline{\\textstyle \\int _G\\!", "}(A,B) (A\\mathrel {} B)$ .", "(The fact that a closed monoidal category becomes self-enriched is often described only for closed symmetric monoidal categories, but it works just as well for closed non-symmetric ones, as long as we use the right hom.)", "Bringing things back down to each a bit, in our specific case with $=\\mathbf {Cat} $ and $G=\\lbrace +,-\\rbrace $ , let us write $= \\int _{\\lbrace +,-\\rbrace }\\!\\mathbf {Cat} $ .", "The underlying category of is just $\\mathbf {Cat} \\times \\mathbf {Cat} $ , but we denote its objects as $A = (A^+,A^-)$ , with $A^+$ the covariant part and $A^-$ the contravariant part.", "The monoidal structure on is the following nonstandard one: $(A\\otimes B)^+&\\big (A^+\\times B^+\\big ) \\amalg \\big (A^-\\times (B^-)^{\\mathrm {op}}\\big )\\\\(A\\otimes B)^-&\\big (A^+\\times B^-\\big ) \\amalg \\big (A^-\\times (B^+)^{\\mathrm {op}}\\big )$ The unit object is $ {1} (1,0) $ where 1 denotes the terminal category and 0 the initial (empty) one.", "The conclusion of our equipment-theoretic digression above is then the following: Theorem 5.1 The 2-category of 2-categories with contravariance, 2-functors preserving contravariance, and 2-natural transformations respecting contravariance is 2-equivalent to the 2-category of -enriched categories.", "This theorem is easy to prove explicitly as well, of course.", "A -category has, for each pair of objects $x,y$ , a pair of hom-categories $(\\underline{A} ^+(x,y),\\underline{A} ^-(x,y))$ , together with composition functors that end up looking just like those in label@name@defn:2cat-contra REF , and so on.", "But I hope that the digression makes this theorem seem less accidental; it also makes it clear how to generalize it to other examples.", "The underlying ordinary category $\\underline{A} _{\\,o}$ of a 2-category $\\underline{A}$ with contravariance, in the usual sense of enriched category theory, consists of its objects and its covariant 1-morphisms (the objects of the categories $\\underline{A} ^+(x,y)$ ).", "It also has an underlying ordinary 2-category, induced by the lax monoidal forgetful functor $(-)^+: \\rightarrow \\mathbf {Cat} $ , whose hom-categories are the categories $\\underline{A} ^+(x,y)$ ; we denote this 2-category by $\\underline{A} ^+$ .", "Of course, there is no 2-category to denote by “$\\underline{A} ^-$ ”, but we could say for instance that $\\underline{A} ^-$ is a profunctor from $\\underline{A} ^+$ to itself." ], [ "Opposites through enrichment", "For most of this section, we let $(,\\otimes ,{1})$ be an arbitrary biclosed monoidal category, not assumed symmetric.", "We are, of course, thinking of our from the last section, or more generally $\\textstyle \\int _G\\!$ .", "Suppose $\\underline{A}$ is a -category, that $x\\in \\operatorname{ob}\\underline{A} $ , and $\\omega \\in \\operatorname{ob}$ .", "A copower (or tensor) of $x$ by $\\omega $ is an object $\\omega \\odot x$ of $\\underline{A} $ together with isomorphisms in : ${}\\underline{A} (\\omega \\odot x,y) \\cong \\omega \\mathrel {} \\underline{A} (x,y)$ for all $y\\in \\operatorname{ob}\\underline{A} $ , which are -natural in the sense that for any $y,z\\in \\operatorname{ob}\\underline{A} $ , the following diagram commutes: ${\\underline{A} (y,z) \\otimes \\underline{A} (\\omega \\odot x,y)[r]^-\\cong [d] &\\underline{A} (y,z) \\otimes (\\omega \\mathrel {} \\underline{A} (x,y)) [r] &\\omega \\mathrel {} (\\underline{A} (y,z)\\otimes \\underline{A} (x,y)) [d]\\\\\\underline{A} (\\omega \\odot x, z)[rr]_\\cong &&\\omega \\mathrel {} \\underline{A} (x,z)}$ $Taking $ =x$ in~(\\ref {eq:copower}), we obtain from$ 1x$ a canonical map $$\\underline{A}$ (x,x)$, which bythe Yoneda lemma determines~(\\ref {eq:copower}) uniquely.", "Of course,this is just the usual definition of copowers in enriched categories,specialized to enrichment over .", "We have spelled it out explicitlyto emphasize that it makes perfect sense even though is notsymmetric, as long as we choose the correct hom $$ and not$ $ (see~\\cite {street:absolute}, which treats the even moregeneral case of enrichment over a \\emph {bicategory}).$ Note that if $\\underline{A} =$ (the category regarded as a -category), then the tensor product $\\omega \\otimes x$ is a copower $\\omega \\odot x$ .", "Moreover, for general $\\underline{A}$ , if $\\omega ,\\varpi \\in $ and the copowers $\\omega \\odot x$ and $\\varpi \\odot (\\omega \\odot x)$ exist, we have $\\underline{A} (\\varpi \\odot (\\omega \\odot x), y)&\\cong \\varpi \\mathrel {} \\underline{A} (\\omega \\odot x,y)\\\\&\\cong \\varpi \\mathrel {} (\\omega \\mathrel {} \\underline{A} (x,y))\\\\&\\cong (\\varpi \\otimes \\omega )\\mathrel {} \\underline{A} (x,y)$ so that $\\varpi \\odot (\\omega \\odot x)$ is a copower $(\\varpi \\otimes \\omega ) \\odot x$ .", "In particular, these observations mandate writing the copower as $\\omega \\odot x$ rather than $x\\odot \\omega $ .", "Frequently one defines a power in a -category $\\underline{A}$ to be a copower in $\\underline{A} ^{\\mathrm {op}}$ , but since our is not symmetric, -categories do not have opposites.", "Thus, we must define directly a power of $x$ by $\\omega $ to be an object $x \\mathrel {\\oslash } \\omega \\in \\operatorname{ob}\\underline{A} $ together with isomorphisms $\\underline{A} (y,x \\mathrel {\\oslash } \\omega ) \\cong \\underline{A} (y,x) \\mathrel {\\!\\!}", "\\omega $ for all $y\\in \\operatorname{ob}\\underline{A} $ , which are -natural in that the following diagram commutes: ${\\underline{A} (y,x \\mathrel {\\oslash } \\omega ) \\otimes \\underline{A} (z,y)[r]^-\\cong [d] &(\\underline{A} (y,x) \\mathrel {\\!\\!}", "\\omega ) \\otimes \\underline{A} (z,y)[r] &(\\underline{A} (y,x) \\otimes \\underline{A} (z,y)) \\mathrel {\\!\\!}", "\\omega [d]\\\\\\underline{A} (z,x \\mathrel {\\oslash } \\omega ) [rr]_-\\cong & &\\underline{A} (z,x) \\mathrel {\\!\\!}", "\\omega }$ $Analogous arguments to those for copowers show that when $ A =$,then $ x $ is a power $ x $, and that in general we have$ (x ) x ()$.", "If boththe copower $ x$ and the power $ x $ exist, then we have{\\begin{@align}{1}{-1}\\underline{A} _{\\,o}(\\omega \\odot x,y)&\\cong ({1},{\\underline{A} (\\omega \\odot x,y)})\\\\&\\cong ({1}, \\omega \\mathrel {} \\underline{A} (x,y))\\\\&\\cong (\\omega ,{\\underline{A} (x,y)})\\\\&\\cong ({1},\\underline{A} (x,y) \\mathrel {\\!\\!}", "\\omega )\\\\&\\cong ({1},\\underline{A} (x,y \\mathrel {\\oslash } \\omega ))\\\\&\\cong \\underline{A} _{\\,o}(x,y \\mathrel {\\oslash } \\omega ).\\end{@align}}so that the endofunctors $ (-)$ and $ (- )$ on theunderlying 1-category $$\\underline{A}$ o$ are adjoint.", "They are \\emph {not}adjoint -functors, even when $$\\underline{A}$ =$: in our motivating example,the isomorphisms~(\\ref {eq:biclosed}) do not even lift from the1-category $ o=$ to the 2-category $ +$.$ Now suppose that $\\omega $ is right dualizable in , i.e.", "that we have an object $\\omega ^\\in $ and morphisms $\\omega ^ \\otimes \\omega \\rightarrow {1} $ and ${1} \\rightarrow \\omega \\otimes \\omega ^$ satisfying the triangle identities.", "Then $(-\\otimes \\omega ^)$ is right adjoint to $(-\\otimes \\omega )$ , hence isomorphic to $(\\omega \\mathrel {} -)$ ; and dually we have $(\\omega \\otimes -) \\cong (- \\mathrel {\\!\\!}", "\\omega ^)$ .", "Thus, a copower $\\omega \\odot x$ in a -category $\\underline{A}$ is equivalently characterized by an isomorphism $\\underline{A} (\\omega \\odot x,-) \\cong \\underline{A} (x,-) \\otimes \\omega ^,{}$ while a power $x \\mathrel {\\oslash } \\omega ^$ is characterized by an isomorphism $\\underline{A} (-,x \\mathrel {\\oslash } \\omega ^) \\cong \\omega \\otimes \\underline{A} (-,x).", "{}$ However, for fixed $x$ , the right-hand sides of () and () are adjoint in the bicategory of -modules.", "Since $\\underline{A} (\\omega \\odot x,-)$ always has an adjoint $\\underline{A} (-,\\omega \\odot x)$ , and likewise $\\underline{A} (-,x \\mathrel {\\oslash } \\omega ^)$ always has an adjoint $\\underline{A} (x \\mathrel {\\oslash } \\omega ^,-)$ , it follows that giving a copower $\\omega \\odot x$ is equivalent to giving a power $x \\mathrel {\\oslash } \\omega ^*$ .", "Now let us specialize to the case of $\\textstyle \\int _G\\!", "$ .", "Then for any $g\\in G$ , we have a twisted unit ${1} ^{g} \\in \\textstyle \\int _G\\!", "$ , defined by ${1} ^{g}(h) = \\delta _{g,h} \\cdot {1} $ .", "By definition of $$ and $\\!\\!\\,$ , we have $({1} ^{h}\\mathrel {} \\underline{A} (x,y))(g) \\;&\\cong \\; \\underline{A} ^{gh}(x,y) \\qquad \\text{and}\\\\(\\underline{A} (x,y) \\mathrel {\\!\\!}", "{1} ^{h})(g) \\;&\\cong \\; {(\\underline{A} ^{hg}(x,y))}^{h^{-1}}.", "$ Thus, ${1} ^{h}\\odot x$ , if it exists, is characterized by isomorphisms $\\underline{A} ^g({1} ^{h}\\odot x,y) \\cong \\underline{A} ^{gh}(x,y)$ that are suitably and jointly natural in $y$ .", "In other words, a copower ${1} ^{h}\\odot x$ is precisely an $h$ -variator of $x$ as defined in label@name@sec:genmulti .", "And in our particular case of $=\\mathbf {Cat} $ , a copower ${1} ^{-}\\odot x$ is precisely an opposite of $x$ as defined in label@name@defn:opposite REF .", "Thus we have: Theorem 6.1 A 2-category with contravariance has opposites, as in label@name@defn:opposite REF , exactly if when regarded as a -category it has all copowers by ${1} ^{-}$ .", "Note that since ${1} ^{h}\\otimes {1} ^{h^{-1}} \\cong {1} $ , in particular ${1} ^{h}$ is dualizable.", "Thus, copowers by ${1} ^{h}$ are equivalent to powers by ${1} ^{h^{-1}}$ .", "In particular, since $-\\in \\lbrace +,-\\rbrace $ is its own inverse, it follows that ${1} ^{-}$ is self-dual, and opposites are also characterized by isomorphisms $\\underline{A} ^+(y,x^\\circ ) \\cong \\underline{A} ^-(y,x)^{\\mathrm {op}}\\qquad \\text{and}\\qquad \\underline{A} ^-(y,x^\\circ ) \\cong \\underline{A} ^+(y,x)^{\\mathrm {op}}.$ This gives another reason why a 2-functor preserving contravariance must preserve opposites: copowers by a dualizable object are absolute colimits [35]." ], [ "Bicategories with contravariance", "We have now reached the top of the right-hand side of the ladder from label@name@sec:introduction .", "It remains to move across to the other side and head down, starting with a bicategorical version of -categories for our $=\\int _{\\lbrace +,-\\rbrace }\\mathbf {Cat} $ .", "In fact, it will be convenient to stay in a more general setting.", "Thus, suppose that our monoidal category is actually a 2-category , and that our group $G$ acts on it by 2-functors.", "In this case, the construction of $\\textstyle \\int _G\\!$ can all be done with 2-categories, obtaining a monoidal 2-category $\\textstyle \\int _G\\!$ (in the strict sense of a monoidal $\\mathbf {Cat}$ -enriched category).", "Since a monoidal 2-category is a fortiori a monoidal bicategory, we can consider $\\textstyle \\int _G\\!$ -enriched bicategories, which we call $g$ -variant -bicategories.", "The most comprehensive extant reference on enriched bicategories seems to be [13], though the basic definition dates back at least to [4], [17].", "The definition of an enriched bicategory is quite simple: we just write out the definition of bicategory and replace all hom-categories by objects of $\\textstyle \\int _G\\!$ , cartesian products of categories by $\\otimes $ , and functors and natural transformations by morphisms and 2-cells in $\\textstyle \\int _G\\!$ .", "If we write this out explicitly, it consists of the following.", "[leftmargin=2em] A collection $\\operatorname{ob}\\underline{A} $ of objects; For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a category $\\underline{A} ^g(x,y)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , a unit morphism $1_x : {1} \\rightarrow \\underline{A} ^1(x,x)$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , composition morphisms $ \\underline{A} ^h(y,z) \\otimes {(\\underline{A} ^g(x,y))}^{h} \\rightarrow \\underline{A} ^{gh}(x,z) $ For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , two natural unitality isomorphisms; For each $x,y,z,w\\in \\operatorname{ob}\\underline{A} $ and $g,h,k\\in G$ , an associativity isomorphism; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , a unitality axiom holds; and For each $x,y,z,w,u\\in \\operatorname{ob}\\underline{A} $ and $g,h,k,\\ell \\in G$ , an associativity pentagon holds.", "Enriched bicategories, of course, come naturally with a notion of enriched functor.", "(In fact, as described in [13] we have a whole tricategory of enriched bicategories, but we will not need the higher structure.)", "Explicitly, a $\\textstyle \\int _G\\!$ -enriched functor $F:\\underline{A} \\rightarrow $ consists of [leftmargin=2em] A function $F:\\operatorname{ob}\\underline{A} \\rightarrow \\operatorname{ob}$ ; and For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a morphism $F:\\underline{A} ^g(x,y)\\rightarrow ^g(Fx,Fy)$ ; For each $x\\in \\operatorname{ob}\\underline{A} $ , an isomorphism $F(1_x) \\cong 1_{F x}$ ; For each $x,y,z\\in \\operatorname{ob}\\underline{A} $ and $g,h\\in G$ , a natural functoriality isomorphism of the form $(Fg)(Ff) \\cong F(gf)$ ; For each $x,y\\in \\operatorname{ob}\\underline{A} $ and $g\\in G$ , a unit coherence diagram commmutes; For each $x,y,z,w\\in \\operatorname{ob}\\underline{A} $ and $g,h,k\\in G$ , an associativity coherence diagram commutes.", "In the case of interest, we have $=\\mathbf {Cat} $ , which is of course enhances to the 2-category $\\mathcal {C}\\mathit {at}$ .", "However, we cannot take $=\\mathcal {C}\\mathit {at} $ , because as we have remarked, $(-)^{\\mathrm {op}}$ is not a 2-functor on $\\mathcal {C}\\mathit {at}$ , so $\\lbrace +,-\\rbrace $ does not act on $\\mathcal {C}\\mathit {at}$ through 2-functors.", "However, $(-)^{\\mathrm {op}}$ is a 2-functor on $\\mathcal {C}\\mathit {at} _g$ , the 2-category of categories, functors, and natural isomorphisms; so this is what we take as our $$ .", "We denote the resulting monoidal 2-category $\\textstyle \\int _G\\!$ by , and make the obvious definition: Definition 5 A bicategory with contravariance is a -enriched bicategory, and a pseudofunctor preserving contravariance is a -enriched functor.", "If we write this out explicitly in terms of covariant and contravariant parts, we see that a bicategory with contravariance has four kinds of composition functors, eight kinds of associativity isomorphisms, and sixteen coherence pentagons.", "Working with an abstract and $G$ thus allows us to avoid tedious case-analyses.", "We now generalize the enriched notion of $g$ -variator (and hence of “opposite”) from label@name@sec:opposites to the bicategorical case.", "For any $\\omega \\in \\textstyle \\int _G\\!", "$ , any $\\textstyle \\int _G\\!$ -bicategory $\\underline{A}$ , and any $x\\in \\underline{A} $ , a copower of $x$ by $\\omega $ is an object $\\omega \\odot x$ together with a map $\\omega \\rightarrow \\underline{A} (x,\\omega \\odot x)$ such that for any $y$ the induced map $\\underline{A} (\\omega \\odot x,y) \\rightarrow \\omega \\mathrel {} \\underline{A} (x,y)$ is an equivalence (not necessarily an isomorphism).", "(This is essentially the special case of [13] when is the unit $\\textstyle \\int _G\\!$ -bicategory.)", "As in label@name@sec:opposites , we are mainly interested in the case when $\\omega $ is one of the twisted units ${1} ^{g}$ .", "In this case we again write ${x}^{g}$ for ${1} ^{g}\\odot x$ , and the map ${1} ^{g} \\rightarrow \\underline{A} (x,{x}^{g})$ is just a $g$ -variant morphism $\\chi _{g,x}\\in \\underline{A} ^g(x,{x}^{g})$ .", "Its universal property says that any $gh$ -variant morphism $x\\xrightarrow[gh]{} y$ factors essentially uniquely through $\\chi _{g,x}$ via an $h$ -variant morphism ${x}^{g} \\xrightarrow[h]{} y$ (and similarly for 2-cells); that is, we have equivalences $ \\underline{A} ^h({x}^{g},y) \\xrightarrow{}\\underline{A} ^{gh}(x,y)$ As before, by Yoneda arguments this is equivalent to having a $g$ -variant morphism $x \\xrightarrow[g]{} {x}^{g}$ and a $g^{-1}$ -variant morphism ${x}^{g} \\xrightarrow[g^{-1}]{} x$ whose composites in both directions are isomorphic to identities; that is, a “$g$ -variant equivalence”.", "In the specific example of $=\\mathcal {C}\\mathit {at} _g$ and $G=\\lbrace +,-\\rbrace $ , we of course call ${x}^{-}$ a (weak) opposite of $x$ , written $x^\\circ $ .", "Our goal now is to show that any weak duality involution on a bicategory $\\mathcal {A}$ gives it the structure of a bicategory with contravariance having weak opposites; but to minimize case analyses, we will work in the generality of and $G$ .", "Thus, we first define a (weak, strictly unital) twisted $G$ -action on a -category $\\mathcal {A}$ to consist of: [leftmargin=2em] For each $g\\in G$ , a -functor ${(-)}^{g}:{\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ .", "(Note that here ${\\mathcal {A}}^{g}$ denotes the hom-wise action, ${\\mathcal {A}}^{g}(x,y) = {\\mathcal {A} (x,y)}^{g}$ .)", "When $g=1$ is the unit element of $G$ , we ask that ${(-)}^{1}$ be exactly equal to the identity functor.", "For each $g,h\\in G$ , a -pseudonatural adjoint equivalence $ {{(\\mathcal {A})}^{gh} [dr]_{{({(-)}^{g})}^{h}} [rr]^{{(-)}^{gh}} && \\mathcal {A}.\\\\& {\\mathcal {A}}^{h}[ur]_{{(-)}^{h}} @{}[u]\|(.6){\\Downarrow }}$ (Note that ${{({(-)}^{g})}^{h}}$ means the homwise endofunctor ${(-)}^{h}$ of $$ -bicategories applied to the action functor ${(-)}^{g}:{\\mathcal {A}}^{g}\\rightarrow \\mathcal {A} $ .)", "When $g$ or $h$ is $1\\in G$ , we ask that $$ be exactly the identity transformation.", "For each $g,h,k\\in G$ , an invertible -modification $ {&{\\mathcal {A}}^{hk} [dr] [dd] \\\\{\\mathcal {A}}^{ghk} [ur] [dr] @{}[r]\|(.6){{_{g,h}}^{k}} &@{}[r]\|(.4){_{h,k}}& \\mathcal {A} \\\\& {\\mathcal {A}}^{k} [ur] }$ $As before, when $ $, $ h$, or $ k$ is $ 1G$, we ask that $$ beexactly the identity.\\item For each $ g,h,k,G$, a 4-simplex diagram of instances of$$ commutes.$ In our motivating example of $=\\mathcal {C}\\mathit {at} _g$ and $G=\\lbrace +,-\\rbrace $ , the strict identity requirements mean that: [leftmargin=2em] The only nontrivial action is ${(-)}^{-}$ , which we write as $(-)^\\circ $ .", "The only nontrivial $$ is $_{-,-}$ , which has the same type as the $$ in label@name@defn:duality-involution REF .", "The only nontrivial $\\zeta $ is $\\zeta _{-,-,-}$ , which has an equivalent type to the $\\zeta $ in label@name@defn:duality-involution REF (since $--=+$ is the identity, $_{-,--}$ and $_{--,-}$ are identities, so the type of $\\zeta $ displayed above has moved one copy of $$ from the codomain to the domain).", "The only nontrivial axiom likewise reduces to the one given in label@name@defn:duality-involution REF .", "Thus, this really does generalize our notion of duality involution.", "Now we will show: Theorem 7.1 Let $\\mathcal {A}$ be a -bicategory with a twisted $G$ -action, and for $x,y\\in \\mathcal {A} $ and $g\\in G$ define $\\underline{A} ^g(x,y) = \\mathcal {A} ({x}^{g},y)$ .", "Then $\\underline{A} $ is a $\\textstyle \\int _G\\!$ -bicategory with copowers by all the twisted units.", "We define the composition morphisms as follows: $\\begin{array}{rcl}\\underline{A} ^h(y,z) \\otimes {(\\underline{A} ^g(x,y))}^{h}&=& \\mathcal {A} ({y}^{h},z) \\otimes {(\\mathcal {A} ({x}^{g},y))}^{h}\\\\&=& \\mathcal {A} ({y}^{h},z) \\otimes {\\mathcal {A}}^{h}({x}^{g},y)\\\\&\\xrightarrow{}& \\mathcal {A} ({y}^{h},z) \\otimes \\mathcal {A} ({({x}^{g})}^{h},{y}^{h})\\\\&\\xrightarrow{}& \\mathcal {A} ({({x}^{g})}^{h}, z)\\\\&\\xrightarrow{}& \\mathcal {A} ({x}^{gh},z)\\\\&=&\\underline{A} ^{gh}(x,z)\\end{array}$ Informally (or, formally, in an appropriate internal “linear type theory” of ), we can say that the composite of $\\beta \\in \\underline{A} ^h(y,z)$ and $\\alpha \\in {(\\underline{A} ^g(x,y))}^{h}$ is $\\beta \\circ {\\alpha }^{h}\\circ _{g,h}$ where $\\circ $ denotes composition in $\\mathcal {A}$ .", "Expressed in the same way, the associator for $\\alpha \\in {(\\underline{A} ^g(x,y))}^{hk}$ , $\\beta \\in {(\\underline{A} ^h(y,z))}^{k}$ , and $\\gamma \\in \\underline{A} ^k(z,w)$ is $(\\gamma \\circ {\\beta }^{k}\\circ _{h,k}) \\circ {\\alpha }^{hk}\\circ _{g,hk}&\\cong \\gamma \\circ {\\beta }^{k}\\circ {({\\alpha }^{h})}^{k} \\circ _{h,k} \\circ _{g,hk}\\\\&\\cong \\gamma \\circ {\\beta }^{k}\\circ {({\\alpha }^{h})}^{k} \\circ {_{g,h}}^{k} \\circ _{gh,k}\\\\&\\cong \\gamma \\circ {(\\beta \\circ {\\alpha }^{h}\\circ _{g,h})}^{k} \\circ _{gh,k}$ using the naturality of , the modification $\\zeta $ , and the functoriality of ${(-)}^{k}$ (and omitting the associativity isomorphisms of $\\mathcal {A}$ , by coherence for bicategories).", "For the unit, since $\\underline{A} ^1(x,y)= \\mathcal {A} (x,y)$ , the unit map ${1} \\rightarrow \\underline{A} ^1(x,y)$ is just the unit of $\\mathcal {A}$ .", "One unit isomorphism is just that of $\\mathcal {A}$ , while the other is that of $\\mathcal {A}$ together with the unit isomorphism of the pseudofunctor ${(-)}^{g}$ .", "And the associator appearing in the unit axiom has $g=k=1$ , so all the $$ 's collapse and it is essentially trivial, and the unit axiom follows immediately from that of $\\mathcal {A}$ .", "To show that $\\underline{A}$ is a $\\textstyle \\int _G\\!$ -bicategory, it remains to consider the pentagon axiom.", "Omitting $\\circ $ from now on, the pentagon axiom is an equality of two morphisms $ \\delta {\\gamma }^{\\ell }_{k,\\ell } {\\beta }^{k\\ell }_{h,k\\ell } {\\alpha }^{hk\\ell }_{g,hk\\ell }\\longrightarrow \\delta {(\\gamma {(\\beta {\\alpha }^{h}_{g,h})}^{k} _{gh,k})}^{\\ell } _{ghk,\\ell }$ By naturality of the functoriality isomorphisms for the actions ${(-)}^{g}$ , we can certainly push all applications of them to the end where they will be equal; thus it suffices to compare the morphisms $ \\delta {\\gamma }^{\\ell }_{k,\\ell } {\\beta }^{k\\ell }_{h,k\\ell } {\\alpha }^{hk\\ell }_{g,hk\\ell }\\longrightarrow \\delta {\\gamma }^{\\ell }{({\\beta }^{k})}^{\\ell } {({({\\alpha }^{h})}^{k})}^{\\ell } {({_{g,h}}^{k})}^{\\ell } {_{gh,k}}^{\\ell } _{ghk,\\ell }$ This is done in Figure REF , where most of the regions are naturality, except for the one at the bottom left which is the 4-simplex axiom for $\\zeta $ .", "Figure: The pentagon axiomNow we must show that $\\underline{A}$ has copowers by the twisted units; of course we will use ${x}^{g}$ as the copower ${1} ^{g} \\odot x$ .", "Since $\\underline{A} ^g(x,{x}^{g}) = \\mathcal {A} ({x}^{g},{x}^{g})$ by definition, for $\\chi _{g,x}$ we can take the identity map of ${x}^{g}$ in $\\mathcal {A}$ .", "By definition of composition in $\\underline{A}$ , the induced precomposition map $\\underline{A} ^h({x}^{g},y) \\rightarrow \\underline{A} ^{gh}(x,y)$ is essentially just precomposition with $$ : $ \\mathcal {A} ({({x}^{g})}^{h},y) \\rightarrow \\mathcal {A} ({x}^{gh},y) $ and hence is an equivalence.", "Thus, $\\underline{A}$ has copowers by the twisted units.", "Inspecting the construction, we also conclude: Scholium 1 If a 2-category $\\mathcal {A}$ has a twisted $G$ -action in the sense of label@name@sec:genmulti , and we regard this as a weak twisted $G$ -action in the sense defined above with the actions being strict functors, $$ strictly natural, and $\\zeta $ an identity, then the $\\textstyle \\int _G\\!$ -bicategory constructed in label@name@thm:bicat REF is actually a strict $\\textstyle \\int _G\\!$ -category, and this construction agrees with the one in §–.", "In particular, if $\\mathcal {A}$ is a 2-category with a strong duality involution, and we regard it as a bicategory with a weak duality involution to construct a bicategory with contravariance $\\underline{A}$ , the result is the 2-category with contravariance we already obtained from it in label@name@sec:genmulti .", "With some more work we could enhance label@name@thm:bicat REF to a whole equivalence of tricategories.", "However, all we will need for our coherence theorem, in addition to label@name@thm:bicat REF and label@name@thm:bicat-2cat REF , is to go backwards on biequivalences.", "Before stating such a theorem, we have to define what we want to get out of it.", "Suppose $\\mathcal {A}$ and are -bicategories with twisted $G$ -action; by a twisted $G$ -functor $F:\\mathcal {A} \\rightarrow $ we mean a functor of -bicategories together with: [leftmargin=2em] For each $g\\in G$ , a -pseudonatural adjoint equivalence ${{\\mathcal {A}}^{g}[d]_{{(-)}^{g}}[r]^-{{F}^{g}} {\\mathfrak {i}} &{}^{g}[d]^{{(-)}^{g}}\\\\\\mathcal {A} [r]_-F &.", "}$ $\\item For each $ ,hG$, an invertible -modification$$@R=3pc@C=3pc{&& {{\\mathcal {A}}^{gh}} [dl]_{{({(-)}^{g})}^{h}} [r]^{{F}^{gh}} {\\mathfrak {i}}& {}^{gh}[dl]\|{{({(-)}^{g})}^{h}} ^{{{(-)}^{gh}}}{}\\\\&{\\mathcal {A}}^{h}[d]_{{(-)}^{h}}[r]\|-{{F}^{h}} {\\mathfrak {i}} &{}^{h}[d]\|{{(-)}^{h}}\\\\&\\mathcal {A} [r]_-F &}$ $(As before, $ ((-)g)h$ denotes the functorial action ofthe homwise endofunctor $ (-)h$ of -bicategories on the givenaction functor $ (-)g: $\\mathcal {A}$ h$\\mathcal {A}$ $.)", "This can bewritten formally as$$ \\mathfrak {i}_h \\circ {\\mathfrak {i}_g}^{h} \\circ ^_{g,h} \\cong ^\\mathcal {A} _{g,h} \\circ \\mathfrak {i}_{gh} $$\\item For all $ g,h,kG$, an axiom holds that can be writtenformally as the commutative diagram shown in Figure~\\ref {fig:theta-ax}.\\begin{figure}\\centering {\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} {({\\mathfrak {i}_g}^{h})}^{k} ^_{h,k} ^_{g,hk} [r] [d]_\\zeta &\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} ^_{h,k} {\\mathfrak {i}_g}^{hk} ^_{g,hk} [r]^\\theta &^\\mathcal {A} _{h,k} \\mathfrak {i}_{hk} {\\mathfrak {i}_g}^{hk} ^_{g,hk} [d]^\\theta \\\\\\mathfrak {i}_k {\\mathfrak {i}_h}^{k} {({\\mathfrak {i}_g}^{h})}^{k} {^_{g,h}}^{k} ^_{gh,k} [d] && ^\\mathcal {A} _{h,k} ^\\mathcal {A} _{g,hk} \\mathfrak {i}_{ghk} [d]^\\zeta \\\\\\mathfrak {i}_k {(\\mathfrak {i}_h {{\\mathfrak {i}_g}^{h}} {^_{g,h}})}^{k} ^_{gh,k} [d]_\\theta & &{(^\\mathcal {A} _{g,h})}^{k} ^\\mathcal {A} _{gh,k} \\mathfrak {i}_{ghk}\\\\\\mathfrak {i}_k {( {^\\mathcal {A} _{g,h}} \\mathfrak {i}_{gh})}^{k} ^_{gh,k} [r] &\\mathfrak {i}_k {(^\\mathcal {A} _{g,h})}^{k} {(\\mathfrak {i}_{gh})}^{k} ^_{gh,k} [r] &{(^\\mathcal {A} _{g,h})}^{k} \\mathfrak {i}_k {(\\mathfrak {i}_{gh})}^{k} ^_{gh,k} [u]_\\theta & }\\caption {The axiom for \\theta }{}\\end{figure}$ Theorem 7.2 Suppose $\\mathcal {A}$ and are -bicategories with twisted $G$ -action, with resulting $\\textstyle \\int _G\\!$ -bicategories $\\underline{A}$ and .", "If $\\underline{A}$ and are biequivalent as $\\textstyle \\int _G\\!$ -bicategories, then $\\mathcal {A}$ and are biequivalent by a twisted $G$ -functor.", "In particular, if two bicategories $\\mathcal {A}$ and with duality involution give rise to biequivalent bicategories-with-contravariance, then $\\mathcal {A}$ and are biequivalent by a duality pseudofunctor.", "Let $F:\\underline{A} \\rightarrow $ be a $\\textstyle \\int _G\\!$ -biequivalence.", "In particular, then, it is a biequivalence on the 1-parts, hence a biequivalence $\\mathcal {A} \\simeq $ .", "Now by label@name@thm:bicat REF , for any $x\\in \\mathcal {A} $ we have a “$g$ -variant equivalence” $x\\xrightarrow[g]{} {x}^{g}$ with inverse ${x}^{g} \\xrightarrow[g^{-1}]{} x$ .", "This structure is preserved by $F$ , so we have a $g$ -variant equivalence between $Fx$ and $F({x}^{g})$ .", "But we also have a $g$ -variant equivalence between $Fx$ and ${(Fx)}^{g}$ , and composing them we obtain an ordinary (1-variant) isomorphism ${(Fx)}^{g}\\cong F({x}^{g})$ .", "These supply the components of $\\mathfrak {i}$ ; their pseudonaturality is straightforward.", "Now, by construction of the copowers by twisted units, it follows that $_{g,h} : {x}^{gh} \\rightarrow {({x}^{g})}^{h}$ is isomorphic to the composite of the variant equivalences $ {x}^{gh} \\xrightarrow[(gh)^{-1}]{} x \\xrightarrow[g]{} {x}^{g} \\xrightarrow[h]{} {({x}^{g})}^{h}$ while $\\zeta $ is obtained by canceling and uncanceling some of these equivalences.", "In particular, when $$ is composed with $\\mathfrak {i}$ , we can simply cancel some inverse variant equivalences to obtain the components of $\\theta $ .", "As for Figure , its source is ${(F(x))}^{ghk}\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[g]{} {(F(x))}^{g}\\xrightarrow[hk]{} {({(F(x))}^{g})}^{hk}\\\\\\xrightarrow[(hk)^{-1}]{} {(F(x))}^{g}\\xrightarrow[h]{} {({(F(x))}^{g})}^{h}\\xrightarrow[k]{} {({({(F(x))}^{g})}^{h})}^{k}\\\\\\xrightarrow[g^{-1}]{} {({(F(x))}^{h})}^{k}\\xrightarrow[g]{} {({(F({{x}^{g}}))}^{h})}^{k}\\\\\\xrightarrow[h^{-1}]{} {(F({{x}^{g}}))}^{k}\\xrightarrow[h]{} {(F({({x}^{g})}^{h}))}^{k}\\xrightarrow[k^{-1}]{} {F({({x}^{g})}^{h})}\\xrightarrow[k]{} F({({({x}^{g})}^{h})}^{k})$ while its target is ${(F(x))}^{ghk}\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[ghk]{} F({x}^{ghk})\\\\\\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[gh]{} F({x}^{gh})\\xrightarrow[k]{} F({({x}^{gh})}^{k})\\\\\\xrightarrow[(gh)^{-1}]{} F({x}^{k})\\xrightarrow[g]{} F({({x}^{g})}^{k})\\xrightarrow[h]{} F({({({x}^{g})}^{h})}^{k})$ Here we have applied functors such as ${(-)}^{k}$ to variant morphisms; we can define this by simply “conjugating” with the variant equivalences $x\\xrightarrow[k]{} {x}^{k}$ .", "We leave it to the reader to apply naturality and cancel all the redundancy in these composites, reducing them both to $ {(F(x))}^{ghk} \\xrightarrow[(ghk)^{-1}]{} F(x)\\xrightarrow[g]{} F({{x}^{g}})\\xrightarrow[h]{} F({({x}^{g})}^{h})\\xrightarrow[k]{} F({({({x}^{g})}^{h})}^{k}) $ so that they are equal.", "Therefore, to strictify a bicategory with duality involution, it will suffice to strictify its corresponding bicategory with contravariance.", "This is the task of the next, and final, section.", "Coherence for enriched bicategories We could continue in the generality of $G$ and , but there seems little to be gained by it any more.", "Theorem 8.1 Any bicategory with contravariance is biequivalent to a 2-category with contravariance.", "Just as there are two ways to prove the coherence theorem for ordinary bicategories, there are two ways to prove this coherence theorem.", "The first is an algebraic one, involving formally adding strings of composable arrows that hence compose strictly associatively.", "This can be expressed abstractly using the same general coherence theorem for pseudo-algebras over a 2-monad that we used in label@name@sec:2-monadic-approach .", "As sketched at the end of [31], this theorem (or a slight generalization of it) applies as soon as we observe that our 2-category is closed monoidal and cocomplete.", "The other method is by a Yoneda embedding.", "To generalize this to the enriched (and non-symmetric) case, first note that for any -bicategory $\\underline{A}$ , by [13] we have a -bicategory $\\underline{A} $ of moderate $\\underline{A}$ -modules, and a Yoneda embedding $\\underline{A} \\rightarrow \\underline{A} $ that is fully faithful.", "Thus, $\\underline{A}$ is biequivalent to its image in $\\underline{A} $ .", "However, since is a strict 2-category that is closed and complete, $\\underline{A} $ is actually a strict -category, and hence so is any full subcategory of it.", "Explicitly, an $\\underline{A}$ -module consists of categories $F^+(x)$ and $F^-(x)$ for each $x\\in \\underline{A} $ together with actions $F^+(y) \\times \\underline{A} ^+(x,y) &\\rightarrow F^+(x)\\\\F^+(y) \\times \\underline{A} ^-(x,y) &\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^+(x,y)^{\\mathrm {op}}&\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^-(x,y)^{\\mathrm {op}}&\\rightarrow F^+(x)$ and coherent associativity and unitality isomorphisms.", "A covariant $\\underline{A}$ -module morphism consists of functors $F^+(x) \\rightarrow G^+(x)$ and $F^-(x)\\rightarrow G^-(x)$ that commute up to coherent natural isomorphism with the actions, while a contravariant one consists similarly of functors $F^+(x)^{\\mathrm {op}}\\rightarrow G^-(x)$ and $F^-(x)^{\\mathrm {op}}\\rightarrow G^+(x)$ .", "Since $\\mathcal {C}\\mathit {at}$ is a strict 2-category, the bicategory-with-contravariance of modules is in fact a strict 2-category with contravariance.", "The Yoneda embedding, of course, sends each $z\\in \\underline{A} $ to the representable $Y_z$ defined by $Y_z^+(x) \\underline{A} ^+(x,z)$ and $Y_z^-(x)\\underline{A} ^-(x,z)$ .", "We have almost completed our trip over the ladder; it remains to make the following observation and then put all the pieces together.", "Theorem 8.2 If $\\underline{A}$ is a 2-category with contravariance that has weak opposites, then it is biequivalent to a 2-category with contravariance having strict opposites.", "Let $\\underline{A}^{\\prime }$ be the free cocompletion of $\\underline{A}$ , as a strict -category, under strict opposites (a strict -weighted colimit).", "It is easy to see that this can be done in one step, by considering the collection of all opposites of representables in the presheaf -category of $\\underline{A}$ .", "Thus, the embedding $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is -fully-faithful, and every object of $\\underline{A}^{\\prime }$ is the strict opposite of something in the image.", "However, $\\underline{A}$ has weak opposites, which are preserved by any -functor, and any strict opposite is a weak opposite.", "Thus, every object of $\\underline{A}^{\\prime }$ is equivalent to something in the image of $\\underline{A}$ , since they are both a weak opposite of the same object.", "Hence $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is bicategorically essentially surjective, and thus a biequivalence.", "Finally, we can prove label@name@thm:main REF .", "Theorem 8.3 If $\\mathcal {A} $ is a bicategory with a weak duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality pseudofunctor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a biequivalence.", "By label@name@thm:bicat REF , we can regard $\\mathcal {A}$ as a bicategory with contravariance $\\underline{A}$ having weak opposites.", "By label@name@thm:bicat-coherence REF , it is therefore biequivalent to a 2-category with contravariance and weak opposites, and therefore by label@name@thm:opposites-weaktostrict REF also biequivalent to a 2-category with contravariance and strict opposites.", "Now by label@name@thm:gm-2cat REF and label@name@thm:contrav-enriched REF , the latter is equivalently a 2-category with a strong duality involution.", "Thus, by label@name@thm:2monad-coherence REF it is equivalent to a 2-category with a strict duality involution, say $\\mathcal {A}^{\\prime }$ .", "So we have a biequivalence $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a pseudofunctor preserving contravariance, and by label@name@thm:biequiv REF , we can also regard it as a duality pseudofunctor.", "As mentioned in label@name@sec:introduction , we could actually dispense with the right-hand side of the ladder as follows.", "Let $\\mathcal {A}^{\\prime }$ be the full sub–bicategory of $\\mathcal {A} $ , as in label@name@thm:bicat-coherence REF , consisting of the modules that are either of the form $Y_z$ or of the form $Y_z^\\circ $ , where $Y_z^\\circ $ is defined by $(Y_z^\\circ )^+(x) \\underline{A} ^-(x,z)^{\\mathrm {op}}$ and $(Y_z^\\circ )^-(x) \\underline{A} ^+(x,z)^{\\mathrm {op}}$ .", "This $\\mathcal {A}^{\\prime }$ is a 2-category with a strict duality involution that interchanges $Y_z$ and $Y_z^\\circ $ , and the Yoneda embedding is a biequivalence for the same reasons.", "However, this quicker argument still depends on the description of weak duality involutions using bicategorical enrichment from label@name@sec:bicat-contra , and thus still depends conceptually on the entire picture.", "R. Blackwell, G. M. Kelly, and A. J.", "Power.", "Two-dimensional monad theory.", "J.", "Pure Appl.", "Algebra, 59(1):1–41, 1989.", "G. J. Bird, G. M. Kelly, A. J.", "Power, and R. H. Street.", "Flexible limits for 2-categories.", "J.", "Pure Appl.", "Algebra, 61(1):1–27, 1989.", "Julia E. Bergner and Charles Rezk.", "Comparison of models for $(\\infty ,n)$ -categories, I. Geometry & Topology, 17:2163–2202, 2013. arXiv:1204.2013.", "S.M.", "Carmody.", "Cobordism Categories.", "PhD thesis, University of Cambridge, 1995.", "G.S.H.", "Cruttwell and Michael Shulman.", "A unified framework for generalized multicategories.", "Theory Appl.", "Categ., 24:580–655, 2010. arXiv:0907.2460.", "Brian Day.", "A reflection theorem for closed categories.", "J.", "Pure Appl.", "Algebra, 2(1):1–11, 1972.", "Brian Day and Ross Street.", "Monoidal bicategories and Hopf algebroids.", "Adv.", "Math., 129(1):99–157, 1997.", "Richard Garner.", "Lawvere theories, finitary monads and Cauchy-completion.", "Journal of Pure and Applied Algebra, 218(11):1973–1988, 2014.", "Richard Garner.", "An embedding theorem for tangent categories.", "Advances in Mathematics, 323:668 – 687, 2018.", "R. Gordon and A. J.", "Power.", "Enrichment through variation.", "J.", "Pure Appl.", "Algebra, 120(2):167–185, 1997.", "Richard Garner and John Power.", "An enriched view on the extended finitary monad–Lawvere theory correspondence.", "arXiv:1707.08694, 2017.", "R. Gordon, A. J.", "Power, and Ross Street.", "Coherence for tricategories.", "Mem.", "Amer.", "Math.", "Soc., 117(558):vi+81, 1995.", "Richard Garner and Michael Shulman.", "Enriched categories as a free cocompletion.", "Adv.", "Math, 289:1–94, 2016. arXiv:1301.3191.", "Nick Gurski.", "Biequivalences in tricategories.", "Theory and Applications of Categories, 26(14):329–384, 2012.", "Claudio Hermida.", "From coherent structures to universal properties.", "J.", "Pure Appl.", "Algebra, 165(1):7–61, 2001.", "A. Joyal.", "Quasi-categories and Kan complexes.", "Journal of Pure and Applied Algebra, 175:207–222, 2002.", "Stephen Lack.", "The algebra of distributive and extensive categories.", "PhD thesis, University of Cambridge, 1995.", "Stephen Lack.", "A coherent approach to pseudomonads.", "Adv.", "Math., 152(2):179–202, 2000.", "Stephen Lack.", "Codescent objects and coherence.", "J.", "Pure Appl.", "Algebra, 175(1-3):223–241, 2002.", "Special volume celebrating the 70th birthday of Professor Max Kelly.", "Fosco Loregian.", "Answer to MathOverflow question “Twisted Day convolution”.", "http://mathoverflow.net/questions/233812/twisted-day-convolution, March 2016.", "Stephen Lack and Michael Shulman.", "Enhanced 2-categories and limits for lax morphisms.", "Advances in Mathematics, 229(1):294–356, 2012. arXiv:1104.2111.", "Jacob Lurie.", "Higher topos theory.", "Number 170 in Annals of Mathematics Studies.", "Princeton University Press, 2009.", "M. Makkai.", "Avoiding the axiom of choice in general category theory.", "J.", "Pure Appl.", "Algebra, 108(2):109–173, 1996.", "A. J.", "Power.", "A general coherence result.", "J.", "Pure Appl.", "Algebra, 57(2):165–173, 1989.", "John Power.", "Three dimensional monad theory.", "In Categories in algebra, geometry and mathematical physics, volume 431 of Contemp.", "Math., pages 405–426.", "Amer.", "Math.", "Soc., Providence, RI, 2007.", "Dorette A. Pronk.", "Etendues and stacks as bicategories of fractions.", "Compositio Math., 102(3):243–303, 1996.", "Charles Rezk.", "A cartesian presentation of weak $n$ -categories.", "Geometry & Topology, 14, 2010. arXiv:0901.3602.", "David M. Roberts.", "Internal categories, anafunctors and localisations.", "Theory and Applications of Categories, 26(29):788–829, 2012. arXiv:1101.2363.", "Emily Riehl and Dominic Verity.", "Fibrations and Yoneda's lemma in an $\\infty $ -cosmos.", "J.", "Pure Appl.", "Algebra, 221(3):499–564, 2017. arXiv:1506.05500.", "Michael Shulman.", "Framed bicategories and monoidal fibrations.", "Theory and Applications of Categories, 20(18):650–738 (electronic), 2008. arXiv:0706.1286.", "Michael Shulman.", "Not every pseudoalgebra is equivalent to a strict one.", "Adv.", "Math., 229(3):2024–2041, 2012. arXiv:1005.1520.", "Michael Stay.", "Compact closed bicategories.", "Theory and Applications of Categories, 31(26):755–798, 2016. arXiv:1301.1053.", "Ross Street.", "The formal theory of monads.", "J.", "Pure Appl.", "Algebra, 2(2):149–168, 1972.", "Ross Street.", "Fibrations and Yoneda's lemma in a 2-category.", "In Category Seminar (Proc.", "Sem., Sydney, 1972/1973), pages 104–133.", "Lecture Notes in Math., Vol.", "420.", "Springer, Berlin, 1974.", "Ross Street.", "Absolute colimits in enriched categories.", "Cahiers Topologie Géom.", "Différentielle, 24(4):377–379, 1983.", "Mark Weber.", "Yoneda structures from 2-toposes.", "Appl.", "Categ.", "Structures, 15(3):259–323, 2007." ], [ "Coherence for enriched bicategories", "We could continue in the generality of $G$ and , but there seems little to be gained by it any more.", "Theorem 8.1 Any bicategory with contravariance is biequivalent to a 2-category with contravariance.", "Just as there are two ways to prove the coherence theorem for ordinary bicategories, there are two ways to prove this coherence theorem.", "The first is an algebraic one, involving formally adding strings of composable arrows that hence compose strictly associatively.", "This can be expressed abstractly using the same general coherence theorem for pseudo-algebras over a 2-monad that we used in label@name@sec:2-monadic-approach .", "As sketched at the end of [31], this theorem (or a slight generalization of it) applies as soon as we observe that our 2-category is closed monoidal and cocomplete.", "The other method is by a Yoneda embedding.", "To generalize this to the enriched (and non-symmetric) case, first note that for any -bicategory $\\underline{A}$ , by [13] we have a -bicategory $\\underline{A} $ of moderate $\\underline{A}$ -modules, and a Yoneda embedding $\\underline{A} \\rightarrow \\underline{A} $ that is fully faithful.", "Thus, $\\underline{A}$ is biequivalent to its image in $\\underline{A} $ .", "However, since is a strict 2-category that is closed and complete, $\\underline{A} $ is actually a strict -category, and hence so is any full subcategory of it.", "Explicitly, an $\\underline{A}$ -module consists of categories $F^+(x)$ and $F^-(x)$ for each $x\\in \\underline{A} $ together with actions $F^+(y) \\times \\underline{A} ^+(x,y) &\\rightarrow F^+(x)\\\\F^+(y) \\times \\underline{A} ^-(x,y) &\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^+(x,y)^{\\mathrm {op}}&\\rightarrow F^-(x)\\\\F^-(y) \\times \\underline{A} ^-(x,y)^{\\mathrm {op}}&\\rightarrow F^+(x)$ and coherent associativity and unitality isomorphisms.", "A covariant $\\underline{A}$ -module morphism consists of functors $F^+(x) \\rightarrow G^+(x)$ and $F^-(x)\\rightarrow G^-(x)$ that commute up to coherent natural isomorphism with the actions, while a contravariant one consists similarly of functors $F^+(x)^{\\mathrm {op}}\\rightarrow G^-(x)$ and $F^-(x)^{\\mathrm {op}}\\rightarrow G^+(x)$ .", "Since $\\mathcal {C}\\mathit {at}$ is a strict 2-category, the bicategory-with-contravariance of modules is in fact a strict 2-category with contravariance.", "The Yoneda embedding, of course, sends each $z\\in \\underline{A} $ to the representable $Y_z$ defined by $Y_z^+(x) \\underline{A} ^+(x,z)$ and $Y_z^-(x)\\underline{A} ^-(x,z)$ .", "We have almost completed our trip over the ladder; it remains to make the following observation and then put all the pieces together.", "Theorem 8.2 If $\\underline{A}$ is a 2-category with contravariance that has weak opposites, then it is biequivalent to a 2-category with contravariance having strict opposites.", "Let $\\underline{A}^{\\prime }$ be the free cocompletion of $\\underline{A}$ , as a strict -category, under strict opposites (a strict -weighted colimit).", "It is easy to see that this can be done in one step, by considering the collection of all opposites of representables in the presheaf -category of $\\underline{A}$ .", "Thus, the embedding $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is -fully-faithful, and every object of $\\underline{A}^{\\prime }$ is the strict opposite of something in the image.", "However, $\\underline{A}$ has weak opposites, which are preserved by any -functor, and any strict opposite is a weak opposite.", "Thus, every object of $\\underline{A}^{\\prime }$ is equivalent to something in the image of $\\underline{A}$ , since they are both a weak opposite of the same object.", "Hence $\\underline{A} \\rightarrow \\underline{A}^{\\prime }$ is bicategorically essentially surjective, and thus a biequivalence.", "Finally, we can prove label@name@thm:main REF .", "Theorem 8.3 If $\\mathcal {A} $ is a bicategory with a weak duality involution, then there is a 2-category $\\mathcal {A}^{\\prime }$ with a strict duality involution and a duality pseudofunctor $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a biequivalence.", "By label@name@thm:bicat REF , we can regard $\\mathcal {A}$ as a bicategory with contravariance $\\underline{A}$ having weak opposites.", "By label@name@thm:bicat-coherence REF , it is therefore biequivalent to a 2-category with contravariance and weak opposites, and therefore by label@name@thm:opposites-weaktostrict REF also biequivalent to a 2-category with contravariance and strict opposites.", "Now by label@name@thm:gm-2cat REF and label@name@thm:contrav-enriched REF , the latter is equivalently a 2-category with a strong duality involution.", "Thus, by label@name@thm:2monad-coherence REF it is equivalent to a 2-category with a strict duality involution, say $\\mathcal {A}^{\\prime }$ .", "So we have a biequivalence $\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime }$ that is a pseudofunctor preserving contravariance, and by label@name@thm:biequiv REF , we can also regard it as a duality pseudofunctor.", "As mentioned in label@name@sec:introduction , we could actually dispense with the right-hand side of the ladder as follows.", "Let $\\mathcal {A}^{\\prime }$ be the full sub–bicategory of $\\mathcal {A} $ , as in label@name@thm:bicat-coherence REF , consisting of the modules that are either of the form $Y_z$ or of the form $Y_z^\\circ $ , where $Y_z^\\circ $ is defined by $(Y_z^\\circ )^+(x) \\underline{A} ^-(x,z)^{\\mathrm {op}}$ and $(Y_z^\\circ )^-(x) \\underline{A} ^+(x,z)^{\\mathrm {op}}$ .", "This $\\mathcal {A}^{\\prime }$ is a 2-category with a strict duality involution that interchanges $Y_z$ and $Y_z^\\circ $ , and the Yoneda embedding is a biequivalence for the same reasons.", "However, this quicker argument still depends on the description of weak duality involutions using bicategorical enrichment from label@name@sec:bicat-contra , and thus still depends conceptually on the entire picture.", "R. Blackwell, G. M. Kelly, and A. J.", "Power.", "Two-dimensional monad theory.", "J.", "Pure Appl.", "Algebra, 59(1):1–41, 1989.", "G. J. Bird, G. M. Kelly, A. J.", "Power, and R. H. Street.", "Flexible limits for 2-categories.", "J.", "Pure Appl.", "Algebra, 61(1):1–27, 1989.", "Julia E. Bergner and Charles Rezk.", "Comparison of models for $(\\infty ,n)$ -categories, I. Geometry & Topology, 17:2163–2202, 2013. arXiv:1204.2013.", "S.M.", "Carmody.", "Cobordism Categories.", "PhD thesis, University of Cambridge, 1995.", "G.S.H.", "Cruttwell and Michael Shulman.", "A unified framework for generalized multicategories.", "Theory Appl.", "Categ., 24:580–655, 2010. arXiv:0907.2460.", "Brian Day.", "A reflection theorem for closed categories.", "J.", "Pure Appl.", "Algebra, 2(1):1–11, 1972.", "Brian Day and Ross Street.", "Monoidal bicategories and Hopf algebroids.", "Adv.", "Math., 129(1):99–157, 1997.", "Richard Garner.", "Lawvere theories, finitary monads and Cauchy-completion.", "Journal of Pure and Applied Algebra, 218(11):1973–1988, 2014.", "Richard Garner.", "An embedding theorem for tangent categories.", "Advances in Mathematics, 323:668 – 687, 2018.", "R. Gordon and A. J.", "Power.", "Enrichment through variation.", "J.", "Pure Appl.", "Algebra, 120(2):167–185, 1997.", "Richard Garner and John Power.", "An enriched view on the extended finitary monad–Lawvere theory correspondence.", "arXiv:1707.08694, 2017.", "R. Gordon, A. J.", "Power, and Ross Street.", "Coherence for tricategories.", "Mem.", "Amer.", "Math.", "Soc., 117(558):vi+81, 1995.", "Richard Garner and Michael Shulman.", "Enriched categories as a free cocompletion.", "Adv.", "Math, 289:1–94, 2016. arXiv:1301.3191.", "Nick Gurski.", "Biequivalences in tricategories.", "Theory and Applications of Categories, 26(14):329–384, 2012.", "Claudio Hermida.", "From coherent structures to universal properties.", "J.", "Pure Appl.", "Algebra, 165(1):7–61, 2001.", "A. Joyal.", "Quasi-categories and Kan complexes.", "Journal of Pure and Applied Algebra, 175:207–222, 2002.", "Stephen Lack.", "The algebra of distributive and extensive categories.", "PhD thesis, University of Cambridge, 1995.", "Stephen Lack.", "A coherent approach to pseudomonads.", "Adv.", "Math., 152(2):179–202, 2000.", "Stephen Lack.", "Codescent objects and coherence.", "J.", "Pure Appl.", "Algebra, 175(1-3):223–241, 2002.", "Special volume celebrating the 70th birthday of Professor Max Kelly.", "Fosco Loregian.", "Answer to MathOverflow question “Twisted Day convolution”.", "http://mathoverflow.net/questions/233812/twisted-day-convolution, March 2016.", "Stephen Lack and Michael Shulman.", "Enhanced 2-categories and limits for lax morphisms.", "Advances in Mathematics, 229(1):294–356, 2012. arXiv:1104.2111.", "Jacob Lurie.", "Higher topos theory.", "Number 170 in Annals of Mathematics Studies.", "Princeton University Press, 2009.", "M. Makkai.", "Avoiding the axiom of choice in general category theory.", "J.", "Pure Appl.", "Algebra, 108(2):109–173, 1996.", "A. J.", "Power.", "A general coherence result.", "J.", "Pure Appl.", "Algebra, 57(2):165–173, 1989.", "John Power.", "Three dimensional monad theory.", "In Categories in algebra, geometry and mathematical physics, volume 431 of Contemp.", "Math., pages 405–426.", "Amer.", "Math.", "Soc., Providence, RI, 2007.", "Dorette A. Pronk.", "Etendues and stacks as bicategories of fractions.", "Compositio Math., 102(3):243–303, 1996.", "Charles Rezk.", "A cartesian presentation of weak $n$ -categories.", "Geometry & Topology, 14, 2010. arXiv:0901.3602.", "David M. Roberts.", "Internal categories, anafunctors and localisations.", "Theory and Applications of Categories, 26(29):788–829, 2012. arXiv:1101.2363.", "Emily Riehl and Dominic Verity.", "Fibrations and Yoneda's lemma in an $\\infty $ -cosmos.", "J.", "Pure Appl.", "Algebra, 221(3):499–564, 2017. arXiv:1506.05500.", "Michael Shulman.", "Framed bicategories and monoidal fibrations.", "Theory and Applications of Categories, 20(18):650–738 (electronic), 2008. arXiv:0706.1286.", "Michael Shulman.", "Not every pseudoalgebra is equivalent to a strict one.", "Adv.", "Math., 229(3):2024–2041, 2012. arXiv:1005.1520.", "Michael Stay.", "Compact closed bicategories.", "Theory and Applications of Categories, 31(26):755–798, 2016. arXiv:1301.1053.", "Ross Street.", "The formal theory of monads.", "J.", "Pure Appl.", "Algebra, 2(2):149–168, 1972.", "Ross Street.", "Fibrations and Yoneda's lemma in a 2-category.", "In Category Seminar (Proc.", "Sem., Sydney, 1972/1973), pages 104–133.", "Lecture Notes in Math., Vol.", "420.", "Springer, Berlin, 1974.", "Ross Street.", "Absolute colimits in enriched categories.", "Cahiers Topologie Géom.", "Différentielle, 24(4):377–379, 1983.", "Mark Weber.", "Yoneda structures from 2-toposes.", "Appl.", "Categ.", "Structures, 15(3):259–323, 2007." ] ]	1606.05058
[ [ "Blue straggler formation at core collapse" ], [ "Abstract Among the most striking feature of blue straggler stars (BSS) is the presence of multiple sequences of BSSs in the colour-magnitude diagrams (CMDs) of several globular clusters.", "It is often envisaged that such a multiple BSS sequence would arise due a recent core collapse of the host cluster, triggering a number of stellar collisions and binary mass transfers simultaneously over a brief episode of time.", "Here we examine this scenario using direct N-body computations of moderately massive star clusters (of order 10^4 Msun ).", "As a preliminary attempt, these models are initiated with approx.", "8-10 Gyr old stellar population and King profiles of high concentrations, being \"tuned\" to undergo core collapse quickly.", "BSSs are indeed found to form in a \"burst\" at the onset of the core collapse and several of such BS-bursts occur during the post-core-collapse phase.", "In those models that include a few percent primordial binaries, both collisional and binary BSSs form after the onset of the (near) core-collapse.", "However, there is as such no clear discrimination between the two types of BSSs in the corresponding computed CMDs.", "We note that this may be due to the less number of BSSs formed in these less massive models than that in actual globular clusters." ], [ "Introduction", "Blue Straggler Stars (hereafter BSS) of a stellar ensemble are main-sequence (hereafter MS) stars that are located on the colour-magnitude-diagram (hereafter CMD) as an extension of the regular MS, beyond its turn-off point.", "The most natural explanation for this stellar subpopulation is that they are stars with rejuvenated hydrogen content so that they are still on the MS despite being more massive than the turnoff mass.", "There are two primary channels considered for this rejuvenation and mass gain, namely, (a) direct stellar collisions [7] and (b) mass transfer in a binary [8], [9].", "Being distinctly identifiable in the CMD and being more massive than the average stellar population, BSSs serve as excellent tracers of dynamical processes in open and globular clusters [12], [5]; especially their radial profile, as governed by their mass segregation, serves as a “dynamical clock” for globular clusters [5].", "The focus of this study is another remarkable feature of the BSSs, namely, the existence of a double sequence of BSSs in several globular clusters.", "Perhaps the best example of this is the BSSs' double sequence observed in the globular cluster M30 [4]; another vivid example is that of the globular NGC 362 [2].", "An intriguing explanation for this is the recent core collapse of the parent cluster that triggers both of the formation channels simultaneously so that the “red” sequence comprise binary (mass-transferring) BSSs and the “blue” sequence comprise the collision products [4].", "Indeed, the presence of a central cusp in the density profile of M30 does indicate its post core collapse state [4].", "More recent and detailed study of mass transfer in binary evolution models only strengthens this notion and constrain the types of binaries that make up the red sequence [16].", "Generally, the red BSSs are found to be more centrally concentrated than the blue ones.", "Motivated by the above possibility, the objective of the present work is to study the BSS formation following core collapse [15], [6] of a model star cluster.", "To allow the stars to collide “naturally” (i.e., without any constructed collision-triggering procedure), a direct N-body approach is followed.", "This is for the first time stellar collisions and BSS formation following core collapse (or, speaking more generally, during the central energy-generation phase of a cluster; [15]) is studied explicitly.", "Figure: The Lagrangian radii (top) and the time sequence of formationof MS collision products (bottom) for the computation withM cl (0)≈1.5×10 4 M ⊙ M_{cl}(0)\\approx 1.5\\times 10^4M_{\\odot }, W 0 =7.5W_0=7.5 and r h (0)=1r_h(0)=1 pcwith initially only single stars." ], [ "N-body calculations", "To obtain a prompt but clear core collapse, systems with a moderate number, $N$ , of stars and having high central concentration needs to be evolved — that way one “tunes” a cluster for core collapse.", "In this work, clusters initially with masses, $M_{cl}(0)$ , between $1-3\\times 10^4M_{\\odot }$ following $W_0=7.5$ [10] density profiles and of half-mass radius $r_h(0)\\approx 1$ pc are computed.", "Such concentrations are common in present-day Galactic and Local-Group globulars.", "To mimic an old stellar population like in globular clusters, low-mass stars between $0.1-1.0M_{\\odot }$ with a [11] mass function is assumed, which are pre-evolved for $\\approx 8.8$ Gyr (this corresponds to the “turn-off” age of $1.0M_{\\odot }$ stars at solar metallicity, when the giant branch on the CMD is just appearing).", "Such models are computed both with initially single stars and with $\\approx 5$ % binaries following a [3] period distribution.", "No initial mass segregation is assumed as consistent with what is typically observed in globulars.", "All the N-body runs are done using the NBODY7 code [1], [13]." ], [ "Results", "Fig.", "REF (top panel) shows the evolution of the Lagrangian radii of such a computed model with $M_{cl}(0)\\approx 1.5\\times 10^4M_{\\odot }$ without initial binaries.", "The occurrence of core collapse is indicated by the abrupt halting of the inner region of the cluster, followed by a slow expansion.", "The high central density at and after the collapse boosts collisions among stars which can be called a “burst”; Fig.", "REF (bottom) shows the timeline for collisions occurring among MS stars where BSSs are those whose final mass (Y-axis) exceeds the turn-off mass (upper horizontal line).", "Figure: CMDs corresponding to the calculation inFig. .", "The colour codingrepresents the different stellar-evolutionary stages where0=lower MS, 1=upper MS, 2=subgiant, 3=red giant, etc.", "A filled squareindicates a single star and a filled triangle implies a binary (combined magnitudesused).Fig.", "REF shows the corresponding computed CMDs at $t=0$ Myr and $t\\approx 165$ Myr evolutionary time, which are obtained from the simulation data using a slightly modified version of the GalevNB program [14].", "The BSS formation rate from stellar collisions is much higher after the core collapse (c.f. Fig.", "REF ) owing to the high central density and the rapid mass segregation (at least locally, close to the cluster's center) of the most massive (MS) stars that occurs nearly simultaneously to the collapse.", "The mass segregation is important here without which there would not have been the marked increase of BSS formation right after the collapse.", "Some BSSs as well get ejected from the cluster due to dynamical interactions; this is likely as they are the most massive members of the cluster and therefore are most likely to participate in close encounters.", "The above condition is to some extent fine tuned as the most massive member ($\\approx 1.0M_{\\odot }$ ) is chosen to be at the MS turn-off point by adjusting the stellar pre-evolution age.", "This has also made the cluster initially free of white dwarfs (WDs) which can be more than or similarly massive as the turn-off mass and might suppress the collision rate of MS stars close to the turn-off, that generates the BSSs.", "To examine the role of the WDs, a $M_{cl}(0)\\approx 3.0\\times 10^4M_{\\odot }$ cluster (no initial binaries) is computed with $0.08-8.0M_{\\odot }$ stars which are pre-evolved for $\\approx 10$ Gyr; that gives $\\approx 1M_{\\odot }$ turn-off mass and a large number of WDs.", "Although the collisions between between WDs and MS stars (resulting in red giants and AGBs) are frequent in this calculation, BSSs continue to form and its rate boosts after the core collapse.", "This is demonstrated in Fig.", "REF .", "Figure: Collision timeline for the calculation withM cl (0)≈3.0×10 4 M ⊙ M_{cl}(0)\\approx 3.0\\times 10^4M_{\\odot }, W 0 =7.5W_0=7.5 and r h (0)=1r_h(0)=1 pcwith initially only single stars.In one final demonstration, a $M_{cl}(0)\\approx 1.5\\times 10^4M_{\\odot }$ cluster with $\\approx 5$ % binaries (see Sec. )", "is computed.", "The corresponding CMD after $t\\approx 143$ Myr evolution is shown in Fig.", "REF .", "A hint of double BSS sequence is apparent although the number of BSSs in the red sequence is much less than that in the blue sequence.", "On closer inspection of the computation, it is found that both the binaries in the red sequence contain a BSS and a red giant in a close binary having residual eccentricity.", "This indicates a recent dynamical origin of these involving encounters with the giant's envelope.", "This is consistent with what has been envisaged so far.", "However, given that only two of such binaries plus a third MS-MS binary (close to the MS turn-off; see Fig.", "REF ) define the red BSS sequence in this computed model, this is only a marginal case of formation of a double BSS sequence.", "This double BSS sequence is found to last for $\\approx 80$ Myr, after which the binary BSSs evolve off.", "Computations of more massive clusters would potentially provide more concrete outcomes.", "The radial distribution of the BSSs in the model with primordial binaries is also inspected where the members of the red BSS sequence is found to be radially more concentrated then the blue ones, as observed.", "Again, given that there are only three red members, this comparison is only marginal.", "Interestingly, in the computations without any initial binaries, the BSSs are much more centrally concentrated than their counterparts in the model with binaries — the super-elastic encounters involving binaries [6] and the resulting kicks make the BSSs more radially spread out in the latter case.", "Figure: The CMD at t≈143t\\approx 143 Myr evolutionfor the calculation withM cl (0)≈3.0×10 4 M ⊙ M_{cl}(0)\\approx 3.0\\times 10^4M_{\\odot }, W 0 =7.5W_0=7.5 and r h (0)=1r_h(0)=1 pcwith initially ≈5\\approx 5% binaries.The meanings of the symbols and the colours are sameas in Fig.", "." ], [ "Conclusions and outlook", "From the above preliminary study suggest that: A “burst” of BSSs appears when an old (GC-like) cluster approaches core collapse.", "This is true despite the presence of primordial binaries (of a few percent) and a significant population of white dwarfs, as long as “some form of” core collapse happens.", "BSSs continue to form (and evolve/get ejected) during post core collapse phase.", "A “second” binary BSS sequence can appear in the presence of primordial binaries which are typically outcomes of recent dynamical interactions.", "Primordial binaries also seem to determine the radial distribution of BSS.", "Computing more massive models with primordial binaries is necessary to consolidate the above results and to better understand the properties of post core collapse BSSs, which will be the forthcoming step of this study." ] ]	1606.05213
[ [ "Universal behavior of the Shannon mutual information in non-integrable\n self-dual quantum chains" ], [ "Abstract An existing conjecture states that the Shannon mutual information contained in the ground state wavefunction of conformally invariant quantum chains, on periodic lattices, has a leading finite-size scaling behavior that, similarly as the von Neumann entanglement entropy, depends on the value of the central charge of the underlying conformal field theory describing the physical properties.", "This conjecture applies whenever the ground state wavefunction is expressed in some special basis (conformal basis).", "Its formulation comes mainly from numerical evidences on exactly integrable quantum chains.", "In this paper the above conjecture was tested for several general non-integrable quantum chains.", "We introduce new families of self-dual $Z(Q)$ symmetric quantum chains ($Q=2,3,\\ldots$).", "These quantum chains contain nearest neighbour as well next-nearest neighbour interactions (coupling constant $p$).", "In the cases $Q=2$ and $Q=3$ they are extensions of the standard quantum Ising and 3-state Potts chains, respectively.", "For $Q=4$ and $Q\\geq 5$ they are extensions of the Ashkin-Teller and $Z(Q)$ parafermionic quantum chains.", "Our studies indicate that these models are interesting on their own.", "They are critical, conformally invariant, and share the same universality class in a continuous critical line.", "Moreover, our numerical analysis for $Q=2-8$ indicate that the Shannon mutual information exhibits the conjectured behaviour irrespective if the conformally invariant quantum chain is exactly integrable or not.", "For completeness we also calculated, for these new families of quantum chains, the two existing generalizations of the Shannon mutual information, which are based on the R\\'enyi entropy and on the R\\'enyi divergence." ], [ "Introduction", "The connection between the quantum correlations and the entanglement properties of quantum many body systems provided us, in recent years, a powerful tool to detect [1], , , , and classify quantum phase transitions (see [6], , and references therein).", "Several measures of the entanglement were proposed along the years, like the von Neumann and Rényi entanglement entropies [8], [9], [10], the concurrence [11], the fidelity [12], , , etc.", "Among these measures the von Neumann and Rényi entanglement entropies are the most popular since in one dimension, where most of the critical chains are conformally invariant, they provide a way to calculate the central charge of the underlying conformal field theory (CFT), identifying the universality class of critical behavior.", "Although interesting proposals were presented [15], [16], [17], [18] it is quite difficult to measure these quantities in the laboratory, and the central charge of a critical chain has never been measured experimentally.", "An interesting measure that is also efficient in detecting quantum phase transitions is the Shannon mutual information.", "This quantity differently from the previous mentioned measures is based on the measurements of observables.", "It measures the shared information among parts of a quantum system.", "Consider a quantum chain with $L$ sites that we split into two subsystems $\\cal A$ and $\\cal B$ , formed by consecutive $\\ell $ and $L-\\ell $ sites, respectively.", "Suppose the quantum chain is in the quantum state given by the wavefunction $\|\\Psi _{{\\cal {A}}U{\\cal {B}}}\\rangle =\\sum _{n,m}c_{n,m}\|\\phi _{\\cal A}^n\\rangle \\otimes \|\\phi _{\\cal B}^m\\rangle $ , where $\\lbrace \|\\phi _{\\cal A}^n\\rangle \\rbrace $ and $\\lbrace \|\\phi _{\\cal B}^m\\rangle \\rbrace $ are the basis spanning the subsets $\\cal A$ and $\\cal B$ .", "The Shannon mutual information of the subsets $\\cal A$ and $\\cal B$ is defined as $I({\\cal A},{\\cal B}) = Sh({\\cal A}) + Sh({\\cal B}) - Sh({\\cal A} U {\\cal B}),$ where $Sh(\\chi ) = - \\sum _x p_x\\ln {p_x}$ is the standard Shannon entropy of the subsystem $\\chi $ with probability $p_x$ of being in the configuration $x$ .", "The probability of the configurations in the subsets $\\cal A$ and $\\cal B$ are given by the marginal probabilities $p_{\|\\phi _{\\scriptsize {\\cal A}}^n\\rangle }=\\sum _m \|c_{n,m}\|^2$ and $p_{\|\\phi _{\\scriptsize {\\cal B}}^m\\rangle }=\\sum _n \|c_{n,m}\|^2$ , respectively.", "It is important to notice that differently from the von Neumann entanglement entropy and the von Neumann mutual information, which are basis independent, the Shannon entropy $Sh$ and the Shannon mutual information $I(\\cal {A},\\cal {B})$ are basis dependent quantities.", "In [19] it was conjectured that, for periodic critical quantum chains in their ground state, the Shannon mutual information shows universal features provided the ground state is expressed in some special bases, called conformal basis.", "A given basis of the Hilbert space of the quantum chain is related to a certain boundary condition in the time direction of the underlying (1+1)-Euclidean CFT.", "In general these time-boundary conditions destroy the conformal invariance in the bulk.", "The conformal basis are related to the boundary conditions that do not destroy the conformal invariance, as happens in the case of Dirichlet and Neumann boundary conditions.", "It was conjectured [19] that whenever the ground state wavefunction is expressed in the conformal basis, the leading finite-size scaling behavior of the Shannon mutual information for large systems and subsystem sizes is given by $I(\\ell ,L-\\ell ) = \\frac{c}{4}\\ln \\left(\\frac{L}{\\pi }\\sin (\\frac{\\ell \\pi }{L})\\right) + \\gamma ,$ where $c$ is the central charge of the underlying CFT and $\\gamma $ is a non-universal constant.", "It is interesting to note that this leading behavior is the same as the Rényi entanglement entropy with Rényi index $n=2$ [20].", "The above conjecture was tested analytically and numerically for a large number of exactly integrable quantum chains [19], [21], [22], namely, a set of coupled harmonic oscillators (Klein Gordon theory), the XXZ quantum chain, the Ashkin-Teller, the spin-1 Fateev-Zamolodchikov, the $Q$ -state Potts models ($Q=2,3,4$ ) and the $Z(Q)$ parafermionic models ($Q=5-8$ ).", "Up to now, except for the chain of coupled harmonic oscillators, this conjecture was only tested numerically.", "Moreover all the tests for this conjecture were done for exactly integrable models.", "Since there is no general analytical results supporting this conjecture it is import to check if the existing numerical agreement is not just a consequence of the exact integrability of all the quantum chains tested so far.", "All the agreements obtained are reasonable taking into account the lattice sizes of the considered quantum chains.", "However there exist a controversy in the case of the Ising quantum chain.", "A numerical analysis due to Stéphan [23] on this quantum chain indicates that the prefactor in (REF ), instead of being the central charge ($c=0.5$ in this case) is a close number $b\\approx 0.4801$ .", "In the conclusions of this paper we present additional discussions about this point.", "In this paper we are going to check the universality feature of the conjecture (REF ) by considering critical chains belonging to several universality classes of critical behavior but being not exactly integrable.", "The ground state eigenfunction can only be calculated numerically for quantum chains of relatively small lattice sizes.", "It will be then interesting to consider non-integrable quantum chains whose critical points are exactly known.", "For this sake we introduce in this paper a set of generalized self-dual non-integrable quantum chains whose exact critical points are given by their self-dual points.", "Moreover, each of these quantum chains seems to share the same symmetries and long-distance physics of an exactly integrable conformally invariant chain whose central charge $c$ is exactly known.", "The validity of the conjecture (REF ) will imply that the Shannon mutual information of these models share the same asymptotic behavior.", "We should also mention some additional studies of the Shannon and Rényi entropies and mutual information in quantum systems [24], , , , , , , [30], and also in two-dimensional spin systems [31], , , , .", "The paper is organized as follows.", "In the next section we introduce the several new quantum chains, and show their self-dual properties.", "In Sec.", "III we present our results for the models in the universality class of the Ising model and 3-state Potts model.", "In Sec.", "IV the results for the models in the universality class of the $Z(Q)$ -parafermionic models, with $Q=4,5,6,7$ and 8 are presented.", "We also consider in this section a numerical analysis for a generalization of the $Z(Q)$ clock models with $Q=5,6,7$ and 8.", "In Sec.", "V we calculate for these new quantum chains the two existing extensions of the Shannon mutual information: the Rényi mutual information and the less known generalized mutual information [36], [22].", "Finally in Sec.", "VI we present our conclusions." ], [ " The $Z(Q)$ generalized self-sual quantum chains", "We introduce initially a special generalization of the nearest-neighbor Ising quantum chain that also contains next-nearest neighbor interactions.", "The Hamiltonian is given by: $&&H^{(2)}(\\lambda ,p) = -\\sum _i\\left[\\sigma _i^z\\sigma _{i+1}^z + \\lambda \\sigma _i^x \\right.\\nonumber \\\\&& \\left.", "-p(\\sigma _i^z\\sigma _{i+2}^z + \\lambda \\sigma _i^x\\sigma _{i+1}^x)\\right],$ where $\\sigma _i^z$ and $\\sigma _i^x$ are spin-$\\frac{1}{2}$ Pauli matrices attached to the lattice sites ($i=1,2,\\ldots $ ), and $\\lambda $ and $p$ are the coupling constants.", "At $p=0$ the Hamiltonian (REF ) reduces to the standard nearest-neighbor quantum Ising chain, which is exactly integrable and critical at $\\lambda =1$ .", "In order to show that $H^{(2)}( \\lambda ,p)$ is self-dual, for any value of $p$ , let us define the new operators $\\rho _{2i}^{(e)} = \\sigma _i^z\\sigma _{i+1}^z \\mbox{ and } \\rho _{2i-1}^{(o)} =\\sigma _i^x, \\quad i=1,2,\\ldots ,$ that obey the following commuting and anti-commuting relations $&&\\left({\\rho _i^{(e)}}\\right)^2=\\left({\\rho _i^{(o)}}\\right)^2=1, \\quad [\\rho _i^{(o)},\\rho _j^{(o)}]=[\\rho _i^{(e)},\\rho _j^{(e)}]=0,\\nonumber \\\\&&[\\rho _i^{(o)},\\rho _j^{(e)}]=0, \\mbox{ unless } \|i-j\|=1, \\nonumber \\\\&&\\lbrace \\rho _i^{(e)},\\rho _j^{(o)}\\rbrace =0, \\mbox{ if } \|i-j\|=1.$ In terms of these new operators the Hamiltonian (REF ) is given by $&&H^{(2)}(\\lambda ,p) = -\\sum _i\\left[\\rho _{2i}^{(e)}+\\lambda \\rho _{2i-1}^{(o)} \\nonumber \\right.\\\\+&&\\left.", "p(\\rho _{2i}^{(e)}\\rho _{2i+2}^{(e)}+ \\lambda \\rho _{2i-1}^{(o)}\\rho _{2i+1}^{(o)})\\right].$ We now make a transformation by defining the new operators: $\\tilde{\\rho }_{2i}^{(e)}=\\rho _{2i+1}^{(o)}, \\quad \\tilde{\\rho }_{2i-1}^{(o)}=\\rho _{2i}^{(e)}.$ It is simple to see that these new operators obey the same commutation relations as the old ones, given in (REF ).", "In terms of these new operators the Hamiltonian (REF ) is now given by $&&H^{(2)}(\\lambda ,p) = -\\lambda \\sum _i[\\tilde{\\rho }_{2i}^{(e)} + \\frac{1}{\\lambda } \\tilde{\\rho }_{2i-1}^{(o)} \\nonumber \\\\&&+p (\\tilde{\\rho }_{2i}^{(e)}\\tilde{\\rho }_{2i+2}^{(e)} + \\frac{1}{\\lambda } \\tilde{\\rho }_{2i-1}^{(o)}\\tilde{\\rho }_{2i+1}^{(o)})].$ Consequently, apart from a boundary term This transformation for finite lattices will produce constraints among the operators $\\lbrace \\rho _i^{(o)},\\rho _i^{(e)}\\rbrace $ and the exact relation for finite chains only relate sectors of the associated Hilbert space.", "[38] that could be neglected as the lattice size increases, the model is self-dual: $H^{(2)}(\\lambda ,p) = \\lambda H^{(2)}(\\frac{1}{\\lambda },p).$ Implying that the low-lying eigenlevels in the eigenspectrum of both sides of (REF ) become identical as the lattice size increases.", "Since we have no reason to expect more than a single $Z(2)$ critical point for a fixed value of $p$ , this model should be critical at $\\lambda =1$ and at least for $p\\le p_c$ (with $p_c$ finite) the model should share the same universality class as the standard quantum Ising chain $H^{(2)}(1,0)$ .", "Actually for $p \\rightarrow \\infty $ the model is $Z(2)\\otimes Z(2)$ symmetric due to the commutations of $H^{(2)}(\\lambda ,p\\rightarrow \\infty )$ with the nonlocal $Z(2)$ operators ${\\cal {P}}^{(e)} = \\prod _{i} \\sigma _{2i}^x$ and ${\\cal {P}}^{(o)} = \\prod _{i} \\sigma _{2i-1}^x$ , and therefore is not in the Ising universality class.", "Similarly as we did for the Ising quantum chain we now introduce the self-dual generalized next-nearest neighbor $Z(Q)$ models ($Q=2,3,\\ldots $ ).", "They describe the dynamics of the $Q\\times Q$ matrices $\\lbrace S_i\\rbrace $ , $\\lbrace R_i\\rbrace $ , attached on the lattice sites $i=1,2,\\ldots $ , and obey the algebraic relations $&&S_i^Q=R_i^Q=1,\\quad [S_i,S_j]=[R_i,R_j]=0, \\nonumber \\\\&&[S_i,R_j]=0 \\mbox{ if } i\\ne j\\mbox{ and } S_iR_i=e^{i\\frac{2\\pi }{Q}}R_iS_i.$ The Hamiltonian we introduce is given by $&&H^{(Q)}(\\lambda ,\\lbrace \\alpha \\rbrace ) = -\\sum _i \\left[\\sum _{n=1}^Q \\alpha _n(S_i^nS_{i+1}^{Q-n} + \\lambda R_i^n) \\right.", "\\nonumber \\\\&& \\left.", "+ p \\sum _{n=1}^Q \\alpha _n(S_i^nS_{i+2}^{Q-n} + \\lambda R_i^nR_{i+1}^n)\\right],$ where $\\lambda $ and $\\lbrace \\alpha _n\\rbrace $ ($n=1,\\ldots ,Q$ ) are coupling constants.", "We chose real coupling constants and $\\alpha _n=\\alpha _{Q-n}$ to ensure the hermiticity of the Hamiltonian.", "This Hamiltonian reduces to (REF ) for $Q=2$ .", "We now consider the $Z(Q)$ operators: $\\rho _{2i}^{(e)}=S_iS_{i+1}^{Q-1} \\mbox{ and } \\rho _{2i-1}^{(o)}=R_i, \\quad i=1,2,\\ldots ,$ that obey the following algebraic relations $&&\\left(\\rho _i^{(e)}\\right)^Q=\\left(\\rho _i^{(o)}\\right)^Q=1,[\\rho _i^{(e)},\\rho _j^{(e)}]=[\\rho _i^{(o)},\\rho _j^{(o)}]=0, \\nonumber \\\\&&[\\rho _i^{(e)},\\rho _j^{(o)}]=0 \\mbox{ unless } \|i-j\|=1, \\nonumber \\\\&&\\rho _i^{(e)}\\rho _{i\\pm 1}^{(o)}=e^{\\mp i\\frac{2\\pi }{Q}}\\rho _{i\\pm 1}^{(o)}\\rho _{i}^{(e)}.$ In terms of these operators we have $&&H^{Q)} (\\lambda ,\\lbrace \\alpha \\rbrace ) = -\\sum _i\\left\\lbrace \\sum _{n=1}^{Q-1}\\alpha _n \\left[(\\rho _{2i}^{(e)})^n + \\lambda (\\rho _{2i-1}^{(o)})^n \\right] \\right.", "\\nonumber \\\\&& \\left.", "+p\\sum _{n=1}^{Q-1} \\alpha _n \\left[(\\rho _{2i}^{(e)} \\rho _{2i+2}^{(e)})^n+\\lambda ( \\rho _{2i-1}^{(o)}\\rho _{2i+1}^{(o)} )^n\\right] \\right\\rbrace .$ We now perform the same canonical transformation $\\rho \\rightarrow \\tilde{\\rho }$ , given by (REF ).", "It is simple to verify that the transformation is canonical since the commutation's relations of the new operators are the same as the old ones.", "The Hamiltonian is now given by: $&& H^{(Q)}(\\lambda ,\\lbrace \\alpha \\rbrace ) = - \\lambda \\left\\lbrace \\sum _{n=1}^{Q-1}\\alpha _n \\left[(\\tilde{\\rho }_{2i}^{(e)})^n+ \\frac{1}{\\lambda }( \\tilde{\\rho }_{2i-1}^{(o)} )^n \\right] \\right.", "\\nonumber \\\\&& \\left.", "+p\\sum _{n=1}^{Q-1}\\alpha _n \\left[(\\tilde{\\rho }_{2i}^{(e)}\\tilde{\\rho }_{2i+2}^{(e)})^n +\\frac{1}{\\lambda } ( \\tilde{\\rho }_{2i-1}^{(o)}\\tilde{\\rho }_{2i+1}^{(o)} )^n \\right] \\right\\rbrace .$ Comparing (REF ) and (REF ), we obtain, apart from a boundary term [18] $H^{(Q)} (\\lambda ,\\lbrace \\alpha \\rbrace ) = \\lambda H^{(Q)} (\\frac{1}{\\lambda },\\lbrace \\alpha \\rbrace ).$ The particular choice $\\alpha _n = \\frac{1}{\\sin (\\frac{\\pi n}{Q})}$ , $n=1,2,\\ldots ,Q-1$ give us an interesting family of quantum chains that we are going to study in the next sections.", "At their self-dual point ($\\lambda =1$ ) these Hamiltonians are given by: $&&H^{(Q)}(p) = -\\sum _i\\left\\lbrace \\sum _{n=1}^{Q-1} \\frac{1}{\\sin (\\frac{\\pi n}{Q})}\\left[S_i^nS_{i+1}^{Q-n} + R_i^n \\right.", "\\right.", "\\nonumber \\\\&& \\left.", "\\left.+ p(S_i^nS_{i+2}^{Q-n} + R_i^n R_{i+1}^n)\\right] \\right\\rbrace .$ These Hamiltonians at $p=0$ are critical, conformal invariant and exactly integrable.", "They correspond for $Q=2,3$ to the 2-state and 3-state Potts models, for $Q=4$ it is the Ashkin-Teller model with a special value of its anisotropy, and for $Q>4$ they correspond to the $Z(Q)$ parafermionic models [39], , .", "For $p\\ne 0$ the models lose their exact integrability but we do expect that, at least for small values of the parameter $p$ , they stay critical and in the same universality class of the related $p=0$ exactly integrable quantum chain.", "For large values of $p$ this may not be true since, as happened in the Ising case, for $p \\rightarrow \\infty $ the symmetry increases from a single $Z(Q)$ to a $Z(Q)\\times Z(Q)$ ." ], [ "Results for the extended Ising and 3-state Potts quantum chains", "We present in this section our numerical results for the generalized self-dual Ising and 3-state Potts quantum chains whose Hamiltonians $H^{(Q)}(p)$ are given by (REF ) with the values $Q=2$ and $Q=3$ , respectively.", "At $p=0$ these models are exactly integrable and conformally invariant, being ruled by a CFT with central charge $c=1/2$ and $c=4/5$ , respectively.", "Our aim is to compute the Shannon mutual information for the values of the parameter ($p\\ne 0$ ) where the models are still critical but not exactly integrable.", "Since we are testing a conjecture we should initially confirm the expectation that the models, for small values of the parameter $p$ are still critical and in the same universality class as the $p=0$ exactly integrable quantum chain.", "A first test of the critical universality for the quantum chains can be done by comparing their central charge $c$ calculated directly from the finite-size behavior of the ground state energy and low-lying energy gaps.", "The ground state energy $E_0(L)$ of a conformally invariant quantum chain with periodic boundary should have the asymptotic behavior [42], : $\\frac{E_0}{L} = e_{\\infty } - v_s \\frac{\\pi c}{6L^2} + o(L^{-2}),$ where $e_{\\infty }$ is the energy per site in the bulk limit and $v_s$ is the sound velocity.", "The sound velocity can be extracted from the leading finite-size behavior of the first energy gap related to a given primary operator of the underlying CFT [44], .", "For example the lowest energies $E_1(p)$ in the eigensector with $Z(Q)$ charge $q=1$ and momentum $P=0,\\frac{2\\pi }{L},\\frac{4\\pi }{L},\\ldots $ are associated to the $Z(Q)$ -magnetic operators of these models.", "We have then the estimate $v_s(L)$ for the sound velocity [46] $v_s(L)= \\frac{L[E_1(\\frac{2\\pi }{L}) - E_1(0)]}{2\\pi } + o(L^{-1}), \\nonumber $ that together with (REF ) give us an estimate for the central charge of the quantum chain: $c_{\\mbox{\\scriptsize {est}}}(L) =-\\frac{\\frac{E_0(L)}{L}-\\frac{E_0(L-1)}{L-1}}{\\frac{1}{L^2}-\\frac{1}{(L-1)^2}} \\frac{12}{L(E_1(\\frac{2\\pi }{L})-E_1(0))} + o(L^{-1}).$ In Fig.", "1 and 2 we illustrate our results for the estimate $c_{\\scriptsize \\mbox{est}}(L)$ in the extended self-dual Ising and 3-state Potts models, respectively.", "We consider the models with the parameter $p=0,0.5,1$ and 1.5, and lattice sizes up to $L_{\\scriptsize \\mbox{max}}=30$ for the Ising case and $L_{\\scriptsize \\mbox{max}}=19$ for the 3-state Potts case.", "We also show in the figures the estimated results $c_{\\scriptsize \\mbox{est}}(L\\rightarrow \\infty )$ for the central charge $c$ .", "They were obtained by considering a simple quadratic fit of $c_{\\scriptsize \\mbox{est}}(L)$ for $30\\le L\\le 20$ in the Ising case and $19\\le L\\le 11$ in the 3-state Potts case.", "The numerical results in these figures indicate that for the parameters $p\\lesssim 1.5$ the extended models stay in the same universality class of the related $p=0$ exactly integrable model, i. e., $c=1/2$ ad $c=8/10$ for the Ising and 3-sate Potts models, respectively.", "Figure: The estimate c est (L)c_{\\scriptsize \\mbox{est}}(L) given by () as a function of1/L1/L for the extended self-dual Ising model given by the Hamiltonian (), and for the values of the parameter p=0,0.5,1p=0,0.5,1 and 1.5.", "The estimated valuesc est (L→∞)=cc_{\\scriptsize \\mbox{est}}(L\\rightarrow \\infty )=c, shown in the figure, were obtainedfrom a quadratic fit by considering the lattice sizes 20≤L≤3020\\le L \\le 30.Figure: The estimate c est (L)c_{\\scriptsize \\mbox{est}}(L) given by (), as a function of1/L1/L, for the extended 3-state Potts quantum chain by the Hamiltonian() and for the values of the parameter p=0,0.5,1p=0,0.5,1 and 1.5.", "The estimated valuesc est (L→∞)=cc_{\\scriptsize \\mbox{est}}(L\\rightarrow \\infty )=c, shown in the figure, were obtainedfrom a quadratic fit by considering the lattice sizes 14≤L≤1914\\le L \\le 19.A second test can be done by calculating the von Neumann entanglement entropy $S_{vN}(\\ell ,L)$ of subsystems with sizes $\\ell $ and $(L-\\ell )$ in the quantum chains.", "Its finite-size scaling behavior, for a periodic chain, is giving by [47], [48], [49] $S_{vN}(\\ell ,L-\\ell ) = \\frac{c}{3}\\ln (\\frac{L}{\\pi }\\sin (\\frac{\\ell \\pi }{L})) + k,$ where $k$ is a constant.", "In order to calculate $S_{vN}(\\ell ,L)$ , from a given ground state wave function, we should fully diagonalize the reduced density matrix of the subsystems (dimension $Q^{\\ell } \\times Q^{\\ell }$ ).", "This brings an extra numerical limitation since we can only handle the complete diagonalization of matrices with dimensions smaller than $\\sim $ 6000.", "We are then restricted for the $Q=2$ ($Q=3$ ) model with sublattices sizes $\\ell \\le 12$ ($\\ell \\le 7$ ).", "In Fig.", "3 (Fig.", "4) we show, for several values of $p$ , $S_{vN}(\\ell ,L)$ as a function of $\\sin (\\frac{L}{\\pi }\\sin (\\frac{\\pi }{L}))/3$ for the $Q=2$ ($Q=3$ ) extended quantum chains with $L=24$ ($L=14$ ) sites.", "It is also shown in these figures the estimated values of the central charge obtained from a linear fit.", "These results clearly indicate that these quantum chains are indeed critical, and share the same universality class of critical behavior as the exactly integrable quantum chain $p=0$ , whose central charge is $c=0.5$ .", "Figure: The von Neumann entropy for the extended self-dual Ising model () withL=24L=24 sites and the parameter values p=0,0.5,1p=0,0.5,1 and 1.5.", "The estimated values for the centralcharge are shown.", "They were obtained from a linear fit (see ()), consideringthesublattice sizes ℓ=5-12\\ell =5-12.Figure: The von Neumann entropy for the extended 3-state Potts model () withL=14L=14 sites and the values of the parameter p=0,0.5,1p=0,0.5,1 and 1.5.", "The estimated values for the centralcharge are shown.", "They were obtained from a linear fit (see ()), considering thesublattice sizes ℓ=4-7\\ell =4-7.Once we have convinced ourselves about the universal behavior of these non-integrable quantum chains for $0\\le p \\lesssim 1.5$ , we can now test the universal behavior (REF ) claimed for the Shannon mutual information $I(\\ell ,L-\\ell )$ of periodic quantum chains in their ground states.", "The Shannon mutual information depends on the particular basis we chose to express the ground state weave function.", "The previous results [19], [21], based on exactly integrable quantum chains, indicate that two good basis, where the universal behavior are shown, are the basis where either the \"kinetic interactions\" or the \"static interactions\" are diagonal.", "In the set of models we are testing these basis are the ones where the operators $\\lbrace S_i\\rbrace $ or $\\lbrace R_i\\rbrace $ are diagonal.", "In Fig.", "5 and Fig.", "6 the Shannon mutual information are shown for the extended Ising chain (REF ) with $L=30$ sites and for values of the parameter $p=0,0.5,1$ and $1.5$ .", "The results of Fig 5 (Fig.", "6) are obtained from the ground state wavefunction given in the $\\lbrace \\sigma ^z\\rbrace $ -basis ($\\lbrace \\sigma ^x\\rbrace $ -basis).", "We clearly see in these figures a linear behavior indicating $\\ln (L\\sin (\\pi \\ell /L))$ as the finite-size scaling function.", "The estimated values of the central charge $c =0.48-0.50$ , are also close to the expected value $c=1/2$ .", "These estimates were obtained from a linear fit by considering all the sublattice sizes.", "Figure: The Shannon mutual information I(ℓ,L-ℓ)I(\\ell ,L-\\ell ), as a function ofln[Lsin(πℓ/L)/π]/4\\ln [L\\sin (\\pi \\ell /L)/\\pi ]/4, for the extended selfdual Ising quantum chain(), with the values of the parameter p=0,0.5,1p=0, 0.5,1 and 1.5.The results are obtained for the ground state wavefunction of the L=30L=30 sitesquantum chain expressed in the basis where {σ i z }\\lbrace \\sigma _i^z\\rbrace are diagonal.The estimatedresults, based on the conjecture () are also shown.", "They were obtainedfrom a linear fit by considering all the sublattices sizes.Figure: Same as in Fig.", "5 but with the ground state wavefunction expressedin the basis where {σ i x }\\lbrace \\sigma _i^x\\rbrace are diagonal.In Fig.", "7 and Fig.", "8 we show the Shannon mutual information for the extended $Z(3)$ models with the values of the parameter $p=0,0.5,1$ and $1.5$ .", "In Fig.", "7 (Fig.", "8) the quantum chain has $L=18$ ($L=19$ ) sites and is in the basis where the matrices $\\lbrace S_i\\rbrace $ ($\\lbrace R_i\\rbrace $ ) are diagonal, respectively.", "The linear fit obtained by using all the sublattice sizes predicts the value for the central charge $c\\approx 0.77-0.79$ .", "These values are close to the predicted value $c=8/10$ , indicating the validity of the conjecture (REF ) even for non-integrable quantum chains.", "It is interesting to notice that differently from the calculation of $S_{vN}(\\ell ,L)$ , it is not necessary to full diagonalize reduced matrices and we could calculate $I(\\ell ,L-\\ell )$ for larger lattice sizes, namely $L=30$ and $L=19$ for the extended Ising and 3-state Potts chains, respectively.", "Figure: The Shannon mutual information I(ℓ,L-ℓ)I(\\ell ,L-\\ell ), as a function ofln[Lsin(πℓ/L)/π]/4\\ln [L\\sin (\\pi \\ell /L)/\\pi ]/4, for the extended Q=3Q=3 selfdual Potts quantum chain(), with the values of the parameter p=0,0.5,1p=0, 0.5,1 and 1.5.The results are obtained for the ground state wavefunction of the L=18L=18 sitesquantum chain, expressed in the basis where {S i z }\\lbrace S_i^z\\rbrace are diagonal.The estimatedresults, based on the conjecture () are also shown.", "They were obtainedfrom a linear fit by considering all the sublattice sizes.Figure: Same as Fig.", "7 but for lattice size L=19L=19 and the results areobtained from the ground state wavefunction expressed in the {R i }\\lbrace R_i\\rbrace basis." ], [ "Results for the extended Z(Q)-parafermionic quantum chains", "We consider in this section the numerical tests of the conjecture (REF ) for the extended non-integrable $Z(Q)$ -parafermionic models (REF ).", "The cases where the parameter $p=0$ reduces to the known exactly integrable $Z(Q)$ -parafermionic quantum chains [39], , , which are critical and conformally invariant with conformal central charges: $c= \\frac{2(Q-1)}{Q+2}, \\quad Q=2,3,\\ldots \\, .$ The cases $Q=2$ and $Q=3$ are the Ising and 3-state Potts models considered in the last section.", "The quantum chain with $Q=4$ corresponds to a particular anisotropy of the $c=1$ critical line of the quantum Ashkin-Teller chain.", "The cases $Q>4$ are the Z(Q)-parafermionic quantum chains with central charge $c>1$ .", "Actually these last models are multicritical points and are expected to be endpoints [41], [50] of critical lines belonging to a massless phases with central charge $c=1$ and belonging to the Berezinskii-Kosterlitz-Thouless universality class [50], [51].", "The Shannon mutual information for the extended $Q=4$ quantum chain with the values of $p=0,0.5,1$ and $1.5$ are shown in Fig. 9.", "The calculations were done by expressing the ground state wavefunction either in the $S$ -basis ($L=14$ ) or in the $R$ -basis ($L=13$ ).", "The linear fit, using all the sublattice sizes, give the estimated values of the central charge shown in the figure $c\\approx 0.97- 1.03$ , which within the numerical accuracy corroborates the conjecture (REF ).", "Figure: The Shannon mutual information I(ℓ,L-ℓ)I(\\ell ,L-\\ell ), as a function ofln[Lsin(πℓ/L)/π]/4\\ln [L\\sin (\\pi \\ell /L)/\\pi ]/4 for the extended Q=4Q=4 self-dual quantumchain (), with the values of the parameters p=0,0.5,1p=0,0.5,1 and 1.5.", "Theresults were obtained for the lattice size L=15L=15 and L=14L=14, when theground state wavefunction spanned in the basis where {S i }\\lbrace S_i\\rbrace and {R i }\\lbrace R_i\\rbrace are diagonal,respectively.", "The estimated values shown in the figure were obtained from alinear fit by considering all the sublattice sizes.Let us now consider the extended models with $Q>4$ .", "Since the $p=0$ models are multicritical it is not clear if the non-integrable quantum chains, although critical, will stay in the same universality class as the integrable model $p=0$ .", "Surprisingly this seems to be the case.", "In Figs.", "10, 11, 12 and 13 we show for some values of $p$ the Shannon mutual information for the quantum chains with $Q=5,6,7$ and 8, respectively.", "The calculation were done for the ground state wavefunction expressed in the basis where either $\\lbrace S_i\\rbrace $ or $\\lbrace R_i\\rbrace $ are diagonal.", "The lattice sizes used are given in the figure captions.", "The estimated values for the central charge are givem in the figure and were obtained from a linear fit, where all the sublattice sizes are considered.", "They are close to the predicted values: $c=8/7=1.14285...$ ($Q=5$ ), $c=5/4=1.25$ ($Q=6$ ), $c=4/3=1.333...$ ($Q=7$ ) and $c=7/5=1.4$ ($Q=8$ ).", "Taking into account the lattice sizes we could calculate, these results indicate that the models are still in the same universality class of the multicritical point ($p=0$ ), at least for the values of parameters $0<p \\lesssim 1$ .", "These results tests the universal character of the conjecture (REF ), corroborating its validity for non-integrable critical quantum chains.", "Figure: The Shannon mutual information I(ℓ,L-ℓ)I(\\ell ,L-\\ell ), as a function ofln[Lsin(πℓ/L)/π]/4\\ln [L\\sin (\\pi \\ell /L)/\\pi ]/4 for the extended Q=5Q=5 self-dual quantumchains (), with the values of the parameters p=0,0.5p=0,0.5 and 1.", "Theresults were obtained for the lattice sizes L=12L=12 and L=13L=13, when theground state wavefunction are in the basis where {S i }\\lbrace S_i\\rbrace and {R i }\\lbrace R_i\\rbrace are diagonal,respectively.", "The estimated values shown in the figure were obtained from alinear fit by considering all the sublattice sizes.Figure: Same as Fig.", "10 for the extended Z(6)Z(6) sef dual quantumchain ().", "The lattice sizes are L=12L=12 and L=13L=13 for the basis where{S i }\\lbrace S_i\\rbrace and {R i }\\lbrace R_i\\rbrace are diagonal, respectively.Figure: Same as Fig.", "10 for the extended Z(7)Z(7) self-dual quantumchain ().", "The lattice sizes are L=11L=11 and L=12L=12 for the basis where{S i }\\lbrace S_i\\rbrace and {R i }\\lbrace R_i\\rbrace are diagonal, respectively.Figure: Same as Fig.", "10 for the extended Q=8Q=8 self-dual quantumchain ().", "The lattice sizes are L=10L=10 and L=11L=11 for the basis where{S i }\\lbrace S_i\\rbrace and {R i }\\lbrace R_i\\rbrace are diagonal, respectively.Before closing this section let us do an additional test for the conjecture (REF ).", "For $Q\\ge 5$ the $Z(Q)$ family of clock quantum chains (which is related to the time-continuum limit of the 2-d classical clock models [52]) is known to have, besides a disordered and ordered phases, an intermediate massless phase belonging to the Berezinskii-Kosterlitz Thouless universality and are expected to be ruled by a CFT with central charge $c=1$ [50], [51].", "These models, although not exactly integrable, are self-dual.", "Their self-dual points belong to the intermediate $c=1$ CFT.", "Exploring the general results of Sec.", "2, similarly as we did for the Z(Q) parafermionic models, we can extend the standard clock models by choosing in (REF ) $\\alpha _n=\\delta _{n,1} + \\delta _{n,Q-1}$ for ($n=1,\\ldots ,Q-1$ ).", "At its self-dual point the extended clock models are given by $&&H_{\\scriptsize \\mbox{clock}}(p) = -\\sum _i\\left[S_iS_{i+1}^+ + S_i^+S_{i+1} + R_i + R_i^+ + \\right.", "\\nonumber \\\\&& \\left.", "p(S_iS_{i+2}^+ + S_i^+S_{i+2} + R_iR_{i+1}+R_i^+R_{i+1}^+)\\right],$ where, as before, $S_i$ and $R_i$ are the $Z(Q)$ matrices with algebraic relations given by (REF ).", "At $p=0$ these Hamiltonians reduce to the standard $Z(Q)$ clock quantum chains.", "Our numerical results indicate that for arbitrary values of $0\\le p \\le 1$ the models share the same $c=1$ CFT.", "In Fig.", "14 we show our tests for the Shannon mutual information $I(\\ell ,L-\\ell )$ for the $Z(Q)$ clock model with $Q=5,6,7$ and 8.", "We only present the results in the case where the ground state wavefunction is expressed in the $\\lbrace R_i\\rbrace $ basis.", "In this figure, for each value of $Q$ the data are for the values of the parameter $p=0,0.5$ and 1.", "We clearly see the linear dependence with $\\ln [L\\sin (\\pi \\ell /L)]/4$ .", "The linear fit, by considering all the values of $p$ , and sublattice sizes for a given $Z(Q)$ model, give us estimates of the central charge in the range $c=1.03-1.04$ , that are close to the expected value $c=1$ , indicating the validity of the conjecture (REF ).", "Figure: The Shannon mutual information for the extended Z(Q)Z(Q) clock models defined in (), for thevalues of Q=5,6,7Q=5,6,7 and 8, and lattice sizes L=13,12,11L=13,12,11 and 10, respectively.", "For eachZ(Q)Z(Q) model the results are for the values of the parameter p=0,0.5p=0,0.5 and 1.", "The calculationswere done for the ground state spanned in the {R i }\\lbrace R_i\\rbrace basis.", "The lines are thelinear fit considering all the points for a given Z(Q)Z(Q) model." ], [ " Generalized mutual informations", "A crucial step in deriving most of the analytical results (e.g.", "[49], [53], ) for the von Neumann entanglement entropy come from two facts.", "The Shannon entropy is obtained from the $n\\rightarrow 1$ limit of the $n$ -Rényi entanglement entropy, and at this limit the replica trick, used for the conformal transformations, is regular.", "There exists two generalizations of the Shannon mutual information considered in the early sections.", "These extensions are based either on the Rényi entropy or on the Rényi divergence [36].", "Previous numerical calculation, on exactly integrable quantum chains [21], [22] show numerical evidence that these quantities, when computed on the ground state wave functions of critical chains expressed in special basis (conformal basis), exhibit some universal features.", "It is then interesting to compute these generalized mutual information for the extended $Z(Q)$ models introduced in this paper and test the universal behavior for those critical non-integrable quantum chains.", "In order to define the generalized mutual informations let us split, as before, the quantum chain $\\cal C$ with $L$ sites in the subsystems $\\cal A$ and $\\cal B$ formed by $\\ell $ and ($L-\\ell $ ) consecutive sites, respectively.", "We now consider the quantum chain in the normalized ground state, with wavefunction $\|\\Psi _{\\cal C}\\rangle = \\sum _{\\lbrace I_{\\cal A},I_{\\cal B}\\rbrace }a_{I_{\\cal A},I_{\\cal B}} \|I_{\\cal A}\\rangle \\otimes \|I_{\\cal B}\\rangle $ , where $\|I_{\\cal A}\\rangle =\|i_1,i_2,\\ldots ,i_{\\ell }\\rangle $ and $\|I_{\\cal B}\\rangle =\|i_{\\ell +1},\\ldots ,i_{L}\\rangle $ are the local basis for the subsystems $\\cal A$ and $\\cal B$ .", "The Rényi entropy for the entire system $\\chi ={\\cal C}$ and the subsystems $\\chi ={\\cal A}$ or $\\chi ={\\cal B}$ are given by: $Sh_n(\\chi ) = \\frac{1}{1-n} \\sum _{\\lbrace I_{\\chi }\\rbrace } \\ln P_{I_{\\chi }}^n, \\quad \\chi ={\\cal A}, {\\cal B}, {\\cal C},$ where for the entire system $P_{I_{\\cal C}} = \|a_{I_{\\cal A},I_{\\cal B}}\|^2$ and for the subsystems $\\cal A$ and $\\cal B$ , $P_{I_{\\cal A}} = \\sum _{I_{\\cal B}} \|a_{I_{\\cal A},I_{\\cal B}}\|^2$ and $P_{I_{\\cal B}} = \\sum _{I_{\\cal A}} \|a_{I_{\\cal A},I_{\\cal B}}\|^2$ , respectively.", "The Rényi mutual information is the shared information among the subsystems measured in terms of the Rényi entropy (REF ), i. e., $I_n(\\ell ,L-\\ell ) = Sh_n(\\ell ) + Sh_n(L-\\ell ) -Sh_n(L),$ where instead of denoting the subsystem, we denote their lattice sizes.", "At the limiting case $n\\rightarrow 1$ the Rényi entropy and the Rényi mutual information reduces to the Shannon entropy and the Shannon mutual information, respectively.", "Previous calculations of $I_n(\\ell ,L-\\ell )$ for the ground state wave functions of several exactly integrable chains show the same finite-size scaling function for arbitrary values of $n$ : $I_n(\\ell ,L-\\ell )= c_n \\ln (\\frac{L}{\\pi }\\sin (\\frac{\\ell \\pi }{L})) + k,$ where $k$ is a $o(1)$ constant.", "As happens with the Shannon mutual information $I(\\ell ,L-\\ell )$ this behavior is not general, it happens only when the ground state wavefunction is expressed on the special basis (conformal basis).", "The coefficients $c_n$ besides its $n$ dependence also depends on the conformal basis considered.", "Under certain plausible assumptions the large-$n$ behavior of $c_n$ is known analytically [55].", "However in the general case the limiting case $n \\rightarrow 1$ is singular, preventing a general analytical calculation of the Shannon mutual information $I_1(\\ell ,L-\\ell )=I(\\ell ,L-\\ell )$ .", "Our numerical analysis for the extended self-dual $Z(Q)$ models introduced in Sec.", "II indicates the same universal finite-size scaling behavior shown in (REF ).", "This confirmation was done for the values of the parameter $p$ that we believe the model share the universality class of critical behavior of the corresponding exactly integrable model ($p=0$ ).", "For brevity we only show the results for the self-dual extended Ising models (REF ).", "In Fig.", "15 and Fig.", "16 the results are for the quantum chain with $L=30$ sites and the ground state wavefunction spanned in the conformal bases where $\\lbrace \\sigma _i^z\\rbrace $ or $\\lbrace \\sigma _i^x\\rbrace $ are diagonal.", "In theses figures we show the coefficient $c_n$ obtained from the linear fit of (REF ), by using all the sublattice sizes.", "We can see that in both basis, apart from some small deviations, most probably due to the finite-size effects, the overall behavior of $I_n(\\ell ,L-\\ell )$ is the same for different values of $p$ , indicating the universal behavior of the models.", "It is clear from this figure that the singular behavior as $n\\rightarrow 1$ , already known [21] for the exactly integrable model ($p=0$ ), also happens for the extended Ising quantum chains with $p\\ne 0$ .", "Figure: The generalized mutual informations for the ground state wavefunction of theextended Ising chain (), with L=30L=30 sites.", "The coefficients c n c_n and c ˜ n \\tilde{c}_n areobtained from the linear fit of () of the Rényi mutual informationI n (ℓ,L-ℓ)I_n(\\ell ,L-\\ell ) ()-() and from the generalized mutual informationI ˜ n (ℓ,L-ℓ)\\tilde{I}_n(\\ell ,L-\\ell ), given by (), respectively.The ground sates of the quantum chains are expressed in the {σ z }\\lbrace \\sigma ^z\\rbrace basis andthe values of the parameter p=0,0.5,1p=0,0.5,1 and 1.5 .Figure: Same as Fig.", "15, but with the ground state wave function spanned in the {σ x }\\lbrace \\sigma ^x\\rbrace basis.Another interesting generalization of the Shannon mutual information, instead of being based in the Rényi entropy is based in the Rényi divergence [36].", "Differently from the Rényi mutual information this generalized mutual information is always a positive function and is a more appropriate measure, from the point of view of information theory, of the shared information among subsystems.", "Using the notations in (REF ) this generalized mutual information is defined by: $\\tilde{I}_n(\\ell ,L-\\ell )= \\frac{1}{n-1}\\ln \\left(\\sum _{\\lbrace I_{\\cal A},I_{\\cal B}\\rbrace }\\frac{P_{I_{\\cal A},I_{\\cal B}}^n}{P_{I_{\\cal A}}^{n-1}P_{I_{\\cal B}}^{n-1}}.\\right)$ Like $I_n(\\ell ,L-\\ell )$ this quantity, in the limiting case $n\\rightarrow 1$ , gives the Shannon mutual information.", "This quantity was measured for several exactly integrable quantum chains [22].", "It shows the same universal finite-size scaling function given in (REF ) for $n\\lesssim 2$ (we denote the linear coefficient as $\\tilde{c}_n$ ).", "We measured this quantity for the extended $Z(Q)$ models introduced in Sec. 2.", "The results for the extended Ising quantum chain are shown in Figs.", "15 and Fig.", "16 for the ground state wavefunction expressed in the $\\lbrace \\sigma ^z\\rbrace $ - and $\\lbrace \\sigma ^x\\rbrace $ -basis, respectively.", "Again for $0<n<2$ we clearly see in both basis the independence of the curves with the parameter $p$ of the non-integrable quantum chain.", "Actually the agreement of this behavior for several values of $p$ is even better as compared with the case of the Rényi mutual information, this indicates that the finite-size scaling corrections in $\\tilde{I}_n(\\ell ,L-\\ell )$ are smaller than the ones in $I_n(\\ell ,L-\\ell )$ .", "It is also clearly shown that the limiting case $n\\rightarrow 1$ is regular for all values of $p$ , differently from the case of the Rényi mutual information.", "This imply that $\\tilde{I}_n(\\ell ,L-\\ell )$ , as compared with $I_n(\\ell ,L-\\ell )$ is a more suitable quantity for an analytical approach towards the proof of the conjecture (REF )." ], [ "Conclusions", "In this paper we made an extensive test of the conjecture (REF ) for the Shannon mutual information $I(\\ell ,L-\\ell )$ of conformally invariant quantum critical chains at their ground states.", "In general the Shannon mutual information depends on the particular basis where the wavefunction is spanned.", "According to the conjecture (REF ) the finite-size scaling function of $I(\\ell ,L-\\ell )$ give us an interesting tool for calculating the central charge $c$ , if the ground state is spanned in the conformal basis.", "These basis corresponds, in the underlying Euclidean CFT, to the boundary condition in the time direction that do no destroy the conformal invariance of the CFT.", "This paper provide us with the first extensive numerical check of the universal character of (REF ).", "The previous tests of (REF ) were done only for exactly integrable quantum chains, and since there is no analytical proof of (REF ) it is important to verify if its validity is not connected to the exact integrability of the critical quantum chains tested previously.", "In order to produce tests for non-integrable models we introduced new families of self-dual quantum chains with nonlocal $Z(Q)$ symmetries.", "Due to their self-duality their critical points are exactly known.", "All these non-integrable quantum chains contains next-nearest neighbor coupling constants $p$ .", "Our numerical analysis concentrated in two special families of models.", "The first family is the generalization of the $Z(Q)$ parafermionic models ($Q=2-8$ ), and the second one is the generalization of the $Z(Q)$ clock models ($Q=5-8$ ).", "The first family at $p=0$ reduces to the exactly integrable parafermionic quantum chains with central charge $c=\\frac{1}{2},\\frac{4}{10},1,\\frac{8}{7},\\frac{5}{4},\\frac{4}{3},\\frac{7}{5}$ , for $Q=2-8$ , respectively.", "The second family reduces at $p=0$ to non-integrable quantum chains in the Beresinzkii-Kosterlitz Thouless universality, whose underlying CFT is expected to have a central charge $c=1$ for $Q \\ge 5$ .", "Exploring the consequences of conformal invariance, our numerical studies of the low-lying energies of these quantum chains, at finite lattice sizes, indicate that at least for a finite range of the couplings $0 \\le p\\le p_c$ the models share the same universal critical behavior, and consequently are ruled by the same CFT.", "The last observation make these introduced quantum chains even more interesting, since as we change continuously the parameter $p$ they give a critical line with a fixed value of the central charge.", "In particular the extended parafermionic quantum chains for $Q\\ge 5$ give us critical lines ruled by an underlying $Z(Q)$ parafermionic CFT with $c> 1$ .", "The extensive calculations of the Shannon mutual information $I(\\ell ,L-\\ell )$ of the ground state wavefunctions of all these quantum chains indicate the validity of the conjecture (REF ) for general critical and conformally invariant quantum chains, irrespective of being exactly integrable or not.", "It is important to mention that Stéphan [23] presented a contradictory prediction for the critical Ising quantum chain.", "In [23], by exploring the free-fermionic nature of the model, $I(\\ell ,L-\\ell )$ was calculated numerically up to lattice sizes $L=36$ , and the results indicate that the pre-factor in (REF ) instead of being the central charge $c=0.5$ , is the close, but distinct number $c=0.4801629(2)$ .", "This would imply that the conjecture (REF ) is not valid and the pre-factor is a universal unknown number whose value is close to the central charge, at least for the Ising case.", "All the numerical results we have obtained so far for the several quantum chains does not have enough precision to discard the possibility that for all the critical chains the pre-factor in the conjecture (REF ) could not be the central charge $c$ , but a number close to it.", "The single exact analytical exact calculation we have is for the set of coupled harmonic oscillators that gives in this case the central charge value $c=1$ [19].", "The result in [23] was obtained by assuming that the finite-size corrections of $I(\\ell ,L-\\ell )$ are given by the power series $\\sum _{p=0}^5 \\alpha _p/ \\ell ^{p}$ , being the fitting quite stable indicating no presence of logarithmic corrections, like $\\frac{ln{\\ell }}{\\ell }$ terms.", "As is well known in order to have a controlled prediction of quantities in the bulk limit, based on finite-size lattice estimators we should know the functional dependence of the finite-size corrections with the lattice size.", "Unfortunately this is not the case for $I(\\ell ,L-\\ell )$ .", "This is an essential point.", "$I(\\ell ,L-\\ell )$ is calculated by combining the probabilities $p_{\\lbrace x\\rbrace }$ of the configuration $\\lbrace x\\rbrace $ in the subsystem of size $\\ell $ .", "The probabilities for special configurations of the Ising quantum chain can be calculated for quite large lattices $L\\sim 1000$ .", "The results for $\\epsilon (\\lbrace x\\rbrace ) = -\\ln p_{\\lbrace x\\rbrace }$ , also called as the formation probabilities, shows that for special commensurable configurations $\\lbrace x\\rbrace $ , like the emptiness formation probability and generalizations (see appendix of [30]), indicate that correction terms $\\frac{\\ln {\\ell }}{\\ell }$ are always present.", "If as a result of the combinations of the several probabilities in $I(\\ell ,L-\\ell )$ these logarithmic corrections are canceled then the prediction of Stéphan [23] is correct and the conjecture has to be modified.", "On the other hand if still these corrections are present in $I(\\ell ,L-\\ell )$ , then we should consider lattice sizes or order $L\\sim 1000$ to discard or to confirm the conjecture (REF ).", "This is indeed a quite interesting point to be settled in the future.", "It is a challenge either to derive analytically $I(\\ell ,L-\\ell )$ or at least to derive the behavior of the finite-size corrections.", "There exist two extensions of the Shannon mutual information, namely The Rényi mutual information $I_n(\\ell ,L-\\ell )$ and the generalized mutual information $\\tilde{I}_n(\\ell ,L-\\ell )$ , based on the Rényi divergence.", "These quantities were calculated previously for several exactly integrable quantum chains in their ground state.", "As the Shannon mutual information they also show some universal features whenever the ground state wavefunction is spanned in a conformal basis.", "We calculate the generalizations $I_n(\\ell ,L-\\ell )$ and $\\tilde{I}_n(\\ell ,L-\\ell )$ for the non-integrable models introduced in this paper.", "Our results indicate that the universal features previously observed [21], [22] does not depend if the quantum chain is exactly integrable or not.", "It is important to mention that, as happen for the exactly integrable cases [22], $\\tilde{I}_n(\\ell ,L-\\ell )$ in general does not have a divergence as $n\\rightarrow 1$ , differently from the generalization $I_n(\\ell ,L-\\ell )$ .", "Since this divergence destroy the analytical continuation $n\\rightarrow 1$ , the quantity $\\tilde{I}(\\ell ,L-\\ell )$ seems to be more appropriate for an analytical derivation for the conjecture (REF ) for the Shannon mutual information $\\tilde{I}_1(\\ell ,L-\\ell )=I(\\ell ,L-\\ell )$ .", "Acknowledgments This work was supported in part by FAPESP and CNPq (Brazilian agencies).", "We thank M. A. Rajabpour and J.", "A. Hoyos for useful discussions." ] ]	1606.04994
[ [ "HEMI: Hyperedge Majority Influence Maximization" ], [ "Abstract In this work, we consider the problem of influence maximization on a hypergraph.", "We first extend the Independent Cascade (IC) model to hypergraphs, and prove that the traditional influence maximization problem remains submodular.", "We then present a variant of the influence maximization problem (HEMI) where one seeks to maximize the number of hyperedges, a majority of whose nodes are influenced.", "We prove that HEMI is non-submodular under the diffusion model proposed." ], [ "Introduction", "Signed networks are networks with both positive and negative interactions between nodes.", "They may occur as explicit edges, as in the Slashdot and Epinions social networks, or be inferred from interactions such as administrator elections on Wikipedia or conversations on Twitter .", "Mathematically, a signed network can be specified as $(V,E^{+},E^{-})$ , where $V$ is the set of vertices, with $E^{+}$ and $E^{-}$ being sets of directed edges of the form $(a,b)$ , denoting $a$ trusts $b$ or $a$ distrusts $b$ , for $E^{+}$ and $E^{-}$ respectively.", "In a social network, a centrality measure assigns each node a value, which denotes its importance within the network.", "The notion of importance may vary based on the application, leading to a wide variety of such measures, based on position , betweenness , and prestige.", "A centrality measure for a signed network needs to incorporate two sources of information and the interplay between them - namely the positive and negative edges.", "This, in addition to the imbalance in real-world signed networks between positive and negative edges, makes defining signed network centrality measures a non-trivial task.", "A simple centrality measure for signed social networks, first proposed in , is the simple net positive in- degree, also called Fans Minus Freaks (FMF) centrality measure - where fans are nodes with positive edges to pointing to the node under consideration, while freaks are the one with negative edges.", "Other measures, such as eigenvector centrality and PageRank , have also been generalized to signed networks.", "A disadvantage of some of these centrality measures is that they consider every node in isolation when computing the centrality.", "This ignores the synergy between nodes, where a node is important by virtue of its combination with other groups of nodes.", "Moreover, ignoring the synergy can also make some of these centrality measures vulnerable to attacks wherein groups of nodes work together, as noted in , to boost their individual centralities or reduce other nodes's centralities.", "One approach to incorporate this synergy is to define a cooperative game, which assigns a value to every possible subset of nodes $C \\subseteq V$ given by the characteristic function $\\nu (C)$ of the game.", "The value assigned to a node is a weighted sum of the marginal contributions it makes to the values of all possible subsets, also known as the Shapley Value (SV).", "SV based centrality provides an intuitive way of capturing a node's centrality in combination with different groups of other nodes in the network." ], [ "Shapley Value Centrality - Preliminaries", "A cooperative game is defined by a set of agents $A=\\lbrace a_1,a_2 \\ldots a_{N}\\rbrace $ and a characteristic function $\\nu () : P(A) \\mapsto R$ , where $P(A)$ represents the power set of $A$ , and $R$ is the set of real numbers.", "$\\nu (C)$ essentially maps every $C \\subseteq A$ to a real number, which represents the payoff of the subset, or the coalition The terms subset and coalition are used here interchangeably.", "Generally, $\\nu (C)$ is defined such that $\\nu (\\phi )=0$ , where $\\phi $ is the null set, which corresponds to the coalition with no agents.", "Shapley value, first proposed in , is a way of distributing the payoff of the grand coalition (the coalition of all agents) amongst each agent.", "It proposes that the individual payoff of an agent should be determined by considering the marginal contribution of the agent to every possible coalition it is a part of.", "Such a scheme of distribution is also found to obey certain desirable criteria.", "Let $\\pi \\in \\Pi (A)$ be a permutation of $A$ , and let $C_\\pi (i)$ denote the coalition of all the predecessors of agent $a_i$ in $\\pi $ , then the Shapley value is defined as $SV(a_i) = \\frac{1}{\|A\|!}", "\\sum _{\\pi \\in \\Pi (A)} (\\nu (C_\\pi (i) \\cup a_i)-\\nu (C_\\pi (i)))$ For defining a SV based centrality measure on a graph $G=(V,E)$ , we consider $A=V$ .", "$\\nu (C)$ is defined such that it represents the a measure of power/centrality/influence of the coalition $C$ .", "The centrality of each node $v_i$ is then given by its Shapley value $SV(v_i)$ ." ], [ "Computing the SV", "Earlier methods for SV based centrality, such as used a Monte-Carlo sampling based approximation to compute $SV(v_i)$ for each $v_i$ .", "This essentially involves uniformly sampling a large number of permutations from $\\Pi (V)$ , and then finding the average marginal contribution of $v_i$ as per the definition above.", "This approach is expensive (since one has to sample a large number of permutations), as well as inexact.", "Moreover, since the number of permutations grows as $O(n!", ")$ , where $n=\|V\|$ is the number of vertices in the graph.", "first proposed that by defining $\\nu (C)$ conveniently, one could compute the SV exactly in closed form.", "This approach is naturally more preferable than the MC sampling based one, although it requires defining $\\nu $ , such that SV can be easily derived using arguments from probability and combinatorics." ], [ "Contributions", "To the best of our knowledge, ours is the first work to define cooperative game theoretic centrality measures for signed social networks.", "Moreover, we are also the first to evaluate such measures for a centrality-based ranking task, in traditional or signed networks.", "Earlier works have evaluated these measures for other tasks such as influence maximization , or selecting gatekeeper nodes .", "Here, we consider the task of ranking users to detect trolls or malicious users in a social network.", "Intuitively, these users are expected to have a a highly negative reputation amongst the users of the network.", "In the availability of ground truth about trolls, one can evaluate a signed network centrality measure by considering how low these “trolls\" rank in a list of users ranked according to the measure." ], [ "Measures", "We now define several Shapley value based centrality measures for directed, signed networks, given by $G = (V,E^+,E^-)$ , where each edge $(a,b) \\in E^{+}$ denotes $a$ trusts $b$ and each edge $(a,b) \\in E^{-}$ denotes $a$ distrusts $b$ .", "We take $G$ to be a signed, directed network, since in the context of Slashdot, a positive/negative edge denotes that user $a$ approves/disapproves or trusts/distrusts user $b$ 's content.", "Hence, the directionality is of importance.", "Each of our measures is based on a different definition of $\\nu (C)$ .", "We denote the positive and negative in-degrees by $d^{+}_{in}(V)$ and $d^{-}_{in}(V)$ , and the corresponding out-degrees by $d^{+}_{out}(V)$ and $d^{-}_{out}(V)$ .", "The positive and negative in-neighbor sets are denoted by $N^{+}_{in}(V)$ and $N^{-}_{in}(V)$ , and the corresponding out-neighbor sets by $N^{+}_{out}(V)$ and $N^{-}_{out}(V)$ ." ], [ "Fans Minus Freaks (FMF)", "This is a simple generalization of the degree centrality measure to directed, signed networks.", "More formally, $FMF(v_i) = d^{+}_{in}(v_i) - d^{-}_{in}(v_i)$ We attempt to generalize this measure to sets of nodes by appropriate definitions of $\\nu (C)$ , and then compute individual centralities of nodes using the $SV$ of $\\nu $ ." ], [ "Net Positive Fringe (NPF)", "We first define the sets $\\nu ^{+}(C)$ and $\\nu ^{-}(C)$ Note the slight overloading of $\\nu $ here.", "$\\nu ^{+}$ and $\\nu ^{-}$ denote sets, not real values.", "$\\nu ^{+}(C)$ is the set of all nodes $v_j$ such that $v_j$ has atleast one positive out-neighbor $v_i \\in C$ OR $v_j \\in C$ The second clause results from the intuitive assumption that $v_i$ always trusts itself.", "$\\nu ^{-}(C)$ is the set of all nodes $v_j$ such that $v_j$ has atleast one negative out-neighbor $v_i \\in C$ .", "Note that $v_j$ itself may be present inside or outside the coalition, and this does not affect $\\nu ^{-}(C)$ .", "The characteristic function $\\nu (C)$ is given by $\\nu (C) = \|\\nu ^{+}(C)\| - \|\\nu ^{-}(C)\|$ We can think of the $\\nu (C)$ as the difference between two characteristic functions, namely $\|\\nu ^{+}(C)\|$ and $\|\\nu ^{-}(C)\|$ .", "Hence the $SV(v_i)$ of $\\nu $ will be the difference between the Shapley values $SV^{+}(v_i)$ and $SV^{-}(v_i)$ for the characteristic functions $\|\\nu ^{+}(C)\|$ and $\|\\nu ^{-}(C)\|$ .", "Let us refer to these games as Game 1 and Game 2 respectively.", "Game 1 Consider a permutation $\\pi $ sampled uniformly from $\\Pi (V)$ .", "Let $C$ be the set of nodes preceding $v_i$ in the coalition.", "Let $v_j$ by any node $\\in V$ .", "Note that $v_j$ could also be $v_i$ itself.", "We denote $B_{v_i,v_j}$ as the random variable denoting the contribution made by $v_i$ through $v_j$ .", "Here, “$v_i$ through $v_j$ \" means that as a result of $v_i$ being added to $C$ , what is the effect on the membership of $v_j$ in $\\nu ^{+}(C)$ .", "The Shapley Value of $SV(v_i)$ of $v_i$ , is then given by $\\sum _{v_j \\in V} E[B_{v_i,v_j}]$ .", "Note that since we are taking the expectation over a permutation drawn uniformly from $\\Pi (V)$ , it is equivalent to averaging over every possible permutation.", "Now, we can easily see that $B_{v_i,v_j}$ can be non-zero only if $v_j \\in v_i \\cup N^{+}_{in}(v_i)$ , since in all other cases, $v_j$ can neither be added or removed from $\\nu ^{+}(C)$ as a result of $v_i$ being added.", "Now, consider the case where $v_j \\in v_i \\cup N^{+}_{in}(v_i)$ .", "$E[B_{v_i,v_j}]$ will be equal to the fraction of permutations in which $v_i$ is able to add $v_j$ to the set $\\nu ^{+}(C)$ .", "This can happen only if $v_j$ is neither itself in $C$ , nor is any other out-neighbor of $v_j$ .", "Note that as a result, the event of $v_i$ adding $v_j$ only depends on the ordering of these $d^{+}_{out}(v_j)+1$ nodes within the permutation.", "Hence , we can directly consider the ordering of these nodes, ignoring the other nodes in our calculation.", "$v_i$ must be the first node amongst these $d^{+}_{out}(v_j)+1$ in the permutation $\\pi $ , for it to contribute $v_j$ .", "Hence, $E[B_{v_i,v_j}] &= \\frac{(d^{+}_{out}(v_j))!}{(d^{+}_{out}(v_j)+1)!}", "\\\\&= \\frac{1}{(d^{+}_{out}(v_j)+1)}$ Now, $SV^{+}(v_i)$ , the Shapley value of Game 1, is consequently given by $SV^{+}(v_i) &= \\sum _{v_j \\in v_i \\cup N^{+}_{in}(v_i)} \\frac{1}{(d^{+}_{out}(v_j)+1)}$ Game 2 Using arguments similar to Game 1, we get $SV^{-}(v_i) &= \\sum _{v_j \\in N^{-}_{in}(v_i)} \\frac{1}{(d^{-}_{out}(v_j))}$ Note that the +1 term in the denominator for the $SV^{+}(v_i)$ expression is not present here, since the node $v_j$ cannot add itself to $\\nu ^{-}(C)$ The final expression of $SV(v_i)$ for the NPF game is thus given by $SV(v_i) &= \\sum _{v_j \\in v_i \\cup N^{+}_{in}(v_i)} \\frac{1}{(d^{+}_{out}(v_j)+1)} - \\sum _{v_j \\in \\cup N^{-}_{in}(v_i)} \\frac{1}{(d^{-}_{out}(v_j))}$ The time taken to compute the NPF, $T_{NPF}$ would be given by $T_{NPF} &= O(\\sum _{v \\in V} (1 + d^{+}_{in}(v) + d^{-}_{in}(v))) \\\\T_{NPF} &= O(V+E)$" ], [ "Fringe Of Absolute Trust (FAT)", "Note that we refer to $\\nu ^{+}(C)$ and $\\nu ^{-}(C)$ , as defined in the previous game.", "In this game, a node is included in the coalition's value if it satisfies both the conditions below It either belongs to the coalition or has a positive out neighbor in the coalition.", "It does not have a negative out neighbor in the coalition.", "The intuition underlying this measure is that every node contributing to the set's value should be such that it does not distrust any node in the set.", "Most signed networks have more positive edges than negative ones.", "For instance, Slashdot has only 23.9% of its edges marked as negative.", "Hence, the negative edges may be interpreted strongly as an explicit “vote of distrust\".", "Note that distrusting even a single member in the coalition removes a node from the coalition's value.", "Previous work proposes similarly motivated solutions to overcome imbalance.", "For instance, in , the authors undersample the positive edges to be equal to the negative edges.", "If node $v_j \\in N^{+}_{in}(v_i)\\cup v_i$ , then $B_{v_i,v_j}$ is +1 if $v_i$ is the first of any of the out-neighbours of $v_j$ (positive or negative) or the node $v_j$ itself to occur in the permutation.", "This is because if a $v_k \\in N^{+}_{out}(v_j) \\cup v_j$ is in $C$ , without any negative out neighbor of $v_j$ being in $C$ , then $v_j$ already is such that $v_j \\in \\nu ^{+}(C)-\\nu ^{-}(C)$ .", "Also, if any negative out neighbor of $v_j \\in C$ , then $v_j$ can never belong to $\\nu ^{+}(C)-\\nu ^{-}(C)$ , hence adding $v_i$ to $C$ would have no effect.", "This argument holds good even if $v_i=v_j$ .", "Therefore, for $v_j \\in N^{+}_{in}(v_i)\\cup v_i$ $E[B_{v_i,v_j}] &= \\frac{1}{d^{+}_{out}(v_j)+d^{-}_{out}(v_j)+1}$ Now consider $v_j \\in N^{-}_{in}(v_i)$ .", "We can see that $B_{v_i,v_j} \\ne 0$ iff A node $v_k \\in N^{+}_{out}(v_j) \\cup v_j$ belongs to C No negative out-neighbor $v_k$ of $v_j$ belongs to C. In fact, $B_{v_i,v_j}=-1$ if both the conditions above are satisfied.", "The expectation is given by $\\tiny E[B_{v_i,v_j}] &= -\\frac{\\sum _{x=1}^{x=d^{+}_{out}(v_j)+1} \\binom{d^{+}_{out}(v_j)+1}{x} x!", "(d^{total}_{out}(v_j)-x)!", "}{(d^{total}_{out}(v_j)+1)!}", "\\\\$ where $d^{total}_{out}(v_j) = d^{+}_{out}(v_j) + d^{-}_{out}(v_j)$ Note that $\\binom{n}{r}$ represents the number of ways of choosing r distinct things from n distinct things.", "Since computing factorials for large values can become infeasible in code due to limits on the value of variables, we simplify the expression into a product of fractions form.", "$E[B_{v_i,v_j}] &= \\frac{-1}{d^{total}_{out}(v_j)+1}\\sum _{x=1}^{z=d^{+}_{out}(v_j)+1} \\frac{\\prod _{\\alpha =1}^{\\alpha =x}(d^{+}_{out}(v_j)-x+\\alpha )}{\\prod _{\\alpha =1}^{\\alpha =x}(d^{total}_{out}(v_j)-x+\\alpha )}$ The final expression for $SV(v_i)$ for NPF is given by $SV(v_i) = \\sum _{v_j \\in v_i \\cup N^{+}_{in}(v_i) \\cup N^{+}_{in}(v_i)} B_{v_i,v_j}$ The complexity of computing FAT, $T_{FAT}$ would be $T_{FAT} &= O(\\sum _{v \\in V} d^{+}_{in}(v_i)+1+d^{-}_{in}(v_i){(\\Delta ^{+}_{out})}^{2}) \\\\T_{FAT} &= O(V+E+E{(\\Delta ^{+}_{out})}^{2}))$ where $\\Delta ^{+}_{out}$ is the maximum positive out degree." ], [ "Negated Fringe Of Absolute Distrust (NFADT)", "This game is in some sense like FAT, but with the roles of distrust and trust reversed.", "$\\nu (C)$ here is given by $-\|\\nu ^{-}(C)-\\nu ^{+}(C)\|$ .", "The negative sign is because $\|\\nu ^{-}(C)-\\nu ^{+}(C)\|$ would be a measure of disrepute (negative reputation).", "We omit the expression here for the sake of brevity.", "The complexity expression of NFADT would be similar to that of $T_{FAT}$ , with $\\Delta ^{+}_{out}$ replaced by $\\Delta ^{-}_{out}$ ." ], [ "Net Trust Votes (NTV)", "The intuition underlying this measure is that the collective importance of a group of nodes is the net number of “votes\" or edges in its favour, by nodes outside the group.", "Given a coalition C, let $E^{+}$ be the set of positive in-edges from a node outside the coalition to a node in the coalition.", "Similarly, $E^{-}$ is the set of negative in-edges from a node outside the coalition into the coalition.", "Note that we do not consider internal edges in either term.", "In the NTV game, $\\nu (C)$ is given by $\|E^{+}\|-\|E^{-}\|$ .", "Let us now consider the derivation of a closed form expression for this game.", "Consider the node $v_i$ being added to the $C$ .", "$v_i$ can contribute to the value of $\|E^{+}\|-\|E^{-}\|$ in four different ways, as stated below Positive out-edges from $v_i$ to nodes $v_k \\in C$ .", "These edges become internal when $v_i$ is added to the coalition, decreasing the value of $\|E^{+}\|-\|E^{-}\|$ by 1.", "Negative out-edges from $v_i$ to nodes $v_k \\in C$ .", "These edges become internal when $v_i$ is added to the coalition, increasing the value of $\|E^{+}\|-\|E^{-}\|$ by 1.", "Positive in-edges from $v_k, v_k \\notin C$ to $v_i$ .", "These edges become a part of $E^{+}$ , increasing $\\nu (C)$ by 1 Negative in-edges from $v_k, v_k \\notin C$ to $v_i$ .", "These edges become a part of $E^{-}$ , decreasing $\\nu (C)$ by 1.", "Consider case 1.", "This case only happens for $v_i$ , $v_j \\in N^{+}_{out}(v_i)$ .", "For this event to happen, $v_j \\in C$ .", "In other words, it should precede $v_i$ in the permutation.", "This will happen in exactly half the permutations, and will result in a contribution of $\\frac{-1}{2}$ .", "The cumulative contribution as a result of case 1 will be $\\sum _{v_j \\in N^{+}_{out}(v_i)} -\\frac{1}{2} = -\\frac{d^{+}_{out}(v_i)}{2}$ .", "Symmetrically, in case 2, the cumulative contribution will be $\\sum _{v_j \\in N^{-}_{out}(v_i)} \\frac{1}{2} = \\frac{d^{-}_{out}(v_i)}{2}$ .", "Case 3 can only happen for $v_i$ , $v_j \\in N^{+}_{in}(v_i)$ .", "Here $v_j$ should follow $v_i$ in the permutation i.e.", "it should not be in $C$ .", "This will happen in exactly half the permutations.", "Hence, the cumulative contribution will be $\\sum _{v_j \\in N^{+}_{in}(v_i)} \\frac{1}{2} = \\frac{d^{+}_{in}(v_i)}{2}$ .", "Symetrically in case 4, we have the cumulative contribution given by $\\sum _{v_j \\in N^{-}_{in}(v_i)} -\\frac{1}{2} = -\\frac{d^{-}_{in}(v_i)}{2}$ .", "Summing over the contributions from the 4 cases we get $SV(v_i) = \\frac{1}{2} (d^{+}_{in}(v_j) - d^{-}_{in}(v_j)) - \\frac{1}{2} (d^{+}_{out}(v_j) - d^{-}_{out}(v_j))$ This expression also exhibits a relation to status theory , where a positive edge from a to b, indicates that b has a higher status than a, while a negative edge indicates the opposite.", "Thus for a node $a$ , there are $d^{+}_{in}(v_j) + d^{-}_{out} (v_j)$ nodes with a status lower than it, and $d^{-}_{in}(v_j) + d^{+}_{out} (v_j)$ nodes with a status higher than it.", "The expression above, is in some sense, an indicator of the node's status relative to its neighbors.", "Since this expression involves only the node degrees, $T_{NTV}=O(V)$ , provided we maintain the in-degrees and out-degrees separately." ], [ "k-Hop NPF", "We can generalize the NPF measure to $k$ hops, considering the sets of in-neighbours $N^{+}_{in}(v_i,k)$ and $N^{-}_{in}(v_i,k)$ to be the set of neighbors within a distance of k-hops using in-edges.", "The notion of a path being positive or negative is determined using the principle “The enemy of my enemy is my friend\", motivated by balance theory .", "However, unlike the traditional balance theory setting, we consider the direction of the edges i.e.", "we only consider paths where edges are pointing from the source to the destination.", "The sign of the path is given by the product of its edge signs.", "Note that $N^{+}_{in}(v_i,k)$ and $N^{-}_{in}(v_i,k)$ may now have a non-zero intersection unlike the simple one-hop neighbor sets.", "Hence, we do not generalize the $FAT$ and $NFADT$ measures to the $k$ hop case, since these measures do not consider the neighbor sets independently." ], [ "Strong Balance Shapley Value (SBSV)", "This measure has its motivation in balance theory , which has also been used in the literature to define energy functions , to characterize a signed graph's stability.", "Balanced triads (where the product of signs is positive) are considered stable and having negative energy, while unbalanced triads are considered unstable.", "Here, we consider the un-normalized We do not normalize the energy by the largest possible number of triads $\\binom{\|C\|}{3}$ , or the total number of triangles, since this makes computing the closed form difficult energy of a set of nodes $C$ as the number of balanced triads minus the number of unbalanced triads.", "$\\nu (C) = \\sum _{\\lbrace v_i,v_j,v_k\\rbrace \\in T(C)} -s_{ij}s_{jk}s_{ik}$ where $T(C)$ is the set of triads in $C$ , and $s_{ij}$ represents the sign of the edge between $i$ and $j$ .", "We can easily see that the marginal contribution of a node $i$ to $C$ would be only dependent on the number of triangles $v_i$ is a part of.", "If the other two nodes of a triad to which $i$ belongs are already present in $C$ , then $i$ can make a marginal contribution through the triad.", "For a given pair of adjacent neighbors j and k of a node i, this will only happen in $\\frac{1}{3}$ of the permutations.", "Thus, the shapley value $SV(v_i)$ will be given by $SV(v_i) = \\sum _{\\lbrace v_j,v_k\\rbrace , v_j \\in N(v_k), v_j,v_k \\in N(v_i)} \\frac{1}{3} (- s_{ij}s_{jk}s_{ik})$ The time complexity $T_{SBSV}$ will be $O(\\sum _{v_i \\in V} (d^{total}_{in}(v_i))^2)$ ." ], [ "Weak Balance Shapley Value (WBSV)", "Weak balance is a variant of balance theory , according to which the triad with three negative signs (all nodes are mutual enemies), is not considered as unbalanced.", "As a result, triads where $s_{ij}$ , $s_{jk}$ and $s_{ik}$ are all $-1$ are not included in the value function, and consequently, in the Shapley Value.", "The time complexity will be the same as that of WBSV." ], [ "Evaluation and Results", "We consider the task of ranking users to detect trolls in the Slashdot signed network http://konect.uni-koblenz.de/networks/slashdot-zoo, with 96 users annotated as trolls.", "The network has 71512 nodes, with 487751 edges.", "About 24% of these edges are negative.", "We evaluate a centrality measure by first ranking the nodes of the graph in ascending order according to them, and then evaluating these ranklists based on how high the ground truth trolls rank in them.", "For evaluation, we consider two metrics The number of trolls in the top $g$ elements of the ranklist, where $g$ is the number of ground truth trolls The average precision (AP) metric from IR , with the trolls corresponding to “relevant documents\".", "The metric is defined as follows $AP = \\sum _{k=1}^{k=n} \\frac{P(k) \\times Mal(k)}{g}$ Here, $P(k)$ is the precision (fraction of ground truth trolls) in the top $k$ elements of the ranklist.", "$Mal(k)$ is 1 if the kth user is a troll, while $g$ is the number of ground truth trolls.", "$n$ is the size of the ranklist.", "Besides computing these for the full graph, we also compute the mean of the AP (MAP) over 50 subgraphs formed by deleting 5 %, 10 % and 20% of the nodes.", "We observe that the NPF, FAT and NFADT measures perform considerably better than FMF, both on the full graph as well as on each of the random subsamples.", "Moreover, the 3-Hop NPF gives the highest average precision amongst all the measures, while the performance of k-Hop FMF decreases when we go to 3 hops.", "Table: Comparison of Measures For Troll Detection" ] ]	1606.05065
[ [ "Quantum phase transitions in a generalized compass chain with three-site\n interactions" ], [ "Abstract We consider a class of one-dimensional compass models with XYZ$-$YZX-type of three-site exchange interaction in an external magnetic field.", "We present the exact solution derived by means of Jordan-Wigner transformation, and study the excitation gap, spin correlations, and establish the phase diagram.", "Besides the canted antiferromagnetic and polarized phases, the three-site interactions induce two distinct chiral phases, corresponding to gapless spinless-fermion systems having two or four Fermi points.", "We find that the $z$ component of scalar chirality operator can act as an order parameter for these chiral phases.", "We also find that the thermodynamic quantities including the Wilson ratio can characterize the liquid phases.", "Finally, a nontrivial magnetoelectric effect is explored, and we show that the polarization can be manipulated by the magnetic field in the absence of electric field." ], [ "Introduction", "The rapid development of spin-orbital physics and quantum information in recent years motivates the search for the realizations of intrinsically frustrated orbital (or pseudospin) interactions.", "Such interactions lead to radically different behavior from Heisenberg SU(2) isotropic exchange, and have been in the focus of very active research in recent years.", "It was realized that the quantum nature of orbital degrees of freedom, that may be released by emerging spin-orbital coupling and spin-orbital entanglement, is interdisciplinary and plays a crucial role in the fields of strongly correlated electrons [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and cold atoms [11], [12], [13], [14], [15].", "The strong frustration of spin-orbital interactions can be best understood by considering generic orbital models, in which the bond-directional interactions provide the building blocks.", "Among them, the two-dimensional (2D) compass model defined on a square lattice [16] and the Kitaev model on a honeycomb lattice [17] can be treated as two quintessential pseudospin models, where the effective moments cannot simultaneously align to satisfy interactions with all neighbors as they favor the quantum states with distinct quantization axes.", "In fact, the latter model is a rare example of an interacting 2D spin model that can be rigorously solved, and was found to support gapped and gapless quantum spin liquids with emergent fractional excitations obeying non-Abelian statistics.", "Otherwise, exact solutions for 2D models with frustrated exchange exist only for classical Ising interactions where a phase transition at finite temperature is found [18].", "Recent studies show that also for the 2D compass model a phase transition to nematic order occurs at finite but much lower temperature [19].", "In low-dimensional magnetic systems collective quantum phenomena are particularly strong since the reduced dimensionality amplifies the consequences of frustrated interactions between individual spins.", "To probe the exotic phases resulting from bond-directional interactions, we introduced a one-dimensional (1D) generalized compass model (GCM) with antiferromagnetic exchange alternating between even and odd bonds [20].", "Such a model may be realized in layered structures of transition metal oxides, with alternating exchange interactions along the bonds parallel to $a$ and $b$ axes along a zigzag chain in an $(a,b)$ plane [21], optical lattices [22], [23], trapped ions [24], [25], and coupled photonic cavities [26], [27].", "On the other hand, the community focuses on two-body interactions in most systems studied, as they contribute to superexchange and are readily accessible experimentally.", "However, the range of the hybridization of the electron wave function will be finite in some realistic bonding geometries, and the effect of such long-range interactions must be addressed.", "Recently three-site interactions received considerable attention in a bit diverse context [28], [29], [31], [32], [35], [30], [36], [34], [33], [37], [38], [39], [40], [41].", "They also occur in an effective spin model in a magnetic field obtained from a 1D plaquette orbital model by an exact transformation, with spin dimers that replace plaquettes.", "Indeed, they are coupled along the chain by three-spin interactions in the Hilbert space reduced by a factor of 2 per plaquette [42].", "Such complex interactions between three subsequent sites essentially enrich the ground state phase diagram of the spin model and open new opportunities for underlying physics.", "Experimentally, it can be realized in NMR quantum simulators [43], [44] or optical lattices [45].", "Three-site spin interactions have been exhibited the multiferroics [46] and the magnetoelectric effect [33], [39].", "The purpose of this paper is to focus on a 1D GCM with three-site interactions.", "We show that this model is exactly solvable and explore the consequences of three-site interactions.", "By investigating spin correlations we identify two chiral phases and demonstrate the existence of a nontrivial magnetoelectric effect.", "The organization of the paper is as follows.", "In Sec.", "we introduce the Hamiltonian of the 1D GCM with three-site interactions in Sec.", "REF and then present the procedure to solve it exactly by employing Jordan-Wigner transformation in Sec.", "REF .", "The ground state and energy gap are retrieved.", "In Sec.", "we use spin correlations to characterize each phase and quantum phase transitions (QPTs).", "The model in the magnetic field is analyzed in Sec.", ", and the complete phase diagram is obtained when the three-site interactions and magnetic fields are varied.", "The obtained exact solution allows us to present the thermodynamic properties including the Wilson ratio in Sec.", ".", "We also point out that the three-site interactions play a role in the magnetoelectric effect in Sec.", ".", "A final discussion and conclusions are given in Sec.", ".", "We consider below a 1D chain of $N$ sites with periodic boundary conditions, with GCM interactions given by $H_{\\rm GCM}(\\theta )= \\sum _{i=1}^{N^{\\prime }}J_{o}\\tilde{\\sigma }_{2i-1}(\\theta )\\tilde{\\sigma }_{2i}(\\theta )+J_{e}\\tilde{\\sigma }_{2i}(-\\theta )\\tilde{\\sigma }_{2i+1}(-\\theta ).\\nonumber \\\\$ Here $N^{\\prime }=N/2$ is the number of two-site unit cells, while $J_o$ and $J_e$ denote the coupling strengths on odd and even bonds, respectively (below we take $J_o$ as the unit of exchange interaction).", "The operator $\\tilde{\\sigma }_i(\\theta )$ (with a tilde) is defined as a linear combination of $\\lbrace \\sigma _{i}^x,\\sigma _{i}^y\\rbrace $ pseudospin components (Pauli matrices), $\\tilde{\\sigma }_{i}(\\theta )&\\equiv & \\cos (\\theta /2)\\,\\sigma _{i}^x+\\sin (\\theta /2)\\,\\sigma _{i}^y.$ These linear combinations imply that Ising-like interactions on an odd/even bond in Eq.", "(REF ) are characterized by the preferential easy axes selected by an arbitrary angle $\\pm \\theta /2$ .", "With increasing angle $\\theta $ , frustration gradually increases when the model Eq.", "(REF ) interpolates between the Ising model at $\\theta =0$ and the quantum compass model (QCM) at $\\theta =\\pi /2$ , in analogy to the 2D compass model [47].", "The model was solved exactly and the ground state is found to have order along the easy axis as long as $\\theta \\ne \\pi /2$ , whereas it becomes a highly disordered spin-liquid ground state at $\\theta =\\pi /2$ [48], [49].", "Here we introduce the XZY$-$ YZX type of three-site interactions in addition, $H_{\\rm 3-site} =J^\\sum _{i=1}^{N}(\\sigma ^x_{i-1}\\sigma ^z_{i}\\sigma ^y_{i+1}-\\sigma ^y_{i-1}\\sigma ^z_{i}\\sigma ^x_{i+1}),$ where $J^$ is its strength.", "Such interactions between three adjacent sites emerge as an energy current of a compass chain in the nonequilibrium steady states, as discussed in the Appendix.", "The complete Hamiltonian of the 1D GCM with the three-site XZY$-$ YZX interaction is ${\\cal H} =H_{\\rm GCM}+H_{\\rm 3-site}.$" ], [ "Exact solution", "We employ the Jordan-Wigner transformation which maps explicitly between quasispin operators and spinless fermion operators through the following relations [50]: $\\sigma _{j}^{z}& =&1-2c_{j}^{\\dagger }c_{j}, \\quad \\sigma _{j}^{y}=i\\sigma _{j}^{x}\\sigma _{j}^{z}, \\\\\\sigma _{j}^{x}& =& \\prod _{i<j}\\,(1-2c_{i}^{\\dagger }c_{i}^{})(c_{j}^{}+c_{j}^{\\dagger }),$ where $c_{j}$ and $c_{j}^{\\dagger }$ are annihilation and creation operators of spinless fermions at site $j$ which obey the standard anticommutation relations, $\\lbrace c_{i},c_{j}\\rbrace =0$ and $\\lbrace c_{i}^{\\dagger },c_{j}\\rbrace =\\delta _{ij}$ .", "By substituting Eqs.", "(REF ) into Eq.", "(REF ), we arrive at a simple bilinear form of the Hamiltonian (REF ) in terms of spinless fermions: $\\cal {H}&=&\\sum _{i=1}^{N^{\\prime }} \\Big [J_{o} e^{i\\theta }c_{2i-1}^{\\dagger } c_{2i}^{\\dagger }+ J_{o} c_{2i-1}^{\\dagger } c_{2i}^{} \\nonumber \\\\& &\\hspace{14.22636pt}+ J_{e}e^{-i\\theta } c_{2i}^{\\dagger } c_{2i+1}^{\\dagger }+ J_{e} c_{2i}^{\\dagger } c_{2i+1}^{} \\nonumber \\\\& &\\hspace{14.22636pt}- 2iJ^(c_{2i-1}^{\\dagger } c_{2i+1}+c_{2i}^{\\dagger }c_{2i+2}^{})+{\\rm H.c.}\\Big ].$ Next discrete Fourier transformation for plural spin sites is introduced by $c_{2j-1}\\!=\\frac{1}{\\sqrt{N^{\\prime }}}\\sum _{k}e^{-ik j}a_{k},\\text{ \\ \\ }c_{2j}\\!=\\frac{1}{\\sqrt{N^{\\prime }}}\\sum _{k}e^{-ik j}b_{k},$ with discrete momenta as $k=\\frac{n\\pi }{ N^\\prime }, \\quad n= -(N^\\prime \\!-1), -(N^\\prime \\!-3),\\ldots , (N^\\prime \\!", "-1).$ The Hamiltonian takes the following form, which is suitable to introduce the Bogoliubov transformation: $\\cal {H}&=& \\sum _{k} \\left[ B_k^{} a_{k}^{\\dagger }b_{-k}^{\\dagger }+ A_k^{} a_{k}^{\\dagger } b_{k}^{}- A_k^ a_{k}^{}b_{k}^{\\dagger }-B_k^* a_{k}^{}b_{-k}^{} \\right.\\nonumber \\\\& & \\left.\\hspace{14.22636pt}- 4J^* \\sin k (a_{k}^{\\dagger } a_{k}^{}+ b_{k}^{\\dagger } b_{k}^{})\\right].$ where $A_k&=& J_{o} + J_{e}+ e^{ik}, \\nonumber \\\\B_k&=& J_o e^{i\\theta }-J_e e^{i(k-\\theta )}.$ To diagonalize the Hamiltonian Eq.", "(REF ), we rewrite it in the Bogoliubov-de Gennes form, ${\\cal H} &=& \\sum _{k}\\,\\Gamma _k^{\\dagger }\\,\\hat{M}_k^{}\\,\\Gamma _k^{},$ where $\\hat{M}_k\\!=\\frac{1}{2}\\!\\left(\\!\\begin{array}{cccc}-G_k & 0 & S_k & P_k+Q_k \\\\0 & -G_k & P_k- Q_k & -S_k \\\\S_k^* & P_k^-Q_k^ & -G_k & 0 \\\\P_k^+Q_k^ & -S_k^* & 0 &-G_k\\end{array}\\!\\right),$ and $\\Gamma _k^{\\dagger }=(a_k^{\\dagger },a_{-k}^{},b_k^{\\dagger },b_{-k}^{})$ .", "In Eq.", "(REF ) the compact notation is introduced: $P_k&=&-i (J_e e^{ik}+J_o)\\sin \\theta , \\nonumber \\\\Q_k&=& (J_e e^{ik}-J_o)\\cos \\theta , \\nonumber \\\\S_k &=&J_o+J_e e^{ik}, \\nonumber \\\\G_k&=& 2J^* \\sin k.$ The diagonalization of Hamiltonian (REF ) is achieved by a four-dimensional Bogoliubov transformation which connects the operators $\\lbrace a_k^{\\dagger },a_{-k}^{},b_k^{\\dagger },b_{-k}^{}\\rbrace $ with four kind of quasiparticles, $\\lbrace \\gamma _{k,1}^{\\dagger },\\gamma _{k,2}^{\\dagger },\\gamma _{k,3}^{\\dagger },\\gamma _{k,4}^{\\dagger }\\rbrace $ , $\\left(\\begin{array}{c}\\gamma _{k,1}^{\\dagger } \\\\\\gamma _{k,2}^{\\dagger } \\\\\\gamma _{k,3}^{ \\dagger }\\\\\\gamma _{k,4}^{\\dagger }\\end{array}\\right)=\\hat{U}_{k} \\left(\\begin{array}{c}a_k^{\\dagger } \\\\a_{-k} \\\\b_k^{\\dagger } \\\\b_{-k}\\end{array}\\right),$ where the rows of $\\hat{U}_{k}$ are eigenvectors of the Bogoliubov-de Gennes equations.", "The diagonalization is readily performed to yield the eigenspectra $\\varepsilon _{k,j}$ ($j=1,\\cdots ,4$ ): $\\varepsilon _{k,1(2)}=-\\frac{1}{2}\\left(\\sqrt{\\xi _k \\pm \\sqrt{\\xi _k^2-\\tau _k^2}}+G_k\\right), \\nonumber \\\\\\varepsilon _{k,3(4)}=\\frac{1}{2}\\left(\\sqrt{\\xi _k \\mp \\sqrt{\\xi _k^2-\\tau _k^2}}-G_k\\right),$ where $\\xi _k&=&\\vert P_k \\vert ^2 + \\vert Q_k \\vert ^2 + \\vert S_k \\vert ^2 ,\\nonumber \\\\\\tau _k&=&\\vert P_k^2 - Q_k^2 + S_k^2 \\vert .$ The eigenenergies for various $J^$ are labeled sequentially from the bottom to the top as $\\varepsilon _{k,1},\\cdots ,\\varepsilon _{k,4}$ in Fig.", "REF .", "One finds that finite $J^$ removes the symmetry of the spectra with respect to $\\varepsilon =0$ energy and they are not invariant with respect to the $k\\rightarrow -k$ transformation, in contrast to the case of the GCM with $J^=0$ shown in Fig.", "REF (a).", "The three-site interactions break both parity (P) symmetry and time reversal (T) symmetry.", "Note that modes $k= 0,\\pm \\pi $ are time reversal invariant and their excitations are independent of $J^$ as a consequence of vanishing $G_{k}$ .", "Instantly, we obtain the diagonal form of the Hamiltonian, ${\\cal H}=\\sum _{k }\\sum _{j=1}^{4} \\varepsilon _{k,j}\\,\\gamma _{k,j}^{\\dagger }\\gamma _{k,j}^{} .$ The most important properties of the 1D quantum system can be explored in the ground state.", "The ground state of any fermion system follows the Fermi-Dirac statistics, and the lowest energy is obtained when all the quasiparticle states with negative energies are filled by fermions.", "More precisely, in the thermodynamic limit ($N\\rightarrow \\infty $ ) the ground state of the system, $\|\\Phi _0\\rangle $ , corresponds to the configuration with chemical potential $\\mu =0$ , where all the states with $\\varepsilon _{k,j}<0$ are occupied and the ones with $\\varepsilon _{k,j}\\ge 0$ are empty.", "This state is realized by means of the corresponding occupation numbers, $n_{k,j}=\\langle \\Phi _0\\vert \\gamma _{k,j}^{\\dagger }\\gamma _{k,j}^{}\\vert \\Phi _0\\rangle = \\left\\lbrace \\begin{array}{l l}0 & \\quad {\\rm for}\\;\\varepsilon _{k,j} \\ge 0,\\\\1 & \\quad {\\rm for}\\;\\varepsilon _{k,j}<0.\\end{array} \\right.$ One recognizes that the Bogoliubov-de Gennes Hamiltonian (REF ) actually acts in an artificially enlarged Nambu-spinor space and it respects an emergent particle-hole symmetry (PHS) ${\\cal C}$ , i.e., ${\\cal C}\\hat{M}_k{\\cal C}=\\hat{M}_{-k}$ , with ${\\cal C}^2=1$ .", "Here in the so-called particle-hole space, the extra degree of freedom $\\textbf {C}^{2}$ leads to two copies of the actual excitation spectrum, a particle and a hole copy emerge simultaneously.", "The PHS implies here that $\\gamma _{k,4}^{\\dagger }$ =$\\gamma _{-k,1}^{}$ , $\\gamma _{k,3}^{\\dagger }$ =$\\gamma _{-k,2}^{}$ , as is evidenced in Fig.", "REF .", "The bands with positive energies correspond to the electron excitations while the negative ones are the corresponding hole excitations.", "When all quasiparticles above the Fermi surface are absent the ground state energy may be expressed as: $E_0 = -\\frac{1}{2} \\sum _{k} \\sum _{j=1}^4\\left\|\\varepsilon _{k,j} \\right\|.$ Accordingly, the gap is determined by the absolute value of the difference between the second and third energy branches, $\\Delta =\\min _{k}\\left\|\\varepsilon _{k,2}-\\varepsilon _{-k,3}\\right\|.$ Figure: The energy spectra ε k,j \\varepsilon _{k,j} (j=1,⋯,4j=1,\\cdots ,4) forincreasing J * J^:(a) J J^* = 0,(b) J * J^* = 0.239,(c) J * J^* = 2, and(d) J * J^* = 5.Parameters are as follows: J o =1J_o = 1, J e =4J_e = 4, θ=π/3\\theta =\\pi /3.One finds that with the increase of $J^$ , the minimum of $\\varepsilon _{k,3}$ bends down until it touches $\\varepsilon =0$ when $J^$ reaches a threshold value $J_{c,1}^$ , i.e., $\\Delta $ = 0; cf.", "Fig.", "REF (b).", "A gapless mode shows up at some incommensurate mode $k_{ic}$ and the spectrum vanishes quadratically.", "Further increase of $J^$ leads to the bands inversion between portions of $\\varepsilon _{k,2}$ and $\\varepsilon _{k,3}$ .", "There is a negative-energy region of $\\varepsilon _{k,3}$ in $k$ space shown in Fig.", "REF (c), and there are two Fermi points across the Fermi surface.", "When $J^$ exceeds another threshold value $J_{c,2}^$ the energy spectrum of spinless fermions may also have two additional Fermi points [31], as observed in Fig.", "REF (d).", "A Lifshitz transition occurs following the topological change of the Fermi surface in the Brillouin zone." ], [ "Correlations and quantum phase transitions", "In order to characterize the QPTs, we studied the nearest neighbor spin correlation function defined by $C^{\\alpha }_{l}&=&-\\frac{2}{N}\\sum _{i=1}^{N^{\\prime }}\\langle \\sigma _{i}^\\alpha \\sigma _{i+l}^\\alpha \\rangle ,$ where $l$ =1(-1) and the superscript $\\alpha =x,y,z$ denotes the cartesian component, and the $z$ component of scalar chirality operator [51] ${\\cal \\chi }^{z} = -\\frac{1}{N}\\sum _{i=1}^{N} \\langle {\\sigma }_{i}^z\\vec{z}\\cdot [\\vec{\\sigma }_{i-1}\\times \\vec{\\sigma }_{i+1}].\\rangle $ The scalar chirality operator can act as a local order parameter for states without PT symmetry.", "As shown in Fig.", "REF , the ground state has finite nearest neighbor correlation functions for $J^=0$ , among which $x$ components $\\lbrace C_l^x\\rbrace $ dominate for $\\theta =\\pi /3$ , implying that the adjacent spins are antiparallel and aligned with a canted angle with respect to the $x$ axis.", "Indeed, the ground state of the GCM is a canted Néel (CN) phase for $\\theta <\\pi /2$ .", "Figure: The nearest neighbor correlations C α C^\\alpha on even bonds and chirality χ α \\chi ^\\alpha by increasing J J^* for h=0h=0.Parameters are as follows: J o =1J_o = 1, J e =4J_e = 4, θ=π/3\\theta =\\pi /3.With the increase of $J^$ , the nearest neighbor correlation functions remain constant.", "After $J^$ surpasses $J_{c,1}^$ , the system stays in a chiral-I phase without finite energy gap, characterized by a nonzero ${\\cal \\chi }^{z}$ .", "In such a chiral-I phase, $x$ components $C_l^x$ decrease while $C_l^y$ and $C_l^z $ grow as $J^$ increases, but they become saturated quickly.", "When $J^>J_{c,2}^$ , the system enters chiral-II phase, where ${\\cal \\chi }^{z}$ grows rapidly and $\\lbrace C_l^\\alpha \\rbrace $ ($\\alpha =x$ ,$y$ , and $z$ ) decreases simultaneously.", "In the fermionic picture different phases correspond to different Fermi-surface topology (different number of Fermi points) for fermions.", "In particular, the two Fermi-point spinless fermions (chiral-I phase) is distinct from the four-Fermi-point spinless fermions (chiral-II phase) [31].", "Both spin-liquid phases have gapless excitations, however, the appearance of new points $k_F$ in the Fermi surface when the controlling parameter crosses a critical value will witness a general feature of the discontinuities in the correlation functions.", "We remark that the number of gapless modes determine the effective central charge and the coefficients of the area-law violating term of bipartite entanglement entropy [52], [53].", "Notably, the chiral-II phase is a dedicated phase of the critical XX model with three-site XZY$-$ YZX interactions added [31], [34], [30], [32], [29], [33], while this phase is absent for anisotropic XY model [38].", "Here we observe the three-site XZY$-$ YZX interactions in the GCM surprisingly triggers both chiral states for arbitrary $\\theta $ , and two different Tomonaga-Luttinger liquids reflect the importance of Fermi surface topology.", "Figure: The critical value of J * J^* as a function of θ\\theta .Parameters are as follows: J o J_o=1, J e J_e=4.The determination of critical values of $J_{c,1}^,J_{c,2}^$ and the corresponding incommensurate momentum $k_{ic}$ can be given by $\\varepsilon _{k_{ic},3(4)}=0 ,\\quad \\partial \\varepsilon _{k_{ic},3(4)}/\\partial k=0.$ This leads to the following quartic equation for $x_{ic}=\\cos k_{ic}$ : $x_{ic}^4 + c_3 x_{ic}^3 + c_2 x_{ic}^2 +c_0 =0,$ where $c_3&=&4(J_o^2 + J_e^2)/(3J_o J_e \\sin ^2\\theta ), \\nonumber \\\\c_2&=&(J_o^2+J_e^2)^2/(3J_o^2 J_e^2 \\sin ^4\\theta )-4\\cot ^4\\theta /3+2/3,\\nonumber \\\\c_0&=&-1/3.", "\\nonumber $ This quartic equation can be solved analytically but the form is rather contrived.", "We plot the critical lines with respect to $\\theta $ in Fig.", "REF .", "One finds that in the Ising limit, i.e., for $\\theta \\rightarrow 0$ , it yields $J^_{c,1} \\rightarrow \\textrm {min} (J_o, J_e) \\quad {\\rm and} \\quad J^_{c,2} \\rightarrow \\textrm {max} (J_o, J_e).$ While in the compass limit, i.e., for $\\theta \\rightarrow \\pi /2$ , we have $J^_{c,1} \\rightarrow 0 \\quad {\\rm and} \\quad J^_{c,2} \\rightarrow \\textrm {max} (J_o, J_e).$ In other words, the system for $\\theta =\\pi /2$ has an emergent $\\mathbb {Z}_2$ symmetry and the ground state can not be ordered.", "Any infinitesimal perturbation of $J^$ will induce the system into gapless chiral-I state.", "For the parameters we choose mostly in this paper, i.e., $J_o=1$ , $J_e=4$ , $\\theta =\\pi /3$ , one finds $J^_{c,1}=0.239$ and $J^_{c,2}=4.048$ ." ], [ "Effect of transverse field", "We now consider the case where the magnetic field $h$ is perpendicular to the easy plane of the spins, i.e., $\\vec{h}=h\\hat{z}$ .", "In this case, the Zeeman term is given by ${\\cal H}_h=h\\hat{z}\\cdot \\sum _{i=1}^{N^{\\prime }}(\\vec{\\sigma }_{2i-1}+\\vec{\\sigma }_{2i}),$ where $h$ is the magnitude of the transverse external field.", "Subsequently, in Nambu representation, the Hamiltonian matrix $\\hat{M}_k$ (REF ) is modified in the following way: $\\hat{M}_k \\rightarrow \\hat{M}_k^{^{\\prime }}=\\hat{M}_k -h \\mathbb {I}_2 \\otimes \\sigma ^z,$ where $\\mathbb {I}_2$ is a ($2\\times 2$ ) unity matrix.", "It is obvious that the external magnetic field plays the role of a chemical potential for spinless fermions.", "Figure: The energy spectra ε k,j \\varepsilon _{k,j} (j=1,⋯,4j=1,\\cdots ,4) for increasingelectric field hh:(a) hh = 1,(b) hh = 2, and(c) hh = 3.The inset in (b) is an amplification of the level crossing at the Fermienergy marked by dashed circle below.", "Parameters are as follows:J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3, and J =0.1J^=0.1.After diagonalization four branches of energies $\\varepsilon _{k,j}$ , with $j=1,\\cdots ,4$ , are given by the following expressions: $\\varepsilon _{k,1(2)}=-\\frac{1}{2}\\left(\\sqrt{\\zeta _k \\pm \\sqrt{\\eta _k }}-G_k\\right),\\nonumber \\\\\\varepsilon _{k,3(4)}=\\frac{1}{2}\\left(\\sqrt{\\zeta _k \\mp \\sqrt{\\eta _k }}-G_k\\right),$ where $\\zeta _k&=&\\vert P_k \\vert ^2 + \\vert Q_k \\vert ^2 + \\vert S_k\\vert ^2+ h^2, \\nonumber \\\\\\eta _k&=& (S_k^ Q_k + S_k Q_k^)^2-(S_k^ P_k - S_k P_k^)^2 \\nonumber \\\\&+&(P_k^ Q_k + P_k Q_k^)^2 + 4 \\vert S_k \\vert ^2 h^2.$ The magnetic field further breaks the T symmetry and polarizes spins along $z$ direction.", "The analytical solution for $J^$ = 0 had been scrutinized recently.", "One finds that increasing transverse field induces finite transverse polarization $\\langle \\sigma _i^z\\rangle $ and drives the system into a saturated polarized phase above the critical field [20].", "The field-induced QPT is of second order for arbitrary angle $\\theta $ and occurs at at the critical value, $h_c = 2 \\sqrt{J_o J_e}\\cos \\theta .$ Figure: The nearest neighbor correlations C α C^\\alpha on even bonds and chirality χ α \\chi ^\\alpha by increasing J * J^* for h=3h = 3.Parameters are as follows: J o =1J_o = 1, J e =4J_e = 4, θ=π/3\\theta =\\pi /3.The field-induced criticality is suited at momentum $k= 0$ , where $G_k$ does not play a role, see Eq.", "(REF ).", "Figure REF shows the energy spectra obtained for three typical values of $h$ and fixed weak $J^=0.1$ .", "We find that a finite gap separates occupied from empty bands except when $h=h_c$ , see Eq.", "(REF ).", "A small value of $J^$ does not modify the critical field and the gap vanishes linearly for $\\theta \\ne \\pi /2$ , see inset in Fig.", "REF (b).", "When $h=h_c$ the gaps opens and grows with increasing $(h-h_c)$ , see Fig.", "REF (c).", "Figure: The gap Δ\\Delta as a function of hh and J * J^.Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3.The nearest neighbor correlation functions $\\lbrace C_l^\\alpha \\rbrace $ ($\\alpha $ =$x$ , $y$ , and $z$ ) and the $z$ component of scalar chirality operator ${\\cal \\chi }^z$ for increasing $J^$ at $h=3$ are shown in Fig.", "REF .", "Finite magnetic field expands the range of CN phase and increases both $J_{c,1}^$ and $J_{c,2}^$ , see Fig.", "REF .", "The $z$ components $\\lbrace C_l^z\\rbrace $ dominate over $x$ components $\\lbrace C_l^x\\rbrace $ for small $J^$ and $\\theta =\\pi /3$ , suggesting that the spins are aligned along the $z$ axis according to the sign of $\\lbrace C_l^z\\rbrace $ .", "The correlation functions are found to be almost independent of $J^$ as long as the system is within the polarized state, but they change in a discontinuous way at phase transitions.", "As $J^$ rises above the critical value $J_{c,1}^$ , a nonzero chirality ${\\cal \\chi }^{z}$ starts to grow and saturates.", "One finds that $C_l^y$ and $C_l^z$ decrease and change sign from negative to positive values upon increasing $J^$ , which is contrast to the trend observed for $C_l^x$ .", "A sharp upturn of ${\\cal \\chi }^{z}$ occurs for $J>J_{c,2}^$ , and it continues to increase with $J^$ .", "Simultaneously, all the correlation functions $\\lbrace C_l^\\alpha \\rbrace $ ($\\alpha =x,y,z$ ) decrease strongly towards zero when the system enters the chiral-II phase.", "To present a three-dimensional panorama of the excitation gap, we display $\\Delta $ for varying $h$ and $J^$ in Fig.", "REF .", "The gap $\\Delta $ diminishes for large value of $J^$ .", "Figure: Magnetic phase diagram of the 1D GCM as a function of transverse fieldhh and three-site XZY--YXZ interaction J J^.Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3.Similarly, we can discriminate the critical lines $J^_{c,1(2)}$ and zero-gap modes $k_{ic}$ using the relations in Eq.", "(REF ).", "The phase diagram is shown in Fig.", "REF .", "The phase diagram at finite three-site XZY$-$ YZX interaction and magnetic field consists of four phases: (i) canted antiferromagnetic, (ii) polarized, (iii) chiral-I, and (iv) chiral-II.", "A tricritical point is determined by the intersection of both critical lines which can be obtained analytically: $h_c=2\\sqrt{J_o J_e}\\cos \\theta , J_c^= J_o J_e \\cos ^2\\theta /(J_o+J_e).$ In the special case of $\\theta =\\pi /2$ , the CN phase is never stable." ], [ "THERMODYNAMIC PROPERTIES", "Since the exact solution of the GCM with three-site interaction and the external field is at hand, it is straightforward to obtain its complete thermodynamic properties at finite temperature.", "All quantum phase transitions of the present 1D GCM are of second order.", "Among many thermodynamic quantities, the specific heat and magnetic susceptibility are easy to to be measured, and both of them are proportional to the electronic density of states at Fermi energy.", "For the particle-hole excitation spectrum (REF ), the free energy of the quantum spin chain at temperature $T$ reads, ${\\cal F}= - k_B T \\sum _k\\sum _{j=1}^4\\ln \\left(2\\cosh \\frac{\\varepsilon _{k,j}}{2k_B T}\\right).$ The low temperature behavior of the heat capacity, $C_V(T)&=&-T\\left(\\frac{\\partial ^2{\\cal F}}{\\partial T^2}\\right)_h\\nonumber \\\\&=& k_B \\sum _k \\sum _{j=1}^{4} \\frac{(\\varepsilon _{k,j}/2k_B T)^2}{ \\cosh ^2 (\\varepsilon _{k,j}/2k_BT)}.$ The magnetic susceptibility is defined as follows, $\\chi (T)&=&-\\left(\\frac{\\partial ^2{\\cal F}}{\\partial h^2}\\right)_T- \\frac{1}{2 }\\sum _k\\sum _{j=1}^{4}\\left\\lbrace \\frac{\\partial ^2\\varepsilon _{k,j}}{\\partial h^2}\\tanh \\left( \\frac{\\varepsilon _{k,j}}{2k_BT}\\right) \\right.", "\\nonumber \\\\&+& \\left.\\left(\\frac{\\partial \\varepsilon _{k,j}}{\\partial h}\\right)^2\\left[2k_BT\\cosh ^2\\left(\\frac{\\varepsilon _{k,j}}{2k_BT}\\right)\\right]^{-1}\\right\\rbrace .$ Figure: The thermodynamic properties for two values of h=1h=1 and h=3h=3at fixed temperature T=0.01T=0.01:(a) the specific heat C V C_V,(b) the magnetic susceptibility χ\\chi .The inset shows the Wilson ratio R W R_W () as a function ofthree-site XZY--YXZ interaction J J^* for h=1h=1 and h=3h=3.Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3.At low temperatures the specific heat has a linear dependence on $T$ in liquid metals due to the contribution from the electrons within the energy interval $k_BT$ near the Fermi surface, while the magnetic susceptibility is independent of temperature owing to the fact that only the electrons within the energy $\\mu _B gH$ near the Fermi surface contribute to magnetization.", "The Sommerfeld-Wilson ratio (Wilson ratio in short) is a parameter which characterizes strongly correlated Fermi liquids.", "It is defined as a dimensionless ratio of the zero-temperature magnetic susceptibility $\\chi $ and the coefficient of the linear term $\\propto T$ in the electronic specific heat $C_V(T)$ [54], $R_{\\rm W}=\\frac{1}{3}\\left(\\frac{2\\pi k_B}{\\mu _B g_{\\rm Lande}} \\right)^2\\frac{T\\chi (T)}{C_V(T)},$ where $k_B$ is Boltzmann's constant, $\\mu _B \\equiv e/(2mc) $ is the Bohr magneton, $g_{\\rm Lande}\\simeq 2$ is the Lande factor.", "Such quantity measures the strength of magnetic fluctuations versus thermal fluctuations.", "Figure REF shows the specific heat $C_V(T)$ and the magnetic susceptibility $\\chi (T)$ for increasing $J^$ , in the range which covers all phases.", "In a 1D antiferromagnet, the zero-temperature magnetic susceptibility exhibits a square-root divergence across critical fields.", "The Wilson ratio (REF ) undergoes an increase due to sudden changes in the density of states near the critical fields [55].", "$R_{\\rm W}=1$ in the free-electron limit when $J^\\rightarrow \\infty $ .", "However, we notice that $R_{\\rm W}$ deviates from 1 in chiral-I phase.", "In particular, $R_{\\rm W}$ is larger here than that in chiral-II phase.", "Furthermore, $R_{\\rm W}$ is enhanced by increasing magnetic field, see inset in Fig.", "REF .", "The Wilson ratio can be measured experimentally as for instance in a recent experiment on a gapped spin-1/2 Heisenberg ladder compound (C$_7$ H$_{10}$ N)$_2$ CuBr$_2$ [56]." ], [ "Magnetoelectric effect ", "Next we consider the magnetoelectric effect (MEE), where the roles of magnetization and polarization can be interchanged.", "A key quantity to characterize the MEE is the linear magnetoelectric susceptibility which defines the dependence of magnetization on the electric field, or the polarization dependence on the magnetic field.", "Figure: Electric polarizations (see legend) as functions of external field hhfor:(a) J * =0J^=0,(b) J =0.5J^=0.5, and(c) J =4.5J^=4.5.Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3.The three-spin interaction was naturally claimed to contribute to the ferroelectricity in the Katsura-Nagaosa-Balatsky (KNB) formula for its particular form [57], in which the local spins (magnetic moments) and the local polarization are coupled, $\\vec{P} = \\gamma \\hat{e}_{ij} \\times (\\vec{\\sigma }_i\\times \\vec{\\sigma }_j),$ where $\\hat{e}_{ij}$ is the unit vector connecting the neighboring spins $\\vec{\\sigma }_i$ and $\\vec{\\sigma }_j$ with a material-dependent coupling coefficient $\\gamma $ .", "Here we place the chain along the $x$ direction in the real space, i.e., $\\hat{e}_{ij}=(1,0,0)$ .", "Considering a particular component ($z$ here, to be specific) of the spin current, $\\frac{d \\sigma _l^z}{dt}= i[{\\cal H}, \\sigma _l^z]=- {\\rm div} j_l^z,$ which defines the current $j_l^z$ and the corresponding $P_l^y$ by Eq.", "(REF ).", "The electric polarization has two sources [39].", "The first term originates from the spin-current model, given by $P_1^y \\propto \\langle \\sigma _l^x\\sigma _{l+1}^y-\\sigma _l^y\\sigma _{l+1}^x\\rangle ,$ which couples with $y$ component of the electric field $\\vec{E}$ induced by the Dzyaloshinskii-Moriya interaction.", "Through the relation $\\vec{P_1}=(\\partial {\\cal H}/\\partial \\vec{E})$ , the absence of external electric field $\\vec{E}$ in Hamiltonian $\\cal {H}$ suggests that it has little contribution to the electric polarization $P_1^y$ .", "However, as shown in Fig.", "REF , $P_1^y$ is induced in the presence of magnetic field $h$ as long as the phases are chiral, and it is larger in chiral-II phase than in chiral-I phase.", "Figure: The evolution of electric polarization contributions P n y P_n^y withincreasing hh at different temperature TT for:(a) P 1 y P_1^y (), and(b) P 2 y P_2^y ().Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3, J =0.5J^=0.5.Figure: The evolution of P 1 y P_1^y and P 2 y P_2^y by reversing the magneticfield hh.", "Parameters are as follows: J o J_o=1, J e J_e=4, θ=π/3\\theta =\\pi /3,J =0.5J^=0.5, T=0.01T=0.01.Another contribution of electric polarization may come from the spin current triggered by the three-site interactions in the following way [39]: $P_2^y\\propto - \\langle \\sigma _l^x\\sigma _{l+1}^z\\sigma _{l+2}^x+\\sigma _l^y\\sigma _{l+1}^z \\sigma _{l+1}^y\\rangle .$ The general form of the current operator is given in the Appendix.", "The form of $P_2^y$ is the well-known XZX$+$ YZY type of three-site interaction and remains solvable in the frame of Jordan-Wigner fermionization [29], [30].", "A little algebra will yield that three-site XZX$+$ YZY interactions acts here as a renormalization (momentum-dependent) of the magnetic field $h$ in the Hamiltonian Eq.", "(REF ).", "The manipulation of $h$ will affect finite $P_2^y$ in an indirect way, as is displayed in Fig.", "REF .", "We find that $P_2^y$ is also induced by $h$ , regardless of their phases.", "It has an opposite sign to $P_1^y$ and almost complements its increase.", "Both $P_1^y$ and $P_2^y$ scale linearly with small $h$ , indicating that they are triggered by the external magnetic field.", "This is in contrast to some models with two-spin interactions only, where the electric polarization can emerge only for finite electric field.", "The compass model with three-site interactions verifies the proposal in Ref.", "[58], and indeed exhibits a nontrivial magnetism-driven ferroelectricity.", "We can observe in Fig.", "REF that the ferroelectricity phenomena are quite stable for moderate temperature.", "An essential feature of the ferroelectric behavior is that the electric polarization can be reversed by the reversal of the magnetic field, as is verified in Fig.", "REF ." ], [ "Summary and Conclusions", "In this paper we have considered the 1D generalized compass model Eq.", "(REF ) which interpolates between the Ising model (at $\\theta =0$ ) and the maximally frustrated quantum compass model (at $\\theta =\\pi /2$ ) and includes three-site XZY$-$ YZX interactions.", "We also investigated this model in the presence of external magnetic field.", "Although the system is quantum and highly frustrated, we have shown that exact solutions of the corresponding model may be obtained through Jordan-Wigner transformation.", "The XZY$-$ YZX type of three-site interactions break both the parity symmetry and the time-reversal symmetry, and then drastically modify the energy spectra, leading to two kind of Tomonaga-Luttinger liquids.", "We find that moderate three-site XZY$-$ YZX interactions will lead to a chiral-I state with two Fermi points in the representation of spinless fermions, and large three-site XZY$-$ YZX interactions transform the system into the four Fermi point spinless fermions.", "Accordingly, this modification of the Fermi surface topology follows some noticeable changes in the central charges, and then affects the ground state properties, such as nearest neighbor correlation functions.", "We find that the $z$ component of scalar chirality operator can well distinguish gapped and gapless phases, and also witness an abrupt change from chiral-I to chiral-II phase.", "In both spin-liquid phases, not only the magnetization is influenced by the magnetic field but the polarization emerges even for $\\vec{E}=0$ and is also affected by the magnetic field.", "To conclude, we emphasize that the advantage of the model considered here is its exact solvability that implies in particular the possibility to calculate accurately various dynamic quantities.", "The reported results may serve to test other approximate techniques used to study more realistic models.", "W.-L.Y.", "acknowledges support by the Natural Science Foundation of Jiangsu Province of China under Grant No.", "BK20141190 and the NSFC under Grant No.", "11474211.", "A.M.O.", "kindly acknowledges support by Narodowe Centrum Nauki (NCN, National Science Center) Project No.", "2012/04/A/ST3/00331.", "" ], [ "Current operator for the compass model", "For a 1D compass chain, the only conserved quantity is the energy.", "We can decompose Eq.", "(REF ) into: $H_{\\rm GCM}(\\theta )= \\sum _{i=1}^{N^{\\prime }} h_i(\\theta ),$ where $h_i(\\theta )\\!=J_{o}\\tilde{\\sigma }_{2i-1}(\\theta )\\tilde{\\sigma }_{2i}(\\theta )+J_{e}\\tilde{\\sigma }_{2i}(-\\theta )\\tilde{\\sigma }_{2i+1}(-\\theta ),$ and $\\tilde{\\sigma }_i(\\theta )$ in defined by Eq.", "(REF ).", "A unit cell contains two bonds.", "Furtheron, one finds the commutation relations: $&&[\\tilde{\\sigma }_{i}(\\theta ), \\tilde{\\sigma }_{j}(\\theta )]=0,\\nonumber \\\\&&[\\tilde{\\sigma }_{i}(\\theta ), \\tilde{\\sigma }_{j}(-\\theta )]=-2i\\sin \\theta \\sigma _i^z \\delta _{ij},\\nonumber \\\\&&[\\tilde{\\sigma }_{i}(-\\theta ), \\tilde{\\sigma }_{j}(\\theta )]=2i\\sin \\theta \\sigma _i^z \\delta _{ij},\\nonumber \\\\&&[\\tilde{\\sigma }_{i}(-\\theta ), \\tilde{\\sigma }_{j}(-\\theta )]=0.$ The energy current $\\hat{J}_l$ of a compass chain in the nonequilibrium steady states is calculated by taking a time derivative of the energy density and follows from the continuity equation [59], 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[ [ "Observation of an Alfv\\'en Wave Parametric Instability in a Laboratory\n Plasma" ], [ "Abstract A shear Alfv\\'en wave parametric instability is observed for the first time in the laboratory.", "When a single finite $\\omega/\\Omega_i$ kinetic Alfv\\'en wave (KAW) is launched in the Large Plasma Device above a threshold amplitude, three daughter modes are produced.", "These daughter modes have frequencies and parallel wave numbers that are consistent with copropagating KAW sidebands and a low frequency nonresonant mode.", "The observed process is parametric in nature, with the frequency of the daughter modes varying as a function of pump wave amplitude.", "The daughter modes are spatially localized on a gradient of the pump wave magnetic field amplitude in the plane perpendicular to the background field, suggesting that perpendicular nonlinear forces (and therefore $k_{\\perp}$ of the pump wave) play an important role in the instability process.", "Despite this, modulational instability theory with $k_{\\perp}=0$ has several features in common with the observed nonresonant mode and Alfv\\'en wave sidebands." ], [ "Observation of an Alfvén Wave Parametric Instability in a Laboratory Plasma S. Dorfman T. A. Carter University of California Los Angeles, Los Angeles, California 90095, USA A shear Alfvén wave parametric instability is observed for the first time in the laboratory.", "When a single finite $\\omega /\\Omega _i$ kinetic Alfvén wave (KAW) is launched in the Large Plasma Device above a threshold amplitude, three daughter modes are produced.", "These daughter modes have frequencies and parallel wave numbers that are consistent with copropagating KAW sidebands and a low frequency nonresonant mode.", "The observed process is parametric in nature, with the frequency of the daughter modes varying as a function of pump wave amplitude.", "The daughter modes are spatially localized on a gradient of the pump wave magnetic field amplitude in the plane perpendicular to the background field, suggesting that perpendicular nonlinear forces (and therefore $k_{\\perp }$ of the pump wave) play an important role in the instability process.", "Despite this, modulational instability theory with $k_{\\perp }=0$ has several features in common with the observed nonresonant mode and Alfvén wave sidebands.", "52.35.Mw, 52.35.Bj Alfvén waves, a fundamental mode of magnetized plasmas, are ubiquitous in space, astrophysical, and laboratory plasmas.", "While the linear behavior of these waves has been extensively studied [1], [2], [3], [4], [5], nonlinear effects are important in many real systems, including the solar wind and solar corona.", "Theoretical predictions show that these Alfvén waves may be unstable to various parametric instabilities (e.g., Refs.", "[6], [7], [8]) even at very low amplitudes ($\\delta {B}/B<10^{-3}$ ).", "Parametric instabilities could contribute to coronal heating [9], the observed spectrum and cross-helicity of solar wind turbulence [10], [11], [12], and damping of fast magnetosonic waves in fusion plasmas [13], [14].", "An abundance of theoretical work [15], [16], [17], [6], [18], [19], [7] has found three types of parametric instabilities for a $k_\\perp =0$ Alfvén wave: decay, modulational, and beat.", "The decay instability is the most widely known and involves the decay of a forward propagating Alfvén wave into a backward propagating Alfvén wave and a forward propagating sound wave.", "By contrast, the modulational instability results in forward propagating upper and lower Alfvénic sidebands as well as well as a nonresonant acoustic mode at the sideband separation frequency.", "To allow the forward propagating waves to interact, the pump wave must be dispersive– therefore the modulational instability at $k_\\perp =0$ requires finite $\\omega /\\Omega _i$ through inclusion of Hall effects [7].", "Ponderomotive coupling between the pump and sideband Alfvén modes self-consistently drives the nonresonant density perturbation parallel to the background magnetic field.", "In this context, “nonresonant” means that the mode does not satisfy a dispersion relation in the absence of the instability drive; this is also called a quasimode in the fusion community [20], [21].", "Both shear Alfvén wave decay and modulational instabilities have been produced in numerical simulations [22], [11], [23], [24], [25], but observational evidence is limited.", "Observations in the ion foreshock region upstream of the bow shock in the Earth's magnetosphere have found cases where a decay instability is possible, but results are not conclusive due to limited available data [26], [27].", "In this Letter, the first laboratory observations of a shear Alfvén wave parametric instability are presented.", "A single finite $\\omega /\\Omega _i$ , finite $k_\\perp $ Alfvén wave is launched, and three daughter waves are observed when the amplitude of the pump is above a threshold: two sideband Alfvén waves copropagating with the pump and a low frequency nonresonant mode.", "Frequency and parallel wave number matching relations are satisfied.", "Although these features of the observed instability are consistent with the $k_\\perp =0$ modulational instability theory, the theoretical growth rate is too small to explain observations.", "The spatial pattern of the daughter modes suggests a perpendicular (to the background magnetic field) nonlinear drive.", "Figure: Experimental setup in LAPD.", "Top: An Alfvén wave antenna on the right end of the device launches the pump wave.", "Magnetic and Langmuir probes used to diagnose the interaction are shown.", "Bottom: Spatial pattern of the pump wave in the xyxy plane measured by a magnetic probe at z=2.6z=2.6 m for the strap antenna (left, B 0 =1135B_0=1135 G) and RMF antenna (right, B 0 =993B_0=993 G).Experiments are conducted using the Large Plasma Device (LAPD) at UCLA, a cylindrical vessel capable of producing a $16.5$ m long, quiescent, magnetized plasma column for wave studies.", "The BaO cathode discharge lasts for $\\sim {10}$ ms, including a several millisecond-long current flattop.", "Typical plasma parameters for the present study are $n_e\\sim 10^{12}$ cm$^{-3}$ , $T_e\\sim 5$ eV, and $B_0\\sim 1000$ G ($\\beta \\sim 10^{-3}$ –$10^{-4}$ ) with a fill gas of helium.", "Extensive prior work has focused on the properties of linear Alfvén waves [28], [29], [5], [30].", "Studies of the nonlinear properties of Alfvén waves have also been performed on the LAPD; in these experiments, two launched Alfvén waves nonlinearly interact to drive a nonresonant mode [31], a drift wave [32], an acoustic mode [33], [34], or an Alfvén wave [35].", "Figure: Observed kinetic Alfvén wave (KAW) parametric instability showing threshold behavior and parametric dependence.", "RMF antenna, RHCP mode, B 0 =993B_0=993 G. (a) Frequency spectrum from a magnetic probe at x=0x=0, y=-6y=-6 cm, z=2.6z=2.6 m for three pump mode amplitudes.", "When the pump amplitude is above threshold for instability, three daughter modes are seen.", "(b) Parametric dependence of the daughter mode frequency as a function of pump amplitude δB 0⊥ /B 0 \\delta {B_{0\\perp }}/B_0.", "The pump amplitude is 0 on the log 10 _{10} color scale.", "White vertical dashed lines represent values of pump amplitude from (a).For the present set of experiments, a single antenna is placed at the far end of the LAPD, as shown in the top panel of Fig.", "REF .", "This is either the 96 cm long strap antenna [36] shown in the diagram or the rotating magnetic field (RMF) antenna described in [37].", "The pump wave is launched at $\\omega _0 \\sim 0.67 \\Omega _i$ , producing the pattern in the plane perpendicular to $B_0$ shown for each antenna in the bottom panel.", "The strap antenna launches a linearly polarized $m=0$ Alfvén wave cone ($k_{\\perp 0}\\rho _s=0.11$ ) in which oscillating magnetic field vectors (white arrows) circle the field-aligned wave current.", "By contrast, the RMF antenna is set up to produce two field-aligned current channels ($k_{\\perp 0}\\rho _s=0.21$ ) rotating around $B_0$ in an $m=1$ pattern [37].", "The rotation direction and hence wave polarization may be controlled by varying the antenna phasing.", "To ensure the launched wave remains nearly monochromatic, the antenna current is digitized (not shown) and found to contain no significant sideband component.", "In the plasma column in front of the antenna, magnetic and Langmuir probes detect the signatures of the pump and daughter modes.", "Each probe is mounted on an automated positioning system that may be used to construct a 2D profile in the $x$ -$y$ plane averaged across multiple discharges.", "When the pump wave amplitude exceeds a threshold value, additional peaks are observed in the frequency spectrum, as shown in Fig.", "REF .", "Panel (a) of the figure shows the appearance of three modes: a low frequency mode ($M1$ ), a lower sideband mode ($M-$ ), and an upper sideband mode ($M+$ ).", "The frequency matching relations $\\omega _{\\pm }\\mp \\omega _1=\\omega _0$ hold.", "However, $M1$ is not purely a density perturbation as predicted by the $k_\\perp =0$ modulational instability theory; as seen in Fig.", "REF , the mode has significant magnetic character.", "A clear parametric dependence of the mode frequencies on pump amplitude is shown in panel (b) of Fig.", "REF .", "As the pump amplitude $\\delta {B_{0\\perp }}/B_0$ increases above threshold, the frequencies of $M1$ and $M+$ increase; there is a corresponding decrease in the frequency of $M-$ such that frequency matching relations are satisfied at all wave powers.", "Figure: Parallel wave number measurement showing daughter modes copropagating with the pump.", "The pump, M-M-, and M+M+ are identified as KAWs while M1M1 is a nonresonant mode.", "Strap antenna, B 0 =1140B_0=1140 G, δB 0⊥ /B 0 =1.9×10 -3 \\delta {B_{0\\perp }}/B_0=1.9\\times 10^{-3}.", "Magnetic probes at z=5.11z=5.11 m, 5.755.75 m, and 6.396.39 m. The fluid dispersion relation for a KAW with the pump k ⊥0 ρ s =0.11k_{\\perp 0}\\rho _s=0.11 and a line with slope ω/k \|\| =0.29V A \\omega /k_{\|\|}=0.29V_A are plotted for comparison.To determine the character of the three observed daughter modes, the parallel wave numbers are measured using a set of three axially separated magnetic probes placed $0.639$ m apart, allowing resolution of wave numbers up to $4.9/m$ .", "As shown in Fig.", "REF , this measurement reveals positive values of $k_{\\parallel }$ for all modes, indicating that all three daughter modes are copropagating with the pump.", "Parallel wave number matching is satisfied, $k_{\\parallel \\pm }\\mp k_{\\parallel 1}=k_{\\parallel 0}$ .", "Based on the measured dispersion relation, the pump, $M-$ , and $M+$ are identified as kinetic Alfvén waves (KAWs) while $M1$ is a nonresonant mode.", "Note that $M1$ falls above the KAW dispersion curve $\\omega =k_{\|\|}V_A\\sqrt{1+{\\left(k_{\\perp }\\rho _s\\right)^2}-\\left({\\omega / \\Omega _i}\\right)^2}$ for all possible values of $k_\\perp $ .", "However, the measured $k_{\|\|1}$ is too small for $M1$ to be an acoustic mode (for these parameters, $C_s=0.012V_A$ ).", "This production of a nonresonant mode is consistent with the modulational instability.", "Figure: Spatial profile of M-M- for the strap antenna suggesting the nonlinearity is perpendicular in nature.", "A cut of δB x \\delta {B_x} is shown on the right.", "Strap antenna pump from Fig., B 0 =1135B_0=1135 G. Color represents fluctuating magnetic field amplitude δB -⊥ \\delta {B_{-\\perp }}; white arrows show relative magnitude and direction.", "The peak in M-M- amplitude occurs on a gradient of the pump mode magnetic field near the current channel center.Measurements in the plane perpendicular to the background field reveal that perpendicular nonlinear forces likely play a role in generating the observed daughter waves.", "This is shown in Fig.", "REF which displays the pattern of a representative daughter mode $M-$ in the strap antenna case; the plot is derived from a magnetic probe scanned spatially over many shots.", "By comparing this figure to the strap pump mode pattern in Fig.", "REF , it can been seen that the amplitude peak of $M-$ occurs near the center of the current channel on a gradient of the pump mode magnetic field.", "By contrast, the parallel ponderomotive force associated with the modulational instability will produce an amplitude peak in the daughter modes at the location where the pump wave magnetic field peaks [33], [38].", "This difference suggests a perpendicular nonlinearity in which perpendicular gradients of the pump mode amplitude (i.e., $k_\\perp $ ) play a key role in the nonlinear terms.", "Figure: Dependence of the observed frequency spectrum on the polarization of the RMF antenna.", "Magnetic probe x=0x=0, y=-6y=-6 cm, z=2.6z=2.6 m. Inset: Polarization of the RMF pump mode from Fig.", "along a cut at x=0x=0.", "B 0 =993B_0=993 G.Figure: Comparison between LAPD data and and k ⊥ =0k_\\perp =0 dispersion relation.", "(a) Solutions to the dispersion relation of and for experimental parameters of Fig. .", "Labeled: s: sound mode, –b: backward propagating lower Alfvénic sideband, –f: forward propagating lower Alfvénic sideband, +f: forward propagating upper Alfvénic sideband.", "Black curves represent stable modes; orange curves representing unstable modes are labeled with the appropriate instability.", "(b) Mode frequency of the modulational instability as a function of pump amplitude for experimental parameters in Fig.", "(blue circles), theoretical predictions (red stars), and strap antenna results with similar parameters (yellow squares).The pump mode polarization also influences the observed instability.", "This is investigated by changing the RMF antenna phasing to produce one of the two polarization patterns shown in the inset panel of Fig.", "REF .", "Polarization is quantified at each spatial point by measuring the ratio of the minor to major radius in the ellipse traced by the rotating magnetic field vector.", "This quantity is signed negative for left-hand rotation and positive for right-hand rotation.", "As shown in Fig.", "REF , left-hand (LHCP) and right-hand (RHCP) pump modes contain opposite polarization mixes that sum to linear polarization.", "Each mix produces a different frequency spectra in the vicinity of the current channel; the sideband separation frequency produced by the LHCP mode is less than half that produced by the RHCP mode.", "As in the linearly polarized strap antenna case, the daughter mode amplitudes peak near the current channel center for the RHCP pump mode.", "The spatial profile and nonlinear physics may be different in the LHCP case and is still under investigation; the LHCP mode also leads to a broadening of the pump mode profile and a corresponding broad spectrum at low frequencies.", "The existence of a polarization dependence is consistent with the theoretical literature on parametric instabilities.", "However, most theoretical work (e.g., Refs.", "[6], [7]) considers uniformly polarized plane waves, making direct comparisons difficult.", "Despite important physical differences with the present work, modulational instability theory with $k_\\perp =0$ still describes some features of the observed process well.", "Figure REF , panel (a) shows the roots of the dispersion relation derived by [6] and [7], solved for LAPD parameters.", "This two-fluid model outputs the dispersion relation of $M1$ given a finite amplitude pump wave propagating parallel to the background field.", "Orange curves for unstable modes reveal the usual decay, beat, and modulational instabilities driven by the parallel ponderomotive force.", "Because the modulational instability involves only forward propogating modes, it is most consistent with the experimental observations.", "An arrow on the figure indicates that the peak growth rate of the modulational instability occurs for daughter nonresonant modes with $\\omega /k_{\\parallel }=0.29V_A$ .", "Comparing this value to the measured dispersion of $M1$ in Fig.", "REF , the line falls just within the upper error bar.", "Therefore, the fact that $M1$ is not a normal mode of the system is well predicted by modulational instability theory with $k_\\perp =0$ .", "The theory also predicts the increase in mode frequency with pump amplitude seen in Fig.", "REF .", "This is shown in panel (b) of Fig.", "REF which plots the frequency of $M1$ for both the experimental case in Fig.", "REF (blue circles) and the $k_\\perp =0$ theoretical prediction [6], [7] (red stars).", "Both theory and experiment follow an upward trend.", "However, the theoretical frequencies are an order of magnitude too low, and the corresponding growth times are longer than the plasma discharge; clearly, the parallel ponderomotive force is too weak to explain the experimental observations.", "Furthermore, changing the $k_\\perp $ spectrum of the pump wave by switching to a different antenna (yellow squares) while keeping other parameters similar results in an increase in the observed $M1$ frequency.", "These observations imply that perpendicular structure plays a key role in the observed instability.", "Further theoretical development is necessary to fully explain the observed daughter modes.", "[6] and [7] predict that the growth rate of the decay instability should be three orders of magnitude larger than that of the modulational instability for the LAPD parameters under investigation.", "Yet parametric decay to sound waves is not observed.", "Possible reasons include (1) the growth rates are modified when finite $k_{\\perp }$ is considered and (2) for the larger values of $k_{\\parallel }$ characteristic of the decay instability ion-neutral collisions present in the experiment significantly reduce the growth rate.", "Concerning the effect of finite $k_{\\perp }$ , very limited theoretical and computational work is available.", "Numerical simulations by [39][40] and [23] show a reduction in the growth rate of the decay instability for oblique pump waves, but do not consider the modulational instability.", "Work by [41], [42] extends the theory to allow the daughter modes to have finite $k_\\perp $ while retaining $k_{\\perp 0}=0$ for the pump.", "This allows for new classes of instabilities at oblique angles.", "In particular, [42] found a magnetoacoustic instability with a very narrow band of unstable wave numbers which is favored at low $\\beta $ and high wave dispersion (i.e., high $\\omega /\\Omega _i$ ).", "The oblique nature of the daughter modes may also explain the Alfvénic character of the observed nonresonant mode $M1$ .", "New insight on the nature of the nonlinear terms may also come from extending theoretical work by [43] which examines copropagating waves, but only with aligned polarizations.", "The applicability of these results to the present Letter is currently under investigation.", "In summary, the first laboratory observations of a shear Alfvén wave parametric instability are presented.", "A single finite $\\omega /\\Omega _i$ , finite $k_\\perp $ Alfvén wave is launched above a threshold amplitude, resulting in three daughter modes: two forward propagating Alfvén wave sidebands and a forward propagating nonresonant mode.", "Frequency and parallel wave number matching relations are satisfied.", "Although these features are consistent with the $k_\\perp =0$ modulational instability theory, the parallel ponderomotive force that drives this process cannot explain the growth or perpendicular spatial profile of the observed daughter modes.", "Future theoretical and computational work will focus on exploring the role of $k_\\perp $ in the instability.", "Experimental data analysis is ongoing to explore variation with plasma parameters.", "The observations reported here open a significant new avenue of research to complement extensive theory [15], [16], [17], [6], [18], [19], [7] and simulation [22], [11], [23], [24], [25] work on this subject.", "Features of the observed instability may provide guidance to future space observation aimed at assessing the role of Alfvén wave parametric instabilities in different regions of the heliosphere, for example, in the ion foreshock region of planetary magnetospheres where large amplitude Alfvén waves are generated by ion beams [26], [27], [44].", "Because the present results are at low $\\beta $ , they may be of particular interest to the upcoming Solar Probe Plus mission aimed at determining what physical processes are most important in the source region of the solar wind.", "The authors thank Y. Lin, R. Sydora, G. Morales, and J. Maggs for insightful discussions, S. Vincena, P. Pribyl, S. K. P. Tripathi, and B.", "Van Compernolle, for insightful discussions and assistance with the experiment, and Z.", "Lucky, M. Drandell, and T. Ly for their excellent technical support.", "S. D. was supported by a NASA Jack Eddy Fellowship.", "This work was performed at the UCLA Basic Plasma Science Facility which is supported by DOE and NSF." ] ]	1606.05055
[ [ "Learning from Non-Stationary Stream Data in Multiobjective Evolutionary\n Algorithm" ], [ "Abstract Evolutionary algorithms (EAs) have been well acknowledged as a promising paradigm for solving optimisation problems with multiple conflicting objectives in the sense that they are able to locate a set of diverse approximations of Pareto optimal solutions in a single run.", "EAs drive the search for approximated solutions through maintaining a diverse population of solutions and by recombining promising solutions selected from the population.", "Combining machine learning techniques has shown great potentials since the intrinsic structure of the Pareto optimal solutions of an multiobjective optimisation problem can be learned and used to guide for effective recombination.", "However, existing multiobjective EAs (MOEAs) based on structure learning spend too much computational resources on learning.", "To address this problem, we propose to use an online learning scheme.", "Based on the fact that offsprings along evolution are streamy, dependent and non-stationary (which implies that the intrinsic structure, if any, is temporal and scale-variant), an online agglomerative clustering algorithm is applied to adaptively discover the intrinsic structure of the Pareto optimal solution set; and to guide effective offspring recombination.", "Experimental results have shown significant improvement over five state-of-the-art MOEAs on a set of well-known benchmark problems with complicated Pareto sets and complex Pareto fronts." ], [ "Introduction", "In practice, a decision maker often requires to consider optimising multiple conflicting objectives.", "This type of optimisation problems are usually referred to as multiobjective optimisation problems (MOPs).", "Since the objectives of the problems usually conflict with each other, there does not exist a unique solution that can optimise all the objectives simultaneously.", "Therefore, a set of Pareto optimal solutions, named as Pareto set (PS), exists for an MOP [1].", "A solution is considered to be `Pareto optimal' if it is impossible to make any one objective better off without making at least another one worse off.", "Finding the PS often challenges greatly on computational capacity and algorithm intelligence [2].", "In the last three decades, extensive research on evolutionary algorithms (EAs) have shown that the EA paradigm is very powerful in handling MOPs, in the sense that a set of solutions that approximates to the PS, named as approximated set, can be obtained in a single run without requiring much computational effort [3][4].", "EAs simulate the genetic evolution of a population of individuals to best fit their living environment [5].", "To design an effective EA, effective recombination for fit offspring generation is a key.", "Research has shown that a problem's domain knowledge, if any, can greatly improve the search efficiency if the knowledge is properly collected or learned during the search process [6].", "For an $m$ -objective optimisation problem, it has been proved that the distribution of the PS exhibits an ($m-1$ )-dimensional manifold structure under mild conditions [7].", "This property is often referred to as the regularity property.", "From the point view of EA design, an effective EA is expected if the manifold structure can be discovered and applied for offspring generation.", "Some EAs have been developed to combine machine learning techniques for the discovery of the intrinsic manifold structure to aid the search for the PS.", "For examples, in regularity model-based estimation of distribution algorithm (RM-MEDA) [6], the local principal component analysis (local PCA) approach is applied at each generation.", "It uses the learned principal components to approximate the manifold structure.", "Some EAs adopted other machine learning techniques to approximate the manifold structure [8].", "All these algorithms apply the machine learning techniques at every generation.", "These learning algorithms often need to visit all data several times (iterations) until converge.", "Thus, a considerable amount of computational resources is consumed on learning.", "To reduce the computational overhead, the multiobjective EA (MOEA) proposed by Zhang et al.", "[4] couples the population evolution and the model inference.", "In their MOEA, only one iteration of the learning algorithm is applied at each generation.", "This scheme provides an important development on saving computational resources.", "The evolution procedure can also be seen as a learning procedure; intrinsic PS's structure of an MOP is expected to be learned dynamically from the changing candidate solutions.", "However, there is a fundamental issue in this scheme.", "As well known, one of the main assumptions in machine learning is that sample observations are assumed to be effectively i.i.d.", "(independent and identically distributed) for the purposes of statistical inference.", "But, under the scheme in [4], along the evolution procedure, the assumption is largely violated.", "First, solutions at adjacent generations have rather different qualities in terms of their respective objectives, which indicate that they might not be sampled from the same underlying distribution (i.e.", "these solutions are not identically distributed).", "Second, the generation of new solutions at present generation depends on collective information from previous generation, which indicates solutions at adjacent generations are not independent.", "Look deeply into the data (i.e.", "offsprings created during the evolution search) we try to learn from, some special characteristics can be observed: 1) the structure to be discovered along evolution is temporal and changing dynamically.", "In other words, these data are produced by a non-stationary processA process is stationary if and only if the joint distribution of the data at different time are the same.", "Specifically, if we let $t = 1,\\cdots $ be the generations of the evolution, and $\\mathbf {y}_t$ be a $n$ -dimensional solution.", "The sequence $\\mathbf {y}_t$ is a stationary stochastic process if the joint probabilistic distribution of $(\\mathbf {y}_{t_1+h}, \\cdots , \\mathbf {y}_{t_N+h})$ and $(\\mathbf {y}_{t_1}, \\cdots , \\mathbf {y}_{t_N})$ are the same for all $h = 0, 1, \\cdots , $ and an arbitrary selection of $t_1, \\cdots , t_N$ .", "This is obviously not the case for the stream of offsprings created during the evolution process.", "; 2) the structure determined by the data is scale-variant.", "On a short time scale, the structure is pseudo-stationary, while on a long time scale, the structure has a sequential and converging property.", "That is, along the evolution process, the underlying structure is similar between adjacent generations, while the structure will finally be converging to the PS's manifold structure of the considered optimisation problem.", "In this paper, we present the first-ever MOEA based on an online machine learningIn computer science, online machine learning methods learn patterns from data which are available in a sequential order as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once.", "from a stream of non-stationary data.", "In our algorithm, a modified algorithm to the online agglomerative clustering algorithm presented in [9] is developed to learn the PS's structure addressing the above mentioned characteristics.", "Obvious advantages of the proposed online clustering based evolutionary algorithm (OCEA) include 1) a perfect match between the search dynamics and the non-stationary structure learning and 2) a significantly reduced computational cost on learning (data need to be visited only once in the context of online learning).", "To successfully implement the proposed algorithm, we need to address three main issues.", "First, how to modify the online agglomerative clustering in accordance with the evolution process to discover the underlying structure?", "Second, how to properly use the learned structure to create offsprings effectively?", "Finally, how to select the fittest individuals to drive the search towards the PS?", "These issues will be discussed in the following sections.", "The rest of the paper is organised as follows.", "The background and previous work on multiobjective evolutionary algorithms is introduced in Section .", "Section presents the proposed algorithm in detail.", "Experimental studies are shown in Section and .", "The analysis of parameters effect to algorithmic performance is discussed in Section .", "Section concludes the paper." ], [ "Background and Previous Work", "A box-constrained continuous MOP can be stated as follows: $\\begin{array}{ll}\\min & \\mathbf {F}(\\mathbf {x})=(f_1(\\mathbf {x}),\\cdots ,f_m(\\mathbf {x}))^\\intercal \\\\\\mathrm {s.t. }", "& \\mathbf {x}=(x_1,\\cdots ,x_n)^\\intercal \\in \\Omega \\end{array}$ where $\\Omega =\\prod _{i=1}^n[a_i, b_i] \\subseteq \\mathbb {R}^n$ defines the decision (search) space; $a_i$ and $b_i$ are the lower and upper boundaries of variable $x_i$ , respectively; $\\mathbf {x}=(x_1, \\cdots , x_n)^\\intercal $ is a vector of decision variable; $\\mathbf {F}: \\Omega \\rightarrow \\mathbb {R}^{m}$ represents the mapping from search space to objective space where $m$ objective functions $f_i(\\mathbf {x}), i=1,\\ldots ,m$ are to be considered.", "Suppose that $\\mathbf {u}=(u_1,\\cdots ,u_m)^\\intercal , \\mathbf {v}=(v_1,\\cdots ,v_m)^\\intercal \\in \\mathbb {R}^m$ are two vectors.", "If $u_i \\le v_i$ for all $i\\in \\lbrace 1,\\cdots , m\\rbrace $ , but there exists at least one index $j$ , such that $u_j < v_j$ , then $\\mathbf {u}$ is said to dominate $\\mathbf {v}$The definition of domination is for minimization.", "“Dominate\" means “be better than\"., denoted by $\\mathbf {u}\\prec \\mathbf {v}$ .", "A solution $\\mathbf {x}^\\in \\Omega $ is called (globally) Pareto optimal if there is no $\\mathbf {x}\\in \\Omega $ such that $\\mathbf {F}(\\mathbf {x}) \\prec \\mathbf {F}(\\mathbf {x^})$ .", "The set of all Pareto optimal solutions, denoted by PS, are named as Pareto set.", "The set of the objective vectors of the Pareto optimal solutions is called Pareto front, denoted by PF.", "The goal of an MOEA for an MOP is to find a set of approximated solutions whose objective vectors (the objective vectors constitute an approximated front) are as close to the PF as possible (i.e.", "the convergence requirement), and distribute along the PF as widely and evenly as possible (i.e.", "the diversity requirement).", "Great efforts have been made to deal with MOPs in the evolutionary computation community [3].", "These developed approaches focus either on establishing a mechanism to balance convergence and diversity, or on developing effective recombination.", "MOEAs concerning the balance between convergence and diversity basically fall into three categories.", "In the first category, the Pareto dominance relationship is applied for promising solution selection.", "The nondominated sorting developed by Deb et al.", "[10] is the most known method.", "Its primary use is to drive the search towards the PF which favours convergence.", "It needs to incorporate other strategies, such as crowding distance [10] and K-nearest neighbor method [11], to preserve the population diversity.", "It has been found out that dominance-based sorting method is not able to provide enough comparability for many-objective ($\\ge 4$ objectives) optimization problems.", "Typical dominance-based MOEAs include NSGA-II [10], SPEA2 [11], PESA-II [12], NSGA-III [13], and others.", "In the second category, MOEAs based on performance metrics, such as hypervolume (HV), R2 and $\\Delta _p$ , were developed.", "The performance metrics embed the convergence and diversity requirements together so that they can be employed to directly guide the selection of solutions for a good balance of convergence and diversity.", "Representative MOEAs include SMS-EMOA [14], HyPE [15], R2-IBEA [16] and DDE [17].", "The computation of the performance metrics becomes much more difficult and time-consuming in dealing with many-objective optimisation problems.", "The third category is the decomposition-based MOEAs.", "In this category, a number of reference vectors in the objective space are used to decompose the problem into a set of single objective subproblems [18], or several simple multiobjective subproblems [19].", "The convergence is controlled by the objective values of the subproblems; while the diversity is managed by computing the distances of the solutions to the reference vectors.", "Representative decomposition-based MOEAs include MOEA/D [20], MOEA/D-DE [18], MOEA/D-STM [21], MOEA/D-M2M [19] and others.", "Regarding MOEAs focusing on effective recombination, they are almost all designed based on the regularity property of MOPs.", "The underlying assumption is that the manifold structure could be used to greatly improve the search efficiency since high-quality offsprings can be generated if the regularity structure is properly modelled and learned.", "The first work on applying the regularity property in designing MOEA, i.e., aforementioned RM-MEDA, was proposed in 2008 [6], where the manifold structure is approximated by the first $(m-1)$ principal components.", "This work was improved later by using help from the modelling on the PF [22].", "Various regularity based MOEAs have been developed since then, such as a reducing redundant cluster based RM-MEDA [8], a RM-MEDA with local learning strategy [23], evolutionary multiobjective optimisation via manifold learning [24], and others.", "Moreover, in [4], a self-organising map method is incorporated within the evolution procedure to search for the manifold PS structure." ], [ "The Algorithm", "As discussed previously, existing regularity based MOEAs usually spend a high computational cost on learning.", "To reduce the consumption of computational resources, we propose to adopt an online machine learning scheme.", "Offsprings are considered as a stream of data since they come in order along the evolution process, and can only be accessed once or a small number of generations.", "Moreover, it is observed that along the evolution process, the stream of solutions is dependent, and non-stationary.", "Therefore, the application of online learning algorithm is able to reduce the number of visits and account for the non-stationary nature.", "This can significantly reduce the computational resources.", "Note that a finite mixture of Gaussian clusters can be used to well approximate the distribution of a set of data points statistically.It is well acknowledged that mixtures of Gaussian distributions are dense in the set of probability distributions with respect to weak topology [25].", "This motives us to approximate the manifold structure by using an online clustering algorithm.", "The cluster statistics, including the number of clusters, cluster mean and variance-covariance, will evolve over time.", "To model this non-stationary process, we propose to modify an online agglomerative clustering algorithm called AddC [9] and use it to dynamically estimate the cluster statistics.", "In the following, we first describe the online agglomerative clustering algorithm developed in [9] and discuss how it should be modified to adapt to the evolution process of MOEAs.", "The other details of the developed algorithm are then presented." ], [ "Online Agglomerative Clustering", "AddC, presented by Guedalia et al.", "[9] in 1998, is developed for clustering a stream of non-stationary data.", "AddC's clustering procedures are shown in Alg.", "REF .", "From line REF to REF , an arriving new data point $\\mathbf {y}$ is assigned to the cluster that is closest to it at first.", "This step attempts to minimise the within cluster variance.", "Afterwards, from line REF to REF , if there are less than $K_{\\max }$ clusters, $\\mathbf {y}$ is employed as a centroid to create a new cluster; otherwise, from line REF to REF , two redundant clusters which are closest to each other are merged, and $\\mathbf {y}$ is also treated as a centroid to create a new cluster for replacing the redundant cluster (i.e.", "$\\mathcal {C}^\\delta $ in line REF ).", "The merging operation is aimed to maximise the distances between the centroids and to remove redundant clusters.", "The creation of new clusters is to consider the temporal changes in the distribution of the data.", "In line REF , if there still exist data points to be clustered, the clustering operations are repeated.", "Otherwise, a post process is conducted to remove clusters with a negligible number ($\\epsilon $ ) of data in line REF .", "The post process is to eliminate outliers if any.", "[htbp] Online Agglomerative Clustering AddC [1] an arriving new data point $\\mathbf {y}$ , centroids $\\mathbf {z}^k$ and counters $c^k$ of $m$ existing clusters $\\mathcal {C}^1, \\cdots , \\mathcal {C}^m$ , $1\\le k \\le m$ , and the maximum number of clusters allowed $K_{\\max }$ .", "a new set of clusters.", "The centroid which is closest to the data point $\\mathbf {y}$ is defined as the winner, $j = \\arg \\min \\limits _{1\\le k \\le m} \|\|\\mathbf {y} - \\mathbf {z}^k\|\|.$ Update the closest centroid and its counter, $\\mathbf {z}^j = \\mathbf {z}^j +\\frac{\\mathbf {y} - \\mathbf {z}^j}{c^j};~c^j = c^j + 1,$ where $c^j$ is the number of data points in $\\mathcal {C}^j$ .", "If $m < K_{\\max }$ , set $m = m + 1$ and $\\delta = m$ .", "Goto step REF .", "Find a pair of closest (redundant) centroids, $(\\gamma , \\delta ) = \\arg \\min \\limits _{\\gamma , \\delta , \\gamma \\ne \\delta } \|\|\\mathbf {z}^\\gamma -\\mathbf {z}^\\delta \|\|.$ Merge redundant clusters and update the cluster statistics, $\\mathbf {z}^\\gamma = \\frac{\\mathbf {z}^\\gamma c^\\gamma + \\mathbf {z}^\\delta c^\\delta }{c^\\gamma + c^\\delta };~c^\\gamma = c^\\gamma + c^\\delta .$ Initialise a new cluster $\\mathcal {C}^\\delta $ , $\\mathbf {z}^\\delta = \\mathbf {y}$ and $c^\\delta = 0$ .", "If there still exist data points to be clustered, take a new point $\\mathbf {y}$ and goto Step REF .", "Post process: $\\forall k$ , if $c^k < \\epsilon $ , perform steps REF and REF ." ], [ "Algorithmic Framework", "The framework of OCEA is presented in Alg.", "REF .", "In line REF to REF , an initial population $\\cal P$ is yielded, an external archive $\\mathcal {A}$ is initialised to be the same as $\\cal P$ .", "In the first generation, each solution is considered as a cluster where itself is initialised to be the centroid $\\mathbf {z}^i = \\mathbf {x}^i$ and counter $c^i = 1$ , $i=1,\\cdots ,N$ .", "Afterwards, at each generation, an offspring $\\mathbf {y}^i$ is generated around each solution $\\mathbf {x}^i$ (lines REF to REF ).", "To generate $\\mathbf {y}^i$ , a mating control parameter $\\beta \\in [0,1]$ is applied to balance exploration and exploitation.", "With $\\beta $ , the solution generation will be in favour of exploitation.", "That is, the reference (or parent) solutions are chosen from the cluster that $\\mathbf {x}^i$ locates.", "With $1-\\beta $ , the reference individuals are chosen from the global mating pool specified in line REF .", "This is to favour exploration.", "After recombination, the generated offspring $\\mathbf {y}^i$ is then used to update external archive and current population by environmental selection, and the clustering information (lines REF and REF ).", "The solution generation and the updating procedures for population and clusters will be described in the following subsections.", "[htbp] OCEA framework [1] population size $N$ , maximum evolutionary generations $T$ , mating control parameter $\\beta $ .", "population $\\cal P$ .", "Intialization $\\mathcal {P} = \\lbrace {\\mathbf {x}^1}, \\cdots ,{\\mathbf {x}^N}\\rbrace $ and an external archive $\\mathcal {A} = \\mathcal {P}$ .", "Take each $\\mathbf {x}^i \\in \\mathcal {P}$ as a cluster ${\\cal C}^i$ with centroid $\\mathbf {z}^i = \\mathbf {x}^i$ and counter $c^i = 1$ .", "$t\\leftarrow 1$ to $T$ Set $m = $ #clusters.", "Construct a global mating pool $\\mathcal {M}$ by randomly choosing a solution from a $\\mathcal {C}^i, 1\\le i \\le m$ .", "$i \\leftarrow 1$ to $N$ Construct a mating pool $\\mathcal {Q}^i$ for each $\\mathbf {x}^i$ as follows: $\\mathcal {Q}^i = \\left\\lbrace {\\begin{array}{ll}{\\mathcal {C}^{k_i}}&{\\mbox{if}{\\hspace{1.0pt}} {\\hspace{1.0pt}} rand() < \\beta }\\\\{\\mathcal {M}}&{\\mbox{otherwise}}\\nonumber \\end{array}} \\right.,$ where $\\mathcal {C}^{k_i}$ represents that $\\mathbf {x}^i$ loactes in $\\mathcal {C}^k$ , $rand()$ is a random number generator in $[0,1]$ .", "Generate ${\\mathbf {y}^i}$ = SolGen $({\\mathcal {Q}^i},{\\mathbf {x}^i})$ .", "Update and clustering $[\\mathcal {A}, \\mathcal {C}]$ = Esoc $(\\mathcal {A}, \\mathbf {y}^i, \\mathcal {C})$ .", "Set $\\mathcal {P}=\\mathcal {A}$ and pass the clustering results of $\\mathcal {A}$ to $\\mathcal {P}$ ." ], [ "New Solution Generation", "In this paper, the differential evolution (DE) and polynomial mutation (PM) operators are adopted to generate offsprings as presented in Alg.", "REF .", "The recombination operator takes the current solution $\\mathbf {x}$ and its mating pool $\\mathcal {Q}$ as input and outputs an offspring $\\mathbf {y}$ .", "DE [26] is firstly used to generate a trial solution (line REF ), a repair mechanism is employed to correct any component that is outside the search boundary of that component (line REF ).", "After repair, the PM [2] operator is applied to generate a new solution (line REF ).", "The new solution is repaired again if necessary and the final solution is returned (line REF ).", "In Alg.", "REF , $F$ and $CR$ are the two control parameters for the DE operator, $p_m$ and $\\eta _m$ are the parameters for the PM operator.", "If $CR=1$ , the DE operator in Alg.", "REF is rotation invariant, which is of advantage to deal with complicated PS [18].", "Therefore DE is selected to generate new offsprings in OCEA.", "Obviously, the use of other recombination operators is not limited; e.g.", "we could use the recombination operators in [27].", "[htbp] Solution generation (SolGen) operator [1] a current solution $\\mathbf {x}$ and its mating pool $\\cal Q$ a trial solution $\\mathbf {y}$ Choose randomly two distinct parent individuals $\\mathbf {x}^1$ and $\\mathbf {x}^2$ from $\\mathcal {Q}$ Generate $\\mathbf {y}^{^{\\prime }}=(y^{^{\\prime }}_1,\\cdots ,y^{^{\\prime }}_n)^\\intercal $ as follows: $y^{^{\\prime }}_i = \\left\\lbrace \\begin{array}{ll}x_i+F \\times (x_i^{1}-x_i^{2})&\\text{if}~rand()\\le CR\\\\x_i&\\text{otherwise}\\\\\\end{array}\\right..$ Repair $\\mathbf {y}^{^{\\prime }}$ , $y^{^{\\prime \\prime }}_i = \\left\\lbrace \\begin{array}{ll}a_i & \\text{if}~y^{^{\\prime }}_i < a_i\\\\b_i & \\text{if}~y^{^{\\prime }}_i > b_i\\\\y^{^{\\prime }}_i & \\text{otherwise}\\end{array}\\right.,$ where $\\mathbf {x}_i \\in [a_i, b_i]$ .", "Mutate $\\mathbf {y}^{^{\\prime \\prime }}$ , $y_i = \\left\\lbrace \\begin{array}{ll}y^{^{\\prime \\prime }}_i+\\delta _i \\times (b_i-a_i) & \\text{if}~rand()< p_m\\\\y^{^{\\prime \\prime }}_i & \\text{otherwise}\\end{array}\\right.,$ where $r=rand()$ if a uniform random generator in [0,1], and $\\delta _i = \\left\\lbrace \\begin{array}{ll}\\left[2r+(1-2r)(\\frac{b_i-y_i^{^{\\prime \\prime }}}{b_i-a_i})^{\\eta _m+1}\\right]^{\\frac{1}{\\eta _m+1}}-1 & \\text{if}~r<0.5,\\\\1-\\left[2-2r+(2r-1)(\\frac{y_i^{^{\\prime \\prime }}-a_i}{b_i-a_i})^{\\eta _m+1}\\right]^{\\frac{1}{\\eta _m+1}} & \\text{otherwise}\\end{array}\\right.$ If necessary, repair $\\mathbf {y}^{^{\\prime \\prime }}\\rightarrow \\mathbf {y}$" ], [ "Updating on Population and Clusters", "In Alg.", "REF line REF , function $\\textsc {Esco}$ is applied to carry out environmental selection and clustering updating.", "OCEA adopts the environmental selection method proposed in SMS-EMOA [14] which is based on the hypervolume metric.", "The hypervolume metric is the only known unitary metric that is Pareto compliant [28].", "It has shown better performance over decomposition-based and Pareto dominance-based environmental selection approaches [15].", "Regarding cluster updating, we modify the online agglomerative clustering algorithm AddC (Alg.", "REF ) so that it can be fitted into the evolutionary search mechanism.", "The modified AddC is fused in OCEA to update/refine the clusters to adaptively learn the PS's structure.", "Alg.", "REF presents the details of Esoc.", "For each new solution $\\mathbf {y}$ , $\\mathcal {A}$ is updated by the hypervolume metric based environmental selection.", "Specifically, the fast non-dominanted sorting approach proposed in NSGA-II [10] is applied to partition the external archive ${\\mathcal {A}} \\cup \\lbrace \\mathbf {y}\\rbrace $ into $L$ non-dominanted fronts $\\lbrace \\mathcal {B}^1,\\cdots ,\\mathcal {B}^L\\rbrace $ , where $\\mathcal {B}^1$ is the best front and $\\mathcal {B}^L$ is the worst one (line REF ).", "$L>1$ which indicates that there are more than one front in ${\\mathcal {A}} \\cup \\lbrace \\mathbf {y}\\rbrace $ .", "If it is the case, the solution $\\mathbf {x}^$ in $\\mathcal {B}^L$ with the largest $d(\\mathbf {x},{\\mathcal {A}} \\cup \\lbrace \\mathbf {y}\\rbrace )$ value is removed, where $d(\\mathbf {x},\\mathcal {A} \\cup \\lbrace \\mathbf {y}\\rbrace )$ denotes the number of solutions in $\\mathcal {A} \\cup \\lbrace \\mathbf {y}\\rbrace $ that dominates $\\mathbf {x}$ .", "Otherwise, if $L=1$ , the solution $\\mathbf {x}^$ that least contributes to the hypervolume, i.e.", "$\\Delta _{\\varphi }(\\mathbf {x},\\mathcal {B}^1)$ (line REF to REF , and REF ), is excluded.", "The calculation of $\\Delta _{\\varphi }$ can be found in [14].", "[ht] The updating procedure (Esoc).", "[1] a new solution $\\mathbf {y}$ , external archive $\\mathcal {A}$ , centroids $\\mathbf {z}^k$ and counters $c^k$ of current existing clusters $\\mathcal {C}^1, \\cdots , \\mathcal {C}^m$ , $1\\le k \\le m$ , and the maximum number of clusters allowed $K_{\\max }$ .", "External archive ${\\mathcal {A}}$ and its cluster information.", "Apply the fast non-dominanted sorting approach on ${\\mathcal {A}} \\cup \\lbrace \\mathbf {y}\\rbrace $ to obtain $L$ fronts $\\lbrace \\mathcal {B}^1,\\cdots ,\\mathcal {B}^L\\rbrace $ .", "$L>1$ Determine the worst solution, $\\mathbf {x}^* = \\arg \\max \\limits _{\\mathbf {x}\\in \\mathcal {B}^L}d(\\mathbf {x},{\\mathcal {A}} \\cup \\lbrace \\mathbf {y}\\rbrace ).$ Determine the worst solution, $\\mathbf {x}^* = \\arg \\min \\limits _{\\mathbf {x}\\in {\\mathcal {A}}\\cup \\lbrace \\mathbf {y} \\rbrace }\\Delta _{\\varphi }(\\mathbf {x},\\mathcal {B}^1).$ $\\mathbf {x}^\\ne \\mathbf {y}$ If $\\mathbf {x}^ \\in \\mathcal {C}^k, k\\in \\lbrace 1,\\cdots ,m\\rbrace $ , then remove $\\mathbf {x}^$ from $\\mathcal {C}^k$ : $\\mathcal {C}^k=\\mathcal {C}^k\\backslash \\lbrace \\mathbf {x}^\\rbrace $ .", "$\\mathcal {C}^k=\\emptyset $ Remove cluster $\\mathcal {C}^k$ , set $m=m-1$ .", "Update cluster $\\mathcal {C}^k$ : $c^k=c^k-1, \\mathbf {z}^k = \\mathbf {z}^k - \\frac{\\mathbf {x}^-\\mathbf {z}^k}{c^k}.$ Delete the worst solution ${\\mathcal {A}}= {\\mathcal {A}}\\cup \\lbrace \\mathbf {y}\\rbrace \\backslash \\lbrace \\mathbf {x}^\\rbrace $ .", "Set $m=m+1$ , construct a new cluster $\\mathcal {C}^m$ , set $c^m=1$ , $\\mathbf {z}^m=\\mathbf {y}$ .", "$m>K_{\\max }$ Find two closest clusters, $(\\gamma , \\delta ) = \\arg \\min \\limits _{\\gamma , \\delta , \\gamma \\ne \\delta } \|\|\\mathbf {z}^\\gamma -\\mathbf {z}^\\delta \\Vert .$ Merge the two clusters, $\\mathbf {z}^\\gamma = \\frac{\\mathbf {z}^\\gamma c^\\gamma + \\mathbf {z}^\\delta c^\\delta }{c^\\gamma + c^\\delta },~c^\\gamma = c^\\gamma + c^\\delta .$ Delete the worst solution in ${\\mathcal {A}}= {\\mathcal {A}}\\cup \\lbrace \\mathbf {y}\\rbrace \\backslash \\lbrace \\mathbf {x}^\\rbrace $ .", "If $\\mathbf {y}$ is kept in $\\mathcal {A}$ after environmental selection, i.e., $\\mathbf {x}^\\ne \\mathbf {y}$ , the online clustering operation is invoked.", "First, $\\mathbf {x}^$ is removed from its cluster ${\\cal C}^$ , and its cluster's centroid and counter are updated following equations in line REF .", "It differs from AddC where no data points are to be removed during the online clustering process.", "Then $\\mathbf {y}$ is taken as a new centroid to construct a new cluster (line REF ).", "If there are more than $K_{\\max }$ clusters in $\\mathcal {A}$ , two clusters that are closest to each other are emerged (lines REF to REF ) to complete the clustering operation." ], [ "Notes on OCEA", "It is necessary to emphasize that: The evolution procedure of OCEA is also an online clustering procedure working on a stream of offsprings which are created and updated during the evolution process.", "We would expect that the clustering structure is to be gradually emerged during evolution and finally gets well shaped at termination.", "Different from the original AddC (Alg.", "REF ), (a) the clustering procedure in OCEA starts from the $N$ initial clusters composed of the $N$ solutions in the initial population (line REF in Alg.", "REF ); (b) During the evolution, some solutions are dominated and need to be removed.", "An extra operation is added to account for the removal of solutions, including the updating of cluster statistics and the discarding of any empty cluster (lines REF to REF in Alg.", "REF ); (c) In our online clustering procedure, $\\mathbf {y}$ is not assigned to its closest cluster as opposed to Alg.", "REF where a new data need to be assigned to its closest cluster (line REF in Alg.", "REF ).", "OCEA incorporates the online clustering tightly within the evolution search.", "The online clustering discovers adaptively the PS structure along with the evolution.", "New solutions are created taking the cluster information into account at each generation.", "As a result, it can be seen that the online clustering closely adapts to the search procedure; and accounts for the non-stationary of the evolution dynamics.", "Different from existing regularity model-based MOEAs in which the learning at each generation has a time complexity linearly to the number of training iterations.", "The number of generations should be large enough to make sure the convergence of the learning algorithm.", "On the contrary, in our scheme, each solution is visited only once.", "This can significantly reduce the computational burden.", "In our scheme, we do not require a post-process which is different from the original AddC algorithm." ], [ "Experimental Study", "To investigate the performance of OCEA, it is compared with two decomposition-based MOEAs (MOEA/D-DE [18] and TMOEA/D [29]), one regularity model based MOEA (RM-MEDA [6]), one popular performance metric based MOEA (SMS-EMOA [14]), and one typical Pareto dominance based MOEA (NSGA-II [10]).", "Among these algorithms, MOEA/D-DE decomposes the MOP into a set of single-objective problems with uniformly distributed weights.", "It might be not able to obtain approximated fronts with good diversity for MOPs with complex PFs.", "TMOEA/D transforms the objective functions into those that are easy to be addressed by MOEA/D.", "This is to make MOEA/D perform well on MOPs with complex PFs.", "RM-MEDA is developed based on the regularity property.", "It learns some local principle components at each generation, and uses the principle components to approximate the manifold structure.", "SMS-MOEA uses the hypervolume metric as the selection criterion.", "NSGA-II, on the other hand, uses the Pareto dominance relationship among individuals and crowding distance to carry out environmental selection.", "These comparison algorithms cover all the main streams of MOEAs in the literatures." ], [ "Test Instances and Performance Metrics", "MOPs with complex PF and complicated PS structures are particularly focused in this paper.", "The GLT test suite from [4] are used in the comparison experiments.", "The test suite includes a variety of problems with various characteristics that challenge MOEAs greatly.", "Those characteristics include disconnected PF, convex PF, nonlinear variable linkage, etc.", "Two commonly-used performance metrics, inverted generational distance (IGD) [6] and hypervolume (HV) [30], are employed to measure the algorithm's performance.", "These two metrics can measure both the convergence and diversity of the final approximated fronts found by MOEAs.", "Lower IGD and larger HV metric values imply better performance of MOEAs.", "To calculate the HV metric value of an approximated front, a reference point which can be dominated by all the objective vectors in the final approximated front need to be set.", "The reference points chosen for the test instances are as follows: for GLT1, $\\mathbf {r}=(2,2)^\\intercal $ , for GLT2 $\\mathbf {r}=(2,11)^\\intercal $ , for GLT3 $\\mathbf {r}=(2,2)^\\intercal $ and for GLT4 $\\mathbf {r}=(2,3)^\\intercal $ , for GLT5-GLT6, $\\mathbf {r}=(2,2,2)^\\intercal $ ." ], [ "Experimental Settings", "It has been well acknowledged that for the GLT test instances, the DE and PM operators are more able to produce promising solutions than other operators [4].", "Therefore, to make a fair comparison, the recombination operators in NSGA-II and SMS-EMOA are replaced by the DE and PM operators used in this paper.", "Furthermore, all parameters in the experiments are adjusted through preliminary experiments for optimal performance on these test instances.", "All algorithms are implemented in Matlab and tested in the same computer.", "The parameter settings for these algorithms are as follows: Common parameters: population size: $N=100$ for bi-objective and 105 for tri-objective instances; search space dimension: $n=10$ for GLT1-GLT6; runs: each algorithms independently runs each test instance for 33 times; termination: maximum evolutionary generation $T=300$ .", "Parameters for OCEA: maximum number of clusters allowed: $K_{\\max }=7$ ; mating control parameter: $\\beta =0.6$ ; DE control parameters: $F=0.6,~CR=1$ ; PM control parameters: $p_m=1/n,~\\eta _m=20$ .", "Parameters for MOEA/D-DE: neighbourhood size: $NS=5$ ; mating control parameter: $\\beta =0.7$ ; maximum number of solutions to be replaced by an offspring: 2; DE control parameters: $F=0.9,~CR=0.6$ ; PM control parameters: $p_m=1/n,~\\eta _m=20$ .", "Parameters for TMOEA/D: neighbourhood size: $NS=30$ ; generations for the first stage: $T1=T/10$ ; generations for the second stage: $T2=\\alpha T$ , $\\alpha = \\lbrace 0.01,~0.02,~\\cdots ,~0.1,~0.1,~0.1,~0.15\\rbrace $ ; DE control parameters: $F=0.5,~CR=1$ .", "Parameters for RM-MEDA: number of clusters in local PCA: 5; maximum iterations used in local PCA: 50; sampling extension ratio: 0.25.", "Parameters for NSGA-II and SMS-EMOA: DE control parameters: $F=0.5,~CR=1$ ; PM control parameters: $p_m=1/n,~\\eta _m=20$ .", "To get statistically sound conclusions in the experiments, each algorithm independently runs 33 times for each instance, and the comparisons are performed based on the statistics of the performance metric values, i.e., mean and standard deviation values.", "In the comparison table, the mean IGD and HV metric values for each instance are sorted in an ascending and descending order, respectively, and the ranks are given in the square brackets of the table.", "The best mean metric values are highlighted in bold face with gray background.", "The Wilcoxon's rank sum test at a 5% significance level is also performed to test the significance of differences between the mean metric values of each instance obtained by each pair of algorithms.", "In the tables, “$\\dag $ \", “$§$ \", and “$\\approx $ \" are used to denote that the mean metric values obtained by OCEA is better than, worse than, or similar to those achieved by the comparison algorithm, respectively." ], [ "Comparison Study", "To study the statistical performance of OCEA, Table REF shows the statistics of IGD and HV metric values obtained by MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA on the GLT test suite averaged over 33 independent runs.", "In general, OCEA obtains 8 out of 12 best mean metric values, while the rest algorithms only obtain 4.", "According to the mean ranks, the algorithms' performance ranked from the best to the worst are OCEA, RM-MEDA, TMOEA/D, SMS-EMOA, NSGA-II and MOEA/D-DE.", "Specifically, according to the Wilcoxon's rank sum test, in the 12 comparisons with each of MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II and SMS-EMOA, OCEA achieves 12, 11, 11, 12, 11 better, 0, 1, 1, 0, 0 worse, and 0, 0, 0, 0, 1 similar mean metric values, respectively.", "Table REF denotes that OCEA performs the best overall on the GLT test suite.", "Table: Statistics (mean(std.", "dev.", ")[rank]) of IGD and HV metric values of final approximated fronts obtained by MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA algorithms over 33 independent runs on the GLT test suiteTo observe the search efficiency of OCEA, Fig.", "REF shows the evolution of the statistics of the IGD metric values obtained by the six algorithms on GLT1-GLT6.", "From the figure, it can be seen that for GLT1 and GLT3-GLT6, OCEA reaches the fastest to the lowest mean IGD metric values.", "For GLT2, OCEA has the slower, similar and faster speed in comparison with RM-MEDA, TMOEA/D and the other algorithms, respectively.", "Moreover, when dealing with GLT2, OCEA actually performs better than RM-MEDA at the early stage compared with RM-MEDA.", "From the evolution of the standard deviations of the metrics, it also can be observed that within 300 generations, OCEA has achieved robust performance on all the instances except for GLT3.", "Fig.", "REF indicates that OCEA approaches the fastest to the PFs and maintains the most diverse populations among the comparison algorithms on average.", "Figure: Evolution of the statistics of IGD metric values obtained by MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA on GLT1-GLT6To reveal the search processes, Fig.", "REF plots the evolution of the approximated fronts obtained by RM-MEDA, NSGA-II, MOEA/D-DE and OCEA on GLT4.", "It is noted that the evolution of the approximated front obtained by each algorithm plotted in the figure is representative.", "The representative evolution of an algorithm here indicates the final approximated front yielded by the evolution is with the median IGD metric value in 33 independent runs.", "It can be seen from the figure that, at the 100th generation, the approximated front yielded by OCEA has reached the PF completely, and almost covered the whole PF.", "After 300 generations, it has reached the approximated front with excellent convergence and diversity.", "On the other hand, after 300 generations, the final approximated fronts obtained by RM-MEDA, NSGA-II, MOEA/D-DE still cannot cover the whole PF, are not distributed unevenly.", "Fig.", "REF shows that OCEA can indeed greatly improve the search efficiency.", "Figure: OCEAFigure: representative frontsTo further investigate the effect of OCEA, Fig.", "REF plots the final approximated fronts obtained by RM-MEDA and OCEA on GLT1-GLT6.", "All the final approximated fronts of each instance obtained by RM-MEDA and OCEA, are plotted in Fig.", "REF and REF .", "The final approximated front of each instance with median IGD metric value (called representative front) obtained by RM-MEDA and OCEA, respectively, over 33 independent runs are plotted in Fig.", "REF and REF .", "From Fig.", "REF and REF , it can be seen that through 33 independent runs, the final approximated fronts of each instance achieved by RM-MEDA and OCEA, respectively, both can cover the whole PF of that instance.", "However, compared with RM-MEDA, OCEA performs more stably.", "From Fig.", "REF and REF , it is observed that the representative fronts of GLT5-GLT6 yielded by RM-MEDA do not reach the PFs.", "For GLT1-GLT4, although the representative fronts yielded by RM-MEDA all reach the PFs, the PFs are not completely covered.", "By contrast, the representative fonts obtained by OCEA for each instance all converge to the PFs and distributed well over them.", "Fig.", "REF implies that for the GLT test instances, OCEA is stable and robust in terms of convergence and diversity.", "In summary, we may conclude that OCEA has shown an excellent performance for dealing with MOPs with complicated PSs and complex PFs." ], [ "Performance on WFG test suite", "To deeply understand the performance of OCEA, OCEA is also applied to the WFG test suite [31] and compared with the five algorithms mentioned above.", "It is well known that the WFG test instances have complex PFs and are with various complicated characteristics, such as nonseparable, multimodal, degenerate, deceptive, etc.", "In this section, 9 bi-objective WFG test instances with 30 dimensional decision variables are taken as the test-bed.", "The maximum evolutionary generation is set as 450.", "Through preliminary optimisation over parameters, part of the parameter settings of these algorithms are listed in Table REF ; while the rest is the same as in Section REF .", "Again 33 independent runs of these algorithms are carried out on each test instance.", "Table REF shows the statistics of the IGD and HV metric values obtained by MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA on the WFG test instances over 33 independent runs.", "Table: Parameter settings for MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA on the WFG test suiteTable: Statistics (mean(std.", "dev.", ")[rank]) of the IGD and HV metric values of final approximated fronts obtained by MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II, SMS-EMOA and OCEA algorithms over 33 independent runs on the WFG test suite.Table REF shows that OCEA achieves 9 out of the 18 best mean metrics.", "The rest five algorithms obtain only 9.", "The performance of these algorithms ranked from the best to the worst is OCEA, MOEA/D-DE, SMS-EMOA, TMOEA/D, NSGA-II and RM-MEDA according to the mean ranks.", "The Wilcoxon's rank sum test suggests that OCEA performs better than MOEA/D-DE, TMOEA/D, RM-MEDA, NSGA-II and SMS-EMOA in 12, 12, 14, 15 and 11 out of the 18 mean metric values; performs worse in 5, 4, 3, 0 and 4; and similar in 1, 2, 1, 3 and 3.", "From Table REF , we may conclude that OCEA performs very well in solving the WFG test instances.", "It also indicates that OCEA is able to deal with MOPs with complex PFs and with complicated problem characteristics." ], [ "Parameter Sensitivity Analysis", "The sensitivity of OCEA to its parameters is analysed in this section.", "The GLT test suite is used for the analysis." ], [ "Maximum Number of Clusters", "To test how $K_{\\max }$ affects the performance of OCEA, $K_{\\max }=\\lbrace $ 4, 5, 7, 10, 20$\\rbrace $ are chosen to do analysis.", "The rest parameters are the same as those in Section REF .", "OCEA was run on each test instances independently 22 times with different $K_{\\max }$ values.", "Fig.", "REF shows the mean and standard deviation values of the IGD metric values obtained by OCEA.", "From Fig.", "REF , it can be seen that for GLT2, GLT5-GLT6, OCEA can always achieve similar performance robustly for different $K_{\\max }$ values.", "But for GLT1, GLT3-GLT4, different $K_{\\max }$ leads to relatively large performance differences.", "Especially, when $K_{\\max }$ is large, the performance of OCEA is not well enough.", "In general, a small $K_{\\max }$ can result in good search results by OCEA on the GLT test instances.", "This implies that OCEA is not very sensitive to the $K_{\\max }$ values on the GLT test instances.", "Therefore, $K_{\\max }=7$ is chosen in Section to carry out the comparison.", "It should be noted that the optimal $K_{\\max }$ depends on individual problem.", "Figure: CRCR" ], [ "Clustering Effectiveness Analysis", "The evolution procedure couples naturally with the online clustering procedure in OCEA.", "It is expected that the approximated set will present a clustering effect when the evolution procedure has converged.", "To justify the effectiveness of the online clustering, Fig.", "REF plots the clustering results in the first 3-dimensional search space on the GLT1-GLT6 test instances.", "In the figure, the solutions in each different cluster are marked with different colors and symbols.", "It can be seen that the final approximated sets are partitioned into 7 clusters clearly (note that $K_{\\max }$ is set as 7).", "This figure indicates that OCEA can indeed approximate the clustering structure effectively.", "Figure: The clusters of final approximated sets obtained by OCEA for GLT1-GLT6" ], [ "Mating Restriction Probability", "To test the sensitivity of the OCEA's performance to the mating control parameter $\\beta $ , $\\beta =\\lbrace $ 0.5, 0.6, 0.7, 0.8, 0.9$\\rbrace $ are used for the analysis.", "The rest parameters are the same as those in Section REF .", "Again, for different $\\beta $ value, OCEA independently run 22 time on the test instances.", "Fig.", "REF shows the statistics of the obtained IGD metric values.", "From Fig.", "REF , it is observable that for GLT5 and GLT6, different $\\beta $ values bring a similar performance for OCEA; but for GLT1-GLT4, OCEA with different $\\beta $ values performs very differently.", "Nevertheless, when $\\beta =0.6$ , OCEA has relatively better performance for all the instances.", "The observation in Fig.", "REF indicates that OCEA is not so sensitive to the setting of $\\beta $ in solving the GLT test instances.", "Therefore, $\\beta =0.6$ is chosen in Section for the controlled comparison experiments.", "Again, it is necessary to point out that an optimal $\\beta $ setting should depend on the problem characteristics." ], [ "Control Parameters of Differential Evolution Operator", "The effect of the DE parameters, i.e., $F$ and $CR$ , are to be evaluated in this section.", "$F~(CR)=\\lbrace $ 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$\\rbrace $ are chosen to proceed analysis.", "The rest parameters are the same as in Section REF .", "When different $F$ ($CR$ ) values are set, $CR$ ($F$ ) is set as 1 (0.6).", "The mean and standard deviation values of the IGD metric values obtained by OCEA with different $F$ or $CR$ over 22 independent runs are shown in Figs.", "REF and REF .", "Fig.", "REF shows that the $F$ value has a crucial effect on the OCEA performance for GLT1, GLT3-GLT4, and a large $F$ value can lead to a good OCEA performance.", "However, for GLT2, GLT5-GLT6, different $F$ settings do not affect the OCEA performance acutely.", "Fig.", "REF shows that the $CR$ value has a significant effect on OCEA for GLT1-GLT4, and a small $CR$ value is better.", "But OCEA performs rather stably for GLT5-GLT6 with different $CR$ values.", "In case $F=\\lbrace 0.6,~0.8\\rbrace $ ($CR=1$ ) and $CR=\\lbrace 0.9,~1\\rbrace $ ($F=0.6$ ), OCEA can always find good IGD metric values for all the GLT test instances.", "In general, Figs.", "REF and REF denote that OCEA is not very sensitive to the $F$ and $CR$ settings." ], [ "Conclusion", "This paper presented a first-ever MOEA that incorporate an online clustering to address the non-stationary nature of the evolutionary search.", "The underlying consideration is 1) to learn the manifold structure of the PS (i.e.", "the so-called regularity property of MOPs) through clustering; and 2) to adapt to the non-stationary search dynamics.", "The online agglomerative clustering approach developed in [9] is modified to accommodate the evolution search dynamics.", "Experimental study has shown that the online clustering can address the non-stationary search process well, and is able to adaptively learn the clustering structure of the PS.", "The comparison against five well-known MOEAs has also shown that the structures learned adaptively by the online clustering can indeed improve the search efficiency (in terms of search speed) and effectiveness (in terms of the quality of the final approximated sets and fronts).", "Future work includes 1) the development of intelligent recombination operators that can be well fitted in the online learning mechanism; 2) the development and/or incorporation of other online learning strategies; and 3) the study of the developed framework for many-objective optimisation problems." ] ]	1606.05169
[ [ "Non-convex balls in the Teichm\\\"uller metric" ], [ "Abstract We prove that the Teichm\\\"uller space of surfaces of genus $\\mathbf{g}$ with $\\mathbf{p}$ punctures contains balls which are not convex in the Teichm\\\"uller metric whenever $3\\mathbf{g}-3+\\mathbf{p} > 1$." ], [ "Introduction", "Let $\\overline{S}$ be a closed oriented surface and $P\\subset \\overline{S}$ a finite set.", "The Teichmüller space of $S=\\overline{S} \\setminus P$ is the set of conformal structures on $\\overline{S}$ up to biholomorphisms homotopic to the identity rel $P$ .", "The Teichmüller metric on this space $\\operatorname{\\mathcal {T}}(S)$ measures how much diffeomorphisms of $\\overline{S}$ homotopic to the identity rel $P$ must distort angles with respect to different conformal structures.", "This metric is complete, uniquely geodesic, Finsler, and agrees with the Kobayashi metric on $\\operatorname{\\mathcal {T}}(S)$ .", "However, its local geometry is quite subtle.", "Indeed, we prove that: Theorem 1.1 There exist non-convex balls in $\\operatorname{\\mathcal {T}}(S)$ whenever its complex dimension is greater than 1.", "Note that for any $X \\in \\operatorname{\\mathcal {T}}(S)$ , the balls of sufficiently small radius centered at $X$ are convex.", "This is true in any Finsler manifold [27] [25]." ], [ "Motivation", "If $S$ is a once-punctured torus or a four-times-punctured sphere, then $\\operatorname{\\mathcal {T}}(S)$ is isometric to $\\mathbb {H}^2$ , the hyperbolic plane with constant curvature $-4$ .", "This led Kravetz to argue that in general, $\\operatorname{\\mathcal {T}}(S)$ is non-positively curved in the sense of Busemann [12].", "However, Linch [13] found a flaw in Kravetz's reasoning and soon after, Masur [15] showed that the result was false: there exist distinct geodesic rays starting from the same point in $\\operatorname{\\mathcal {T}}(S)$ and staying a bounded distance apart whenever $\\dim _\\mathbb {C}\\operatorname{\\mathcal {T}}(S) > 1$ .", "In particular, Teichmüller space is not $\\operatorname{CAT}(0)$ nor $\\delta $ –hyperbolic.", "In any proper geodesic metric space $\\mathbb {X}$ , we have the implications $& \\mathbb {X} \\text{ is non-positively curved in the sense of Busemann} \\\\\\Rightarrow \\quad & \\text{the distance to any point is strictly convex along any} \\\\& \\text{geodesic not containing that point}\\\\\\Rightarrow \\quad & \\text{closed balls are strictly convex} \\\\\\Rightarrow \\quad & \\text{the convex hull of any finite set is compact.", "}$ The question of whether the third statement held for Teichmüller space was originally motivated by the Nielsen realization problem, which Kravetz thought he had solved with his erroneous result.", "Masur's paper [15] rendered the problem open again.", "If balls had been strictly convex, then a positive solution to the Nielsen realization problem would have followed immediately.", "In light of Theorem REF , this approach was doomed to fail.", "Thankfully, Kerckhoff solved Nielsen's problem many years ago using the convexity of hyperbolic length along earthquake paths [10].", "See also [28] for a solution using Weil–Petersson geometry.", "Whether the fourth statement holds for Teichmüller space is an open question of Masur.", "Given $X \\in \\operatorname{\\mathcal {T}}(S)$ and a simple closed curve $\\alpha \\subset S$ , the extremal length $\\operatorname{EL}(\\alpha ,X)$ is the smallest $c$ such that a Euclidean cylinder of height 1 and circumference $c$ embeds conformally in $X$ in the homotopy class of $\\alpha $ .", "Similarly, the extremal length of a multicurve is the least possible sum of circumferences of disjoint embedded cylinders of height 1 (see Section ).", "The first step of the proof of Theorem REF is to reduce it to a statement about extremal length.", "Lemma 1.2 If every ball in $\\operatorname{\\mathcal {T}}(S)$ is convex, then for every multicurve $\\gamma \\subset S$ and every Teichmüller geodesic $t \\mapsto Z_t$ in $\\operatorname{\\mathcal {T}}(S)$ the function $t \\mapsto \\operatorname{EL}(\\gamma , Z_t)$ has no local maximum.", "It was shown in [14] that extremal length of a curve is not necessarily convex along Teichmüller geodesics.", "Indeed, the authors of that paper constructed an example where the function $t \\mapsto \\operatorname{EL}(\\alpha , X_t)$ increases by a definite amount at first and then stays nearly constant on a later interval.", "The idea of our construction is to take such a pair $(\\alpha ,X_t)$ with the surface having a puncture, then another copy $(\\beta ,Y_t)$ of the same curve and surface but where the time parameter has been reversed and shifted, and to form a connected sum $Z_t = X_t \\# Y_t$ via a small slit at the puncture.", "This is done in such a way that $t \\mapsto Z_t$ is still a Teichmüller geodesic.", "We then show that $\\operatorname{EL}(\\alpha +\\beta ,Z_t)$ converges to $\\operatorname{EL}(\\alpha , X_t)+\\operatorname{EL}(\\beta , Y_t)$ as the size of the slit shrinks.", "If we arrange the time parameter of $Y_t$ so that $\\operatorname{EL}(\\beta , Y_t)$ is nearly constant when $\\operatorname{EL}(\\alpha , X_t)$ increases, and decreases when $\\operatorname{EL}(\\alpha ,X_t)$ is nearly constant, then their sum increases on the first interval and decreases on the second interval.", "By the convergence of $\\operatorname{EL}(\\alpha +\\beta ,Z_t)$ to the sum, that quantity also increases during the first interval and decreases later, provided that the slit is small enough.", "This forces a local maximum in between, and thereby proves the existence of a non-convex ball.", "This proof requires the surface $S$ to be the connected sum of two surfaces each of which is sufficiently complicated.", "It does not work when the complex dimension of $\\operatorname{\\mathcal {T}}(S)$ is less than 4.", "For those lower complexity cases, our proof is based on rigorous numerical calculations.", "In [14], Lenzhen and Rafi proved that balls in $\\operatorname{\\mathcal {T}}(S)$ are quasi-convex.", "More precisely, they showed that there exists a constant $c=c(S)$ such that for any ball $B\\subset \\operatorname{\\mathcal {T}}(S)$ , every geodesic segment with endpoints in $B$ stays within distance $c$ of $B$ .", "In other words, balls cannot fail to be convex arbitrarily badly.", "Theorem REF indicates that the Teichmüller metric is positively curved locally, where balls fail to be convex.", "There are also large-scale manifestations of positive curvature.", "Namely, there are unbounded regions in Teichmüller space where the Teichmüller metric looks like a sup metric on a product [17].", "On the other hand, there is a sense in which $\\operatorname{\\mathcal {T}}(S)$ is hyperbolic relative to its thin parts [19].", "We refer the reader to [16] for a survey on curvature aspects of the Teichmüller metric and to [21] for a coarse description of the Teichmüller metric and its geodesics.", "Lastly, Theorem REF should be put in contrast with previous convexity results: $\\operatorname{\\mathcal {T}}(S)$ is holomorphically convex [3]; hyperbolic length of a curve is convex along earthquake paths [10] and Weil–Petersson geodesics [28]; hyperbolic length [29] and extremal length [18] of a curve are log-plurisubharmonic.", "Section starts with some background on Teichmüller theory.", "We then reformulate of the convexity problem in terms of extremal length in Section .", "Section proves the convergence of extremal length under pinching deformations.", "Examples of local maxima for extremal length are constructed in Section for surfaces with enough topology.", "Finally, Section presents the numerical results which settle the lower complexity cases.", "Acknowledgements The authors thank Jeremy Kahn for suggesting the proof of Lemma REF , Curtis McMullen for useful comments, and Vincent Delecroix and David Dumas for advice on computer-assisted proofs.", "The first author was partially supported by the Fonds de recherche du Québec – Nature et technologies.", "The second author was partially supported by NSERC grant # 435885.", "A point in Teichmüller space $\\operatorname{\\mathcal {T}}(S)$ is a marked analytically finite complex structure on $S$ .", "This means a Riemann surface $X$ together with an orientation-preserving homeomorphism $f: S \\rightarrow X$ which extends to a homeomorphism $\\overline{f} : \\overline{S} \\rightarrow \\overline{X}$ , where $\\overline{X}$ is a closed Riemann surface containing $X$ .", "Two points $(X,f)$ and $(Y,g)$ are identified if there exists a conformal isomorphism $h:X \\rightarrow Y$ homotopic to $g\\circ f^{-1}$ .", "We will write $X \\in \\operatorname{\\mathcal {T}}(S)$ , keeping the marking $f$ implicit.", "A linear map $\\mathbb {R}^2 \\rightarrow \\mathbb {R}^2$ is $K$ –quasiconformal if it preserves signed angles up to a factor $K\\ge 1$ .", "Equivalently, a linear map is $K$ –quasiconformal if it has positive determinant and sends circles to ellipses with major axis to minor axis ratio at most $K$ .", "A homeomorphism between Riemann surfaces is $K$ –quasiconformal if its distributional partial derivatives are locally square-integrable and if its matrix of partial derivatives is $K$ –quasiconformal almost everywhere.", "The dilatation $\\operatorname{Dil}(h)$ of a quasiconformal homeomorphism $h$ is the smallest $K$ for which it is $K$ –quasiconformal.", "All quasiconformal homeomorphisms considered in this paper will be piecewise smooth.", "Given $X$ and $Y$ in $\\operatorname{\\mathcal {T}}(S)$ with markings $f$ and $g$ , the Teichmüller distance between them is defined as $d(X,Y) = \\inf _h \\frac{1}{2} \\log \\operatorname{Dil}(h)$ where the infimum is taken over all quasiconformal homeomorphisms $h: X \\rightarrow Y$ homotopic to $g\\circ f^{-1}$ .", "A half-translation in $\\mathbb {C}$ is either a translation or a rotation of angle $\\pi $ about a point, i.e., a map of the form $z \\mapsto \\pm z + c$ .", "A half-translation surface $\\Phi $ is a collection of polygons in $\\mathbb {C}$ with sides identified in pairs via half-translations, with at most finitely many points removed.", "The Euclidean metric descends to a metric on $\\Phi $ , which is flat except perhaps at finitely many singularities where the cone angle is a positive integer multiple of $\\pi $ .", "We require that there be no $\\pi $ –angle cone points, i.e., if such singularities arise, they should be removed.", "This is to make the surface non-positively curved.", "A half-translation structure on $S$ is an orientation-preserving homeomorphism $f: S \\rightarrow \\Phi $ where $\\Phi $ is a half-translation surface.", "Two half-translation structures $f:S\\rightarrow \\Phi $ and $g:S \\rightarrow \\Psi $ are equivalent if there is an isometry $h: \\Phi \\rightarrow \\Psi $ homotopic to $g \\circ f^{-1}$ which preserves the horizontal direction.", "There is a natural projection $\\pi $ from the space $\\operatorname{\\mathcal {QD}}(S)$ of half-translation structures on $S$ to $\\operatorname{\\mathcal {T}}(S)$ since half-translation structures are in particular complex structures.", "A half-translation structure on a Riemann surface $X \\in \\operatorname{\\mathcal {T}}(S)$ is one that projects to $X$ under $\\pi $ .", "The set $\\pi ^{-1}(X)$ of half-translation structures on $X$ is in bijection with the set of non-zero integrable holomorphic quadratic differentials on $X$ .", "Given a quadratic differential $q$ on $X$ , one obtains half-translation charts by integrating the 1–form $\\sqrt{q}$ .", "Conversely, given a half-translation structure, the differential $dz^2$ in $\\mathbb {C}$ descends to a holomorphic quadratic differential on the underlying Riemann surface.", "See [23] for the definition and basic properties of quadratic differentials.", "We will switch back and forth between the two terminologies as convenient.", "The group $\\operatorname{GL}^+(2,\\mathbb {R})$ of orientation-preserving linear automorphisms of $\\mathbb {R}^2$ acts on $\\operatorname{\\mathcal {QD}}(S)$ since it conjugates the group of half-translations to itself.", "For every $t \\in \\mathbb {R}$ , the linear map $\\mathcal {G}_t = \\begin{pmatrix} e^t & 0 \\\\ 0 & e^{-t} \\end{pmatrix}$ is $e^{2t}$ –quasiconformal.", "The action of the diagonal subgroup $\\left\\lbrace \\mathcal {G}_t \\,\\mid \\, t \\in \\mathbb {R}\\right\\rbrace $ on $\\operatorname{\\mathcal {QD}}(S)$ is called the Teichmüller flow.", "A Teichmüller line is the projection to $\\operatorname{\\mathcal {T}}(S)$ of the $\\mathcal {G}_t$ –orbit of a half-translation structure $\\Phi $ , parametrized by $t \\mapsto \\pi \\left( \\mathcal {G}_t \\Phi \\right)$ .", "Teichmüller proved that every Teichmüller line is a distance-minimizing geodesic for the Teichmüller distance.", "He also proved that through any two distinct points in $\\operatorname{\\mathcal {T}}(S)$ passes a unique Teichmüller line.", "A conformal metric on a Riemann surface $X$ is a Borel measurable function $\\rho : TX \\rightarrow \\mathbb {R}_{\\ge 0}$ such that $\\rho (\\lambda v) = \|\\lambda \| \\rho (v)$ for every $\\lambda \\in \\mathbb {C}$ and every tangent vector $v \\in TX$ .", "In other words, it is a choice of scale at each point.", "Let $\\Gamma $ be a set of 1–manifolds in a Riemann surface $X$ .", "The length of the set $\\Gamma $ with respect to a conformal metric $\\rho $ is $\\ell _\\rho (\\Gamma ) = \\ell (\\Gamma ,\\rho ) = \\inf _{\\gamma \\in \\Gamma } \\int _\\gamma \\rho $ and the area of $\\rho $ is $\\int _X \\rho ^2$ .", "The extremal length of $\\Gamma $ in $X$ is defined as $ \\operatorname{EL}(\\Gamma ,X) = \\sup _\\rho \\frac{\\ell _\\rho (\\Gamma )^2}{\\operatorname{area}(\\rho )}$ where the supremum is over all conformal metrics $\\rho $ of finite positive area.", "Typically, one takes $\\Gamma $ to be the free homotopy class of a simple closed curve $\\alpha $ in $X$ .", "We will abuse notation and write length or extremal length of a curve to mean the length or extremal length of its homotopy class.", "The basic example is when $X$ is an upright Euclidean cylinder of circumference $c$ and height $h$ , and $\\alpha $ is the curve wrapping once around $X$ .", "In this case, the optimal metric $\\rho $ is the Euclidean one and we get that $\\operatorname{EL}(\\alpha ,X) = c/h.$ We will write $\\operatorname{EL}(X)$ instead of $\\operatorname{EL}(\\alpha ,X)$ since the core curve $\\alpha $ is unique up to homotopy.", "Pulling-back metrics shows that extremal length does not increase under conformal embeddings.", "Thus if $X$ is any Riemann surface and $C \\subset X$ is an embedded cylinder, then $\\operatorname{EL}(C)\\ge \\operatorname{EL}(\\alpha ,X)$ where $\\alpha $ is the core curve in $C$ .", "If $X$ is analytically finite and $\\alpha $ is essential, meaning that it is not homotopic to a point or a puncture in $X$ , then the equality $\\operatorname{EL}(C)=\\operatorname{EL}(\\alpha ,X) $ is achieved for a unique embedded annulus $C \\subset X$ homotopic to $\\alpha $ .", "Furthermore, there exists a unique half-translation structure $\\Phi \\in \\pi ^{-1}(X)$ in which $C$ is an upright Euclidean cylinder of height 1 and the equality $\\operatorname{EL}(\\alpha , X)=\\ell _\\rho (\\alpha )^2 / \\operatorname{area}(\\rho )$ holds if and only if $\\rho $ is equal almost everywhere to a positive constant multiple of the Euclidean metric on $\\Phi $ .", "These results are due to Jenkins [8].", "Let $\\mathcal {C}_0(S)$ be the 0–skeleton of the curve complex of $S$ , i.e., the set of homotopy classes of essential simple closed curves in $S$ .", "Kerckhoff's formula [9] states that for any two points $X, Y \\in \\operatorname{\\mathcal {T}}(S)$ we have $ d(X,Y) = \\sup _{\\alpha \\in \\mathcal {C}_0(S)} \\frac{1}{2} \\log \\frac{\\operatorname{EL}(\\alpha ,Y)}{\\operatorname{EL}(\\alpha ,X)}.$ That the Teichmüller distance is at least as large as the right-hand side follows from the fact that extremal length does not increase by more than a factor $K$ under $K$ –quasiconformal homeomorphisms.", "This property (for all sets of curves $\\Gamma $ ) is often taken as the definition of quasiconformal maps.", "A (weighted) multicurve in $S$ is a formal positive linear combination of essential simple closed curves on $S$ that are pairwise disjoint and pairwise homotopically distinct.", "The set of homotopy classes of multicurves in $S$ will be denoted $\\mathbb {R}_+\\times \\mathcal {C}(S)$ ; it is the cartesian product of $\\mathbb {R}_+$ with the curve complex.", "The length of a multicurve $\\alpha = \\sum _{j\\in J} w_j \\alpha _j$ with respect to a conformal metric $\\rho $ is the weighted sum of the lengths of its components: $\\ell _\\rho (\\alpha )=\\sum _{j\\in J} w_j \\ell _\\rho (\\alpha _j).$ The definition REF of extremal length as a supremum of length squared divided by area then extends verbatim to multicurves.", "Given $X \\in \\operatorname{\\mathcal {T}}(S)$ and a multicurve $\\alpha = \\sum _{j\\in J} w_j \\alpha _j$ , we also have $ \\operatorname{EL}(\\alpha , X) = \\inf \\sum _{j\\in J} w_j^2 \\operatorname{EL}(C_j)$ where the infimum is taken over all collections of cylinders $C_j$ embedded conformally and disjointly in $X$ , with $C_j$ homotopic to $\\alpha _j$ [11].", "Again, the infimum is achieved by a unique collection of cylinders $C_j$ and there is a half-translation structure on $X$ in which each $C_j$ is foliated by horizontal trajectories and has height $w_j$ [22].", "Such a half-translation structure obtained by gluing cylinders along their boundaries is known as a Jenkins-Strebel differential.", "There is a topology on weighted multicurves defined using intersection numbers.", "With respect to this topology, weighted simple closed curves are dense, and for every $X \\in \\operatorname{\\mathcal {T}}(S)$ the map $\\alpha \\mapsto \\operatorname{EL}(\\alpha , X)$ is continuous.", "In fact, Kerckhoff showed that this map extends continuously to all measured foliations [9]." ], [ "Horoballs", "The goal of this section is to rephrase the problem of the convexity of balls in terms of extremal length.", "To this end, we look at sublevel sets of extremal length functions, which we call horoballs.", "Definition 3.1 Given $\\alpha \\in \\mathcal {C}_0(S)$ and $c>0$ , we define the associated open horoball as $H(\\alpha , c) = \\lbrace \\, X \\in \\operatorname{\\mathcal {T}}(S) : \\operatorname{EL}(\\alpha , X) < c \\,\\rbrace $ and the associated closed horoball as $\\overline{H}(\\alpha , c) = \\lbrace \\, X \\in \\operatorname{\\mathcal {T}}(S) : \\operatorname{EL}(\\alpha , X) \\le c \\,\\rbrace .$ Remark One can define horoballs for any measured foliation.", "We emphasize that we only consider horoballs associated with simple closed curves here.", "Note that the closure of an open horoball is the corresponding closed horoball, and the interior of a closed horoball is the corresponding open horoball.", "This follows from the fact that the extremal length of a curve $\\alpha $ is continuous in the second variable and does not have local minima in $\\operatorname{\\mathcal {T}}(S)$ .", "Indeed, every point $X \\in \\operatorname{\\mathcal {T}}(S)$ lies on a geodesic along which the extremal length of $\\alpha $ increases exponentially, given by the Jenkins-Strebel differential on $X$ with a single cylinder homotopic to $\\alpha $ .", "In fact, the boundary of any horoball is a hypersurface in $\\operatorname{\\mathcal {T}}(S)$ homeomorphic to Euclidean space [6].", "It follows directly from Kerckhoff's formula that closed balls are intersections of closed horoballs.", "Lemma 3.2 Every closed ball in $\\operatorname{\\mathcal {T}}(S)$ is a countable intersection of closed horoballs.", "Let $X \\in \\operatorname{\\mathcal {T}}(S)$ , let $r\\ge 0$ and let $\\overline{B}(X,r)$ be the closed ball of radius $r$ centered at $X$ .", "By equation REF , $d(X,Y) \\le r$ if and only if $\\operatorname{EL}(\\alpha , Y) \\le e^{2r}\\operatorname{EL}(\\alpha ,X)$ for every $\\alpha \\in \\mathcal {C}_0(S)$ , which shows that $\\overline{B}(X,r)= \\bigcap _{\\alpha \\in \\mathcal {C}_0(S)} \\overline{H}(\\alpha ,e^{2r} \\operatorname{EL}(\\alpha ,X)).$ In hyperbolic space, horoballs are limits of larger and larger balls with centers escaping to infinity.", "More precisely, any open horoball is the union of all the open balls that share a given normal vector.", "The same description holds in Teichmüller space.", "Lemma 3.3 Every open horoball in $\\operatorname{\\mathcal {T}}(S)$ is a nested union of open balls.", "Let $\\alpha \\in \\mathcal {C}_0(S)$ and let $c > 0$ .", "Pick an arbitrary point $X \\in \\partial H(\\alpha ,c)$ and consider the geodesic $X_t$ defined by the half-translation structure $\\Phi $ on $X$ in which almost all vertical trajectories are homotopic to $\\alpha $ , so that $\\operatorname{EL}(\\alpha ,X_t)=e^{-2t}c$ .", "We will show that $H(\\alpha ,c) = \\bigcup _{t>0} B(X_t,t).$ If $d(X_t,Y)< t$ then $\\operatorname{EL}(\\alpha ,Y) < e^{2t} \\operatorname{EL}(\\alpha ,X_t) = c$ by Kerckhoff's formula, which shows that $B(X_t,t) \\subset H(\\alpha ,c)$ for every $t>0$ .", "The triangle inequality implies that $B(X_s,s) \\subset B(X_t,t)$ whenever $0<s<t$ , i.e., the union is nested.", "Let $Y \\in H(\\alpha ,c)$ and let $b=\\operatorname{EL}(\\alpha ,Y)$ .", "We need to show that $Y \\in B(X_t,t)$ when $t$ is large enough, which amounts to constructing a $K$ –quasiconformal homeomorphism between $Y$ and $X_t$ with $K < e^{2t}$ .", "The construction is essentially the same as the one in [15] showing that certain geodesic rays in Teichmüller space stay a bounded distance apart.", "Let $Y_s$ be the geodesic through $Y$ corresponding to the half-translation structure in which almost all vertical trajectories are homotopic to $\\alpha $ , but with the time parameter shifted so that $\\operatorname{EL}(\\alpha ,Y_s)=e^{-2s}c$ .", "Then fix an $s<0$ such that $\\frac{c}{b}-e^{2s} > 1.$ Let $\\alpha _{Y}$ (respectively $\\alpha _{X}$ ) be the closed vertical trajectory in the half-translation structure on $Y_s$ (respectively $X_s$ ) that splits the $\\alpha $ –cylinder in two equal parts.", "By [15], there exists a quasiconformal homeomorphism $f: Y_s \\rightarrow X_s$ that respects the markings and sends $\\alpha _{Y}$ isometrically onto $\\alpha _{X}$ .", "Let $L$ be the quasiconformal dilatation of $f$ .", "We rescale the flat metric on $Y$ and $Y_s$ so that the circumference of the $\\alpha $ –cylinder is 1.", "After rescaling, the Teichmüller map $Y_s \\rightarrow Y$ becomes a horizontal stretch by some factor bigger than 1.", "Now $Y_s$ is a cylinder with boundary identifications, and stretching a cylinder lengthwise is the same as cutting it in the middle and inserting another piece of cylinder to make it longer.", "In other words, $Y$ can be obtained by cutting $Y_s$ open along the core curve $\\alpha _{Y}$ and gluing back a cylinder of modulusThe modulus of a Euclidean cylinder is the reciprocal of its extremal length, i.e., the distance between its boundary components once it has been rescaled to have circumference 1.", "$\\frac{1}{\\operatorname{EL}(\\alpha ,Y)} - \\frac{1}{\\operatorname{EL}(\\alpha ,Y_s)}=\\frac{1}{b} - \\frac{e^{2s}}{c}$ without twisting.", "Similarly $X_t$ can be obtained from $X_s$ by inserting a cylinder of modulus $\\frac{1}{\\operatorname{EL}(\\alpha ,X_t)} - \\frac{1}{\\operatorname{EL}(\\alpha ,X_s)}= \\frac{e^{2t}}{c}-\\frac{e^{2s}}{c}$ in the middle.", "From this cut-and-paste decomposition of $Y$ and $X_t$ , we can define a marking-preserving homeomorphism $g_t: Y \\rightarrow X_t$ by using $f$ on the complement of the middle cylinder and using the horizontal stretch map of magnitude $\\frac{e^{2t}-e^{2s}}{\\frac{c}{b} - e^{2s}} < e^{2t}-e^{2s} < e^{2t}$ on the middle cylinder.", "Then $g_t$ is $K_t$ –quasiconformal where $K_t= \\max \\left\\lbrace \\frac{e^{2t}-e^{2s}}{\\frac{c}{b} - e^{2s}}, L \\right\\rbrace .$ If $t$ is large enough, then $L < e^{2t}$ and hence $d(X_t, Y)\\le \\frac{1}{2} \\log K_t < t$ .", "The previous lemmata imply that the convexity of balls is equivalent to the convexity of horoballs.", "Theorem 3.4 The following are equivalent in $\\operatorname{\\mathcal {T}}(S)$ : every closed horoball is convex; every closed ball is convex; every open ball is convex; every open horoball is convex.", "If every closed horoball is convex, then every closed ball is convex by Lemma REF , since an arbitrary intersection of convex sets is convex.", "If every closed ball is convex, then every open ball is convex.", "Indeed, if an open ball is not convex, then there is a smaller closed ball with the same center which is non-convex.", "If every open ball is convex, then every open horoball is convex by Lemma REF , since nested unions of convex sets are convex.", "Suppose that a closed horoball is non-convex.", "Then the open horoballs of slightly higher level for the same simple closed curve are also non-convex.", "Thus if every open horoball is convex, then every closed horoball is convex.", "Therefore, to show the existence of a non-convex ball, we need to find a non-convex horoball.", "More explicitly, we need to find a simple closed curve $\\alpha \\in \\mathcal {C}_0(S)$ and three points $X,Y,Z \\in \\operatorname{\\mathcal {T}}(S)$ appearing in that order along a geodesic such that the extremal length of $\\alpha $ in $Y$ is strictly larger than in both $X$ and $Z$ .", "We can weaken this criterion by allowing $\\alpha $ to be a multicurve and replacing the 3–point condition by a 4–point condition.", "This will be useful later.", "Lemma 3.5 Suppose that there exists four points $X,Y,Z,W$ appearing in that order along a geodesic in $\\operatorname{\\mathcal {T}}(S)$ and a multicurve $\\alpha \\in \\mathbb {R}_+ \\times \\mathcal {C}(S)$ such that $\\operatorname{EL}(\\alpha , X)<\\operatorname{EL}(\\alpha , Y) \\quad \\mbox{and} \\quad \\operatorname{EL}(\\alpha , Z)>\\operatorname{EL}(\\alpha , W).$ Then there exists a non-convex ball in $\\operatorname{\\mathcal {T}}(S)$ .", "Since extremal length depends continuously on the first variable and since weighted simple closed curves are dense in $\\mathbb {R}_+ \\times \\mathcal {C}(S)$ , there exists a weighted simple closed curve $w \\beta $ such that $\\operatorname{EL}(w\\beta , X)<\\operatorname{EL}(w\\beta , Y) \\quad \\mbox{and} \\quad \\operatorname{EL}(w\\beta , Z)>\\operatorname{EL}(w\\beta , W).$ As extremal length is homogeneous of degree 2 in the first variable, we also have $\\operatorname{EL}(\\beta , X)<\\operatorname{EL}(\\beta , Y) \\quad \\mbox{and} \\quad \\operatorname{EL}(\\beta , Z)>\\operatorname{EL}(\\beta , W).$ Let $c = \\max (\\operatorname{EL}(\\beta , X), \\operatorname{EL}(\\beta , W))$ .", "Then the geodesic segment from $X$ to $W$ has its endpoints in $\\overline{H}(\\beta , c)$ and passes through $Y$ and $Z$ .", "At least one of $Y$ or $Z$ lies outside $\\overline{H}(\\beta ,c)$ , so that the closed horoball $\\overline{H}(\\beta ,c)$ is non-convex.", "By Theorem REF , this implies the existence of a non-convex ball.", "If extremal length of a multicurve has a strict local maximum along a geodesic, then we can clearly find 4 points satisfying the hypotheses of the above lemma, so that non-convex balls exist.", "It turns out that every local maximum is a strict local maximum (see below), which implies Lemma REF .", "In practice, the hypotheses of Lemma REF are easier to check than the existence of a local maximum, but Lemma REF was simpler to state for the introduction.", "Lemma 3.6 Any local maximum of extremal length along a geodesic is a strict local maximum.", "Suppose there is a multicurve $\\alpha \\in \\mathbb {R}_+ \\times \\mathcal {C}(S)$ and a geodesic $t \\mapsto X_t$ in $\\operatorname{\\mathcal {T}}(S)$ such that the function $f(t)=\\operatorname{EL}(\\alpha , X_t)$ has a local maximum at $T$ and let $M=f(T)$ .", "If the local maximum is not strict, then $f^{-1}(M)$ accumulates at $T$ .", "Since $f$ is real-analytic [18], it must be constant by the identity principle.", "But this is impossible.", "Indeed, take $\\rho _t$ to be the Euclidean metric defining the geodesic $X_t$ .", "Then the area of $\\rho _t$ is constant, whereas the length $\\ell (\\alpha ,\\rho _t)$ is unbounded in at least one direction.", "Indeed, $\\ell (\\alpha ,\\rho _t)$ is bounded below by the intersection number between $\\alpha $ and the vertical foliation on $X_t$ as well as by the intersection number with the horizontal foliation.", "These intersection numbers depend exponentially on $t$ and at least one of them is non-zero, so that it diverges as $t \\rightarrow \\infty $ or as $t \\rightarrow -\\infty $ .", "Since $f(t) \\ge \\ell (\\alpha ,\\rho _t)^2 / \\operatorname{area}(\\rho _t)$ , this is a contradiction.", "Rather than exhibiting 4 collinear points verifying the inequalities of Lemma REF , we will construct a sequence of collinear points $X_n$ , $Y_n$ , $Z_n$ and $W_n$ that degenerate in a controlled way as $n \\rightarrow \\infty $ and such that the desired inequalities hold in the limit.", "We thus need to show that extremal length behaves well under mild degeneration." ], [ "Convergence of extremal length under pinching", "Let $R = \\sqcup _{j \\in J} R_j$ be a subsurface of $S$ obtained by cutting $S$ along a multicurve and possibly forgetting some of the pieces.", "Each connected component $R_j$ of $R$ is homeomorphic to a punctured surface $R_j^{\\prime }$ .", "The Teichmüller space $\\operatorname{\\mathcal {T}}(R)$ is defined as the Cartesian product $\\Pi _{j\\in J} \\operatorname{\\mathcal {T}}(R_j^{\\prime })$ .", "Definition 4.1 Let $X_n \\in \\operatorname{\\mathcal {T}}(S)$ and $Y \\in \\operatorname{\\mathcal {T}}(R)$ .", "We say that $X_n$ converges conformally to $Y$ as $n\\rightarrow \\infty $ if there exist nested surfaces $Y_n \\subset Y$ exhausting $Y$ and $K_n$ –quasiconformal embeddings $f_n:Y_n \\rightarrow X_n$ homotopic to the inclusion map $R \\subset S$ such that $K_n \\rightarrow 1$ as $n\\rightarrow \\infty $ .", "We emphasize that in this definition, the ends of $Y$ are all required to be punctures.", "Informally speaking, $X_n$ converges conformally to $Y$ if there is a multicurve in $X_n$ that gets pinched and in the process, some pieces survive to form $Y$ .", "We don't care about the other pieces, meaning that they don't need to stabilize as $n \\rightarrow \\infty $ .", "Thus conformal convergence in the above sense is more general than convergence in the augmented Teichmüller space, where every piece is required to stabilize [1].", "There are different equivalent ways to formulate conformal convergence.", "We can say that $X_n$ converges conformally to $Y$ if for every simple closed curve $\\alpha $ in $R$ (including peripheral ones), the hyperbolic length of $\\alpha $ in $X_n$ converges to the hyperbolic length of $\\alpha $ in $Y$ ; for every $j\\in J$ , the covering space of $X_n$ associated with the subsurface $R_j$ , equipped with its hyperbolic metric, converges in the Gromov-Hausdorff topology to the corresponding component $Y_j$ of $Y$ with respect to some choices of basepoints; for every $j\\in J$ , $\\rho _n^j$ converges up to conjugacy to $\\rho ^j$ , where $\\rho _n:\\pi _1(S) \\rightarrow \\operatorname{PSL}(2,\\mathbb {R})$ is the representation defining $X_n$ , $\\rho _n^j$ is its restriction to $\\pi _1(R_j)$ coming from the inclusion $R_j \\subset S$ and $\\rho _j:\\pi _1(R_j) \\rightarrow \\operatorname{PSL}(2,\\mathbb {R})$ is the representation defining $Y_j$ .", "For our purposes, the definition in terms of nearly conformal embeddings is the most convenient.", "The statement we want to prove is that conformal convergence implies convergence of extremal length for multicurves supported on the limiting surface.", "Extremal length on a disconnected surface is defined in the usual way, as the supremum of weighted length squared divided by area over all conformal metrics.", "A standard argument shows this equals the sum of the extremal lengths on connected components (cf.", "[2]).", "Lemma 4.2 Let $Y = \\sqcup _{j \\in J} Y_j$ be a disjoint union of Riemann surfaces, let $\\alpha $ be a multicurve on $Y$ and let $\\alpha ^j=\\alpha \\cap Y_j$ .", "Then $\\operatorname{EL}(\\alpha , Y) = \\sum _{j \\in J} \\operatorname{EL}(\\alpha ^j, Y_j).$ First observe that the extremal length of $\\alpha $ on $Y$ is the same as its extremal length on the union $Z$ of the components which it intersects.", "Indeed, given a metric $\\rho $ on $Y$ , the ratio $\\ell (\\alpha ,\\rho )^2 / \\operatorname{area}(\\rho )$ does not decrease if we modify $\\rho $ to be zero outside $Z$ .", "This shows that $\\operatorname{EL}(\\alpha ,Y) \\le \\operatorname{EL}(\\alpha , Z)$ and the reverse inequality follows by extending any conformal metric on $Z$ to be zero on $Y \\setminus Z$ .", "If $\\alpha ^j$ is empty, then clearly $\\operatorname{EL}(\\alpha ^j, Y_j)=0$ .", "In proving the above formula, we may therefore assume that $\\alpha ^j$ is non-empty for each $j \\in J$ .", "For each $j \\in J$ , let $\\rho _j$ be any metric on $Y_j$ such that $\\ell (\\alpha ^j, \\rho _j)$ and $\\operatorname{area}(\\rho _j)$ are finite and positive.", "By rescaling $\\rho _j$ , we may assume that $\\ell (\\alpha ^j, \\rho _j)=\\operatorname{area}(\\rho _j)$ .", "Let $\\rho $ be the metric on $Y$ which is equal to $\\rho _j$ on $Y_j$ .", "Then $\\ell (\\alpha ,\\rho )=\\sum _{j \\in J} \\ell (\\alpha ^j, Y_j) = \\sum _{j \\in J} \\operatorname{area}(\\rho _j)=\\operatorname{area}(\\rho )$ which implies that $\\operatorname{EL}(\\alpha , Y) \\ge \\frac{\\ell (\\alpha ,\\rho )^2}{\\operatorname{area}(\\rho )} = \\sum _{j \\in J} \\frac{\\ell (\\alpha ,\\rho _j)^2}{\\operatorname{area}(\\rho _j)}.$ We can replace the right-hand side by its supremum over all non-degenerate metrics $\\rho _j$ to get $\\operatorname{EL}(\\alpha , Y) \\ge \\sum _{j \\in J} \\operatorname{EL}(\\alpha ^j, Y_j).$ Conversely, let $\\sigma $ be any conformal metric on $Y$ and let $\\sigma _j$ be its restriction to $Y_j$ .", "Then for each $j\\in J$ we have $\\operatorname{EL}(\\alpha ^j, Y_j) \\ge \\frac{\\ell (\\alpha ^j, \\sigma _j)^2}{\\operatorname{area}(\\sigma _j)}.$ Summing over all $j$ yields $\\sum _{j\\in J}\\operatorname{EL}(\\alpha ^j, Y_j) \\ge \\sum _{j \\in J} \\frac{\\ell (\\alpha ^j, \\sigma _j)^2}{\\operatorname{area}(\\sigma _j)} \\ge \\frac{\\left(\\sum _{j \\in J} \\ell (\\alpha ^j, \\sigma _j)\\right)^2}{\\sum _{j \\in J} \\operatorname{area}(\\sigma _j)}$ where the second inequality follows from the Cauchy-Schwarz inequality.", "Finally, observe that $\\sum _{j \\in J} \\ell (\\alpha ^j, \\sigma _j) = \\ell (\\alpha ,\\sigma ) \\quad \\text{and} \\quad \\sum _{j \\in J} \\operatorname{area}(\\sigma _j) = \\operatorname{area}(\\sigma )$ so that $\\sum _{j\\in J}\\operatorname{EL}(\\alpha ^j, Y_j) \\ge \\frac{\\ell (\\alpha ,\\sigma )^2}{\\operatorname{area}(\\sigma )}.$ Since the inequality holds for any conformal metric $\\sigma $ , it holds for the supremum as well and we have $\\sum _{j\\in J}\\operatorname{EL}(\\alpha ^j, Y_j) &\\ge \\operatorname{EL}(\\alpha , Y).", "$ This lemma implies that equation REF , which says that extremal length is the infimum of weighted sums of extremal lengths of embedded cylinders, still holds for disconnected surfaces.", "We use this in the proof of convergence of extremal length under degeneration.", "Theorem 4.3 Let $\\alpha $ be a multicurve in $R$ and suppose that $X_n$ converges conformally to $Y$ as $n\\rightarrow \\infty $ .", "Then $\\operatorname{EL}(\\alpha , X_n) \\rightarrow \\operatorname{EL}(\\alpha , Y)$ as $n \\rightarrow \\infty $ .", "Let $K>1$ .", "We will show that if $n$ is large enough, then $\\frac{1}{K^2} \\operatorname{EL}(\\alpha , Y) \\le \\operatorname{EL}(\\alpha , X_n) \\le K^2 \\operatorname{EL}(\\alpha , Y),$ starting with the second inequality.", "Write $\\alpha $ as a weighted sum of simple closed curves $\\sum _{i\\in I} w_i \\alpha _i$ and let $C = \\sqcup _{i \\in I} C_i$ be the collection of cylinders in $Y$ such that $\\operatorname{EL}(\\alpha , Y) = \\sum _{i\\in I} w_i^2 \\operatorname{EL}(C_i)$ .", "For each $i \\in I$ , let $A_i \\subset C_i$ be a compactly contained essential cylinder (for example a straight subcylinder) such that $\\operatorname{EL}(A_i) \\le K \\operatorname{EL}(C_i)$ .", "Let $Y_n$ be a nested exhaustion of $Y$ and let $f_n: Y_n \\rightarrow X_n$ be quasiconformal embeddings as in Definition REF .", "If $n$ is large enough, then $Y_n$ contains $\\cup _{i \\in I} A_i$ and $f_n$ is $K$ –quasiconformal.", "Then by equation REF we have $\\operatorname{EL}(\\alpha , X_n) & \\le \\sum _{i \\in I} w_i^2 \\operatorname{EL}(f_n(A_i)) \\\\&\\le \\sum _{i \\in I} w_i^2 K \\operatorname{EL}(A_i) \\\\& \\le K^2 \\sum _{i \\in I} w_i^2 \\operatorname{EL}(C_i) = K^2 \\operatorname{EL}(\\alpha , Y).$ For the reverse inequality, let $\\rho $ be the conformal metric realizing $\\operatorname{EL}(\\alpha , Y)$ .", "Our goal is to construct a good enough conformal metric on $X_n$ from $\\rho $ .", "We may assume that $\\alpha $ intersects every component of $Y$ (otherwise ignore the superfluous components).", "By the previous lemma and Renelt's theorem [22], $\\rho =\\sqrt{\|q\|}$ for a holomorphic quadratic differential $q$ on $Y$ with at most simple poles at the punctures.", "Let $\\overline{Y}$ be the completion of $Y$ in the metric $\\rho $ and let $Q = \\overline{Y} \\setminus Y$ be the set of punctures of $Y$ .", "Since $Q$ is finite, there exists a $\\delta _0>0$ such that the $\\delta _0$ –balls around the points of $Q$ are embedded and pairwise disjoint.", "Here we are using the fact that $\\rho $ has isolated singularities so that the distance it induces on $\\overline{Y}$ via path integrals is compatible with the underlying topology.", "Let $\\mu = 2 \\delta _0$ .", "Then any homotopically non-trivial arc from $Q$ to itself has $\\rho $ –length at least $\\mu $ .", "For $\\delta >0$ , let $\\mathcal {N}_\\delta (Q)$ be the open $\\delta $ –neighborhood of $Q$ in the metric $\\rho $ and let $\\rho _\\delta $ be the metric on $Y$ which agrees with $\\rho $ outside of $\\overline{\\mathcal {N}_\\delta (Q)}$ and is identically zero on $\\overline{\\mathcal {N}_\\delta (Q)} \\cap Y$ .", "Claim 4.4 For every $\\beta \\in \\mathcal {C}_0(R)$ the length $\\ell ( \\beta ,\\rho _\\delta )$ converges to $\\ell (\\beta , \\rho )$ as $\\delta \\rightarrow 0$ .", "[Proof of claim] It is clear that $\\ell ( \\beta ,\\rho _\\delta )\\le \\ell ( \\beta ,\\rho )$ since $\\rho _\\delta \\le \\rho $ .", "We will show that $\\ell ( \\beta ,\\rho ) \\le \\frac{\\mu +\\delta }{\\mu -2\\delta }\\, \\ell ( \\beta ,\\rho _\\delta )$ whenever $\\delta < \\delta _0$ .", "Let $\\gamma $ be a piecewise smooth curve homotopic to $\\beta $ on $Y$ .", "Our goal is to find a curve $\\widetilde{\\gamma }$ homotopic to $\\gamma $ such that $\|\\widetilde{\\gamma }\| \\le \\frac{\\mu +\\delta }{\\mu -2\\delta }\\, \\left\|\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}\\right\|$ where the length $\|\\cdot \|$ is measured with respect to $\\rho $ .", "We modify $\\gamma $ in two steps, each time making it shorter outside $\\overline{\\mathcal {N}_\\delta (Q)}$ .", "We estimate its total length at the end.", "To avoid unnecessary notation, we denote the modified curve by $\\gamma $ again instead of $\\widetilde{\\gamma }$ .", "We may assume that each component of $\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}$ is homotopically non-trivial rel endpoints after collapsing each component of $\\overline{\\mathcal {N}_\\delta (Q)}$ .", "Otherwise, we can homotope those trivial subarcs to the boundary of $\\overline{\\mathcal {N}_\\delta (Q)}$ , which shortens $\\gamma $ .", "We can also assume that $\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}$ has only finitely many components, because each such component has length at least $\\mu - 2 \\delta $ .", "If $\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}$ has infinite length, then there is nothing to show.", "Thus $\\gamma \\cap \\overline{\\mathcal {N}_\\delta (Q)}$ has finitely many components as well.", "Each such component can be homotoped, keeping the endpoints fixed, to a path in $\\overline{\\mathcal {N}_\\delta (Q)}$ of $\\rho $ –length at most $3\\delta $ .", "This does not change the length of the portion of $\\gamma $ lying outside $\\overline{\\mathcal {N}_\\delta (Q)}$ .", "If $\\gamma $ is disjoint from $\\overline{\\mathcal {N}_\\delta (Q)}$ , then $\|\\gamma \| = \\left\|\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}\\right\|$ and we are done.", "Otherwise $\\gamma $ breaks up into $\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}$ and $\\gamma \\cap \\overline{\\mathcal {N}_\\delta (Q)}$ and these two sets have the same number of components.", "Let $\\sigma $ be a component of $\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)}$ and $\\tau $ a component of $\\gamma \\cap \\overline{\\mathcal {N}_\\delta (Q)}$ .", "Then $\|\\sigma \| \\ge \\mu - 2\\delta $ and $\|\\tau \| \\le 3 \\delta $ , so that $\|\\sigma \|+\|\\tau \| \\le \\frac{\\mu + \\delta }{ \\mu - 2 \\delta } \\, \|\\sigma \|.$ Adding these inequalities over distinct $\\sigma $ -$\\tau $ pairs whose union is $\\gamma $ yields $\|\\gamma \| \\le \\frac{\\mu + \\delta }{ \\mu - 2 \\delta }\\, \\left\|\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)} \\right\|,$ which shows that $\\ell ( \\beta ,\\rho ) \\le \\frac{\\mu +\\delta }{\\mu -2\\delta }\\,\\left\|\\gamma \\setminus \\overline{\\mathcal {N}_\\delta (Q)} \\right\|.$ Since $\\gamma $ was arbitrary, the right-hand side can be replaced with the infimum $\\frac{\\mu +\\delta }{\\mu -2\\delta }\\, \\ell ( \\beta ,\\rho _\\delta )$ .", "The analogous result for multicurves follows by linearity.", "Thus there exists a $\\delta >0$ such that $\\ell (\\alpha ,\\rho _\\delta )^2 \\ge \\frac{1}{K} \\ell (\\alpha ,\\rho )^2.$ We fix such a $\\delta $ for the rest of the proof.", "Let $n$ be large enough so that $Y_n$ contains $Y \\setminus \\mathcal {N}_\\delta (Q)$ and so that $f_n:Y_n \\rightarrow X_n$ is $K$ –quasiconformal.", "We may assume that $f_n$ is smooth except at finitely many points.", "We define a conformal metric $\\rho _n$ on $X_n$ by $\\rho _n(v)= \\max _{\\theta \\in [0,2\\pi ]} \\rho (\\mathrm {d}f_n^{-1}(e^{i\\theta } v))$ if $v$ is a tangent vector based at a point in $f_n(Y \\setminus \\mathcal {N}_\\delta (Q))$ and $\\rho _n(v)=0$ otherwise.", "Claim 4.5 For every $\\beta \\in \\mathcal {C}_0(R)$ , we have $\\ell ( \\beta ,\\rho _n)\\ge \\ell (\\beta , \\rho _\\delta )$ .", "[Proof of claim] Let $\\gamma $ be a curve homotopic to $\\beta $ in $X_n$ .", "If $\\gamma $ is contained in the image $f_n(Y \\setminus \\mathcal {N}_\\delta (Q))$ , then $\\int \\rho _n(\\gamma ^{\\prime }(t)) \\,\|dt\| &\\ge \\int \\rho ((\\mathrm {d}f_n^{-1} (\\gamma ^{\\prime }(t))) \\,\|dt\| \\\\& =\\int \\rho ((f_n^{-1}\\circ \\gamma )^{\\prime }(t))) \\,\|dt\| \\\\& \\ge \\ell (\\beta , \\rho ) \\\\& \\ge \\ell (\\beta , \\rho _\\delta ).$ Otherwise, we can homotope $\\gamma $ to a curve $\\widetilde{\\gamma }$ which is not longer, yet is contained in $f_n(Y \\setminus \\mathcal {N}_\\delta (Q))$ .", "Once again, the analogous result for multicurves on $R$ follows immediately.", "Claim 4.6 The areas satisfy $\\operatorname{area}(\\rho _n) \\le K \\operatorname{area}(\\rho )$ .", "[Proof of claim] Since $f_n$ is $K$ –quasiconformal, we have $\\max _{\\theta \\in [0,2\\pi ]} \\rho (\\mathrm {d}f_n^{-1}(e^{i\\theta } v)) \\le K \\min _{\\theta \\in [0,2\\pi ]} \\rho (\\mathrm {d}f_n^{-1}(e^{i\\theta } v)).$ This shows that $\\rho _n^2 \\le K (f_n)_* \\rho ^2$ on $f_n(Y \\setminus \\mathcal {N}_\\delta (Q))$ , which means that $\\operatorname{area}(\\rho _n) &= \\int _{X_n} \\rho _n^2 \\\\ &\\le K \\int _{f_n(Y \\setminus \\mathcal {N}_\\delta (Q))} (f_n)_* \\rho ^2 \\\\ &= K \\int _{Y \\setminus \\mathcal {N}_\\delta (Q)} \\rho ^2 \\\\ &\\le K \\operatorname{area}(\\rho ).$ Combining the above inequalities yields $\\operatorname{EL}(\\alpha ,X_n) \\ge \\frac{\\ell (\\alpha ,\\rho _n)^2}{\\operatorname{area}(\\rho _n)} \\ge \\frac{1}{K} \\frac{\\ell (\\alpha ,\\rho _\\delta )^2}{\\operatorname{area}(\\rho )} \\ge \\frac{1}{K^2} \\frac{\\ell (\\alpha ,\\rho )^2}{\\operatorname{area}(\\rho )} = \\frac{1}{K^2} \\operatorname{EL}(\\alpha , Y)$ for all large enough $n$ , which is what we wanted to show.", "We now possess all the necessary tools to construct Teichmüller geodesics along which extremal length of a multicurve increases first and then decreases later." ], [ "The sphere with seven punctures", "Given non-negative lengths $l_1,\\ldots ,l_n$ and heights $h_1,\\ldots ,h_n$ , consider the staircase-shaped polygon $P(l_1,h_1,\\ldots ,l_n,h_n) \\subset \\mathbb {R}^2$ with $j$ –th step of length $l_j$ and height $h_j$ .", "This is illustrated in Figure REF for $n=4$ .", "We allow either $l_1$ or $h_n$ to be infinite, in which case $P$ is a horizintal or vertical half-infinite strip ending in a staircase.", "If all lengths and heights are finite, then we put the bottom-left corner of $P$ at the origin.", "Figure: The staircase-shaped polygon PP and the half-translation surface Φ a \\Phi _a.For $0<a<1$ , let $P_a=P(1,1,1,1/a^3,1/a^2,1/a,a,1/a)$ and let $\\overline{\\Phi }_a$ be the double of $P_a$ across its boundary.", "More precisely, take $P_a$ and its image $P_a^$ under a horizontal reflection $\\sigma $ and glue them along their boundary using $\\sigma $ .", "It helps to think of $P_a$ as the front of $\\overline{\\Phi }_a$ and $P_a^$ as the back.", "The sphere $\\overline{\\Phi }_a$ has 7 cone points of angle $\\pi $ (coming from the 7 interior right angles in $P_a$ ), which we remove in order to get the half-translation surface $\\Phi _a$ .", "Let $\\alpha $ be the simple closed curve that separates the two highest punctures on $\\Phi _a$ from the rest, i.e., the double of the middle horizontal line in the highest step of the staircase $P_a$ .", "Similarly, let $\\beta $ be the double of the middle vertical line in the step furthest to the right in $P_a$ (see Figure REF ).", "As we apply Teichmüller flow to the half-translation surface $\\Phi _a$ , the extremal length of $\\alpha $ increases rapidly at first and then stays nearly constant for a long time.", "The extremal length of $\\beta $ does the opposite: it remains nearly constant for a long time then decreases rapidly.", "Also, since $\\alpha $ and $\\beta $ are separated by cylinders of very large modulus, the extremal length of $\\alpha +\\beta $ is roughly equal to the sum of the individual extremal lengths.", "The net effect is that $\\operatorname{EL}(\\alpha +\\beta , \\mathcal {G}_t \\Phi _a)$ increases at first and decreases some time later.", "Let us be more precise.", "Consider the points $X_a &= \\begin{pmatrix} 1/e & 0 \\\\ 0 & e \\end{pmatrix}\\Phi _a & Y_a &= \\Phi _a \\\\Z_a &= \\begin{pmatrix} 1/a & 0 \\\\ 0 & a \\end{pmatrix}\\Phi _a & W_a &= \\begin{pmatrix} e/a & 0 \\\\ 0 & a/e \\end{pmatrix} \\Phi _a.$ appearing at times $-1$ , 0, $\\log (1/a)$ and $\\log (1/a)+1$ along the Teichmüller line $t \\mapsto \\mathcal {G}_t \\Phi _a$ .", "Observe that $Z_a = \\tau Y_a$ and $W_a = \\tau X_a$ , where $\\tau $ is the reflection about the line $y=x$ in the plane.", "Indeed, the definition of $P_a$ was arranged so that $\\begin{pmatrix} 1/a & 0 \\\\ 0 & a \\end{pmatrix} P_a =P(1/a,a,1/a,1/a^2,1/a^3,1,1,1) = \\tau P_a.$ We claim that from the point of view of the multicurve $\\alpha + \\beta $ , the surfaces $X_a$ , $Y_a$ , $Z_a$ and $W_a$ all have conformal limits as $a\\rightarrow 0$ .", "Let $\\Psi ^\\alpha $ be the double of $P(1,1,1,\\infty )$ minus the $\\pi $ –angle singularities, let $\\Psi ^\\beta $ be the double of $P(\\infty ,1,0,1)$ minus the three distinguished vertices, and let $\\Psi = \\Psi ^\\alpha \\sqcup \\Psi ^\\beta $ (see Figure REF ).", "We consider $\\Psi $ as a half-translation surface with infinite area.", "The conformal limits of $X_a$ , $Y_a$ , $Z_a$ and $W_a$ as $a\\rightarrow 0$ are $X_0 = \\mathcal {G}_{-1} \\Psi , \\quad Y_0=\\Psi , \\quad Z_0 = \\tau \\Psi \\quad \\mbox{and} \\quad W_0 = \\tau \\mathcal {G}_{-1} \\Psi $ respectively.", "Figure: The conformal limit Ψ\\Psi at time t=0t=0.", "Observe that Ψ β \\Psi ^\\beta is conformally invariant under Teichmüller flow.Lemma 5.1 For each $\\Lambda \\in \\lbrace X,Y,Z,W\\rbrace $ , the surface $\\Lambda _a$ converges conformally to $\\Lambda _0$ as $a\\rightarrow 0$ .", "We prove that $Y_a=\\Phi _a$ converges conformally to $Y_0=\\Psi $ as $a \\rightarrow 0$ .", "The proof of conformal convergence at other times is similar.", "It is clear that the top half of $P_a$ converges to $P(1,1,1,\\infty )$ as $a \\rightarrow 0$ .", "Indeed, for any $L>0$ the top portion of $P(1,1,1,\\infty )$ of height $L$ embeds isometrically into $P_a$ provided that $1/a^2 > L$ .", "Moreover, the isometric embedding maps vertices to vertices.", "By doubling this embedding, we obtain a conformal embedding of a large portion of $\\Psi ^\\alpha $ into $\\Phi _a$ .", "If we apply an homothety of factor $a$ to $P_a$ , we get the polygon $Q_a=P(a,a,a,1/a^2,1/a,1,a^2,1).$ We will show that the right half of $Q_a$ converges conformally to the unbounded polygon $P(\\infty ,1,0,1)$ .", "For this, we will use a theorem of Radó which says that if a sequence of parametrized Jordan curves $\\gamma _n: S^1 \\rightarrow \\widehat{\\mathbb {C}}$ converges uniformly to a Jordan curve $\\gamma _\\infty : S^1 \\rightarrow \\widehat{\\mathbb {C}}$ , then the corresponding (appropriately normalized) Riemann maps converge uniformly on the closed unit disk [20] [7].", "For $a\\ge 0$ , let $R_a=P(\\infty ,1,a^2,1)$ .", "For concreteness, we take the finite vertices of $R_a$ to be located at 0, $i$ , $-a^2+i$ and $-a^2 + 2i$ in $\\mathbb {C}$ .", "Let $\\gamma _a: \\mathbb {R}\\cup \\lbrace \\infty \\rbrace \\rightarrow \\widehat{\\mathbb {C}}$ be the Jordan curve $\\partial R_a \\cup \\lbrace \\infty \\rbrace $ parametrized counter-clockwise by arclength with $\\gamma _a(0)=0$ and $\\gamma _a(\\infty )=\\infty $ .", "Then $\\gamma _a$ converges uniformly to $\\gamma _0$ as $a \\rightarrow 0$ .", "Let $f_a : R_0 \\rightarrow R_a$ be the unique conformal homeomorphism such that $f_a(0)=0$ , $f_a(2i) = -a^2 + 2i$ and $f_a(\\infty )=\\infty $ .", "Then $f_a$ converges uniformly to the identity by Radó's theorem.", "In particular, $f_a^{-1}(i)=i v_a$ converges to $i$ as $a \\rightarrow 0$ .", "Define $g_a: R_0 \\rightarrow R_0$ by $g_a(x+iy) = {\\left\\lbrace \\begin{array}{ll} x+ i v_a y & \\mbox{if } y \\in [0,1] \\\\ x +i ((2-v_a)(y-1)+v_a) & \\mbox{if } y\\in (1,2].", "\\end{array}\\right.", "}$ This map is piecewise linear, fixes 0, $2i$ and $\\infty $ and sends $i$ to $f_a(i)$ .", "The quasiconformal dilatation of $g_a$ is equal to $\\max \\left\\lbrace v_a, \\frac{1}{v_a}, 2-v_a, \\frac{1}{2-v_a} \\right\\rbrace $ , which tends to 1 as $a \\rightarrow 0$ .", "Thus when $a$ is small, $f_a\\circ g_a$ is a nearly conformal homeomorphism from $R_0$ to $R_a$ taking 0, $i$ , $2i$ and $\\infty $ to 0, $i$ , $-a^2+2i$ and $\\infty $ respectively.", "Let $L<0$ and let $C = \\lbrace z \\in R_0 \\mid \\operatorname{Re}z > L \\rbrace $ .", "The images $f_a \\circ g_a(C)$ stay bounded away from $\\infty $ since $f_a \\circ g_a$ converges uniformly to the identity.", "Let's say that $\\operatorname{Re}z \\ge u$ for all $z \\in f_a \\circ g_a(C)$ and all $a\\ge 0$ .", "Then $f_a \\circ g_a(C)$ embeds isometrically in $Q_a$ in the obvious way provided that $1/a > \|u\|$ .", "By doubling all these objects and maps, we obtain a quasiconformal embedding of the double of $C$ into $\\Phi _a$ with dilatation arbitrarily close to 1 when $a$ is small.", "It only remains to prove estimates for the extremal length on these limiting surfaces.", "Lemma 5.2 $\\operatorname{EL}(\\alpha , X_0)$ and $\\operatorname{EL}(\\beta , W_0)$ are bounded above by $2/e^{2}.$ Recall that the component of $X_0$ containing $\\alpha $ is $\\mathcal {G}_{-1} \\Psi ^\\alpha $ .", "Take the top $1\\times 1$ square in $P(1,1,1,\\infty )$ without its horizontal sides.", "Its double is an open Euclidean cylinder of circumference 2 and height 1 homotopic to $\\alpha $ in $\\Psi ^\\alpha $ .", "This cylinder gets stretched to one of circumference $2/e$ and height $e$ , hence extremal length $2/e^2$ , under the map $\\mathcal {G}_{-1}$ .", "The inequality $\\operatorname{EL}(\\alpha , X_0)\\le 2/e^2$ thus follows from the monotonicity of extremal length under conformal embeddings.", "Now, the reflection $\\tau $ maps $X_0$ anti-conformally onto $W_0$ and sends $\\alpha $ to $\\beta $ so that $\\operatorname{EL}(\\alpha , X_0)=\\operatorname{EL}(\\beta , W_0)$ .", "Lemma 5.3 $\\operatorname{EL}(\\alpha ,Y_0)$ and $\\operatorname{EL}(\\beta ,Z_0)$ are bounded below by $2/3.$ Take $\\rho $ to be the Euclidean metric on the top part $T$ of height 2 in the component of $Y_0$ containing $\\alpha $ (this is a union of 6 unit squares, 3 in the front, 3 in the back) extended to be identically zero elsewhere.", "Then $\\rho $ has area 6.", "Moreover, any curve $\\gamma $ homotopic to $\\alpha $ on $Y_0$ has length at least 2 in the metric $\\rho $ .", "If $\\gamma $ is not contained in $T$ , then some point $p$ on $\\gamma $ is at height less than $-2$ .", "But some point $q$ on $\\gamma $ has to be at height at least $-1$ since it is homotopic to $\\alpha $ (it has to cross the seam between the front and back of $Y_0$ joining the two punctures on the top right).", "In this case, the length of $\\gamma $ is at least twice the height difference between $q$ and the bottom of $T$ (because there is a subarc from $p$ to $q$ then from $q$ to $p$ ), i.e., at least 2.", "A similar argument also applies if $\\gamma $ is contained in $T$ (it then has to cross the left seam in addition to the other one).", "We conclude that the extremal length of $\\alpha $ on $Y_0$ is at least $2^2 / 6 = 2/3$ .", "The extremal length of $\\beta $ on $Z_0$ is the same as the extremal length of $\\alpha $ on $Y_0$ by symmetry.", "Lemma 5.4 We have $\\operatorname{EL}(\\beta , X_0) = \\operatorname{EL}(\\beta , Y_0)$ and $\\operatorname{EL}(\\alpha ,Z_0)=\\operatorname{EL}(\\alpha ,W_0)$ .", "The equality $\\operatorname{EL}(\\beta , X_0) = \\operatorname{EL}(\\beta , Y_0)$ is due to the fact that the component $\\mathcal {G}_{-1}\\psi ^\\beta $ of $X_0$ containing $\\beta $ is conformally equivalent to the corresponding component $\\Psi ^\\beta $ of $Y_0$ .", "Indeed, recall that $\\Psi ^\\beta $ is the double of $P(\\infty ,1,0,1)$ .", "The image of the latter by $\\mathcal {G}_{-1}$ is $P(\\infty ,e,0,e)$ which is homothetic to the first polygon by a factor $e$ .", "This homothety doubles to a conformal isomorphism between $\\Psi ^\\beta $ and $\\mathcal {G}_{-1}\\Psi ^\\beta $ preserving the marked points and the curve $\\beta $ .", "Similarly, $\\operatorname{EL}(\\alpha ,Z_0)=\\operatorname{EL}(\\alpha ,W_0)$ since the connected component of $Z_0$ containing $\\alpha $ is conformally invariant under Teichmüller flow.", "These are all the ingredients we need to prove the desired behavior for the extremal length of $\\alpha + \\beta $ along the geodesic $\\mathcal {G}_t \\Phi _a$ .", "Theorem 5.5 If $a$ is small enough, then $\\operatorname{EL}(\\alpha +\\beta , X_a)<\\operatorname{EL}(\\alpha +\\beta , Y_a) \\quad \\mbox{and} \\quad \\operatorname{EL}(\\alpha +\\beta , Z_a)>\\operatorname{EL}(\\alpha +\\beta , W_a).$ For each $\\Lambda \\in \\lbrace X,Y,Z,W\\rbrace $ we have that $\\operatorname{EL}(\\alpha + \\beta , \\Lambda _a) \\rightarrow \\operatorname{EL}(\\alpha + \\beta , \\Lambda _0)$ as $a \\rightarrow 0$ by Lemma REF and Theorem REF .", "Since each $\\Lambda _0$ is disconnected, we also have $\\operatorname{EL}(\\alpha + \\beta , \\Lambda _0) = \\operatorname{EL}(\\alpha , \\Lambda _0) + \\operatorname{EL}(\\beta , \\Lambda _0).$ by Lemma REF .", "By the previous three lemmata we have $\\operatorname{EL}(\\alpha +\\beta , X_0) &=\\operatorname{EL}(\\alpha , X_0)+\\operatorname{EL}(\\beta ,X_0) \\\\&\\le \\frac{2}{e^2}+\\operatorname{EL}(\\beta ,X_0) \\\\&< \\frac{2}{3} + \\operatorname{EL}(\\beta ,Y_0) \\\\&\\le \\operatorname{EL}(\\alpha , Y_0)+\\operatorname{EL}(\\beta ,Y_0)\\\\&=\\operatorname{EL}(\\alpha +\\beta , Y_0)$ and $\\operatorname{EL}(\\alpha +\\beta , Z_0) &=\\operatorname{EL}(\\alpha , Z_0)+\\operatorname{EL}(\\beta ,Z_0) \\\\&\\ge \\frac{2}{3}+\\operatorname{EL}(\\beta ,Z_0) \\\\&> \\frac{2}{e^2} + \\operatorname{EL}(\\beta ,W_0) \\\\&\\ge \\operatorname{EL}(\\alpha , W_0)+\\operatorname{EL}(\\beta ,W_0)\\\\&=\\operatorname{EL}(\\alpha +\\beta , W_0).$ The analogous inequalities must hold for small enough $a>0$ by convergence.", "By Lemma REF , this implies the existence of non-convex balls in $\\operatorname{\\mathcal {T}}(S_{0,7})$ , where $S_{\\mathbf {g},\\mathbf {p}}$ is the closed surface of genus $\\mathbf {g}$ with $\\mathbf {p}$ points removed." ], [ "Increasing the genus", "We modify the above construction to get a surface of genus 1 with 4 punctures.", "As before, we start with the polygon $P_a=P(1,1,1,1/a^3,1/a^2,1/a,a,1/a)$ for $0<a<1$ and take a copy $P_a^$ of $P_a$ with reverse orientation.", "We think of $P_a$ as the front of the surface to be constructed and $P_a^$ as the back.", "We glue each the side of $P_a$ to the corresponding side of $P_a^$ except for the highest two horizontal sides.", "Call these sides $A$ and $B$ and let $A^$ and $B^$ be the corresponding sides of $P_a^$ .", "Then we glue $A$ to $B^$ and $B$ to $A^$ to obtain $\\overline{\\Phi }_a$ .", "In other words, we glue the circle $A\\cup A^$ to $B\\cup B^$ in an orientation-reversing manner but with a half-twist.", "This creates a handle and a singularity of angle $4\\pi $ .", "Then we remove the 4 singularities of angle $\\pi $ from $\\overline{\\Phi }_a$ to obtain the half-translation surface $\\Phi _a$ .", "The curves $\\alpha $ and $\\beta $ are as before.", "We can also cut the top squares in $P_a$ and $P_a^$ along their diagonal, rotate and glue $A\\cup A^$ to $B\\cup B^$ to obtain another useful representation of $\\Phi _a$ .", "See Figure REF .", "Figure: Left: adding a handle to Φ a \\Phi _a.", "Right: another representation of Φ a \\Phi _a obtained by cut-and-paste.As in the previous subsection, we let $X_a = \\mathcal {G}_{-1} \\Phi _a$ , $Y_a = \\Phi _a$ , $Z_a = \\mathcal {G}_{\\log (1/a)} \\Phi _a$ and $W_a = \\mathcal {G}_{\\log (1/a)+1} \\Phi _a$ .", "The claim is that all of these have conformal limits as $a \\rightarrow 0$ .", "Let $\\Upsilon $ be two copies of the polygon $P(1,1,1,\\infty )$ glued in the same pattern as described above, i.e., as in Figure REF .", "Also let two copies of the polygon $P(1,0,1,\\infty )$ with corresponding vertical sides glued together, the segment $[0,1]$ on the front glued to $[1,2]$ on the back, and vice versa.", "Denote the resulting surface $\\Omega $ .", "Let $Y_0 = \\Upsilon \\sqcup \\Psi ^\\beta $ , $X_0 = \\mathcal {G}_{-1} Y_0$ , $Z_0= \\Omega \\sqcup \\tau \\Psi ^\\alpha $ and $W_0 = \\mathcal {G}_1 Z_0$ , where $\\Psi ^\\alpha $ , $\\Psi ^\\beta $ and $\\tau $ are as in the previous subsection.", "Lemma 5.6 For each $\\Lambda \\in \\lbrace X,Y,Z,W\\rbrace $ , the surface $\\Lambda _a$ converges conformally to $\\Lambda _0$ as $a\\rightarrow 0$ .", "For each $\\Lambda $ , the convergence from the point of view of the bottom right subsurface containing $\\beta $ holds for the same reasons as before.", "From the point of view of $\\alpha $ , it is clear that $Y_a$ converges conformally to $\\Upsilon $ as $a \\rightarrow 0$ since any compact subset of $\\Upsilon $ eventually embeds isometrically into $Y_a$ .", "Similarly, the top left portion of $X_a$ converges conformally to $ \\mathcal {G}_{-1} \\Upsilon $ as $a \\rightarrow 0$ .", "The only part left to prove is that $Z_a$ and $W_a$ converge to $\\Omega $ from the point of view of $\\alpha $ .", "We prove this for $Z_a$ , the other case being similar.", "For $a\\ge 0$ and $L>0$ , let $T_a^L=\\lbrace \\, (x,y, \\varepsilon ) \\in \\mathbb {R}^2 \\times \\lbrace +,-\\rbrace : \|x\|\\le 1, -L < y\\le a^2\|x\| \\,\\rbrace / \\sim $ where $(x,a^2\|x\|,+)\\sim (-x,a^2\|x\|,-)$ for every $x \\in [-1,1]$ .", "This is a torus with one hole obtained by gluing two $M$ –shapes together.", "Note that $T_0^\\infty = \\Omega $ .", "If we rescale $Z_a$ by a factor $a$ so that its left vertical chimney has circumference 4, we see that $T_a^L$ embeds conformally into $Z_a$ provided that $1/a \\ge L$ .", "This uses the alternative gluing pattern for $\\Phi _a$ with diagonal lines.", "Consider the piecewise linear homeomorphism $f_a : T_0^\\infty \\rightarrow T_a^\\infty $ defined by $f_a(x,y,\\varepsilon ) = (x,a^2\|x\|+y,\\varepsilon ).$ On each piece of $T_0^\\infty $ where $x$ and $\\varepsilon $ have constant sign, the map $f_a$ is a vertical shear.", "Its dilatation tends to 1 as $a \\rightarrow 0$ .", "Let $L\\in (0,\\infty )$ .", "If $1/a \\ge L$ , then the restriction of $f_a$ to $T_0^L$ followed by the conformal embedding of $T_a^L$ into $Z_a$ provides a quasiconformal embedding with dilatation arbitrarily close to 1.", "Since the subsurfaces $T_0^L$ exhaust $\\Omega $ , we are done.", "We leave it to the reader to check that the extremal length estimates of Lemma REF , Lemma REF and Lemma REF hold for this example as well.", "In the same way as before, we deduce that $\\operatorname{EL}(\\alpha +\\beta , X_a)<\\operatorname{EL}(\\alpha +\\beta , Y_a) \\quad \\mbox{and} \\quad \\operatorname{EL}(\\alpha +\\beta , Z_a)>\\operatorname{EL}(\\alpha +\\beta , W_a)$ provided that $a$ is small enough.", "Hence there exist non-convex balls in $\\operatorname{\\mathcal {T}}(S_{1,4})$ .", "In the same fashion, we can further replace the 3 punctures on the bottom right of $\\Phi _a$ by a handle, which shows that $\\operatorname{\\mathcal {T}}(S_{2,1})$ contains non-convex balls.", "We can also produce examples in any higher topological complexity as follows.", "Suppose that $3\\mathbf {g}-3+\\mathbf {p}> 4$ and let $\\mathbf {h}=\\min (2,\\mathbf {g})$ and $\\mathbf {q} = 7 - 3\\mathbf {h}$ .", "Let $\\Phi _a$ be the half-translation surface constructed above of genus $\\mathbf {h}$ with $\\mathbf {q}$ punctures.", "In the bottom left corner of $\\Phi _a$ , we may remove $\\mathbf {p}- \\mathbf {q}$ points, cut $\\mathbf {g}-\\mathbf {h}$ horizontal slits, and glue each one back to itself in an $ABA^{-1}B^{-1}$ pattern to form a handle.", "The resulting half-translation surface $\\widetilde{\\Phi }_a$ has genus $\\mathbf {g}$ and $\\mathbf {p}$ punctures.", "Moreover, the conformal limits of $\\mathcal {G}_{-1} \\widetilde{\\Phi }_a$ , $\\widetilde{\\Phi }_a$ , $\\mathcal {G}_{\\log (1/a)} \\widetilde{\\Phi }_a$ and $\\mathcal {G}_{\\log (1/a)+1} \\widetilde{\\Phi }_a$ for the top left and bottom right subsurfaces are all unchanged.", "Indeed, the images of the nearly conformal embeddings used to prove conformal convergence were all disjoint from the bottom left corner.", "The same proof carries over and we obtain: Theorem 5.7 There exist non-convex balls in $\\operatorname{\\mathcal {T}}(S_{\\mathbf {g},\\mathbf {p}})$ whenever $3\\mathbf {g}- 3 + \\mathbf {p}\\ge 4$ .", "This leaves out 5 cases with $\\dim _\\mathbb {C}\\operatorname{\\mathcal {T}}(S_{\\mathbf {g},\\mathbf {p}}) = 3 \\mathbf {g}- 3 + \\mathbf {p}> 1$ : $S_{0,5}$ , $S_{0,6}$ , $S_{1,2}$ , $S_{1,3}$ and $S_{2,0}$ .", "Note that the above strategy of proof cannot be applied to $S_{0,5}$ .", "Indeed, we would need to split $S_{0,5}$ into two subsurfaces each containing an essential simple closed curve.", "But no matter how we cut $S_{0,5}$ , one component is a sphere with at most 3 holes, hence has no essential simple closed curve.", "Although the limiting argument does not carry over, the idea of playing a horizontal curve against a vertical curve is still fruitful." ], [ "The Schwarz-Christoffel formula", "Consider the polygon $L_a=P(1,a,a,1)$ where $a>0$ and $P$ is the staircase-shaped polygon from section REF .", "This is an $L$ -shape obtained by removing the top right $a$ by $a$ square from a $(1+a)$ by $(1+a)$ square.", "We mark each of the 5 internal right angles in $L_a$ .", "Let $\\alpha $ be the arc crossing the vertical leg in $L_a$ , let $\\beta $ be the arc crossing the horizontal leg, and let $\\gamma =\\alpha +\\beta $ .", "Figure: The polygon L a L_a.Since all the sides in $L_a$ are either horizontal or vertical, its double $\\Phi _a$ —topologically a sphere with 5 marked points—is a half-translation surface.", "We want to study the behavior of the extremal length of the double $\\widehat{\\gamma }$ of $\\gamma $ in $\\Phi _a$ under the Teichmüller flow.", "Lemma 6.1 There exists a conformal homeomorphism $h: \\mathcal {G}_t L_a \\rightarrow M$ where $M=R_\\alpha \\cup R_\\beta $ is a stack of two rectangles of height 1 that line up on their right side such that the inverse images of corners of $M$ with interior angle $\\pi /2$ are marked points in $\\mathcal {G}_t L_a$ ; $h(\\alpha )$ joins the left side of $R_\\alpha $ to the right side of $M$ ; $h(\\beta )$ joins the left side of $R_\\beta $ to the right side of $M$ .", "Moreover, we have $\\operatorname{EL}(\\widehat{\\gamma }, \\mathcal {G}_t \\Phi _a) = 2 \\operatorname{EL}(\\gamma , \\mathcal {G}_t L_a) = 2 \\operatorname{area}(M).$ As mentioned in Section , the extremal length $\\operatorname{EL}(\\widehat{\\gamma },\\mathcal {G}_t \\Phi _a)$ is realized by a unique Jenkins-Strebel half-translation structure $\\mathcal {G}_t \\Phi _a \\rightarrow \\Psi $ partitioned into two horizontal cylinders $C_\\alpha $ and $C_\\beta $ of height 1 each, homotopic to the doubles $\\widehat{\\alpha }$ and $\\widehat{\\beta }$ of the arcs $\\alpha $ and $\\beta $ .", "Then $\\operatorname{EL}(\\widehat{\\gamma }, \\mathcal {G}_t \\Phi _a) = \\operatorname{EL}(C_\\alpha )+\\operatorname{EL}(C_\\beta ) = \\operatorname{area}(C_\\alpha )+\\operatorname{area}(C_\\beta ) = \\operatorname{area}(\\Psi ).$ Let $J : \\mathcal {G}_t \\Phi _a \\rightarrow \\mathcal {G}_t \\Phi _a$ be the anti-conformal involution exchanging $\\mathcal {G}_t L_a$ with its mirror image.", "Then $J(C_\\alpha )$ and $J(C_\\beta )$ are disjoint cylinders homotopic to $\\widehat{\\alpha }$ and $\\widehat{\\beta }$ respectively having the same extremal length as $C_\\alpha $ and $C_\\beta $ .", "By uniqueness of the extremal cylinders, the latter are invariant under $J$ .", "It follows that $\\Psi $ is also symmetric with respect to $J$ .", "Indeed, $\\overline{J^ \\Psi }$ is a half-translation structure on $\\mathcal {G}_t \\Phi _a$ partitioned into two horizontal cylinders of height 1 homotopic to $\\widehat{\\alpha }$ and $\\widehat{\\beta }$ , and is thus equal to $\\Psi $ by uniqueness.", "Any anti-conformal involution of a Euclidean cylinder $S^1 \\times I$ which reverses the orientation of its core curve comes from a reflection of $S^1$ about a diameter.", "Thus $R_\\alpha =C_\\alpha \\cap \\mathcal {G}_t L_a$ and $R_\\beta =C_\\beta \\cap \\mathcal {G}_t L_a$ are Euclidean rectangles of height 1 in the half-translation structure $\\Psi $ .", "Let $h: \\mathcal {G}_t L_a \\rightarrow M$ be the restriction of the conformal isomorphism $\\mathcal {G}_t \\Phi _a \\rightarrow \\Psi $ .", "Then $M=R_\\alpha \\cup R_\\beta $ with $R_\\alpha $ and $R_\\beta $ glued isometrically along some part of their horizontal boundary.", "The Gauss-Bonnet theorem tells us that $\\Psi $ an angle defect of $4\\pi $ .", "Since $\\Psi $ has at most 5 cone points of angle $\\pi $ , it has at most one cone point of angle $3\\pi $ .", "Such a cone point has to lie on the circle of symmetry of $\\Psi $ , otherwise there would be two.", "Thus $M$ has no singularities in its interior, which means that it is really a polygon.", "The preimages of the right angles in $M$ by $h$ have to be marked points in $\\mathcal {G}_t L_a$ , for after doubling $M$ the right angles give rise to $\\pi $ -angle singularities of $\\Psi $ .", "Since there are only 5 marked points in $\\mathcal {G}_t L_a$ , the rectangles have to line up on one side.", "If we rotate $M$ so that $R_\\alpha $ is on top, then they line up on the right side, where there is no marked point separating $\\alpha $ from $\\beta $ .", "Let $\\rho $ be the Euclidean metric on $M$ .", "Then $\\operatorname{EL}(\\gamma , \\mathcal {G}_t L_a) &= \\operatorname{EL}(h(\\gamma ), M) \\\\ & \\ge \\frac{\\ell (h(\\gamma ),\\rho )^2}{\\operatorname{area}(\\rho )} = \\frac{(\\ell (R_\\alpha )+\\ell (R_\\beta ))^2}{\\operatorname{area}(M)} = \\operatorname{area}(M).$ On the other hand, if the ratio $\\ell (\\gamma , \\sigma )^2 / \\operatorname{area}(\\sigma )$ was strictly bigger than $\\operatorname{area}(M)$ for some conformal metric $\\sigma $ on $\\mathcal {G}_t L_a$ , then by doubling we would get $\\operatorname{EL}(\\widehat{\\gamma }, \\mathcal {G}_t \\Phi _a) \\ge \\frac{\\ell (\\widehat{\\gamma }, \\widehat{\\sigma })^2}{\\operatorname{area}(\\widehat{\\sigma })} = \\frac{(2\\ell (\\gamma , \\sigma ))^2}{2\\operatorname{area}(\\sigma )}> 2 \\operatorname{area}(M) = \\operatorname{area}(\\Psi ),$ a contradiction.", "Alternatively, one can prove that $\\rho $ is extremal using the standard length-area argument [11].", "Figure: The conformal homeomorphism in Lemma If we show that for some $a>0$ the function $t \\mapsto \\operatorname{EL}(\\gamma ,\\mathcal {G}_t L_a)$ increases and later decreases, then the same holds for the function $t \\mapsto \\operatorname{EL}(\\widehat{\\gamma },\\mathcal {G}_t \\Phi _a)$ and this implies the existence of non-convex balls in $\\operatorname{\\mathcal {T}}(S_{0,5})$ .", "Another relevant observation is that the reflection $\\tau $ in the diagonal line $y=x$ maps $\\mathcal {G}_t L_a$ to $\\mathcal {G}_{-t} L_a$ anti-conformally and sends the homotopy class of $\\gamma $ to itself so that the function $t \\mapsto \\operatorname{EL}(\\gamma ,\\mathcal {G}_t L_a)$ is even.", "Therefore, all we have to show is that there exists positive $a$ and $t$ such that $\\operatorname{EL}(\\gamma ,\\mathcal {G}_0 L_a)>\\operatorname{EL}(\\gamma ,\\mathcal {G}_t L_a)$ .", "Let $f: \\overline{\\mathbb {H}^2} \\rightarrow \\mathcal {G}_t L_a$ be a conformal homeomorphism.", "Then $f$ extends by Schwarz reflection to a conformal homeomorphism $ f : \\widehat{\\mathbb {C}}\\rightarrow \\mathcal {G}_t \\overline{\\Phi _a}$ .", "The pull-back $q={f}^* dz^2$ is a meromorhic quadratic differential on $\\widehat{\\mathbb {C}}$ with a simple pole at the preimage of each marked point and a simple zero at the preimage of the inward corner in $\\mathcal {G}_t L_a$ .", "Moreover, $q$ is symmetric with respect to complex conjugation.", "We thus have $q= \\frac{A(z-b)}{\\Pi _{j=0}^4(z-z_j)} dz^2 = (f^{\\prime }(z))^2 dz^2$ for some $A$ , $b$ and $z_j$ in $\\mathbb {R}$ .", "It follows that $f(z) = \\sqrt{A} \\int _0^z \\sqrt{\\frac{(\\zeta -b)}{\\Pi _{j=0}^4(\\zeta -z_j)}} d\\zeta + f(0).$ This is a special case of the Schwarz-Christoffel formula for conformal maps onto polygons [5].", "For the formula to make sense, one has to pick a consistent choice of square root, which we can do on $\\overline{\\mathbb {H}^2}$ .", "Let $g=h \\circ f: \\overline{\\mathbb {H}^2} \\rightarrow M$ where $h: \\mathcal {G}_t L_a \\rightarrow M$ is as in Lemma REF .", "By the same reasoning as above, $g$ has the form $g(z) = C \\int _0^z \\sqrt{\\frac{(\\zeta -p)}{\\Pi _{j=0}^4(\\zeta -z_j)}} d\\zeta + D$ for some constants $p$ , $C$ and $D$ .", "The area of $M$ can then be recovered from its side lengths, which are integrals of the above form.", "Remark One can use the Schwarz-Christoffel formula to prove the first part of Lemma REF .", "Indeed, for any choice of $p \\in \\mathbb {R}$ , the map $G_p(z) = \\int _0^z \\sqrt{\\frac{(\\zeta -p)}{\\Pi _{j=0}^4(\\zeta -z_j)}} d\\zeta $ is a conformal homeomorphism from $\\overline{\\mathbb {H}^2}$ to a polygon with angle $\\pi /2$ at each vertex $G_p(z_j)$ and angle $3\\pi /2$ at $G_p(p)$ .", "Suppose that $ \|G_{z_1}(z_0)-G_{z_1}(z_1)\|\\le \|G_{z_1}(z_1)-G_{z_1}(z_2)\|.$ Then by the intermediate value theorem, there exists a point $p$ between $z_1$ and $z_2$ such that $\|G_{p}(z_0)-G_{p}(z_1)\|=\|G_{p}(p)-G_{p}(z_2)\|.$ Indeed, $\|G_{p}(z_0)-G_{p}(z_1)\|$ is bounded away from zero for $p \\in [z_1,z_2]$ whereas $\|G_{p}(p)-G_{p}(z_2)\|$ tends to zero as $p \\rightarrow z_2$ .", "If the reverse of inequality (REF ) holds, then there is a $p$ between $z_0$ and $z_1$ such that $\|G_{p}(z_0)-G_{p}(p)\|=\|G_{p}(z_1)-G_{p}(z_2)\|.$ In either case, after rescaling we get that $G_p(\\overline{\\mathbb {H}^2})$ is a stack of two rectangles of height 1.", "The problem of calculating $\\operatorname{EL}(\\gamma , \\mathcal {G}_t L_a)$ has now been reduced to finding the correct parameters $z_0,...,z_4$ , $b$ and $p$ (all of which depend on $a$ and $t$ ).", "The Schwarz-Christoffel Toolbox [4] for MATLAB is designed to solve this parameter problem numerically.", "We used this to compute $\\operatorname{EL}(\\gamma , \\mathcal {G}_t L_a)$ for $a=1/4$ at $10^3+1$ equally spaced values of $t$ in the interval $[-0.275,0.275]$ and obtained Figure REF .", "Figure: Graph of t↦EL(γ,𝒢 t L a )t \\mapsto \\operatorname{EL}(\\gamma , \\mathcal {G}_t L_a) for a=1/4a=1/4.The figure clearly shows a decrease from time $t=0$ to $t\\approx 0.159$ .", "However, the Schwarz-Christoffel Toolbox does not come with any certified error estimates.", "Moreover, the apparent decrease of extremal length is rather small: it drops from about $3.87$ to $3.856$ , which represents less than $1\\%$ decrease.", "In order to turn this into a rigorous proof, we do the following.", "We take the approximate parameters provided by the SC Toolbox, then compute the corresponding Schwarz-Christoffel integrals numerically but with certified precision.", "Since the initial parameters are inexact, the images of the Schwarz-Christoffel maps are not the polygons we expect, but we can estimate how far away they are from the correct polygons and deduce bounds for extremal length.", "One way to get rigorous bounds on a numerical result is to use interval arithmetic.", "Roughly speaking, interval arithmetic means that instead of rounding to the nearest representable number, the computer keeps track of correct lower and upper bounds for every operation, yielding a true interval in which the result of a calculation lies.", "There exist packages that do numerical integration using interval arithmetic.", "However, we did not find any that can handle improper integrals.", "We thus wrote a program in Sage [26] to compute lower and upper bounds on the integrals needed using interval arithmetic.", "The Sage worksheet and its output are available at http://github.com/maxforbou/non-convex-balls." ], [ "Rigorous bounds", "Let $k=5.27110734472$ , let $f(z)=\\int _0^z \\frac{d\\zeta }{\\sqrt{\\zeta (\\zeta ^2-1)(\\zeta ^2-k^2)}}$ and let $X=f(\\overline{\\mathbb {H}^2})$ with marked points at $0=f(0)$ , $f(\\pm 1)$ and $f(\\pm k)$ .", "The polygon $X$ is an $L$ -shape with angle $\\pi /2$ at the marked points and angle $3\\pi /2$ at $f(\\infty )$ .", "Furthermore, $X$ is symmetric about the diagonal line $y=x$ since the function under the square root is odd.", "Thus, $X$ is a rescaled copy of $L_a$ , where $a=\\frac{\|f(0)-f(1)\|}{\|f(1)-f(k)\|}-1.$ We want to get rigorous bounds on both the shape of $X$ and the extremal length of $\\gamma $ in $X$ .", "The first thing we need to compute is the integral $\|f(0)-f(1)\|=\\left\|\\int _0^1 \\frac{dx}{\\sqrt{x(x^2-1)(x^2-k^2)}}\\right\|=\\int _0^1 \\frac{dx}{\\sqrt{\|x(x^2-1)(x^2-k^2)\|}}.$ The main observation is that the integrand $F(x)=\\frac{1}{\\sqrt{\|x(x^2-1)(x^2-k^2)\|}}$ is logarithmically convex (hence convex) on $(0,1)$ .", "Lemma 6.2 Suppose that $z_0 < z_1 < z_2 < z_3 < z_4$ .", "Then the function $F(x)= \\prod _{j=0}^4\|x-z_j\|^{-1/2}$ is log-convex between any two consecutive $z_j$ 's.", "We compute $(\\log F)^{\\prime }(x) = -\\frac{1}{2} \\left(\\sum _{j=0}^4 \\frac{1}{x-z_j} \\right)$ and $(\\log F)^{\\prime \\prime }(x) = \\frac{1}{2} \\left(\\sum _{j=0}^4 \\frac{1}{(x-z_j)^2} \\right)>0.$ Therefore, for any compact subtinterval $I \\subset (0,1)$ and any partition $\\lbrace x_{-n},\\ldots ,x_n\\rbrace $ of $I$ we have $\\sum _{j=-n}^{n-1} (x_{j+1} - x_j)F\\left(\\frac{x_j+x_{j+1}}{2}\\right) &\\le \\int _I F(x)\\,dx \\\\\\le &\\sum _{j=-n}^{n-1} (x_{j+1} - x_j)\\left(\\frac{F(x_j)+F(x_{j+1})}{2}\\right)$ by the trapezoid rule.", "We choose the partition $\\lbrace x_{-n},\\ldots ,x_n\\rbrace $ using the tanh-sinh quadrature [24] which is well-adapted for this type of singular integral.", "On a bounded interval $(a,b)$ the quadrature points are defined as $x_j = \\frac{(a+b)}{2} + \\frac{(b-a)}{2}\\tanh \\left(\\frac{\\pi }{2}\\sinh (j\\Delta )\\right)$ where $\\Delta >0$ is a step size to be determined together with $n$ .", "In this case we took $\\Delta =2^{-13}$ and $n=2^{15}$ .", "Let $\\delta =x_{-n}=1-x_n$ where the $x_j$ 's are sample points for the interval $(0,1)$ .", "An elementary calculation shows that $0 \\le \\int _0^\\delta \\frac{dx}{\\sqrt{x(1-x^2)(k^2-x^2)}} \\le \\frac{2 \\sqrt{\\delta }}{\\sqrt{(1-\\delta ^2)(k^2-\\delta ^2)}}.$ and $0 \\le \\int _{1-\\delta }^1 \\frac{dx}{\\sqrt{x(1-x^2)(k^2-x^2)}} \\le \\frac{2\\sqrt{\\delta }}{\\sqrt{(1-\\delta )(2-\\delta )(k^2-1)}}.$ Adding the lower bounds for each of the three subintervals $[0,\\delta ]$ , $[\\delta , 1-\\delta ]$ and $[1-\\delta , 1]$ yields a certified lower bound on $\|f(0)-f(1)\|$ , and similarly for upper bounds.", "We use the same method to estimate $\|f(1)-f(k)\|$ .", "In order to compute the extremal length $\\operatorname{EL}(\\gamma ,X)$ , we consider the conformal homeomorphism $g(z)=-i \\int _0^z \\frac{d\\zeta }{\\sqrt{(\\zeta ^2-1)(\\zeta ^2-k^2)}}$ between $\\overline{\\mathbb {H}^2}$ and a rectangle $X^{\\prime }$ with marked points at $0=g(0)$ , $g(\\pm 1)$ and $g(\\pm k)$ .", "Then $g\\circ f^{-1} : X \\rightarrow X^{\\prime }$ is a conformal homeomorphism preserving the marked points so that $\\operatorname{EL}(\\gamma ,X)=\\operatorname{EL}(\\gamma ,X^{\\prime })$ .", "Since the above integrand is even, $g(0)$ subdivides $X^{\\prime }$ into two congruent rectangles.", "After rescaling $X^{\\prime }$ to have height 2, the extremal length is given by area.", "This means that $\\operatorname{EL}(\\gamma ,X)=\\operatorname{EL}(\\gamma ,X^{\\prime })=2\\frac{\|g(1)-g(k)\|}{\|g(0)-g(1)\|}.$ We get rigorous bounds on $\|g(0)-g(1)\|$ and $\|g(1)-g(k)\|$ with the same method as for $f$ .", "The results are compiled in Table REF .", "Table: Certified bounds on the side lengths of XX and X ' X^{\\prime }.", "The last column shows the corresponding value calculated with Sage's nintegral routine.We now estimate extremal length after stretching $X$ .", "Let the parameters $z_0$ , $z_1$ , $z_2$ , $p$ , $z_3$ and $z_4$ be equal to $-3.33297982345$ , $-0.26873921366$ , 0, $0.17317940636$ , 1 and $2.94288195633$ respectively.", "Then let $\\phi (z)=\\int _{0}^z \\prod _{j=0}^4 (\\zeta -z_j)^{-1/2}\\, d\\zeta ,$ $\\psi (z)=-i \\int _{0}^z (\\zeta -p)^{1/2}\\prod _{j=0}^4 (\\zeta -z_j)^{-1/2}\\, d\\zeta ,$ $Y=\\phi (\\overline{\\mathbb {H}^2})$ and $Y^{\\prime }=\\psi (\\overline{\\mathbb {H}^2})$ .", "The polygon $Y$ is meant to be close to a rescaled version of $\\mathcal {G}_t X$ for $t\\approx 0.159$ whereas $Y^{\\prime }$ is a stack of two rectangles of nearly the same height, which we use to estimate $\\operatorname{EL}(\\gamma ,Y)$ .", "Since the integrand $\\prod _{j=0}^4 \|x-z_j\|^{-1/2}$ is convex, we may use the trapezoid rule to compute the side lengths of $Y$ .", "There are also elementary estimates near the poles like before.", "For $Y^{\\prime }$ the integrand is convex on each interval of continuity not adjacent to $p$ .", "Indeed, if $G(x)= \|x-p\|^{1/2}\\prod _{j=0}^4 \|x-z_j\|^{-1/2}$ then $2(\\log G)^{\\prime \\prime }(x)= \\sum _{j=0}^4 \\frac{1}{(x-z_j)^2} - \\frac{1}{(x-p)^2}$ which is positive when $x<z_2$ and when $x>z_3$ .", "We can thus apply the trapezoid rule with tanh-sinh quadrature to bound the side lengths of $Y^{\\prime }$ not adjacent to $\\psi (p)$ .", "The length $\|\\psi (z_4)-\\psi (z_0)\| = \|\\psi (z_4)-\\psi (\\infty )\| + \|\\psi (\\infty )-\\psi (z_0)\|$ is a little bit different since we need to compute integrals over two half-infinite intervals.", "We use another doubly exponential quadrature on these intervals given by $x_j = \\exp \\left(\\frac{\\pi }{2}\\sinh (j \\Delta )\\right)$ for the interval $(0,\\infty )$ .", "To estimate the area lost by truncating away from infinity, note that for $x> 2z_4-p$ we have $\|x-p\|< 2\|x-z_4\|$ as well as $\|x-z_j\|\\ge \|x-z_4\|$ for each $j$ .", "It follows that $G(x) \\le \\sqrt{2} \|x-z_4\|^{-2}$ and hence $\\int _a^\\infty G(x)\\,dx \\le \\sqrt{2} \|a-z_4\|^{-1}$ provided that $a \\ge 6$ .", "Similarly, we have $\\int _{-\\infty }^b G(x)\\,dx \\le \\sqrt{2} \|b-z_0\|^{-1}$ provided that $b \\le -7$ .", "The polygon $Y^{\\prime }$ is not exactly a stack of two rectangles of the same height, but we can still use it to estimate $\\operatorname{EL}(\\gamma ,Y)= \\operatorname{EL}(\\gamma ,Y^{\\prime })$ .", "Using the Euclidean metric on $Y^{\\prime }$ yields the lower bound $\\operatorname{EL}(\\gamma ,Y^{\\prime }) \\ge \\frac{\\ell (\\gamma )^2}{\\operatorname{area}(Y^{\\prime })} = \\frac{(\|\\psi (z_0)-\\psi (z_1)\|+\|\\psi (z_3)-\\psi (z_4)\|)^2}{\\operatorname{area}(Y^{\\prime })}.$ Moreover, the sum of the extremal lengths of the horizontal rectangles $R_\\alpha $ and $R_\\beta $ in $Y^{\\prime }$ is an upper bound for the extremal length: $\\operatorname{EL}(\\gamma ,Y^{\\prime }) \\le \\frac{\|\\psi (z_0)-\\psi (z_1)\|}{\|\\psi (z_1)-\\psi (z_2)\|}+\\frac{\|\\psi (z_3)-\\psi (z_4)\|}{\|\\psi (p)-\\psi (z_3)\|}.$ The last caveat is that $Y$ does not lie exactly along the Teichmüller geodesic through $X$ .", "Let $a=\\frac{\|f(0)-f(1)\|}{\|f(1)-f(k)\|}-1$ and $K=\\frac{\|\\phi (z_0)-\\phi (z_1)\|}{\|\\phi (z_3)-\\phi (z_4)\|}$ and consider the polygon $Z=P(K,a,Ka,1)$ .", "Then up to rescaling $Z = \\mathcal {G}_t X$ for $t=\\frac{1}{2}\\log K$ .", "Divide each of $Y$ and $Z$ into three rectangles with sides parallel to the coordinate axes and let $h:Y \\rightarrow Z$ be the homeomorphism which is affine on each subrectangle.", "Then $h$ preserves the marked points and $\\frac{1}{C}\\operatorname{EL}(\\gamma ,Y) \\le \\operatorname{EL}(\\gamma ,Z) \\le C \\operatorname{EL}(\\gamma ,Y)$ where $C\\ge \\exp (2 d(Y,Z))$ is the dilatation of $h$ .", "Note that $C$ can be expressed in terms of the aspect ratios of the three subrectangles in $Y$ and $Z$ .", "The resulting bounds are shown in Table REF .", "Table: Certified integrals after stretchingWe thus have $\\operatorname{EL}(\\gamma , \\mathcal {G}_t X) = \\operatorname{EL}(\\gamma , Z) < 3.8557 < 3.8698 < \\operatorname{EL}(\\gamma , X),$ from which we conclude that $\\operatorname{\\mathcal {T}}(S_{0,5})$ contains non-convex balls." ], [ "Remaining cases", "Adding an artificial marked point on the boundary of $X$ between $f(1)$ and $f(k)$ (the right-most side of $X$ ) does not change the extremal length of $\\gamma $ at any time.", "After doubling, this shows the existence of a non-convex ball in $\\operatorname{\\mathcal {T}}(S_{0,6})$ .", "Recall that there are isometries $\\operatorname{\\mathcal {T}}(S_{0,5}) \\cong \\operatorname{\\mathcal {T}}(S_{1,2})$ and $\\operatorname{\\mathcal {T}}(S_{0,6}) \\cong \\operatorname{\\mathcal {T}}(S_{2,0})$ arising from the hyperelliptic involutions on $S_{1,2}$ and $S_{2,0}$ .", "This shows that there exist non-convex balls in those two cases as well.", "To treat the torus with 3 punctures, we can cut horizontal slits of length $s>0$ at two punctures in the double of $X$ then glue the two slits together to form a handle.", "As $s \\rightarrow 0$ , the extremal length of the double $\\widehat{\\gamma }$ of $\\gamma $ on the 3 times punctured torus converges to its value on the double of $X$ .", "The same is true after applying the Teichmüller flow $\\mathcal {G}_t$ for any $t$ .", "It follows that if $s$ is small enough, then the resulting geodesic in $\\operatorname{\\mathcal {T}}(S_{1,3})$ exhibits an increase of extremal length followed by a decrease.", "This completes the proof of Theorem REF ." ] ]	1606.05170
[ [ "Smart Reply: Automated Response Suggestion for Email" ], [ "Abstract In this paper we propose and investigate a novel end-to-end method for automatically generating short email responses, called Smart Reply.", "It generates semantically diverse suggestions that can be used as complete email responses with just one tap on mobile.", "The system is currently used in Inbox by Gmail and is responsible for assisting with 10% of all mobile responses.", "It is designed to work at very high throughput and process hundreds of millions of messages daily.", "The system exploits state-of-the-art, large-scale deep learning.", "We describe the architecture of the system as well as the challenges that we faced while building it, like response diversity and scalability.", "We also introduce a new method for semantic clustering of user-generated content that requires only a modest amount of explicitly labeled data." ], [ "shapes,arrows 2016 rightsretained KDD '16August 13-17, 2016, San Francisco, CA, USA 978-1-4503-4232-2/16/08 http://dx.doi.org/10.1145/2939672.2939801 Smart Reply: Automated Response Suggestion for Email 6 Anjuli KannanEqual contribution.", "Karol Kurach[1] Sujith Ravi[1] Tobias Kaufmann[1] Andrew Tomkins Balint Miklos Greg Corrado László Lukács Marina Ganea Peter Young Vivek Ramavajjala Table: Acknowledgments" ] ]	1606.04870
[ [ "Pseudo-Kan Extensions and Descent Theory" ], [ "Abstract There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits.", "We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen \"Facets of Descent II\", such as B\\'{e}nabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms.", "In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad theory, studying idempotent pseudomonads and proving a $2$-dimensional version of the adjoint triangle theorem.", "Also, we work out the concept of pointwise pseudo-Kan extension, used as a framework to talk about bilimits, commutativity and the descent object.", "As a subproduct, this formal approach can be an alternative perspective/guiding template for the development of higher descent theory." ], [ "Introduction", "Descent theory is a generalization of a solution given by Grothendieck to a problem related to modules over rings [17], [24], [22].", "There is a pseudofunctor ${\\rm \\mathsf {M}od}:{\\rm \\mathsf {R}ing}\\rightarrow {\\rm \\mathsf {C}AT}$ which associates each ring $ \\mathcal {R}$ with the category ${\\rm \\mathsf {M}od}( \\mathcal {R})$ of right $\\mathcal {R}$ -modules.", "The original problem of descent is the following: given a morphism $ f: \\mathcal {R}\\rightarrow \\mathcal {S}$ of rings, we wish to understand what is the image of ${\\rm \\mathsf {M}od}( f ): {\\rm \\mathsf {M}od}( \\mathcal {R})\\rightarrow {\\rm \\mathsf {M}od}( \\mathcal {S})$ .", "The usual approach to this problem in descent theory is somewhat indirect: firstly, we characterize the morphisms $f$ in $ {\\rm \\mathsf {R}ing}$ such that ${\\rm \\mathsf {M}od}( f ) $ is a functor that forgets some “extra structure”.", "Then, we would get an easier problem: verifying which objects of ${\\rm \\mathsf {M}od}( \\mathcal {S})$ could be endowed with such extra structure (see, for instance, [24]).", "Given a category ${C}$ with pullbacks and a pseudofunctor $ \\mathcal {A}: {C}^{{\\rm op}} \\rightarrow {\\rm \\mathsf {C}AT}$ , for each morphism $p : E\\rightarrow B$ of $ {C}$ , the descent data plays the role of such “extra structure” in the basic problem (see [22], [23], [45]).", "More precisely, in this context, there is a natural construction of a category ${\\mathcal {D}esc}_ \\mathcal {A}(p)$ , called descent category, such that the objects of ${\\mathcal {D}esc}_ \\mathcal {A}(p) $ are objects of $\\mathcal {A}(E)$ endowed with descent data, which encompasses the 2-dimensional analogue for equality/1-dimensional descent: one invertible 2-cell plus coherence.", "This construction comes with a comparison functor and a factorization; that is to say, we have the commutative diagram below, in which ${\\mathcal {D}esc}_ \\mathcal {A}(p)\\rightarrow \\mathcal {A}(E) $ is the functor which forgets the descent data (see [22], [23]).", "${ \\mathcal {A}(B) [r]^-{\\phi _ p}[rd]_{ \\mathcal {A}( p )}& {{\\mathcal {D}esc}}_\\mathcal {A}( p ) [d]\\\\&\\mathcal {A}(E)}\\qquad \\mathrm {(Descent Factorization)}$ Therefore the problem is reduced to investigating whether the comparison functor $ \\phi _ p $ is an equivalence.", "If it is so, $p$ is is said to be of effective $\\mathcal {A}$ -descent and the image of $\\mathcal {A}(p) $ are the objects of $\\mathcal {A}(E) $ that can be endowed with descent data.", "Pursuing this strategy, it is also usual to study cases in which $ \\phi _ p $ is fully faithful or faithful: in these cases, $p$ is said to be, respectively, of $\\mathcal {A}$ -descent or of almost $\\mathcal {A}$ -descent.", "Furthermore, we may consider that the descent problem (in dimension 2) is, in a broad context, the characterization of the image (up to isomorphism) of any given functor $ F:\\mathtt {b}\\rightarrow \\mathtt {e} $ .", "In this case, using the strategy described above, we investigate if $\\mathtt {b} $ can be viewed as a category of objects in $\\mathtt {e} $ with some extra structure (plus coherence).", "Thereby, taking into account the original basic problem, we can ask, hence, if $F$ is (co)monadic.", "Again, we would get a factorization, the Eilenberg-Moore factorization: ${\\mathtt {b} [r]^-{\\phi }[rd]_{F}& (Co){\\rm \\mathsf {A}lg}[d]\\\\&\\mathtt {e} }$ This approach leads to what is called “monadic descent theory”.", "Bénabou and Roubaud proved that, if $F = \\mathcal {A}(p) $ in which $\\mathcal {A}: {C}^{{\\rm op}} \\rightarrow {\\rm \\mathsf {C}AT}$ is a pseudofunctor satisfying the Beck-Chevalley condition, then “monadic $\\mathcal {A}$ -descent theory” coincides with “Grothendieck $\\mathcal {A}$ -descent theory”.", "More precisely, in this case, $p$ is of effective $\\mathcal {A}$ -descent if and only if $\\mathcal {A}(p )$ is monadic [4], [22], [19], [32].", "Thereby, in the core of classical descent theory, there are two constructions: the category of algebras and the descent category.", "These constructions are known to be examples of 2-categorical limits (see [45], [46]).", "Also, in a 2-categorical perspective, we can say that the general idea of category of objects with “extra structure (plus coherence)” is, indeed, captured by the notion of 2-dimensional limits.", "Not contradicting such point of view, Street considered that (higher) descent theory is about the higher categorical notion of limit [45].", "Following this posture, we investigate whether pure formal methods and commuting properties of bilimits are useful to prove classical and new theorems in the classical context of descent theory of [22], [23], [24], [14].", "Willing to give such formal approach, we employ the following perspective: the problems of descent theory are usually reduced to the study of the image of a (pseudo)monadic (pseudo)functor.", "We restrict our attention to idempotent pseudomonads and prove formal results on pseudoalgebra structures, such as a biadjoint triangle theorem and lifting theorems.", "In order to apply such formal approach to get theorems on commutativity of bilimits, we employ a bicategorical analogue of the concept of (pointwise) Kan extension: (pointwise) pseudo-Kan extension, introduced in [34].", "By successive applications of these formal results, we get results within the context of [22], [23], such as the Bénabou-Roubaud Theorem, embedding results and theorems on effective descent morphisms of bilimits of categories.", "We also apply this approach to get results on effective descent morphisms of categories of small enriched categories $V\\textrm {-}{\\rm \\mathsf {C}at}$ provided that $V$ satisfies suitable hypotheses.", "In this direction, the fundamental standpoint on “classical descent theory” of this paper is the following: the “descent object” of a (pseudo)cosimplicial object in a given context is the image of the initial object of the appropriate notion of Kan extension of such cosimplicial object.", "More precisely, in our context of dimension 2 (which is the same context of [22], [23]), we get the following result (Theorem REF ): The descent category of a pseudocosimplicial object $\\mathcal {A}: \\Delta \\rightarrow {\\rm \\mathsf {C}AT}$ is equivalent to ${\\rm Ps}{\\mathcal {R}an}_ {\\rm j}\\mathcal {A}(\\mathsf {0}) $, in which ${\\rm j}: \\Delta \\rightarrow \\dot{\\Delta } $ is the full inclusion of the category of finite nonempty ordinals into the category of finite ordinals and order preserving functions, and ${\\rm Ps}{\\mathcal {R}an}_ {\\rm j}\\mathcal {A}$ denotes the right pseudo-Kan extension of $\\mathcal {A}$ along ${\\rm j}$ .", "In particular, we show abstract features of the “classical theory of descent” as a theory (of pseudo-Kan extensions) of pseudocosimplicial objects or pseudofunctors $\\dot{\\Delta } \\rightarrow {\\rm \\mathsf {C}AT}$ .", "This work was motivated by three main aims.", "Firstly, to get formal proofs of classical results of descent theory.", "Secondly, to prove new results in the classical context – for instance, formal ways of getting sufficient conditions for a morphism to be of effective descent.", "Thirdly, to get proofs of descent theorems that could be recovered in other contexts, such as in the development of higher descent theory (see, for instance, the work of Hermida [18] and Street [45] in this direction).", "In Section , we give an idea of our scope within the context of [22], [23]: we show the main results classically used to deal with the problem of characterization of effective descent morphisms and we present classical results, which are proved using results on commutativity in Sections and .", "Namely, the embedding results (Theorems REF and REF ) and the Bénabou-Roubaud Theorem (Theorem REF ).", "At the end of Section , we establish a theorem on pseudopullbacks of categories (Theorem REF ) which is proved in Section .", "Section contains most of the abstract results of our formal approach to descent via pseudomonad theory.", "We start by establishing our main setting: the tricategory of 2-categories, pseudofunctors and pseudonatural transformations.", "In REF , we define and study basic aspects of idempotent pseudomonads.", "Then, in REF , we study pseudoalgebra structures w.r.t.", "idempotent pseudomonads, proving a Biadjoint Triangle Theorem (Theorem REF ) and giving a result related to the study of pseudoalgebra structures in commutative squares (Corollary REF ).", "We deal with the technical situation of considering objects that cannot be endowed with pseudoalgebra structures but have comparison morphisms belonging to a special class of morphisms in REF .", "Section explains why we do not use the usual enriched Kan extensions to study commutativity of the 2-dimensional limits related to descent theory: the main point is that we like to have results which work for bilimits in general (not only flexible ones).", "In REF , we define pseudo-Kan extensions and, then, we give the associated factorizations in REF .", "Particular cases of these factorizations are the Eilenberg-Moore factorization of an adjunction and the descent factorization described above.", "We give further background material in REF , studying weighted bilimits and proving the first result that relates pseudo-Kan extensions and weighted bilimits.", "Then, in REF , we introduce pseudoends and prove basic results such as a version of Fubini's theorem for pseudoends (Theorem REF ) and the fundamental equivalence for pseudoends (Proposition REF ).", "In REF , we finally introduce the appropriate notion of pointwise pseudo-Kan extension (Definition REF ) and, using the results on pseudoends, we prove that a pointwise pseudo-Kan extension exists if and only if suitable weighted bilimits exist (Theorem REF and Corollary REF ).", "In REF , REF and REF , we fit the study of pseudo-Kan extensions into the perspective of Section .", "We apply the results of to the special case of weighted bilimits and pseudo-Kan extensions: we get, then, results on commutativity of weighted bilimits/pseudo-Kan extensions and exactness/(almost/effective) descent diagrams.", "Section studies descent objects.", "We prove that the classical descent object (category) is given by the pseudo-Kan extension of a pseudocosimplicial object (as explained above).", "In particular, this means that descent objects are conical bilimits of pseudocosimplicial objects.", "We adopt this description as our definition of descent object of a pseudocosimplicial object.", "We finish Section presenting also the strict version of a descent object, which is given by a Kan extension of a special type of 2-diagram.", "We get, then, the strict factorization of descent theory.", "Section gives elementary examples of our context of effective descent diagrams.", "Every weighted bilimit can be seen as an example, but we focus in examples that we use in applications.", "As mentioned above, the most important examples of bilimits in descent theory are descent objects and Eilenberg-Moore objects: thereby, Section is dedicated to explain how Eilenberg-Moore objects fit in our context, via the free adjunction 2-category of [44].", "In Section , we study the Beck-Chevalley condition: by doctrinal adjunction [26], this is the necessary and sufficient condition to guarantee that a pointwise adjunction between pseudoalgebras can be, actually, extended to an adjunction between such pseudoalgebras.", "We show how it is related to commutativity of weighted bilimits, giving our first version of a Bénabou-Roubaud Theorem (Theorem REF ).", "We apply our results to the usual context [22], [23] of descent theory in Section : we prove a general version (Theorem REF ) of the embedding results (Theorem REF ), we prove another Bénabou-Roubaud Theorem (Theorem REF ) and, finally, we give a weak version of Theorem REF .", "We finish the paper in Section : there, we give a stronger result on commutativity (Theorem REF ) and we apply our results to descent theory, proving Theorem REF and the Galois result of [20] (Theorem REF ).", "Finally, we prove that $V$ -${\\rm \\mathsf {C}at}$ can be nicely embedded in the category of internal categories ${\\rm \\mathsf {C}at}(V) $ provided that $V$ satisfies suitable hypotheses.", "In this situation, we apply Theorem REF to get effective descent morphisms of the category of enriched categories $V$ -${\\rm \\mathsf {C}at}$ .", "We give instances of this result, getting effective descent morphisms of ${\\rm \\mathsf {T}op}$ -${\\rm \\mathsf {C}at}$ and ${\\rm \\mathsf {C}at}$ -${\\rm \\mathsf {C}at}$ .", "This work was realized during my PhD program at University of Coimbra.", "I am grateful to my supervisor Maria Manuel Clementino for her precious help, support and attention.", "I also thank all the speakers of our informal seminar on descent theory for their insightful talks: Maria Manuel Clementino, George Janelidze, Andrea Montoli, Dimitri Chikhladze, Pier Basile and Manuela Sobral.", "Finally, I wish to thank Stephen Lack for our brief conversations which helped me to understand aspects related to this work about 2-dimensional category theory, Kan extensions and coherence." ], [ "Basic Problem", "In the context of [19], [22], [23], [24], [32], [43], [9], the very basic problem of descent is the characterization of effective descent morphisms w.r.t.", "the basic fibration.", "As a consequence of Bénabou-Roubaud Theorem [4], this problem is trivial for suitable categories (for instance, for locally cartesian closed categories).", "However there are remarkable examples of nontrivial characterizations.", "The topological case, solved by Tholen and Reiterman [43] and reformulated by Clementino and Hofmann [8], [10], is an important example.", "Below, we present some theorems classically used as a framework to deal with this basic problem.", "In this paper, we show that most of these theorems are consequences of a formal theorem presented in Section , while others are consequences of theorems about bilimits.", "Firstly, the most fundamental features of descent theory are the descent category and its related factorization.", "Assuming that ${C}$ is a category with pullbacks, if $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ is a pseudofunctor, the REF is described by Janelidze and Tholen in [23].", "We show in Section that the concept of pseudo-Kan extension encompasses these features.", "In fact, the comparison functor and the REF (up to isomorphism) come from the unit and the triangular invertible modification of the (bi)adjunction $\\left[{\\rm t}, {\\rm \\mathsf {C}AT}\\right]_{({PS})}\\dashv ({\\rm Ps}){\\mathcal {R}an}_ {\\rm t}$ .", "Secondly, for the nontrivial problems, the usual approach to study (basic/universal) effective/almost descent morphisms is the embedding in well behaved categories, in which “well behaved category” means just that we know which are the effective descent morphisms of this category.", "For this matter, there are some theorems in [22], [19].", "We state below examples of these results: Theorem 1.1 Let $U: {C}\\rightarrow {D}$ be a pullback preserving functor between categories with pullbacks.", "If $U$ is faithful, then $U$ reflects almost descent morphisms; If $U$ is fully faithful, then $U$ reflects descent morphisms.", "The result on descent morphisms above can be seen as a consequence of Proposition of 2.6 of [22].", "Theorem 1.2 ([22], [19]) Let ${C}$ and $ {D}$ be categories with pullbacks.", "If $U: {C}\\rightarrow {D}$ is a fully faithful pullback preserving functor and $U(p)$ is of effective descent in $ {D}$ , then $p$ is of effective descent if and only if it satisfies the following property: whenever the diagram below is a pullback in ${D}$ , there is an object $C$ in ${C}$ such that $U(C)\\cong A$ .", "${ {U(P)}[r]^-{}[d]_{}&{A}[d]^{}\\\\{U(E)}[r]_-{U(p)}&{U(B)}}$ We show in Section that Theorem REF is a very easy consequence of formal and commuting properties of pseudo-Kan extensions (Corollary REF and Corollary REF ) that follow directly from results of Section , while we show in Section that Theorem REF is a consequence of a theorem on bilimits (Theorem REF ) which also implies the generalized Galois Theorem of [20].", "It is interesting to note that, since Theorems REF and REF are just formal properties, they can be applied in other contexts – for instance, for morphisms between pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ and $\\mathcal {B}: {D}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ , as it is explained in Section .", "Finally, Bénabou-Roubaud Theorem [4], [22] is a celebrated result of Descent Theory which allows us to understand some problems via monadicity: it says that monadic $\\mathcal {A}$ -descent theory is equivalent to Grothendieck $\\mathcal {A}$ -descent theory in suitable cases, such as the basic fibration.", "We demonstrate in Section that it is also a corollary of formal results of Section .", "Theorem 1.3 (Bénabou-Roubaud [4], [22]) Let ${C}$ be a category with pullbacks.", "If $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ is a pseudofunctor such that, for every morphism $p: E\\rightarrow B$ of ${C}$ , $A(p)$ has left adjoint $A(p)!$ and the invertible 2-cell induced by $\\mathcal {A}$ below satisfies the Beck-Chevalley condition, then the REF is pseudonaturally equivalent to the Eilenberg-Moore factorization.", "In other words, assuming the hypotheses above, Grothendieck $\\mathcal {A}$ -descent theory is equivalent to monadic descent theory.", "${ {\\mathcal {A}(B) }[d]_-{\\mathcal {A}(p) }[r]^-{\\mathcal {A}(p) }&\\mathcal {A}(E) [d]^-{}@{}\|{\\cong }[dl]\\\\\\mathcal {A}(E) [r]_-{} & \\mathcal {A}( E\\times _ p E)}$ Clementino and Hofmann [9] studied the problem of characterization of effective descent morphisms for $(T,V)$ -categories provided that $V$ is a lattice.", "To deal with this problem, they used the embedding $(T,V)\\textrm {-}{\\rm \\mathsf {C}at}\\rightarrow (T,V)\\textrm {-}{\\rm \\mathsf {G}rph}$ and Theorems REF and REF .", "However, for more general monoidal categories $V$ , such inclusion is not fully faithful and the characterization of effective descent morphisms still is an open problem even for the simpler case of the category of enriched categories $V\\textrm {-}{\\rm \\mathsf {C}at}$ .", "As an application, we give some results about effective descent morphisms of $V$ -${\\rm \\mathsf {C}at}$ .", "They are consequences of formal results given in this paper on effective descent morphisms of categories constructed from other categories: more precisely, 2-dimensional limits of categories.", "More precisely, firstly we prove Theorem REF in Section .", "Then, we prove that, if $V$ is a cartesian closed category satisfying suitable hypotheses and ${\\rm \\mathsf {C}at}( V ) $ is the category of internal categories, there is a full inclusion $V$ -${\\rm \\mathsf {C}at}\\rightarrow {\\rm \\mathsf {C}at}( V ) $ which is the pseudopullback of a suitable fully faithful functor ${\\rm \\mathsf {S}et}\\rightarrow V $ along the projection of the underlying object of objects ${\\rm \\mathsf {C}at}( V )\\rightarrow V $ .", "In this case, we conclude that the inclusion reflects effective descent morphisms by Theorem REF .", "Since the characterization of effective descent morphisms for the category of internal categories in our setting was already done by Le Creurer [32], we actually get effective descent morphisms for the category of $V$ -enriched categories.", "Theorem 1.4 Assume that the diagram of categories with pullbacks ${ {B}[r]^S [d]_ Z & {C}[d]^F\\\\@{}\|{\\cong }[ru]{D}[r]_G &{E}}$ is a pseudopullback such that all the functors are pullback preserving functors.", "If $p$ is a morphism in ${B}$ such that $ S(p), Z(p) $ are of effective descent and $FS(p)$ is a descent morphism, then $p$ is of effective descent." ], [ "Formal Results", "Our perspective herein is that, instead of considering the problem of understanding the image of a generic (pseudo)functor, the main theorems of descent theory usually deal with the problem of understanding the pseudoalgebras of (fully) property-like (pseudo)monads [25].", "It is easier to study these pseudoalgebras: they are just the objects that can be endowed with a unique pseudoalgebra structure (up to isomorphism), or, more appropriately, the effective descent points/objects.", "Thereby results on pseudoalgebra structures are in the core of our formal approach.", "In this section, we give the main results of this paper in this direction, restricting the scope to idempotent pseudomonads.", "This setting is sufficient to deal with the classical descent problem of [22], [23].", "We start by recalling basic results of bicategory theory [3], [47], [48].", "To fix notation, we give the definition of the tricategory of 2-categories, pseudofunctors, pseudonatural transformations and modifications, denoted by 2-${\\rm \\mathsf {C}AT}$ .", "We refer to [34] for the omitted coherence axioms of to and for the proof of Lemma REF .", "Henceforth, in a given 2-category, we always denote by $\\cdot $ the vertical composition of 2-cells and by $\\ast $ their horizontal composition.", "[Pseudofunctor] Let $\\mathfrak {A}, \\mathfrak {B}$ be 2-categories.", "A pseudofunctor $\\mathcal {A}:\\mathfrak {A}\\rightarrow \\mathfrak {B}$ is a pair $(\\mathcal {A}, \\mathfrak {a}) $ with the following data: Function $\\mathcal {A}: {\\rm obj}(\\mathfrak {A})\\rightarrow {\\rm obj}(\\mathfrak {B}) $ ; Functors $\\mathcal {A}_{{}_{XY}}: \\mathfrak {A}(X,Y)\\rightarrow \\mathfrak {B}(\\mathcal {A}(X), \\mathcal {A}(Y)) $ ; For each pair $g: X\\rightarrow Y , h: Y\\rightarrow Z $ of 1-cells in $\\mathfrak {A}$ , an invertible 2-cell in $\\mathfrak {B}$ : $\\mathfrak {a}_ {{}_{hg}}: \\mathcal {A}(h) \\mathcal {A}(g)\\Rightarrow \\mathcal {A}(hg) $ ; For each object $X$ of $\\mathfrak {A}$ , an invertible 2-cell $\\mathfrak {a}_ {{}_{X}}: {\\rm Id}_{{}_{\\mathcal {A}X}}\\Rightarrow \\mathcal {A}({\\rm Id}_ {{}_X} )$ in $\\mathfrak {B}$ ; subject to associativity, identity and naturality coherence axioms.", "If $\\mathcal {A}= (\\mathcal {A}, \\mathfrak {a}): \\mathfrak {A}\\rightarrow \\mathfrak {B}$ and $(\\mathcal {B}, \\mathfrak {b}): \\mathfrak {B}\\rightarrow \\mathfrak {C}$ are pseudofunctors, we define the composition as follows: $ \\mathcal {B}\\circ \\mathcal {A}:= \\left( \\mathcal {B}\\mathcal {A}, \\left(\\mathfrak {b}\\mathfrak {a}\\right)\\right)$ , in which $(\\mathfrak {b}\\mathfrak {a}) _ {{}_{hg}}:= \\mathcal {B}(\\mathfrak {a}_ {{}_{hg}})\\cdot \\mathfrak {b}_ {{}_{\\mathcal {A}(h)\\mathcal {A}(g)}} $ and $(\\mathfrak {b}\\mathfrak {a}) _ {{}_{X}}:=\\mathcal {B}(\\mathfrak {a}_ {{}_{X}})\\cdot \\mathfrak {b}_ {{}_{\\mathcal {A}(X)}} $ .", "This composition is associative and it has trivial identities.", "A pseudonatural transformation between pseudofunctors $\\mathcal {A}\\longrightarrow \\mathcal {B}$ is a natural transformation in which the usual (natural) commutative squares are replaced by invertible 2-cells plus coherence.", "[Pseudonatural transformation] If $\\mathcal {A}, \\mathcal {B}:\\mathfrak {A}\\rightarrow \\mathfrak {B}$ are pseudofunctors, a pseudonatural transformation $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ is defined by: For each object $X$ of $\\mathfrak {A}$ , a 1-cell $\\alpha _{{}_X}: \\mathcal {A}(X)\\rightarrow \\mathcal {B}(X) $ of $\\mathfrak {B}$ ; For each 1-cell $g:X\\rightarrow Y $ of $\\mathfrak {A}$ , an invertible 2-cell $\\alpha _{{}_g}: \\mathcal {B}(g) \\alpha _{{}_X}\\Rightarrow \\alpha _{{}_Y}\\mathcal {A}(g) $ of $\\mathfrak {B}$ ; such that coherence axioms of associativity, identity and naturality hold.", "Firstly, the vertical composition, denoted by $\\beta \\alpha $ , of two pseudonatural transformations $\\alpha : \\mathcal {A}\\Rightarrow \\mathcal {B}$ , $\\beta : \\mathcal {B}\\Rightarrow \\mathcal {C}$ is defined by $(\\beta \\alpha ) _ {{}_W} :=\\beta _{{}_W}\\alpha _ {{}_W} $ ${\\mathcal {A}(W)[r]^{\\beta _{{}_W}\\alpha _ {{}_W}}[d]_{\\mathcal {A}(f) }@{}[dr]\|{{(\\beta \\alpha ) _{{}_f}} }&\\mathcal {C}(W)@{}[drr]\|{:=}[d]^{\\mathcal {C}(f)}&&\\mathcal {A}(W)[rr]^{\\alpha _{{}_W}}[d]^{\\mathcal {A}(f)}@{}[drr]\|{{\\alpha _{{}_f}} }&& \\mathcal {B}(W)[d]_{\\mathcal {B}(f) } [r]^{\\beta _ {{}_W}}@{}[dr]\|{{\\beta _{{}_f}} }& \\mathcal {C}(W) [d]^{\\mathcal {C}(f)}\\\\\\mathcal {A}(X)[r]_{\\beta _ {{}_X}\\alpha _ {{}_X}} & \\mathcal {C}(X)&& \\mathcal {A}(X)[rr]_ {\\alpha _ {{}_X}} && \\mathcal {B}(X)[r]_{\\beta _ {{}_X}} &\\mathcal {C}(X)}$ Secondly, let $(\\mathcal {U},\\mathfrak {u}), (\\mathcal {L}, \\mathfrak {l}) : \\mathfrak {B}\\rightarrow \\mathfrak {C}$ and $ \\mathcal {A}, \\mathcal {B}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ be pseudofunctors.", "If $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ , $\\lambda :\\mathcal {U}\\longrightarrow \\mathcal {L}$ are pseudonatural transformations, then the horizontal composition of $\\mathcal {U}$ with $\\alpha $ , denoted by $\\mathcal {U}\\alpha $ , is defined by: $(\\mathcal {U}\\alpha ) _{{}_W} := \\mathcal {U}(\\alpha _{{}_W})$ and $(\\mathcal {U}\\alpha )_ {{}_f} := \\left(\\mathfrak {u}_{{}_{\\alpha _{{}_X}\\mathcal {A}(f)}}\\right) ^{-1} \\cdot \\mathcal {U}(\\alpha _{{}_f})\\cdot \\mathfrak {u}_{{}_{\\mathcal {B}(f)\\alpha _{{}_W}}}$ , while the composition $\\lambda \\mathcal {A}$ is defined trivially.", "Thereby, we get the definition of the horizontal composition $\\left( \\lambda \\ast \\alpha \\right) := (\\lambda \\mathcal {B})(\\mathcal {U}\\alpha )\\cong (\\mathcal {L}\\alpha )(\\lambda \\mathcal {A}).$ Similarly, we get the three types of compositions of modifications.", "[Modification] Let $\\mathcal {A}, \\mathcal {B}:\\mathfrak {A}\\rightarrow \\mathfrak {B}$ be pseudofunctors.", "If $\\alpha , \\beta : \\mathcal {A}\\Rightarrow \\mathcal {B}$ are pseudonatural transformations, a modification $ \\Gamma : \\alpha \\Longrightarrow \\beta $ is defined by the following data: For each object $X$ of $\\mathfrak {A}$ , a 2-cell $\\Gamma _{{}_X}: \\alpha _ {{}_X}\\Rightarrow \\beta _{{}_X} $ of $\\mathfrak {B}$ subject to one coherence axiom of naturality.", "It is straightforward to verify that $2\\textrm {-}{\\rm \\mathsf {C}AT}$ is a tricategory which is locally a 2-category.", "In particular, we denote by $[\\mathfrak {A}, \\mathfrak {B}]_{PS} $ the 2-category of pseudofunctors $\\mathfrak {A}\\rightarrow \\mathfrak {B}$ , pseudonatural transformations and modifications.", "Also, we have the bicategorical Yoneda lemma [47] and, hence, the bicategorical Yoneda embedding $\\mathcal {Y}:\\mathfrak {A}\\rightarrow [\\mathfrak {A}^{{\\rm op}}, {\\rm \\mathsf {C}AT}]_ {PS}$ is locally an equivalence (i.e.", "it induces equivalences between the hom-categories).", "Lemma 2.1 (Bicategorical Yoneda Lemma [47]) The Yoneda embedding $\\mathcal {Y}:\\mathfrak {A}\\rightarrow [\\mathfrak {A}^{{\\rm op}}, {\\rm \\mathsf {C}AT}]_ {PS} : \\qquad X\\mapsto \\mathfrak {A}(- , X) $ is locally an equivalence.", "[Bicategorically representable] A pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ is called bicategorically representable if there is an object $W$ of $\\mathfrak {A}$ such that $\\mathcal {A}$ is pseudonaturally equivalent to $ \\mathfrak {A}(W,-): \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ .", "In this case, $W$ endowed with a pseudonatural equivalence $\\mathcal {A}\\simeq \\mathfrak {A}(W,-) $ is called the bicategorical representation of $\\mathcal {A}$ .", "By the bicategorical Yoneda lemma, if it exists, a bicategorical representation of a pseudofunctor is unique up to equivalence.", "[Bicategorical reflection] Let $\\mathcal {L}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ be a pseudofunctor and $X$ an object of $\\mathfrak {B}$ such that $\\mathfrak {B}(\\mathcal {L}-, X): \\mathfrak {A}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ has a bicategorical representation $\\mathcal {U}(X) $ .", "If $\\varepsilon _ {{}_ {X}}:\\mathcal {L}\\mathcal {U}(X)\\rightarrow X $ denotes the image of the identity on $\\mathcal {U}(X) $ by the equivalence $\\mathfrak {A}(\\mathcal {U}(X), \\mathcal {U}(X)) \\simeq \\mathfrak {B}(\\mathcal {L}\\mathcal {U}(X), X), $ the pair $(\\mathcal {U}(X), \\varepsilon _ {{}_ {X}} ) $ is called the right bicategorical reflection of $X$ along $\\mathcal {L}$ .", "In this case, we often omit the morphism and say that $\\mathcal {U}(X) $ is the right bicategorical reflection and $\\varepsilon _ {{}_{X}}$ is the universal arrow or counit of the right bicategorical reflection.", "Since bicategorical representations are unique up to equivalence, right bicategorical reflections are unique up to equivalence as well.", "This means that, whenever $(\\mathcal {U}(X)^{\\prime }, \\varepsilon _ {{}_ {X}}^{\\prime } ) $ and $(\\mathcal {U}(X), \\varepsilon _ {{}_ {X}} ) $ are right bicategorical reflections of $X$ along $\\mathcal {L}$ , there exists an equivalence $\\underline{v}: \\mathcal {U}(X)^{\\prime }\\simeq \\mathcal {U}(X) $ such that there is an invertible 2-cell $\\varepsilon _ {{}_ {X}} \\mathcal {L}(\\underline{v} )\\cong \\varepsilon _ {{}_ {X}}^{\\prime } $ .", "In the context of the definition above, it is easy to verify that $(\\mathcal {U}(X), \\varepsilon _ {{}_ {X}} ) $ is a right bicategorical reflection of $X$ along $\\mathcal {L}$ if and only if $\\mathfrak {A}(-, \\mathcal {U}(X)) \\rightarrow \\mathfrak {B}(\\mathcal {L}-, X), \\quad f\\mapsto \\varepsilon _ {{}_ {X}}\\, \\mathcal {L}(f) $ defines a pseudonatural equivalence.", "The dual notion is that of left bicategorical reflection.", "Namely, if it exists, the left bicategorical reflection of $X$ along $\\mathcal {L}$ is the right bicategorical reflection of $\\mathcal {L}^{\\rm op}: \\mathfrak {A}^{\\rm op}\\rightarrow \\mathfrak {B}^{\\rm op}.$ Hence, if it exists, it consists of a pair $(X _ \\mathcal {L}, \\rho _ {{}_{X}} ) $ in which $X _ \\mathcal {L}$ is an object of $\\mathfrak {A}$ and $\\rho _ {{}_X}: X\\rightarrow \\mathcal {L}(X_ \\mathcal {L}) $ is a morphism in $\\mathfrak {B}$ such that $\\mathfrak {A}(X_\\mathcal {L}, -) \\rightarrow \\mathfrak {B}(X, \\mathcal {L}- ), \\quad g\\mapsto \\mathcal {L}(g)\\, \\rho _ {{}_ {X}} $ is a pseudonatural equivalence.", "We say that $\\mathcal {L}$ is left biadjoint to $\\mathcal {U}: \\mathfrak {B}\\rightarrow \\mathfrak {A}$ if, for every object $X$ of $\\mathfrak {B}$ , $\\mathcal {U}(X) $ is the right bicategorical reflection of $X$ along $\\mathcal {L}$ .", "In this case, we say that $\\mathcal {U}$ is right biadjoint to $\\mathcal {L}$ .", "This definition of biadjunction is equivalent to Definition .", "[Biadjunction] A pseudofunctor $\\mathcal {L}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ is left biadjoint to $\\mathcal {U}$ if there exist pseudonatural transformations $\\eta :{\\rm Id}_ {\\mathfrak {A}} \\longrightarrow \\mathcal {U}\\mathcal {L}$ and $\\varepsilon :\\mathcal {L}\\mathcal {U}\\longrightarrow {\\rm Id}_ { \\mathfrak {B}}$ invertible modifications $s : {\\rm Id}_{\\mathcal {L}} \\Longrightarrow (\\varepsilon \\mathcal {L}) \\cdot (\\mathcal {L}\\eta )$ and $t : (\\mathcal {U}\\varepsilon ) \\cdot (\\eta \\mathcal {U}) \\Longrightarrow {\\rm Id}_{\\mathcal {U}}$ satisfying coherence equations.", "In this case, $(\\mathcal {L}\\dashv \\mathcal {U}, \\eta , \\varepsilon , s, t ): \\mathfrak {A}\\rightarrow \\mathfrak {B}$ is a biadjunction.", "Sometimes we omit the invertible modifications, denoting a biadjunction by $(\\mathcal {L}\\dashv \\mathcal {U}, \\eta , \\varepsilon ) $ .", "By the bicategorical Yoneda lemma, if $\\mathcal {L}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ is left biadjoint, its right biadjoint $\\mathcal {U}: \\mathfrak {B}\\rightarrow \\mathfrak {A}$ is unique up to pseudonatural equivalence.", "Furthermore, if $\\mathcal {L}$ is left 2-adjoint, it is left biadjoint.", "A pseudofunctor $\\mathcal {U}$ is a local equivalence if it induces equivalences between the hom-categories.", "Lemma 2.2 A right biadjoint $\\mathcal {U}$ is a local equivalence if and only if the counit of the biadjunction is a pseudonatural equivalence." ], [ "Idempotent Pseudomonads", "Since we deal only with idempotent pseudomonads, we give an elementary approach focusing on them.", "The main benefit of this approach is that idempotent pseudomonads have only free pseudoalgebras.", "For this reason, assuming that $\\eta $ is the unit of an idempotent pseudomonad $\\mathcal {T}$ , an object $X$ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure if and only if $\\eta _ {{}_X}: X\\rightarrow \\mathcal {T}(X) $ is an equivalence.", "Recall that a pseudomonad $\\mathcal {T}$ on a 2-category $\\mathfrak {H}$ consists of a sextuple $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ , in which $\\mathcal {T}:\\mathfrak {H}\\rightarrow \\mathfrak {H}$ is a pseudofunctor, $\\mu : \\mathcal {T}^2\\longrightarrow \\mathcal {T}, \\eta : {\\rm Id}_ {{}_\\mathfrak {H}}\\longrightarrow \\mathcal {T}$ are pseudonatural transformations and ${ \\mathcal {T}@/_4ex/@{=}[dr][r]^-{\\eta _ {{}_\\mathcal {T}}}@{}[dr]\|-{{\\hspace{3.00003pt}\\Lambda \\hspace{3.00003pt}}}& \\mathcal {T}^2[d]\|-{\\mu }& \\mathcal {T}[l]_-{\\mathcal {T}\\eta }@/^4ex/@{=}[dl]@{}[dl]\|-{{\\hspace{3.00003pt}\\rho \\hspace{3.00003pt}}} &&\\mathcal {T}^3[r]^{\\mathcal {T}\\mu }[d]_{\\mu _ \\mathcal {T}}@{}[dr]\|{{\\hspace{3.99994pt}\\Gamma \\hspace{3.99994pt}}}&\\mathcal {T}^2[d]^{\\mu }\\\\&\\mathcal {T}& &&\\mathcal {T}^2 [r]_ {\\mu } &\\mathcal {T}}$ are invertible modifications satisfying the following coherence equations [39], [34]: Identity: ${ &\\mathcal {T}^2[dl]_-{\\mathcal {T}\\eta \\mathcal {T}}[dr]^-{\\mathcal {T}\\eta \\mathcal {T}}[dd]\|-{{\\rm Id}_ {{}_{\\mathcal {T}^2}}}&&&&\\mathcal {T}^2[d]\|-{\\mathcal {T}\\eta \\mathcal {T}}&\\\\\\mathcal {T}^3[dr]_-{\\mu \\mathcal {T}}@{}[r]\|-{{\\rho \\mathcal {T}} }&&\\mathcal {T}^3@{}[l]\|-{{\\widehat{\\mathcal {T}\\Lambda }}}[dl]^-{\\mathcal {T}\\mu }&&&\\mathcal {T}^3[dl]\|-{\\mu \\mathcal {T}}[dr]\|-{\\mathcal {T}\\mu }&\\\\&\\mathcal {T}^2[d]\|-{\\mu }&&=&\\mathcal {T}^2@{}[rr]\|-{{\\hspace{5.69046pt}\\Gamma \\hspace{5.69046pt}} }[dr]\|-{\\mu }&&\\mathcal {T}^2[dl]\|-{\\mu }\\\\&\\mathcal {T}&&&&\\mathcal {T}&}$ Associativity: ${ \\mathcal {T}^4[r]^-{\\mathcal {T}^2 \\mu }[dr]\|-{\\mathcal {T}\\mu \\mathcal {T}}[d]_{\\mu \\mathcal {T}^2}&\\mathcal {T}^3[dr]^-{\\mathcal {T}\\mu }@{}[d]\|-{{\\widehat{\\mathcal {T}\\Gamma } }}&&&\\mathcal {T}^4[r]^-{\\mathcal {T}^2\\mu }@{}[dr]\|-{{\\mu _ {{}_{{}_{\\mu } }}^{-1}}}[d]_-{\\mu \\mathcal {T}^2}&\\mathcal {T}^3[d]\|-{\\mu \\mathcal {T}}[dr]^-{\\mathcal {T}\\mu }&\\\\\\mathcal {T}^3[dr]_-{\\mu \\mathcal {T}}@{}[r]\|{{\\Gamma \\mathcal {T}}}&\\mathcal {T}^3[r]\|-{\\mathcal {T}\\mu }[d]\|-{\\mu \\mathcal {T}}@{}[dr]\|-{ {\\hspace{2.84544pt}\\Gamma \\hspace{2.84544pt}} }&\\mathcal {T}^2[d]^-{\\mu }&=&\\mathcal {T}^3[r]\|-{\\mathcal {T}\\mu }[dr]_-{\\mu \\mathcal {T}}&\\mathcal {T}^2@{}[r]\|-{{\\hspace{2.84544pt}\\Gamma \\hspace{2.84544pt}}}[dr]@{}[d]\|-{{\\hspace{2.84544pt}\\Gamma \\hspace{2.84544pt}} }&\\mathcal {T}^2[d]^{\\mu }\\\\&\\mathcal {T}^2[r]_{\\mu }&\\mathcal {T}&&&\\mathcal {T}^2[r]_ {\\mu }&\\mathcal {T}}$ in which $\\widehat{\\mathcal {T}\\Lambda } := \\left( \\mathfrak {t}_ {{}_{\\mathcal {T}}} \\right)^{-1} \\left(\\mathcal {T}\\Lambda \\right) \\left(\\mathfrak {t}_ {{}_{(\\mu )(\\eta \\mathcal {T}) }}\\right),\\qquad \\qquad \\widehat{\\mathcal {T}\\Gamma } := \\left( \\mathfrak {t}_{{}_{(\\mu ) (\\mu \\mathcal {T}) }}\\right)^{-1}\\left(\\mathcal {T}\\Gamma \\right) \\left( \\mathfrak {t}_{{}_{(\\mu ) (\\mathcal {T}\\mu ) }}\\right) .$ [Idempotent pseudomonad] A pseudomonad $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ is idempotent if there is an invertible modification $\\eta \\mathcal {T}\\cong \\mathcal {T}\\eta $ .", "Similarly to 1-dimensional monad theory, the name idempotent pseudomonad is justified by Lemma REF , which says that the multiplication of an idempotent pseudomonad is a pseudonatural equivalence.", "Lemma 2.3 A pseudomonad $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ is idempotent if and only if the multiplication $\\mu $ is a pseudonatural equivalence.", "In this case, $\\eta \\mathcal {T}$ is an inverse equivalence of $\\mu $ .", "Since $\\mu (\\eta \\mathcal {T})\\cong {\\rm Id}_ {{}_\\mathcal {T}} \\cong \\mu (\\mathcal {T}\\eta ) $ , it is obvious that, if $\\mu $ is a pseudonatural equivalence, then $\\eta \\mathcal {T}\\cong \\mathcal {T}\\eta $ .", "Therefore $\\mathcal {T}$ is idempotent and $\\eta \\mathcal {T}$ is an equivalence inverse of $\\mu $ .", "Reciprocally, assume that $\\mathcal {T}$ is idempotent.", "By the definition of pseudomonads, there is an invertible modification $\\mu (\\eta \\mathcal {T})\\cong {\\rm Id}_ {{}_\\mathcal {T}} $ .", "And, since $\\eta \\mathcal {T}\\cong \\mathcal {T}\\eta $ , we get the invertible modifications $(\\eta \\mathcal {T})\\mu \\cong (\\mathcal {T}\\mu ) (\\eta \\mathcal {T}^2)\\cong (\\mathcal {T}\\mu ) (\\mathcal {T}\\eta \\mathcal {T})\\cong \\mathcal {T}(\\mu (\\eta \\mathcal {T}) )\\cong {\\rm Id}_ {{}_{\\mathcal {T}^2}} $ which prove that $\\mu $ is a pseudonatural equivalence and $\\eta \\mathcal {T}$ is a pseudonatural equivalence inverse.", "The reader familiar with lax-idempotent/Kock-Zöberlein pseudomonads will notice that an idempotent pseudomonad is just a Kock-Zöberlein pseudomonad whose adjunction $\\mu \\dashv \\eta \\mathcal {T}$ is actually an adjoint equivalence.", "Hence, idempotent pseudomonads are fully property-like pseudomonads [25].", "Every biadjunction induces a pseudomonad [29], [34].", "In fact, we get the multiplication $\\mu $ from the counit, and the invertible modifications $\\Lambda , \\rho , \\Gamma $ come from the invertible modifications of Definition .", "Of course, a biadjunction $\\mathcal {L}\\dashv \\mathcal {U}$ induces an idempotent pseudomonad if and only if its unit $\\eta $ is such that $\\eta \\mathcal {U}\\mathcal {L}\\cong \\mathcal {U}\\mathcal {L}\\eta $ .", "As a consequence of this characterization, we have Lemma REF which is necessary to give the Eilenberg-Moore factorization for idempotent pseudomonads.", "Lemma 2.4 A biadjunction $(\\mathcal {L}\\dashv \\mathcal {U}, \\eta , \\varepsilon )$ induces an idempotent pseudomonad if and only if $\\eta \\mathcal {U}: \\mathcal {U}\\longrightarrow \\mathcal {U}\\mathcal {L}\\mathcal {U}$ is a pseudonatural equivalence.", "By the triangle invertible modifications of Definition , if $\\varepsilon $ is the counit of the biadjunction $\\mathcal {L}\\dashv \\mathcal {U}$ , $(\\mathcal {U}\\varepsilon ) (\\eta \\mathcal {U})\\cong {\\rm Id}_ {{}_{\\mathcal {U}}}.", "$ Also, since $\\mathcal {U}\\mathcal {L}\\eta \\cong \\eta \\mathcal {U}\\mathcal {L}$ , we have the following invertible modifications $(\\eta \\mathcal {U})\\cdot (\\mathcal {U}\\varepsilon ) \\cong (\\mathcal {U}\\mathcal {L}\\mathcal {U}\\varepsilon ) (\\eta \\mathcal {U}\\mathcal {L}\\mathcal {U})\\cong (\\mathcal {U}\\mathcal {L}\\mathcal {U}\\varepsilon ) (\\mathcal {U}\\mathcal {L}\\eta \\mathcal {U})\\cong \\mathcal {U}\\mathcal {L}( {\\rm Id}_ {{}_{\\mathcal {U}}} )\\cong {\\rm Id}_ {{}_{\\mathcal {U}\\mathcal {L}\\mathcal {U}}} $ Therefore $\\eta \\mathcal {U}$ is a pseudonatural equivalence.", "Reciprocally, if $\\eta \\mathcal {U}$ is a pseudonatural equivalence, so is $\\eta \\mathcal {U}\\mathcal {L}$ .", "Therefore the multiplication of the induced pseudomonad is an inverse equivalence of $\\eta \\mathcal {U}\\mathcal {L}$ and, by Lemma REF , we conclude that the induced pseudomonad is idempotent.", "We can avoid the coherence equations [39], [29], [34] used to define the 2-category of pseudoalgebras of a pseudomonad $\\mathcal {T}$ when assuming that $\\mathcal {T}$ is idempotent.", "[Pseudoalgebras] Let $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ be an idempotent pseudomonad on a 2-category $\\mathfrak {H}$ .", "We define the 2-category of $\\mathcal {T}$ -pseudoalgebras $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ as follows: Objects: the objects of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ are the objects $X$ of $\\mathfrak {H}$ such that $\\eta _ {{}_X}: X\\rightarrow \\mathcal {T}(X) $ is an equivalence; The inclusion ${\\rm obj}(\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg})\\rightarrow {\\rm obj}(\\mathfrak {H}) $ extends to a full inclusion 2-functor $\\mathcal {I}:\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}\\rightarrow \\mathfrak {H}$ In other words, the inclusion $\\mathcal {I}: \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}\\rightarrow \\mathfrak {H}$ is defined to be final among the full inclusions $\\widehat{\\mathcal {I}}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ such that $\\eta \\widehat{\\mathcal {I}} $ is a pseudonatural equivalence.", "If $\\eta _ {{}_X}: X\\rightarrow \\mathcal {T}(X) $ is an equivalence, $X$ can be endowed with a pseudoalgebra structure and the left adjoint $a:\\mathcal {T}(X)\\rightarrow X $ to $\\eta _ {{}_X}: X\\rightarrow \\mathcal {T}(X) $ is called a pseudoalgebra structure to $X$ .", "Because we could describe $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ by means of pseudoalgebras/pseudoalgebra structures, we often denote the objects of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ by small letters $a, b $ .", "Theorem 2.5 (Eilenberg-Moore biadjunction) Let $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ be an idempotent pseudomonad on a 2-category $\\mathfrak {H}$ .", "There is a unique pseudofunctor $\\mathcal {L}^{{}^\\mathcal {T}}$ such that ${ \\mathfrak {H}[rr]\|-{\\mathcal {T}}[dr]\|-{\\mathcal {L}^{{}^\\mathcal {T}}}&&\\mathfrak {H}\\\\&\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}[ur]\|-{\\mathcal {I}} &}$ is a commutative diagram.", "Furthermore, $\\mathcal {L}^{{}^\\mathcal {T}} $ is left biadjoint to $\\mathcal {I}$ .", "Firstly, we define $\\mathcal {L}^{{}^\\mathcal {T}} (X): = \\mathcal {T}(X) $ .", "On the one hand, it is well defined, since, by Lemma REF , $\\eta \\mathcal {T}: \\mathcal {T}\\longrightarrow \\mathcal {T}^2 $ is a pseudonatural equivalence.", "On the other hand, the uniqueness of $\\mathcal {L}^{{} ^\\mathcal {T}} $ is a consequence of $\\mathcal {I}$ being a monomorphism.", "Now, it remains to show that $\\mathcal {L}^{{}^\\mathcal {T}} $ is left biadjoint to $\\mathcal {I}$ .", "By abuse of language, if $a$ is an object of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ , we denote by $a$ its pseudoalgebra structure (of Definition REF ).", "Then we define the mutually inverse equivalences below $\\begin{aligned}\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}(\\mathcal {T}(X), b) &\\rightarrow \\mathfrak {H}( X, \\mathcal {I}(b))&\\\\f&\\mapsto f\\eta _ {{}_X}&\\\\\\alpha &\\mapsto \\alpha \\ast {\\rm Id}_ {{}_{\\eta _ {{}_X}}} &\\end{aligned}\\qquad \\begin{aligned} \\mathfrak {H}( X, \\mathcal {I}(b)) &\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}(\\mathcal {T}(X), b)&\\\\g &\\mapsto bT(g)&\\\\\\beta &\\mapsto {\\rm Id}_ {{}_b}\\ast T(\\beta )&\\end{aligned}$ It completes the proof that $\\mathcal {L}^{{}^\\mathcal {T}}\\dashv \\mathcal {I}$ .", "Theorem REF shows that this biadjunction $\\mathcal {L}^{{}^\\mathcal {T}} \\dashv \\mathcal {I}$ satisfies the expected universal property [29] of the 2-category of pseudoalgebras, which is the Eilenberg-Moore factorization.", "In other words, we prove that our definition of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ for idempotent pseudomonads $\\mathcal {T}$ agrees with the usual definition [33], [29], [39], [47] of pseudoalgebras for a pseudomonad.", "Theorem 2.6 (Eilenberg-Moore) If $\\mathcal {L}\\dashv \\mathcal {U}$ is a biadjunction which induces an idempotent pseudomonad $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ , then we have a unique comparison pseudofunctor $\\mathcal {K}:\\mathfrak {B}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ such that ${\\mathfrak {B}[r]^-{\\mathcal {K}}[rd]_-{\\mathcal {U}}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}[d]^-{\\mathcal {I}}& \\mathfrak {A}[r]^-{\\mathcal {L}^{{}^\\mathcal {T}} }[rd]_-{\\mathcal {L}}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}\\\\&\\mathfrak {A}&&\\mathfrak {B}[u]_-{\\mathcal {K}} }$ commute.", "It is enough to define $\\mathcal {K}(X) = \\mathcal {U}(X) $ and $\\mathcal {K}(f) = \\mathcal {U}(f) $ .", "This is well defined, since, by Lemma REF , $\\eta \\mathcal {U}: \\mathcal {U}\\longrightarrow \\mathcal {T}\\mathcal {U}$ is a pseudonatural equivalence.", "Actually, in $2\\textrm {-}{\\rm \\mathsf {C}AT}$ , every biadjunction $\\mathcal {L}\\dashv \\mathcal {U}$ induces a comparison pseudofunctor and an Eilenberg-Moore factorization [33] as above, in which $\\mathcal {T}= \\mathcal {U}\\mathcal {L}$ denotes the induced pseudomonad.", "When the comparison pseudofunctor $\\mathcal {K}$ is a biequivalence, we say that $\\mathcal {U}$ is pseudomonadic.", "Although there is the Beck's theorem for pseudomonads [33], [18], [34], the setting of idempotent pseudomonads is simpler.", "Theorem 2.7 Let $\\mathcal {L}\\dashv \\mathcal {U}$ be a biadjunction.", "The pseudofunctor $\\mathcal {U}$ is a local equivalence (or, equivalently, the counit is a pseudonatural equivalence) if and only if $\\mathcal {U}$ is pseudomonadic and the induced pseudomonad is idempotent.", "Firstly, if the counit $\\varepsilon $ of the biadjunction of $\\mathcal {L}\\dashv \\mathcal {U}$ is a pseudonatural equivalence, then $\\mu : = \\mathcal {U}\\varepsilon \\mathcal {L}$ is a pseudonatural equivalence as well.", "And, thereby, the induced pseudomonad is idempotent.", "Now, if $a:\\mathcal {T}(X)\\rightarrow X $ is a pseudoalgebra structure to $X$ , we have that ${\\mathcal {K}(\\mathcal {L}(X)) = \\mathcal {T}(X) [r]_-{\\simeq }^-{a} & X.", "}$ Thereby $\\mathcal {U}$ is pseudomonadic.", "Reciprocally, if $\\mathcal {L}\\dashv \\mathcal {U}$ induces an idempotent pseudomonad and $\\mathcal {U}$ is pseudomonadic, then we have that $\\mathcal {I}\\circ \\mathcal {K}= \\mathcal {U}$ , $\\mathcal {K}$ is a biequivalence and $\\mathcal {I}$ is a local equivalence.", "Thereby $\\mathcal {U}$ is a local equivalence and $\\varepsilon $ is a pseudonatural equivalence.", "In descent theory, one needs conditions to decide if a given object can be endowed with a pseudoalgebra structure.", "Idempotent pseudomonads provide the following simplification.", "Theorem 2.8 Let $\\mathcal {T}= (\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ be an idempotent pseudomonad on $\\mathfrak {H}$ .", "Given an object $X$ of $\\mathfrak {H}$ , the following conditions are equivalent: The object $X$ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure; $\\eta _ {{}_X}:X\\rightarrow \\mathcal {T}(X) $ is a pseudosection, i.e.", "there is $a:\\mathcal {T}(X)\\rightarrow X $ such that $ a\\eta _ {{}_X}\\cong {\\rm Id}_ {{}_X} $ ; $\\eta _ {{}_X}:X\\rightarrow \\mathcal {T}(X) $ is an equivalence.", "Assume that $\\eta _ {{}_X}:X\\rightarrow \\mathcal {T}(X) $ is a pseudosection.", "By hypothesis, there is $a:\\mathcal {T}(X)\\rightarrow X $ such that $ a\\eta _ {{}_X}\\cong {\\rm Id}_ {{}_X}$ .", "Thereby $\\eta _ {{}_X}a \\cong \\mathcal {T}(a)\\eta _{{}_{\\mathcal {T}(X)}}\\cong \\mathcal {T}(a)\\mathcal {T}(\\eta _ {{}_X})\\cong \\mathcal {T}(a\\eta _ {{}_X})\\cong {\\rm Id}_ {{}_{\\mathcal {T}(X)}} .$ Hence $\\eta _ {{}_X} $ is an equivalence." ], [ "Biadjoint Triangle Theorem", "The main result of this formal approach is somehow related to distributive laws of pseudomonads [39], [40].", "However, we choose a more direct approach, avoiding some technicalities of distributive laws unnecessary to our setting.", "To give such direct approach, we use the Biadjoint Triangle Theorem REF .", "Precisely, we give a bicategorical analogue (for idempotent pseudomonads) of an adjoint triangle theorem [11], [1], [41].", "It is important to note that this bicategorical version holds for more general biadjoint triangles [34], [35], [37], so that our restriction to the idempotent version is due to our scope.", "Lemma 2.9 Let $(\\mathcal {L}\\dashv \\mathcal {U}, \\eta , \\varepsilon )$ and $(\\widehat{\\mathcal {L}} \\dashv \\widehat{\\mathcal {U}}, \\widehat{\\eta } , \\widehat{\\varepsilon } ) $ be biadjunctions.", "Assume that $\\widehat{\\mathcal {L}}\\dashv \\widehat{\\mathcal {U}}$ induces an idempotent pseudomonad and that there is a pseudonatural equivalence ${ \\mathfrak {A}@{}[drr]\|{\\simeq } &&\\mathfrak {B}[ll]\|-{\\mathtt {E}}\\\\&\\mathfrak {C}[ul]\|-{\\mathcal {L}}[ur]\|-{\\widehat{\\mathcal {L}}}&}$ If $\\eta _ {{}_X} $ is a pseudosection, then $\\widehat{\\eta } _ {{}_X} $ is an equivalence.", "Let $X$ be an object of $\\mathfrak {C}$ such that $ \\eta _ {{}_X} : X\\rightarrow \\mathcal {U}\\mathcal {L}(X) $ is pseudosection.", "By Theorem REF , it is enough to prove that $\\widehat{\\eta } _ {{}_X} $ is a pseudosection, because the pseudomonad induced by $\\widehat{\\mathcal {L}}\\dashv \\widehat{\\mathcal {U}} $ is idempotent.", "To prove that $\\widehat{\\eta } _ {{}_X} $ is a pseudosection, we construct a pseudonatural transformation $\\alpha : \\widehat{\\mathcal {U}}\\widehat{\\mathcal {L}}\\longrightarrow \\mathcal {U}\\mathcal {L}$ such that there is an invertible modification ${ &{\\rm Id}_ {{}_{\\mathfrak {C}}}@{}[d]\|{\\cong }[ld]\|{\\widehat{\\eta } }[rd]\|{\\eta }&\\\\\\widehat{\\mathcal {U}}\\widehat{\\mathcal {L}}[rr]\|{\\alpha } && \\mathcal {U}\\mathcal {L}}$ Without losing generality, we assume that $\\mathtt {E}\\circ \\widehat{\\mathcal {L}}=\\mathcal {L}$ .", "Then we define $\\alpha := (\\mathcal {U}\\mathtt {E}\\widehat{\\varepsilon }\\widehat{\\mathcal {L}}) (\\eta \\widehat{\\mathcal {U}}\\widehat{\\mathcal {L}}) $ .", "Indeed, $\\alpha \\widehat{\\eta } = (\\mathcal {U}\\mathtt {E}\\widehat{\\varepsilon }\\widehat{\\mathcal {L}}) \\left( \\eta \\widehat{\\mathcal {U}}\\widehat{\\mathcal {L}}\\right) (\\widehat{\\eta })\\cong (\\mathcal {U}\\mathtt {E}\\widehat{\\varepsilon }\\widehat{\\mathcal {L}}) \\left( \\mathcal {U}\\mathcal {L}\\widehat{\\eta }\\right) (\\eta )\\cong (\\mathcal {U}\\mathtt {E}\\widehat{\\varepsilon }\\widehat{\\mathcal {L}}) \\left( \\mathcal {U}\\mathtt {E}\\widehat{\\mathcal {L}}\\widehat{\\eta }\\right) (\\eta )\\cong \\eta $ Therefore, if $\\eta _ {{}_X} $ is a pseudosection, so is $\\widehat{\\eta } _ {{}_X} $ .", "And, as mentioned, by Theorem REF , if $\\widehat{\\eta } _ {{}_X} $ is a pseudosection, it is an equivalence.", "Let $\\widehat{\\mathcal {T}}$ be the idempotent pseudomonad induced by $\\widehat{\\mathcal {L}}\\dashv \\widehat{\\mathcal {U}} $ and $\\mathcal {T}$ the pseudomonad induced by $\\mathcal {L}\\dashv \\mathcal {U}$ .", "Then Lemma REF could be written as follows: If $X$ is an object of $\\mathfrak {C}$ that can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure, then $X$ can be endowed with a $\\widehat{\\mathcal {T}} $ -pseudoalgebra structure, provided that there is a pseudonatural equivalence $\\mathtt {E}\\widehat{\\mathcal {L}}\\simeq \\mathcal {L}$ .", "Theorem 2.10 Let $(\\mathcal {L}\\dashv \\mathcal {U}, \\eta , \\varepsilon )$ and $(\\widehat{\\mathcal {L}} \\dashv \\widehat{\\mathcal {U}}, \\widehat{\\eta } , \\widehat{\\varepsilon } ) $ be biadjunctions such that their right biadjoints are local equivalences.", "If there is a pseudonatural equivalence ${ \\mathfrak {A}@{}[drr]\|{\\simeq } &&\\mathfrak {B}[ll]\|-{\\mathtt {E}}\\\\&\\mathfrak {C}[ul]\|-{\\mathcal {L}}[ur]\|-{\\widehat{\\mathcal {L}}}&}$ then $\\mathtt {E}$ is left biadjoint to a pseudofunctor $\\mathtt {R}$ which is a local equivalence.", "It is enough to define $\\mathtt {R}:= \\widehat{\\mathcal {L}} \\mathcal {U}$ .", "By Lemma REF , $\\left(\\widehat{\\eta } {\\mathcal {U}}\\right): \\mathcal {U}\\longrightarrow \\widehat{\\mathcal {U}}\\widehat{\\mathcal {L}}\\mathcal {U}= \\widehat{ \\mathcal {U}} \\mathtt {R}$ is a pseudonatural equivalence.", "Thereby we get $\\mathfrak {A}(\\mathtt {E}(b), a) \\simeq \\mathfrak {A}(\\mathtt {E}\\widehat{\\mathcal {L}}\\widehat{\\mathcal {U}} (b) , a)\\simeq \\mathfrak {A}(\\mathcal {L}\\widehat{\\mathcal {U}} (b), a)\\simeq \\mathfrak {C}(\\widehat{ \\mathcal {U}} (b) , \\mathcal {U}(a) )\\simeq \\mathfrak {C}( \\widehat{ \\mathcal {U}} (b) , \\widehat{\\mathcal {U}}\\mathtt {R}(a) )\\simeq \\mathfrak {B}(b, \\mathtt {R}(a) ).$ This completes the proof that $\\mathtt {R}$ is right biadjoint to $\\mathtt {E}$ .", "Assume that $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ and $\\mathcal {B}:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ are pseudomonadic pseudofunctors, and their induced pseudomonads are idempotent.", "Then it is obvious that $\\mathcal {B}\\circ \\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {C}$ is also pseudomonadic and induces an idempotent pseudomonad.", "Indeed, by Theorem REF , this statement is equivalent to: compositions of right biadjoint local equivalences are right biadjoint local equivalences as well.", "Corollary 2.11 Assume that there is a pseudonatural equivalence ${ \\mathfrak {A}@{}[dr]\|-{\\simeq } &\\mathfrak {H}[l]\|-{\\mathtt {E}}\\\\\\mathfrak {B}[u]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {A}}}} } &\\mathfrak {C}[l]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}}[u]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}}}$ such that $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {A}}}}\\dashv \\mathcal {A}$ , $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}\\dashv \\mathcal {B}$ and $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}\\dashv \\mathcal {C}$ are pseudomonadic biadjunctions inducing idempotent pseudomonads $\\mathcal {T}_ {{}_ \\mathcal {A}}, \\mathcal {T}_ {{}_ \\mathcal {B}}, \\mathcal {T}_ {{}_ \\mathcal {C}}$ .", "Then $\\mathtt {E}\\dashv \\mathtt {R}$ and $\\mathtt {R}$ is a local equivalence.", "In particular, if $(X, a) $ is a $\\mathcal {T}_ {{}_ \\mathcal {B}}$ -pseudoalgebra that can be endowed with a $\\mathcal {T}_ {{}_ \\mathcal {A}}$ -pseudoalgebra structure, then $X$ can be endowed with a $\\mathcal {T}_ {{}_ \\mathcal {C}}$ -pseudoalgebra structure as well.", "Lemma REF and Corollary REF are results on our formal approach to descent theory, i.e.", "they give conditions to decide whether a given object can be endowed with a pseudoalgebra structure.", "In fact, most of the theorems proved in this paper are consequences of successive applications of these results, including Bénabou-Roubaud Theorem and other theorems within the context of [22], [23].", "However it does not deal with the technical “almost descent” aspects, which follow from the results on $\\mathfrak {F}$ -comparisons below." ], [ "$\\mathfrak {F}$ -comparisons", "Instead of restricting attention to objects that can be endowed with a pseudoalgebra structure, we often are interested in almost descent and descent objects as well.", "In the context of idempotent pseudomonads, these are objects that possibly do not have pseudoalgebra structure but have comparison 1-cells belonging to special classes of morphisms.", "In this subsection, every 2-category $\\mathfrak {H}$ is assumed to be endowed with a special subclass of morphisms $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ satisfying the following properties: Every equivalence of $\\mathfrak {H}$ belongs to $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ ; $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ is closed under composition; If there is an invertible 2-cell $f\\Rightarrow h$ in $\\mathfrak {H}$ such that $ f\\in \\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ , then $h\\in \\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ ; (Left) cancellation property: if $fg$ and $f$ belong to $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ , $g$ belongs to $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ as well.", "If $f$ is a morphism of $\\mathfrak {H}$ that belongs to $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ , we say that $f$ is an $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ -morphism.", "Recall that a morphism in a 2-category is faithful/fully faithful if its images by the representable 2-functors are faithful/fully faithful.", "Given any 2-category $\\mathfrak {H}$ , there are at least three important examples of subclasses of morphisms satisfying the properties above.", "The first class is the class of equivalences of $\\mathfrak {H}$ .", "The others are respectively the classes of faithful and fully faithful morphisms of $\\mathfrak {H}$ .", "Let $(\\mathcal {T}, \\mu , \\eta , \\Lambda , \\rho , \\Gamma )$ be an idempotent pseudomonad on a 2-category $\\mathfrak {H}$ .", "An object $X$ is an $(\\mathfrak {F}_ {{}_{\\mathfrak {H}}}, \\mathcal {T})$ -object if the comparison $\\eta _ {{}_X} : X\\rightarrow \\mathcal {T}(X) $ is an $\\mathfrak {F}_ {{}_{\\mathfrak {H}}}$ -morphism.", "We say that a pseudofunctor $\\mathcal {E}:\\mathfrak {H}\\rightarrow \\mathfrak {H}$ preserves $(\\mathfrak {F}_ {{}_{\\mathfrak {H}}}, \\mathcal {T})$ -objects if it takes $(\\mathfrak {F}_ {{}_{\\mathfrak {H}}}, \\mathcal {T})$ -objects to $(\\mathfrak {F}_ {{}_{\\mathfrak {H}}}, \\mathcal {T})$ -objects.", "Theorem REF follows from the construction given in the proof of Lemma REF .", "Similarly to Corollary REF , Theorem REF is a commutativity result for $(\\mathfrak {F}_ {{}_{\\mathfrak {H}}}, \\mathcal {T})$ -objects.", "Theorem 2.12 Let ${ \\mathfrak {A}@{}[dr]\|-{\\simeq } &\\mathfrak {H}[l]\|-{\\mathtt {E}}\\\\\\mathfrak {B}[u]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {A}}}} } &\\mathfrak {C}[l]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}}[u]\|-{\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}}}$ be a pseudonatural equivalence such that $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {A}}}}\\dashv \\mathcal {A}$ , $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}\\dashv \\mathcal {B}$ and $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}\\dashv \\mathcal {C}$ are biadjunctions inducing pseudomonads $\\mathcal {T}_ {{}_ \\mathcal {A}}, \\mathcal {T}_ {{}_ \\mathcal {B}}, \\mathcal {T}_ {{}_ \\mathcal {C}}$ .", "Also, we denote by $\\mathcal {T}$ the pseudomonad induced by the biadjunction $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {A}}}}\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}\\dashv \\mathcal {B}\\mathcal {A}$ .", "Assume that all the right biadjoints are local equivalences, $\\mathcal {B}$ takes $\\mathfrak {F}_ {{}_{\\mathfrak {B}}}$ -morphisms to $\\mathfrak {F}_ {{}_{\\mathfrak {C}}}$ -morphisms and $\\mathcal {T}_ {{}_ \\mathcal {C}}$ preserves $(\\mathfrak {F}_ {{}_{\\mathfrak {C}}}, \\mathcal {T}) $ -objects.", "If $X$ is a $(\\mathfrak {F}_ {{}_{\\mathfrak {C}}}, \\mathcal {T}_ {{}_{\\mathcal {B}}} ) $ -object of $\\mathfrak {C}$ and $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}(X) $ is a $(\\mathfrak {F} _ {{}_{\\mathfrak {B}}}, \\mathcal {T}_ {{}_\\mathcal {A}} )$ -object, then $X$ is a $(\\mathfrak {F} _ {{}_{\\mathfrak {C}}}, \\mathcal {T}_ {{}_\\mathcal {C}})$ -object as well.", "By the composition of biadjunctions, the unit $\\eta $ of $\\mathcal {T}$ is such that, for each object $Y$ of $\\mathfrak {C}$ , $\\eta _ {{}_Y} \\cong \\left(\\mathcal {B}\\eta ^{{}^\\mathcal {A}}\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}\\right)_ {{}_Y}\\, \\, \\, \\eta ^{{}^\\mathcal {B}} _ {{}_Y}.$ Let $X$ be an object satisfying the hypotheses of the theorem.", "We have that $\\left(\\mathcal {B}\\eta ^{{}^\\mathcal {A}}\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {B}}}}\\right)_ {{}_X}$ and $\\eta ^{{}^\\mathcal {B}} _ {{}_X}$ are $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphisms.", "Hence, by the closure under composition and by the invertible 2-cell above, we conclude that $\\eta _ {{}_X}$ is an $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphism.", "By the proof of Lemma REF , there is a pseudonatural transformation $\\alpha :\\mathcal {T}_ {{}_\\mathcal {C}}\\longrightarrow \\mathcal {T}$ such that we have in particular an invertible 2-cell ${ &X@{}[d]\|{\\cong }[ld]\|{\\eta ^{{}^\\mathcal {C}}_{{}_X} }[rd]\|{\\eta _ {{}_X} }&\\\\\\mathcal {T}_ {{}_\\mathcal {C}}(X) [rr]\|-{\\alpha _ {{}_X}} && \\mathcal {T}(X).", "}$ By the left cancellation property of the subclass $\\mathfrak {F} _ {{}_{\\mathfrak {C}}} $ and by the invertible 2-cell above, we only need to prove that $\\alpha _ {{}_X} $ is an $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphism to complete our proof that $\\eta ^{{}^\\mathcal {C}}_{{}_X} $ is an $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphism.", "Recall that $\\alpha _ {{}_X}$ is defined by $\\alpha _ {{}_X}:= (\\mathcal {B}\\mathcal {A}\\varepsilon ^{{}^\\mathcal {C}} \\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}) _ {{}_X}\\,\\,\\, (\\eta \\mathcal {T}_{{}_\\mathcal {C}} ) _ {{}_X}, $ in which $\\varepsilon ^{{}^\\mathcal {C}}$ is the counit of the biadjunction $\\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}\\dashv \\mathcal {C}$ .", "Firstly, $(\\mathcal {B}\\mathcal {A}\\varepsilon ^{{}^\\mathcal {C}} \\mathcal {L}_ {{}_{{}_{{}_\\mathcal {C}}}}) _ {{}_ X}$ is an equivalence.", "Secondly, since $\\mathcal {T}_{{}_\\mathcal {C}}$ preserves $(\\mathfrak {F}_ {{}_{\\mathfrak {C}}}, \\mathcal {T}) $ -objects, $(\\eta \\mathcal {T}_{{}_\\mathcal {C}}) _ {{}_X}\\cong (\\mathcal {T}_{{}_\\mathcal {C}}\\eta ) _ {{}_X}$ is an $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphism.", "Therefore $\\alpha _ {{}_X}$ is a composition of $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphisms and, hence, an $\\mathfrak {F}_{{}_{\\mathfrak {C}}}$ -morphism as well.", "The result above can be seen as a generalization of Corollary REF , since we get that corollary from Theorem REF by defining the classes $\\mathfrak {F}_ {{}_{\\mathfrak {A}}}$ , $ \\mathfrak {F}_ {{}_{\\mathfrak {B}}}$ , $\\mathfrak {F}_ {{}_{\\mathfrak {C}}}$ to be the classes of equivalences.", "It is known that the descent category and the category of algebras are 2-categorical limits (see, for instance, [46], [47], [21]).", "Thereby, our standpoint is to deal with the context of [22], [23] strictly guided by bilimits results.", "For the sake of this aim, we focus our study on the pseudomonads coming from the bicategorical analogue of the notion of right Kan extension.", "Actually, since the concept of “right Kan extension” plays the leading role in this work, “(pseudo-)Kan extension” means always right (pseudo-)Kan extension, while we always make the word “left” explicit when we refer to the dual notion.", "We explain below why we need to use a bicategorical notion of Kan extension, instead of employing the fully developed theory of enriched Kan extensions.", "The natural setting of (classical) descent theory is $2\\textrm {-}{\\rm \\mathsf {C}AT}$ .", "Although we can construct the bilimits related to descent theory as (enriched/strict) Kan extensions of 2-functors in the 3-category of 2-categories, 2-functors, 2-natural transformations and modifications (see [46], [42], [34], [35]), the necessary replacements [28], [34] do not make computations and formal manipulations any easier.", "Furthermore, most of the transformations between 2-functors that are necessary in the development of the theory are pseudonatural.", "Thus, to work within the “strict world” without employing repeatedly coherence theorems (such as the general coherence result of [28]), we would need to add hypotheses to assure that usual Kan extensions of pseudonaturally equivalent diagrams are pseudonaturally equivalent.", "This is not true in most of the cases: it is easy to construct examples of pseudonaturally isomorphic diagrams such that their usual Kan extensions are not pseudonaturally equivalent.", "For instance, consider the 2-category $\\mathfrak {A}$ below.", "${ \\mathsf {1} @<0.3 ex>[r]^{d^0 } @<-0.3 ex>[r]_{d^1 } & \\mathsf {2} }$ The 2-category $\\mathfrak {A}$ has no nontrivial 2-cells.", "Assume that $\\dot{\\mathfrak {A}} $ is the 2-category obtained from $\\mathfrak {A}$ adding an initial object $\\mathsf {0}$ , with full inclusion ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ .", "Now, if $\\ast $ is the terminal category and $\\nabla \\mathsf {2} $ is the category with two objects and one isomorphism between them (i.e.", "$\\nabla \\mathsf {2} $ is the localization of the preorder $\\mathsf {2}$ w.r.t.", "all morphisms), then there are two 2-natural isomorphism classes of diagrams $\\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ of the type below, while all such diagrams are pseudonaturally isomorphic.", "${ \\ast @<0.3 ex>[r] @<-0.3 ex>[r] & \\nabla \\mathsf {2} }$ These 2-natural isomorphism classes give pseudonaturally nonequivalent Kan extensions along ${\\rm t}$ .", "More precisely, if $\\mathcal {X},\\mathcal {Y}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ are such that $\\mathcal {X}(\\mathsf {1} )=\\mathcal {Y}(\\mathsf {1} )= \\ast $ , $\\mathcal {X}(\\mathsf {2} )= \\mathcal {Y}(\\mathsf {2} )= \\nabla \\mathsf {2} $ , $\\mathcal {X}( d^0 ) \\ne \\mathcal {X}( d^1 ) $ and $\\mathcal {Y}( d^0 ) = \\mathcal {Y}( d^1 ) $ ; then ${\\mathcal {R}an}_ {\\rm t}\\mathcal {X}(\\mathsf {0} ) = \\emptyset $ , while ${\\mathcal {R}an}_{\\rm t}\\mathcal {Y}(\\mathsf {0} ) = \\ast $ .", "Therefore ${\\mathcal {R}an}_{\\rm t}\\mathcal {X}$ and ${\\mathcal {R}an}_ {\\rm t}\\mathcal {Y}$ are not pseudonaturally equivalent, while $\\mathcal {X}$ is pseudonaturally isomorphic to $\\mathcal {Y}$ .", "The usual Kan extensions behave well if we add extra hypotheses related to flexible diagrams (see [5], [6], [42], [28], [34]).", "However, we do not give such restrictions and technicalities.", "Thereby we deal with the problems natively in the tricategory 2-${\\rm \\mathsf {C}AT}$ , without employing further coherence results.", "The first step is, hence, to understand the appropriate notion of Kan extension in this tricategory." ], [ "The Definition", "In a given tricategory, if ${\\rm t}: a\\rightarrow b$ , $f: a\\rightarrow c $ are 1-cells, we might consider that the formal right Kan extension of $f$ along ${\\rm t}$ is the right 2-reflection of $f$ along the 2-functor $[{\\rm t}, c]:[b,c]\\rightarrow [a,c] $ .", "That is to say, if it exists for all $f:a\\rightarrow c $ , the (formal) global Kan extension along ${\\rm t}: a\\rightarrow b $ would be a 2-functor $ [a,c]\\rightarrow [b,c] $ right 2-adjoint to $ [{\\rm t}, c] : [b, c]\\rightarrow [a,c] $ .", "But, in important cases, such concept is very restrictive, because it does not take into account the bicategorical structure of the hom-2-categories of the tricategory.", "Hence, it is possible to consider other notions of Kan extension, corresponding to the two other important notions of adjunctions between 2-categories [16], that is to say, lax adjunction and biadjunction.", "For instance, Gray [15] studied the notion of lax-Kan extension.", "We also consider an alternative notion of Kan extension in our tricategory 2-${\\rm \\mathsf {C}AT}$ , that is to say, the notion of pseudo-Kan extension, introduced in [34].", "In our case, the need of this concept comes from the fact that, even with many assumptions, the (formal) Kan extension of a pseudofunctor may not exist.", "Furthermore, we prove in Section that the descent object (descent category) and the Eilenberg-Moore object (Eilenberg-Moore category) can be easily described using our language.", "Henceforth, $\\mathfrak {A}, \\mathfrak {B}$ always denote small 2-categories.", "If ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ and $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ are pseudofunctors, a right pseudo-Kan extension of $\\mathcal {A}$ along ${\\rm t}$ , denoted by ${\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}$ , is, if it exists, a right bicategorical reflection of $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ along the pseudofunctor $\\left[ {\\rm t}, \\mathfrak {H}\\right] _{PS} : \\left[ \\mathfrak {B},\\mathfrak {H}\\right] _ {PS} \\rightarrow \\left[ \\mathfrak {A},\\mathfrak {H}\\right] _ {PS} $ .", "Although it is omitted in our notation, every right pseudo-Kan extension comes with a universal arrow $ \\varepsilon _ {{}_{\\mathcal {A}}} : \\left({\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}\\right) \\circ {\\rm t}\\longrightarrow \\mathcal {A}$ by Definition of right bicategorical reflection.", "Furthermore, by Remark we could actually give the definition of pseudo-Kan extension via the property of this universal arrow.", "That is to say, $({\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}, \\varepsilon _ {{}_{\\mathcal {A}}}) $ is the right pseudo-Kan extension of $\\mathcal {A}$ along ${\\rm t}$ if and only if $ \\left[ \\mathfrak {B},\\mathfrak {H}\\right] _ {PS} (-, {\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A})\\rightarrow \\left[ \\mathfrak {A},\\mathfrak {H}\\right] _ {PS} (- \\circ {\\rm t}, \\mathcal {A}) : \\qquad \\alpha \\mapsto \\varepsilon _ {{}_{\\mathcal {A}}}\\, \\left( \\alpha {\\rm t}\\right)$ defines a pseudonatural equivalence.", "By uniqueness (up to equivalence) of bicategorical reflections, pseudo-Kan extensions are unique up to pseudonatural equivalence.", "The global right pseudo-Kan extension along ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ w.r.t.", "a 2-category $\\mathfrak {H}$ is the right biadjoint of $\\left[ {\\rm t}, \\mathfrak {H}\\right] _{PS} $ , provided that it exists.", "That is to say, a pseudofunctor ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}:\\left[ \\mathfrak {A},\\mathfrak {H}\\right] _ {PS} \\rightarrow \\left[ \\mathfrak {B},\\mathfrak {H}\\right] _ {PS} $ such that $\\left[ {\\rm t}, \\mathfrak {H}\\right] _{PS}\\dashv {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}$ .", "Herein, the expression Kan extension refers to the usual notion of Kan extension in ${\\rm \\mathsf {C}AT}$ -enriched category theory.", "That is to say, if ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ and $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ are 2-functors, the (right) Kan extension of $\\mathcal {A}$ along ${\\rm t}$ , denoted by ${\\mathcal {R}an}_ {\\rm t}\\mathcal {A}: \\mathfrak {B}\\rightarrow \\mathfrak {H}$ , is (if it exists) the right 2-reflection of $\\mathcal {A}$ along the 2-functor $\\left[ {\\rm t}, \\mathfrak {H}\\right] $ .", "And the global Kan extension is a right 2-adjoint of $\\left[ {\\rm t}, \\mathfrak {H}\\right]: \\left[ \\mathfrak {B},\\mathfrak {H}\\right] \\rightarrow \\left[ \\mathfrak {A},\\mathfrak {H}\\right] ,$ in which $\\left[ \\mathfrak {B},\\mathfrak {H}\\right]$ denotes the 2-category of 2-functors $\\mathfrak {B}\\rightarrow \\mathfrak {H}$ , ${\\rm \\mathsf {C}AT}$ -natural transformations and modifications.", "If ${\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists, it is not generally true that ${\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ is pseudonaturally equivalent to ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ .", "This is a coherence problem, related to flexible diagrams [6], [28], [5], [34] and to the construction of bilimits via strict 2-limits [46], [47].", "For instance, in particular, using the results of [34], we can easily prove that, for a given pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ and a 2-functor ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ , we can replace $\\mathcal {A}$ by a pseudonaturally equivalent 2-functor $\\mathcal {A}^{\\prime }:\\mathfrak {A}\\rightarrow \\mathfrak {H}$ such that ${\\mathcal {R}an}_ {\\rm t}\\mathcal {A}^{\\prime } $ is equivalent to ${\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}^{\\prime }\\simeq {\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}$ , provided that $\\mathfrak {H}$ satisfies some completeness conditions (for instance, if $\\mathfrak {H}$ is ${\\rm \\mathsf {C}AT}$ -complete).", "In Section we show that the descent category, as defined and studied in [48], [22], [23], of a pseudocosimplicial object $D :\\Delta \\rightarrow {\\rm \\mathsf {C}AT}$ is equivalent to ${\\rm Ps}{\\mathcal {R}an}_{\\rm j}D (0) $ , in which ${\\rm j}: \\Delta \\rightarrow \\dot{\\Delta } $ is the inclusion of the category of nonempty finite ordinals into the category of finite ordinals.", "Observe that the Kan extension of a cosimplicial object does not give the descent object: it gives an equalizer (which is the notion of descent for dimension 1), although we might give the descent object via a Kan extension after replacing the (pseudo)cosimplicial objects by suitable strict versions of pseudocosimplicial objects as it is done in REF ." ], [ "Factorization", "Our setting often reduces to the study of right pseudo-Kan extensions of pseudofunctors $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ along ${\\rm t}$ , in which ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is the full inclusion of a small 2-category $\\mathfrak {A}$ into a small 2-category $\\dot{\\mathfrak {A}} $ which has only one extra object $\\mathsf {a} $ .", "[$\\mathsf {a}$ -inclusion] A 2-functor ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is called an $\\mathsf {a}$ -inclusion, if $\\mathsf {a} $ is an object of $\\dot{\\mathfrak {A}} $ and ${\\rm t}$ is a fully faithful functor between small 2-categories in which ${\\rm obj}(\\dot{\\mathfrak {A}} ) = {\\rm obj}(\\mathfrak {A})\\cup \\left\\lbrace \\mathsf {a}\\right\\rbrace $ is a disjoint union.", "The terminology established in the definition above makes reference to the extra object.", "Hence, using this terminology, a full inclusion $\\mathfrak {B}\\rightarrow \\dot{\\mathfrak {B}} $ between small 2-categories in which ${\\rm obj}(\\dot{\\mathfrak {B}} ) = {\\rm obj}(\\mathfrak {B})\\cup \\left\\lbrace \\mathsf {b}\\right\\rbrace $ is called a $\\mathsf {b} $ -inclusion.", "Theorem REF shows that a right pseudo-Kan extension of a pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ along an $\\mathsf {a}$ -inclusion ${\\rm t}$ is precisely $\\mathcal {A}$ extended with a weighted bilimit ${\\rm bilim}(\\dot{\\mathfrak {A}} (\\mathsf {a} , {\\rm t}- ), \\mathcal {A}) $ whenever such weighted bilimit exists.", "Thereby $\\mathsf {a} $ -inclusions are precisely what we need to give statements and proofs on (weighted) bilimits via pseudo-Kan extensions.", "In this setting, we have factorizations for pseudo-Kan extensions along $\\mathsf {a}$ -inclusions, which follow formally from the biadjunction $\\left[ {\\rm t}, \\mathfrak {H}\\right] _ {PS}\\dashv {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}$ .", "Theorem 3.1 (Factorization) Assume that $(\\left[ {\\rm t}, \\mathfrak {H}\\right] _ {PS}\\dashv {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}, \\eta , \\varepsilon ) $ is a biadjunction and ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is an $\\mathsf {a}$ -inclusion.", "If $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is a pseudofunctor, $\\mathsf {a}\\ne b $ and $f:b\\rightarrow \\mathsf {a}$ , $g:\\mathsf {a}\\rightarrow b $ are morphisms of $\\dot{\\mathfrak {A}} $ , we get induced “factorizations” (actually, invertible 2-cells): $@C=1em{ \\mathcal {A}(b)[rr]\|{\\mathcal {A}(f)}[dr]\|{f_{{}_{{}_\\mathcal {A}}} }&& \\mathcal {A}(\\mathsf {a})[ld]\|{\\eta ^\\mathsf {a}_ {{}_{{}_{\\mathcal {A}} }}} &\\mathcal {A}(\\mathsf {a})[rr]\|{\\mathcal {A}(g)}[dr]\|{\\eta ^\\mathsf {a}_ {{}_{{}_{\\mathcal {A}} }} }&& \\mathcal {A}(b)\\\\&{\\rm Ps}{\\mathcal {R}an}_{\\rm t}(\\mathcal {A}\\circ {\\rm t}) (\\mathsf {a})@{}[u]\|{\\cong } & & &{\\rm Ps}{\\mathcal {R}an}_{\\rm t}(\\mathcal {A}\\circ {\\rm t}) (\\mathsf {a})@{}[u]\|{\\cong }[ru]\|{g_{{}_{{}_\\mathcal {A}}}} &}$ in which $f_{{}_{{}_\\mathcal {A}}} : = {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t})(f)\\circ \\eta ^b_ {{}_{{}_{\\mathcal {A}} }}\\qquad \\qquad g_{{}_{{}_\\mathcal {A}}} := \\varepsilon _ {{}_{{}_{(\\mathcal {A}\\circ {\\rm t}) } }}^b\\circ {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t})(g)$ and $\\eta ^\\mathsf {a}_ {{}_{{}_{\\mathcal {A}} }}$ , $\\varepsilon _ {{}_{{}_{(\\mathcal {A}\\circ {\\rm t})} }}^{b} $ are the 1-cells induced by the components of $\\eta $ and $\\varepsilon $ .", "By the triangular invertible modifications of Definition , $g_{{}_{{}_\\mathcal {A}}}\\circ \\eta ^\\mathsf {a}_ {{}_{{}_{\\mathcal {A}} }} = \\varepsilon _ {{}_{{}_{(\\mathcal {A}\\circ {\\rm t})} }}^{b}\\circ {\\rm Ps}{\\mathcal {R}an}_{\\rm t}(\\mathcal {A}\\circ {\\rm t})(g)\\circ \\eta ^\\mathsf {a}_ {{}_{{}_{\\mathcal {A}} }}\\cong \\varepsilon _ {{}_{{}_{(\\mathcal {A}\\circ {\\rm t})} }}^{b}\\circ \\eta ^{b}_ {{}_{{}_{\\mathcal {A}} }}\\circ \\mathcal {A}(g)\\cong \\mathcal {A}(g) $ The factorization involving $\\mathcal {A}(f) $ follows from the pseudonaturality of $\\eta $ .", "Using the results of REF below, we find the factorizations above to be properties of (weighted) bilimits as we show in Section .", "For instance, we get the usual REF and the Eilenberg-Moore factorization, respectively, in Sections and ." ], [ "Weighted bilimits", "Similarly to the usual approach for (enriched) Kan extensions, we define what should be called pointwise pseudo-Kan extension.", "We prove that pseudo-Kan extensions exist and are pointwise whenever the codomain has suitable weighted bilimits.", "Pointwise (left) pseudo-Kan extensions are constructed with weighted bi(co)limits [34], [42], [47], [48], the bicategorical analogue of (enriched) weighted (co)limits [27], [12].", "Thereby we list some needed results on weighted bilimits.", "[Weighted bilimit] Let $\\mathcal {W}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ and $ \\mathcal {A}:\\mathfrak {A}\\rightarrow \\mathfrak {H}$ be pseudofunctors.", "If it exists, a weighted bilimit of $\\mathcal {A}$ with weight $\\mathcal {W}$ is an object of $\\mathfrak {H}$ , denoted by $\\left\\lbrace \\mathcal {W}, \\mathcal {A}\\right\\rbrace _ {{\\rm bi}} $ or by ${\\rm bilim}(\\mathcal {W}, \\mathcal {A})$ , endowed with an equivalence (pseudonatural in $X$ ) $\\mathfrak {H}(X,{\\rm bilim}(\\mathcal {W}, \\mathcal {A}) ) \\simeq \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right]_{PS}(\\mathcal {W}, \\mathfrak {H}(X, \\mathcal {A}-) ),$ which means that ${\\rm bilim}(\\mathcal {W}, \\mathcal {A})$ is a bicategorical representation of $\\mathfrak {H}^{\\rm op}\\rightarrow {\\rm \\mathsf {C}AT}: \\qquad X\\mapsto \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right]_{PS}(\\mathcal {W}, \\mathfrak {H}(X, \\mathcal {A}-) ).$ In other words, a weighted bilimit is, if it exists, the left bicategorical reflection of $\\mathcal {W}$ along the 2-functor $ \\mathfrak {H}^{\\rm op}\\rightarrow \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right]_{PS} : \\qquad X\\mapsto \\mathfrak {H}(X, \\mathcal {A}-), \\quad f\\mapsto \\mathfrak {H}(f, \\mathcal {A}-).", "$ By the uniqueness of the right bicategorical reflection, ${\\rm bilim}(\\mathcal {W}, \\mathcal {A}) $ is unique up to equivalence.", "We refer to it as the ($\\mathcal {W}$ -weighted) bilimit (of $\\mathcal {A}$ ).", "By Remark , in the context of the definition above, the weighted bilimit is a pair $({\\rm bilim}(\\mathcal {W}, \\mathcal {A}), \\rho _{{}_{\\mathcal {A}}})$ in which ${\\rm bilim}(\\mathcal {W}, \\mathcal {A})$ is an object of $\\mathfrak {H}$ and $\\rho _{{}_{{\\rm bilim}(\\mathcal {W}, \\mathcal {A}) }}: \\mathcal {W}\\longrightarrow \\mathfrak {H}( {\\rm bilim}(\\mathcal {W}, \\mathcal {A}) , \\mathcal {A}-) $ is a pseudonatural transformation such that $ \\mathfrak {H}(X,{\\rm bilim}(\\mathcal {W}, \\mathcal {A}) )\\rightarrow \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right]_{PS}(\\mathcal {W}, \\mathfrak {H}(X, \\mathcal {A}-) ), \\quad f\\mapsto \\mathfrak {H}(f, \\mathcal {A}-)\\, \\rho _{{}_{{\\rm bilim}(\\mathcal {W}, \\mathcal {A}) }} $ is an equivalence pseudonatural in $X$ .", "[Weighted bicolimit] The dual notion is called weighted bicolimit.", "If it exists, given a pseudofunctor $\\mathcal {W}^{\\prime } : \\mathfrak {A}^{\\rm op}\\rightarrow {\\rm \\mathsf {C}AT}$ , the $\\mathcal {W}^{\\prime } $ -weighted bicolimit of $\\mathcal {A}$, denoted by $\\mathcal {W}^{\\prime }_ {{\\rm bi}} \\mathcal {A}$ or by ${\\rm bicolim}(\\mathcal {W}^{\\prime },\\mathcal {A}) $ , is an object of $\\mathfrak {H}$ endowed with an equivalence (pseudonatural in $X$ ) $[\\mathfrak {A}, {\\rm \\mathsf {C}AT}]_{PS}(\\mathcal {W}^{\\prime }, \\mathfrak {H}( \\mathcal {A}- , X ) )\\simeq \\mathfrak {H}( {\\rm bicolim}(\\mathcal {W}^{\\prime } , \\mathcal {A}) , X ) .$ [Conical Bilimit] Analogously to the enriched case, denoting by $\\top $ the appropriate 2-functor constantly equal to the terminal category, the $\\top $ -weighted bi(co)limit of a pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ is the conical bi(co)limit of $\\mathcal {A}$ provided that it exists.", "The 2-category ${\\rm \\mathsf {C}AT}$ is bicategorically complete, that is to say, it has all (small) weighted bilimits.", "Indeed, if $\\mathcal {W}, \\mathcal {A}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ are pseudofunctors, we have that ${\\rm bilim}( \\mathcal {W}, \\mathcal {A}) \\simeq [\\mathfrak {A}, {\\rm \\mathsf {C}AT}]_{PS}(\\mathcal {W}, \\mathcal {A}) .$ Moreover, from the bicategorical Yoneda lemma, we get the strong bicategorical Yoneda lemma.", "Lemma 3.2 (Yoneda Lemma) Let $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ be a pseudofunctor.", "There is an equivalence ${\\rm bilim}( \\mathfrak {A}(X, -), \\mathcal {A})\\simeq \\mathcal {A}(X) $ pseudonatural in $X$ .", "Before giving results on pointwise pseudo-Kan extensions, the following result, which is mainly used in Section , already gives a glimpse of the relation between weighted bilimits and pseudo-Kan extensions.", "Theorem 3.3 Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ , $\\mathcal {W}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ be pseudofunctors.", "If the left pseudo-Kan extension ${\\rm Ps}{\\mathcal {L}an}_ {\\rm t}\\mathcal {W}$ exists and $\\mathcal {A}: \\mathfrak {B}\\rightarrow \\mathfrak {H}$ is a pseudofunctor, then there is an equivalence ${\\rm bilim}( \\mathcal {W}, \\mathcal {A}\\circ {\\rm t}) \\simeq {\\rm bilim}( {\\rm Ps}{\\mathcal {L}an}_ {{\\rm t}} \\mathcal {W}, \\mathcal {A}) $ either side existing if the other does.", "Assuming that ${\\rm Ps}{\\mathcal {L}an}_ {\\rm t}\\mathcal {W}$ exists, we have a pseudonatural equivalence between the 2-functors $X\\mapsto \\left[ \\mathfrak {B}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathcal {W}, \\mathfrak {H}(X, \\mathcal {A}\\circ {\\rm t}-)) \\quad \\mbox{and}\\quad X\\mapsto \\left[ \\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS} ({\\rm Ps}{\\mathcal {L}an}_ {\\rm t}\\mathcal {W}, \\mathfrak {H}(X , \\mathcal {A}-)).", "$ Therefore, assuming that any of the 2-functors above has a bicategorical representation $Y$ , we get that $Y$ is indeed a bicategorical representation of both 2-functors." ], [ "Pseudoends", "There is one important notion remaining to study weighted bilimits: the bicategorical analogue of the end [27], [12].", "Below and in [34], we give a direct definition of pseudoend, avoiding the unnecessary work on bicategorical analogues of the extranatural transformations of the classical enriched case.", "In order to do this, we consider the usual characterization of the end of a 2-functor $T: \\mathfrak {A}^{\\rm op}\\times \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ in the strict/enriched case given by: $\\left[ \\mathfrak {A}^{\\rm op}\\times \\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] (\\mathfrak {A}(-,-), T) .$ Herein, we do not work with ends and, hence, we reserve the integral sign to denote pseudoends.", "[Pseudoend] The pseudoend of a pseudofunctor $T: \\mathfrak {A}^{\\rm op}\\times \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ is $ \\int _ \\mathfrak {A}T := [\\mathfrak {A}^{\\rm op}\\times \\mathfrak {A}, {\\rm \\mathsf {C}AT}] _ {PS} ( \\mathfrak {A}(-,-), T)\\simeq {\\rm bilim}(\\mathfrak {A}(-,-), T) .$ In order to avoid possible confusions when stating results that involve iterated pseudoends, we often adopt a terminology in which the pseudoend is indexed by the variable as well.", "That is to say, we use the following terminology: $ \\int _ {a\\in \\mathfrak {A}} T(a,a) : = \\int _ \\mathfrak {A}T .", "$ This definition allows us to get analogues of the usual fundamental results on ends of the strict/enriched case [27], [12].", "We start by: Proposition 3.4 (Fundamental equivalence for pseudoends) Let $\\mathcal {A}, \\mathcal {B}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ be pseudofunctors.", "There is a pseudonatural equivalence $ \\int _ {a\\in \\mathfrak {A}} \\mathfrak {H}(\\mathcal {A}(a), \\mathcal {B}(a) ) \\simeq [\\mathfrak {A}, \\mathfrak {H}] _ {PS}( \\mathcal {A}, \\mathcal {B}) .$ Firstly, observe that a pseudonatural transformation $\\alpha : \\mathfrak {A}(-,-)\\longrightarrow \\mathfrak {H}(\\mathcal {A}-, \\mathcal {B}- ) $ corresponds to a collection of 1-cells $\\alpha _{{}_{(W,X)}}: \\mathfrak {A}(W,X)\\rightarrow \\mathfrak {H}(\\mathcal {A}(W), \\mathcal {B}(X) ) $ and collections of invertible 2-cells $\\alpha _ {{}_{(Y,f)}} : \\mathfrak {H}(\\mathcal {A}(Y) , \\mathcal {B}(f))\\alpha _ {{} _ {(Y,W)}}&\\cong & \\alpha _ {{} _ {(Y,X)}}\\mathfrak {A}( Y , f)\\\\\\alpha _ {{}_{(f,Y)}} : \\mathfrak {H}(\\mathcal {A}(f) , \\mathcal {B}(Y))\\alpha _ {{} _ {(X,Y)}}&\\cong & \\alpha _ {{} _ {(W,Y)}}\\mathfrak {A}( f , Y )$ such that, for each object $Y$ of $\\mathfrak {A}$ , $\\alpha _{{}_{(Y,-)}} $ and $\\alpha _{{}_{(-,Y)}} $ (with the invertible 2-cells above) are pseudonatural transformations.", "In other words, pseudonatural transformations are transformations which are pseudonatural in each variable.", "By the bicategorical Yoneda lemma, we get what we want: such a pseudonatural transformation corresponds (up to isomorphism) to a collection of 1-cells $\\gamma _ {{}_W}:= \\alpha _ {{}_{W,W}}({\\rm Id}_ {{}_W}) : \\mathcal {A}(W)\\rightarrow \\mathcal {B}(W) $ with (coherent) invertible 2-cells $\\mathcal {B}(f)\\circ \\gamma _ {{}_W}\\cong \\gamma _ {{}_W}\\circ \\mathcal {A}(f) $ .", "Hence, the original bicategorical Yoneda lemma may be reinterpreted.", "Given a pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ , we have an equivalence $ \\int _ {a\\in \\mathfrak {A}} {\\rm \\mathsf {C}AT}(\\mathfrak {A}(X, a), \\mathcal {A}(a) ) \\simeq \\mathcal {A}(X) $ pseudonatural in $X$ .", "Theorem REF is the bicategorical analogue of the Fubini's Theorem in the enriched context.", "Theorem 3.5 Given a pseudofunctor $T: \\mathfrak {A}^{{\\rm op}}\\times \\mathfrak {B}^{{\\rm op}}\\times \\mathfrak {A}\\times \\mathfrak {B}^{\\rm op}\\rightarrow {\\rm \\mathsf {C}AT}$ , there are pseudofunctors $T^{{}^\\mathfrak {B}}: \\mathfrak {A}^{{\\rm op}}\\times \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ and $ T ^{{}^\\mathfrak {A}}: \\mathfrak {B}^{{\\rm op}}\\times \\mathfrak {B}\\rightarrow {\\rm \\mathsf {C}AT}$ such that $ T^{{}^\\mathfrak {B}}(X,Y) \\cong \\int _ {b\\in \\mathfrak {B}} T(X, b, Y, b)\\quad \\mbox{and}\\quad T^{{}^\\mathfrak {A}}(W, Z)\\cong \\int _ {a\\in \\mathfrak {A}} T(a, W, a, Z).", "$ Furthermore, $ \\displaystyle \\int _ {\\mathfrak {A}\\times \\mathfrak {B}} T \\simeq \\int _ {\\mathfrak {A}}T^{{}^\\mathfrak {B}} \\simeq \\int _ {\\mathfrak {B}}T^{{}^\\mathfrak {A}} $ .", "We usually denote the iterated pseudoends of the result above by: $ \\int _{a\\in \\mathfrak {A}} \\int _ {b\\in \\mathfrak {B}} T(a,b,a,b) := \\int _ {\\mathfrak {A}}T^{{}^\\mathfrak {B}} \\quad \\mbox{and}\\quad \\int _ {b\\in \\mathfrak {B}}\\int _ {a\\in \\mathfrak {A}} T(a,b,a,b) := \\int _ {\\mathfrak {B}}T^{{}^\\mathfrak {A}}.", "$" ], [ "Pointwise pseudo-Kan extensions", "If we consider the full 2-subcategory $ \\mathfrak {H}_ \\mathcal {Y}$ of $[\\mathfrak {B}^{{\\rm op}}, {\\rm \\mathsf {C}AT}]_ {PS} $ such that the objects of $ \\mathfrak {H}_ \\mathcal {Y}$ are the bicategorically representable pseudofunctors of a 2-category $\\mathfrak {H}$ , the Yoneda embedding $\\mathcal {Y}: \\mathfrak {H}\\rightarrow \\mathfrak {H}_ \\mathcal {Y}$ is a biequivalence: that is to say, we can choose a pseudofunctor $ I: \\mathfrak {H}_ \\mathcal {Y}\\rightarrow \\mathfrak {H}$ and pseudonatural equivalences $\\mathcal {Y}I\\simeq {\\rm Id}$ and $I\\mathcal {Y}\\simeq {\\rm Id}$ .", "Therefore if $\\mathfrak {H}$ has all the weighted bilimits of a pseudofunctor $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ , there is a pseudofunctor ${\\rm bilim}(-, \\mathcal {A}):\\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}^{ {\\rm op}}\\rightarrow \\mathfrak {H}$ which is unique up to pseudonatural equivalence and which gives the bilimits of $\\mathcal {A}$ [46], [34].", "More precisely, in this case, we are actually assuming that the pseudofunctor $F: \\left[ \\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}^{{\\rm op}} \\rightarrow \\left[ \\mathfrak {H}^{{\\rm op}}, {\\rm \\mathsf {C}AT}\\right] _ {PS}$ , in which $F(\\mathcal {W}): \\mathfrak {B}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}: \\qquad X \\mapsto \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\mathcal {W}, \\mathfrak {H}(X , \\mathcal {A}- ))$ is such that $F(\\mathcal {W})$ has a bicategorical representation for every weight $\\mathcal {W}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ .", "Therefore $F$ can be lifted to a pseudofunctor $ F: \\left[ \\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}^{{\\rm op}} \\rightarrow \\mathfrak {H}_ \\mathcal {Y}$ .", "Hence we can take ${\\rm bilim}( -,\\mathcal {A}) := IF $ .", "Theorem 3.6 Assume that $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}, {\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ are pseudofunctors.", "If $\\mathfrak {H}$ has the weighted bilimit ${\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) $ for every object $b$ of $\\mathfrak {B}$ , then ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists.", "Furthermore, there is an equivalence ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b)\\simeq {\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A})$ pseudonatural in $b$ .", "In this proof, we denote by ${\\mathcal {R}AN}_ {\\rm t}\\mathcal {A}$ the pseudofunctor defined by ${\\mathcal {R}AN}_ {\\rm t}\\mathcal {A}(b) := {\\rm bilim}(\\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) .$ By the propositions presented in this section, we have the following pseudonatural equivalences (in $S$ ): $\\left[\\mathfrak {B}, \\mathfrak {H}\\right] _ {PS}(S, {\\mathcal {R}AN}_ {\\rm t}\\mathcal {A})&\\simeq & \\int _ {b\\in \\mathfrak {B}} \\mathfrak {H}(S(b), {\\mathcal {R}AN}_ {\\rm t}\\mathcal {A}(b))\\\\&\\simeq & \\int _{b\\in \\mathfrak {B}} \\mathfrak {H}(S(b), {\\rm bilim}(\\mathfrak {B}(b, {\\rm t}-), \\mathcal {A})) \\\\& \\simeq & \\int _ {b\\in \\mathfrak {B}} \\left[ \\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\mathfrak {B}(b, {\\rm t}-), \\mathfrak {H}(S(b), \\mathcal {A}-))\\\\& \\simeq & \\int _ {b\\in \\mathfrak {B}} \\int _ {a\\in \\mathfrak {A}} {\\rm \\mathsf {C}AT}(\\mathfrak {B}(b, {\\rm t}(a)), \\mathfrak {H}(S(b), \\mathcal {A}(a)))\\\\& \\simeq & \\int _ {a\\in \\mathfrak {A}} \\int _ {b\\in \\mathfrak {B}} {\\rm \\mathsf {C}AT}(\\mathfrak {B}(b, {\\rm t}(a)), \\mathfrak {H}(S(b), \\mathcal {A}(a)))\\\\&\\simeq & \\int _ {a\\in \\mathcal {A}} \\mathfrak {H}(S\\circ {\\rm t}(a), \\mathcal {A}(a))\\\\&\\simeq & \\left[ \\mathcal {A}, \\mathfrak {H}\\right]_ {PS} (S\\circ {\\rm t}, \\mathcal {A}).$ More precisely, the first, fourth, sixth and seventh pseudonatural equivalences come from the fundamental equivalence for pseudoends (Proposition REF ), while the second and third are, respectively, the definitions of ${\\mathcal {R}AN}_ {\\rm t}\\mathcal {A}$ and the definition of weighted bilimit.", "The remaining pseudonatural equivalence follows from Theorem REF .", "These pseudonatural equivalences show that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists and ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}\\simeq {\\mathcal {R}AN}_ {\\rm t}\\mathcal {A}$ .", "Corollary 3.7 Assume that $\\mathcal {A}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}, {\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ are pseudofunctors.", "Then ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists and ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b) \\simeq \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathfrak {B}(b, {\\rm t}-) , \\mathcal {A}).$ This result follows from Theorem REF and from the fact that ${\\rm bilim}(\\mathfrak {B}(b, {\\rm t}-) , \\mathcal {A}) \\simeq \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathfrak {B}(b, {\\rm t}-) , \\mathcal {A}).$ It is clear that Theorem REF has a dual.", "That is to say, we have an equivalence ${\\rm Ps}{\\mathcal {L}an}_ {\\rm t}\\mathcal {A}(b) \\simeq {\\rm bicolim}(\\mathfrak {B}({\\rm t}-,b), \\mathcal {A}) $ pseudonatural in $b$ whenever ${\\rm bicolim}(\\mathfrak {B}({\\rm t}-,b), \\mathcal {A})$ exists for each $b$ in $\\mathfrak {B}$ .", "By Remark REF and Theorem REF , if $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ is a pseudofunctor, if it exists, the conical bilimit of $\\mathcal {A}$ is equivalent to ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}) (\\mathsf {a})\\simeq {\\rm bilim}( \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}- ), \\mathcal {A})$ in which ${\\rm t}:\\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is the $\\mathsf {a}$ -inclusion such that $\\mathsf {a}$ is the initial object added to $\\mathfrak {A}$ .", "[Preservation of pseudo-Kan extensions] Let $U : \\mathfrak {H}\\rightarrow \\mathfrak {H}^{\\prime } $ , $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ and ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ be pseudofunctors.", "Assume that ${\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}$ exists and has the universal arrow $\\varepsilon _ {{}_{\\mathcal {A}}}$ .", "We say that $U$ preserves this pseudo-Kan extension if ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(U\\circ \\mathcal {A}) \\simeq U\\circ {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}\\mbox{ and }U\\varepsilon _ {{}_{\\mathcal {A}}} : (U\\circ {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A})\\circ {\\rm t}\\longrightarrow U\\circ \\mathcal {A}$ is the universal arrow of ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(U\\circ \\mathcal {A})$ .", "Lemma 3.8 If $U$ has a left biadjoint, then it preserves all the existing right pseudo-Kan extensions.", "[Pointwise pseudo-Kan extension] Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}, \\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}$ be pseudofunctors such that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists.", "We say that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ is a pointwise (right) pseudo-Kan extension of $\\mathcal {A}$ along ${\\rm t}$ if ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ is preserved by all representable pseudofunctors.", "The definition of pointwise pseudo-Kan extension is motivated by the usual definition of pointwise Kan extension in the strict/enriched case.", "See page 240 of [38] or page 52 and 54 of [12].", "Analogously to the enriched case (see [12]), pointwise pseudo-Kan extensions are pointwise constructed as in Theorem REF by weighted bilimits.", "Corollary REF makes this statement precise.", "Corollary 3.9 Assume that $\\mathcal {A}: \\mathfrak {A}\\rightarrow \\mathfrak {H}, {\\rm t}: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ are pseudofunctors.", "The pseudofunctor $\\mathcal {A}$ has a pointwise pseudo-Kan extension ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ if and only if $\\mathfrak {H}$ has the weighted bilimit ${\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) $ for every object $b$ of $\\mathfrak {B}$ .", "In this case, there is an equivalence ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b)\\simeq {\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A})$ pseudonatural in $b$ .", "Firstly, assume that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ is pointwise.", "Since it is pointwise, we have an equivalence $\\mathfrak {H}(X, {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b) ) \\simeq {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathfrak {H}(X, \\mathcal {A}- ) (b) $ pseudonatural in $b$ , while, by the bicategorical Yoneda lemma and the universal property of ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathfrak {H}(X, \\mathcal {A}- )$ , we have an equivalence ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathfrak {H}(X, \\mathcal {A}- ) (b)&\\simeq & \\left[\\mathfrak {B}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\mathfrak {B}(b, -) , {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathfrak {H}(X, \\mathcal {A}- ) )\\\\&\\simeq & \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\mathfrak {B}(b, {\\rm t}-) , \\mathfrak {H}(X, \\mathcal {A}- ) )$ pseudonatural in $b$ .", "This completes the proof that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b)$ is the left bicategorical reflection of $\\mathfrak {B}(b, {\\rm t}-)$ along $\\mathfrak {H}^{\\rm op}\\rightarrow \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right]_{PS} : \\qquad X\\mapsto \\mathfrak {H}(X, \\mathcal {A}- ) ,$ which means that ${\\rm bilim}(\\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) $ exists and ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b)\\simeq {\\rm bilim}(\\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) $ pseudonatural in $b $ .", "Reciprocally, assume that ${\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}) $ exists for every $b $ in $\\mathfrak {B}$ .", "In this case, by Theorem REF ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}$ exists and there is an equivalence ${\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A})\\simeq {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b)$ pseudonatural in $b $ .", "Given any $X $ in $\\mathfrak {H}$ , we have equivalences $\\mathfrak {H}(X, {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}(b))\\simeq \\mathfrak {H}(X, {\\rm bilim}( \\mathfrak {B}(b, {\\rm t}-), \\mathcal {A}))\\simeq \\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathfrak {B}(b, {\\rm t}-) , \\mathfrak {H}(X, \\mathcal {A}- ) ) $ pseudonatural in $b$ .", "Since by Corollary REF we have an equivalence $\\left[\\mathfrak {A}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathfrak {B}(b, {\\rm t}-) , \\mathfrak {H}(X, \\mathcal {A}- ) ) \\simeq {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathfrak {H}(X, \\mathcal {A}- ) (b) $ pseudonatural in $b $ , the proof is complete.", "In the enriched case [12], [27], in the presence of weighted limits, pointwise Kan extensions are constructed pointwise by equalizers of products of cotensor products, since every weighted limit can be seen as such.", "In the bicategorical case, weighted bilimits are descent objects of bicategorical products and cotensor products whenever they exist and, therefore, the result above shows that, in suitable cases, pointwise pseudo-Kan extensions can be constructed pointwise by them (see [34]).", "In this paper, for simplicity, we always assume that $\\mathfrak {H}$ is a bicategorically complete 2-category, or at least $\\mathfrak {H}$ has enough bilimits to construct the considered (right) pseudo-Kan extensions as pointwise pseudo-Kan extensions.", "The pointwise pseudo-Kan extension was studied originally in [34] using the Biadjoint Triangle Theorem proved therein.", "The construction presented above is similar to the usual approach of the enriched case [27], [12], while the argument via biadjoint triangles of [34] is not." ], [ "The pseudomonad $\\left\\langle {\\rm t}\\right\\rangle : = {\\rm Ps}{\\mathcal {R}an}_{\\rm t}(-\\circ {\\rm t})$", "By the (bicategorical) Yoneda lemma, whenever ${\\rm t}$ is a local equivalence, if the pseudo-Kan extension ${\\rm Ps}{\\mathcal {R}an}_{\\rm t}\\mathcal {A}$ exists, it is actually a pseudoextension.", "More precisely: Theorem 3.10 If ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is a local equivalence and there is a biadjunction $\\left[ {\\rm t}, \\mathfrak {H}\\right] _{PS}\\dashv {\\rm Ps}{\\mathcal {R}an}_{\\rm t},$ its counit is a pseudonatural equivalence.", "Thereby ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}: \\left[ \\mathfrak {A}, \\mathfrak {H}\\right] _ {PS}\\rightarrow \\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS} $ is a local equivalence and, hence, pseudomonadic and the induced pseudomonad is idempotent.", "It follows from the bicategorical Yoneda lemma.", "By Lemma REF , if $X$ is an object of $\\mathfrak {A}$ , ${\\rm bilim}( \\dot{\\mathfrak {A}} ( {\\rm t}(X), {\\rm t}-), \\mathcal {A})\\simeq {\\rm bilim}( \\mathfrak {A}(X , -), \\mathcal {A})\\simeq \\mathcal {A}(X) $ .", "In the context of the result above, we denote the idempotent pseudomonad induced by the biadjunction $\\left[ {\\rm t}, \\mathfrak {H}\\right] _{PS}\\dashv {\\rm Ps}{\\mathcal {R}an}_{\\rm t}$ by ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(-\\circ {\\rm t})$ or, for short, $\\left\\langle {\\rm t}\\right\\rangle $ .", "Our interest is to study the objects of $\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS} $ that can be endowed with $\\left\\langle {\\rm t}\\right\\rangle $ -pseudoalgebra structure.", "Assuming that ${\\rm t}$ is a local equivalence, this means that our interest is to study the image of the forgetful Eilenberg-Moore 2-functor $\\left\\langle {\\rm t}\\right\\rangle \\textrm {-}{\\rm \\mathsf {A}lg}\\rightarrow \\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}.$ [Effective Diagrams] Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} , \\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ be pseudofunctors.", "$\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ is of effective ${\\rm t}$ -descent if $\\mathcal {A}$ can be endowed with a $\\left\\langle {\\rm t}\\right\\rangle $ -pseudoalgebra structure.", "We now can apply the results of Section on idempotent pseudomonads.", "Firstly, by Theorem REF , we can easily study the $\\left\\langle {\\rm t}\\right\\rangle $ -pseudoalgebra structures on diagrams, using the unit of the pseudomonad.", "Theorem 3.11 Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ be a local equivalence and $\\mathcal {A}:\\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ a pseudofunctor.", "The following conditions are equivalent: $\\mathcal {A}$ is of effective ${\\rm t}$ -descent; The component of the unit on $\\mathcal {A}$ /comparison $\\eta _ {{}_\\mathcal {A}} : \\mathcal {A}\\rightarrow {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t}) $ is a pseudonatural equivalence; The comparison $\\eta _ {{}_\\mathcal {A}} : \\mathcal {A}\\rightarrow {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t}) $ is a pseudonatural pseudosection.", "The component of the unit $\\eta _ {{}_\\mathcal {A}} : \\mathcal {A}\\longrightarrow {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t}) $ is a pseudonatural equivalence if and only if all components of $\\eta _ {{}_\\mathcal {A}} $ are equivalences.", "By Theorem REF , assuming that ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is an $\\mathsf {a}$ -inclusion, $\\eta _ {{}_\\mathcal {A}}^{b} $ is an equivalence for all ${b}$ in $\\mathfrak {A}$ .", "Thereby we get: Lemma 3.12 Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ be an $\\mathsf {a}$ -inclusion.", "A pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is of effective ${\\rm t}$ -descent if and only if $\\eta _ {{}_\\mathcal {A}} ^\\mathsf {a} : \\mathcal {A}(\\mathsf {a} )\\rightarrow {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t}) (\\mathsf {a}) $ is an equivalence." ], [ "Commutativity", "Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}}$ and ${\\rm h}: \\mathfrak {B}\\rightarrow \\dot{\\mathfrak {B}} $ be, respectively, an $\\mathsf {a}$ -inclusion and a $\\mathsf {b}$ -inclusion (see Definition REF and Remark REF ).", "Unless we explicitly state otherwise, henceforth we always consider right pseudo-Kan extensions along such type of inclusions.", "In general, we have that (see [48]): $\\left[\\dot{\\mathfrak {A}}\\times \\dot{\\mathfrak {B}}, \\mathfrak {H}\\right] _ {PS}\\approx \\left[\\dot{\\mathfrak {A}}, \\left[ \\dot{\\mathfrak {B}}, \\mathfrak {H}\\right] _ {PS}\\right] _ {PS}\\cong \\left[\\dot{\\mathfrak {B}}, \\left[ \\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}\\right] _ {PS}$ .", "Thereby every pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}}\\times \\dot{\\mathfrak {B}}\\rightarrow \\mathfrak {H}$ can be seen (up to pseudonatural equivalence) as a pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\left[ \\dot{\\mathfrak {B}}, \\mathfrak {H}\\right] _ {PS} $ .", "Also, $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\left[ \\dot{\\mathfrak {B}}, \\mathfrak {H}\\right] _ {PS} $ can be seen as a pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {B}} \\rightarrow \\left[ \\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS} $ .", "Applying our formal approach of Section to our context of pseudo-Kan extensions, we get theorems on commutativity as we show below.", "Theorem 3.13 If $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is an effective ${\\rm t}$ -descent pseudofunctor and $\\mathcal {T}$ is an idempotent pseudomonad on $\\mathfrak {H}$ such that $\\mathcal {A}\\circ {\\rm t}$ can be factorized through $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}\\rightarrow \\mathfrak {H}$ , then $\\mathcal {A}(\\mathsf {a}) $ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure.", "Let $\\mathcal {L}\\dashv \\mathcal {U}$ be the biadjunction induced by $\\mathcal {T}$ and $\\widehat{\\mathfrak {H}}:=\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\mathsf {A}lg}$ (see Definition REF and Theorem REF ).", "Observe that the pseudonatural equivalence ${ \\left[ \\mathfrak {A},\\widehat{\\mathfrak {H}}\\right] _ {PS}@{}[ddrr]\|-{ = }&&\\left[ \\dot{\\mathfrak {A}} ,\\widehat{\\mathfrak {H}}\\right] _ {PS}[ll]\|-{\\left[ {\\rm t},\\widehat{\\mathfrak {H}}\\right] _ {PS} }\\\\&&\\\\\\left[ \\mathfrak {A},\\mathfrak {H}\\right] _ {PS}[uu]\|-{\\left[ \\mathfrak {A},\\mathcal {L}\\right] _ {PS} } &&\\left[ \\dot{\\mathfrak {A}} ,\\mathfrak {H}\\right] _ {PS}[ll]\|-{\\left[ {\\rm t},\\mathfrak {H}\\right] _ {PS}}[uu]\|-{\\left[ \\dot{\\mathfrak {A}} ,\\mathcal {L}\\right] _ {PS}}}$ satisfies the hypotheses of Corollary REF .", "If $\\mathcal {A}:\\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is an effective ${\\rm t}$ -descent pseudofunctor such that all the objects of the image of $\\mathcal {A}\\circ {\\rm t}$ have $\\mathcal {T}$ -pseudoalgebra structure, it means that $\\mathcal {A}$ satisfies the hypotheses of Corollary REF .", "That is to say, $\\mathcal {A}$ is a $\\left\\langle {\\rm t}\\right\\rangle $ -pseudoalgebra that can be endowed with a $\\left[ \\mathfrak {A}, \\mathcal {T}\\right]_ {PS}$ -pseudoalgebra structure.", "Thereby, by Corollary REF , $\\mathcal {A}$ can be endowed with a $\\left[ \\dot{\\mathfrak {A}}, \\mathcal {T}\\right] _ {PS} $ -pseudoalgebra structure.", "Corollary 3.14 Let $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\left[ \\dot{\\mathfrak {B}} , \\mathfrak {H}\\right] _ {PS}$ be an effective ${\\rm t}$ -descent pseudofunctor such that the diagrams in the image of $\\mathcal {A}\\circ {\\rm t}$ are of effective ${\\rm h}$ -descent, then $\\mathcal {A}(\\mathsf {a} ) $ is of effective ${\\rm h}$ -descent as well.", "Since $\\left\\langle {\\rm h}\\right\\rangle $ is idempotent, this result is Theorem REF applied to the case $\\mathcal {T}= \\left\\langle {\\rm h}\\right\\rangle $ .", "Corollary 3.15 Assume that the pseudofunctors $\\widehat{\\mathcal {A}}: \\dot{\\mathfrak {A}}\\rightarrow \\left[ \\dot{\\mathfrak {B}} , \\mathfrak {H}\\right] _ {PS}$ and $\\bar{\\mathcal {A}}: \\dot{\\mathfrak {B}}\\rightarrow \\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}$ are mates such that the diagrams in the image of $\\widehat{\\mathcal {A}}\\circ {\\rm t}$ and $ \\bar{\\mathcal {A}}\\circ {\\rm h}$ are respectively of effective ${\\rm h}$ - and ${\\rm t}$ -descent.", "We have that $\\widehat{\\mathcal {A}}(\\mathsf {a} ) $ is of effective ${\\rm h}$ -descent if and only if $\\bar{\\mathcal {A}} (\\mathsf {b} ) $ is of effective ${\\rm t}$ -descent." ], [ "Almost descent pseudofunctors", "Recall that a 1-cell in a 2-category $\\mathfrak {H}$ is called faithful/fully faithful if its images by the (covariant) representable 2-functors are faithful/fully faithful.", "Let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ be an $\\mathsf {a}$ -inclusion.", "A pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is of almost ${\\rm t}$ -descent/${\\rm t}$ -descent if $\\eta _ {{}_\\mathcal {A}} ^\\mathsf {a} : \\mathcal {A}(\\mathsf {a} )\\rightarrow {\\rm Ps}{\\mathcal {R}an}_ {\\rm t}(\\mathcal {A}\\circ {\\rm t}) (\\mathsf {a}) $ is faithful/fully faithful.", "Consider the class $\\mathfrak {F} _ {{}_{\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}}}$ of pseudonatural transformations in $\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}$ whose components are faithful.", "This class satisfies the properties described in REF .", "Also, a pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ is of almost descent if and only if $\\mathcal {A}$ is a $(\\mathfrak {F} _ {{}_{\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}}}, \\left\\langle {\\rm t}\\right\\rangle )$ -object.", "Analogously, if we take the class $\\mathfrak {F} _ {{}_{\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}}}^{\\prime }$ of objectwise fully faithful pseudonatural transformations, $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ is of descent if and only if $\\mathcal {A}$ is a $(\\mathfrak {F} _ {{}_{\\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}}}^{\\prime }, \\left\\langle {\\rm t}\\right\\rangle )$ -object.", "Since in our context of right pseudo-Kan extensions along local equivalences the hypotheses of Theorem REF hold, we get the corollaries below.", "Again, we are considering full inclusions ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}}$ , ${\\rm h}: \\mathfrak {B}\\rightarrow \\dot{\\mathfrak {B}} $ as in REF .", "Corollary 3.16 Let $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\left[ \\dot{\\mathfrak {B}} , \\mathfrak {H}\\right] _ {PS}$ be an almost ${\\rm t}$ -descent pseudofunctor such that the pseudofunctors in the image of $\\mathcal {A}\\circ {\\rm t}$ are of almost ${\\rm h}$ -descent.", "In this case, $\\mathcal {A}(\\mathsf {a} ) $ is also of almost ${\\rm h}$ -descent.", "Similarly, if $\\mathcal {A}$ is of ${\\rm t}$ -descent and the pseudofunctors of the image of $\\mathcal {A}\\circ {\\rm t}$ are of ${\\rm h}$ -descent, then $\\mathcal {A}(\\mathsf {a} ) $ is of ${\\rm h}$ -descent as well.", "In order to show that both cases fit in the technical conditions of the hypothesis of REF , we only need to observe that any poitwise (right) pseudo-Kan extension pseudofunctor preserves pointwise faithful and pointwise fully faithful pseudonatural transformations.", "In order to verify that, it is enough to see that, given any small 2-category $\\mathfrak {C}$ , we have that $\\left[ \\mathfrak {C}, {\\rm \\mathsf {C}AT}\\right]_{PS} (\\mathcal {B}, \\alpha ), $ is pointwise (fully) faithful whenever $\\alpha $ is so.", "Corollary 3.17 Assume that the mates $\\widehat{\\mathcal {A}}: \\dot{\\mathfrak {A}}\\rightarrow \\left[ \\dot{\\mathfrak {B}} , \\mathfrak {H}\\right] _ {PS}$ and $\\bar{\\mathcal {A}}: \\dot{\\mathfrak {B}}\\rightarrow \\left[ \\dot{\\mathfrak {A}} , \\mathfrak {H}\\right] _ {PS}$ are such that the diagrams in the image of $\\widehat{\\mathcal {A}}\\circ {\\rm t}$ and $\\bar{\\mathcal {A}}\\circ {\\rm h}$ are respectively of almost ${\\rm h}$ - and ${\\rm t}$ -descent.", "In this case, $\\widehat{\\mathcal {A}}(\\mathsf {a} ) $ is of almost ${\\rm h}$ -descent if and only if $\\bar{\\mathcal {A}} (\\mathsf {b} ) $ is of almost ${\\rm t}$ -descent.", "If, furthermore, the pseudofunctors in the image of $\\widehat{\\mathcal {A}}\\circ {\\rm t}$ and $\\bar{\\mathcal {A}}\\circ {\\rm h}$ are respectively of ${\\rm h}$ - and ${\\rm t}$ -descent, then: $\\widehat{\\mathcal {A}}(\\mathsf {a} ) $ is of ${\\rm h}$ -descent if and only if $\\bar{\\mathcal {A}} (\\mathsf {b} ) $ is of ${\\rm t}$ -descent." ], [ "Descent Objects", "In this section, we give a description of the descent category, as defined in classical descent theory, via pseudo-Kan extensions.", "The results of the first part of this section is hence important to fit the context of [22], [23] within our framework.", "Let ${\\rm j}: \\Delta \\rightarrow \\dot{\\Delta } $ be the full inclusion of the category of finite nonempty ordinals into the category of finite ordinals and order preserving functions.", "Recall that $\\dot{\\Delta } $ is generated by its degeneracy and face maps.", "That is to say, $\\dot{\\Delta } $ is generated by the diagram ${ \\mathsf {0} [rr]^-{d=d^0} && \\mathsf {1}@<2ex>[rr]\|-{d^0}@<-2ex>[rr]\|-{d^1} && \\mathsf {2}[ll]\|-{s^0}@<2 ex>[rr]\|-{d ^0}[rr]\|-{d ^1}@<-2ex>[rr]\|-{d ^2} && \\mathsf {3}@/_4ex/@<-2 ex>[ll]\|-{s^0}@/^4ex/@<2ex>[ll]\|-{s^1}@<1.5ex>[rr]@<0.5ex>[rr]@<-0.5ex>[rr]@<-1.5ex>[rr]&& \\cdots @/^2ex/@<1.5ex>[ll]@/^5ex/@<1.5ex>[ll]@/_3ex/@<-1.5 ex>[ll] } $ with the following relations: $\\begin{aligned}d ^k d ^i&=& d^{i}d^{k-1}, &\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i<k ;& \\\\s^ks^i &=& s^is^{k+1}, &\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i\\le k ;& \\\\s^k d^i &=& d^i s^{k-1}, &\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i< k ;&\\end{aligned}\\qquad \\qquad \\qquad \\begin{aligned}d^0 d & = d^1 d ;\\\\s^k d^i &= {\\rm id},\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i=k\\mbox{ and } i=k+1 ; \\\\s^k d^i &= d^{i-1}s^k, \\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i>k+1 .\\end{aligned}$ The category $\\dot{\\Delta } $ has an obvious strict monoidal structure $(+,\\mathsf {0} )$ that turns $(\\dot{\\Delta }, +, \\mathsf {0}, \\mathsf {1}) $ into the initial object of the category of monoidal categories with a chosen monoid [31].", "There is a full inclusion $\\dot{\\Delta }\\rightarrow {\\rm \\mathsf {C}AT}$ such that the image of each $\\mathsf {n} $ is the corresponding ordinal.", "This is the reason why we may consider that $\\dot{\\Delta } $ is precisely the full subcategory of ${\\rm \\mathsf {C}AT}$ of the finite ordinals (considered as partially ordered sets).", "In this context, the object $\\mathsf {n}$ is often confused with its image which is the category $\\mathsf {0}\\rightarrow \\mathsf {1} \\rightarrow \\mathsf {2}\\rightarrow \\cdots \\mathsf {n-1}.", "$ It is important to keep in mind that $\\dot{\\Delta } $ is a category, but we often consider it inside the tricategory $2\\textrm {-}{\\rm \\mathsf {C}AT}$ .", "More precisely, by abuse of language, $\\dot{\\Delta } $ and $\\Delta $ denote respectively the images of the categories $\\dot{\\Delta } $ and $\\Delta $ by the inclusion ${\\rm \\mathsf {C}AT}\\rightarrow 2\\textrm {-}{\\rm \\mathsf {C}AT}$ .", "Hence $\\dot{\\Delta } $ is locally discrete and is not a full sub-2-category of ${\\rm \\mathsf {C}AT}$ .", "In fact, it is clear that $\\Delta (\\mathsf {1}, \\mathsf {n})$ is the image of $\\mathsf {n} $ by the comonad induced by the right adjoint forgetful functor between the category of small categories and the category of sets, the counit of which is denoted by $\\varepsilon ^\\textrm {d}$ .", "A pseudofunctor $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {H}$ is called a pseudocosimplicial object of $\\mathfrak {H}$ .", "The descent object of such a pseudocosimplicial object $\\mathcal {A}$ is ${\\rm Ps}{\\mathcal {R}an}_ {\\rm j}\\mathcal {A}(\\mathsf {0}) $ .", "Since $\\mathsf {0}$ is the initial object of $\\dot{\\Delta } $ , the weight $\\dot{\\Delta }(\\mathsf {0}, {\\rm j}- )$ is terminal.", "By Remark REF and Theorem REF , we get that the descent object of $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {H}$ is by Definition the conical bilimit of $\\mathcal {A}$ .", "Theorem REF shows that our definition of descent object agrees with Definition , which is the usual definition of the descent object [48], [22], [21].", "The category $\\dot{\\Delta } _ {{}_{3}}$ is generated by the diagram: ${ \\mathsf {0} [rr]^-d && \\mathsf {1}@<1.7 ex>[rrr]^-{d^0}@<-1.7ex>[rrr]_-{d^1} &&& \\mathsf {2}[lll]\|-{s^0}@<1.7 ex>[rrr]^{\\partial ^0}[rrr]\|-{\\partial ^1}@<-1.7ex>[rrr]_{\\partial ^2} &&& \\mathsf {3} }$ such that: $d^1d = d^0d ;\\qquad \\qquad \\partial ^k d ^i= \\partial ^{i}d^{k-1}\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i<k; \\qquad \\qquad s^0 d^0 = s^0d ^1 = {\\rm id}.$ We denote by ${\\rm j}_ {{}_3} : \\Delta _{{}_3}\\rightarrow \\dot{\\Delta } _ {{}_3} $ the full inclusion of the subcategory $\\Delta _ {{}_{3}}$ in which ${\\rm obj}(\\Delta _ {{}_{3}}) = \\left\\lbrace \\mathsf {1}, \\mathsf {2}, \\mathsf {3}\\right\\rbrace $ .", "Still, there are obvious inclusions: $\\dot{{\\rm t}} _ {{}_{3}}:\\dot{\\Delta } _ {{}_3} \\rightarrow \\dot{\\Delta }$ and ${\\rm t}_ {{}_{3}}:\\Delta _ {{}_3} \\rightarrow \\Delta .$ Again, $\\dot{\\Delta } _ {{}_{3}}$ herein usually denotes the respective locally discrete 2-category.", "We denote by ${W}: \\Delta _ {{}_{3}}\\rightarrow {\\rm \\mathsf {C}AT}$ the weight below (defined in [48]), in which $\\nabla \\mathsf {n} $ denotes the localization of the category/finite ordinal $\\mathsf {n} $ w.r.t all the morphisms.", "${ \\nabla \\mathsf {1}@<2ex>[rr]@<-2ex>[rr] && \\nabla \\mathsf {2}[ll]@<2 ex>[rr][rr]@<-2ex>[rr] && \\nabla \\mathsf {3} }$ Following [48], if $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {H}$ is a pseudofunctor, we define ${\\mathcal {D}esc}(\\mathcal {A}):= {\\rm bilim}( {W}, \\mathcal {A}\\circ {\\rm t}_ {{}_3} ) .$ The weight ${W}$ is pseudonaturally equivalent to the terminal weight.", "Therefore, ${\\mathcal {D}esc}(\\mathcal {A})$ is by definition (equivalent to) the conical bilimit of $\\mathcal {A}\\circ {\\rm t}_ {{}_3} $ .", "In order to prove Theorem REF , we need: Proposition 4.1 Let $Y$ be any category and $\\underline{Y}: \\Delta _ {{}_{3}}^{\\rm op}\\rightarrow {\\rm \\mathsf {C}AT}$ the constant 2-functor $\\mathsf {n}\\mapsto Y $ .", "Given any (strict) 2-functor $\\mathcal {B}: \\Delta _ {{}_{3}} ^{\\rm op}\\rightarrow \\mathfrak {H}$ and a pseudonatural transformation $\\alpha : \\mathcal {B}\\longrightarrow \\underline{Y}$ , the following equations hold: $\\begin{split}{&Y@{<->}[rr]^{{\\rm Id}_{{}_Y} }&&Y&&&Y&\\\\\\mathcal {B}(\\mathsf {1})@{}[r]\|-{{\\alpha _{{}_{d ^1} }^{-1} } }[ru]^{\\alpha _{{}_\\mathsf {1} } }&&\\mathcal {B}(\\mathsf {1})@{}[l]\|-{{\\alpha _{{}_{d ^0} } } }@{}[r]\|-{{\\alpha _{{}_{d ^1} }^{-1} } }[lu]\|-{\\alpha _{{}_\\mathsf {1} } }[ru]\|-{\\alpha _{{}_\\mathsf {1} } }@{}[u]\|-{=}@{}[d]\|-{=}&&\\mathcal {B}(\\mathsf {1})@{}[r]\|-{=}@{}[l]\|-{{\\alpha _{{}_{d ^0} } } }[lu]_-{\\alpha _{{}_\\mathsf {1} } }&\\mathcal {B}(\\mathsf {1})@{}[r]\|-{{\\alpha _{{}_{d ^1} }^{-1} } }[ru]^{\\alpha _{{}_\\mathsf {1} } }&&\\mathcal {B}(\\mathsf {1})@{}[l]\|-{{\\alpha _{{}_{d ^0} } } }[lu]_-{\\alpha _{{}_\\mathsf {1} } }\\\\&\\mathcal {B}(\\mathsf {2})[lu]^{\\mathcal {B}(d ^1 ) }[ru]\|-{\\mathcal {B}(d^0 ) }[uu]\|-{\\alpha _{{}_\\mathsf {2} } }&\\mathcal {B}(\\mathsf {3} )[l]^-{\\mathcal {B}(\\partial ^2) }[r]_-{\\mathcal {B}(\\partial ^0) }&\\mathcal {B}(\\mathsf {2})[lu]\|-{\\mathcal {B}(d ^1 ) }[ru]_-{\\mathcal {B}(d^0 ) }[uu]\|-{\\alpha _{{}_\\mathsf {2} } }&&&\\mathcal {B}(\\mathsf {2})[lu]^-{\\mathcal {B}(d ^1 ) }[ru]\|-{\\mathcal {B}(d^0 ) }[uu]\|-{\\alpha _{{}_\\mathsf {2} } }&\\mathcal {B}(\\mathsf {3})[l]^{\\mathcal {B}(\\partial ^ 1 ) }}\\end{split}\\qquad \\mathrm {(associativity codescent equation)}$ $\\begin{split}{&Y&&&Y\\\\\\mathcal {B}(\\mathsf {1})[ru]^{\\alpha _{{}_\\mathsf {1} }}@{}[r]\|-{{\\alpha _{{}_{d^1} }^{-1} } }&&\\mathcal {B}(\\mathsf {1} )@{}[l]\|-{{\\alpha _{{}_{d^0} } } }[lu]_{\\alpha _{{}_\\mathsf {1} }}&&\\\\&\\mathcal {B}(\\mathsf {2})[lu]^{\\mathcal {B}(d ^1 )}[ru]_{\\mathcal {B}(d ^0 ) }[uu]\|-{\\alpha _{{}_{\\mathsf {2} } } }&\\\\&\\mathcal {B}(\\mathsf {1} )[u]\|{\\mathcal {B}(s ^0 ) }&&&\\mathcal {B}(\\mathsf {1})@{}[uuull]^-{=}@{}[uuu]\|-{=}@/_3ex/[uuu]_-{\\alpha _{{}_{\\mathsf {1} } } }@/^3ex/[uuu]^-{\\alpha _{{}_{\\mathsf {1} } } }}\\end{split}\\qquad \\mathrm {(identity of codescent)}$ We start by proving the REF .", "Indeed, by Definition of pseudonatural transformation (see [34]), since $d ^0s ^0= d ^1 s^0 ={\\rm id}_ {{}_{\\mathsf {1} }}$ , $\\mathcal {B}$ is a 2-functor and $\\underline{Y}$ is constant equal to $Y$ , we have that $\\alpha _{{}_{d^0 s^0 } }={\\rm Id}_{{}_{ \\alpha _{{}_{\\mathsf {1} } } } } = \\alpha _{{}_{d^1 s^0 } } $ which implies in particular that ${\\mathcal {B}(\\mathsf {1})@{}[rd]\|-{{\\alpha _{{}_{s^0 } } } }[r]^-{\\mathcal {B}(s ^0)}[rdd]_-{\\alpha _{{}_{\\mathsf {1} } } }&\\mathcal {B}(\\mathsf {2})[dd]\|-{\\alpha _{{}_{\\mathsf {2} } } }[r]^{\\mathcal {B}(d ^0 )}&\\mathcal {B}(\\mathsf {1})@{}[ld]\|-{{\\alpha _{{}_{d^0 } } } }[ldd]^-{\\alpha _{{}_{\\mathsf {1} } } }@/^2ex/@{{ }{ }}[dd]\|-{=}&\\mathcal {B}(\\mathsf {1})@{}[dd]\|-{=}@/^2ex/[dd]^{\\alpha _{{}_{\\mathsf {1} } }}@/_2ex/[dd]_{\\alpha _{{}_{\\mathsf {1} } } }&\\mathcal {B}(\\mathsf {1})@{}[rd]\|-{{\\alpha _{{}_{s^0 } } } }[r]^-{\\mathcal {B}(s ^0)}[rdd]_-{\\alpha _{{}_{\\mathsf {1} } } }@/_2ex/@{{ }{ }}[dd]\|-{=}&\\mathcal {B}(\\mathsf {2})[dd]\|-{\\alpha _{{}_{\\mathsf {2} } } }[r]^{\\mathcal {B}(d ^1 )}&\\mathcal {B}(\\mathsf {1})@{}[ld]\|-{{\\alpha _{{}_{d^1 } } } }[ldd]^-{\\alpha _{{}_{\\mathsf {1} } } }\\\\&&&&&&\\\\&Y&&Y&&Y&}$ and therefore: ${\\mathcal {B}(\\mathsf {1})@{}[rd]\|-{{\\alpha _{{}_{d^1 } }^{-1} } }[rdd]_-{\\alpha _{{}_{\\mathsf {1} } } }&\\mathcal {B}(\\mathsf {2})[l]_-{\\mathcal {B}(d ^1 )}[dd]\|-{\\alpha _{{}_{\\mathsf {2} } } }&\\mathcal {B}(\\mathsf {1})[l]_-{\\mathcal {B}(s ^0 )}@{}[ld]\|-{{\\alpha _{{}_{s^0 } }^{-1} } }[ldd]^-{\\alpha _{{}_{\\mathsf {1} } } }@{}[rd]\|-{{\\alpha _{{}_{s^0 } } } }[r]^-{\\mathcal {B}(s ^0)}[rdd]_-{\\alpha _{{}_{\\mathsf {1} } } }@{{ }{ }}[dd]\|-{=}&\\mathcal {B}(\\mathsf {2})[dd]\|-{\\alpha _{{}_{\\mathsf {2} } } }[r]^{\\mathcal {B}(d ^0 )}&\\mathcal {B}(\\mathsf {1})@{}[ld]\|-{{\\alpha _{{}_{d^0 } } } }[ldd]^-{\\alpha _{{}_{\\mathsf {1} } } }@/^4ex/@{{ }{ }}[dd]\|-{=}&&\\mathcal {B}(\\mathsf {2} )[dd]\|-{\\alpha _{{}_{\\mathsf {2} } } }[rd]\|-{\\mathcal {B}(d^0) }[ld]\|-{\\mathcal {B}(d ^1) }&\\mathcal {B}(\\mathsf {1})[l]_-{\\mathcal {B}(s ^0) }\\\\&&&&&\\mathcal {B}(\\mathsf {1})@{{ }{ }}[r]\|-{{\\alpha _{{}_{d^1 } }^{-1} }}[rd]\|-{\\alpha _{{}_{\\mathsf {1} } } }&&\\mathcal {B}(\\mathsf {1})[ld]\|-{\\alpha _{{}_{\\mathsf {1} } } }@{}[l]\|-{{\\alpha _{{}_{d^0 } } } }&\\\\&Y@{<->}[rr]_{{\\rm Id}_{{}_{Y}} }&&Y&&&Y&&}$ is equal to the identity on $\\alpha _{{}_{\\mathsf {1} } } $ .", "This proves that the REF holds.", "It remains to prove that the REF holds.", "Since by the definition of pseudonatural transformation we have that $ \\left(\\alpha _{{}_{d^0 } }\\ast {\\rm Id}_ {{}_{\\mathcal {B}(\\partial ^2 ) }}\\right)\\cdot \\alpha _{{}_{\\partial ^2 } }=\\alpha _{{}_{d^0\\partial ^2 }} = \\alpha _{{}_{d^1\\partial ^0 }} =\\left(\\alpha _{{}_{d^1 } }\\ast {\\rm Id}_ {{}_{\\mathcal {B}(\\partial ^0 ) }}\\right)\\cdot \\alpha _{{}_{\\partial ^0 } },$ we conclude that ${Y@{<->}[rr]^{{\\rm Id}_ {{}_{Y}} }&&Y@{{ }{ }}[rdd]\|-{=}&Y[rr]^{{\\rm Id}_ {{}_{Y}} }@/^4ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^2 } }^{-1} } }&&Y@/_3ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^0 } }} }\\\\&\\mathcal {B}(\\mathsf {1})@{}[l]\|-{{\\alpha _{{}_{d^0 } } } }@{}[u]\|-{=}@{}[d]\|-{=}@{}[r]\|-{{\\alpha _{{}_{d^1 } }^{-1} } }[ru]\|-{\\alpha _ {{}_{\\mathsf {1} }} }[lu]\|-{\\alpha _ {{}_{\\mathsf {1} }} }&&&&\\\\\\mathcal {B}(\\mathsf {2})[uu]^{\\alpha _ {{}_{\\mathsf {2} }} }[ru]\|-{\\mathcal {B}(d ^0)}&\\mathcal {B}(\\mathsf {3})[l]^{\\mathcal {B}(\\partial ^2) }[r]_{\\mathcal {B}(\\partial ^0) }&\\mathcal {B}(\\mathsf {2})[uu]_-{\\alpha _ {{}_{\\mathsf {2} }} }[lu]\|-{\\mathcal {B}(d ^1)}&\\mathcal {B}(\\mathsf {2})[uu]^{\\alpha _ {{}_{\\mathsf {2} }} }&\\mathcal {B}(\\mathsf {3})[l]^-{\\mathcal {B}(\\partial ^2) }@/_2ex/[luu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }[r]_-{\\mathcal {B}(\\partial ^0)}@/^2ex/[ruu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }@{}[uu]\|-{=}&\\mathcal {B}(\\mathsf {2})[uu]_{\\alpha _ {{}_{\\mathsf {2} }} }&}$ holds.", "Since $\\alpha _{{}_{d^0\\partial ^0 }} = \\alpha _{{}_{d^1\\partial ^0 }}$ , $\\alpha _{{}_{d^1\\partial ^2 }} = \\alpha _{{}_{d^1\\partial ^1 }}$ , by the equality above, the left side of the REF is equal to $@C=1em{&Y[rr]^{{\\rm Id}_ {{}_{Y}} }@/^3ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^2 } }^{-1} } }&&Y@/_3ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^0 } }} }&&&Y[rr]^{{\\rm Id}_ {{}_{Y}} }@/^3ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^1 } }^{-1} } }&&Y@/_3ex/@{{ }{ }}[dd]\|-{{\\alpha _{{}_{\\partial ^1 } }} }&\\\\\\mathcal {B}(\\mathsf {1})[ru]^{\\alpha _ {{}_{\\mathsf {1} }}}@{{ }{ }}[r]\|-{{\\alpha _{{}_{d^1 } }^{-1} } }&&&&\\mathcal {B}(\\mathsf {1})[lu]_-{\\alpha _ {{}_{\\mathsf {1} }} }@{}[l]\|-{{\\alpha _{{}_{d^0 } }}}@{{ }{ }}[r]\|-{=}&\\mathcal {B}(\\mathsf {1})[ru]^{\\alpha _ {{}_{\\mathsf {1} }}}@{{ }{ }}[r]\|-{{\\alpha _{{}_{d^1 } }^{-1} } }&&&&\\mathcal {B}(\\mathsf {1})[lu]_-{\\alpha _ {{}_{\\mathsf {1} }} }@{}[l]\|-{{\\alpha _{{}_{d^0 } }}}\\\\&\\mathcal {B}(\\mathsf {2})[lu]^{\\mathcal {B}(d^1) }[uu]\|-{\\alpha _ {{}_{\\mathsf {2} }} }&\\mathcal {B}(\\mathsf {3})[l]^-{\\mathcal {B}(\\partial ^2) }@/_2ex/[luu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }[r]_-{\\mathcal {B}(\\partial ^0)}@/^2ex/[ruu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }@{}[uu]\|-{=}&\\mathcal {B}(\\mathsf {2})[uu]\|-{\\alpha _ {{}_{\\mathsf {2} }} }[ru]_-{\\mathcal {B}(d ^0 ) }&&&\\mathcal {B}(\\mathsf {2})[lu]^{\\mathcal {B}(d^1) }[uu]\|-{\\alpha _ {{}_{\\mathsf {2} }} }&\\mathcal {B}(\\mathsf {3})[l]^-{\\mathcal {B}(\\partial ^1) }@/_2ex/[luu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }[r]_-{\\mathcal {B}(\\partial ^1)}@/^2ex/[ruu]\|-{\\alpha _ {{}_{\\mathsf {3} }} }@{}[uu]\|-{=}&\\mathcal {B}(\\mathsf {2})[uu]\|-{\\alpha _ {{}_{\\mathsf {2} }} }[ru]_-{\\mathcal {B}(d ^0 ) }&}$ which is clearly equal to the right side of the REF .", "One important difference between (pointwise) pseudo-Kan extensions (weighted bilimits) and (pointwise) Kan extensions (strict 2-limits) is the following: if we consider the inclusion ${\\rm t}_ {{}_2}:\\Delta _ {{}_{2}}\\rightarrow \\Delta $ of the full subcategory with only $\\mathsf {1}$ and $\\mathsf {2}$ as objects into the category $\\Delta $ , then ${\\mathcal {L}an}_{{\\rm t}_ {{}_2} } \\top \\cong \\top $ while ${\\rm Ps}{\\mathcal {L}an}_{{\\rm t}_ {{}_2} } \\top \\lnot \\simeq \\top $ , where, by abuse of language, $\\top $ always denotes the appropriate 2-functor constantly equal to the terminal category.", "Actually, ${\\rm Ps}{\\mathcal {L}an}_{{\\rm t}_ {{}_2} } \\top (\\mathsf {3}) $ is equivalent to the category with only one object and one nontrivial automorphism.", "Theorem 4.2 Let $\\top : \\Delta _ {{}_{3}}\\rightarrow {\\rm \\mathsf {C}AT}$ and $\\top :\\Delta \\rightarrow {\\rm \\mathsf {C}AT}$ be the terminal weights.", "We have that ${\\rm Ps}{\\mathcal {L}an}_{{\\rm t}_ {{}_3} } \\top \\simeq \\top $ .", "We prove below that, given a constant 2-functor $ \\underline{ Y}: \\Delta _ {{}_{3}} \\rightarrow {\\rm \\mathsf {C}AT}$ , $\\left[ \\Delta _ {{}_{3}} ^{\\rm op}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n}), \\underline{ Y} )\\simeq {\\rm \\mathsf {C}AT}(\\nabla \\mathsf {n}, Y) $ which, by the dual of Theorem REF given in Remark REF , completes our argument since it proves that ${\\rm bicolim}(\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n}), \\top )\\simeq \\nabla \\mathsf {n}\\simeq \\top (\\mathsf {n})$ .", "Let $\\varepsilon ^\\textrm {d}$ be the counit of the discrete comonad on the category of small categories (see ), we define the functor ${\\rm \\mathsf {C}AT}(\\nabla \\mathsf {n}, Y)\\rightarrow \\left[ \\Delta _ {{}_{3}} ^{\\rm op}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n}), \\underline{ Y} ),\\quad A \\mapsto \\xi ^A, \\quad \\left( \\mathfrak {x}: A\\rightarrow B\\right) \\mapsto \\left(\\xi ^\\mathfrak {x}: \\xi ^A\\Longrightarrow \\xi ^B\\right)$ in which, given a functor $A: \\nabla \\mathsf {n} \\rightarrow Y$ and a natural transformation $\\mathfrak {x}: A\\rightarrow B $ , $\\xi ^A$ and $\\xi ^\\mathfrak {x}$ are defined by: $\\begin{aligned}\\xi ^A _{{}_{\\mathsf {1}}} &: = A\\circ \\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }},&\\\\\\xi ^A _{{}_{\\mathsf {2}}} &: = A\\circ \\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }} \\circ \\Delta ({\\rm t}_ {{}_3}( d^1 ), \\mathsf {n}),& \\\\\\xi ^A _{{}_{\\mathsf {3}}}& : = A\\circ \\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }} \\circ \\Delta ({\\rm t}_ {{}_3}( d^1\\partial ^2), \\mathsf {n}),& \\end{aligned}\\qquad \\begin{aligned}\\xi ^A _{{}_{d^1}} &: = {\\rm Id}_ {{}_{\\xi _ {{}_{\\mathsf {2} }} }}, &\\\\\\xi ^A _{{}_{s^0}}& : = {\\rm Id}_ {{}_{\\xi _ {{}_{\\mathsf {2} }} }}, & \\\\\\xi ^A _{{}_{\\partial ^ 1 }}& : = {\\rm Id}_ {{}_{\\xi _ {{}_{\\mathsf {2} }} }}, &\\end{aligned}\\qquad \\begin{aligned}\\left(\\xi ^A _{{}_{d^0}}\\right) _{{}_{f:\\mathsf {2}\\rightarrow \\mathsf {n} }}&: = A(f(\\mathsf {0})\\le f(\\mathsf {1}) ),&\\\\\\xi ^A _{{}_{\\partial ^0}}& : = {\\rm Id}_ {{}_{\\Delta ({\\rm t}_ {{}_3}(\\partial ^2), \\mathsf {n}) }}\\ast \\xi ^A _{{}_{d^0}}.", "&\\end{aligned}$ $\\xi ^\\mathfrak {x}_{{}_{\\mathsf {1}}} : = \\mathfrak {x}\\ast {\\rm Id}_ {{}_{\\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }}}},\\quad \\xi ^\\mathfrak {x}_{{}_{\\mathsf {2}}} : = \\mathfrak {x}\\ast {\\rm Id}_ {{}_{\\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }} \\circ \\Delta ({\\rm t}_ {{}_3}( d^1 ), \\mathsf {n})}}, \\quad \\xi ^\\mathfrak {x}_{{}_{\\mathsf {3}}} : = \\mathfrak {x}\\ast {\\rm Id}_ {{}_{\\varepsilon ^\\textrm {d}_{{}_{\\mathsf {n} }} \\circ \\Delta ({\\rm t}_ {{}_3}( d^1\\partial ^2), \\mathsf {n})}}.", "$ We prove that this functor is actually an equivalence.", "Firstly, we define the inverse equivalence $\\left[ \\Delta _ {{}_{3}} ^{\\rm op}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n}), \\underline{ Y} )\\rightarrow {\\rm \\mathsf {C}AT}(\\nabla \\mathsf {n}, Y),\\quad \\alpha \\mapsto \\wp ^\\alpha ,\\quad \\left( \\mathfrak {y}: \\alpha \\Longrightarrow \\beta \\right) \\mapsto \\left(\\wp ^\\mathfrak {y}: \\wp ^\\alpha \\Longrightarrow \\wp ^\\beta \\right)$ where $\\left(\\wp ^\\mathfrak {y}\\right)_{{}_{\\mathsf {j} }}:=\\left(\\mathfrak {y}_{{}_{\\mathsf {1} }}\\right) _ {{}_{\\mathsf {j} }}$ and $\\wp ^\\alpha (\\mathsf {i}\\le \\mathsf {j}) $ is the component of the natural transformation below on the object $(\\mathsf {i}, \\mathsf {j} ): \\mathsf {2}\\rightarrow \\mathsf {n} $ of $\\Delta ({\\rm t}_ {{}_3}(\\mathsf {2}), \\mathsf {n})$ .", "${&Y&\\\\\\Delta ({\\rm t}_ {{}_3}(\\mathsf {1}), \\mathsf {n})@{}[r]\|-{{\\alpha _{{}_{d ^1} }^{-1} } }[ru]^{\\alpha _{{}_\\mathsf {1} } }&&\\Delta ({\\rm t}_ {{}_3}(\\mathsf {1}), \\mathsf {n} )@{}[l]\|-{{\\alpha _{{}_{d ^0} } } }[lu]_-{\\alpha _{{}_\\mathsf {1} } }\\\\&\\Delta ({\\rm t}_ {{}_3}(\\mathsf {2}), \\mathsf {n} )[lu]^-{\\Delta (d ^1, \\mathsf {n} ) }[ru]_-{\\Delta (d^0, \\mathsf {n} ) }[uu]\|-{\\alpha _{{}_\\mathsf {2} } }&}$ It remains to show that $\\wp ^\\alpha $ defines a functor $ \\nabla \\mathsf {n}\\rightarrow Y $ .", "Indeed, this follows from the REF and the REF of Proposition REF .", "More precisely, $\\alpha $ satisfies the equations of this proposition, since $\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n} )$ is a 2-functor.", "Given $\\mathsf {i}\\le \\mathsf {j}\\le \\mathsf {k} $ of $\\nabla \\mathsf {n} $ , by the definition of $\\wp ^\\alpha $ , $\\wp ^\\alpha (\\mathsf {j}\\le \\mathsf {k} )\\wp ^\\alpha (\\mathsf {i}\\le \\mathsf {j} )$ is the component of the natural transformation of the left side of the REF on $(\\mathsf {i}, \\mathsf {j}, \\mathsf {k} ): \\mathsf {3}\\rightarrow \\mathsf {n} $ , while the component of the right side on $(\\mathsf {i}, \\mathsf {j}, \\mathsf {k} )$ is equal to $\\wp ^\\alpha (\\mathsf {i}\\le \\mathsf {k} )$ .", "Analogously, the REF implies that $\\wp ^\\alpha ({\\rm id}_{{}_{\\mathsf {i} }}) = {\\rm id}_{{}_{\\wp ^\\alpha (\\mathsf {i}) }} $ .", "Finally, since it is clear that $\\wp ^{\\xi ^{(-)}} = {\\rm Id}_{{}_{{\\rm \\mathsf {C}AT}(\\nabla \\mathsf {n}, Y)}}$ , the proof is completed by showing the natural isomorphism $: \\xi ^{\\wp ^{(-)}}\\Longrightarrow {\\rm Id}_{{}_{\\left[ \\Delta _ {{}_{3}} ^{\\rm op}, {\\rm \\mathsf {C}AT}\\right] _ {PS}(\\Delta ({\\rm t}_ {{}_3}-, \\mathsf {n}), \\underline{ Y} )}} $ where each component is the invertible modification defined by: $\\left(_{{}_{\\alpha }}\\right)_{{}_{\\mathsf {1}}}: ={\\rm Id}_{{}_{\\alpha _{{}_{\\mathsf {1} } }}}, \\quad \\left(_{{}_{ \\alpha }}\\right) _{{}_{\\mathsf {2} } }: = \\alpha _{{}_{ d ^1 } }, \\quad \\left(_{{}_{ \\alpha }}\\right) _{{}_{ \\mathsf {3} } } : = \\alpha _{{}_{ d ^1 \\partial ^2 } }.$ Theorem 4.3 (Descent Objects) Let $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {H}$ be a pseudofunctor.", "We have that ${\\mathcal {D}esc}(\\mathcal {A})\\simeq {\\rm Ps}{\\mathcal {R}an}_ {\\rm j}\\mathcal {A}(\\mathsf {0}) $ .", "By Remarks and , we need to prove that the conical bilimit of $\\mathcal {A}$ is equivalent to the conical bilimit of $\\mathcal {A}\\circ {\\rm t}_ {{}_3} $ .", "Indeed, by Theorems REF and REF , ${\\rm bilim}( \\top , \\mathcal {A}\\circ {\\rm t}_ {{}_3} )\\simeq {\\rm bilim}( {\\rm Ps}{\\mathcal {L}an}_{{\\rm t}_ {{}_3} }\\top , \\mathcal {A})\\simeq {\\rm bilim}( \\top , \\mathcal {A}).", "$ Observe that, by Theorem REF , if $\\mathcal {A}: \\dot{\\Delta } \\rightarrow \\mathfrak {H}$ is a pseudofunctor, then $\\mathcal {A}$ is of (almost/effective) ${\\rm j}$ -descent if and only if $\\mathcal {A}\\circ \\dot{{\\rm t}} _ {{}_{3}} $ is of (almost/effective) ${\\rm j}_ {{}_3}$ -descent." ], [ "Strict Descent Objects", "To finish this section, we show how we can see descent objects via (strict/enriched) Kan extensions of 2-diagrams.", "Although this construction gives a few strict features of descent theory (such as the strict factorization), we do not use the results of this part in the rest of the paper.", "For this reason, the reader can skip this part and consider it to be technical observations on strict results.", "Clearly, since the point of these observations is to consider strict results, unlike the general viewpoint of this paper, we have to deal closely with coherence theorems.", "The coherence replacements used here follow from the 2-monadic approach to general coherence results [28], [6], [34].", "Also, to formalize some observations of free 2-categories, we use the concept of computad, defined in [46].", "Therefore it is clear that this part assumes knowledge on coherence [28], [34], icons [30], [36], computads [46], [36] and flexible weighted limits [5].", "Moreover, we omit most of the proofs of this last part of this section.", "The first step is actually older than the general coherence results: the strict replacement of a bicategory.", "Consider the locally full inclusion ${\\mathcal {I}con}\\rightarrow {\\mathcal {B}icat}$ of the 2-category ${\\mathcal {I}con}$ of 2-categories and 2-functors into the 2-category ${\\mathcal {B}icat}$ of 2-categories, pseudofunctors and icons.", "By the general coherence result [28], [34], this inclusion has a left 2-adjoint $\\mathrm {Str}: {\\mathcal {B}icat}\\rightarrow {\\mathcal {I}con}$ , and the unit of this 2-adjunction is a pseudonatural equivalence (which means that it is pointwise an equivalence in ${\\mathcal {B}icat}$ ).", "Lemma 4.4 If there is an equivalence $\\mathrm {Str}(\\mathfrak {A})\\rightarrow \\mathfrak {A}_{{}_{\\mathrm {Str}}}$ in ${\\mathcal {I}con}$ , the inclusion $\\left[ \\mathfrak {A}_{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] \\rightarrow \\left[ \\mathfrak {A}_{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] _ {PS}$ is essentially surjective (which means that every pseudofunctor $\\mathfrak {A}_{{}_{\\mathrm {Str}}}\\rightarrow \\mathfrak {H}$ is pseudonaturally isomorphic to a 2-functor $\\mathfrak {A}_{{}_{\\mathrm {Str}}}\\rightarrow \\mathfrak {H}$ ).", "This follows from the fact the composition of the equivalences ${\\mathcal {I}con}(\\mathrm {Str}(\\mathfrak {A}) , \\mathfrak {H})\\cong {\\mathcal {B}icat}(\\mathfrak {A}, \\mathfrak {H})\\simeq {\\mathcal {B}icat}(\\mathrm {Str}(\\mathfrak {A}) , \\mathfrak {H}), $ in which ${\\mathcal {B}icat}(\\mathfrak {A}, \\mathfrak {H})\\simeq {\\mathcal {B}icat}(\\mathrm {Str}(\\mathfrak {A}) , \\mathfrak {H})$ is the precomposition of the component of the unit on $\\mathfrak {A}$ , gives the inclusion ${\\mathcal {I}con}(\\mathrm {Str}(\\mathfrak {A}) , \\mathfrak {H})\\rightarrow {\\mathcal {B}icat}(\\mathrm {Str}(\\mathfrak {A}) , \\mathfrak {H}) $ .", "Therefore this inclusion is an equivalence of categories.", "In particular, for each pseudofunctor $\\mathcal {A}: \\mathrm {Str}(\\mathfrak {A})\\rightarrow \\mathfrak {H}$ , there are a 2-functor $\\mathcal {A}^{\\prime } : \\mathrm {Str}(\\mathfrak {A})\\rightarrow \\mathfrak {H}$ and an invertible icon $\\mathcal {A}\\longrightarrow \\mathcal {A}^{\\prime }$ .", "This property clearly holds for any $\\mathfrak {A}_{{}_{\\mathrm {Str}}}$ such that there is an equivalence $\\mathfrak {A}_{{}_{\\mathrm {Str}}}\\rightarrow \\mathrm {Str}(\\mathfrak {A}) $ in ${\\mathcal {I}con}$ .", "Since invertible icons are pseudonatural isomorphisms with identity 1-cell components, this fact proves that $\\left[ \\mathfrak {A}_{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] \\rightarrow \\left[ \\mathfrak {A}_{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] _ {PS}$ is indeed essentially surjective.", "Herein, given a small 2-category $\\mathfrak {A}$ , a strict replacement of $\\mathfrak {A}$ is a 2-category $\\mathfrak {A}_{{}_{\\mathrm {Str}}}$ such that there is an equivalence $\\mathfrak {A}_{{}_{\\mathrm {Str}}}\\rightarrow \\mathrm {Str}(\\mathfrak {A}) $ in ${\\mathcal {I}con}$ .", "Thus strict replacements are clearly unique up to equivalence and choices of strict replacements define a left biadjoint to the inclusion ${\\mathcal {I}con}\\rightarrow {\\mathcal {B}icat}$ .", "A 2-category $\\mathfrak {A}$ is locally groupoidal if every hom-category $\\mathfrak {A}(a,b) $ is a groupoid.", "Moreover, $\\mathfrak {A}$ is locally thin if there is at most one 2-cell $f\\Rightarrow g $ for every ordered pair of 1-cells $(f,g) $ of $\\mathfrak {A}$ .", "Finally, $\\mathfrak {A}$ is locally thin groupoidal, or, for short, locally t.g., if it is locally groupoidal and locally thin.", "We denote by $\\dot{\\Delta }_{{}_{\\mathrm {Str}}} $ the locally t.g.", "2-category freely generated by the diagram ${ \\mathsf {0} [rr]^-d && \\mathsf {1}@<1.7 ex>[rrr]^-{d^0}@<-1.7ex>[rrr]_-{d^1} &&& \\mathsf {2}[lll]\|-{s^0}@<1.7 ex>[rrr]^{\\partial ^0}[rrr]\|-{\\partial ^1}@<-1.7ex>[rrr]_{\\partial ^2} &&& \\mathsf {3} }$ with the 2-cells: $\\begin{aligned}\\sigma _{01} &:& \\partial ^1 d ^0\\Rightarrow \\partial ^{0}d^{0},\\\\\\sigma _{02} &:& \\partial ^2 d ^0\\Rightarrow \\partial ^{0}d^{1},\\\\\\sigma _{12} &:& \\partial ^2 d ^1\\Rightarrow \\partial ^{1}d^{1}, \\end{aligned}\\qquad \\qquad \\qquad \\begin{aligned} n_0 &:& s^0d^0\\Rightarrow {\\rm id}_{{}_\\mathsf {1}}, \\\\n_1 &:& s^{0}d^{1}\\Rightarrow {\\rm id}_{{}_\\mathsf {1}},\\\\\\vartheta &:& d^1d\\Rightarrow d^0d.\\end{aligned}$ We consider the full inclusion ${\\rm j}_{{}_{\\mathrm {Str}}} : \\Delta _{{}_{\\mathrm {Str}}} \\rightarrow \\dot{\\Delta }_{{}_{\\mathrm {Str}}} $ in which ${\\rm obj}(\\Delta _{{}_{\\mathrm {Str}}} ) = \\left\\lbrace \\mathsf {1}, \\mathsf {2}, \\mathsf {3}\\right\\rbrace $ .", "Observe that the diagram and the invertible 2-cells described above define a computad [46] (or, more appropriately, a groupoidal computad [36]) which we denote by $\\dot{\\mathfrak {\\Delta }}$ .", "Thereby Definition REF is precise in the following sense: there is a forgetful functor between the category of locally t.g.", "2-categories and the category of (groupoidal) computads.", "This forgetful functor has a left adjoint which gives the locally t.g.", "2-category freely generated by each computad.", "The (locally groupoidal) 2-category $\\dot{\\Delta }_{{}_{\\mathrm {Str}}} $ is, by definition, the image of the computad $\\dot{\\mathfrak {\\Delta }}$ by this left adjoint functor.", "The 2-categories $\\dot{\\Delta }_{{}_{\\mathrm {Str}}} $ and $\\Delta _{{}_{\\mathrm {Str}}}$ are strict replacements of the 2-categories $\\dot{\\Delta } _ {{}_3}$ and $\\Delta _ {{}_3}$ respectively.", "Actually, ${\\rm j}_{{}_{\\mathrm {Str}}}$ is a strict replacement of ${\\rm j}_ {{}_3} $ .", "By this fact and by the result that descent objects are flexible [5], we get: Proposition 4.5 There are obvious biequivalences $\\Delta _{{}_{\\mathrm {Str}}}\\approx \\Delta _ {{}_{3}}$ and $\\dot{\\Delta }_{{}_{\\mathrm {Str}}}\\approx \\dot{\\Delta } _ {{}_{3}}$ which are bijective on objects.", "Also, if $\\mathfrak {H}$ is any 2-category, $\\left[ \\Delta _{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] \\rightarrow \\left[ \\Delta _{{}_{\\mathrm {Str}}}, \\mathfrak {H}\\right] _ {PS}$ is essentially surjective.", "Moreover, for any 2-functor $\\mathcal {C}: \\Delta _{{}_{\\mathrm {Str}}}\\rightarrow {\\rm \\mathsf {C}AT}$ , we have an equivalence $\\left[ \\Delta _{{}_{\\mathrm {Str}}}, {\\rm \\mathsf {C}AT}\\right] (\\dot{\\Delta }_{{}_{\\mathrm {Str}}}(\\mathsf {0},{\\rm j}_{{}_{\\mathrm {Str}}}(-) ), \\mathcal {C})\\simeq \\left[ \\Delta _{{}_{\\mathrm {Str}}}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\dot{\\Delta }_{{}_{\\mathrm {Str}}}(\\mathsf {0},{\\rm j}_{{}_{\\mathrm {Str}}}(-) ), \\mathcal {C}).$ The last part of the result follows from the fact that the descent object is a flexible weighted limit (see [5]).", "The rest follows from the fact that $\\Delta _{{}_{\\mathrm {Str}}}$ is the strict replacement of $\\Delta _ {{}_3}$ (see Lemma REF ).", "Corollary 4.6 If $\\mathcal {A}: \\Delta _ {{}_{\\mathrm {Str}}}\\rightarrow \\mathfrak {H}$ is a 2-functor, ${\\rm Ps}{\\mathcal {R}an}_ {{\\rm j}_ {{}_3}} \\check{\\mathcal {A}}\\simeq {\\rm Ps}{\\mathcal {R}an}_ {{\\rm j}_{{}_{\\mathrm {Str}}}} \\mathcal {A}\\simeq {\\mathcal {R}an}_ {{\\rm j}_{{}_{\\mathrm {Str}}}} \\mathcal {A}$ provided that the pointwise Kan extension ${\\mathcal {R}an}_ {{\\rm j}_{{}_{\\mathrm {Str}}}} \\mathcal {A}$ exists, in which $\\check{\\mathcal {A}}$ is the composition of $\\mathcal {A}$ with the biequivalence $\\Delta _ {{}_{3}}\\approx \\Delta _{{}_{\\mathrm {Str}}}$ .", "Assuming that the pointwise Kan extension ${\\mathcal {R}an}_ {{\\rm j}_{{}_{\\mathrm {Str}}}} \\mathcal {A}$ exists, ${\\mathcal {R}an}_ {{\\rm j}_{{}_{\\mathrm {Str}}}} \\mathcal {A}(\\mathsf {0}) $ is called the strict descent diagram of $\\mathcal {A}$ .", "By Corollary REF , the descent object of $\\mathcal {A}$ is equivalent to its strict descent object provided that $\\mathcal {A}$ has a strict descent object.", "Using the strict descent object, we can construct the “strict” factorization described in Section .", "If $\\mathcal {A}: \\dot{\\Delta } _{{}_{\\mathrm {Str}}} \\rightarrow \\mathfrak {H}$ is a 2-functor and $\\mathfrak {H}$ has strict descent objects, we get the factorization from the universal property of the right Kan extension of $\\mathcal {A}\\circ {\\rm j}_{{}_{\\mathrm {Str}}}: \\Delta _{{}_{\\mathrm {Str}}}\\rightarrow \\mathfrak {H}$ along ${\\rm j}_{{}_{\\mathrm {Str}}} $ .", "More precisely, since ${\\rm j}_ {{}_{\\mathrm {Str}}} $ is fully faithful, we can consider that ${\\mathcal {R}an}_ {{}_{{\\rm j}_{{}_{\\mathrm {Str}}}}}\\mathcal {A}\\circ {\\rm j}_{{}_{\\mathrm {Str}}} $ is actually a strict extension of $\\mathcal {A}\\circ {\\rm j}_ {{}_{\\mathrm {Str}}}$ .", "Thereby we get the factorization ${ & & {\\mathcal {R}an}_ {{}_{{\\rm j}_{{}_{\\mathrm {Str}}}}} (\\mathcal {A}\\circ {\\rm j}_{{}_{\\mathrm {Str}}}) (\\mathsf {0}) [dd]^{{\\mathcal {R}an}_ {{}_{{\\rm j}_{{}_{\\mathrm {Str}}}}} (\\mathcal {A}\\circ {\\rm j}_{{}_{\\mathrm {Str}}}) (d)}\\\\&&\\\\\\mathcal {A}(\\mathsf {0}) [rruu]^{\\eta ^\\mathsf {0}_ {{}_{\\mathcal {A}}} }[rr]_{ \\mathcal {A}( d )} &&\\mathcal {A}(\\mathsf {1})}$ in which $\\eta ^\\mathsf {0}_ {{}_{\\mathcal {A}}} $ is the comparison induced by the unit/comparison $\\eta _ {{}_{\\mathcal {A}}}: \\mathcal {A}\\longrightarrow {\\mathcal {R}an}_ {{}_{{\\rm j}_{{}_{\\mathrm {Str}}}}} (\\mathcal {A}\\circ {\\rm j}_{{}_{\\mathrm {Str}}}) $ .", "As observed in Section REF , the right Kan extension of a 2-functor $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {H}$ along ${\\rm j}$ gives the equalizer of $\\mathcal {A}(d^0 ) $ and $\\mathcal {A}(d^1) $ .", "This is a consequence of the isomorphism ${\\mathcal {L}an}_ {{}_{{\\rm t}_ {{}_2} }} \\top \\cong \\top $ of Remark .", "We get a glimpse of the explicit nature of the (strict) descent object at Theorem REF which gives a presentation to $\\dot{\\Delta }_{{}_{\\mathrm {Str}}}$ .", "We denote by $\\mathcal {F}_{g} (\\dot{\\mathfrak {\\Delta }})$ the locally groupoidal 2-category freely generated by the diagram and 2-cells described in Definition REF .", "It is important to note that $\\mathcal {F}_{g} (\\dot{\\mathfrak {\\Delta }})$ is not locally thin.", "Moreover, there is an obvious 2-functor $\\mathcal {F}_{g} (\\dot{\\mathfrak {\\Delta }})\\rightarrow \\dot{\\Delta }_{{}_{\\mathrm {Str}}}$ , induced by the unit of the adjunction between the category of locally groupoidal 2-categories and the category of locally t.g.", "2-categories.", "Theorem 4.7 ([36]) Let $\\mathfrak {H}$ be a 2-category.", "There is a bijection between 2-functors $\\mathcal {A}: \\dot{\\Delta }_{{}_{\\mathrm {Str}}}\\rightarrow \\mathfrak {H}$ and 2-functors $\\mathsf {A}:\\mathcal {F}_{g} (\\dot{\\mathfrak {\\Delta }})\\rightarrow \\mathfrak {H}$ satisfying the following equations: Associativity: $@C=3em@R=1.5em{ \\mathsf {A}(\\mathsf {0})[r]^-{\\mathsf {A}(d)}[d]_-{\\mathsf {A}(d)}@{}[rd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\vartheta ) \\hspace{1.00006pt}} }&\\mathsf {A}(\\mathsf {1})[d]\|-{\\mathsf {A}(d^0)}[r]^-{\\mathsf {A}(d^0)}@{}[rd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\sigma _{01}) \\hspace{1.00006pt}}}&\\mathsf {A}(\\mathsf {2})[d]^-{\\mathsf {A}(\\partial ^0)}@{}[rrdd]\|-{=}&&\\mathsf {A}(\\mathsf {3})@{}[rd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\sigma _{02}) \\hspace{1.00006pt}}}&\\mathsf {A}(\\mathsf {2})[l]_-{\\mathsf {A}(\\partial ^0)}@{}[rdd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\vartheta )\\hspace{1.00006pt}}}@{=}[r]&\\mathsf {A}(\\mathsf {2})\\\\\\mathsf {A}(\\mathsf {1})[r]\|-{\\mathsf {A}(d^1)}[d]_-{\\mathsf {A}(d^1)}@{}[rrd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\sigma _ {12}) \\hspace{1.00006pt}}}&\\mathsf {A}(\\mathsf {2})[r]\|-{\\mathsf {A}(\\partial ^1)}&\\mathsf {A}(\\mathsf {3})[d]^-{\\mathsf {A}({\\rm id}_ {{}_\\mathsf {3}})}&&\\mathsf {A}(\\mathsf {2})@{}[rd]\|-{{\\hspace{1.00006pt}\\mathsf {A}(\\vartheta ) \\hspace{1.00006pt}}}[u]^-{\\mathsf {A}(\\partial ^2)}&\\mathsf {A}(\\mathsf {1})[l]\|-{\\mathsf {A} (d^0)}[u]\|-{\\mathsf {A}(d^1)}&\\\\\\mathsf {A}(\\mathsf {2})[rr]_-{\\mathsf {A}(\\partial ^2)}&&\\mathsf {A}(\\mathsf {3})&&\\mathsf {A}(\\mathsf {1})[u]^-{\\mathsf {A}(d^1)}&\\mathsf {A}(\\mathsf {0})[l]^-{\\mathsf {A}(d)}[u]\|-{\\mathsf {A}(d)}[r]_ {\\mathsf {A}(d)}&\\mathsf {A}(\\mathsf {1})[uu]_-{\\mathsf {A}(d^0)}}$ Identity: $@C=0.7em@R=1.2em{ \\mathsf {A}(\\mathsf {0} )[rr]^{\\mathsf {A}(d)}[dd]_{\\mathsf {A}(d)}&&{\\mathsf {A}(\\mathsf {1})}[dd]\|-{\\mathsf {A}(d^1)}@{=}@/^4ex/[dddr]@{}[dddr]\|{{\\mathsf {A}(n_1)}} &&&\\mathsf {A}(\\mathsf {0})@/_3ex/[ddd]\|-{\\mathsf {A}(d)}@{}[ddd]\|=@/^3ex/[ddd]\|-{\\mathsf {A}(d)}\\\\& {\\hspace{1.00006pt}\\mathsf {A}(\\vartheta ) \\hspace{1.00006pt}} && &\\\\{\\mathsf {A}(\\mathsf {1})} [rr]\|-{\\mathsf {A}(d^0)}@{=}@/_4ex/[drrr]@{}[drrr]\|{{\\mathsf {A}(n_0)}} &&{\\mathsf {A}(\\mathsf {2})}[dr]\|{\\mathsf {A}(s^0)} \\\\&&& {\\mathsf {A}(\\mathsf {1})}@{}[uuur]\|-{=} && {\\mathsf {A}(\\mathsf {1})} }$ [[36]] $\\Delta _{{}_{\\mathrm {Str}}}$ is the locally groupoidal 2-category freely generated by the corresponding diagram and 2-cells $\\sigma _{01}$ , $\\sigma _{02}$ , $\\sigma _{12}$ , $n_0$ , $n_1$ .", "[[34]] The 2-category ${\\rm \\mathsf {C}AT}$ is ${\\rm \\mathsf {C}AT}$ -complete.", "In particular, ${\\rm \\mathsf {C}AT}$ has strict descent objects.", "More precisely, if $\\mathcal {A}: \\Delta _ {{}_{\\mathrm {Str}}}\\rightarrow {\\rm \\mathsf {C}AT}$ is a 2-functor, then $\\mathrm {lim}(\\dot{\\Delta }_ {{}_{\\mathrm {Str}}}(\\mathsf {0}, {\\rm j}_{{}_{\\mathrm {Str}}}(-) ), \\mathcal {A})\\cong \\left[ \\Delta _ {{}_{\\mathrm {Str}}} , {\\rm \\mathsf {C}AT}\\right] \\left( \\dot{\\Delta }_ {{}_{\\mathrm {Str}}}(\\mathsf {0}, {\\rm j}_{{}_{\\mathrm {Str}}}(-)), \\mathcal {A}\\right) .$ Thereby, we can describe the category the strict descent object of $\\mathcal {A}:\\Delta _ {{}_{\\mathrm {Str}}}\\rightarrow {\\rm \\mathsf {C}AT}$ explicitly as follows: Objects are 2-natural transformations $\\mathsf {W}: \\dot{\\Delta }_ {{}_{\\mathrm {Str}}}(\\mathsf {0}, -)\\longrightarrow \\mathcal {A}$ .", "We have a bijective correspondence between such 2-natural transformations and pairs $(W, \\varrho _ {{}_{\\mathsf {W}}})$ in which $W$ is an object of $ \\mathcal {A}(\\mathsf {1}) $ and $\\varrho _ {{}_{\\mathsf {W}}}: \\mathcal {A}(d^1)(W)\\rightarrow \\mathcal {A}(d^0)(W) $ is an isomorphism in $ \\mathcal {A}(\\mathsf {2}) $ satisfying the following equations: Associativity: $\\left(\\mathcal {A}(\\partial ^0)(\\varrho _ {{}_{\\mathsf {W} }} )\\right) \\left( \\mathcal {A}(\\sigma _ {{}_{02}}) _ {{}_{W}}\\right)\\left(\\mathcal {A}(\\partial ^2)(\\varrho _ {{}_{\\mathsf {W}}} )\\right)\\left(\\mathcal {A}(\\sigma _ {{}_{12}} ) ^{-1}_ {{}_W}\\right) = \\left(\\mathcal {A}(\\sigma _ {{}_{01}}) _ {{}_{W}}\\right)\\left(\\mathcal {A}(\\partial ^1)(\\varrho _ {{}_{\\mathsf {W}}})\\right) $ Identity: $\\left(\\mathcal {A}(n_0) _ {{}_W}\\right)\\left(\\mathcal {A}(s^0) (\\varrho _ {{}_{\\mathsf {W}}}) \\right)\\left(\\mathcal {A}(n_1) _ {{}_W}\\right) = {\\rm id}_ {{}_{W}} $ If $\\mathsf {W}: \\dot{\\Delta }(\\mathsf {0}, -)\\longrightarrow \\mathcal {A}$ is a 2-natural transformation, we get such pair by the correspondence $\\mathsf {W}\\mapsto (\\mathsf {W} _ {{}_{\\mathsf {1} }}(d), \\mathsf {W} _ {{}_{\\mathsf {2} }}(\\vartheta )) $ .", "The morphisms are modifications.", "In other words, a morphism $\\mathsf {m} : \\mathsf {W}\\rightarrow \\mathsf {X} $ is determined by a morphism $\\mathfrak {m}: W\\rightarrow X $ such that $\\mathcal {A}(d^0)(\\mathfrak {m} )\\varrho _ {{}_\\mathsf {W}} = \\varrho _ {{}_X}\\mathcal {A}(d^1)(\\mathfrak {m} ) $ ." ], [ "Elementary Examples", "We use some particular elementary examples of inclusions ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ for which we can study the $\\left\\langle {\\rm t}\\right\\rangle $ -pseudoalgebras/effective ${\\rm t}$ -descent diagrams in the setting of Section .", "These examples are given herein.", "Let $\\mathfrak {H}$ be a 2-category with enough bilimits to construct our pseudo-Kan extensions as global pointwise pseudo-Kan extensions.", "The most simple example is taking the final category $ \\mathsf {1} $ and the inclusion $\\mathsf {0}\\rightarrow \\mathsf {1} $ of the empty category/empty ordinal.", "In this case, a pseudofunctor $\\mathcal {A}: \\mathsf {1}\\rightarrow \\mathfrak {H}$ is of effective descent if and only if this pseudofunctor (which corresponds to an object of $\\mathfrak {H}$ ) is equivalent to the pseudofinal object of $\\mathfrak {H}$ .", "If, instead, we take the inclusion $ d^0: \\mathsf {1}\\rightarrow \\mathsf {2} $ of the ordinal $ \\mathsf {1} $ into the ordinal $\\mathsf {2}$ such that $d^0$ is the inclusion of the codomain object, then a pseudofunctor $\\mathcal {A}: \\mathsf {2}\\rightarrow \\mathfrak {H}$ corresponds to a 1-cell of $\\mathfrak {H}$ and $\\mathcal {A}$ is of effective $ d^0$ -descent if and only if its image is an equivalence 1-cell.", "Moreover, $\\mathcal {A}$ is almost $ d^0$ -descent/$d^0$ -descent if and only if its image is faithful/fully faithful.", "Precisely, the comparison morphism would be the image $\\mathcal {A}(\\mathsf {0}\\stackrel{d}{\\rightarrow } \\mathsf {1})$ of the only nontrivial 1-cell of $\\mathsf {2}$ .", "Furthermore, we may consider the following 2-categories $ \\dot{\\mathfrak {B}}$ .", "The first one corresponds to the bilimit notion of comma object, while the second corresponds to the notion of pseudopullback.", "${ \\mathsf {b} [r] [d] & e [d] && \\mathsf {b} [r] [d] & e[d] \\\\@{}\|{\\Rightarrow }[ru]c[r] &o &&c[r] &o }$ As explained in Remark REF , all the examples above but the comma object are conical bilimits: it is clear that we can get every conical bilimit via a pseudo-Kan extension.", "Actually, we can study the exactness of any weighted bilimit in our setting.", "More precisely, if $\\mathcal {W}: \\mathfrak {A}\\rightarrow {\\rm \\mathsf {C}AT}$ is a weight, we can define $\\dot{\\mathfrak {A}} $ adding an extra object $\\mathsf {a}$ and defining $\\dot{\\mathfrak {A}}(\\mathsf {a}, \\mathsf {a}):= \\ast \\qquad \\qquad \\dot{\\mathfrak {A}}(\\mathsf {a}, b):= \\mathcal {W}(b) \\qquad \\qquad \\dot{\\mathfrak {A}}(b, \\mathsf {a}):= \\emptyset $ for each object $b$ of $\\mathfrak {A}$ .", "Hence, it remains just to define the unique nontrivial composition, that is to say, we define the functor composition $ \\circ : \\dot{\\mathfrak {A}}(b, c)\\times \\dot{\\mathfrak {A}}(\\mathsf {a}, b)\\rightarrow \\dot{\\mathfrak {A}}(\\mathsf {a}, c) $ for each pair of objects $b, c $ of $\\mathfrak {A}$ to be the “mate” of $ \\mathcal {W}_{{}_{bc}} : \\dot{\\mathfrak {A}}(b, c)\\rightarrow {\\rm \\mathsf {C}AT}(\\mathcal {W}(b), \\mathcal {W}(c)).", "$ Thereby, a pseudofunctor $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ is of effective ${\\rm t}$ -descent/${\\rm t}$ -descent/almost ${\\rm t}$ -descent if the canonical comparison 1-cell $\\mathcal {A}(\\mathsf {a})\\rightarrow {\\rm bilim}( \\mathcal {W}, \\mathcal {A}\\circ {\\rm t})$ is an equivalence/fully faithful/faithful." ], [ "Eilenberg-Moore Objects", "Let $\\mathfrak {H}$ be a 2-category as in .", "The 2-category $\\mathsf {Adj}$ such that an adjunction in a 2-category corresponds to a 2-functor $\\mathsf {Adj}\\rightarrow \\mathfrak {H}$ is described in [44].", "There is a full inclusion ${\\rm m}: \\mathsf {Mnd}\\rightarrow \\mathsf {Adj}$ such that monads in $\\mathfrak {H}$ correspond to 2-functors $\\mathsf {Mnd}\\rightarrow \\mathfrak {H}$ .", "We describe this 2-category below, and we show how it (still) works in our setting.", "The 2-category $\\mathsf {Adj}$ has two objects: $\\mathsf {alg} $ and $\\mathtt {b}$ .", "The hom-categories are defined as follows: $\\mathsf {Adj}(\\mathtt {b}, \\mathtt {b}):= \\dot{\\Delta }\\qquad \\mathsf {Adj}(\\mathsf {alg}, \\mathtt {b}):= \\Delta _- \\qquad \\mathsf {Adj}(\\mathsf {alg}, \\mathsf {alg}):= \\Delta _ -^+ \\qquad \\mathsf {Adj}(\\mathtt {b}, \\mathsf {alg}):= \\Delta ^+$ in which $\\Delta _ - $ denotes the subcategory of $\\Delta $ with the same objects such that its morphisms preserve initial objects and, analogously, $\\Delta _ + $ is the subcategory of $\\Delta $ with the same objects and last-element-preserving arrows.", "Finally, $\\Delta _ -^+$ is just the intersection of both $\\Delta _-$ and $\\Delta ^+$ .", "Then the composition of $\\mathsf {Adj} $ is such that $\\mathsf {Adj}(\\mathtt {b}, w)\\times \\mathsf {Adj}(c, \\mathtt {b})\\rightarrow \\mathsf {Adj}(c, w)$ is given by the usual “ordinal sum” $+$ (given by the usual strict monoidal structure of $\\Delta $ ) for all objects $c,w $ of $\\mathsf {Adj} $ and $&\\mathsf {Adj}(\\mathsf {alg}, w)\\times \\mathsf {Adj}(c, \\mathsf {alg})&\\rightarrow \\mathsf {Adj}(c, w)\\\\&(x,y)&\\mapsto x+y-\\mathsf {1}\\\\&(\\phi : x\\rightarrow x^{\\prime } , \\upsilon : y\\rightarrow y^{\\prime } ) & \\mapsto \\phi \\oplus \\upsilon $ in which $\\phi \\oplus \\upsilon (i) :={\\left\\lbrace \\begin{array}{ll}\\upsilon (i),& \\text{if } i < y\\\\\\phi (i-m) - 1+y^{\\prime } & \\text{otherwise}.\\end{array}\\right.", "}$ It is straightforward to verify that $\\mathsf {Adj}$ is a 2-category.", "We denote by $u$ the 1-cell $\\mathsf {1}\\in \\mathsf {Adj}(\\mathsf {alg}, \\mathtt {b}) $ and by $l$ the 1-cell $\\mathsf {1}\\in \\mathsf {Adj}(\\mathtt {b}, \\mathsf {alg})$ .", "Also, we consider the following 2-cells $\\dot{\\Delta }(\\mathsf {0}, \\mathsf {1})\\ni n: {\\rm id}_ {{}_{\\mathtt {b}}}\\Rightarrow ul,\\qquad \\qquad \\qquad \\Delta _-^+(\\mathsf {1}, \\mathsf {2})\\ni e: lu\\Rightarrow {\\rm id}_{{}_{\\mathsf {alg}}}.$ The 2-category $\\mathsf {Mnd} $ is defined to be the full sub-2-category of $\\mathsf {Adj} $ with the unique object $\\mathtt {b} $ .", "As mentioned above, we denote its full inclusion by ${\\rm m}: \\mathsf {Mnd}\\rightarrow \\mathsf {Adj}$ .", "Firstly, observe that $(l\\dashv u, n, e)$ is an adjunction in $\\mathsf {Adj}$ , therefore the image of $(l\\dashv u, n, e)$ by a 2-functor is an adjunction.", "Also, if $(L\\dashv U , \\eta , \\varepsilon ) $ is an adjunction in $\\mathfrak {H}$ , then there is a unique 2-functor $\\mathcal {A}: \\mathsf {Adj}\\rightarrow \\mathfrak {H}$ such that $\\mathcal {A}(u) : = U $ , $\\mathcal {A}(l) : = L $ , $\\mathcal {A}(e):= \\varepsilon $ and $\\mathcal {A}(u) := \\eta $ .", "Thereby, it gives a bijection between adjunctions in $\\mathfrak {H}$ and 2-functors $ \\mathsf {Adj}\\rightarrow \\mathfrak {H}$ [44].", "Secondly, as observed in [44], there is a similar bijection between 2-functors $\\mathsf {Mnd}\\rightarrow \\mathfrak {H}$ and monads in the 2-category $\\mathfrak {H}$ .", "Also, if the pointwise (enriched) Kan extension of a 2-functor $ \\mathsf {Mnd}\\rightarrow \\mathfrak {H}$ along ${\\rm m}$ exists, it gives the usual Eilenberg-Moore adjunction.", "Moreover, given a 2-functor $\\mathcal {A}: \\mathsf {Adj}\\rightarrow \\mathfrak {H}$ , if the pointwise Kan extension ${\\mathcal {R}an}_ {\\rm m}\\left( \\mathcal {A}\\circ {\\rm m}\\right) $ exists, the usual comparison $\\mathcal {A}(\\mathsf {alg})\\rightarrow {\\mathcal {R}an}_ {\\rm m}\\left( \\mathcal {A}\\circ {\\rm m}\\right)(\\mathsf {alg}) $ is the Eilenberg-Moore comparison 1-cell.", "If, instead, $\\mathcal {A}:\\mathsf {Adj}\\rightarrow \\mathfrak {H}$ is a pseudofunctor, we also get that $\\mathcal {A}(l)\\dashv \\mathcal {A}(u) $ and $\\left( \\mathcal {A}(l)\\dashv \\mathcal {A}(u), \\mathfrak {a}_{{}_{ul}}^{-1}\\mathcal {A}(n) \\mathfrak {a}_ {{}_{\\mathtt {b}}}, \\mathfrak {a}_ {{}_{\\mathsf {alg} }}^{-1} \\mathcal {A}(e) \\mathfrak {a}_{{}_{lu}} \\right)\\qquad \\mathrm {(strict adjunction)}$ is an adjunction in $\\mathfrak {H}$ .", "It is straightforward to verify that the unique 2-functor $\\mathcal {A}^{\\prime }: \\mathsf {Adj}\\rightarrow \\mathfrak {H}$ corresponding to this adjunction is pseudonaturally isomorphic to $\\mathcal {A}$ .", "Furthermore, the Eilenberg-Moore object is a flexible limit as it is shown in [5].", "Proposition 6.1 If $\\mathfrak {H}$ is any 2-category, $\\left[ \\mathsf {Adj}, \\mathfrak {H}\\right] \\rightarrow \\left[ \\mathsf {Adj}, \\mathfrak {H}\\right] _ {PS}$ is essentially surjective.", "Moreover, for any 2-functor $\\mathcal {C}: \\mathsf {Adj}\\rightarrow {\\rm \\mathsf {C}AT}$ , we have an equivalence $\\left[ \\mathsf {Mnd}, {\\rm \\mathsf {C}AT}\\right] (\\mathsf {Adj} (\\mathsf {alg},{\\rm m}(-) ), \\mathcal {C})\\simeq \\left[ \\mathsf {Mnd}, {\\rm \\mathsf {C}AT}\\right] _ {PS} (\\mathsf {Adj} (\\mathsf {alg},{\\rm m}(-) ), \\mathcal {C}).$ In order to prove the first part, as mentioned above, it is enough to show that there is a pseudonatural isomorphism between $\\mathcal {A}$ and $\\mathcal {A}^{\\prime } $ .", "The 1-cell components of this pseudonatural isomorphism $\\alpha $ are identities, while the component 2-cells are induced by the structure of pseudofunctor of $\\mathcal {A}$ (the constraints/invertible 2-cells).", "The second part follows from the fact that Eilenberg-Moore objects are flexible weighted limits [5].", "Corollary 6.2 If $\\mathcal {A}: \\mathsf {Mnd}\\rightarrow \\mathfrak {H}$ is a pseudofunctor, ${\\rm Ps}{\\mathcal {R}an}_ {{{\\rm j}_ {{}_3}} } \\mathcal {A}\\simeq {\\rm Ps}{\\mathcal {R}an}_ {{\\rm m}} \\check{\\mathcal {A}}\\simeq {\\mathcal {R}an}_ {{\\rm m}} \\check{\\mathcal {A}}$ provided that the pointwise Kan extension ${\\mathcal {R}an}_ {{\\rm m}} \\check{\\mathcal {A}} $ exists, in which $\\check{\\mathcal {A}}$ is a 2-functor pseudonaturally isomorphic to $\\mathcal {A}$ .", "Therefore, if $\\mathfrak {H}$ has Eilenberg-Moore objects, a pseudofunctor $\\mathcal {A}: \\mathsf {Adj}\\rightarrow \\mathfrak {H}$ is of effective ${\\rm m}$ -descent/${\\rm m}$ -descent/almost ${\\rm m}$ -descent if and only if $\\mathcal {A}(u)$ is monadic/premonadic/almost monadic.", "Also, the “factorizations” $@C=1em{ \\mathcal {A}(\\mathtt {b})[rr]\|{\\mathcal {A}(l)}[dr]\|{l_{{}_{{}_\\mathcal {A}}} }&& \\mathcal {A}(\\mathsf {alg})&\\mathcal {A}(\\mathsf {alg})[rr]\|{\\mathcal {A}(u)}[dr]\|{\\eta ^\\mathsf {alg}_ {{}_{{}_{\\mathcal {A}} }} }&& \\mathcal {A}(\\mathtt {b})\\\\&{\\rm Ps}{\\mathcal {R}an}_ {\\rm m}(\\mathcal {A}\\circ {\\rm m}) (\\mathsf {alg})@{}[u]\|{\\cong }[ru]\|{\\eta ^\\mathsf {alg}_ {{}_{{}_{\\mathcal {A}} }}} & & &{\\rm Ps}{\\mathcal {R}an}_ {\\rm m}(\\mathcal {A}\\circ {\\rm m}) (\\mathsf {alg})@{}[u]\|{\\cong }[ru]\|{u_{{}_{{}_\\mathcal {A}}}} &}$ described in Theorem REF are pseudonaturally equivalent to the usual Eilenberg-Moore factorizations.", "Henceforth, these factorizations are called Eilenberg-Moore factorizations (even if the 2-category $\\mathfrak {H}$ does not have the strict version of it)." ], [ "The Beck-Chevalley Condition", "With these elementary examples, we can already give generalizations of Theorems REF and REF .", "We keep our setting in which ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is an $\\mathsf {a}$ -inclusion as in REF .", "Let $\\mathcal {T}$ be an idempotent pseudomonad over the 2-category $\\mathfrak {H}$ .", "The most obvious consequence of the commutativity results of Section is the following: if an object $X$ of $\\mathfrak {H}$ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure and there is an equivalence $X\\rightarrow W$ , then $W$ can be endowed with a $\\mathcal {T}$ -pseudoalgebra as well.", "In the case of pseudo-Kan extensions, we have the following: let $\\mathcal {A}, \\mathcal {B}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ be pseudofunctors.", "A pseudonatural transformation $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ can be seen as a pseudofunctor $\\mathcal {C}_ \\alpha : \\mathsf {2}\\rightarrow \\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}$ .", "By Corollaries REF and REF , we get the following: if $\\mathcal {C}_\\alpha (\\mathsf {1}) $ is of effective ${\\rm t}$ -descent/${\\rm t}$ -descent/almost ${\\rm t}$ -descent and the images of the mate $\\dot{\\mathfrak {A}}\\rightarrow \\left[\\mathsf {2}, \\mathfrak {H}\\right] _ {PS}$ of $\\mathcal {C}_ \\alpha $ are of effective $d ^0 $ -descent/$d^0$ -descent/almost $d^0$ -descent as well, then $\\mathcal {C}_\\alpha (\\mathsf {0}) $ is also of effective ${\\rm t}$ -descent/${\\rm t}$ -descent/almost ${\\rm t}$ -descent.", "In Section , we show that Theorem REF is a particular case of: Proposition 7.1 Let $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ be a pseudonatural transformation.", "If $\\mathcal {B}$ is of effective ${\\rm t}$ -descent/${\\rm t}$ -descent/almost ${\\rm t}$ -descent and $\\alpha $ is a pseudonatural equivalence/objectwise fully faithful/objectwise faithful, then $\\mathcal {A}$ is of effective ${\\rm t}$ -descent/${\\rm t}$ -descent/almost ${\\rm t}$ -descent as well.", "[Beck-Chevalley condition] A pseudonatural transformation $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ satisfies the Beck-Chevalley condition if every 1-cell component of $\\alpha $ is left adjoint and, for each 1-cell $f:w\\rightarrow c $ of the domain of $\\mathcal {A}$ , the mate of the invertible 2-cell $\\alpha _ {{}_{f}} : \\mathcal {B}(f)\\alpha _{{}_w}\\Rightarrow \\alpha _ {{}_{c}}\\mathcal {A}(f) $ w.r.t.", "the adjunctions $\\widehat{\\alpha }^{{}^{w}}\\dashv \\alpha _ {{}_w}$ and $\\widehat{\\alpha }^{{}^{c}}\\dashv \\alpha _ {{}_c}$ is invertible.", "By doctrinal adjunction [26], $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ satisfies the Beck-Chevalley condition if and only if $\\alpha $ is itself a right adjoint in the 2-category $\\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}$ .", "In other words, we get: Lemma 7.2 Let $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ be a pseudonatural transformation and $\\mathcal {C}_ \\alpha : \\mathsf {2}\\rightarrow \\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}$ the corresponding pseudofunctor.", "Consider the inclusion $\\mathtt {u}: \\mathsf {2}\\rightarrow \\mathsf {Adj} $ of the morphism $u$ .", "There is a pseudofunctor $\\widehat{\\mathcal {C}} _ \\alpha : \\mathsf {Adj}\\rightarrow \\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}$ such that $ \\widehat{\\mathcal {C}} _ \\alpha \\circ \\mathtt {u} = \\mathcal {C}_ \\alpha $ if and only if $\\alpha $ satisfies the Beck-Chevalley condition.", "Thereby, as straightforward consequences of Corollaries REF and REF , using the terminology of Lemma REF , we get what can be called a generalized version of Bénabou-Roubaud Theorem: Theorem 7.3 Assume that $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ is a pseudonatural transformation satisfying the Beck-Chevalley condition and all components of $\\alpha {\\rm t}= \\alpha \\ast Id _{{}_{{\\rm t}}} $ are monadic.", "If $\\mathcal {B}$ is of almost ${\\rm t}$ -descent, then: $\\alpha _ {{}_{\\mathsf {a} }}$ is of almost ${\\rm m}$ -descent if and only if $\\mathcal {A}$ is of almost ${\\rm t}$ -descent; If $\\mathcal {B}$ is of ${\\rm t}$ -descent, then: $\\alpha _ {{}_{\\mathsf {a} }}$ is premonadic if and only if $\\mathcal {A}$ is of ${\\rm t}$ -descent; If $\\mathcal {B}$ is of effective ${\\rm t}$ -descent, then: $\\alpha _ {{}_{\\mathsf {a} }}$ is monadic if and only if $\\mathcal {A}$ is of effective ${\\rm t}$ -descent.", "Indeed, by the hypotheses, for each item, there is a pseudofunctor $\\widehat{\\mathcal {C}} _ \\alpha : \\mathsf {Adj}\\rightarrow \\left[\\dot{\\mathfrak {A}}, \\mathfrak {H}\\right] _ {PS}$ satisfying the hypotheses of Corollary REF or Corollary REF .", "It is important to observe that the hypothesis of the theorem obviously does not include the monadicity of $\\alpha _{{}_{\\mathsf {a} }}$ , since ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ is an $\\mathsf {a}$ -inclusion." ], [ "Descent Theory", "In this section, we establish the setting of [22], [23] and prove all the classical results mentioned in Section for pseudocosimplicial objects, except Theorem REF which is postponed to Section .", "Henceforth, let ${C}, {D}$ be categories with pullbacks and $\\mathfrak {H}$ be a 2-category with the weighted bilimits whenever needed as in the previous sections.", "In the context of [23], given a pseudofunctor $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ , the morphism $p: E\\rightarrow B $ of ${C}$ is of effective $\\mathcal {A}$ -descent/$\\mathcal {A}$ -descent/almost $\\mathcal {A}$ -descent if $\\mathcal {A}_p: \\dot{\\Delta }\\rightarrow \\mathfrak {H}$ is of effective ${\\rm j}$ -descent/${\\rm j}$ -descent/almost ${\\rm j}$ -descent, where $\\mathcal {A}_ p $ is the composition of the diagram $\\mathcal {D}_ p :\\dot{\\Delta }^{{\\rm op}}\\rightarrow {C}$ ${ \\cdots @<0.3ex>[r]@<-0.3ex>[r]@<0.9ex>[r]@<-0.9ex>[r] &E\\times _ p E\\times _ p E @<0.9 ex>[r][r]@<-0.9ex>[r]@/_2ex/[l]@/^2ex/[l]@/_3ex/[l] & E\\times _ p E@/_2ex/[l]@/^2ex/[l]@<0.9 ex>[r]@<-0.9ex>[r] & E[l][r]^-p& B } $ with the pseudofunctor $\\mathcal {A}$ , in which the diagram above is given by the pullbacks of $p$ along itself, its projections and diagonal morphisms.", "By the results of Section , for $\\mathfrak {H}={\\rm \\mathsf {C}AT}$ , this definition of effective $\\mathcal {A}$ -descent morphism coincides with the classical one in the context of [22], [23].", "We get the usual factorizations of (Grothendieck) $\\mathcal {A}$ -descent theory [22], [23] from Theorem REF , although the usual strict factorization comes from Remark REF .", "More precisely, if $p:E\\rightarrow B $ is a morphism of ${C}$ , we get: ${ \\mathcal {A}_ p(\\mathsf {0}) = \\mathcal {A}(B)[rr]\|{\\mathcal {A}(p)}[dr]\|{\\eta ^\\mathsf {0}_ {{}_{{}_{\\mathcal {A}\\circ \\mathcal {D}_ p } }} }&& \\mathcal {A}_ p(\\mathsf {1}) = \\mathcal {A}(E)\\\\&{\\mathcal {D}esc}_ \\mathcal {A}(p)\\simeq {\\rm Ps}{\\mathcal {R}an}_ {\\rm j}(\\mathcal {A}_ p\\circ {\\rm j}) (\\mathsf {0})@{}[u]\|{\\cong }[ru]\|{d_{{}_{{}_{\\mathcal {A}_ p} }}} &}$ In descent theory, a morphism $(U, \\alpha )$ between pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ and $\\mathcal {B}:{D}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ is a pullback preserving functor $U: {C}\\rightarrow {D}$ with a pseudonatural transformation $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}\\circ U $ .", "Such a morphism is called faithful/fully faithful if $\\alpha $ is objectwise faithful/fully faithful.", "For each morphism $p:E\\rightarrow B $ of ${C}$ , a morphism $(U, \\alpha )$ between pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ and $\\mathcal {B}:{D}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ induces a pseudonatural transformation $\\alpha ^{{}^p} : \\mathcal {A}_ p \\longrightarrow \\mathcal {B}_{U(p)} $ .", "Of course, $\\alpha ^{{}^p} $ is objectwise faithful/fully faithful if $(U, \\alpha ) $ is faithful/fully faithful.", "We say that such a morphism $(U, \\alpha )$ between pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ and $\\mathcal {B}:{D}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ reflects almost descent/descent/effective descent morphisms if, whenever $U(p) $ is of almost $\\mathcal {B}$ -descent/$\\mathcal {B}$ -descent/effective $\\mathcal {B}$ -descent, $p$ is of almost $\\mathcal {A}$ -descent/$\\mathcal {A}$ -descent/effective $\\mathcal {A}$ -descent.", "Consider the pseudofunctor given by the basic fibration $(\\mbox{ })^\\ast : {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ in which $(p)^\\ast : {C}/B\\rightarrow {C}/E $ is the change of base functor, given by the pullback along $p:E\\rightarrow B $ .", "For short, we say that a morphism $p:E\\rightarrow B $ is of effective descent if $p$ is of effective $(\\mbox{ })^{\\ast }$ -descent.", "In this case, a pullback preserving functor $U:{C}\\rightarrow {D}$ induces a morphism $(U, \\mathtt {u} ) $ between the basic fibrations $(\\mbox{ })^\\ast : {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ and $(\\mbox{ })^\\ast : {D}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ in which, for each object $B$ of ${C}$ , $\\mathtt {u} _ {{}_B}:{C}/B\\rightarrow {D}/U(B) $ is given by the evaluation of $U$ .", "If $U$ is faithful/fully faithful, so is the induced morphism $(U, \\mathtt {u} ) $ between the basic fibrations.", "We study pseudocosimplicial objects $\\mathcal {A}: \\dot{\\Delta }\\rightarrow \\mathfrak {H}$ and verify the obvious implications within the setting described above.", "We start with the embedding results (which are particular cases of REF ): Theorem 8.1 (Embedding Results) Let $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ be a pseudonatural transformation.", "If $\\alpha $ is objectwise faithful and $\\mathcal {B}$ is of almost ${\\rm j}$ -descent, then so is $\\mathcal {A}$ .", "Furthermore, if $\\mathcal {B}$ is of ${\\rm j}$ -descent and $\\alpha $ is objectwise fully faithful, then $\\mathcal {A}$ is of ${\\rm j}$ -descent as well.", "Of course, we have that, if $\\mathcal {A}\\simeq \\mathcal {B}$ , then $\\mathcal {A}$ is of almost ${\\rm j}$ -descent/${\\rm j}$ -descent/effective ${\\rm j}$ -descent if and only if $\\mathcal {B}$ is of almost ${\\rm j}$ -descent/${\\rm j}$ -descent/effective ${\\rm j}$ -descent as well.", "Corollary 8.2 Let $(U, \\alpha )$ be a morphism between the pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ and $\\mathcal {B}:{D}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ (as defined above).", "If $(U, \\alpha )$ is faithful, it reflects almost descent morphisms; If $(U, \\alpha )$ is fully faithful, it reflects descent morphisms; If $\\alpha $ is a pseudonatural equivalence, $(U, \\alpha )$ reflects and preserves effective descent morphisms, descent morphisms and almost descent morphisms.", "We finish this section by proving Bénabou-Roubaud Theorems.", "A functor $F$ is a pseudosection if there is $G$ such that $G\\circ F$ is naturally isomorphic to the identity.", "We use the following straightforward result: Lemma 8.3 (Monadicity of pseudosections) If a pseudosection is right adjoint, then it is monadic.", "In particular, if $\\mathcal {A}$ is a pseudocosimplicial object, then $\\mathcal {A}(d ^i: \\mathsf {n}\\rightarrow \\mathsf {n}+\\mathsf {1} ) $ is monadic whenever it has a left adjoint.", "Assume that $F\\circ G$ is isomorphic to the identity.", "Given an absolute colimit diagram $G\\circ D$ , it follows that $F\\circ G\\circ D\\cong D $ is an absolute colimit diagram.", "The result follows, then, from the monadicity theorem [2].", "The second part of the lemma follows from the fact that $d ^i$ is a retraction and, hence, since $\\mathcal {A}$ is a pseudofunctor, $\\mathcal {A}(d ^i: \\mathsf {n}\\rightarrow \\mathsf {n}+\\mathsf {1} ) $ is a pseudosection for any $i\\le \\mathsf {n}$ .", "Recall that $\\mathsf {1}$ is a monoid in $\\dot{\\Delta }$ , as explained in Remark .", "On the one hand, the monad induced by this monoid, considered, for instance, in [47] and [31], is denoted by ${\\rm suc}: = (\\mathsf {1}+-)$ on $\\dot{\\Delta }$ .", "On the other hand, this monad induces a pseudomonad ${\\rm Suc}: = \\left[{\\rm suc}, \\mathfrak {H}\\right] _{PS} $ on the 2-category $\\left[\\dot{\\Delta } , \\mathfrak {H}\\right] _{PS}$ of pseudocosimplicial objects of $\\mathfrak {H}$ .", "This is the 2-dimensional (dual) analogue of the notion of décalage of simplicial sets as in [13].", "In particular, for each $\\mathcal {A}: \\dot{\\Delta }\\rightarrow \\mathfrak {H}$ the component of the unit of ${\\rm Suc}$ on $\\mathcal {A}$ gives a pseudonatural transformation ${\\rm Suc}^\\mathcal {A}: \\mathcal {A}\\longrightarrow \\mathcal {A}\\circ {\\rm Suc}$ whose correspondent pseudofunctor is denoted by $\\mathcal {C}_ {{}_\\mathcal {A}}:\\mathsf {2}\\rightarrow \\left[ \\dot{\\Delta }, \\mathfrak {H}\\right] _ {PS} $ .", "Observe that, $\\mathcal {C}_ {{}_\\mathcal {A}}:\\mathsf {2}\\rightarrow \\left[ \\dot{\\Delta }, \\mathfrak {H}\\right] _ {PS} $ is given by the mate of $\\mathcal {A}\\circ {\\rm n}: \\mathsf {2}\\times \\dot{\\Delta } \\rightarrow \\mathfrak {H}$ , where ${\\rm n}$ is the mate of the unit of ${\\rm suc}$ viewed as a functor $\\mathsf {2}\\rightarrow \\left[\\dot{\\Delta } , \\dot{\\Delta }\\right]$ , defined by ${\\rm n}: \\mathsf {2}\\times \\dot{\\Delta } \\rightarrow \\dot{\\Delta }$ $(a,b)\\mapsto b + a \\qquad \\qquad (d,{\\rm id}_ {{}_{b}}) \\mapsto \\left( d^0: b\\rightarrow (b+1) \\right) $ $({\\rm id}_ {{}_{a}}, d^i ) \\mapsto {\\left\\lbrace \\begin{array}{ll}d^{i}:b\\rightarrow (b+1),& \\text{if } a=\\mathsf {0}\\\\d^{i+1}: (b+1)\\rightarrow (b+2), & \\text{otherwise}\\end{array}\\right.", "}$ $({\\rm id}_ {{}_{a}}, s^i ) \\mapsto {\\left\\lbrace \\begin{array}{ll}s^{i}:b\\rightarrow (b+1),& \\text{if } a=\\mathsf {0}\\\\s^{i+1}: (b+1)\\rightarrow (b+2), & \\text{otherwise.}\\end{array}\\right.}", "$ ${ \\mathsf {0} [d]_{d}[rr]^{d} && \\mathsf {1} [d]_ {d^0}@<1.2 ex>[rr]@<-1.2ex>[rr] && \\mathsf {2}[d]_{d^0} [ll]@<1.2 ex>[rr][rr]@<-1.2ex>[rr] && \\mathsf {3} [d]_ {d^0}@/_2ex/@<-1 ex>[ll]\|-{s^0}@/_4ex/@<-1ex>[ll]\|-{s^1}@<1.2 ex>[rr]@<0.4ex>[rr]@<-0.4ex>[rr]@<-1.2ex>[rr]&& \\cdots @/_1ex/@<-1 ex>[ll]\|-{s^0}@/_3ex/@<-1ex>[ll]\|-{s^1}@/_5ex/@<-1ex>[ll]\|-{s^2}[d]_{d^0} \\\\\\mathsf {1} [rr]_-{d^1} && \\mathsf {2} @<1.2 ex>[rr]\|{d^1}@<-1.2ex>[rr]\|{d^2} && \\mathsf {3}[ll]\|-{s^1}@<1.2 ex>[rr]\|-{d^1}[rr]\|-{d^2}@<-1.2ex>[rr]\|-{d^3} && \\mathsf {4}@/^2ex/@<1 ex>[ll]\|-{s^1}@/^4ex/@<1ex>[ll]\|-{s^2}@<1.2 ex>[rr]@<0.4ex>[rr]@<-0.4ex>[rr]@<-1.2ex>[rr]&&\\cdots @/^1ex/@<1 ex>[ll]\|-{s^1}@/^3ex/@<1ex>[ll]\|-{s^2}@/^5ex/@<1ex>[ll]\|-{s^3} }$ We say that a pseudofunctor $\\mathcal {A}: \\dot{\\Delta }\\rightarrow \\mathfrak {H}$ satisfies the descent shift property (or just shift property for short) if $\\mathcal {A}\\circ {\\rm Suc}$ is of effective ${\\rm j}$ -descent.", "We get, then, a version of Bénabou-Roubaud Theorem for pseudocosimplicial objects: Theorem 8.4 Let $\\mathcal {A}: \\dot{\\Delta }\\rightarrow \\mathfrak {H}$ be a pseudofunctor satisfying the shift property.", "If the pseudonatural transformation ${\\rm Suc}^\\mathcal {A}$ satisfies the Beck-Chevalley condition, then the Eilenberg-Moore factorization of $\\mathcal {A}(d) $ is pseudonaturally equivalent to its usual factorization of ${\\rm j}$ -descent theory.", "In particular, $\\mathcal {A}$ is of effective ${\\rm j}$ -descent iff $\\mathcal {A}(d) $ is monadic; $\\mathcal {A}$ is of ${\\rm j}$ -descent iff $\\mathcal {A}(d) $ is premonadic; $\\mathcal {A}$ is of almost ${\\rm j}$ -descent iff $\\mathcal {A}(d) $ is almost monadic.", "By Lemma REF , the components of ${\\rm Suc}^\\mathcal {A}{\\rm j}= ({\\rm Suc}^\\mathcal {A})\\ast {\\rm Id}_ {{}_{{\\rm j}}} $ are monadic.", "It is known that in the context of [22], [23] introduced in this section, the natural morphism $ E\\times _p E\\rightarrow E$ is always of effective $\\mathcal {A}$ -descent.", "It follows from this fact that $\\mathcal {A}_p $ always satisfies the shift property.", "More precisely: Lemma 8.5 Let $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ be a pseudofunctor, in which ${C}$ is a category with pullbacks.", "If $p$ is a morphism of ${C}$ , $\\mathcal {A}_ p$ (defined above as $\\mathcal {A}_ p:=\\mathcal {A}\\circ \\mathcal {D}_ p $ ) satisfies the shift property.", "This follows from the fact that, for any pseudofunctor $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow {\\rm \\mathsf {C}AT}$ , given a morphism $p:E\\rightarrow B$ of ${C}$ , the natural morphism $ E\\times _p E\\rightarrow E$ between the pullback of $p$ along $p$ and $E$ (being a split epimorphism) is of effective $\\mathcal {A}$ -descent.", "In particular, $\\mathcal {A}_ p\\circ {\\rm Suc}\\simeq \\mathcal {A}_ {E\\times _B E\\rightarrow E}$ is of effective ${\\rm j}$ -descent.", "The usual Bénabou-Roubaud Theorem (Theorem REF ) follows from Theorem REF , as it is shown below.", "Assuming that $\\mathcal {A}:{C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ satisfies the hypotheses of Theorem REF , we have in particular that ${\\rm Suc}^{\\mathcal {A}_p} $ satisfies the Beck Chevalley condition.", "Therefore, since $\\mathcal {A}_ p$ satisfies the shift property, $\\mathcal {A}_p (d) = \\mathcal {A}(p) $ is monadic/premonadic/almost monadic iff $\\mathcal {A}_p $ is of effective ${\\rm j}$ -descent/${\\rm j}$ -descent/almost ${\\rm j}$ -descent.", "Finally, the most obvious consequence of the commutativity properties is that bilimits of effective ${\\rm j}$ -descent diagrams are effective ${\\rm j}$ -descent diagrams.", "For instance, taking into account Remark and realizing that pseudopullbacks of functors induce pseudopullback of overcategories we already get a weak version of Theorem REF .", "Next section, we study stronger results on bilimits and apply them to descent theory." ], [ "Further on Bilimits and Descent", "Henceforth, let ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}}, {\\rm h}: \\mathfrak {B}\\rightarrow \\dot{\\mathfrak {B}} $ be inclusions as in REF and let $\\mathfrak {H}$ be a bicategorically complete 2-category.", "[Pure Structure] A morphism $f:\\mathsf {a}\\rightarrow b$ of $\\dot{\\mathfrak {A}} $ is called a ${\\rm t}$ -irreducible morphism if $b\\ne \\mathsf {a}$ and $f$ is not in the image of $\\circ : \\dot{\\mathfrak {A}}(c,b)\\times \\dot{\\mathfrak {A}}(\\mathsf {a},c)\\rightarrow \\dot{\\mathfrak {A}}(\\mathsf {a}, b), $ for every $b\\ne c $ in $\\mathfrak {A}$ .", "An object $c$ of $\\mathfrak {A}$ is called a ${\\rm t}$ -pure structure object if each 1-cell $g$ of $\\dot{\\mathfrak {A}}(\\mathsf {a}, c) $ can be factorized through some ${\\rm t}$ -irreducible morphism $f:\\mathsf {a}\\rightarrow b $ such that $b\\ne c $ .", "That is to say, $c$ is a ${\\rm t}$ -pure structure object if, for all $ g\\in \\dot{\\mathfrak {A}}(\\mathsf {a}, c)$ , there are a morphism $g^{\\prime }$ and a ${\\rm t}$ -irreducible morphism $f$ such that $g^{\\prime }f = g$ .", "The full sub-2-category of the ${\\rm t}$ -pure structure objects of $\\mathfrak {A}$ is denoted by $\\mathfrak {S}_{{}_{{\\rm t}}}$ , while the full sub-2-category of $\\dot{\\mathfrak {A}} $ of the objects that are not in $\\mathfrak {S}_{{}_{{\\rm t}}}$ (including $\\mathsf {a}$ ) is denoted by $\\mathfrak {I} _{{}_{{\\rm t}}} $ .", "We have the full inclusion ${\\rm i}_{{}_{{\\rm t}}}:\\mathfrak {I}_{{}_{{\\rm t}}}\\rightarrow \\dot{\\mathfrak {A}} $ .", "In particular, if $f:\\mathsf {a}\\rightarrow b $ is a ${\\rm t}$ -irreducible morphism of $\\dot{\\mathfrak {A}} $ , then $b$ is an object of $ \\mathfrak {I}_{{}_{{\\rm t}}}$ .", "We denote by $\\mathfrak {g}_ {{}_{{\\rm t}}} :\\overline{\\mathfrak {I} _{{}_{{\\rm t}}}\\times \\mathsf {2} }\\rightarrow \\mathfrak {I} _{{}_{{\\rm t}}}\\times \\mathsf {2} $ the full inclusion in which ${\\rm obj}\\left(\\overline{\\mathfrak {I} _{{}_{{\\rm t}}}\\times \\mathsf {2} } \\right):={\\rm obj}\\left( \\mathfrak {I} _{{}_{{\\rm t}}}\\times \\mathsf {2}\\right)-\\left\\lbrace (\\mathsf {a}, \\mathsf {0})\\right\\rbrace .", "$ Theorem 9.1 Let $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ be an objectwise fully faithful pseudonatural transformation such that $\\mathcal {B}$ is of effective ${\\rm t}$ -descent.", "We consider the mate of $\\alpha $ , denoted by $\\mathcal {C}_ \\alpha : \\dot{\\mathfrak {A}}\\times \\mathsf {2}\\rightarrow \\mathfrak {H}$ .", "The pseudofunctor $\\mathcal {A}$ is of effective ${\\rm t}$ -descent if and only if $\\mathcal {C}_ \\alpha \\circ \\left({\\rm i}_{{}_{{\\rm t}}}\\times {\\rm Id}_ {{}_{\\mathsf {2} }}\\right) : \\mathfrak {I}_{{}_{{\\rm t}}}\\times \\mathsf {2}\\rightarrow \\mathfrak {H}$ is of effective $\\mathfrak {g}_ {{}_{{\\rm t}}}$ -descent.", "Without losing generality, we prove it to $\\mathfrak {H}= {\\rm \\mathsf {C}AT}$ and get the general result via representable 2-functors.", "We just need to prove that ${\\rm Ps}{\\mathcal {R}an}_ {\\rm t}\\mathcal {A}\\circ {\\rm t}(\\mathsf {a}) $ is equivalent to ${\\rm Ps}{\\mathcal {R}an}_{{\\mathfrak {g}_ {{}_{{\\rm t}}} }} \\left(\\mathcal {C}_ \\alpha \\circ \\left({\\rm i}_{{}_{{\\rm t}}}\\times {\\rm Id}_ {{}_{\\mathsf {2} }}\\right) \\circ \\mathfrak {g}_ {{}_{{\\rm t}}} \\right)(\\mathsf {a}, \\mathsf {0}) $ .", "The category of pseudonatural transformations $\\varrho ^{\\prime } : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\rightarrow \\mathcal {A}\\circ {\\rm t}$ is equivalent to the category of pseudonatural transformations $\\varrho : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\longrightarrow \\mathcal {B}\\circ {\\rm t}$ that can be factorized through $\\alpha {\\rm t}$ , since $\\alpha {\\rm t}$ is objectwise fully faithful.", "Also, given $\\varrho : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\longrightarrow \\mathcal {B}\\circ {\\rm t}$ , there exists $\\varrho ^{\\prime } : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\rightarrow \\mathcal {A}\\circ {\\rm t}$ such that $\\varrho \\cong (\\alpha {\\rm t}) \\varrho ^{\\prime }$ if and only if the image of $(\\alpha {\\rm t}) _ {{}_{b}}$ is essentially surjective in the image of $\\varrho _ {{}_{b}}$ for every $b$ of $\\mathfrak {A}$ .", "Also, if such $\\varrho ^{\\prime }$ exists, it is unique up to isomorphism: it is the pseudopullback of $\\varrho $ along $(\\alpha {\\rm t})$ .", "Actually, we claim that, for the existence of such $\\varrho ^{\\prime }$ , it is (necessary and) sufficient $(\\alpha {\\rm t}) _ {{}_{b}}$ be essentially surjective onto the image of $\\varrho _ {{}_{b}}$ for every object $b$ of $\\mathfrak {I}_{{}_{{\\rm t}}}$ .", "That is to say, we just need to verify the lifting property for the objects in $\\mathfrak {I}_{{}_{{\\rm t}}}$ .", "Indeed, assume that $\\varrho {\\rm i}_{{}_{{\\rm t}}}$ can be lifted by $\\alpha {\\rm t}{\\rm i}_{{}_{{\\rm t}}}$ .", "Given an object $c$ of $\\mathfrak {S}_{{}_{{\\rm t}}}$ and a morphism $g: \\mathsf {a}\\rightarrow c $ , we prove that $\\varrho _ {{}_{c}}(g) $ is in the image of $\\left(\\alpha {\\rm t}\\right) _ {{}_{c}} $ up to isomorphism.", "Actually, there is a ${\\rm t}$ -irreducible morphism $f:\\mathsf {a}\\rightarrow b $ such that $g^{\\prime }f = f $ for some $g^{\\prime }:b\\rightarrow c $ morphism of $\\mathfrak {A}$ , and, by hypothesis, there is an object $u$ of $\\mathcal {A}(b) $ such that $\\left(\\alpha {\\rm t}\\right) _ {{}_{b}}(u)\\cong \\varrho _ {{}_{b}} (f) $ , thereby: $\\varrho _ {{}_{c}}(g) = \\varrho _ {{}_{c}}\\cdot \\left(\\dot{\\mathfrak {A}} (\\mathsf {a}, {\\rm t}(g^{\\prime }))\\right)(f)\\cong \\mathcal {B}(g^{\\prime }) \\varrho _ {{}_{b}}(f)\\cong \\mathcal {B}(g^{\\prime }) \\left(\\alpha {\\rm t}\\right) _ {{}_{b}}(u)\\cong \\left(\\alpha {\\rm t}\\right) _ {{}_{c}}(\\mathcal {A}(g^{\\prime })(u)) .$ This completes the proof that it is enough to test the lifting property for the objects in $\\mathfrak {I}_{{}_{{\\rm t}}}$ .", "Now, one should observe that, since $\\mathcal {B}$ is of effective ${\\rm t}$ -descent, a pseudonatural transformation $\\mathfrak {I} _{{}_{{\\rm t}}}\\times \\mathsf {2}((\\mathsf {a}, \\mathsf {0}), \\mathfrak {g}_ {{}_{{\\rm t}}} - ) \\longrightarrow \\mathcal {C}_ \\alpha \\circ \\left({\\rm i}_{{}_{{\\rm t}}}\\times {\\rm Id}_ {{}_{\\mathsf {2} }}\\right)\\circ \\mathfrak {g}_ {{}_{{\\rm t}}} $ is precisely determined (up to isomorphism) by a pseudonatural transformation $\\varrho : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\longrightarrow \\mathcal {B}\\circ {\\rm t}.$ (i.e., an object of $\\mathcal {B}(\\mathsf {a}) $ ), such that $\\varrho {\\rm i}_{{}_{{\\rm t}}}$ can be lifted by $\\alpha {\\rm t}{\\rm i}_{{}_{{\\rm t}}}$ .", "That is to say, as we proved, this is just a pseudonatural transformation $\\varrho ^{\\prime } : \\dot{\\mathfrak {A}}(\\mathsf {a}, {\\rm t}(-) )\\rightarrow \\mathcal {A}\\circ {\\rm t}.$ Definition and Theorem REF are part of a general perspective over generalizations of classical theorems of cubes and pullbacks.", "The exhaustive exposition of such is outside the scope of this paper.", "We return to the context of Section .", "Let $\\mathcal {T}$ be an idempotent pseudomonad on a 2-category $\\mathfrak {H}$ and $X$ be an object of $\\mathfrak {H}$ .", "We say that $X$ is of $\\mathcal {T}$ -descent if the comparison $\\eta _ {{}_{X}}: X\\rightarrow \\mathcal {T}(X) $ is fully faithful.", "It is important to note that, if $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ is of ${\\rm t}$ -descent (following Definition REF ), then $\\mathcal {A}$ is of ${\\rm Ps}{\\mathcal {R}an}_ {{\\rm t}}(-\\circ {\\rm t}) $ -descent.", "Corollary 9.2 Let $\\mathcal {T}$ be an idempotent pseudomonad on $\\mathfrak {H}$ and $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\mathfrak {H}$ a pseudofunctor such that all the objects in the image of $\\mathcal {A}\\circ {\\rm t}$ are $\\mathcal {T}$ -descent objects.", "Assume that both $\\mathcal {A}, \\mathcal {T}\\circ \\mathcal {A}$ are of effective ${\\rm t}$ -descent.", "We assume that $\\mathcal {A}(b) $ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure for every object $b\\notin \\mathfrak {S}_{{}_{{\\rm t}}}$ in $\\mathfrak {A}$ .", "Then $\\mathcal {A}(\\mathsf {a}) $ can be endowed with a $\\mathcal {T}$ -pseudoalgebra structure.", "Corollary 9.3 Let $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow \\left[ \\dot{\\mathfrak {B}} , \\mathfrak {H}\\right] _ {PS}$ be an effective ${\\rm t}$ -descent pseudofunctor such that all the pseudofunctors in the image of $\\mathcal {A}\\circ {\\rm t}$ are of ${\\rm h}$ -descent.", "Furthermore, we assume that $\\mathcal {A}(b) $ is of effective ${\\rm h}$ -descent for every $b\\notin \\mathfrak {S}_{{}_{{\\rm t}}}$ in $\\mathfrak {A}$ .", "Then $\\mathcal {A}(\\mathsf {a}) $ is of effective ${\\rm h}$ -descent.", "Recall the following full inclusion of 2-categories ${\\rm h}: \\mathfrak {B}\\rightarrow \\dot{\\mathfrak {B}}$ described in Section .", "${ & e[d]@{}[rd]\|-{\\mapsto } & \\mathsf {b}[d][r]&e[d] \\\\c[r]&o & c[r]&o}\\qquad \\mathrm {(\\mathfrak {P})}$ As explained there, a diagram $\\dot{\\mathfrak {B}}\\rightarrow \\mathfrak {H}$ is of effective ${\\rm h}$ -descent if and only if it is a pseudopullback.", "In this case, the unique object in $\\mathfrak {S} _ {{}_{{\\rm h}}} $ is $o$ .", "Thereby we get: Corollary 9.4 Assume that $\\mathcal {A}: \\dot{\\mathfrak {B}}\\rightarrow [\\dot{\\mathfrak {A}}, \\mathfrak {H}] _ {PS}$ is a pseudopullback diagram.", "If $\\mathcal {A}(c) , \\mathcal {A}(e): \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ are of effective ${\\rm t}$ -descent and $\\mathcal {A}(o): \\dot{\\mathfrak {A}} \\rightarrow \\mathfrak {H}$ is of ${\\rm t}$ -descent, then $\\mathcal {A}(\\mathsf {b} ) $ is of effective ${\\rm t}$ -descent.", "Taking into account Remark and realizing that pseudopullbacks of functors induce pseudopullback of overcategories, we get Theorem REF as a corollary." ], [ "Applications", "In this subsection, we finish the paper giving applications of our results and proving the remaining theorems presented in Section .", "Firstly, considering our inclusion ${\\rm j}: \\Delta \\rightarrow \\dot{\\Delta } $ , it is important to observe that $\\mathsf {1}\\notin \\mathfrak {S} _ {{}_{{\\rm j}}} $ , while all the other objects of $\\Delta $ belong to $\\mathfrak {S} _ {{}_{{\\rm j}}} $ .", "We start proving Theorem 4.2 of [20], which is presented therein as a generalized Galois Theorem.", "Theorem 9.5 (Galois) Let $\\mathcal {A}, \\mathcal {B}:\\dot{\\Delta }\\rightarrow {\\rm \\mathsf {C}AT}$ be pseudofunctors and $\\alpha : \\mathcal {A}\\longrightarrow \\mathcal {B}$ be an objectwise fully faithful pseudonatural transformation.", "We assume that $\\mathcal {B}$ is of effective ${\\rm j}$ -descent.", "The pseudofunctor $\\mathcal {A}$ is also of effective ${\\rm j}$ -descent if and only if the diagram below is a pseudopullback.", "${ \\mathcal {A}(\\mathsf {0})[d]_{\\alpha _ {{}_{\\mathsf {0} }} }[r]^{\\mathcal {A}(d) }@{}[rd]\|-{{\\alpha _ {{}_{d}}} }& \\mathcal {A}(\\mathsf {1})[d]^-{\\alpha _ {{}_{\\mathsf {1} }}}\\\\\\mathcal {B}(\\mathsf {0})[r]_ {\\mathcal {B}(d) }&\\mathcal {B}(\\mathsf {1})}$ Since, in this case, $\\mathfrak {I}_{{}_{{\\rm j}}} = \\mathsf {2} $ and the inclusion $\\mathfrak {g}_ {{}_{{\\rm j}}} :\\overline{\\mathfrak {I} _{{}_{{\\rm j}}}\\times \\mathsf {2} }\\rightarrow \\mathfrak {I} _{{}_{{\\rm j}}}\\times \\mathsf {2} $ is precisely equal to the inclusion described in the diagram REF , by Theorem REF , the proof is complete.", "As a consequence of Theorem REF , we get a generalization of Theorem REF .", "More precisely, in the context of Section and using the definitions presented there, we get: Corollary 9.6 Let $(U, \\alpha )$ be a fully faithful morphism between pseudofunctors $\\mathcal {A}: {C}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ and $\\mathcal {B}:{D}^{{\\rm op}}\\rightarrow \\mathfrak {H}$ , in which ${C}$ and ${D}$ are categories with pullbacks.", "Assume that $U(p)$ is an effective $\\mathcal {B}$ -descent morphism of ${D}$ .", "Then $p: E\\rightarrow B$ is of effective $\\mathcal {A}$ -descent if and only if, whenever there are $u\\in \\mathcal {B}(B), v\\in \\mathcal {A}(E) $ such that $\\alpha ^{{}^{p}} _{{}_{\\mathsf {1} }} (u)\\cong \\mathcal {B}_ {{}_{U(p) }}(d) (v) $ , there is $w\\in \\mathcal {A}(B) $ such that $\\alpha ^{{}^{p}} _{{}_{\\mathsf {0} }}(w)\\cong u $ .", "Recall the definitions of $\\mathcal {A}_ {{}_{p}}, \\mathcal {B}_ {{}_{U(p)}}, \\alpha ^{{}^{p}}$ .", "Since we already know that $\\mathcal {A}_ {{}_{p}}$ is ${\\rm j}$ -descent, the condition described is precisely the condition necessary and sufficient to conclude that the diagram of Theorem REF is a pseudopullback.", "Indeed, taking into account Remark , we conclude that Theorem REF is actually a immediate consequence of last corollary.", "Given a category with pullbacks $V$ , we denote by ${\\rm \\mathsf {C}at}(V) $ the category of internal categories in $V$ .", "If $V$ is a category with products, we denote by $V$ -${\\rm \\mathsf {C}at}$ the category of small categories enriched over $V$ .", "We give a simple application of the Theorem REF below.", "Lemma 9.7 If $(V, \\times , I) $ is an infinitary lextensive category such that $J: {\\rm \\mathsf {S}et}& \\rightarrow & V\\\\A &\\mapsto & \\sum _{a\\in A} I_a$ is fully faithful, then the pseudopullback of the projection of the object of objects $U_0: {\\rm \\mathsf {C}at}(V)\\rightarrow V$ along $J$ is the category $V$ -${\\rm \\mathsf {C}at}$ .", "We denote by $\\textit {Span}(V)$ the usual bicategory of objects of $V$ and spans between them and by $V$ -$\\mathrm {Mat}$ the usual bicategory of sets and $V$ -matrices between them.", "Let $\\textit {Span} _{{}_{{\\rm \\mathsf {S}et}}}(V) $ be the full sub-bicategory of $\\textit {Span}(V)$ in which the objects are in the image of ${\\rm \\mathsf {S}et}$ .", "Assuming our hypotheses, we have that $\\textit {Span} _{{}_{{\\rm \\mathsf {S}et}}}(V) $ is biequivalent to $V$ -$\\mathrm {Mat}$ .", "Indeed, we define “identity” on the objects and, if $A, B $ are sets, take a matrix $M: A\\times B\\rightarrow {\\rm obj}(V) $ to the obvious span given by the coproduct $\\displaystyle \\sum _ {(x,y)\\in A\\times B} M(x,y) $ , that is to say, the morphism $\\sum _ {(x,y)\\in A\\times B} M(x,y)\\rightarrow A $ is induced by the morphisms $M(x,y)\\rightarrow I_x $ and the morphism $\\sum _ {(x,y)\\in A\\times B} M(x,y)\\rightarrow B $ is analogously defined.", "Since $V$ is lextensive, this defines a biequivalence.", "Thereby this completes our proof.", "Corollary 6.2.5 of [32] says in particular that, for lextensive categories, effective descent morphisms of ${\\rm \\mathsf {C}at}(V)$ are preserved by the projection $U_0: {\\rm \\mathsf {C}at}(V)\\rightarrow V$ to the objects of objects.", "Thereby, by Theorem REF , we get: Theorem 9.8 If $(V, \\times , I) $ is an infinitary lextensive category such that each arrow of $V$ can be factorized as a regular epimorphism followed by a monomorphism and $J: {\\rm \\mathsf {S}et}& \\rightarrow & V\\\\A &\\mapsto & \\sum _{a\\in A} I_a$ is fully faithful, then $I: V\\textrm {-}{\\rm \\mathsf {C}at}\\rightarrow {\\rm \\mathsf {C}at}(V) $ reflects effective descent morphisms.", "We denote by $U: V\\textrm {-}{\\rm \\mathsf {C}at}\\rightarrow {\\rm \\mathsf {S}et}$ the forgetful functor and by $U_0: {\\rm \\mathsf {C}at}(V)\\rightarrow V$ the projection defined above.", "We have that $U_0, U, J $ and $I$ are pullback preserving functors.", "If $p:E\\rightarrow B $ is a morphism of $V$ -${\\rm \\mathsf {C}at}$ such that $I(p)$ is of effective descent, then $U_0 I(p) $ is of descent (by Corollary 5.2.1 of [32]).", "Therefore $JU(p)$ is of descent.", "Since $J$ is fully faithful, by Theorem REF , $U(p) $ is of descent.", "Therefore, since descent morphisms of ${\\rm \\mathsf {S}et}$ are of effective descent, we conclude that $U(p)$ is of effective descent.", "This completes the proof.", "For instance, Theorem 6.2.8 of [32] and Proposition REF can be applied to the cases of $V={\\rm \\mathsf {C}at}$ or $V = {\\rm \\mathsf {T}op}$ : Corollary 9.9 A 2-functor $F$ between ${\\rm \\mathsf {C}at}$ -categories is of effective descent in ${\\rm \\mathsf {C}at}$ -${\\rm \\mathsf {C}at}$ , if $F$ is surjective on objects; $F$ is surjective on composable triples of 2-cells; $F$ induces a functor surjective on composable pairs of 2-cells between the categories of composable pairs of 1-cells; $F$ induces a functor surjective on 2-cells between the categories of composable triples of 1-cells.", "Corollary 9.10 A ${\\rm \\mathsf {T}op}$ -functor $F$ between ${\\rm \\mathsf {T}op}$ -categories is of effective descent in ${\\rm \\mathsf {T}op}$ -${\\rm \\mathsf {C}at}$ , if $F$ induces effective descent morphisms between the discrete spaces of objects and between the spaces of morphisms in ${\\rm \\mathsf {T}op}$ ; a descent continuous map between the spaces of composable pairs of morphisms in ${\\rm \\mathsf {T}op}$ ; an almost descent continuous map between the spaces of composable triples of morphisms in ${\\rm \\mathsf {T}op}$ .", "Since the characterization of (effective/almost) descent morphisms in ${\\rm \\mathsf {T}op}$ is known [43], [10], [8], the result above gives effective descent morphisms of ${\\rm \\mathsf {T}op}$ -${\\rm \\mathsf {C}at}$ .", "We can give further formal results on (basic) effective descent morphisms (context of Remark ).", "The main technique in this case is to understand our overcategory as a bilimit of other overcategories.", "For instance, we study below the categories of morphisms of a given category ${C}$ with pullbacks.", "Consider the full inclusion of 2-categories ${\\rm t}: \\mathfrak {A}\\rightarrow \\dot{\\mathfrak {A}} $ ${\\mathsf {0}[d]\|-{d}@{}[rd]\|{\\mapsto }& \\mathsf {a}[rd]\|-{pro_{{}_{0}} }[rr]\|-{pro_{{}_{1}}}&@{}[d]\|-{{\\xi } } &\\mathsf {0}[ld]\|-{d}\\\\\\mathsf {1}& &\\mathsf {1}&} $ Given a morphism of ${C}$ , i.e.", "a functor $F:\\mathsf {2}\\rightarrow {C}$ , we take the overcategory $\\textrm {Fun}(\\mathsf {2}, {C})/F $ and define $\\mathcal {A}: \\dot{\\mathfrak {A}}\\rightarrow {\\rm \\mathsf {C}AT}$ in which $\\begin{aligned}\\mathcal {A}(\\mathsf {a} ):= &\\textrm {Fun}(\\mathsf {2}, {C})/F\\end{aligned}\\qquad \\begin{aligned}\\mathcal {A}(\\mathsf {0} ):= & {C}/F(\\mathsf {1})\\end{aligned}\\qquad \\begin{aligned}\\mathcal {A}(\\mathsf {1} ):= & {C}/F(\\mathsf {0}).\\end{aligned}$ Finally, $\\mathcal {A}(pro_{{}_{0}}), \\mathcal {A}(pro_{{}_{1}}) $ are given by the obvious projections, $\\mathcal {A}(d) := F(d) ^\\ast $ and the component $\\mathcal {A}(\\xi ) $ in a morphism $\\varpi : H\\rightarrow F $ is given by the induced morphism from $H(\\mathsf {0} ) $ to the pullback.", "Observe that $\\mathcal {A}$ is of effective ${\\rm t}$ -descent, that is to say, we have that the overcategory $\\textrm {Fun}(\\mathsf {2}, {C})/F $ is a bilimit constructed from overcategories ${C}/F(\\mathsf {0})$ and ${C}/F(\\mathsf {1})$ .", "Also, given a natural transformation $\\varpi : F\\rightarrow G $ between functors $\\mathsf {2}\\rightarrow {C}$ , i.e.", "a morphism of $\\textrm {Fun}(\\mathsf {2}, {C})$ , taking Remark , we can extend $\\mathcal {A}$ to a 2-functor $\\overline{\\mathcal {A}}:\\dot{\\mathfrak {A}}\\rightarrow [\\dot{\\Delta }, {\\rm \\mathsf {C}AT}] $ in which $\\overline{\\mathcal {A}}(\\mathsf {a}) : = (\\mbox{ })^\\ast _\\varpi $ , $\\overline{\\mathcal {A}}(\\mathsf {0}) : = (\\mbox{ })^\\ast _{\\varpi _\\mathsf {1}} $ and $\\overline{\\mathcal {A}}(\\mathsf {1}) : = (\\mbox{ })^\\ast _{\\varpi _\\mathsf {0}} $ .", "The 2-functor $\\overline{\\mathcal {A}}$ is also of effective ${\\rm t}$ -descent.", "Therefore, by our results, we conclude that, if the components $\\varpi _\\mathsf {1}, \\varpi _\\mathsf {0} $ are of (basic) effective descent, so is $\\varpi $.", "Analogously, considering the category of spans in ${C}$ , the morphisms between spans which are objectwise of effective descent are of effective descent." ] ]	1606.04999
[ [ "Comments on the slope function" ], [ "Abstract The exact slope function was first proposed in $SL(2)$ sector and generalized to $SU(2)$ sector later.", "In this note, we consider the slope function in $SU(1\|1)$ sector of ${\\cal N}=4$ SYM.", "We derive the quantity through the method invented by N. Gromov and discuss about its validity.", "Further, we give comments on the slope function in deformed SYM." ], [ "Introduction", "Solving interacting QFT is generally very difficult because we still lack general techniques beyond perturbation.", "However, if we restrict our interest to a highly symmetric but still interesting model, one can utilize various methods to treat nonperturbative field theory.", "For instance, the sine(sinh)-Gordon model in $(1+1)$ -dimension is a prototypical QFT which could be exactly solved through integrability [1].", "Beyond two dimensional theories, such an example seems to be the maximally supersymmetric, four-dimensional Yang-Mills theory since the theory shows integrable structures in itself and also in holographic dual description.", "Integrability structures were found in various observables of $N=4$ SYM such as conformal dimensions of gauge invariant operators, spacetime scattering amplitudes and correlation functions.Although there exist many literatures for integrability of $N=4$ SYM, the best exposition would be [2] written by leading experts in this area.", "Very recently, pedagogical lecture notes were written and available in [3].", "Among those, the spectral problem is the most developed and is now formulated in a beautiful algebraic structure, the so-called quantum spectral curve [4].", "Historically, several key concepts, quantities and equations led to obtain the QSC.", "For example, the Beiset-Staudacher Bethe ansatz equation [5], exact $S$ -matrix [6], [7], $Y,T,Q$ -systems [8], [9], [10], [11], [12] are of those.", "One of such important developments was the exact slope function [13].", "In [13], Basso gave a conjecture for the slope function of composite operators in $sl(2)$ sector.", "The slope function is defined as a coefficient of the leading anomalous dimension in the small spin $S$ limit such as $\\Delta -J = S+ \\gamma = \\alpha _J S +{\\cal O}(S^2), $ where $J$ is the number of $Z$ fields consisted of the BPS vacuum and $S$ is the number of the covariant derivatives ${\\cal D}$ where $S$ is clearly an integer but can be thought as any real number in a kind of analytical continuation.", "Very remarkable facts of the conjecture are its exactness for all-loops of the coupling constant $\\lambda $ and that the quantity is independent of wrapping effects.", "On the other hand, the slope function in ABJM model depends on wrapping contributions.", "Thus, since simple derivation through asympototic BAE or the Baxter equation have been not enough, we needed a formulation beyond asymptotic formalism.", "Later, the quantum spectral curve allows to calculate the slope function [14].", "After Basso's conjecture, the slope function was formally derived in [15], [16] and was also generalized from $SL(2)$ sector to $SU(2)$ sector [15].", "Till now, the conjecture have passed some nontrivial tests [18], [19].", "Finally, the slope and the curvature which are separately the leading and the next to leading terms of small spin expansion were delicately calculated through the quantum spectral curve [17].", "In this letter, we would like to obtain the expression of the slope function in the $SU(1\|1)$ sector.", "It will provide a complete table for slope functions in all rank-one sectors.", "Further, it would give another source to check the quantum spectral curve as comparing results in this note.", "We derive the anomalous dimension and consider the small impurity limit which would be a kind of analytic continuation.", "Moreover, we give some comments on the slope function in deformed SYM.", "As the simplest choices, we consider the slope of $SU(2)_{\\beta }$ and $SL(2)_{\\beta }$ We conclude this letter with some discussions." ], [ "Comments on the slope function", "The anomalous dimension $\\gamma $ in (REF ) is defined as the second conserved charge ${\\cal Q}_2$ such as $\\gamma = \\frac{i \\sqrt{\\lambda }}{2\\pi } \\sum _{j=1}^{S} \\left(\\frac{1}{x_{j}^{+}}-\\frac{1}{x_{j}^{-}} \\right) \\equiv {\\cal Q}_{2}$ among infinitely many conserved charges.", "The $\\gamma $ can be expanded in both of large and small 't Hooft coupling constant $\\lambda $ and generally depends on complicated wrapping corrections.", "Surprisingly, Basso conjectured that it could be computed only from the asymptotic BAE since the slope function of $N=4$ SYM is independent of wrapping effects.", "Such an exact slope is given as $\\alpha _{J} = 1 + \\frac{\\Lambda }{J} \\frac{I_{J+1}(\\Lambda )}{I_{J}(\\Lambda )},$ where $\\Lambda $ is defined as $n\\sqrt{\\lambda }$ with the mode number $n$ and the function $I_{n}(x)$ is the modified Bessel function of the first kind.", "After this work, Gromov elegantly derived the slope function by considering a quite special limit [15].", "With this derivation method, we first analyse the slope function in $SU(1\|1)$ sector of $N=4$ SYM and discuss about the expression.", "Next, we give comments on the slope function in $\\beta $ -deformed SYM." ], [ "Slope function in $SU(1\|1)$ sector", "With the following notation, $&& x+\\frac{1}{x} = \\frac{u}{g}, \\quad x_{k}^{\\pm } = x\\left(u_{k} \\pm \\frac{i}{2} \\right) \\\\&& x(u) = \\frac{u}{2g} + \\frac{u}{2g} \\sqrt{1- \\frac{4g^2}{u^2}} ,$ the $su(1\|1)$ BAE for the composite operator of one type of boson and fermion such as ${\\rm tr}(\\psi ^{J-S} Z^{S})$ can be written as followsWe used a little different but physically equivalent convention for spectral parameters compared with [5].", ": $\\left(\\frac{x_{k}^{+}}{x_{k}^{-}}\\right)^{J} = \\prod _{j \\ne k}^{S} \\frac{1-\\frac{1}{x_k^{+} x_j^{-}}}{1-\\frac{1}{x_k^{-} x_j^{+}}} \\sigma ^2 (x_k , x_j), \\quad (k=1,\\cdots , S) $ Taking the logarithm and dividing by $i$ , one gets $\\frac{J}{i}\\log {\\left(\\frac{x_{k}^{+}}{x_{k}^{-}}\\right)} - \\frac{1}{i}\\sum _{j \\ne k}^{S} \\log {\\left(\\frac{1-\\frac{1}{x_k^{+} x_j^{-}}}{1-\\frac{1}{x_k^{-} x_j^{+}}} \\sigma ^2 (x_k , x_j ) \\right)} = 2\\pi n_{k} .", "$ Now we will consider the limit which was used in [15] such as $n_k = n \\rightarrow 0 \\quad {\\rm for} \\,\\ \\rm {all} \\,\\ k , \\quad \\Lambda = n \\sqrt{\\lambda } = {\\rm fixed}.$ We should give an important remark for this limit.", "The mode numbers $n_{k}$ should usually be different because of the fermionic nature of the $SU(1\|1)$ sector.", "Nevertheless, it was known that roots with different mode numbers do not interact in $n_{k} \\rightarrow 0$ limit [13], [15].", "Thus, one can reintroduce the different mode numbers in the final expression of anomalous dimension.", "Since this limit is definitely a large $\\lambda $ limit, one can make use of strong coupling expressions for Zhukowski variables and the dressing phase such as $\\log \\sigma (x_k ,x_{j}) &\\simeq & {\\frac{i(x_k -x_{j}) }{g(-1+x_k^2) (-1+x_k x_{j}) (-1+x_{j}^2)} }, \\\\x^{\\pm }(z) &=& x(z) \\pm \\frac{i}{2 g} \\frac{x^2(z)}{x^2(z) -1} + O(1/g^2), \\\\\\gamma &\\simeq & 2 \\sum _{j=1}^{S} \\frac{1}{x_j^2 -1} = G(-1) - G(1)$ where we defined the resolvent $G(x)$ such as $G(x) = \\sum _{j=1}^{S} \\frac{1}{x-x_j }.", "$ With (REF ), by multiplying $\\frac{\\sqrt{\\lambda }(x_k^2 -1)}{4 \\pi x_k^2}$ and using the above identities, we get the following form of the BAE : $\\frac{2J+\\gamma }{2x_k} - \\sum _{j=1}^{S} \\left( \\frac{x_j}{x_j^2 -1} \\right) \\frac{1}{x_k^2} = \\frac{\\Lambda (x_k^2 -1)}{2 x_k^2} .", "$ Next, what we have to do is to multiply $\\frac{1}{x- x_k}$ to (REF ) and sum over $k$ .", "Then, the first term of the l.h.s.", "of (REF ) becomes $\\left(\\frac{2J+\\gamma }{2x} \\right) (G(x) - G(0)).$ Also, the second term of the l.h.s.", "of (REF ) can be expressed as $\\left(\\sum _j \\frac{x_j }{x_j^2 -1 }\\right) \\left(\\sum _k \\frac{1}{x_k^2 } \\frac{1}{x-x_k} \\right) = -\\frac{1}{2} (G(1)+G(-1)) \\times \\frac{G(x)-G(0) - G^{\\prime }(0)x }{x^2},$ where we used $G(0) = - \\sum _k \\frac{1}{x_k }, \\quad G^{\\prime }(0) = - \\sum _k \\frac{1}{x_k^2 }.$ Lastly, the r.h.s of (REF ) can be also written similarly such as $\\sum _{k} \\frac{\\Lambda (x_k^2 -1)}{2 x_k^2 } \\frac{1}{x-x_{k} } = \\frac{\\Lambda }{2} G(x) -\\frac{\\Lambda }{2} \\left( \\frac{G(x)-G(0) - G^{\\prime }(0)x }{x^2} \\right).$ Thus, (REF ) can be rewritten such as $G(x) = \\frac{(G(1)+G(-1)+\\Lambda )G(0)+[(G(1)+G(-1)+\\Lambda )G^{\\prime }(0)+(2J+\\gamma )G(0)]x }{(G(1)+G(-1)+\\Lambda ) + (2J+\\gamma )x - \\Lambda x^2}.", "$ Now let us consider large $x$ limit of (REF ) to replace $G^{\\prime }(0)$ by non-derivative quantities such as $G(1)$ , $G(-1)$ and $G(0)$ .", "In this limit, the resolvent $G(x)$ behaves as $G \\sim \\frac{S}{x}$ from (REF ).", "Thus, by comparing leading contributions from each sides, we get $G^{\\prime }(0) = - \\frac{\\Lambda S + (2J+\\gamma )G(0)}{\\Lambda + G(1) + G(-1)}.", "$ Substituting (REF ) in (REF ), we can write the following : $G(x) = \\frac{(G(1)+G(-1)+\\Lambda )G(0)- \\Lambda S x }{(G(1)+G(-1)+\\Lambda ) + (2J+\\gamma )x - \\Lambda x^2}.", "$ At this stage, we still have some unknown quantities such as $G(1)$ , $G(-1)$ and $G(0)$ .", "However, they can be eliminated by checking the consistency of (REF ) at $x = \\pm 1$ .", "They are determined such as $G(1)= -\\frac{\\Lambda S+\\gamma J}{2J}, \\quad G(-1)= \\frac{-\\Lambda S+\\gamma J}{2J}, \\quad G(0)= -\\frac{2 J^3 \\gamma +J^2 \\gamma ^2-S^2 \\Lambda ^2}{2 J^2 \\Lambda -2 J S \\Lambda }.$ We finally have $G(x) = \\frac{\\left(-2 J^3 \\gamma -J^2 \\gamma ^2+S^2 \\Lambda ^2\\right)-2 \\left(J^2 S \\Lambda \\right) x}{\\left(2 J^2 \\Lambda -2 J S \\Lambda \\right)+\\left(4 J^3+2 J^2 \\gamma \\right) x-2 \\left(J^2 \\Lambda \\right) x^2}.$ Note that the resolvent $G(x)$ is analytic.", "So, if we require vanishing residues at poles of $G(x)$ , one can obtain the following possibilities for $\\gamma $ : $\\gamma = -2J - \\frac{S \\Lambda }{J}, \\quad -2J + \\frac{S \\Lambda }{J}, \\quad -S - S \\sqrt{1+\\frac{\\Lambda ^2}{J^2}} , \\quad -S + S \\sqrt{1+\\frac{\\Lambda ^2}{J^2}}.$ The first and second expressions would be inconsistent because they would not become zero in the $S \\rightarrow 0$ limit.", "Besides, the third one seems unphysical because it means that energy of non-BPS state is lower than that of a BPS.", "Therefore, we could take the last expression as a correct expression of the anomalous dimension : $\\gamma = -S + S \\sqrt{1+\\frac{\\Lambda ^2}{J^2}} \\rightarrow \\sum _{n_{k} =1}^{S} \\left(-1 + \\sqrt{1+ \\frac{\\lambda }{J^2}n_{k}^{2}} \\right) + {\\cal O}(S^2 ),$ where we reintroduced the mode number $n_{k}$ .", "Actually, this dispersion is matched with the result from pp-wave limit of $SU(1\|1)$ string [20].", "Further, as $\\gamma $ can be expanded in small and large $\\frac{\\lambda }{J^2}\\equiv \\lambda ^{\\prime }$ limit, we have $&& \\alpha _J =1-\\frac{1}{S}\\sum _{k=1}^{S} n_{k} + \\frac{\\lambda }{2 J^2}\\frac{1}{S}\\sum _{k=1}^{S} n^2_{k}+O(\\lambda ^{\\prime 2} ) , \\\\&& \\alpha _J =-\\frac{1}{S}\\sum _{k=1}^{S} n_{k} + \\frac{J}{2\\sqrt{\\lambda }S}\\sum _{k=1}^{S}\\frac{1}{ n_{k}}+O\\left(\\frac{1}{\\lambda ^{\\prime 3/2}} \\right) , $ where $\\alpha _{J} \\equiv 1+ \\frac{\\gamma }{S}$ .", "Moreover, one can get expressions for the higher conserved charges which is defined as ${\\cal Q}_r = i \\frac{\\sqrt{\\lambda }}{2\\pi } \\sum _{j=1}^{S} \\left[\\frac{\\left(x_{j}^{+}\\right)^{1-r}}{r-1} - \\frac{\\left(x_{j}^{-}\\right)^{1-r}}{r-1} \\right] ,$ through their generating function $H(x)$ and $H_{0}(x)$ defined by $n \\rightarrow 0$ limit of $H(x)$ [15]: $H(x) = \\sum _{j=1}^{S} \\frac{\\sqrt{\\lambda }}{4\\pi i} \\log {\\left(\\frac{1-\\frac{x}{x_{j}^{-}}}{1-\\frac{x}{x_{j}^{+}}} \\right)} = -\\frac{1}{2} \\sum _{r=1} {\\cal Q}_{r+1} x^{r}.$ The $H_{0}(x)$ in $su(1\|1)$ is given as $H_{0} (x) = \\frac{\\Lambda ^2 S x^2 + \\Lambda (J-S)\\gamma x}{2 J \\Lambda x^2 - 2J(2J+\\gamma )x -2\\Lambda (J-S)}$ Thus, by expanding $H_{0}(x)$ in polynomials of $x$ with the slope function $\\alpha _{J}$ , we have the following leading order of conserved charges : ${\\cal Q}_{3} &=& \\frac{(2 J^2 \\gamma + J \\gamma ^2 - S \\Lambda ^2 )x^2 }{2\\Lambda (J-S)}, \\\\{\\cal Q}_{4} &=& \\frac{(4J^3 \\gamma + 4J^2\\gamma ^2 - 2S\\gamma )J x^3 }{2\\Lambda ^2 (J-S)^2}.$ Note that this is just a formal expression and we need to substitute $\\gamma $ and $\\Lambda $ with certain precision into (REF ) and () for an explicit value." ], [ "Slope function in $SU(2)_{\\beta }$", "Before going to the deformed case, let us recall how to generalize from the $SL(2)$ result to the $SU(2)$ one.", "In [15], the corresponding expression in $SU(2)$ sector was obtained from the slope function in $SL(2)$ by replacing $J$ and $\\Lambda $ by $-L$ and $-\\Lambda $ with $M=S$ : $\\alpha _{L} = 1- \\frac{\\Lambda }{L} \\frac{I_{-L+1}(\\Lambda )}{I_{-L}(\\Lambda )}, $ It was further pointed out that the expression (REF ) is meaningful only for first considering small $\\Lambda $ and setting $L$ to integers.", "As the small $\\Lambda $ expansion of the modified Bessel function gives the gamma function, one can get the expression (36) of [15].", "For the $SU(2)_{\\beta }$ sector of $\\beta $ -deformed SYM, the asymptotic Bethe equation is given as $\\left(\\frac{x^{+}_{k}}{x^{-}_{k}}\\right)^L = \\prod _{\\stackrel{j=1}{j \\ne k}}^{M}\\Bigg \\lbrace \\left( \\frac{x^{+}_{k} - x^{-}_{j}}{x^{-}_{k} - x^{+}_{j}}\\right)\\left(\\frac{1- \\frac{1}{x^{+}_{k} x^{-}_{j}}}{1-\\frac{1}{x^{-}_{k} x^{+}_{j}}} \\right)\\sigma ( u_{k}, u_{j} )^{2} \\Bigg \\rbrace e^{2\\pi i \\beta L} $ where $M$ is the number of magnons in length $L$ .", "The deformation effect to the BAE is just $e^{2\\pi i \\beta L}$ .", "Therefore, if we repeat similar calculations with (REF ), one can use the same expression (REF ) by redefining $\\Lambda $ such as $\\Lambda \\rightarrow \\Lambda + {\\hat{\\beta }}L$ with ${\\hat{\\beta }}\\equiv \\beta \\sqrt{\\lambda }$ .", "Thus, we simply get the slope function such as $\\gamma _{\\beta } = -M \\left( \\frac{\\Lambda }{L} +{\\hat{\\beta }} \\right) \\frac{I_{-L+1}(\\Lambda + {\\hat{\\beta }} L)}{I_{-L}(\\Lambda + {\\hat{\\beta }} L)}.$ Notice that we do not consider any wrapping effects since we just used the asymptotic BAE.", "Thus, (REF ) would be valid asymptotically.", "In the regime of $\\Lambda ^{\\prime }=\\Lambda +{\\hat{\\beta }} L << 1$ , we would have exactly same expression as in undeformed $SU(2)$ : $\\gamma _{\\beta } = \\frac{(\\Lambda +{\\hat{\\beta }} L)^2}{2L(L-1)} - \\frac{(\\Lambda +{\\hat{\\beta }} L)^4 }{8 L (L-1)^2 (L-2)} + \\cdots $ If we further expand $\\gamma _{\\beta }$ in the small ${\\hat{\\beta }}$ limit, we can easily express the difference between anomalous dimensions in deformed and undeformed theories as below.", "$\\delta \\gamma \\equiv \\gamma _{\\beta }-\\gamma = M {\\hat{\\beta }} \\left( \\frac{\\Lambda }{L-1}-\\frac{\\Lambda ^3 }{2(L-1)^2 (L-2)} \\right) + {\\cal O}({\\hat{\\beta }}^2)$ It still remains to be understood if the expression (REF ) is valid at large ${\\hat{\\beta }}$ regime.", "Because the generalized slope is delicately defined as we mentioned in the underformed $SU(2)$ sector." ], [ "Slope function in $SL(2)_{\\beta }$", "As the $\\beta $ -deformation of SYM does not break its conformality and only affect to $SO(6)$ part, we do not have any twisted effects in $SL(2)$ subsector of $SO(4,2)$ .", "However, the quantum string Bethe equation for the TsT-transformations of $AdS_5$ could be studied [21].", "Although the dual gauge theory for this deformation is still unclear, its string theory is well-defined [22].", "Here, because the string Bethe equation in $SL(2)_\\beta $ sector could be expressed from undeformed $SL(2)$ Bethe equation by considering the shift of the mode number such as $n_{k} \\rightarrow n_{k} +\\beta J, \\nonumber $ the fixed parameter $\\Lambda $ is also shifted under the mode number shift.", "Therefore, the slope function for $sl(2)_\\beta $ sector can be written as $\\alpha = 1+ \\left(\\frac{\\Lambda +{\\hat{\\beta }}J}{J} \\right) \\frac{I_{J+1}(\\Lambda +{\\hat{\\beta }}J)}{I_{J}(\\Lambda +{\\hat{\\beta }}J)}.$ This can be expanded at the string coupling regime.", "However, its weak coupling analysis is unclear since it is not obtained from gauge theory." ], [ "Concluding remarks", "In this letter, we studied the slope function in the $SU(1\|1)$ sector of $N=4$ SYM and in integrable twisted models.", "We note that we followed the derivation of [15] for computation of the slope function in $SU(1\|1)$ sector, there exists an alternative derivation method based on Baxter equation as in [16].", "In this derivation, we need to know the long-ranged $su(1\|1)$ Baxter equation and its linearized form in the small spin limit.", "It would be nice to examine this derivation and compare with our results.", "It would be interesting to derive the results in this letter from string theory.", "The light cone gauged string theory in $SU(1\|1)$ have been intensively studied in [20].", "Also, the short string was exposited via algebraic curve in [23].", "If we can utilize algebraic curve for $SU(1\|1)$ sector and consider the short string limit, it may be possible to get the slope function.", "Also, note that the deformation parameter naturally became $\\hat{\\beta }$ in the slope for $SU(2)_{\\beta }$ .", "Interestingly, the metric and various fluxes in Lunin-Maldacena background which is the dual spacetime of $\\beta $ -deformed SYM are all written in terms of $\\hat{\\beta }$ [24].", "So, it would be nice to obtain the slope from direct string computation on the background with $\\hat{\\beta }$ .", "We further remark that we ignored the wrapping effects when we obtained the slope in $SU(2)_{\\beta }$ and $SL(2)_{\\beta }$ .", "It is known that the slope which is the leading coefficient in small spin limit does not depend on the wrapping in $N=4$ SYM.", "However, there is no guarantee that such a wrapping independent nature also appears in deformed models.", "Thus, our results for $SU(2)_{\\beta }$ and $SL(2)_{\\beta }$ is asymptotically correct.", "It would be very interesting to check if the wrapping contribution gives any correction.", "It should be possible since the all-loop formulation is available through the twisted quantum spectral curve [25]." ], [ "Acknowledgements", "We thank Z. Bajnok for valuable comments.", "This work was supported by a postdoctoral fellowship of the Hungarian Academy of Sciences, a Lendület grant and OTKA 116505." ] ]	1606.05141
[ [ "Exclusive diffractive production of $\\pi^{+}\\pi^{-}\\pi^{+}\\pi^{-}$ via\n the intermediate $\\sigma\\sigma$ and $\\rho\\rho$ states in proton-proton\n collisions within tensor pomeron approach" ], [ "Abstract We present first predictions of the cross sections and differential distributions for the central exclusive reaction $pp \\to pp \\pi^{+} \\pi^{-} \\pi^{+} \\pi^{-}$ being studied at RHIC and LHC.", "The amplitudes for the processes are formulated in terms of the tensor pomeron and tensor $f_{2 R}$ reggeon exchanges with the vertices respecting the standard crossing and charge-conjugation relations of Quantum Field Theory.", "The $\\sigma \\sigma$ and $\\rho \\rho$ contributions to the $\\pi^{+} \\pi^{-} \\pi^{+} \\pi^{-}$ final state are considered focussing on their specificities.", "The correct inclusion of the pomeron spin structure seems crucial for the considered sequential mechanisms in particular for the $\\rho\\rho$ contribution which is treated here for the first time.", "The mechanism considered gives a significant contribution to the $pp \\to pp \\pi^{+} \\pi^{-} \\pi^{+} \\pi^{-}$ reaction.", "We adjust parameters of our model to the CERN-ISR experimental data and present several predictions for the STAR, ALICE, ATLAS and CMS experiments.", "A measurable cross section of order of a few $\\mu b$ is obtained including the experimental cuts relevant for the LHC experiments.", "We show the influence of the experimental cuts on the integrated cross section and on various differential distributions." ], [ "Introduction", "Last years there was a renewed interest in exclusive production of two meson pairs (mostly $\\pi ^+ \\pi ^-$ pairs) at high energies related to planned experiments at RHIC [1], Tevatron [2], [3], and LHC [4], [5], [6].", "From the experimental point of view the exclusive processes are important in the context of resonance production, in particular, in searches for glueballs.", "The experimental data on central exclusive $\\pi ^{+}\\pi ^{-}$ production measured at the energies of the ISR, RHIC, Tevatron, and the LHC collider all show visible structures in the $\\pi ^{+}\\pi ^{-}$ invariant mass.", "It is found that the pattern of these structures seems to depend on experiment.", "But, as we advocated in Ref.", "[7], this dependence could be due to the cuts used in a particular experiment (usually these cuts are different for different experiments).", "So far theoretical studies concentrated on two-pion continuum production.", "Some time ago two of us have formulated a Regge-type model with parameters fixed from phenomenological analysis of total and elastic $NN$ and $\\pi N$ scattering [8].", "The model was extended to include also absorption effects due to proton-proton interaction [9], [10].", "In Ref.", "[9] the exclusive reaction $pp \\rightarrow pp \\pi ^{+}\\pi ^{-}$ constitutes an irreducible background to the scalar $\\chi _{c0}$ meson production.", "These model studies were extended also to $K^{+}K^{-}$ production [10].", "For a related work on the exclusive reaction $pp \\rightarrow nn \\pi ^{+}\\pi ^{+}$ , see [11].", "A revised view of the absorption effects including the $\\pi N$ nonperturbative interactions was presented very recently [12].", "Such an approach gives correct order of magnitude cross sections, however, does not include resonance contributions which interfere with the continuum.", "It was known for a long time that the frequently used vector-pomeron model has problems considering a field theory point of view.", "Taken literally it gives opposite signs for $pp$ and $\\bar{p}p$ total cross sections.", "A way out of these problems was already shown in [13] where the pomeron was described as a coherent superposition of exchanges with spin 2 + 4 + 6 + ... .", "The same idea is realised in the tensor-pomeron model formulated in [14].", "In this model pomeron exchange can effectively be treated as the exchange of a rank-2 symmetric tensor.", "The corresponding couplings of the tensorial object to proton and pion were worked out.", "In Ref.", "[15] the model was applied to the diffractive production of several scalar and pseudoscalar mesons in the reaction $p p \\rightarrow p p M$ .", "In [16] an extensive study of the photoproduction reaction $\\gamma p \\rightarrow \\pi ^{+} \\pi ^{-} p$ in the framework of the tensor-pomeron model was presented.", "The resonant ($\\rho ^0 \\rightarrow \\pi ^{+}\\pi ^{-}$ ) and non-resonant (Drell-Söding) photon-pomeron/reggeon $\\pi ^{+} \\pi ^{-}$ production in $pp$ collisions was studied in [17].", "The exclusive diffractive production of $\\pi ^{+} \\pi ^{-}$ continuum together with the dominant scalar $f_{0}(500)$ , $f_{0}(980)$ , and tensor $f_{2}(1270)$ resonances was studied by us very recently in Ref. [7].", "The past program of central production of pairs of mesons was concentrated on the discussion of mesonic resonances.", "The low-energy program of studying meson excitations can be repeated at the LHC, where we expect dominance of one production mechanism only, two-pomeron exchange.", "The identification of glueballs can be very difficult.", "The partial wave analyses of future experimental data of the STAR, ALICE, ATLAS and CMS Collaborations could be used in this context.", "Also the studies of different decay channels in central exclusive production would be very valuable.", "One of the possibilities is the $p p \\rightarrow p p \\pi ^+ \\pi ^- \\pi ^+ \\pi ^-$ reaction being analysed by the ATLAS, CMS and ALICE Collaborations at the LHC.", "Identification of the glueball-like states in this channel requires calculation/estimation of the four-pion background from other sources.", "Pairs of $\\rho ^{0} \\rho ^{0}$ (giving four pions) can be produced also in photon-hadron interactions in a so-called double scattering mechanism.", "In Ref.", "[18] double vector meson production in photon-photon and photon-hadron interactions in $pp$ /$pA$ /$AA$ collisions was studied.", "In heavy ion collisions the double scattering mechanism is very important [19], [18].", "In proton-proton collisions, for instance at the center-of-mass energy of $\\sqrt{s} = 7$ TeV, total cross sections for the double $\\rho ^{0}$ meson production, taking into account the $\\gamma \\gamma $ and double scattering mechanisms, were estimated [18] to be of 182 $pb$ and 4 $pb$ , respectively.", "In the present paper we wish to concentrate on the four charged pion continuum which is a background for future studies of diffractively produced resonances.", "We shall present a first evaluation of the four-pion continuum in the framework of the tensor pomeron model consistent with general rules of Quantum Field Theory.", "Here we shall give explicit expressions for the amplitudes of $\\rho \\rho \\equiv \\rho (770) \\rho (770)$ and $\\sigma \\sigma \\equiv f_0(500) f_0(500)$ production with the $\\rho $ and $\\sigma $ decaying to $\\pi ^+ \\pi ^-$ .", "We shall discuss their specificity and relevance for the $p p \\rightarrow p p \\pi ^+ \\pi ^- \\pi ^+ \\pi ^-$ reaction.", "In the Appendix we present the formulas of the double-pomeron exchange mechanism for the exclusive production of scalar resonances decaying into $\\sigma \\sigma $ and/or $\\rho \\rho $ pairs." ], [ "Exclusive diffractive production of four pions", "In the present paper we consider the $2 \\rightarrow 6$ processes from the diagrams shown in Fig.", "REF , $\\begin{split}& p p \\rightarrow p p \\,\\sigma \\sigma \\rightarrow p p \\, \\pi ^{+} \\pi ^{-} \\pi ^{+} \\pi ^{-}\\,, \\\\&p p \\rightarrow p p \\, \\rho \\rho \\rightarrow p p\\, \\pi ^{+} \\pi ^{-} \\pi ^{+} \\pi ^{-}\\,.\\end{split}$ That is, we treat effectively the $2 \\rightarrow 6$ processes (REF ) as arising from $2 \\rightarrow 4$ processes, the central diffractive production of two scalar $\\sigma $ mesons and two vector $\\rho $ mesons in proton-proton collisions.", "To calculate the total cross section for the $2 \\rightarrow 4$ reactions one has to calculate the 8-dimensional phase-space integral numerically [8] In the integration over four-body phase space the transverse momenta of the produced particles ($p_{1t}$ , $p_{2t}$ , $p_{3t}$ , $p_{4t}$ ), the azimuthal angles of the outgoing protons ($\\phi _{1}$ , $\\phi _{2}$ ) and the rapidity of the produced mesons ($y_{3}$ , $y_{4}$ ) were chosen as integration variables over the phase space..", "Some modifications of the $2 \\rightarrow 4$ reaction are needed to simulate the $2 \\rightarrow 6$ reaction with $\\pi ^{+} \\pi ^{-} \\pi ^{+} \\pi ^{-}$ in the final state.", "For example, one has to include in addition a smearing of the $\\sigma $ and $\\rho $ masses due to their instabilities.", "Then, the general cross-section formula can be written approximately as ${\\sigma }_{2 \\rightarrow 6} = \\int _{2 m_{\\pi }}^{{\\rm max}\\lbrace m_{X_{3}}\\rbrace } \\int _{2 m_{\\pi }}^{{\\rm max}\\lbrace m_{X_{4}}\\rbrace }{\\sigma }_{2 \\rightarrow 4}(...,m_{X_{3}},m_{X_{4}})\\,f_{M}(m_{X_{3}})\\, f_{M}(m_{X_{4}}) \\,dm_{X_{3}}\\, dm_{X_{4}}\\,.$ Here we use for the calculation of the decay processes $M \\rightarrow \\pi ^{+} \\pi ^{-}$ with $M =\\sigma , \\rho $ the spectral function $f_{M}(m_{X_{i}}) =\\left( 1-\\dfrac{4 m_{\\pi }^{2}}{m_{X_{i}}^{2}} \\right)^{n/2}\\frac{\\frac{2}{\\pi }{m_{M}^{2}} \\Gamma _{M,tot}}{(m_{X_{i}}^{2}-m_{M}^{2})^{2} + m_{M}^{2} \\Gamma _{M,tot}^{2}}\\,N_{I}\\,,$ where $i = 3, 4$ .", "In (REF ) $n = 3$ , $N_{I} = 1$ for $\\rho $ meson and $n = 1$ , $N_{I} = \\frac{2}{3}$ for $\\sigma $ meson, respectively.", "The quantity $\\left( 1-4 m_{\\pi }^{2}/m_{X_{i}}^{2} \\right)^{n/2}$ smoothly decreases the spectral function when approaching the $\\pi ^{+}\\pi ^{-}$ threshold, $m_{X_{i}} \\rightarrow 2 m_{\\pi }$ .", "Figure: The “Born level” diagrams for double-pomeron/reggeoncentral exclusive σσ\\sigma \\sigma (left diagram) and ρρ\\rho \\rho (right diagram) productionand their subsequent decays into π + π - π + π - \\pi ^+ \\pi ^- \\pi ^+ \\pi ^- in proton-proton collisions." ], [ "$pp \\rightarrow pp \\sigma \\sigma $", "Here we discuss the exclusive production of $\\sigma \\sigma \\equiv f_{0}(500) f_{0}(500)$ pairs in proton-proton collisions, $p(p_{a},\\lambda _{a}) + p(p_{b},\\lambda _{b}) \\rightarrow p(p_{1},\\lambda _{1}) + \\sigma (p_{3}) + \\sigma (p_{4}) + p(p_{2},\\lambda _{2}) \\,,$ where $p_{a,b}$ , $p_{1,2}$ and $\\lambda _{a,b}$ , $\\lambda _{1,2} = \\pm \\frac{1}{2}$ denote the four-momenta and helicities of the protons and $p_{3,4}$ denote the four-momenta of the mesons, respectively.", "The diagram for the $\\sigma \\sigma $ production with an intermediate $\\sigma $ meson is shown in Fig.", "REF (left diagram).", "The amplitude for this process can be written as the following sum: ${\\cal M}^{(\\sigma \\mathrm {-exchange})}_{pp \\rightarrow pp \\sigma \\sigma } &=&{\\cal M}^{(\\mathbb {P}\\mathbb {P}\\rightarrow \\sigma \\sigma )} +{\\cal M}^{(\\mathbb {P}f_{2 \\mathbb {R}} \\rightarrow \\sigma \\sigma )} +{\\cal M}^{(f_{2 \\mathbb {R}} \\mathbb {P}\\rightarrow \\sigma \\sigma )} +{\\cal M}^{(f_{2 \\mathbb {R}} f_{2 \\mathbb {R}} \\rightarrow \\sigma \\sigma )}\\,.$ For instance, the $\\mathbb {P}\\mathbb {P}$ -exchange amplitude can be written as ${\\cal M}^{(\\mathbb {P}\\mathbb {P}\\rightarrow \\sigma \\sigma )} ={\\cal M}^{({\\hat{t}})}_{\\lambda _{a} \\lambda _{b} \\rightarrow \\lambda _{1} \\lambda _{2} \\sigma \\sigma }+{\\cal M}^{({\\hat{u}})}_{\\lambda _{a} \\lambda _{b} \\rightarrow \\lambda _{1} \\lambda _{2} \\sigma \\sigma }$ with the $\\hat{t}$ - and $\\hat{u}$ -channel amplitudes $\\begin{split}& {\\cal M}^{({\\hat{t}})}_{\\lambda _{a} \\lambda _{b} \\rightarrow \\lambda _{1} \\lambda _{2} \\sigma \\sigma }= \\\\& \\quad (-i)\\bar{u}(p_{1}, \\lambda _{1})i\\Gamma ^{(\\mathbb {P}pp)}_{\\mu _{1} \\nu _{1}}(p_{1},p_{a})u(p_{a}, \\lambda _{a})\\,i\\Delta ^{(\\mathbb {P})\\, \\mu _{1} \\nu _{1}, \\alpha _{1} \\beta _{1}}(s_{13},t_{1}) \\,i\\Gamma ^{(\\mathbb {P}\\sigma \\sigma )}_{\\alpha _{1} \\beta _{1}}(p_{t},-p_{3}) \\,i\\Delta ^{(\\sigma )}(p_{t}) \\\\& \\quad \\times i\\Gamma ^{(\\mathbb {P}\\sigma \\sigma )}_{\\alpha _{2} \\beta _{2}}(p_{4},p_{t})\\,i\\Delta ^{(\\mathbb {P})\\, \\alpha _{2} \\beta _{2}, \\mu _{2} \\nu _{2}}(s_{24},t_{2}) \\,\\bar{u}(p_{2}, \\lambda _{2})i\\Gamma ^{(\\mathbb {P}pp)}_{\\mu _{2} \\nu _{2}}(p_{2},p_{b})u(p_{b}, \\lambda _{b}) \\,,\\end{split}$ $\\begin{split}& {\\cal M}^{({\\hat{u}})}_{\\lambda _{a} \\lambda _{b} \\rightarrow \\lambda _{1} \\lambda _{2} \\sigma \\sigma }= \\\\& \\quad (-i)\\,\\bar{u}(p_{1}, \\lambda _{1})i\\Gamma ^{(\\mathbb {P}pp)}_{\\mu _{1} \\nu _{1}}(p_{1},p_{a})u(p_{a}, \\lambda _{a}) \\,i\\Delta ^{(\\mathbb {P})\\, \\mu _{1} \\nu _{1}, \\alpha _{1} \\beta _{1}}(s_{14},t_{1}) \\,i\\Gamma ^{(\\mathbb {P}\\sigma \\sigma )}_{\\alpha _{1} \\beta _{1}}(p_{4},p_{u})\\,i\\Delta ^{(\\sigma )}(p_{u}) \\\\& \\quad \\times i\\Gamma ^{(\\mathbb {P}\\sigma \\sigma )}_{\\alpha _{2} \\beta _{2}}(p_{u},-p_{3})\\,i\\Delta ^{(\\mathbb {P})\\, \\alpha _{2} \\beta _{2}, \\mu _{2} \\nu _{2}}(s_{23},t_{2}) \\,\\bar{u}(p_{2}, \\lambda _{2})i\\Gamma ^{(\\mathbb {P}pp)}_{\\mu _{2} \\nu _{2}}(p_{2},p_{b})u(p_{b}, \\lambda _{b}) \\,,\\end{split}$ where $p_{t} = p_{a} - p_{1} - p_{3}$ , $p_{u} = p_{4} - p_{a} + p_{1}$ , $s_{ij} = (p_{i} + p_{j})^{2}$ , $t_1 = (p_{1} - p_{a})^{2}$ , $t_2 = (p_{2} - p_{b})^{2}$ .", "Here $\\Delta ^{(\\mathbb {P})}$ and $\\Gamma ^{(\\mathbb {P}pp)}$ denote the effective propagator and proton vertex function, respectively, for the tensorial pomeron.", "The effective propagators and vertex functions for the tensorial pomeron/reggeon exchanges respect the standard crossing and charge-conjugation relations of Quantum Field Theory.", "For the explicit expressions of these terms see Sect.", "3 of [14].", "We assume that $\\Gamma ^{(\\mathbb {P}\\sigma \\sigma )}$ has the same form as $\\Gamma ^{(\\mathbb {P}\\pi \\pi )}$ (see (3.45) of [14]) but with the $\\mathbb {P}\\sigma \\sigma $ coupling constant $g_{\\mathbb {P}\\sigma \\sigma }$ instead of the $\\mathbb {P}\\pi \\pi $ one $2 \\beta _{\\mathbb {P}\\pi \\pi }$ .", "The scalar meson propagator $\\Delta ^{(\\sigma )}$ is taken as in (4.7) and (4.8) of [7] with the running (energy-dependent) width.", "In a similar way the $\\mathbb {P}f_{2 \\mathbb {R}}$ , $f_{2 \\mathbb {R}} \\mathbb {P}$ and $f_{2 \\mathbb {R}} f_{2 \\mathbb {R}}$ amplitudes can be written.", "For the $f_{2 \\mathbb {R}} \\sigma \\sigma $ vertex our ansatz is as for $f_{2 \\mathbb {R}} \\pi \\pi $ in (3.53) of [14] but with $g_{f_{2 \\mathbb {R}} \\pi \\pi }$ replaced by $g_{f_{2 \\mathbb {R}} \\sigma \\sigma }$ .", "In the high-energy small-angle approximation we can write the $2 \\rightarrow 4$ amplitude (REF ) as $\\begin{split}& {\\cal M}^{(\\sigma \\mathrm {-exchange})}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\sigma \\sigma }\\simeq \\;2 (p_1 + p_a)_{\\mu _{1}} (p_1 + p_a)_{\\nu _{1}}\\,\\delta _{\\lambda _{1} \\lambda _{a}} \\, F_{1}(t_{1}) \\,F_{M}(t_{1})\\\\& \\quad \\times \\bigg \\lbrace {V}^{\\mu _{1} \\nu _{1}}(s_{13}, t_{1}, p_{t}, -p_{3})\\;\\Delta ^{(\\sigma )}(p_{t})\\;{V}^{\\mu _{2} \\nu _{2}}(s_{24}, t_{2}, p_{t}, p_{4})\\, \\left[ \\hat{F}_{\\sigma }(p_{t}^{2}) \\right]^{2}\\\\& \\qquad +{V}^{\\mu _{1} \\nu _{1}}(s_{14}, t_{1}, p_{u}, p_{4})\\;\\Delta ^{(\\sigma )}(p_{u})\\;{V}^{\\mu _{2} \\nu _{2}}(s_{23}, t_{2}, p_{u}, -p_{3})\\, \\left[ \\hat{F}_{\\sigma }(p_{u}^{2}) \\right]^{2}\\bigg \\rbrace \\\\& \\quad \\times 2 (p_2 + p_b)_{\\mu _{2}} (p_2 + p_b)_{\\nu _{2}}\\,\\delta _{\\lambda _{2} \\lambda _{b}} \\, F_{1}(t_{2}) \\,F_{M}(t_{2})\\,.\\end{split}$ The function ${V}_{\\mu \\nu }$ has the form ($M_{0} \\equiv 1$ GeV) $\\begin{split}{V}_{\\mu \\nu }(s,t,k_{2},k_{1})= &(k_{1}+k_{2})_{\\mu }(k_{1}+k_{2})_{\\nu }\\;\\frac{1}{4s}\\bigg [3 \\beta _{\\mathbb {P}NN} \\, g_{\\mathbb {P}\\sigma \\sigma }(- i s \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t)-1}\\\\& + \\frac{1}{2 M_{0}^{2}} g_{f_{2 \\mathbb {R}}pp} \\, g_{f_{2 \\mathbb {R}} \\sigma \\sigma }(- i s \\alpha ^{\\prime }_{f_{2 \\mathbb {R}}})^{\\alpha _{f_{2 \\mathbb {R}}}(t) -1}\\bigg ]\\,,\\end{split}$ where $\\beta _{\\mathbb {P}NN}$ = 1.87 GeV$^{-1}$ and $g_{f_{2 \\mathbb {R}} pp}$ = 11.04 from (6.53) and (6.55) of [14], respectively.", "If the $\\sigma $ meson has substantial gluon content or some $q\\bar{q}q\\bar{q}$ component its coupling to $\\mathbb {P}$ and $f_{2 \\mathbb {R}}$ may be larger than for the pion.", "To illustrate effects of this possibility we take in the calculation two sets of the coupling constants $&&{\\rm set\\;A}: \\;g_{\\mathbb {P}\\sigma \\sigma } = 2 \\beta _{\\mathbb {P}\\pi \\pi }\\,, \\quad g_{f_{2 \\mathbb {R}} \\sigma \\sigma } = g_{f_{2 \\mathbb {R}} \\pi \\pi } \\,, \\\\&&{\\rm set\\;B}: \\;g_{\\mathbb {P}\\sigma \\sigma } = 4 \\beta _{\\mathbb {P}\\pi \\pi }\\,, \\quad g_{f_{2 \\mathbb {R}} \\sigma \\sigma } = 2 g_{f_{2 \\mathbb {R}} \\pi \\pi } \\,,$ where $\\beta _{\\mathbb {P}\\pi \\pi }$ = 1.76 GeV$^{-1}$ and $g_{f_{2 \\mathbb {R}} \\pi \\pi }$ = 9.30 from (7.15) and (7.16) of [14], respectively.", "The form of the off-shell meson form factor $\\hat{F}_{\\sigma }(k^{2})$ in (REF ) and of the analogous form factor for $\\rho $ mesons $\\hat{F}_{\\rho }(k^{2})$ (see the next section) is unknown.", "We write generically $\\hat{F}_{M}(k^{2})$ $(M = \\sigma , \\rho )$ for these form factors which we normalize to unity at the on-shell point, $\\hat{F}_{M}(m_{M}^{2}) = 1$ , and parametrise here in two ways: $&&\\hat{F}_{M}(k^{2})=\\exp \\left(\\frac{k^{2}-m_{M}^{2}}{\\Lambda ^{2}_{off,E}}\\right) \\,, \\\\&&\\hat{F}_{M}(k^{2})=\\dfrac{\\Lambda ^{2}_{off,Mp} - m_{M}^{2}}{\\Lambda ^{2}_{off,Mp} - k^{2}} \\,, \\quad \\Lambda _{off,Mp}>m_{M} \\,.$ The cut-off parameters $\\Lambda _{off,E}$ for the exponential form or $\\Lambda _{off,Mp}$ for the monopole form of the form factors can be adjusted to experimental data.", "A factor $\\frac{1}{2}$ due to the identity of the two $\\sigma $ mesons in the final state has to be taken into account in the phase-space integration in (REF )." ], [ "$pp \\rightarrow pp \\rho \\rho $", "Here we focus on exclusive production of $\\rho \\rho \\equiv \\rho (770) \\rho (770)$ in proton-proton collisions, see Fig.", "REF (right diagram), $p(p_{a},\\lambda _{a}) + p(p_{b},\\lambda _{b}) \\rightarrow p(p_{1},\\lambda _{1}) + \\rho (p_{3},\\lambda _{3}) + \\rho (p_{4},\\lambda _{4}) + p(p_{2},\\lambda _{2}) \\,,$ where $p_{3,4}$ and $\\lambda _{3,4} = 0, \\pm 1$ denote the four-momenta and helicities of the $\\rho $ mesons, respectively.", "We write the amplitude as $\\begin{split}{\\cal M}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho } =\\left(\\epsilon ^{(\\rho )}_{\\rho _{3}}(\\lambda _{3})\\right)^\\left(\\epsilon ^{(\\rho )}_{\\rho _{4}}(\\lambda _{4})\\right)^{\\cal M}^{\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\,,\\end{split}$ where $\\epsilon ^{(\\rho )}_{\\rho }(\\lambda )$ are the polarisation vectors of the $\\rho $ meson.", "Then, with the expressions for the propagators, vertices, and form factors, from [14] ${\\cal M}^{\\rho _{3} \\rho _{4}}$ can be written in the high-energy approximation as $\\begin{split}& {\\cal M}^{(\\rho \\mathrm {-exchange})\\,\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\simeq \\;2 (p_1 + p_a)_{\\mu _{1}} (p_1 + p_a)_{\\nu _{1}}\\,\\delta _{\\lambda _{1} \\lambda _{a}} \\, F_{1}(t_{1}) \\,F_{M}(t_{1})\\\\& \\times \\bigg \\lbrace {V}^{\\rho _{3} \\rho _{1} \\mu _{1} \\nu _{1}}(s_{13}, t_{1}, p_{t}, p_{3})\\;\\Delta ^{(\\rho )}_{\\rho _{1}\\rho _{2}}(p_{t})\\;{V}^{\\rho _{4} \\rho _{2} \\mu _{2} \\nu _{2}}(s_{24}, t_{2}, -p_{t}, p_{4})\\, \\left[ \\hat{F}_{\\rho }(p_{t}^{2}) \\right]^{2}\\\\& \\quad \\quad \\; +{V}^{\\rho _{4} \\rho _{1} \\mu _{1} \\nu _{1}}(s_{14}, t_{1}, -p_{u}, p_{4})\\;\\Delta ^{(\\rho )}_{\\rho _{1}\\rho _{2}}(p_{u})\\;{V}^{\\rho _{3} \\rho _{2} \\mu _{2} \\nu _{2}}(s_{23}, t_{2}, p_{u}, p_{3})\\, \\left[ \\hat{F}_{\\rho }(p_{u}^{2}) \\right]^{2}\\bigg \\rbrace \\\\& \\times 2 (p_2 + p_b)_{\\mu _{2}} (p_2 + p_b)_{\\nu _{2}}\\,\\delta _{\\lambda _{2} \\lambda _{b}} \\, F_{1}(t_{2}) \\,F_{M}(t_{2}) \\,,\\end{split}$ where ${V}_{\\mu \\nu \\kappa \\lambda }$ reads as $\\begin{split}{V}_{\\mu \\nu \\kappa \\lambda }(s,t,k_{2},k_{1})= &2 \\Gamma _{\\mu \\nu \\kappa \\lambda }^{(0)}(k_{1},k_{2})\\frac{1}{4s}\\bigg [3 \\beta _{\\mathbb {P}NN} \\, a_{\\mathbb {P}\\rho \\rho }(- i s \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t)-1}\\\\& \\qquad \\qquad \\qquad \\quad \\; + \\frac{1}{M_{0}} g_{f_{2 \\mathbb {R}}pp} \\, a_{f_{2 \\mathbb {R}} \\rho \\rho }(- i s \\alpha ^{\\prime }_{f_{2 \\mathbb {R}}})^{\\alpha _{f_{2 \\mathbb {R}}}(t) -1}\\bigg ] \\\\& -\\Gamma _{\\mu \\nu \\kappa \\lambda }^{(2)}(k_{1},k_{2})\\frac{1}{4s}\\bigg [3 \\beta _{\\mathbb {P}NN} \\, b_{\\mathbb {P}\\rho \\rho }(- i s \\alpha ^{\\prime }_{\\mathbb {P}} )^{\\alpha _{\\mathbb {P}}(t)-1}\\\\& \\qquad \\qquad \\qquad \\quad + \\frac{1}{M_{0}} g_{f_{2 \\mathbb {R}}pp} \\, b_{f_{2 \\mathbb {R}} \\rho \\rho }(- i s \\alpha ^{\\prime }_{f_{2 \\mathbb {R}}})^{\\alpha _{f_{2 \\mathbb {R}}}(t) -1}\\bigg ] \\,.\\end{split}$ The explicit tensorial functions $\\Gamma _{\\mu \\nu \\kappa \\lambda }^{(i)}(k_{1},k_{2})$ , $i$ = 0, 2, are given in Ref.", "[14], see formulae (3.18) and (3.19), respectively.", "In our calculations the parameter set A of coupling constants $a$ and $b$ from [17] was used, see Eq.", "(2.15) there.", "We consider in (REF ) unpolarised protons in the initial state and no observation of polarizations in the final state.", "In the following we are mostly interested in the invariant mass distributions of the $4\\pi $ system and in distributions of the parent $\\rho $ mesons.", "Therefore, we have to insert in (REF ) the cross section $\\sigma _{2 \\rightarrow 4}$ summed over the $\\rho $ meson polarizations.", "The spin sum for a $\\rho $ meson of momentum $k$ and squared mass $k^{2}=m_{X}^{2}$ is $\\begin{split}\\sum _{\\lambda = 0, \\pm 1}\\epsilon ^{(\\rho )\\,\\mu }(\\lambda )\\left(\\epsilon ^{(\\rho )\\,\\nu }(\\lambda )\\right)^* =-g^{\\mu \\nu } + \\dfrac{k^{\\mu }k^{\\nu }}{m_{X}^{2}}\\,.\\end{split}$ But the $k^{\\mu }k^{\\nu }$ terms do not contribute since we have the relations $\\begin{split}p_{3\\, \\rho _{3}} {\\cal M}^{\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho } =0\\,, \\qquad p_{4\\, \\rho _{4}} {\\cal M}^{\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho } =0\\,.\\end{split}$ These follow from the properties of $\\Gamma ^{(0,2)}_{\\mu \\nu \\kappa \\lambda }$ in (REF ); see (3.21) of [14].", "Taking also into account the statistical factor $\\frac{1}{2}$ due to the identity of the two $\\rho $ mesons we get for the amplitudes squared (to be inserted in $\\sigma _{2 \\rightarrow 4}$ in (REF )) $\\begin{split}&\\frac{1}{2} \\frac{1}{4} \\sum _{\\rm {spins}}\\Big \|\\left(\\epsilon ^{(\\rho )}_{\\rho _{3}}(\\lambda _{3})\\right)^\\left(\\epsilon ^{(\\rho )}_{\\rho _{4}}(\\lambda _{4})\\right)^{\\cal M}^{\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\Big \|^{2}\\\\&=\\frac{1}{8} \\sum _{\\lambda _{a},\\lambda _{b},\\lambda _{1},\\lambda _{2}}\\left({\\cal M}^{\\sigma _{3} \\sigma _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\right)^{*}{\\cal M}^{\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\,g_{\\sigma _{3}\\rho _{3}} \\,g_{\\sigma _{4}\\rho _{4}}\\,.\\end{split}$ So far we have treated the exchanged mesonic object for $\\rho ^0 \\rho ^0$ production as spin-1 particle.", "However, we should take into account the fact that the exchanged intermediate object is not a simple meson but may correspond to a whole family of daughter exchanges, that is, the reggeization of the intermediate $\\rho $ meson is necessary.", "For related works, where this effect was included in practical calculations, see e.g.", "[20], [15].", "The “reggeization” of the amplitude given in Eq.", "(REF ) is included here for $\\sqrt{s_{34}} \\geqslant 2 m_{\\rho }$ , only approximately, by replacing the $\\rho $ propagator both in the $\\hat{t}$ - and $\\hat{u}$ -channel amplitudes by $&&\\Delta ^{(\\rho )}_{\\rho _{1}\\rho _{2}}(p) \\rightarrow \\Delta ^{(\\rho )}_{\\rho _{1}\\rho _{2}}(p)\\left( \\frac{s_{34}}{s_{0}} \\right)^{\\alpha _{\\rho }(p^{2})-1} \\,,$ where we take $s_{0} = 4 m_{\\rho }^{2}$ and $\\alpha _{\\rho }(p^{2}) = 0.5 + 0.9 \\,t$ with the momentum transfer $t = p^{2}$ .", "To give the full physical amplitudes we should add absorptive corrections to the Born amplitudes (REF ) and (REF ) for the $pp \\rightarrow pp \\sigma \\sigma $ and $pp \\rightarrow pp \\rho \\rho $ reactions, respectively.", "For the details how to include the $pp$ -rescattering corrections in the eikonal approximation for the four-body reaction see Sect.", "3.3 of [17]." ], [ "Preliminary results", "In this section we wish to present first results for the $pp \\rightarrow pp \\sigma \\sigma $ and $pp \\rightarrow pp \\rho \\rho $ processes corresponding to the diagrams shown in Fig.", "REF .", "We start from a discussion of the $\\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}$ invariant mass distribution.", "In Fig.", "REF we compare the $\\sigma \\sigma $ - and $\\rho \\rho $ -contributions to the CERN-ISR data [21] at $\\sqrt{s} = 62$ GeV.", "Here, the four pions are restricted to lie in the rapidity region $\|y_{\\pi }\| < 1.5$ and the cut The Feynman-$x$ variable was defined as $x_{p} = 2 p_{z,p}/\\sqrt{s}$ in the center-of-mass frame with $p_{z,p}$ the longitudinal momentum of the outgoing proton.", "$\|x_{p}\| > 0.9$ is imposed on the outgoing protons.", "In Ref.", "[21] five contributions to the four-pion spectrum were identified.", "A $4 \\pi $ phase-space term with total angular momentum $J = 0$ , two $\\rho \\pi \\pi $ terms with $J = 0$ and $J = 2$ , and two $\\rho \\rho $ terms with $J = 0$ and $J = 2$ .", "In the following we will compare the $4 \\pi $ phase-space term with our $\\sigma \\sigma $ result, the $\\rho \\rho $ terms with our $\\rho \\rho $ result.", "The theoretical results correspond to the calculation including absorptive corrections related to the $pp$ nonperturbative interaction in the initial and final state.", "The ratio of full and Born cross sections $\\langle S^{2}\\rangle $ (the gap survival factor) is approximately $\\langle S^{2}\\rangle = 0.4$ .", "In our calculation both the $\\mathbb {P}\\mathbb {P}$ and the $\\mathbb {P}f_{2 \\mathbb {R}}$ , $f_{2 \\mathbb {R}} \\mathbb {P}$ , $f_{2 \\mathbb {R}} f_{2 \\mathbb {R}}$ exchanges were included.", "At the ISR energy the $f_{2 \\mathbb {R}}$ exchanges, including their interference terms with the $\\mathbb {P}\\mathbb {P}$ one, give about $50\\%$ to the total cross section.", "In the left panel of Fig.", "REF we compare our $\\sigma \\sigma $ contribution assuming the coupling constants () with the $4 \\pi $ ($J=0$ , phase space) ISR data (marked as full data points).", "We present results for two different forms of off-shell meson form factor, the exponential type (REF ), $\\Lambda _{off,E} = 1.6$ GeV, and the monopole type (), $\\Lambda _{off,Mp} = 1.6$ GeV, see the black lower line and the red upper line, respectively.", "There is quite a good agreement between our $\\sigma \\sigma $ result with a monopole form factor and the $4 \\pi $ ($J=0$ , phase space) data.", "Note that this implies that the set B of $\\mathbb {P}\\sigma \\sigma $ and $f_{2 \\mathbb {R}} \\sigma \\sigma $ couplings, which are larger than the corresponding pion couplings, seems to be preferred.", "In the right panel of Fig.", "REF we compare our result for the $\\rho \\rho $ contribution to corresponding ISR data Here we plotted the sum of the experimental cross section $\\sigma = \\sigma _{J=0} + \\sigma _{J=2}$ for the $J=0$ and $J=2$ $\\rho \\rho $ terms (see Figs.", "3c and 3e in [21], respectively) and the corresponding error is approximated as $\\delta \\sigma = \\sqrt{\\delta _{J=0}^{2} + \\delta _{J=2}^{2}}$ ..", "In the calculation of the $\\rho \\rho $ contribution we take into account the intermediate $\\rho $ meson reggeization.", "The reggeization leads to an extra strong damping of the large $M_{4 \\pi }$ cross section.", "The effect of reggeization is expected only when the separation in rapidity between the two produced resonances is large.", "We will return to this issue in the further part of this section.", "We note that our model is able to give a qualitative account of the ISR $\\rho \\rho $ data for $M_{4 \\pi } \\gtrsim 1.4$ GeV within the large experimental errors.", "The total $4 \\pi $ experimental data (marked as open data points in the left panel of Fig.", "REF ) are also shown for comparison.", "In Ref.", "[21] an integrated (total) cross section of 46 $\\mu b$ at $\\sqrt{s} = 62$ GeV was estimated.", "There are other processes besides the ones of (REF ) contributing to the $4 \\pi $ final state, such as resonance production shown in the diagrams in Fig.", "REF of Appendix .", "Thus, the $\\sigma \\sigma $ and the $\\rho \\rho $ contributions considered here should not be expected to fit the ISR data precisely.", "In addition, the ISR $4 \\pi $ data includes also a large $\\rho ^{0} \\pi ^{+} \\pi ^{-}$ ($J=0$ and $J=2$ ) component (see Figs.", "3b and 3d of [21]) with an enhancement in the $J=2$ term which was interpreted there as a $f_{2}(1720)$ state.", "Also the $\\rho ^{0} \\rho ^{0}$ ($J=2$ ) term indicates a signal of $f_{2}(1270)$ state; see Fig.", "3e of [21].", "Therefore, a consistent model for the resonance and continuum contributions, including the interference effects between them, would be required to better describe the ISR data.", "We leave this interesting problem for future studies.", "Figure: Invariant mass distributions for the central π + π - π + π - \\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-} systemcompared to the CERN-ISR data at s=62\\sqrt{s} = 62 GeV.In the left panel the lines represent results for theσσ\\sigma \\sigma contribution only andwith the enhanced pomeron/reggeon-σ\\sigma -σ\\sigma couplings ().We used two forms of the off-shell meson form factor,the exponential form ()with Λ off,E =1.6\\Lambda _{off,E} = 1.6 GeV (the black lines)and the monopole form ()with Λ off,Mp =1.6\\Lambda _{off,Mp} = 1.6 GeV (the red thin lines).In the right panel the lines represent results for the ρρ\\rho \\rho contributionwithout (the dotted lines) and with (the solid lines) the inclusionof the intermediate ρ\\rho meson reggeization.For comparison, the upper blue solid line was obtainedwith the monopole form factor and Λ off,Mp =1.8\\Lambda _{off,Mp} = 1.8 GeV.The absorption effects were included here.In Fig.", "REF we show our preliminary four-pion invariant mass distributions for experimental cuts relevant for the RHIC and LHC experiments.", "In the calculation of the $\\sigma \\sigma $ and the $\\rho \\rho $ contributions the pomeron and $f_{2 \\mathbb {R}}$ exchanges were included.", "Imposing limitations on pion (pseudo)rapidities, e.g.", "$\|\\eta _{\\pi }\| < 1$ , and going to higher energies strongly reduces the role of subleading $f_{2 \\mathbb {R}}$ exchanges.", "The gap survival factors $\\langle S^{2}\\rangle $ estimated within the eikonal approximation are 0.30, 0.21, 0.23 for $\\sqrt{s}$ = 0.2, 7, 13 TeV, respectively.", "In the case of $\\sigma \\sigma $ contribution we use two sets of the coupling constants; standard (REF ) and enhanced ones ().", "Figure: Four-pion invariant mass distributions for differentcenter-of-mass energies s\\sqrt{s} and experimental kinematical cuts.The black lines represent results for the ρρ\\rho \\rho contribution,the blue lines for the σσ\\sigma \\sigma contribution.The exponential off-shell meson form factors ()with Λ off,E =1.6\\Lambda _{off,E} = 1.6 GeV were used.For the case of σσ\\sigma \\sigma contribution onlythe red dot-dashed line was obtainedwith the monopole form factor and Λ off,Mp =1.6\\Lambda _{off,Mp} = 1.6 GeV.The absorption effects were included here.The correlation in rapidity of pion pairs $({\\rm Y}_{3},{\\rm Y}_{4})$ (e.g., ${\\rm Y}_3$ means ${\\rm Y}_{\\pi ^{+} \\pi ^{-}}$ where the pions are produced from a meson decay) for both the $\\sigma \\sigma $ and the $\\rho \\rho $ contributions is displayed in Fig.", "REF for $\\sqrt{s} = 200$ GeV.", "For the $\\sigma \\sigma $ contribution, see the top panel, we observe a strong correlation ${\\rm Y}_3 \\approx {\\rm Y}_4$ .", "For the $\\rho \\rho $ case the $({\\rm Y}_3, {\\rm Y}_4)$ distribution extends over a much broader range of ${\\rm Y}_3 \\ne {\\rm Y}_4$ which is due to the exchange of the spin-1 particle.", "However, as discussed in the section devoted to the formalism we may include, at least approximately, the effect of reggeization of the intermediate $\\rho $ meson.", "In the left and right bottom panels we show the results without and with the $\\rho $ meson reggeization, respectively.", "As shown in the right panel this effect becomes crucial when the separation in rapidity between the two $\\rho $ mesons increases.", "After the reggeization is performed the two-dimensional distribution looks very similar as for the $\\sigma \\sigma $ case.", "The reggeization effect discussed here is also closely related to the damping of the four-pion invariant mass distribution discussed already in Fig.", "REF .", "Figure: The distributions in (Y 3 ,Y 4 )({\\rm Y}_{3},{\\rm Y}_{4}) spacefor the reaction pp→ppπ + π - π + π - pp \\rightarrow pp \\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}via fusion of two tensor pomerons and f 2ℝ f_{2 \\mathbb {R}} reggeonsat s=200\\sqrt{s} = 200 GeV.Plotted is the ratioR Y 3 Y 4 =d 2 σ dY 3 dY 4 /∫dY 3 dY 4 d 2 σ dY 3 dY 4 R_{{\\rm Y}_{3}{\\rm Y}_{4}} = \\frac{d^{2}\\sigma }{d{\\rm Y}_{3}d{\\rm Y}_{4}}/ \\int {d{\\rm Y}_{3}d{\\rm Y}_{4} \\frac{d^{2}\\sigma }{d{\\rm Y}_{3}d{\\rm Y}_{4}}}.We show the σσ\\sigma \\sigma contribution (top panel)and the ρρ\\rho \\rho contribution (bottom panels)without (left panel) and with (right panel) the ρ\\rho mesonreggeization included.Here Λ off,E =1.6\\Lambda _{off,E} = 1.6 GeV was used.In Table REF we have collected integrated cross sections in $\\mu b$ with different experimental cuts for the exclusive $\\pi ^{+}\\pi ^{-} \\pi ^{+}\\pi ^{-}$ production including only the contributions shown in Fig.", "REF .", "The collected results were obtained in the calculations with the tensor pomeron and reggeon exchanges.", "In the calculations the off-shell-meson form factor (REF ) with $\\Lambda _{off,E} = 1.6$ GeV was used.", "No absorption effects were included in the quoted numbers.", "The full cross section can be obtained by multiplying the Born cross section by the corresponding gap survival factor $\\langle S^{2}\\rangle $ .", "These factors depend on the kinematic cuts and are 0.40 (ISR), 0.46 (STAR, lower $\|t\|$ ), 0.30 (STAR, higher $\|t\|$ ), 0.21 ($\\sqrt{s}$ = 7 TeV), 0.19 ($\\sqrt{s}$ = 13 TeV), 0.23 ($\\sqrt{s}$ = 13 TeV, with cuts on $\|t\|$ ).", "Table: The integrated “Born level” (no gap survival factors) cross sections in μb\\mu bfor the central exclusive π + π - π + π - \\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-} productionin pppp collisions via the σσ\\sigma \\sigma and ρρ\\rho \\rho mechanismsgiven in Fig.", "for some typical experimental cuts.The σσ\\sigma \\sigma contribution was calculated using the couplingconstants ()while the ρρ\\rho \\rho contribution withoutand with (in the parentheses) the inclusion of the intermediate ρ\\rho meson reggeization.The cross sections for the $\\rho \\rho $ final state found here are more than three orders of magnitude larger than the cross sections for the $\\gamma \\gamma \\rightarrow \\rho \\rho $ and double scattering mechanisms considered recently in [18].", "Finally, in Fig.", "REF , we discuss some observables which are very sensitive to the absorptive corrections.", "Quite a different pattern can be seen for the Born case and for the case with absorption included.", "The absorptive corrections lead to significant modification of the shape of the $\\phi _{pp}$ distribution ($\\phi _{pp}$ is the azimuthal angle between the $p_{t}$ vectors of the outgoing protons) and lead to an increase of the cross section for the proton four-momentum transfer $t$ = $t_{1}$ = $t_{2}$ at large $\|t\|$ .", "This effect could be verified in future experiments when both protons are measured which should be possible for ATLAS-ALFA [5] and CMS-TOTEM.", "Figure: Distributions in proton-proton relative azimuthal angle (the left panel) andin four-momentum squared of one of the protons (the right panel)for the central σσ→π + π - π + π - \\sigma \\sigma \\rightarrow \\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}contribution at s=13\\sqrt{s} = 13 TeV with the kinematical cutsspecified in the legend.The solid line corresponds to the Born calculationsand the long-dashed line corresponds to the result includingthe pppp absorptive corrections.Here the enhanced pomeron/reggeon-σ\\sigma -σ\\sigma couplings ()and the exponential form of off-shell meson form factor ()with Λ off,E =1.6\\Lambda _{off,E} = 1.6 GeV were used." ], [ "Conclusions", "In the present paper we have presented first estimates of the contributions with the intermediate $f_{0}(500)f_{0}(500)$ , $\\rho (770) \\rho (770)$ resonance pairs to the reaction $p p \\rightarrow p p \\pi ^+ \\pi ^- \\pi ^+ \\pi ^-$ which is being analyzed experimentally by the STAR, ALICE, CMS, and ATLAS Collaborations.", "The results were obtained within a model where the pomeron and $f_{2 \\mathbb {R}}$ reggeon are treated as effective tensor exchanges.", "The results for processes with the exchange of heavy (compared to pion) mesons strongly depend on the details of the hadronic form factors.", "By comparing the theoretical results and the cross sections found in the CERN-ISR experiment [21] we fixed the parameters of the off-shell meson form factor and the $\\mathbb {P}\\sigma \\sigma $ and $f_{2 \\mathbb {R}} \\sigma \\sigma $ couplings.", "The corresponding values of parameters can be verified by future experimental results obtained at RHIC and LHC.", "We have made estimates of the integrated cross sections for different experimental situations as well as shown several differential distributions.", "The pion-pair rapidities of the two $\\sigma $ mesons are strongly correlated (${\\rm Y}_{3} \\approx {\\rm Y}_{4}$ ).", "This is due to a strong interference effect between the $\\hat{t}$ - and $\\hat{u}$ -channel amplitudes.", "For the $\\rho \\rho $ contribution the situation is the following.", "If we take for the propagator of the exchanged $\\rho $ in Fig.", "REF , right panel, the standard particle propagator the ${\\rm Y}_{3} - {\\rm Y}_{4}$ correlation is very weak.", "But for a reggeized $\\rho $ propagator we get again a strong ${\\rm Y}_{3} - {\\rm Y}_{4}$ correlation, similar to that found in the $\\sigma \\sigma $ case; see Fig.", "REF .", "This is understandable since the reggeization suppresses contributions where the two produced $\\rho $ mesons have large subsystem energies, i.e.", "where there is a large rapidity distance between the two $\\rho $ mesons.", "We have found in this paper that the diffractive mechanism in proton-proton collisions considered by us leads to the cross section for the $\\rho \\rho $ final state more than three orders of magnitude larger than the corresponding cross section for $\\gamma \\gamma \\rightarrow \\rho \\rho $ and double scattering photon-pomeron (pomeron-photon) mechanisms considered recently in [18].", "Closely related to the reaction $pp \\rightarrow pp \\pi ^{+} \\pi ^{-} \\pi ^{+} \\pi ^{-}$ studied by us here is the $4 \\pi $ production in ultra-peripheral nucleus-nucleus collisions.", "A phenomenological study of the reaction mechanism of $AA \\rightarrow AA \\rho ^{0} \\rho ^{0}$ was performed in Ref. [19].", "The application of our methods, based on the tensor-pomeron concept, to collisions involving nuclei is an interesting problem which goes, however, beyond the scope of the present work.", "To summarize: we have given a consistent treatment of the $\\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}$ production via two scalar $\\sigma $ mesons and two vector $\\rho $ mesons in an effective field-theoretic approach.", "A measurable cross section of order of a few $\\mu b$ was obtained for the $pp \\rightarrow pp\\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}$ process which should give experimentalists interesting challenges to check and explore it." ], [ "Four-pion production through $f_{0} \\rightarrow \\sigma \\sigma $ and {{formula:7c9459a8-81a7-43c7-a305-6c4ba87e23fe}} mechanisms", "Here we discuss the diffractive production of the scalar $f_{0}(1370)$ , $f_{0}(1500)$ , and $f_{0}(1710)$ resonances decaying at least potentially into the $\\pi ^+ \\pi ^- \\pi ^+ \\pi ^-$ final state.", "We present relevant formulas for the resonance contributions that could be used in future analyses.", "At present a precise calculation of the resonance contributions to the four-pion channel is not possible as some details of the relevant decays are not well understood.", "The production and decay properties of the scalar mesons in the $4 \\pi $ channel, such as the $f_{0}(1370)$ and $f_{0}(1500)$ states, have been investigated extensively in central diffractive production by the WA102 Collaboration at $\\sqrt{s} = 29.1$ GeV [22], [23], [24] and in $p \\bar{p}$ and $\\bar{p}n$ annihilations by the Crystal Barrel Collaboration [25].", "In central production, see Fig.", "3 of [23], there is a very clear signal from $f_{0}(1500)$ in the $4 \\pi $ spectra, especially in the $\\sigma \\sigma $ channel, and some evidence of the broad $f_{0}(1370)$ resonance in the $\\rho \\rho $ channel.", "In Fig.", "1 of Ref.", "[24] the $J^{PC}= 0^{++}$ $\\rho \\rho $ wave from the $\\pi ^+ \\pi ^- \\pi ^+ \\pi ^-$ channel in four different $\\phi _{pp}$ intervals (each of $45^{o}$ ) was shown.", "A peak below $M_{4 \\pi } \\simeq 1500$ MeV was clearly seen which can be interpreted as the interference effect of the $f_{0}(1370)$ state, the $f_{0}(1500)$ state and the broad $4 \\pi $ background.", "In principle, also the contributions from the $f_0(1710)$ and $f_0(2020)$ are not excluded.", "In Table 1 of [24] the percentage of each resonance in three intervals of the so-called “glueball filter variable” ($dP_{T}$ ) was shown.", "The idea being that for small differences in the transverse momentum vectors between the two exchanged “particles” an enhancement in the production of glueballs relative to $q \\bar{q}$ states may occur.", "The $dP_{T}$ dependence and the $\\phi _{pp}$ distributions presented there are similar to what was found in the analysis of the $\\pi ^+ \\pi ^-$ channel [26].", "In Refs.", "[27], [28] it was shown that also the $f_{0}(1710)$ state has a similar behaviour in the azimuthal angle $\\phi _{pp}$ and in the $dP_{T}$ variable as the $f_{0}(1500)$ state.", "That is, all the undisputed $q \\bar{q}$ states are observed to be suppressed at small $dP_{T}$ , but the glueball candidates $f_{0}(1500)$ , $f_{0}(1710)$ , together with the enigmatic $f_{0}(980)$ , survive.", "It was shown in [24] that the $f_{0}(1370)$ and $f_{0}(2000)$ have similar $dP_{T}$ and $\\phi _{pp}$ dependences.", "The fact that $f_{0}(1370)$ and $f_{0}(1500)$ states have different $dP_{T}$ and $\\phi _{pp}$ dependences confirms that these are not simply $J$ dependent phenomena Some essential discrepancy for $f_{0}(1370)$ and $f_{0}(1500)$ states in the different decay channels was discussed, e.g., in Refs.", "[29], [30].", "In Ref.", "[31], by using a three-flavor chiral effective approach, the authors found that $f_{0}(1710)$ is predominantly the gluonic state and the $\\rho \\rho \\rightarrow 4 \\pi $ decay channel is strongly suppressed..", "This is also true for the $J =2$ states, where the $f_{2}(1950)$ state has different dependences compared to the $f_{2}(1270)$ and $f^{\\prime }_{2}(1520)$ states [26].", "We wish to emphasize that in [15] we obtained a good description of the WA102 experimental distributions [32], [26], [33] for the scalar and pseudoscalar mesons within the framework of the tensor pomeron approach.", "The $dP_{T}$ and $\\phi _{pp}$ effects can be understood as being due to the fact that in general more than one pomeron-pomeron-meson coupling structure is possible [15].", "The behaviour of the tensor $f_{2}(1270)$ state was discussed recently in [7]; see Figs.", "4 and 5 there.", "In the following we present our analytic expressions for the diagrams of Fig.", "REF for $\\mathbb {P}\\mathbb {P}$ fusion only.", "The extension to include also $\\mathbb {P}f_{2 \\mathbb {R}}$ , $f_{2 \\mathbb {R}} \\mathbb {P}$ and $f_{2 \\mathbb {R}} f_{2 \\mathbb {R}}$ fusion is straightforward.", "Figure: The “Born level” diagrams for double-pomeron/reggeoncentral exclusive production of π + π - π + π - \\pi ^+ \\pi ^- \\pi ^+ \\pi ^-through f 0 →σσf_{0} \\rightarrow \\sigma \\sigma (left diagram) andf 0 →ρρf_{0} \\rightarrow \\rho \\rho (right diagram)in proton-proton collisions." ], [ "$pp \\rightarrow pp (f_{0} \\rightarrow \\sigma \\sigma )$", "Here we consider the amplitude for the reaction (REF ) through an $s$ -channel scalar resonance $f_{0} \\rightarrow \\sigma \\sigma $ ; see Fig.", "REF (left diagram).", "Here $f_{0}$ stands, for one of the $f_{0}(1370)$ , $f_{0}(1500)$ , $f_{0}(1710)$ states.", "In the high-energy small-angle approximation we can write this amplitude as $\\begin{split}& {\\cal M}^{(f_{0} \\rightarrow \\sigma \\sigma )}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\sigma \\sigma }\\simeq 3 \\beta _{\\mathbb {P}NN} \\, 2(p_1 + p_a)_{\\mu _{1}} (p_1 + p_a)_{\\nu _{1}}\\,\\delta _{\\lambda _{1} \\lambda _{a}}\\, F_1(t_1) \\;\\frac{1}{4 s_{1}} (- i s_{1} \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t_{1})-1} \\\\& \\quad \\quad \\times \\Gamma ^{(\\mathbb {P}\\mathbb {P}f_{0})\\,\\mu _{1} \\nu _{1}, \\mu _{2} \\nu _{2}}(q_{1},q_{2})\\,\\Delta ^{(f_{0})}(p_{34})\\,\\Gamma ^{(f_{0} \\sigma \\sigma )}(p_{3},p_{4})\\\\& \\quad \\quad \\times \\frac{1}{4 s_{2}} (- i s_{2} \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t_{2})-1}\\,3 \\beta _{\\mathbb {P}NN} \\, 2 (p_2 + p_b)_{\\mu _{2}} (p_2 + p_b)_{\\nu _{2}}\\,\\delta _{\\lambda _{2} \\lambda _{b}}\\, F_1(t_2) \\,,\\end{split}$ where $s_{1} = (p_{1} + p_{3} + p_{4})^{2}$ , $s_{2} = (p_{2} + p_{3} + p_{4})^{2}$ , $q_{1} = p_{a} - p_{1}$ , $q_{2} = p_{b} - p_{2}$ , $t_{1} = q_{1}^{2}$ , $t_{2} = q_{2}^{2}$ , and $p_{34} = p_{3} + p_{4}$ .", "The effective Lagrangians and the vertices for $\\mathbb {P}\\mathbb {P}$ fusion into an $f_{0}$ meson are discussed in Appendix A of [15].", "As was shown there the tensorial $\\mathbb {P}\\mathbb {P}f_{0}$ vertex corresponds to the sum of two lowest values of $(l,S)$ , that is $(l,S) = (0,0)$ and $(2,2)$ with coupling parameters $g_{\\mathbb {P}\\mathbb {P}M}^{\\prime }$ and $g_{\\mathbb {P}\\mathbb {P}M}^{\\prime \\prime }$ , respectively.", "The vertex, including a form factor, reads then as follows ($p_{34} = q_{1} + q_{2}$ ) $i\\Gamma _{\\mu \\nu ,\\kappa \\lambda }^{(\\mathbb {P}\\mathbb {P}f_{0})} (q_{1},q_{2}) =\\left( i\\Gamma _{\\mu \\nu ,\\kappa \\lambda }^{\\prime (\\mathbb {P}\\mathbb {P}f_{0})}\\mid _{bare} +i\\Gamma _{\\mu \\nu ,\\kappa \\lambda }^{\\prime \\prime (\\mathbb {P}\\mathbb {P}f_{0})} (q_{1}, q_{2})\\mid _{bare} \\right)\\tilde{F}^{(\\mathbb {P}\\mathbb {P}f_{0})}(q_{1}^{2},q_{2}^{2},p_{34}^{2}) \\,;$ see (A.21) of [15].", "Unfortunately, the pomeron-pomeron-meson form factor is not well known as it is due to nonperturbative effects related to the internal structure of the respective meson.", "In practical calculations we take the factorized form for the $\\mathbb {P}\\mathbb {P}f_{0}$ form factor $\\tilde{F}^{(\\mathbb {P}\\mathbb {P}f_{0})}(q_{1}^{2},q_{2}^{2},p_{34}^{2}) =F_{M}(q_{1}^{2}) F_{M}(q_{2}^{2}) F^{(\\mathbb {P}\\mathbb {P}f_{0})}(p_{34}^{2})\\,$ normalised to $\\tilde{F}^{(\\mathbb {P}\\mathbb {P}f_{0})}(0,0,m_{f_{0}}^{2}) = 1$ .", "We will further set $F^{(\\mathbb {P}\\mathbb {P}f_{0})}(p_{34}^{2}) =\\exp { \\left( \\frac{-(p_{34}^{2}-m_{f_{0}}^{2})^{2}}{\\Lambda _{f_{0}}^{4}} \\right)}\\,,\\quad \\Lambda _{f_{0}} = 1\\;{\\rm GeV}\\,.$ For the $f_{0} \\sigma \\sigma $ vertex we have $i\\Gamma ^{(f_{0} \\sigma \\sigma )}(p_{3},p_{4}) =i g_{f_{0} \\sigma \\sigma } M_{0}\\, F^{(f_{0} \\sigma \\sigma )}(p_{34}^{2})\\,,$ where $g_{f_{0} \\sigma \\sigma }$ is an unknown parameter.", "We assume $g_{f_{0} \\sigma \\sigma }>0$ and $F^{(f_{0} \\sigma \\sigma )}(p_{34}^{2})$ = $F^{(\\mathbb {P}\\mathbb {P}f_{0})}(p_{34}^{2})$ ; see Eq.", "(REF )." ], [ "$pp \\rightarrow pp (f_{0} \\rightarrow \\rho \\rho )$", "Now we consider the amplitude for the reaction (REF ) through $f_{0}$ exchange in the $s$ -channel as shown in Fig.", "REF (right diagram).", "In the high-energy approximation we can write the amplitude as shown in (REF ) with $\\begin{split}& {\\cal M}^{(f_{0} \\rightarrow \\rho \\rho )\\,\\rho _{3} \\rho _{4}}_{\\lambda _{a}\\lambda _{b}\\rightarrow \\lambda _{1}\\lambda _{2}\\rho \\rho }\\simeq 3 \\beta _{\\mathbb {P}NN} \\, 2(p_1 + p_a)_{\\mu _{1}} (p_1 + p_a)_{\\nu _{1}}\\,\\delta _{\\lambda _{1} \\lambda _{a}}\\, F_1(t_1) \\;\\frac{1}{4 s_{1}} (- i s_{1} \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t_{1})-1} \\\\& \\quad \\quad \\times \\Gamma ^{(\\mathbb {P}\\mathbb {P}f_{0})\\,\\mu _{1} \\nu _{1}, \\mu _{2} \\nu _{2}}(q_{1},q_{2})\\,\\Delta ^{(f_{0})}(p_{34})\\,\\Gamma ^{(f_{0} \\rho \\rho )\\, \\rho _{3} \\rho _{4}}(p_{3},p_{4})\\\\& \\quad \\quad \\times \\frac{1}{4 s_{2}} (- i s_{2} \\alpha ^{\\prime }_{\\mathbb {P}})^{\\alpha _{\\mathbb {P}}(t_{2})-1}\\,3 \\beta _{\\mathbb {P}NN} \\, 2 (p_2 + p_b)_{\\mu _{2}} (p_2 + p_b)_{\\nu _{2}}\\,\\delta _{\\lambda _{2} \\lambda _{b}}\\, F_1(t_2) \\,,\\end{split}$ where we take for the $f_{0} \\rho \\rho $ vertex the following ansatz $\\begin{split}&\\Gamma ^{(f_{0} \\rho \\rho )\\,\\rho _{3} \\rho _{4}}(p_{3},p_{4}) =\\\\&g_{f_{0} \\rho \\rho }^{\\prime } \\,\\frac{2}{M_{0}^{3}}\\,\\bigg [ p_{3}^{2} p_{4}^{2} g^{\\rho _{3} \\rho _{4}}- p_{4}^{2} p_{3}^{\\rho _{3}} p_{3}^{\\rho _{4}}- p_{3}^{2} p_{4}^{\\rho _{3}} p_{4}^{\\rho _{4}}+ \\left( p_{3} \\cdot p_{4} \\right) p_{3}^{\\rho _{3}} p_{4}^{\\rho _{4}} \\bigg ]\\,F^{\\prime (f_{0} \\rho \\rho )}(p_{3}^{2},p_{4}^{2},p_{34}^{2})\\\\&+ g_{f_{0} \\rho \\rho }^{\\prime \\prime } \\,\\dfrac{2}{M_{0}} \\,\\bigg [ p_{4}^{\\rho _{3}} p_{3}^{\\rho _{4}}-(p_{3} \\cdot p_{4}) g^{\\rho _{3}\\rho _{4}} \\bigg ]\\,F^{\\prime \\prime (f_{0} \\rho \\rho )}(p_{3}^{2},p_{4}^{2},p_{34}^{2})\\end{split}$ with $g_{f_{0} \\rho \\rho }^{\\prime }$ and $g_{f_{0} \\rho \\rho }^{\\prime \\prime }$ being free parameters.", "Different form factors $F^{\\prime }$ and $F^{\\prime \\prime }$ are allowed a priori.", "The vertex in Eq.", "(REF ) fulfils the following relations: $\\begin{split}p_{3\\, \\rho _{3}} \\Gamma ^{(f_{0} \\rho \\rho ) \\rho _{3} \\rho _{4}}(p_{3},p_{4}) =0\\,, \\qquad p_{4\\, \\rho _{4}} \\Gamma ^{(f_{0} \\rho \\rho ) \\rho _{3} \\rho _{4}}(p_{3},p_{4}) =0\\,.\\end{split}$ Once evidence for one or more of the $f_{0}$ resonances discussed here is obtained from RHIC and/or LHC experiments the formulae given in this appendix should be useful.", "Then, it will hopefully be possible to determine empirically the coupling parameters of the relevant vertices: $\\mathbb {P}\\mathbb {P}f_{0}$ , $f_{0} \\sigma \\sigma $ and $f_{0} \\rho \\rho $ .", "We are indebted to Leszek Adamczyk, Lidia Görlich, Radosław Kycia, Wolfgang Schäfer and Jacek Turnau for useful discussions.", "This research was partially supported by the MNiSW Grant No.", "IP2014 025173 (Iuventus Plus), the Polish National Science Centre Grant No.", "DEC-2014/15/B/ST2/02528 (OPUS) and by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów." ] ]	1606.05126
[ [ "Automatic Pronunciation Generation by Utilizing a Semi-supervised Deep\n Neural Networks" ], [ "Abstract Phonemic or phonetic sub-word units are the most commonly used atomic elements to represent speech signals in modern ASRs.", "However they are not the optimal choice due to several reasons such as: large amount of effort required to handcraft a pronunciation dictionary, pronunciation variations, human mistakes and under-resourced dialects and languages.", "Here, we propose a data-driven pronunciation estimation and acoustic modeling method which only takes the orthographic transcription to jointly estimate a set of sub-word units and a reliable dictionary.", "Experimental results show that the proposed method which is based on semi-supervised training of a deep neural network largely outperforms phoneme based continuous speech recognition on the TIMIT dataset." ], [ "Introduction", "The three principal resources typically required for developing a phoneme based automatic speech recognizer (ASR) are: transcribed acoustic data for acoustic model estimation, text data for language model estimation, and a pronunciation dictionary to map words to sequences of sub-word units.", "Manual preparation of such resources requires significant investment and expertise.", "Therefore, an automatic generation of pronunciation dictionary from the data is clearly required for many dialects and languages.", "Developing ASRs for dialects and under-resourced languages has attracted growing attention over the past few years [1], [2], [3].", "A main challenge to develop ASR for under-resourced domains is to produce a reliable pronunciation dictionary from limited available resources.", "For major languages, however, a canonical pronunciation dictionary is usually already available.", "However, such dictionaries may be error-prone due to the fact that they are manually generated and in most cases do not cover pronunciation variants.", "There were several attempts to tackle these problems [4], [5], [6], [7].", "Lu et al.", "[8] proposed a data-driven dictionary generator to include new pronunciations based on newly coming acoustic evidence.", "Goel et al.", "in [9] use a grapheme-to-phoneme approach to guess the pronunciation and iteratively refine the acoustic model and the dictionary.", "However, these methods still require a high-quality initial pronunciation dictionary created by an expert.", "In modern ASRs words are represented by smaller sub-word units such as phonemes and the pronunciation dictionary maps words to sequences of sub-word units.", "However, sub-word units do not essentially need to be linguistically motivated elements.", "In fact, given a set of acoustic samples, the linguistically defined units are most probably not the optimal ones for speech recognition [10].", "For instance telephony speech, where high frequency components have been filtered out, requires a modified dictionary with slightly different set of fricatives than full-bandwidth speech.", "Over the past few years, there have been several attempts to move beyond phoneme based sub-word units by jointly learn a set of sub-word units and their corresponding dictionary directly from the given data [11], [12], [8].", "Bacchiani and Ostendorf [12] proposed an iterative acoustic segmentation and clustering approach to build sub-word units from speech signals and subsequently construct the dictionary based on the estimated sub-word units.", "Singh et al.", "[8] introduced a divide-and-conquer strategy to recursively update sub-word units and dictionary.", "The dictionary computation was done by means of an n-best type algorithm which is known to produce sub-optimal solutions.", "Although their approach demonstrates some promising results, the performance is still not comparable with a phoneme based ASR.", "The main focus of this paper is to design an ASR based on an automatically generated dictionary that outperforms commonly used phoneme based ASRs.", "While most of the solutions proposed to find a pronunciation based on multiple utterances of a word are n-best type heuristics [8], [13], [14], in this paper, we employ an approximation of the K-dimensional Viterbi algorithm proposed in our previous works [15], [10].", "This approach gives us the maximum-likelihood estimates of the pronunciations.", "These high-quality pronunciations are one of the key factors to outperform phoneme based ASRs.", "Moreover, to learn proper sub-word units, we combine the strength of Gaussian mixture models (GMM) and deep neural network (DNN) based acoustic modeling.", "We formulate this problem as an instance of a semi-supervised self-learning process.", "By taking advantage of the robustness of hidden Markov models (HMM) with GMM based observation probability distribution against labeling errors, we train the first set of sub-word units and output the first set of pronunciations.", "We then use this dictionary to re-label the data and employ the higher expressiveness of DNNs to improve the modeling of sub-word units and the dictionary in an iterative process.", "In each iteration round, a new dictionary is generated and by means of this new dictionary the data is re-labeled.", "This data is again used to train the DNN.", "As shown in the experiments, the proposed results achieves more than 10% absolute improvement over the phoneme based approach on TIMIT data in a continuous speech recognition task.", "The reminder of this paper is organized as follows.", "The proposed framework and its components for joint sub-word units and dictionary learning are introduced in Section .", "In Section the experimental results are demonstrated and finally, conclusions are summarized in Section ." ], [ "Framework", "In the rest of this paper, we refer to data-driven sub-word units as abstract acoustic elements (AAEs) in contrast to phones.", "Our goal is to jointly learn the pronunciation dictionary $d^* = \\lbrace \\omega _1,\\cdots ,\\omega _L\\rbrace $ of $L$ pronunciations $\\omega _i$ and $N$ AAE models $\\lambda ^=\\lbrace A_1,\\cdots ,A_N\\rbrace $ that maximize the joint likelihood: $\\lambda ^,d^* = \\mathop {\\rm arg~max}\\limits _{\\Lambda ,D} P(\\textbf {X}\|\\textbf {T},\\Lambda ,D)$ where $\\textbf {X}=(X_1, \\cdots ,X_M)$ is the set of training utterances, $\\textbf {T}=(T_1, \\cdots ,T_M)$ is the set of corresponding orthographic transcriptions, $M$ is the number of utterances, $\\Lambda $ is the universe of all possible sets of $N$ AAEs and $D$ is the universe of all the dictionaries which map words to AAEs sequences.", "It is hard to find the optimal solution for the optimization problem in (REF ) due to its complex non-linear nature.", "It is thus decomposed into two simpler optimization problems which can be solved iteratively.", "$d^i = \\mathop {\\rm arg~max}\\limits _{D} P(X\|T,\\lambda ^i,D) \\\\\\lambda ^{i+1} = \\mathop {\\rm arg~max}\\limits _{\\Lambda } P(X\|T,\\Lambda ,d^i) $ Since the pronunciation of each word can be estimated independently from other words, the dictionary estimation in (REF ) can be decomposed into $L$ maximum likelihood estimations as follows: $\\begin{split}\\omega _l = &\\mathop {\\rm arg~max}\\limits _{\\omega } \\prod _{j\\in \\Omega _l} \\max _{\\textbf {S}_j} P(X_j,\\textbf {S}_j\|\\lambda )\\\\&\\text{subject to: }\\textbf {S}_j \\in \\mathbb {S}_\\omega \\end{split}$ where $\\Omega _l$ is the set of indices of utterances of word $W_l$ , $\\textbf {S}_j$ is a sequence of AAEs and $\\mathbb {S}_\\omega $ denotes a set of all possible AAE sequences of the pronunciation $\\omega $ .", "For instance in $\\mathbb {S}_\\omega $ , if the pronunciation is $\\omega =A_1A_2A_3$ , some samples in $\\mathbb {S}_\\omega $ may be $A_1A_1A_1A_2A_3, A_1A_2A_2A_3A_3$ and $A_1A_1A_2A_3A_3A_3$ .", "The constraint in (REF ) implies that all AAE sequences should be samples of the same pronunciation.", "For the case where $\\lambda $ is modeled by a left-to-right HMM without skips, which is the most common topology in HMM based ASRs, a solution of (REF ) has been proposed in [15] (Details are in Section REF .", ").", "In (), since the dictionary is fixed, the problem results in a common acoustic model estimation given the dictionary.", "However, the labels re-assigned by using the estimated dictionary are very noisy since the dictionary is automatically estimated from data without any expert supervision.", "Therefore, a robust model is required at early stage of the training iteration while a more expressive and powerful model such as a DNN [16], [17] can be used after the reliable dictionary is obtained.", "The joint dictionary and AAE learning framework is illustrated in Figure REF and summarized as follows: Semi-supervised joint AAEs and dictionary learning [1] $i=0$ // Initialize AAE models $\\lambda ^0$ (Section REF ) Clustering the acoustic space.", "Model each cluster by GMM and set as $\\lambda ^0$ .", "// Start joint AAEs and dictionary learning ( Performance is improved ) Given AAE models $\\lambda ^i$ , update dictionary $d^i$ by maximizing joint likelihood multiple utterances (Section REF ).", "Given dictionary $d^i$ , double the number of mixtures and update AAE models $\\lambda ^{i+1}$ (Section REF ).", "$i \\leftarrow i + 1$ Replace GMM by DNN and train AAE model using labels obtained by HMM-GMM (Section REF ).", "( Performance is improved ) Given AAE models $\\lambda ^i$ , update dictionary $d^i$ by maximizing joint likelihood multiple utterances.", "Given dictionary $d^i$ , re-train DNN based AAE models $\\lambda ^{i+1}$ (Section REF ).", "$i \\leftarrow i + 1$" ], [ "Acoustic Model Initialization", "Initial AAE models can simply be obtained by clustering the acoustic space.", "The acoustic space can be described by any feature as long as it is informative enough to discriminate between different words.", "We employed the Linde-Buzo-Gray (LBG) algorithm [18] with a squared-error distortion measure to cluster the acoustic feature vectors.", "The LBG clustering algorithm tends to assign more codebook vectors to high-density areas which is a useful property in order to obtain discriminative initial AAEs.", "Each cluster is then modeled by a GMM with a single Gaussian component.", "These models are used as the initial models for AAEs." ], [ "Dictionary Generation", "The solution of (REF ) proposed in [15] is an extension of the standard one-dimensional Viterbi algorithm to $K$ dimensions.", "The K-dimensional Viterbi algorithm calculates the most probable HMM state sequence which is common to $K$ given utterances.", "While this algorithm is rigorous, its complexity grows exponentially with the number of utterances, which consequently makes it infeasible to apply it to more than a few utterances.", "An efficient approximation of the K-dimensional Viterbi algorithm has been proposed in [10] where the problem to find the joint alignment and the optimal common sequence for $K$ utterances is decomposed into $K{-}1$ applications of two-dimensional Viterbi algorithm.", "This approximation starts with finding the best alignment between two utterances.", "Then, while keeping the alignment between the already processed utterances fixed, the next utterance is aligned with this master utterance.", "The AAE sequence of the final master utterance is the approximation of the K-dimensional Viterbi pronunciation." ], [ "Acoustic Modeling", "Once the dictionary is updated, all utterances are decoded based on the new pronunciation of the words in the dictionary and the AAEs are re-estimated according to the new labels.", "The AAEs can be modeled by commonly used models such as HMM/GMM or HMM/DNN.", "However, at the beginning of the training iteration, the model and dictionary are not accurate enough and more probable to get stuck in a bad local optimum if the model's degree of freedom is too high.", "In order to avoid this situation, we start the training with a simple model, namely one Gaussian component for each AAE with a diagonal covariance matrix.", "In each iteration, the dictionary gets more accurate.", "Thus, the number of mixture components are doubled in order to increase the modeling power.", "Once the performance is saturated the GMM is replaced with the DNN in order to utilize more expressive modeling capability.", "This process makes the semi-supervised DNN training feasible and prevents it to get stuck in a bad local optimum.", "The HMM state-level transcription is obtained by force-aligned decoding with optimised HMM-GMM and dictionary.", "This transcription provides labels for DNN training.", "The DNN is trained to estimate HMM posterior states by minimizing the cross entropy loss $L$ with $l_1$ regularization using back propagation: $\\mathop {\\rm arg~min}\\limits _{W} \\sum _{i,j} L(\\textbf {x}^i_j,y^i_j,W)+\\rho \\Vert W\\Vert _1 $ where $\\textbf {x}^i_j\\in X_i$ is the $j$ th feature vector of the $i$ th utterance, $y_j^i$ is the corresponding label and $W$ is the set of network parameters, respectively.", "$\\rho $ is a constant parameter which is set to $10^{-6}$ in this work." ], [ "Experiments", "We conducted several sets of experiments on the TIMIT corpus [19].", "The TIMIT corpus provides a manually prepared dictionary and phone-level transcriptions with 61 phones.", "As a baseline, 61 phone models were trained using the TIMIT dictionary and the provided transcriptions.", "We used 12 mel frequency cepstral coefficients (MFCCs) and energy with their deltas and delta-deltas as descriptors of the acoustic space.", "The speech data was analyzed using a 25 ms Hamming window with a 10 ms frame shift.", "We evaluated phone based DNN-HMM, GMM-HMM and AAE based GMM-HMM model as baselines.", "The DNN architecture was comprised of 7 hidden layers.", "The first hidden layer had 2048 nodes, next 5 layers had 1024 nodes and the number of nodes at the last layer was equal to the number of HMM states to be predicted.", "All hidden layers were equipped with the Rectified Linear Unit (ReLU) non-linearity [20].", "The input to the network was 11 contiguous frames of MFCCs.", "The networks were trained using mini-batch gradient descent based on back propagation with momentum.", "We applied dropout [16] to all hidden layers with dropout probability $0.5$ .", "The batch size was set to 128.", "HMMs had left-to-right, no-skipping topology with three states for each phoneme as opposed to one state for each AAE.", "HMMs were trained using a modified version of HTK [21] and DNNs were implemented using Lasagne [22]." ], [ "Isolated Word Recognition", "The first set of experiments were on the isolated word recognition to test the performance of the proposed methods and investigate the effects of hyper parameters such as the number of mixture components and the number of AAEs.", "For joint pronunciation estimation and acoustic models training, we collected a pronunciation training set comprising of words with more than 10 utterances from the TIMIT training set.", "The total number of utterances in the pronunciation training set was 12800.", "After excluding words with less than 4 characters (e.g., a and the), 339 distinct words were collected from the TIMIT test set for the isolated word speech recognition task, resulting in 3900 utterances in total.", "The baseline GMM based phone models were trained with 32 mixture components.", "During the GMM based AAE model training the number of mixtures was doubled for each iteration until it reached 128 mixtures as described in Section REF ." ], [ "Comparison with phonetic approach", "The word error rates (WER) of each method are shown in Table REF .", "The results show that the proposed data-driven method clearly outperforms the baseline methods.", "The proposed AAE-DNN method achieved 10.3% and 2.4% improvement over GMM and DNN based phonetic acoustic models, respectively.", "This suggests that a more accurate dictionary and better acoustic models can be obtained directly from training data without any human expertise.", "Moreover, AAE-DNN method improves the performance by 3.2% over the AAE-GMM method.", "This indicates that the DNN was successfully trained in the semi-supervised manner and the final model could effectively use the its expressive modeling power.", "Table: Comparison of word error rates of each method on 339 words isolated word recognition (%).", "Baseline phone models are trained by using the TIMIT dictionary." ], [ "Number of AAEs", "Our second experiment focused on the effects of the number of AAEs, i.e.", "$N$ .", "We trained the dictionary and AAE models with $N = 64, 128, 192, 256, 320, 384, 448$ .", "The word error rates of DNN and GMM based AAE models are illustrated in Figure REF .", "The number of mixtures of the GMMs were determined experimentally as shown in Table REF .", "For DNN based AAE models, the best result are obtained with 384 AAEs in contrast to with 320 AAEs for the GMM based models.", "Interestingly, the optimal number of AAE states is far higher than the number of states of the phone models (61 phonemes $\\times $ 3 states = 183 states).", "This is an indication that the proposed data-driven approach to jointly generate the sub-word units and dictionary models the acoustic space more precisely than the linguistically motivated phonetic units and the manually designed dictionary.", "It is also worthwhile to mention that the optimal number of DNN based AAE models was higher than that of GMM based models.", "This is perhaps due to the fact that the DNN was trained discriminatively, allowing to efficiently model the interaction between higher number of AAEs.", "Table: Word error rates in % of AAE based recognizers with different number of AAEs and GMM mixture.", "The best performance for each number of AAE is plotted in Figure .Figure: Performance of AAE based recognizers with different number of AAEs on test set with 339 words." ], [ "Continuous Speech Recognition", "Unlike phoneme based ASRs, the proposed AAE based approach does not depend on linguistic knowledge.", "It is therefore interesting to compare these approaches on a real-world continues speech recognition (CSR) task.", "For this purpose, we used the SX records of the TIMIT corpus which contains 450 sentences spoken by 7 speakers, i.e.", "3150 utterances in total.", "We prepared the test set by randomly selecting and putting aside one speaker for each sentence from the SX recordings and used the remaining samples as the training set (450 sentences $\\times $ 6 speaker = 2700 utterances).", "We also included the SA and SI recordings of the TIMIT corpus in the training set.", "The number of AAEs was 384.", "The number of mixture components in the GMM based phone models was 64.", "The performance was evaluated in two scenarios: with and without language model.", "The language model employed in the baseline and the proposed methods is a simple bigram model.", "Table REF shows that the proposed AAE-DNN based approach significantly outperforms baseline methods in both scenarios.", "The performance improvements over the phone based HMM-DNN method in with and without the language model scenarios were 10.68% and 5.11%, respectively.", "The results suggest that the proposed data-driven dictionary and the AAE models are also useful for CSR and a more accurate representation of speech signals can be learned automatically.", "We observed that all 384 AAEs were actually used in the trained dictionary, and the dictionary tend to assign 39% more HMM states on average to each word as compare with the TIMIT phonetic dictionary.", "This means that in AAEs, the stay-in-state probability is smaller resulting in more frequent state transitions.", "This suggests that by using AAEs, the acoustic space was modeled at a higher resolution.", "This consequently increased the precision of the word pronunciations.", "Table: Comparison of word error rate of each method on continuous speech recognition.", "In column ”No LM”, no language model was used." ], [ "Conclusions", "In this work we proposed a novel joint dictionary and sub-word unit learning framework for ASRs.", "The proposed method does not require linguistic expertise, and can automatically create the set of sub-word units and the corresponding pronunciation dictionary.", "In our method, reliable pronunciations are estimated from multiple utterances by an efficient approximation of K-dimensional Viterbi algorithm which estimates the most probable HMM state sequence common to multiple utterances of a word.", "Experimental results show that the proposed method significantly outperforms the phone based methods which even get manually prepared dictionary and hand crafted transcriptions as inputs.", "We further investigated the effects of the number of data-driven sub-word units and showed that the optimal number of sub-word units is much higher than the total number of HMM states of the 61 phones.", "The future works will be directed towards applying the proposed method to speech recognition for under-resourced languages and large vocabulary continuous speech recognition tasks." ] ]	1606.05007
[ [ "Improving Variational Inference with Inverse Autoregressive Flow" ], [ "Abstract The framework of normalizing flows provides a general strategy for flexible variational inference of posteriors over latent variables.", "We propose a new type of normalizing flow, inverse autoregressive flow (IAF), that, in contrast to earlier published flows, scales well to high-dimensional latent spaces.", "The proposed flow consists of a chain of invertible transformations, where each transformation is based on an autoregressive neural network.", "In experiments, we show that IAF significantly improves upon diagonal Gaussian approximate posteriors.", "In addition, we demonstrate that a novel type of variational autoencoder, coupled with IAF, is competitive with neural autoregressive models in terms of attained log-likelihood on natural images, while allowing significantly faster synthesis." ], [ "=1 pdfinfo= Title=Improved Variational Inference with Inverse Autoregressive Flow, Author=Diederik P. Kingma, Tim Salimans, Rafal Jazefowicz, Xi Chen, Ilya Sutskever, Max Welling [pages=1-last]main.pdf" ] ]	1606.04934
[ [ "Assessing and tuning brain decoders: cross-validation, caveats, and\n guidelines" ], [ "Abstract Decoding, ie prediction from brain images or signals, calls for empirical evaluation of its predictive power.", "Such evaluation is achieved via cross-validation, a method also used to tune decoders' hyper-parameters.", "This paper is a review on cross-validation procedures for decoding in neuroimaging.", "It includes a didactic overview of the relevant theoretical considerations.", "Practical aspects are highlighted with an extensive empirical study of the common decoders in within-and across-subject predictions, on multiple datasets --anatomical and functional MRI and MEG-- and simulations.", "Theory and experiments outline that the popular \" leave-one-out \" strategy leads to unstable and biased estimates, and a repeated random splits method should be preferred.", "Experiments outline the large error bars of cross-validation in neuroimaging settings: typical confidence intervals of 10%.", "Nested cross-validation can tune decoders' parameters while avoiding circularity bias.", "However we find that it can be more favorable to use sane defaults, in particular for non-sparse decoders." ], [ "Introduction: decoding needs model evaluation", "Decoding, ie predicting behavior or phenotypes from brain images or signals, has become a central tool in neuroimage data processing [21], [20], [24], [42], [58], [61].", "In clinical applications, prediction opens the door to diagnosis or prognosis [40], [11], [9].", "To study cognition, successful prediction is seen as evidence of a link between observed behavior and a brain region [19] or a small fraction of the image [28].", "Decoding power can test if an encoding model describes well multiple facets of stimuli [38], [41].", "Prediction can be used to establish what specific brain functions are implied by observed activations [53], [48].", "All these applications rely on measuring the predictive power of a decoder.", "Assessing predictive power is difficult as it calls for characterizing the decoder on prospective data, rather than on the data at hand.", "Another challenge is that the decoder must often choose between many different estimates that give rise to the same prediction error on the data, when there are more features (voxels) than samples (brain images, trials, or subjects).", "For this choice, it relies on some form of regularization, that embodies a prior on the solution [18].", "The amount of regularization is a parameter of the decoder that may require tuning.", "Choosing a decoder, or setting appropriately its internal parameters, are important questions for brain mapping, as these choice will not only condition the prediction performance of the decoder, but also the brain features that it highlights.", "Measuring prediction accuracy is central to decoding, to assess a decoder, select one in various alternatives, or tune its parameters.", "The topic of this paper is cross-validation, the standard tool to measure predictive power and tune parameters in decoding.", "The first section is a primer on cross-validation giving the theoretical underpinnings and the current practice in neuroimaging.", "In the second section, we perform an extensive empirical study.", "This study shows that cross-validation results carry a large uncertainty, that cross-validation should be performed on full blocks of correlated data, and that repeated random splits should be preferred to leave-one-out.", "Results also yield guidelines for decoder parameter choice in terms of prediction performance and stability." ], [ "A primer on cross-validation", "This section is a tutorial introduction to important concepts in cross validation for decoding from brain images." ], [ "Cross-validation: estimating predictive power", "In neuroimaging, a decoder is a predictive model that, given brain images $\\mathbf {X}$ , infers an external variable $\\mathbf {y}$ .", "Typically, $\\mathbf {y}$ is a categorical variable giving the experimental condition or the health status of subjects.", "The accuracy, or predictive power, of this model is the expected error on the prediction, formally: $\\text{accuracy} = \\mathbb {E}\\bigl [\\mathcal {E}(\\mathbf {y}^\\text{pred}, \\mathbf {y}^\\text{ground truth})\\bigr ]$ where $\\mathcal {E}$ is a measure of the error, most oftenFor multi-class problems, where there is more than 2 categories in $\\mathbf {y}$ , or for unbalanced classes, a more elaborate choice is advisable, to distinguish misses and false detections for each class.", "the fraction of instances for which $\\mathbf {y}^\\text{pred} \\ne \\mathbf {y}^\\text{ground truth}$ .", "Importantly, in equation (REF ), $\\mathbb {E}$ denotes the expectation, ie the average error that the model would make on infinite amount of data generated from the same experimental process.", "In decoding settings, the investigator has access to labeled data, ie brain images for which the variable to predict, $\\mathbf {y}$ , is known.", "These data are used to train the model, fitting the model parameters, and to estimate its predictive power.", "However, the same observations cannot be used for both.", "Indeed, it is much easier to find the correct labels for brain images that have been seen by the decoder than for unknown imagesA simple strategy that makes no errors on seen images is simply to store all these images during the training and, when asked to predict on an image, to look up the corresponding label in the store..", "The challenge is to measure the ability to generalize to new data.", "The standard approach to measure predictive power is cross-validation: the available data is split into a train set, used to train the model, and a test set, unseen by the model during training and used to compute a prediction error (figure REF ).", "Chapter 7 of [18] contains a reference on statistical aspects of cross-validation.", "Below, we detail important considerations in neuroimaging.", "Cross-validation relies on independence between the train and test sets.", "With time-series, as in fMRI, the autocorrelation of brain signals and the temporal structure of the confounds imply that a time separation is needed to give truly independent observations.", "In addition, to give a meaningful estimate of prediction power, the test set should contain new samples displaying all confounding uncontrolled sources of variability.", "For instance, in multi-session data, it is harder to predict on a new session than to leave out part of each session and use these samples as a test set.", "However, generalization to new sessions is useful to capture actual invariant information.", "Similarly, for multi-subject data, predictions on new subjects give results that hold at the population level.", "However, a confound such as movement may correlate with the diagnostic status predicted.", "In such a case the amount of movement should be balanced between train and test set." ], [ "Sufficient test data", "Large test sets are necessary to obtain sufficient power for the prediction error for each split of cross-validation.", "As the amount of data is limited, there is a balance to strike between achieving such large test sets and keeping enough training data to reach a good fit with the decoder.", "Indeed, theoretical results show that cross-validation has a negative bias on small dataset [2] as it involves fitting models on a fraction of the data.", "On the other hand, large test sets decrease the variance of the estimated accuracy [2].", "A good cross-validation strategy balances these two opposite effects.", "Neuroimaging papers often use leave one out cross-validation, leaving out a single sample at each split.", "While this provides ample data for training, it maximizes test-set variance and does not yield stable estimates of predictive accuracyOne simple aspect of the shortcomings of small test sets is that they produce unbalanced dataset, in particular leave-one-out for which there is only one class represented in the test set.. From a decision-theory standpoint, it is preferable to leave out 10% to 20% of the data, as in 10-fold cross-validation [18] [5], [27].", "Finally, it is also beneficial to increase the number of splits while keeping a given ratio between train and test set size.", "For this purpose k-fold can be replaced by strategies relying on repeated random splits of the data (sometimes called repeated learning-testingAlso related is bootstrap CV, which may however duplicate samples inside the training set of the test set.", "[2] or ShuffleSplit [43]).", "As discussed above, such splits should be consistent with the dependence structure across the observations (using eg a LabelShuffleSplit), or the training set could be stratified to avoid class imbalance [49].", "In neuroimaging, good strategies often involve leaving out sessions or subjects.", "Figure: Cross-validation: the data is splitmultiple times into a trainset, used to train the model, and a test set, used to computepredictive power." ], [ "A necessary evil: one size does not fit all", "In standard statistics, fitting a simple model on abundant data can be done without the tricky choice of a meta-parameter: all model parameters are estimated from the data, for instance with a maximum-likelihood criterion.", "However, in high-dimensional settings, when the number of model parameters is much larger than the sample size, some form of regularization is needed.", "Indeed, adjusting model parameters to best fit the data without restriction leads to overfit, ie fitting noise [18].", "Some form of regularization or prior is then necessary to restrict model complexity, e.g.", "with low-dimensional PCA in discriminant analysis [7], or by selecting a small number of voxels with a sparse penalty [60], [6].", "If too much regularization is imposed, the ensuing models are too constrained by the prior, they underfit, ie they do not exploit the full richness of the data.", "Both underfitting and overfitting are detrimental to predictive power and to the estimation of model weights, the decoder maps.", "Choosing the amount of regularization is a typical bias-variance problem: erring on the side of variance leads to overfit, while too much bias leads to underfit.", "In general, the best tradeoff is a data-specific choice, governed by the statistical power of the prediction task: the amount of data and the signal-to-noise ratio.", "Figure: Nested cross-validation: two cross-validation loopsare run one inside the other." ], [ "Nested cross-validation", "Choosing the right amount of regularization can improve the predictive power of a decoder and controls the appearance of the weight maps.", "The most common approach to set it is to use cross-validation to measure predictive power for various choices of regularization and to retain the value that maximizes predictive power.", "Importantly, with such a procedure, the amount of regularization becomes a parameter adjusted on data, and thus the predictive performance measured in the corresponding cross-validation loop is not a reliable assessment of the predictive performance of the model.", "The standard procedure is then to refit the model on the available data, and test its predictive performance on new data, called a validation set.", "Given a finite amount of data, a nested cross-validation procedure can be employed: the data are repeatedly split in a validation set and a decoding set to perform decoding.", "The decoding set itself is split in multiple train and test sets with the same validation set, forming an inner “nested” cross-validation loop used to set the regularization hyper-parameter, while the external loop varying the validation set is used to measure prediction performance –see figure REF ." ], [ "Model averaging", "Choosing the best model in a family of good models is hard.", "One option is to average the predictions of a set of suitable models [44], [29], [8], [23] –see [18] for a description outside of neuroimaging.", "A simple version of this idea is bagging [4]: using bootstrap, random resamplings of the data, to generate many train sets and corresponding models, the predictions of which are then averaged.", "The benefit of these approaches is that if the errors of each model are sufficiently independent, they average out: the average model performs better and displays much less variance.", "With linear models often used as decoders in neuroimaging, model averaging is appealing as it boils down to averaging weight maps.", "To benefit from the stabilizing effect of model averaging in parameter tuning, we can use a variant of both cross-validation and model averagingThe combination of cross-validation and model averaging is not new (see eg [23]), but it is seldom discussed in the neuroimaging literature.", "It is commonly used in other areas of machine learning, for instance to set parameters in bagged models such as trees, by monitoring the out-of-bag error (eg in the scikit-learn library [43])..", "In a standard cross-validation procedure, we repeatedly split the data in train and test set and for each split, compute the test error for a grid of hyper-parameter values.", "However, instead of selecting the hyper-parameter value that minimizes the mean test error across the different splits, we select for each split the model that minimizes the corresponding test error and average these models across splits." ], [ "Model selection for neuroimaging decoders", "Decoding in neuroimaging faces specific model-selection challenges.", "The main challenge is probably the scarcity of data relative to their dimensionality, typically hundreds of observationsWhile in imaging neuroscience, hundreds of observations seems acceptably large, it is markedly below common sample sizes in machine learning.", "Indeed, data analysis in brain imaging has historically been driven by very simple models while machine learning has tackled rich models since its inception.. Another important aspect of decoding is that, beyond predictive power, interpreting model weights is relevant." ], [ "Common decoders and their regularization", "Both to prefer simpler models and to facilitate interpretation, linear models are ubiquitous in decoding.", "In fact, their weights form the common brain maps for visual interpretation.", "Figure: Regularization with SVM-ℓ 2 \\ell _2:blue and brown points are training samples of each class.", "The SVMlearns a separating line between the two classes.", "In a weaklyregularized setting (large CC, this line is supported by fewobservations –called support vectors–, circled in black on the figure, while in astrongly-regularized case (small CC), it issupported by the whole data.Figure: Varying amount of regularization on the face vshouse discrimination in the Haxby 2001 data .Left: with a log-reg ℓ 1 \\ell _1, more regularization (small C)induces sparsity.", "Right: with an SVM ℓ 2 \\ell _2, small C means thatweight maps are a combination of a larger numberof original images, although this has only a small visual impact on thecorresponding brain maps.The classifier used most often in fMRI is the support vector machine (SVM) [40], [7], [31].", "However, logistic regressions (Log-Reg) are also often used [52], [57], [7], [60], [50].", "Both of these classifiers learn a linear model by minimizing the sum of a loss $\\mathcal {L}$ –a data-fit term– and a penalty $p$ –the regularizing energy term that favors simpler models: $\\hat{\\mathbf {w}} = \\underset{\\mathbf {w}}{\\text{argmin}}\\bigl (\\mathcal {L}(\\mathbf {w}) + \\frac{1}{C} \\, p(\\mathbf {w})\\bigr )\\qquad \\smash{\\mathcal {L} ={\\left\\lbrace \\begin{array}{ll}\\text{\\raisebox {-1ex}{\\includegraphics [height=3ex]{hinge_loss}}}& \\hspace{-6.375pt}\\text{SVM} \\\\\\text{\\raisebox {-1ex}{\\includegraphics [height=3ex]{logistic_loss}}}& \\hspace{-6.375pt}\\text{logistic}\\end{array}\\right.", "}}$ where $C$ is the regularization parameter that controls the bias-variance tradeoff: small $C$ means strong regularization.", "The SVM and logistic regression model differ only by the loss used.", "For the SVM the loss is a hinge loss: flat and exactly zero for well-classified samples and with a misclassification cost increasing linearly with distance to the decision boundary.", "For the logistic regression, it is a logistic loss, which is a soft, exponentially-decreasing, version of the hinge [18].", "By far the most common regularization is the $\\ell _2$ penalty.", "Indeed, the common form of SVM uses $\\ell _2$ regularization, which we will denote SVM-$\\ell _2$ .", "Combined with the large zero region of the hinge loss, strong $\\ell _2$ penalty implies that SVMs build their decision functions by combining a small number of training images (see figure REF ).", "Logistic regression is similar: the loss has no flat region, and thus every sample is used, but some very weakly.", "Another frequent form of penalty, $\\ell _1$ , imposes sparsity on the weights: a strong regularization means that the weight maps $\\mathbf {w}$ are mostly comprised of zero voxels (see Fig.", "REF )." ], [ "Parameter-tuning in neuroimaging", "In neuroimaging, many publications do not discuss their choice of decoder hyper-parameters; while others state that they use the default value, eg $C = 1$ for SVMs.", "Standard machine learning practice advocates setting the parameters by nested cross-validation [18].", "For non sparse, $\\ell _2$ -penalized models, the amount of regularization often does not have a strong influence on the weight maps of the decoder (see figure REF ).", "Indeed, regularization in these models changes the fraction of input maps supporting the hyperplane (see REF ).", "As activation maps for the same condition often have similar aspects, this fraction impacts weakly decoders' maps.", "For sparse models, using the $\\ell _1$ penalty, sparsity is often seen as a means to select relevant voxels for prediction [6], [52].", "In this case, the amount of regularization has a very visible consequence on weight maps and voxel selection (see figure REF ).", "Neuroimaging studies often set it by cross-validation [6], though very seldom nested (exceptions comprise [8], [57]).", "Voxel selection by $\\ell _1$ penalty on brain maps is unstable because neighboring voxels respond similarly and $\\ell _1$ estimators will choose somewhat randomly few of these correlated features [57], [51].", "Hence various strategies combining sparse models are used in neuroimaging to improve decoding performance and stability.", "Averaging weight maps across cross-validation folds [23], [57], as described above, is interesting, as it stays in the realm of linear models.", "Relatedly, [16] report the median of weight maps, thought it does not correspond to weights in a predictive model.", "Consensus between sparse models over data perturbations gives theoretically better feature selection [35].", "In fMRI, it has been used to screen voxels before fitting linear models [51], [57] or to interpret selected voxels [60].", "For model selection in neuroimaging, prediction performance is not the only relevant metric and some control over the estimated model weights is also important.", "For this purpose, [30], [55], [50] advocate using a tradeoff between prediction performance and stability of decoder maps.", "Stability is a proxy for estimation error on these maps, a quantity that is not accessible without knowing the ground truth.", "While very useful it gives only indirect information on estimation error: it does not control whether all the predictive brain regions were found, nor whether all regions found are predictive.", "Indeed, a decoder choosing its maps independently from the data would be very stable, though likely with poor prediction performance.", "Hence the challenge is in finding a good prediction-stability tradeoff [56], [50]." ], [ "Empirical studies: cross-validation at work", "Here we highlight practical aspects of cross-validation in brain decoding with simple experiments.", "We first demonstrate the variability of prediction estimates on MRI, MEG, and simulated data.", "We then explore how to tune decoders parameters." ], [ "A variety of decoding datasets", "To achieve reliable empirical conclusions, it is important to consider a large number of different neuroimaging studies.", "We investigate cross-validation in a large number of 2-class classification problems, from 7 different fMRI datasets (an exhaustive list can be found in ).", "We decode visual stimuli within subject (across sessions) in the classic Haxby dataset [19], and across subjects using data from [10].", "We discriminate across subjects i) affective content with data from [59], ii) visual from narrative with data from [39], iii) famous, familiar, and scrambled faces from a visual-presentations dataset [22], and iv) left and right saccades in data from [26].", "We also use a non-published dataset, ds009 from openfMRI [47].", "All the across-subject predictions are performed on trial-by-trial response (Z-score maps) computed in a first-level GLM.", "Finally, beyond fMRI, we perform prediction of gender from VBM maps using the OASIS data [34].", "Note that all these tasks cover many different settings, range from easy discriminations to hard ones, and (regarding fMRI) recruit very different systems with different effect size and variability.", "The number of observations available to the decoder varies between 80 (40 per class) and 400, with balanced classes.", "The results and figures reported below are for all these datasets.", "We use more inter-subject than intra-subject datasets.", "However 15 classification tasks out of 31 are intra-subject (see Tab.", "REF ).", "In addition, when decoding is performed intra-subject, each subject gives rise to a cross-validation.", "Thus in our cross-validation study, 82% of the data points are for intra-subject settings.", "All MR data but [26] are openly available from openfMRI [47] or OASIS [34].", "Standard preprocessing and first-level analysis were applied using SPM, Nipype and Nipy (details in REF ).", "All MR data were variance-normalizedDivision of each time series voxel/MEG sensor by its standard deviation and spatially-smoothed at 6 mm FWHM for fMRI data and 2 mm FWHM for VBM data." ], [ "MEG data", "Beyond MR data, we assess cross-validation strategies for decoding of event-related dynamics in neurophysiological data.", "We analyze magneteoencephalography (MEG) data from a working-memory experiment made available by the Human Connectome Project [33].", "We perform intra-subject decoding in 52 subjects with two runs, using a temporal window on the sensor signals (as in [54]).", "Here, each run serves as validation set for the other run.", "We consider two-class decoding problems, focusing on either the image content (faces vs tools) or the functional role in the working memory task (target vs low-level and high-level distractors).", "This yields in total four classification analyzes per subject.", "For each trial, the feature set is a time window constrained to 50 ms before and 300 ms after event onset, emphasizing visual components.", "We use the cleaned single-trial outputs from the HCP “tmegpreproc” pipeline.", "MEG data analysis was performed with the MNE-Python software [13], [14].", "Full details on the analysis are given in REF ." ], [ "Experimental setup", "Our experiments make use of nested cross-validation for an accurate measure of prediction power.", "As in figure REF , we repeatedly split the data in a validation set and a decoding set passed on to the decoding procedure (including parameter-tuning for experiments in REF and REF ).", "To get a good measure of predictive power, we choose large validation sets of 50% of the data, respecting the sample dependence structure (leaving out subjects, or sessions).", "We use 10 different validation sets that each contribute a data point in results.", "We follow standard decoding practice in fMRI [45].", "We use univariate feature selection on the training set to select the strongest 20% of voxels and train a decoder on the selected features.", "As a choice of decoder, we explore classic linear models: SVM and logistic regression with $\\ell _1$ and $\\ell _2$ penaltySimilar decoders adding a regularization that captures spatio-temporal correlations among the voxels are well suited for neuroimaging [15], [36], [16], [25].", "Also, random forests, based on model averaging discussed above, have been used in fMRI [32], [29].", "However, this review focuses on the most common practice.", "Indeed, these decoders entail computational costs that are intractable given the number of models fit in the experiments.. We use scikit-learn for all decoders [43], [1].", "In a first experiment, we compare decoder performance estimated by cross-validation on the decoding set, with performance measured on the validation set.", "In a second experiment, we investigate the use of cross-validation to tune the model's regularization parameter, either using the standard refitting approach, or averaging as described in section REF , as well as using the default $C=1$ choice of parameter, and a value of $C = 1000$ ." ], [ "Reliability of the cross-validation measure", "Considering that prediction error on the large left-out validation set is a good estimate of predictive power, we use it to assess the quality of the estimate given by the nested cross-validation loop.", "Figure REF shows the prediction error measured by cross-validation as a function of the validation-set error across all datasets and validation splits.", "It reveals a small negative bias: as predicted by the theory, cross-validation is pessimistic compared to a model fit on the complete decoding set.", "However, models that perform poorly are often reported with a better performance by cross-validation.", "Additionally, cross-validation estimates display a large variance: there is a scatter between estimates in the nested cross-validation loop and in the validation set.", "Figure: Cross-validation error: different strategies.Difference between accuracy measured by cross-validation and on thevalidation set, in intra and inter-subject settings, for differentcross-validation strategies: leave one sample out,leave one block of samples out (where the block is the naturalunit of the experiment: subject or session), orrandom splits leaving out 20% of the blocks as test data, with 3, 10, or50 random splits.For inter-subject settings, leave one sample out corresponds toleaving a session out.The box gives the quartiles, while the whiskers givethe 5 and 95 percentiles." ], [ "Different cross-validation strategies", "Figure REF summarizes the discrepancy between prediction accuracy measured by cross validation and on the validation set for different cross-validation strategies: leaving one sample out, leaving one block of data out –where blocks are the natural units of the experiment, sessions or subjects– and random splits leaving out 20% of the blocks of data with 3, 10, and 50 repetitions.", "Ideally, a good cross-validation strategy would minimize this discrepancy.", "We find that leave-one-sample out is very optimistic in within-subject settings.", "This is expected, as samples are highly correlated.", "When leaving out blocks of data that minimize dependency between train and test set, the bias mostly disappears.", "The remaining discrepancy appears mostly as variance in the estimates of prediction accuracy.", "For repeated random splits of the data, the larger the number of splits, the smaller the variance.", "Performing 10 to 50 splits with 20% of the data blocks left out gives a better estimation than leaving successively each blocks out, at a fraction of the computing cost if the number of blocks is large.", "While intra and inter subject settings do not differ strongly when leaving out blocks of data, intra-subject settings display a larger variance of estimation as well as a slight negative bias.", "These are likely due to non-stationarity in the time-series, eg scanner drift or loss of vigilance.", "In inter-subject settings, heterogeneities may hinder prediction [56], yet a cross-validation strategy with multiple subjects in the test set will yield a good estimate of prediction accuracyThe probability of correct classification for each subject is also an interesting quantity, though it is not the same thing as the prediction accuracy measured by cross-validation [18].", "It can be computed by non-parametric approaches such as bootstrapping the train set [46], or using a posterior probability, as given by certain classifiers.." ], [ "Other modalities: MEG and simulations", "We run the experiments on the MEG decoding tasks and the simulations.", "We generate simple simulated data that mimic brain imaging to better understand trends and limitations of cross-validation.", "Briefly, we generate data with 2 classes in 100 dimensions with Gaussian noise temporally auto-correlated and varying the separation between the class centers (more details in ).", "We run the experiments on a decoding set of 200 samples.", "Figure: Cross-validation error: non-MRI modalities.Difference between accuracy measured by cross-validation and on thevalidation set, for MEG data and simulated data, with differentcross-validation strategies.Detailed simulation results in .The results, displayed in figure REF , reproduce the trends observed on MR data.", "As the simulated data is temporally auto-correlated, leave-one-sample-out is strongly optimistic.", "Detailed analysis varying the separability of the classes () shows that cross-validation tends to be pessimistic for high-accuracy situations, but optimistic when prediction is low.", "For MEG decoding, the leave-one-out procedure is on trials, and thus does not suffer from correlations between samples.", "Cross-validation is slightly pessimistic and display a large variance, most likely because of inhomogeneities across samples.", "In both situations, leaving blocks of data out with many splits (e.g.", "50) gives best results.", "Figure: Tuning curves for SVM ℓ 2 \\ell _2, logisticregression ℓ 2 \\ell _2, SVM ℓ 1 \\ell _1, and logisticregression ℓ 1 \\ell _1, on the scissors / scramble discrimination forthe Haxby dataset .", "The thin colored lines are testscores for each of the internal cross-validation folds, the thickblack line is the average of these test scores on all folds, and thethick dashed line is the score on left-out validation data.", "The verticaldashed line is the parameter selected on the inner cross-validation score." ], [ "Results on cross-validation for parameter tuning", "We now evaluate cross-validation as a way of setting decoder hyperparameters." ], [ "Tuning curves: opening the black box", "Figure REF is a didactic view on the parameter-selection problem: it gives, for varying values of the meta-parameter $C$ , the cross-validated error and the validation error for a given split of validation dataFor the figure, we compute cross-validated error with a leave-one-session-out on the first 6 sessions of the scissor / scramble Haxby data, and use the last 6 sessions as a validation set..", "The validation error is computed on a large sample size on left out data, hence it is a good estimate of the generalization error of the decoder.", "Note that the parameter-tuning procedure does not have access to this information.", "The discrepancy between the tuning curve, computed with cross-validation on the data available to the decoder, and the validation curve, is an indication of the uncertainty on the cross-validated estimate of prediction power.", "Test-set error curves of individual splits of the nested cross-validation loop show plateaus and a discrete behavior.", "Indeed, each individual test set contains dozens of observations.", "The small combinatorials limit the accuracy of error estimates.", "Figure REF also shows that non-sparse –$\\ell _2$ -penalized– models are not very sensitive to the choice of the regularization parameter C: the tuning curves display a wide plateauThis plateau is due to the flat, or nearly flat, regions of their loss that renders them mostly dependent only on whether samples are well classified or not..", "However, for sparse models ($\\ell _1$ models), the maximum of the tuning curve is a more narrow peak, particularly so for SVM.", "A narrow peak in a tuning curve implies that a choice of optimal parameter may not alway carry over to the validation set.", "Figure: Prediction accuracy: impact of the parameter-tuning strategy.", "For each strategy, difference to the mean predictionaccuracy in a given validation split." ], [ "Impact of parameter tuning on prediction accuracy", "Cross-validation is often used to select regularization hyper-parameters, eg to control the amount of sparsity.", "On figure REF , we compare the various strategies: refitting with the best parameters selected by nested cross-validation, averaging the best models in the nested cross-validation, or simply using either the default value of C or a large one, given that tuning curves can plateau for large C. For non-sparse models, the figure shows that tuning the hyper-parameter by nested cross validation does not lead in general to better prediction performance than a default choice of hyper-parameter.", "Detailed investigations (figure REF ) show that these conclusions hold well across all tasks, though refitting after nested cross-validation is beneficial for good prediction accuracies, ie when there is either a large signal-to-noise ratio or many samples.", "For sparse models, the picture is slightly different.", "Indeed, high values of C lead to poor performance –presumably as the models are overly sparse–, while using default value $C=1$ , refitting or averaging models tuned by cross-validation all perform well.", "Investigating how these compromises vary as a function of model accuracy (figure REF ) reveals that for difficult decoding situations (low prediction) it is preferable to use the default $C=1$ , while in good decoding situations, in is beneficial to tune C by nested cross-validation and rely on model averaging, which tends to perform well and displays less variance." ], [ "Impact of parameter tuning on stability", "The choice of regularization parameter also affects the stability of the weight maps of the classifier.", "Strongly regularized maps underfit, thus depending less on the train data, which may lead to increased stability.", "We measure stability of the decoder maps by computing their correlation across different choices of validation split for a given task.", "Figure REF summarizes the results on stability.", "For all models, sparse and non-sparse, model averaging does give more stable maps, followed by refitting after choosing parameters by nested cross-validation.", "Sparse models are much less stable than non-sparse ones [57]." ], [ "Prediction power – stability tradeoff", "The choice of decoder with the best predictive performance might not give the most stable weight maps, as seen by comparing figures REF and REF .", "Figure REF shows the prediction–stability tradeoff for different decoders and different parameter-tuning strategies.", "Overall, SVM and logistic-regression perform similarly and the dominant effect is that of the regularization: non-sparse, $\\ell _2$ -penalized, models are much more stable than sparse, $\\ell _1$ -penalized, models.", "For non-sparse models, averaging stands out as giving a large gain in stability albeit with a decrease of a couple of percent in prediction accuracy compared to using $C=1$ or $C=1000$ , which gives good prediction and stability (figures REF and REF ).", "With sparse models, averaging offers a slight edge for stability and, for SVM performs also well in prediction.", "$C=1000$ achieves low stability (figure REF ), low prediction power (figure REF ), and a poor tradeoff.", "REF shows trends on datasets where the prediction is easy or not.", "For non-sparse models, averaging brings a larger gain in stability when prediction accuracy is large.", "Conversely, for sparse models, it is more beneficial to average in case of poor prediction accuracy.", "Note that these experimental results are for common neuroimaging settings, with variance-normalization and univariate feature screening." ], [ "Discussion and conclusion: lessons learned", "Decoding seeks to establish a predictive link between brain images and behavioral or phenotypical variables.", "Prediction is intrinsically a notion related to new data, and therefore it is hard to measure.", "Cross-validation is the tool of choice to assess performance of a decoder and to tune its parameters.", "The strength of cross-validation is that it relies on few assumptions and probes directly the ability to predict, unlike other model-selection procedures –eg based on information theoretic or Bayesian criteria.", "However, it is limited by the small sample sizes typically available in neuroimagingThere is a trend in inter-subject analysis to acquire databases with a larger number of subjects, eg ADNI, HCP.", "Conclusions of our empirical study might not readily transfer to these settings.." ], [ "An imprecise assessment of prediction", "The imprecision on the estimation of decoder performance by cross-validation is often underestimated.", "Empirical confidence intervals of cross-validated accuracy measures typically extend more than 10 points up and down (figure REF and REF ).", "Experiments on MRI (anatomical and functional), MEG, and simulations consistently exhibit these large error bars due to data scarcity.", "Such limitations should be kept in mind for many MVPA practices that use predictive power as a form of hypothesis testing –eg searchlight [28] or testing for generalization [26]– and it is recommended to use permutation to define the null hypothesis [28].", "In addition, in the light of cross-validation variance, methods publications should use several datasets to validate a new model." ], [ "Guidelines on cross-validation", "Leave-one-out cross-validation should be avoided, as it yields more variable results.", "Leaving out blocks of correlated observations, rather than individual observations, is crucial for non-biased estimates.", "Relying on repeated random splits with 20% of the data enables better estimates with less computation by increasing the number of cross-validations without shrinking the size of the test set." ], [ "Parameter tuning", "Selecting optimal parameters can improve prediction and change drastically the aspects of weight maps (Fig.", "REF ).", "However, our empirical study shows that for variance-normalized neuroimaging data, non-sparse decoders ($\\ell _2$ -penalized) are only weakly sensitive to the choice of their parameter, particularly for the SVM.", "As a result, relying on the default value of the parameter often outperforms parameter tuning by nested cross-validation.", "Yet, such parameter tuning tends to improve the stability of the maps.", "For sparse decoders ($\\ell _1$ -penalized), default parameters also give good prediction performance.", "However, parameter tuning with model averaging increases stability and can lead to better prediction.", "Note that it is often useful to variance normalize the data (see )." ], [ "Concluding remarks", "Evaluating a decoder is hard.", "Cross-validation should not be considered as a silver bullet.", "Neither should prediction performance be the only metric.", "To assess decoding accuracy, best practice is to use repeated learning-testing with 20% of the data left out, while keeping in mind the large variance of the procedure.", "Any parameter tuning should be performed in nested cross-validation, to limit optimistic biases.", "Given the variance that arises from small samples, the choice of decoders and their parameters should be guided by several datasets.", "Our extensive empirical validation (31 decoding tasks, with 8 datasets and almost 1 000 validation splits with nested cross-validation) shows that sparse models, in particular $\\ell _1$ SVM with model averaging, give better prediction but worst weight-maps stability than non-sparse classifiers.", "If stability of weight maps is important, non-sparse SVM with $C=1$ appears to be a good choice.", "Further work calls for empirical studies of decoder performance with more datasets, to reveal factors of the dataset that could guide better the choice of a decoder for a given task." ], [ "Acknowledgments", "This work was supported by the EU FP7/2007-2013 under grant agreement no.", "604102 (HBP).", "Computing resource were provided by the NiConnect project (ANR-11-BINF-0004_NiConnect) and an Amazon Webservices Research Grant.", "The authors would like to thank the developers of nilearnhttps://github.com/nilearn/nilearn/graphs/contributors, scikit-learnhttps://github.com/scikit-learn/scikit-learn/graphs/contributors and MNE-Pythonhttps://github.com/mne-tools/mne-python/graphs/contributors for continuous efforts in producing high-quality tools crucial for this work.", "In addition, we acknowledge useful feedback from Russ Poldrack on the manuscript." ], [ "Dataset simulation", "We generate data with samples from two classes, each described by a Gaussian of identity covariance in 100 dimensions.", "The classes are centered respectively on vectors $(\\mu , \\dots , \\mu )$ and $(-\\mu , \\dots , -\\mu )$ where $\\mu $ is a parameter adjusted to control the separability of the classes.", "With larger $\\mu $ the expected predictive accuracy would be higher.", "In addition, to mimic the time dependence in neuroimaging data we apply a Gaussian smoothing filter in the sample direction on the noise ($\\sigma = 2$ ).", "Code to reproduce the simulations can be found on https://github.com/GaelVaroquaux/cross_val_experiments.", "We produce different datasets with predefined separability by varyingthe values we explore for $\\mu $ were chosen empirically to vary classification accuracy from 60% to 90%.", "$\\mu $ in $(.05,\\;.1,\\; .2)$ .", "Figure REF shows two of these configurations.", "Figure: Simulated data for different levels of separabilitybetween the two classes (red and blue circles).", "Here, to simplifyvisualization, the data are generated in 2D (2 features), unlike the actualexperiments, which are performed on 100 features.Top: The feature space.Bottom: Time series of the first feature.", "Note that the noise iscorrelated timewise, hence successive data points show similar shifts." ], [ "Experiments: error varying separability", "Unlike with a brain imaging datasets, simulations open the door to measuring the actual prediction performance of a classifier, and therefore comparing it to the cross-validation measure.", "For this purpose, we generate a pseudo-experimental data with 200 train samples, and a separate very large test set, with 10 000 samples.", "The train samples correspond to the data available during a neuroimaging experiment, and we perform cross-validation on these.", "We apply the decoder on the test set.", "The large number of test samples provides a good measure of prediction power of the decoder [2].", "We use the same decoders as for brain-imaging data and repeat the whole procedure 100 times.", "For cross-validation strategies that rely on sample blocks –as with sessions–, we divide the data in 10 continuous blocks.", "Figure: Cross-validation measures on simulations.Prediction accuracy, as measured by cross-validation (box plots)and on a very large test set (vertical lines) for differentseparability on the simulated data and for differentcross-validation strategies: leave one sample out,leave one block of samples out (where the block is the naturalunit of the experiment: subject or session), orrandom splits leaving out 20% of the blocks as test data, with 3, 10, or50 random splits.The box gives the quartiles, while the whiskers givethe 5 and 95 percentiles.Note that here leave-one-block-out is similar of 10 splits of 10% ofthe data." ], [ "Results", "Figure REF summarizes the cross-validation measures for different values of separability.", "Beyond the effect of the cross-validation strategy observed on other figures, the effect of the separability, ie the true prediction accuracy is also visible.", "Setting aside the leave-one-sample-out cross-validation strategy, which is strongly biased by the correlations across the samples, we see that all strategies tend to be biased positively for low accuracy and negatively for high accuracy.", "This observation is in accordance with trends observed on figure REF ." ], [ "Comparing parameter-tuning strategies", "Figure REF shows pairwise comparisons of parameter-tuning strategies, in sparse and non-sparse situations, for the best-performing options.", "In particular, it investigates when different strategies should be preferred.", "The trends are small.", "Yet, it appears that for low predictive power, setting C=1 in non-sparse models is preferable to cross-validation while for high predictive power, cross-validation is as efficient.", "This is consistent with results in figure REF showing that cross-validation is more reliable to measure prediction error in situations with a good accuracy than in situations with a poor accuracy.", "Similar trends can be found when comparing to C=1000.", "For sparse models, model averaging can be preferable to refitting.", "We however find that for low prediction accuracy it is favorable to use C=1, in particular for logistic regression.", "Note these figures show points related to different studies and classification tasks.", "The trends observed are fairly homogeneous and there are not regions of the diagram that stand out.", "Hence, the various conclusions on the comparison of decoding strategies are driven by all studies." ], [ "Results without variance-normalization", "Results without variance normalization of the voxels are given in figure REF for the correspondence between error measured in the inner cross-validation loop, figure REF for the effect of the choice of a parameter-tuning strategy on the prediction performance, and figure REF for the effect on the stability of the weights.", "Cross-validation on non variance-normalized neuroimaging data is not more reliable than on variance-normalized data (figure REF ).", "However, parameter tuning by nested cross-validation is more important than on variance-normalized data for good prediction (figure REF ).", "This difference can be explained by the fact that variance normalizing makes dataset more comparable to each others, and thus a default value of parameters is more likely to work well.", "In conclusion, variance normalizing the data can be important, in particular with non-sparse SVM.", "Figure: Impact of the parameter-tuning strategy on stability ofweights without feature standardization: for each strategy, difference to the mean stability ofthe model weights across validation splits." ], [ "Details on stability–prediction results", "Figure REF summarizes the effects of the decoding strategy on the prediction – stability tradeoff.", "On figure REF , we give the data points that underly this summary." ], [ "Prediction and stability interactions", "Figure REF captures the effects of the decoding strategy across all datasets.", "However, some classification tasks are easier or more stable than others.", "We give an additional figure, figure REF , showing this interaction between classification performance and the best decoding strategy in terms of weight stability.", "The main factor of variation of the prediction accuracy is the choice of dataset, ie the difficulty of the prediction task.", "Here again, we see that the most important choice is that of the penalty: logistic regression and SVM have overall the same behavior.", "In terms of stability of the weights, higher prediction accuracy does correspond to more stability, except for overly-penalized sparse model (C=1000).", "For non-sparse models, model averaging after cross-validation is particularly beneficial in good prediction situations." ], [ "Details on datasets used", "Table REF lists all the studies used in our experiments, as well as the specific prediction tasks.", "In the Haxby dataset [19] we use various pairs of visual stimuli, with differing difficulty.", "We excluded pairs for which decoding was unsuccessful, such as scissors versus bottle)." ], [ "Intra-subject prediction", "For intra-subject prediction, we use the Haxby dataset [19] as provided from the PyMVPA [17] website –http://dev.pymvpa.org/datadb/haxby2001.html.", "Details of the preprocessing are not given in the original paper, beyond the fact that no spatial smoothing was performed.", "We have not performed additional preprocessing on top of this publicly-available dataset, aside from spatial smoothing with an isotropic Gaussian kernel, FWHM of 6 mm (nilearn 0.2, Python 2.7)." ], [ "Inter-subject prediction", "For inter-subject prediction, we use different datasets available on openfMRI [47].", "For all the datasets, we performed standard preprocessing with SPM8Wellcome Department of Cognitive Neurology, http://www.fil.ion.ucl.ac.uk/spm: in the following order, slice-time correction, motion correction (realign), corregistration of mean EPI on subject's T1 image, and normalization to template space with unified segmentation on the T1 image.", "The preprocessing pipeline was orchestrated through the Nipype processing infrastructure [12].", "For each subject, we then performed session-level GLM with a design according to the individual studies, as described in the openfMRI files, using Nipy (version 0.3, Python version 2.7) [37]." ], [ "Structural MR data", "For prediction from structural MR data, we perform Voxel Based Morphometry (VBM) on the Oasis dataset [34].", "We use SPM8 with the following steps: segmentation of the white matter/grey matter/CSF compartiments and estimation of the deformation fields with DARTEL [3].", "The inputs for predictive models is the modulated grey-matter intensity.", "The corresponding maps can be downloaded with the dataset-downloading facilities of the nilearn software (function nilearn.datasets.fetch_oasis_vbm)." ], [ "MEG data", "The magneteoencephalography (MEG) data is from an N-back working-memory experiment made available by the Human Connectome Project [33].", "Data from 52 subjects and two runs was analyzed using a temporal window approach in which all magnetic fields sampled by the sensor array in a fixed time interval yield one variable set (see for example [54] for event related potentials in electroencephalography).", "Here, each of the two runs served as validation set for the other run.", "For consistency, two-class decoding problems were considered, focussing on either the image content (faces VS tools) or the functional role in the working memory task (target VS low-level and high-level distractors).", "This yielded in total four classification analyses per subject.", "For each trial, the time window was then constrained to 50 millisecond before and 300 millisecond after event onset, emphasizing visual components.", "All analyses were based on the cleaned single-trial outputs obtained from the HCP “tmegpreproc” pipeline which provides cleaned segmented sensor space data.", "The MEG data that were recorded with a wholehead MAGNES 3600 (4D Neuroimaging, San Diego, CA) magnetometer system in a magnetically shielded room.", "Contamination by environmental magnetic fields was accounted for by computing the residual MEG signal from concomitant recordings of reference gradiometers and magnetometers located remotely from the main sensor array.", "Data were bandpass filtered between 1.3 and 150Hz using zero-phase forward and everse Butterworth filters.", "Notch filters were then applied at (59-61/119-121 Hz) to attenuate line noise artefacts.", "Data segments contaminated by remaining environmental or system artifacts were detected using a semi-automatic HCP pipeline that takes into account the local and global variation as well as the correlation structure of the data.", "Independent component analysis based on the FastICA algorithm was then used to estimate and suppress spatial patterns of cardiac and ocular artifacts.", "Artifact related components were identified in a semi-automatic fashion assisted by comparisons with concomitantly recorded electrocardiogram (ECG) and electrooculogram (EOG).", "These components were then projected out from the data.", "Depending on the classification of bad channels performed by the HCP pipelines, the data contained fewer than 248 sensors.", "For details on the HCP pipelines see [33] and the HCP reference manual.", "The MEG data were accessed through the MNE-Python software [13], [14] and the MNE-HCP library ." ], [ "Performance on each classification task", "The prediction accuracy results presented in the various figures are differential effects removing the contribution of the dataset.", "In figure REF , we present for each decoding strategy the prediction accuracy on all datasets.", "We can see that the variations of prediction accuracy from one decoding strategy to another are mostly reported across datasets: the various lines are roughly parallel.", "One notable exception is SVM $\\ell _1$ with $C=1000$ for which some datasets show a strong decrease.", "Another, weaker, variation is the fact that $\\ell _1$ models tend to perform better on the Haxby dataset (our source of intra-subject classification tasks).", "This good performance of sparse models could be due to the intra-subject settings: sparse maps are less robust to inter-subject variability.", "However, the core messages of the paper relative to which parameter-tuning strategy to use are applicable to intra and inter-subject settings.", "For non-sparse models, using a large value of C without parameter tuning is an overall safe choice, and for sparse models, model averaging, refitting, and a choice of $C=1$ do not offer a clear win, although model averaging is comparatively less variable." ] ]	1606.05201
[ [ "A New Three-Dimensional Track Fit with Multiple Scattering" ], [ "Abstract Modern semiconductor detectors allow for charged particle tracking with ever increasing position resolution.", "Due to the reduction of the spatial hit uncertainties, multiple Coulomb scattering in the detector layers becomes the dominant source for tracking uncertainties.", "In this case long distance effects can be ignored for the momentum measurement, and the track fit can consequently be formulated as a sum of independent fits to hit triplets.", "In this paper we present an analytical solution for a three-dimensional triplet(s) fit in a homogeneous magnetic field based on a multiple scattering model.", "Track fitting of hit triplets is performed using a linearization ansatz.", "The momentum resolution is discussed for a typical spectrometer setup.", "Furthermore the track fit is compared with other track fits for two different pixel detector geometries, namely the Mu3e experiment at PSI and a typical high-energy collider experiment.", "For a large momentum range the triplets fit provides a significantly better performance than a single helix fit.", "The triplets fit is fast and can easily be parallelized, which makes it ideal for the implementation on parallel computing architectures." ], [ "Motivation", "The trajectory of a free charged particle in a homogeneous magnetic field is described by a helix.", "The non-linear nature of the helix makes the reconstruction of the three-dimensional trajectory from tracking detector hits one of the main computational challenges in particle physics.", "To simplify the problem, the reconstruction is often factorized into a two-dimensional circle fit in the plane transverse to the magnetic field and a two-dimensional straight line fit in the longitudinal planeIn the right-handed coordinate system we define the B-field orientation along the $z$ -axis; the azimuthal angle $\\varphi $ is defined in the transverse $x$ -$y$ plane and the polar angle $\\vartheta $ is defined in the longitudinal $z$ -$s$ plane where $s$ is the track length parameter.. A non-iterative solution to this problem was given by Karimäki [1].", "This simplified treatment however does not make full use of the available detector information and ignores correlations between the two planes, which can be large especially for small helix radii (low momentum particles) at small (large) polar angles $\\vartheta \\approx 0~ (\\pi )$ .", "A further complication of the track reconstruction problem is the treatment of multiple Coulomb scattering (MS) in the detector material, which introduces correlations between the measurement points.", "This problem is addressed by Kálmán filters [2], [3], [4] and broken line fits [5], [6], [7] which both give a correct description of the track parameter error matrix.", "The methods however require computationally expensive matrix inversions and potentially multiple passes.", "In modern semiconductor pixel trackers, extremely precise three-dimensional position information is available and tracking uncertainties are dominated by MS except at the very highest momenta.", "Usually most of the material causing the scattering is located in the sensors or very close to them (services, cooling, mechanics etc.", "); therefore the scattering planes usually coincide with the detection planes.", "This is our motivation for developing a new three-dimensional helix fit which treats MS in the detector as the only uncertainty.", "The resulting algorithm is based on triplets of hits which can be fit in parallel.", "The final result is then obtained by combining all triplets.", "The algorithm is computationally efficient and well suited for track finding.", "The first application of the algorithm is the all-pixel silicon tracker [8] of the Mu3e experiment [9]." ], [ "Triplet Track Fit", "The basic unit of the track fit is a triplet of hits in successive detector layers.", "In the absence of MS and energy losses, the description of a helix through three points requires eight parameters, namely a starting point (three parameters), an initial direction (two parameters), the curvature (one parameter) and the distances to the second and third point (two parameters).", "MS in the central plane requires two additional parameters to describe the change in track directionTwo more parameters, describing a possible position offset at the central plane due to MS inside the material, can be ignored for typical silicon trackers, where the sensor thicknesses are much smaller than the distances between the detector layers.. Three space points, which we assume to be measured without uncertainties, do however only provide a total of nine coordinates; additional constraints are thus needed to obtain the track parameters and scattering angles.", "These constraints can be obtained from MS theory since the scattering angles depend statistically on the particle type and momentum, and the material of the detector.", "Starting from a hit triplet, see Figure REF , a trajectory consisting of two arcs connecting the three-dimensional space points is constructed.", "It is assumed that the middle point $\\mathbf {x_1}$ lies in a scattering plane which deflects the particle and thus creates a kink in the trajectory.", "The corresponding scattering angles in the transverse and longitudinal plane are denoted by $\\Phi _{MS}$ and $\\Theta _{MS}$ respectively.", "We assume that the particle momentum (and thus its three-dimensional radius $R_{3D}$ ) is conservedEnergy loss due to ionization is usually small and can be either neglected or corrected for..", "The scattering angles $\\Phi _{MS}$ and $\\Theta _{MS}$ have a mean of zero and variances $\\sigma _\\vartheta ^2 = \\sigma _{MS}^2$ and $\\sigma _\\phi ^2 = \\sigma _{MS}^2 / \\sin ^2\\vartheta $ , which can be calculated from MS theory, using e.g.", "the Highland approximation [10].", "The task is thus to find a unique $R_{3D}$ which minimizes the scattering angles, explicitly the following $\\chi ^2$ function: $\\chi ^2(R_{3D}) = \\frac{\\Phi _{MS}(R_{3D})^2}{\\sigma _\\phi ^2} +\\frac{\\Theta _{MS}(R_{3D})^2}{\\sigma _\\vartheta ^2} \\ .$ For weak MS effects the momentum dependence of the scattering uncertainty is negligible; the case of large MS effects is discussed in more detail in section REF .", "Assuming $\\frac{d \\sigma _{MS}}{dR_{3D}}=0$ , the minimization of $\\chi ^2(R_{3D})$ is thus equivalent to solving the equation $\\sin ^2\\vartheta \\; \\frac{d \\Phi _{MS}(R_{3D})}{dR_{3D}} \\;\\Phi _{MS}(R_{3D})\\; + \\; \\frac{d \\Theta _{MS}(R_{3D})}{dR_{3D}} \\;\\Theta _{MS}(R_{3D}) = 0$ for $R_{3D}$ .", "The scattering angle in the transverse plane $\\Phi _{MS}$ is given by $\\Phi _{MS} = (\\varphi _{12} - \\varphi _{01}) - \\frac{\\Phi _1(R_{3D}) +\\Phi _2(R_{3D})}{2} \\ ,$ where the bending angles $\\Phi _1$ and $\\Phi _2$ are the solutions of the transcendent equations $\\sin ^2 \\frac{\\Phi _1}{2} &= \\frac{d_{01}^2}{4 R_{3D}^2} +\\frac{z_{01}^2}{R_{3D}^2} \\frac{\\sin ^2 \\frac{\\Phi _1}{2}}{\\Phi _1^2} \\ , \\nonumber \\\\\\sin ^2 \\frac{\\Phi _2}{2} &= \\frac{d_{12}^2}{4 R_{3D}^2} +\\frac{z_{12}^2}{R_{3D}^2} \\frac{\\sin ^2 \\frac{\\Phi _2}{2}}{\\Phi _2^2} \\ .", "$ These equations have several solutions depending on the number of half-turns of the track.", "However, for most practical cases it is sufficient to consider the first two solutions.", "Similarly, the scattering angle in the longitudinal plane is given by $\\Theta _{MS} = \\vartheta _2 -\\vartheta _1$ where the polar angles $\\vartheta _1$ and $\\vartheta _2$ can be calculated from the azimuthal bending angles using the relations $\\sin \\vartheta _1 &= \\frac{d_{01}}{2 R_{3D}} \\mathrm {cosec} \\left(\\frac{z_{01}}{2R_{3D} \\cos \\vartheta _1} \\right) \\ , \\nonumber \\\\\\sin \\vartheta _2 &= \\frac{d_{12}}{2 R_{3D}} \\mathrm {cosec} \\left(\\frac{z_{12}}{2R_{3D} \\cos \\vartheta _2} \\right) \\ .$ Alternatively the relations $\\Phi _1 &= \\frac{z_{01}}{R_{3D}\\cos \\vartheta _1} \\ , \\nonumber \\\\\\Phi _2 &= \\frac{z_{12}}{R_{3D}\\cos \\vartheta _2} $ between the azimuthal bending angles and the polar angles can be exploited.", "Equations REF and REF have no algebraic solutions; they can either be solved by numerical iteration or by using a linearization around an approximate solution; the second approach is discussed in the following." ], [ "Taylor expansion around the circle solution", "The circle solution describes the case of constant curvature in the plane transverse to the magnetic field $r_1 = r_2$ and no scattering in that plane, $\\Phi _{MS} = 0$ .", "This solution exists for any hit triplet and is thus a good starting point for the linearization.", "The radius $R_C$ of the circle in the transverse plane going through three points is given by $R_C = \\frac{d_{01} \\; d_{12} \\; d_{02}}{2 \\;[(\\mathbf {x_1}-\\mathbf {x_0}) \\times (\\mathbf {x_2}-\\mathbf {x_1})]_z},$ where $d_{ij}$ is the transverse distance between the hits $i$ and $j$ of the triplet, see Figure REF .", "The bending angles for the circle solution are $\\Phi _{1C} &= 2 \\arcsin \\frac{d_{01}}{2 R_C} \\ , \\nonumber \\\\\\Phi _{2C} &= 2 \\arcsin \\frac{d_{12}}{2 R_C} \\ .$ Note that the above equations have in general two solutions ($\\Phi _{iC} < \\pi $ and $\\Phi _{iC} > \\pi $ ) and care is needed to select the physical one, especially for highly bent tracks.", "The corresponding three-dimensional radii of the arcs are calculated as $R_{3D,1C}^2 &= R_C^2 +\\frac{z_{01}^2}{\\Phi _{1C}^2} \\ , \\nonumber \\\\R_{3D,2C}^2 &= R_C^2 +\\frac{z_{12}^2}{\\Phi _{2C}^2} \\ .$ In general $\\Theta _{MS}\\ne 0$ such that the two radii are not identical.", "Using equation REF , polar angles for the circle solution are obtained: $\\vartheta _{1C} &= \\arccos \\frac{z_{01}}{\\Phi _{1C} R_{3D,1C}} \\ , \\nonumber \\\\\\vartheta _{2C} &= \\arccos \\frac{z_{12}}{\\Phi _{2C} R_{3D,2C}} \\ .$ Starting from this special circle solution with no scattering in the transverse plane, we calculate the general solution $\\Phi _{MS} \\ne 0$ which minimizes equation REF and for which momentum conservation is fulfilled, i.e.", "$R_{3D}$ does not change between the segments.", "With the positions of the three hits given, the arc lengths and the polar angles depend only on the radius, i.e.", "$\\Phi _{1,2}=\\Phi _{1,2}(R_{3D})$ and $\\vartheta _{1,2}=\\vartheta _{1,2}(R_{3D})$ (equations REF and REF ).", "We can therefore perform a Taylor expansion to first order around the circle solution which is described by the parameters $R_{3D,1C}$ , $R_{3D,2C}$ , $\\Phi _{1C}$ , $\\Phi _{2C}$ , $\\vartheta _{1C}$ and $\\vartheta _{2C}$ : $\\Phi _{1}(R_{3D}) &\\approx \\Phi _{1C} + (R_{3D} - R_{3D,1C})\\left.\\frac{\\operatorname{d}\\!\\Phi _1}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{1C}}, \\nonumber \\\\\\Phi _{2}(R_{3D}) &\\approx \\Phi _{2C} + (R_{3D} - R_{3D,2C})\\left.\\frac{\\operatorname{d}\\!\\Phi _2}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{2C}}$ and $\\vartheta _{1}(R_{3D}) &\\approx \\vartheta _{1C} + (R_{3D} - R_{3D,1C})\\left.\\frac{\\operatorname{d}\\!\\vartheta _1}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\vartheta _{1C}} \\ , \\nonumber \\\\\\vartheta _{2}(R_{3D}) &\\approx \\vartheta _{2C} + (R_{3D} - R_{3D,2C})\\left.\\frac{\\operatorname{d}\\!\\vartheta _2}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\vartheta _{2C}} \\ .$ The derivatives $\\left.\\frac{\\operatorname{d}\\!\\Phi _1}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{1C}}$ and $\\left.\\frac{\\operatorname{d}\\!\\Phi _2}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{2C}}$ can be represented by index parameters: $\\left.\\frac{\\operatorname{d}\\!\\Phi _1}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{1C}} &= - \\alpha _1 \\frac{\\Phi _{1C}}{R_{3D,1C}} \\ , \\nonumber \\\\\\left.\\frac{\\operatorname{d}\\!\\Phi _2}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\Phi _{2C}} &= - \\alpha _2 \\frac{\\Phi _{2C}}{R_{3D,2C}} \\ ,$ which are calculated from equation REF as $\\alpha _1 & = \\frac{R_C^2 \\Phi _{1C}^2 +z_{01}^2}{\\frac{1}{2}R_C^2 \\Phi _{1C}^3 \\cot \\frac{\\Phi _{1C}}{2} + z_{01}^2} \\ , \\nonumber \\\\\\alpha _2 & = \\frac{R_C^2 \\Phi _{2C}^2 +z_{12}^2}{\\frac{1}{2}R_C^2 \\Phi _{2C}^3 \\cot \\frac{\\Phi _{2C}}{2} + z_{12}^2} \\ .$ The derivatives of the polar angles are obtained from equation REF and can be expressed by the same index parameters: $\\left.\\frac{\\operatorname{d}\\!\\vartheta _1}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\vartheta _{1C}} = \\frac{\\cot \\vartheta _{1C}}{R_{3D,1C}} (1-\\alpha _1), \\nonumber \\\\\\left.\\frac{\\operatorname{d}\\!\\vartheta _2}{\\operatorname{d}\\!R_{3D}}\\right\|_{\\vartheta _{2C}} = \\frac{\\cot \\vartheta _{2C}}{R_{3D,2C}} (1-\\alpha _2).$" ], [ "Linearization of the scattering angles", "The above relations can now be used to calculate a linearized expression for the MS angles.", "For $\\Phi _{MS}$ we obtain: $\\Phi _{MS} &= \\varphi _{12} - \\varphi _{01} - \\frac{\\Phi _1(R_{3D})}{2} - \\frac{\\Phi _2(R_{3D})}{2} \\nonumber \\\\&= \\tilde{\\Phi } + {\\eta }\\;R_{3D} \\ ,$ where we have introduced two new parameters: $\\tilde{\\Phi } &= - \\frac{1}{2}(\\Phi _{1C} \\alpha _1 + \\Phi _{2C} \\alpha _2) \\ ,\\\\{\\eta } &= \\frac{\\operatorname{d}\\!\\Phi _{MS}}{\\operatorname{d}\\!R_{3D}} = \\frac{\\Phi _{1C}\\; \\alpha _1}{2 R_{3D,1C}} + \\frac{\\Phi _{2C}\\; \\alpha _2}{2 R_{3D,2C}} \\ .$ And similarly for the polar angle $\\Theta _{MS}$ we obtain: $\\Theta _{MS} & = \\vartheta _2 - \\vartheta _1 \\nonumber \\\\& = \\tilde{\\Theta } +{\\beta }\\; R_{3D} \\ ,$ with the new parameters $\\tilde{\\Theta } & = \\vartheta _{2C} - \\vartheta _{1C} - \\Bigl (\\left(1-\\alpha _2\\right)\\cot \\vartheta _{2C} - \\left(1-\\alpha _1\\right)\\cot \\vartheta _{1C}\\Bigr ) \\ ,\\\\{\\beta } &= \\frac{\\operatorname{d}\\!\\Theta _{MS}}{\\operatorname{d}\\!R_{3D}} = \\frac{(1-\\alpha _2)\\cot \\vartheta _{2C}}{R_{3D,2C}} - \\frac{(1-\\alpha _1)\\cot \\vartheta _{1C}}{R_{3D,1C}} \\ .$" ], [ "Linearized triplet track fit", "We can now minimize the $\\chi ^2$ -function by inserting the derivatives and the expressions for the scattering angles obtained from the linearization in equation REF .", "For the three-dimensional radius we obtain $R_{3D}^{min} = - \\frac{{\\eta }\\;\\tilde{\\Phi }\\;\\sin ^2\\vartheta +{\\beta }\\;\\tilde{\\Theta }}{{\\eta }^2\\sin ^2\\vartheta + {\\beta }^2} \\ .$ Here is $\\vartheta $ the polar angle at the scattering layer, which can be taken as the average of $\\vartheta _{1C}$ and $\\vartheta _{2C}$ .", "The minimum $\\chi ^2$ value is $\\chi ^{2}_{min} = \\frac{1}{\\sigma _{MS}^2}\\frac{({\\beta }\\;\\tilde{\\Phi }- {\\eta }\\;\\tilde{\\Theta })^2}{{\\eta }^2 + {\\beta }^2/\\sin ^2\\vartheta }$ and for the uncertainty of the three-dimensional radius we get $\\sigma (R_{3D}) = \\sigma _{MS}\\sqrt{\\frac{1}{{\\eta }^2\\sin ^2\\vartheta +{\\beta }^2}} \\ .$ The scattering angles are finally given by: $\\Phi _{MS} &= {\\beta } \\ \\frac{{\\beta } \\; \\tilde{\\Phi } - {\\eta } \\;\\tilde{\\Theta } }{{\\eta }^2\\sin ^2\\vartheta +{\\beta }^2} \\ , \\\\\\Theta _{MS} &= - {\\eta } \\sin ^2\\vartheta \\ \\frac{{\\beta } \\; \\tilde{\\Phi } -{\\eta } \\; \\tilde{\\Theta }}{{\\eta }^2\\sin ^2\\vartheta +{\\beta }^2} \\ .$ It is straight-forward to calculate further track parameters using the linearization described above.", "Note that in this approach the fitted track parameters are independent of the momentum and the MS uncertainty.", "The latter can be calculated after fitting the track parameters which allows for an elegant treatment of the material effects.", "We have thus obtained a non-iterative solution to the triplet problem with multiple scattering." ], [ "Strong Multiple Scattering and Weak Bending", "The regime where the MS uncertainty is of similar size as the sum of the bending angles, $\\sigma _{MS} \\approx \\Phi _1+\\Phi _2$ , we define as strong MS or weak bending.", "This corresponds to cases with either a large amount of material at the scattering layer or weak magnetic field strength.", "In this regime the momentum dependence of the scattering uncertainty leads to a systematic shift (bias) of the fitted radius towards larger values.", "This bias can be compensated by including the momentum dependence in the minimization of the $\\chi ^2$ function, given in equation REF , using the ansatz $\\sigma _{MS} = b/R_{3D}$ which is motivated by the Highland formula [10].", "Here $b$ is an effective scattering parameter which is approximately given by $b \\approx \\frac{4.5~\\textrm {cm T}}{B} \\; \\sqrt{X/X_0}$ and assumed to vary only weakly within the parameter range of the fit.", "The so obtained unbiased result $R_{3D}^{unbiased} = - \\; \\frac{{\\eta } \\; \\tilde{\\Phi } \\; \\sin ^2{\\vartheta } + {\\beta }\\; \\tilde{\\Theta } }{{\\eta }^2 \\; \\sin ^2{\\vartheta } + {\\beta }^2}\\ \\left(\\frac{3}{4} + \\; \\frac{\\sqrt{1-8 \\; \\delta ^2 \\sin ^2{\\vartheta } }}{4}\\right)$ with $\\delta = \\frac{{\\beta } \\; \\tilde{\\Phi } - {\\eta } \\; \\tilde{\\Theta }}{{\\eta } \\; \\tilde{\\Phi } \\; \\sin ^2{\\vartheta } + {\\beta }\\; \\tilde{\\Theta } }$ has only a solution if $8 \\; \\delta ^2 \\sin ^2{\\vartheta }\\le 1 \\ .$ For small bias parameters, $\\delta \\approx 0$ , equation REF is restored.", "The bias parameter $\\delta $ is just given by the hit triplet geometry but it can also be expressed by fitted parameters: $\\delta ^2 \\sin ^2{\\vartheta }\\ = \\ \\frac{\\sigma (R_{3D}^{min})^2}{{R_{3D}^{min}}^2} \\; \\chi ^{2}_{min}$ using equations REF , REF and REF .", "The bias term is thus proportional to the sum of the squared scattering angles: $\\delta ^2 \\sin ^2{\\vartheta }\\ \\propto \\ \\Phi _{MS}^2 + \\Theta _{MS}^2 \\sin ^2{\\vartheta } \\ .$ From equation REF a condition on the minimum significance for the radius measurement can be derived: $\\frac{R_{3D}}{\\sigma (R_{3D})} > \\sqrt{8 \\; \\chi ^2} \\ .$ If the radius significance is not large enough, $\\frac{R_{3D}}{\\sigma (R_{3D})} \\lesssim 10$ , significant bias corrections apply.", "For small bending angles (weak bending region!)", "the relation $\|{\\beta }\| \\ll \|{\\eta }\|$ holds and the relative resolution of the three-dimensional radius (momentum) is approximately given by $\\frac{\\sigma (R_{3D})}{R_{3D}}= \\frac{\\sigma (p)}{p} = \\frac{2 \\;b}{s}\\ .$ We can then rewrite equation REF as $s^2 > 32 \\; b^2 \\chi ^2$ where $s=s_{01}+s_{12}$ defines the length of the triplet trajectory.", "This relation should be respected, for example in the design of detectors, to allow for a decent momentum measurement." ], [ "Example Spectrometer", "The resolution of the triplet track fit is investigated for a simple spectrometer configuration with three detector layers for which the spatial hit uncertainties are negligible.", "The first two layers are spaced closely together and the third layer is placed further apart, i.e.", "we assume $\\Phi _1 \\ll \\Phi _2$ for the sweep angles defined in fig:triplet.", "The relative momentum resolution is then calculated using the previously derived expressions for the fitted radius and associated variances as follows: $\\frac{\\sigma _{R_{3D}}}{{R_{3D}}} =2 \\; \\sigma _{MS} \\; \\left( \\Phi _2^2 \\alpha _2^2 \\; \\sin ^2 \\vartheta _2 \\; + \\; 4 (1-\\alpha _2)^2 \\;\\cot ^2 \\vartheta _2 \\right)^{-\\frac{1}{2}},$ with $\\alpha _2$ given by: $\\alpha _2^{-1} & =& \\cos ^2{\\vartheta _2} \\; + \\; \\frac{\\Phi _2}{2}\\cot {\\left( \\frac{\\Phi _2}{2} \\right)} \\; \\sin ^2{\\vartheta _2} \\ .$ The resulting resolution as a function of $\\Phi _2$ and $\\vartheta _2$ is shown in figure REF .", "Note that for some special cases, if $1/\\alpha _2 \\rightarrow 0$ , the momentum resolution approaches zero.", "For transverse going tracks ($\\vartheta _2 = \\pi /2$ ) this is the case if $\\Phi _2 = \\pi $ (i.e.", "semi-circles).", "The geometry of track detectors in a regime where MS dominates can thus be optimized for an almost perfect measurement at certain specific momenta." ], [ "Combining Triplets", "Several triplets can be combined to form longer tracks.", "As MS in each sensor layer is independent of all other layers the following global $\\chi ^2$ function is minimized: $\\chi _{global}^2 = \\sum _{i}^{n_{hit}-2} \\chi ^2_i \\ ,$ where $\\chi _i$ is the minimization function for the $i$ -th triplet number previously defined in equation REF .", "The total number of triplets is given by $n_{hit}-2$ .", "The minimization of equation REF is equivalent to a weighted average of the resulting radii $R_{3D,i}$ of the individual fits: $\\overline{R_{3D}} = \\sum _{i}^{n_{hit}-2}\\frac{R_{3D,i}^3}{\\sigma (R_{3D,i})^2} / \\sum _{i}^{n_{hit}-2}\\frac{R_{3D,i}^2}{\\sigma (R_{3D,i})^2} \\ .$ with the corresponding uncertainty given by $\\sigma (\\overline{R_{3D}}) =\\frac{\\overline{R_{3D}}}{\\sqrt{\\sum _{i}^{n_{hit}-2}\\frac{R_{3D,i}^2}{\\sigma (R_{3D,i})^2}}} \\ .$ This averaging formula is free of any bias if unbiased three-dimensional radii from equation REF are used as input and if $\\frac{\\sigma (R_{3D,i})}{R_{3D,i}}$ is constant, which is a very good assumption for most cases.", "Fitting of multiple hits, $n_{hit} > 3$ , is performed in a three step procedure: first all triplets of consecutive hits are fitted individually, second a weighted mean of the three-dimensional track radii is calculated which is then used in the third step to recalculate all other track parameters.", "It is worth to note that the expected variance of the MS angle for each triplet only enters at the averaging step, where it can be calculated to very good accuracy from the locally fitted triplet track parameters.", "Effects from energy loss can also be incorporated at this step.", "We can now use the average radius $\\overline{R_{3D}}$ together with our linearization to obtain globally fitted (updated) values for the sweep angles $\\Phi _i$ and the polar angles $\\vartheta _i$ : $\\Phi _i^\\prime &= \\Phi _i -(\\overline{R_{3D}} - R_{3D,i})\\; \\frac{\\Phi _i}{R_{3D,i}}\\; \\alpha _i \\ ,\\\\\\vartheta _i^\\prime &= \\vartheta _i - (\\overline{R_{3D}} - R_{3D,i})\\; \\frac{\\cot \\vartheta _i}{R_{3D,i}} \\; (1 - \\alpha _i) \\ .$ Alternatively, $\\vartheta _i^\\prime $ can also be obtained from the relation $\\vartheta _i^\\prime = \\arccos \\frac{{\\eta } \\;z_{\\lbrace i-1,i\\rbrace }}{\\overline{R_{3D}}\\; \\Phi _i^\\prime }$ if the sweep angle is known.", "We have thus obtained a non-iterative solution to the MS problem which is especially suitable for implementation on massively parallel architectures such as graphics processors (GPUs) as the triplets can be fit in parallel." ], [ "Track Fit Comparisons", "To compare the performance of the triplets fit with other fit algorithms we simulate particle tracks in different detector geometries using a toy Monte Carlo.", "Tracks are then reconstructed using the triplets fit, a single helix fit [1], and the general broken lines (GBL) fit [7].", "For the comparison study we choose two exemplary geometries.", "Detector layers are modeled as cylindrical high resolution pixel sensors centered around the origin and aligned along the direction of the homogeneous magnetic field.", "Tracks are generated at the origin and propagated in the magnetic field to the detector layers.", "MS is simulated by smearing the track direction at each layer with kink angles drawn from a Gaussian distribution with a width according to the Highland formula [10], [11].", "The position resolution of the detector is simulated by smearing the registered hit positions with a Gaussian distribution along both sensitive directions.", "The triplets fit, which includes only MS uncertainties, is performed as described in the previous section.", "The single helix fit, which is another example for a direct track fit, only takes into account the spatial measurement uncertainties.", "Here, the transverse track parameters are obtained from the Karimäki circle fit [1] and the longitudinal parameters result from a linear regression to the points in the projected arclength-$z$ plane.", "The GBL fit is an extended track fit that takes into account both, scattering effects and spatial uncertainties.", "It has been shown [7] to be equivalent to the Kálmán filter [3] and uses a track model consistent with all simulated uncertainties.", "The GBL algorithm performs a linearized fit by varying positions and kink angles at selected points around a reference trajectory.", "This reference can be derived from any direct track fit.", "Here, we use the helix fit and the triplets fit for comparison.", "In the first case, the track parameters are refitted by introducing non-vanishing kink angles.", "In the second case, the kink angles are optimized by introducing residuals to the measured positions.", "In this study we use only one iteration and ideally, the GBL converges to the same optimized trajectory from either of the two initial estimates in that single step.", "For the comparison study, the fitted track parameters are calculated at the inner-most detector layer at which the track parameters are maximally uncorrelated.", "The first simulated configuration is the silicon pixel tracker of the Mu3e experiment [8].", "The geometry and example trajectories to illustrate the uncertainties are shown in figure REF .", "The detector is placed in a homogeneous magnetic field of $B={1}{T}$ , has four layers with radii at approximately 2.2;2.8;7.0;7.8, and is optimized for low momentum electrons in a momentum range of 1553c.", "The spatial resolution on the detector plane is $80/\\sqrt{12}$ and the layer thickness is 0.1 radiation lengths.", "The resulting parameter resolution for the four different fits is shown in figure REF .", "The triplets fit has a consistently better resolution than the single helix fit in all track parameters.", "The GBL-fit with the triplets as reference (GBL-T) shows no significant improvement at very small momentum ($\\lesssim $ 25c).", "In this region MS uncertainties dominate the track uncertainties.", "Above 25c the GBL-T fit allows for small improvements of the angular resolutions as spatial uncertainties start to contribute.", "The GBL-fit with the single helix reference (GBL-H) reproduces for $\\gtrsim $ 30c the result of GBL-T.", "Although GBL-H improves with respect to the single helix reference fit also for $\\lesssim $ 30c, the momentum and azimuthal angle resolution is worse than for triplets fit and GBL-T.", "In this region the single helix parameterization is not a good reference and linearization point for the GBL.", "With the large non-linearity of the strongly curved tracks the single GBL-H step is insufficient.", "The behaviour of the position resolution (not shown) follows the behaviour of the angle resolution, i.e.", "the GBL is slightly better than the triplets fit and the helix fit is significantly worse than other fits.", "To extend this study beyond the unique Mu3e configuration, a generic pixel tracker design similar to existing or planned trackers for high-energy collider detectors, e.g.", "ATLAS, CMS or ILC, is evaluated.", "It comprises five equidistant detector layers at radii between 40340 with a spatial resolution of $50/\\sqrt{12}$ and a sensor thickness corresponding to 2 of radiation length.", "Particles are simulated in the momentum range between 5005000c, a region where MS significantly contributes to the track uncertainties.", "The track parameter resolution of the four algorithms is shown in figure REF .", "All fits show a similar momentum resolution.", "At low momentum the triplets fit provides the better resolution, at higher momentum the single helix fit, with a crossover at around 3000c.", "The GBL-fits give the optimal resolution over the full range.", "The polar angle resolution of the triplets fit is constant, its value fully determined by the spatial hit uncertainties.", "Interestingly, the triplets fit gives a significantly better resolution than the single helix fit even at high momenta.", "For the GBL-fits an improvement of the polar angle resolution for $\\gtrsim $ 3000c is visible.", "The azimuthal angle resolution shows a similar crossover behaviour as the momentum resolution at about 3000c.", "Again both GBL-fits lead to an improvement of the resolution.", "But they show a small difference at low momentum.", "Interestingly, the first iteration step of GBL-H yields a better azimuthal resolution than GBL-T.After sufficient iteration the GBL-H resolution converges to the GBL-T resolution, which in turn agrees with the resolution of the triplets fit.", "Note that the position of the crossover point depends on the geometry, the material, and the spatial resolution of the detector.", "We have compared execution times and the number of floating point operations for several implementations of the triplets fit and the single helix fit.", "The number of cycles required varies greatly depending on how many geometric quantities are pre-calculated and cached and whether (and where) a covariance matrix is calculated.", "With ideal caching and no calculation of the covariance matrix, the triplets fit outperforms the single helix fit by almost a factor of 2; if all track parameters and the full covariance matrix is calculated at each hit position, the single helix fit (with its global covariance matrix) needs about a factor 2 (5) less cycles for three (eight) hits." ], [ "Conclusions", "We presented a new track fit algorithm, the triplets fit, that is only based on MS uncertainties to determine global momentum and local direction parameters.", "The triplets fit is motivated by the excellent position resolution of modern silicon pixel sensors which create track fitting problems with dominating MS uncertainties.", "Although developed initially for reconstructing very low momentum electrons in the Mu3e experiment the triplets fit exhibits good performance for pixel trackers at the high energy experiments at LHC where MS uncertainties dominate or significantly contribute up to around 10c.", "In this regime, the performance of the triplets fit is as good as for GBL-fits.", "The triplets fit enables a very fast computation of track parameters and provides a natural scheme for track finding and linking via the combination of single triplets.", "This makes the triplets fit ideally suited for fast online reconstruction, as a reference for extended track fits, and as fast algorithm for pattern recognition problems." ], [ "Acknowledgements", "N. Berger and A. Kozlinskiy would like to thank the Deutsche Forschungsgemeinschaft for support through an Emmy Noether grant and the PRISMA cluster of excellence at Johannes Gutenberg University Mainz.", "M. Kiehn acknowledges support by the International Max Planck Research School for Precision Tests of Fundamental symmetries." ] ]	1606.04990
[ [ "Physical aging in article page views" ], [ "Abstract Statistics of article page views is useful for measuring the impact of individual articles.", "Analyzing the temporal evolution of article page views, we find that article page views usually decay over time after reaching a peak, especially exhibiting relaxation with nonexponentiality.", "This finding suggests that relaxation in article page views resembles physical aging as frequently found in complex systems." ], [ "Introduction", "Nowadays, many scientific journals promote online publication, which is efficient in distributing research impact through web-based communities.", "Once a research article is published online on a journal website, the public can access the article immediately [1].", "Counting article page views is an alternative method to measure the impact of individual articles, instead of the journal impact factor [2], [3], [4], [5].", "Empirically, when an article receives attention, its daily page views will reach a peak and then should decay with time, which is similar to a physical relaxation process.", "A recent study raised the possibility of a universal decay pattern in article page views, based on observations from six different PLoS journals [1].", "However, it remains unclear what mathematical model is appropriate for statistics in article page views.", "Physical aging is the spontaneous temporal evolution of out-of-equilibrium systems [6].", "Glasses, for instance, usually show relaxation toward equilibrium, which is commonly referred to as physical aging [7].", "Nonexponential relaxation is ubiquitous in complex systems.", "Nonexponentiality in relaxation is frequently described by the stretched exponential [8], [9], [10] or the Kohlrausch-Williams-Watts (KWW) decay function [11], [12] (also known as the Weibull function [13]): $s = \\exp [-(t/\\alpha )^{\\beta }]$ , where the characteristic lifetime $\\alpha $ corresponds to the specific lifetime for $s = \\exp (-1)$ and the stretched exponent $\\beta $ reflects the nonexponentiality.", "The simple exponential decay corresponds to $\\beta = 1$ and the classical stretched exponential decay to $0 < \\beta < 1$ (classically $\\beta $ is invariant).", "Particularly, the stretched exponent is associated with a cascade mechanism of relaxation [7].", "In this study, we utilize reliable statistical data of article page views that are available in Scientific Reports for articles published after 1 January 2012.", "Data for daily page views including HTML views and PDF downloads are offered 48 hours after online publication and updated daily.", "Here, we suggest a useful methodology for evaluating the nonexponentiality in article page views by adopting a modified stretched exponential function [14], [15], [16], [17], [18].", "This study shows the possibility that relaxation in article page views resembles physical aging as frequently found in complex systems.", "Figure: Statistics of article page views after online publication.The first dataset (A,C,E) contains 42 articles selected from Scientific Reports published in 2012 and the second dataset (B,D,F) contains 19 articles published between January and May in 2015, showing more than 1,000 views at the early stage of online publication.", "The normalized page views over time are described by normal scales (A,B) and logarithmic scales (C,D).", "The temporal evolution of stretched exponents is obtained from the normalized page views, showing nonexponentiality in relaxation for both cases.", "As a guide, srep00223 (A,C,D) and srep07971 (B,E,F) are marked with bold solid lines." ], [ "Results", "We selected two datasets: the first comprised 42 articles from Scientific Reports published in 2012 and the second comprised 19 articles published between January and May in 2015, showing more than 1,000 daily page views in a few days after online publication.", "First, we counted the number of page views ($p_{t}$ ) at time $t$ (days) normalized by the number of initially maximized page views ($p_{0}$ ).", "The attention rate can be defined as $s = p_{t}/p_{0}$ and usually decreases with time from the initial peak.", "Empirically, there would be three types of public attention: initial attention at time $t_{0}$ , irregular attention at time $t$ , and no attention (for insignificant $p_{0}$ ).", "For demonstration, the normalized page views for the two datasets are illustrated by the normal scales in Fig REF A-B.", "As illustrated, the maximum peak can exist at the origin of time, indicating the initial attention, and a few peaks can exist in the middle stage, indicating irregular attention.", "To examine whether the normalized page views have power-law time dependencies, we depicted the log-log plots in Fig REF C-D by simply rescaling Fig REF A-B.", "Here the straightness in the log-log plots is not manifested, which means that the power-law scaling of $s$ vs $t$ is invalid.", "Such a temporal behavior in $s$ vs $t$ is repeatable for both datasets.", "Next, we tested the nonexponentiality in the normalized page views.", "A convenient methodology is the adoption of a modified stretched exponential function.", "This function is described as $s = \\exp [-(t/\\alpha )^{\\beta }]$ with the characteristic lifetime $\\alpha $ and the stretched exponent $\\beta $ , especially where $\\beta $ varies with time [14], [15].", "This function was originally adopted to describe complicated biological survival curves [16], [17], [18].", "The stretched exponent is calculated as $\\beta = \\ln [-\\ln (s)]/\\ln (t/\\alpha )$ from the $\\alpha $ values.", "The characteristic lifetime $\\alpha $ can be measured by detecting the interception point between $s(t)$ and $s(\\alpha ) = \\exp (-1)$ .", "The rough $\\alpha $ estimate can be obtained from a linear regression from two data points that exist just above (+1p) and just below ($-$ 1p) the $\\exp (-1)$ point for each $s$ vs $t$ curve [18].", "Here the stretched exponent is a good measure for testing the nonexponentiality of a decay curve and is relevant to an asymmetrical broadening of the relaxation time distribution [9].", "As illustrated in Fig REF E-F, we are able to see the nonexponentiality of the normalized page views for the two different datasets published in Scientific Reports.", "Interestingly, the stretched exponents behave irregularly over a short period (shorter than 10 days approximately) and then gradually decrease with time over a long period (longer than 10 days)." ], [ "Discussion", "Our finding from the page view statistics suggests that public attention in research is similar to physical relaxation processes in complex systems.", "Social media are indeed complex systems that actively connect people every day.", "Interestingly, the temporal evolution of the stretched exponents in the normalized page views is identical to that of luminescence decays: the stretched exponents become smaller than unity and gradually decrease with time (see refs.", "[14], [15]).", "Public attention and physical perturbation would be alike in terms of the time evolution of impact and information propagation.", "A stochastic model for information propagation would be relevant to the nonexponentiality of page view statistics [1].", "Further studies would be necessary on statistical analyses for a universal decay pattern in page views with wide datasets.", "In summary, we present a novel statistical approach for article page views for online published articles.", "A significant finding is obtained: article page views usually decay over time after reaching a peak, especially exhibiting nonexponentiality.", "A feasible methodology is suggested for evaluation of the nonexponentiality.", "Our study shows that article page views follow nonexponential decay as public attention propagates through web-based communities, as frequently found in complex systems.", "Acknowledgments This work was supported by Sungkyun Research Fund, Sungkyunkwan University, 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[ [ "A necessary and sufficient condition for minimum phase and implications\n for phase retrieval" ], [ "Abstract We give a necessary and sufficient condition for a function $E(t)$ being of minimum phase, and hence for its phase being univocally determined by its intensity $\|E(t)\|^2$.", "This condition is based on the knowledge of $E(t)$ alone and not of its analytic continuation in the complex plane, thus greatly simplifying its practical applicability.", "We apply these results to find the class of all band-limited signals that correspond to distinct receiver states when the detector is sensitive to the field intensity only and insensitive to the field phase, and discuss the performance of a recently proposed transmission scheme able to linearly detect all distinguishable states." ], [ "Introduction", "Phase retrieval is a longstanding problem in many fields of physics and applied sciences [1], [2], [3], [4].", "Sufficient conditions ensuring that the phase of the signal and hence the full $E(t)$ can be reconstructed from the knowledge of the intensity profile only $\|E(t)\|^2$ are well known [2], [3], [5], [6].", "A necessary and sufficient condition is also well known, and it is based on the position in the complex plane of the zeros of $E(z)$ , the analytic continuation of $E(t)$ .", "Such a condition however is of limited practical use, because the analytic continuation of a function is an ill posed problem and hence far from being amenable to simple numerical solutions [7].", "Even when analytic continuation is possible, like for instance for bandwidth limited $E(t)$ , finding the zeros in the complex plane is a difficult numerical task.", "Being instead the numerical evaluation of $E(t)$ for $t$ real relatively trivial, a necessary and sufficient condition based on the properties of $E(t)$ for $t$ real only would be a very useful tool.", "To the best of the author's knowledge, however, such a condition has not been reported yet.", "The purpose of this paper is to derive such a condition, which we will see is very similar to the well-established Nyqvist stability criterion [8] of control theory.", "We will consider for simplicity to band-limited signals only, and leave generalizations to wider classes of signals to future studies.", "The outline of this paper is the following.", "After presenting the derivation of a necessary and sufficient condition ensuring that the phase of a band-limited field can be retrieved from its intensity profile, we apply this condition to find the class of band-limited signals corresponding to distinct states when the receiver is insensitive to the signal phase and sensitive to the field intensity only.", "The implications for the capacity of an optical system using square-law detection at the receiver are finally discussed.", "While emphasis will be given to examples belonging to the field of telecommunications only, the new condition discussed here can be of help in all fields of physics and applied sciences where the problem of phase reconstruction from the intensity only is an important one [4]." ], [ "A necessary and sufficient condition for minimum phase", "In this paper, we define the Fourier transform of functions $F(t) \\in L^1 \\bigcap L^2$ as $\\tilde{F}(\\omega ) = \\int _{-\\infty }^\\infty {\\rm d}t \\exp (i \\omega t) F(t).", "$ The fact that $F(t) \\in L^1$ insures that $F(\\omega )$ is uniformly continuous for $\\omega \\in \\mathbb {R}$ [9].", "Lemma 1 Assume a function $E_s(t) \\in L^1 \\bigcap L^2$ , such that its Fourier transform $\\tilde{E}(\\omega )$ is $\\tilde{E}_s(\\omega ) = 0$ , $\\forall \\omega < 0$ .", "Then we have $E_s(t) = \\frac{i}{\\pi }\\, \\mathrm {p.v.", "}\\int _{-\\infty }^\\infty \\frac{E_s(t^{\\prime })}{t^{\\prime }-t} {\\rm d}t^{\\prime } , $ where with p.v.", "we refer to the Cauchy's principal value of the integral.", "Condition $\\tilde{E}_s(\\omega ) = 0$ , $\\forall \\omega < 0$ implies that $\\tilde{E}_s(\\omega ) = u(\\omega ) \\tilde{E}_s(\\omega )$ , where $u(\\omega )$ is a Heaviside unit step function.", "If we write $\\tilde{E}_s(\\omega ) = \\lim _{\\epsilon \\rightarrow 0^+} u(\\omega ) \\exp (-\\epsilon \\omega ) \\tilde{E}_s(\\omega ), $ inverse Fourier transformation gives $E_s(t) = \\frac{i}{2 \\pi } \\int _{-\\infty }^\\infty \\lim _{\\epsilon \\rightarrow 0^+} \\frac{1}{t^{\\prime }-t+ i \\epsilon } E_s(t^{\\prime }) {\\rm d}t^{\\prime }, $ that is $E_s(t) = \\frac{i}{2 \\pi } \\int _{-\\infty }^\\infty \\lim _{\\epsilon \\rightarrow 0^+} \\frac{t^{\\prime }-t - i \\epsilon }{(t^{\\prime }-t)^2 + \\epsilon ^2} E_s(t^{\\prime }) {\\rm d}t^{\\prime }.", "$ Being $\\lim _{\\epsilon \\rightarrow 0^+} \\epsilon /[(t^{\\prime }-t)^2 + \\epsilon ^2] = \\pi \\delta (t^{\\prime }-t)$ where $\\delta (\\cdot )$ is the Dirac delta distribution, we obtain $E_s(t) = \\frac{i}{\\pi } \\int _{-\\infty }^\\infty \\lim _{\\epsilon \\rightarrow 0^+} \\frac{t^{\\prime }-t}{(t^{\\prime }-t)^2 + \\epsilon ^2} E_s(t^{\\prime }) {\\rm d}t^{\\prime }, $ that is relation (REF ).", "Consequence of Eq.", "(REF ) is that the real and imaginary parts of $E_s(t) = E_{s,r}(t) + i E_{s,i}(t)$ are the Hilbert transform of one another $E_{s,i} (t) &=& \\frac{1}{\\pi }\\, \\mathrm {p.v.", "}\\int _{-\\infty }^\\infty {\\rm d}t^{\\prime } \\frac{E_{s,r}(t^{\\prime })}{t^{\\prime }-t}, \\\\E_{s,r}(t) &=& - \\frac{1}{\\pi }\\, \\mathrm {p.v.", "}\\int _{-\\infty }^\\infty {\\rm d}t^{\\prime } \\frac{E_{s,i}(t^{\\prime })}{t^{\\prime }-t}.", "$ These relations are known in spectroscopy as Kramers Kronig relations [10], [11].", "Let $\\beta $ be a strictly positive constant with $0<\\beta \\le 1$ and let us define $B = (1+\\beta )/T$ .", "Let $\\tilde{\\mathcal {}{C}_\\beta (0,B) be the class of functions E_{s}(t) of the form\\begin{equation}E_s(t) = \\sum _{n=-\\infty }^\\infty a_n \\exp \\left[- i (1+\\beta ) \\frac{\\pi (t-nT) }{T} \\right] H_\\beta (t-nT), \\end{equation}where we assume that the sequence of a_n \\in \\mathbb {C} has a compact support, namely for any E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B), exists an N \\in \\mathbb {Z} such that a_n = 0 for \|n\| > N. The orthogonal set of functions \\lbrace H_\\beta (t-nT), n \\in \\mathbb {Z}\\rbrace are defined as\\begin{equation}H_\\beta (t) = \\frac{1}{\\sqrt{T}} \\frac{\\sin \\left[\\pi \\frac{t}{T} (1-\\beta )\\right] + 4 \\beta \\frac{t}{T} \\cos \\left[\\pi \\frac{t}{T} (1+\\beta )\\right]}{\\pi \\frac{t}{T} \\left[1-\\left(4 \\beta \\frac{t}{T} \\right)^2\\right]}.", "\\end{equation}The functions in \\tilde{\\mathcal {}{C}_\\beta (0,B) belong to L^1 \\bigcap L^2.", "Their Fourier transform is\\begin{equation}\\tilde{E}_{s}(\\omega ) = \\sum _{n=-N}^N a_n \\exp \\left(i n T \\omega \\right) \\tilde{H}_\\beta \\left[\\omega - (1+\\beta ) \\pi /T \\right], \\end{equation}where\\begin{equation}\\tilde{H}_\\beta (\\omega ) = \\sqrt{T} \\hspace{2.84544pt} C_\\beta \\left(\\frac{\\omega T}{2 \\pi }\\right), \\end{equation}with\\begin{equation}C_\\beta (x) = \\left\\lbrace \\begin{array}{ll}1, & \|x\| \\le \\frac{1-\\beta }{2}; \\\\\\cos \\left[ \\frac{\\pi }{2 \\beta } \\left(\|x\|-\\frac{1-\\beta }{2} \\right) \\right], & \\frac{1-\\beta }{2} < \|x\| \\le \\frac{1+\\beta }{2}; \\\\0, & \|x\| > \\frac{1+\\beta }{2}.", "\\end{array} \\right.\\end{equation}The functions \\lbrace H_\\beta (t-nT), n \\in \\mathbb {Z}\\rbrace have a square-root raised cosine spectrum, and their orthogonality can be directly verified in the Fourier domain\\begin{equation}\\int _{-\\infty }^\\infty \|H_\\beta (\\omega )\|^2 \\exp \\left[i (n^{\\prime }-n) T \\omega \\right] \\frac{{\\rm d}\\omega }{2 \\pi }= \\delta _{n,n^{\\prime }}, \\end{equation}where the integral is facilitated by noting that \\sum _{n} \|H_\\beta (\\omega - 2 \\pi T n)\|^2 = T and using the periodicity of \\exp (i n T \\omega ).", "Being E_s(t) \\in L^1 \\bigcap L^2, its spectrum is a continuous function of \\omega .", "In addition, it is zero for \\omega \\notin (0, 2 \\pi B ).", "Finally, it is easy to verify that E_s(z), the analytic continuation of E_s(t) in the complex plane, is an entire function.", "}If we define \\tilde{\\mathcal {}{C} (0,B) = \\lim _{\\beta \\rightarrow 0^+} \\tilde{\\mathcal {}{C}_\\beta (0,B) , any function E_s(t) \\in \\tilde{\\mathcal {}{C}(0,B) can be written in the form\\begin{equation}E_{s}(t) = \\hspace{-5.69046pt} \\sum _{n=-N}^N a_n \\exp \\left[- i \\frac{\\pi (t-nT) }{T} \\right] H_0(t-nT) \\end{equation}where\\begin{equation}H_0(t) = \\lim _{\\beta \\rightarrow 0} H_\\beta (t) = \\frac{1}{\\sqrt{T}} \\hspace{2.84544pt} \\mathrm {sinc}\\left(\\frac{\\pi t}{T}\\right).", "\\end{equation}The Fourier transform of (\\ref {mod}) is\\begin{equation}\\tilde{E}_{s}(\\omega ) = \\sum _{n=-\\infty }^\\infty a_n \\exp \\left(i n T \\omega \\right) \\tilde{H}_0\\left(\\omega - \\pi /T \\right), \\end{equation}where\\begin{equation}\\tilde{H}_0(\\omega ) = \\sqrt{T} \\hspace{2.84544pt} C_0\\left(\\frac{\\omega T}{2 \\pi }\\right), \\end{equation}and C_0(x) = \\lim _{\\beta \\rightarrow 0} C_\\beta (x) is a function equal to 1 in the open interval (-1/2, 1/2), equal to 1/\\sqrt{2} for x=\\pm 1/2 and zero elsewhere.", "Being for \\omega , \\omega ^{\\prime } \\in (0, 2\\pi B)\\begin{equation}\\sum _n \\tilde{H}_0(\\omega ) \\tilde{H}_0(\\omega ^{\\prime }) \\exp \\left[-i n T (\\omega -\\omega ^{\\prime }) \\right] \\nonumber \\\\= 2 \\pi \\delta \\left(\\omega -\\omega ^{\\prime } \\right), \\end{equation}any function E_s(t) \\in L^2 band-limited to the interval (0,2 \\pi B) can be expressed in the form (\\ref {mod}), where the a_n are given by\\begin{equation}a_n = \\int \\tilde{H}_0(\\omega ^{\\prime }) \\exp \\left(-i n \\frac{2 \\pi }{T}\\right) \\tilde{E}_s(\\omega ^{\\prime }) \\frac{{\\rm d}\\omega ^{\\prime }}{2 \\pi }.", "\\end{equation}Although the functions of the form (\\ref {mod}) belonging to \\tilde{\\mathcal {}{C}(0,B) are in L^2, they are in general not in L^1.", "For this reason, in the following where we refer to the class of band-limited functions \\tilde{\\mathcal {}{C}(0,B), we will assume that its components are members of the class \\tilde{\\mathcal {}{C}_\\beta (0,B) for \\beta >0, hence always in the class L^1 \\bigcap L^2, approaching arbitrarily close the limit \\beta = 0.", "}\\begin{Th} Let us assume E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B), and a constant \\bar{E} \\ne 0, and define E(t) = E_s(t) + \\bar{E} with \\bar{E} \\ne 0, such that E(t) \\ne 0, \\forall t \\in \\mathbb {R} and the trajectory of E(t) never encircles the origin for t \\in (-\\infty , \\infty ).", "Then, the number of zeros of E(t+i\\tau ) with \\tau <0 is equal to the winding number (i.e, the number of windings) around the origin of the curve described by E(t) when t runs over the entire real axis from t \\rightarrow -\\infty to t \\rightarrow \\infty .", "}\\end{Th}\\begin{proof}The function E(z) is an entire function.", "Let \\Gamma be a contour encompassing the lower complex plane z = t + i \\tau , which incorporates the real axis from t = -\\infty to \\infty and returns to -\\infty from the lower complex half-plane by a semicircle C with radius \\rho \\rightarrow \\infty .", "By the Cauchy^{\\prime }s argument principle, we have\\begin{equation}I_\\Gamma = \\frac{1}{2 \\pi i} \\oint _{\\Gamma } \\frac{\\dot{E}(z)}{E(z)} {\\rm d}z = N_\\mathrm {zeros} - N_\\mathrm {poles}, \\end{equation}where \\dot{E}(t) = {\\rm d}E(t)/{\\rm d}t, N_\\mathrm {zeros} is the number of zeros and N_\\mathrm {poles} is the number of poles of E(z) encircled by \\Gamma .", "The function E(t + i \\tau ) does not have poles for \\tau <0, N_\\mathrm {poles}=0.", "Using the substitution E(z) = v, we obtain\\begin{equation}I_\\Gamma = \\frac{1}{2 \\pi i} \\int _{v_\\Gamma } \\frac{{\\rm d}v}{v} = N_\\mathrm {zeros}.", "\\end{equation}Let us now split the integral I_\\Gamma in (\\ref {argument}) into two parts, I_\\Gamma = I_r+I_C, the first\\begin{equation}I_r = \\frac{1}{2 \\pi i} \\int _{v_r} \\frac{{\\rm d}v}{v} \\end{equation}being the contribution of E(z) when z belongs to the real axis, the second\\begin{equation}I_C = \\frac{1}{2 \\pi i} \\int _{v_C} \\frac{{\\rm d}v}{v}, \\end{equation}being the contribution of E(z) when z belongs to the infinite semicircle C. The integration path v_C is made of the trajectory described by E(z) for z = R \\exp (i \\phi ) when \\phi \\in [\\pi , 2 \\pi ].", "Being E(z) \\rightarrow \\overline{E} on the infinite semicircle, the length of the integration path v_C tends to zero.", "In addition, being \\bar{E} \\ne 0, the modulus of the integrand tends to a finite constant on the infinite semicircle, \|1/v\| \\rightarrow 1/\|\\bar{E}\|, so that I_C = 0.", "Consequently, only the first term survives,\\begin{equation}I_r = \\frac{1}{2 \\pi i} \\int _{v_r} \\frac{{\\rm d}v}{v} = N_\\mathrm {zeros}.", "\\end{equation}The curve v_r is made of the trajectory described by E(t) when t runs over the entire time axis.", "Such curve is closed because E(t) tends to \\overline{E} for both t \\rightarrow \\infty and t \\rightarrow -\\infty .", "The left hand side of (\\ref {IR}) is then equal to the number of times the curve v_r encircles the origin.", "This shows that the winding number of the curve described by E(t) when t runs from t \\rightarrow -\\infty to t \\rightarrow \\infty is equal N_\\mathrm {zeros}.", "\\end{proof}}Signals such that \\tilde{E}(\\omega ) = 0 for \\omega < 0 are called \\textit {minimum phase signals} if, beside having no poles of E(t+i \\tau ) with \\tau <0, they have also no zeros with \\tau <0.", "Theorem \\ref {Th1} shows that a necessary and sufficient condition for a signal E(t) to be of minimum phase is that the curve described by E(t) when t runs from t \\rightarrow -\\infty to t \\rightarrow \\infty does not encircle the origin.", "This theorem permits to decide whether a signal is of minimum phase or not by numerically evaluating the values of E(t) for t real only, avoiding the difficult, ill-conditioned \\cite {Gallicchio}, analytic continuation of E(t) in the complex plane.", "Notice that, although non zero, the constant bias \\overline{E} can be arbitrarily small.", "}\\begin{Th} Let us assume E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B), and a constant \\bar{E} \\ne 0, and define E(t) = E_s(t) + \\bar{E} with \\bar{E} \\ne 0, such that E(t) \\ne 0, \\forall t \\in \\mathbb {R} and the trajectory of E(t) never encircles the origin for t \\in (-\\infty , \\infty ).", "Then, it is possible to define\\begin{equation}G(z) = \\log \\left[ \\frac{E(z)}{\\overline{E}} \\right], \\end{equation}with the determination of the logarithm chosen such that a) \\lim _{t \\rightarrow -\\infty } G(t+0i) = 0 and b) G(z) is a holomorphic function in the lower half plane including the real axis.", "With this choice, if we define G(t) as the restriction of G(z) on the real axis, G(t) \\in L^1 \\bigcap L^2, and its Fourier transform \\tilde{G}(\\omega ) is such that G(\\omega ) = 0 for \\omega < 0.", "}\\end{Th}\\begin{proof} The continuation of E(t) = E_s(t) + \\bar{E} in the complex plane, E(z) is an entire function, because E_s(z) is an entire function.", "It is possible to define a function G(z) holomorphic in the lower half plane including the real axis and such that E(z) = \\bar{E} \\exp [G(z)] by the following procedure.", "Let us construct G(z) for any z in the lower half plane by analytic continuation from an arbitrary point z_0 in the lower half plane using\\begin{equation}G(z) = G(z_0) + \\int _{z_0}^z \\frac{\\dot{E}(z^{\\prime })}{E(z^{\\prime })} {\\rm d}z^{\\prime }, \\end{equation}where the integration path from z_0 = to z is contained entirely in the lower half plane but otherwise arbitrary.", "The poles of \\dot{E}(z)/E(z) are the poles and the zeros of E(z), E(z) is an entire function and no zeros of E(z) exist in the lower half plane by virtue of theorem \\ref {Th1} and on the real axis by assumption, so that \\dot{E}(z)/E(z) is a holomorphic function in the lower half plane including the real axis and hence G(z) is also holomorphic in the same domain.", "The function G(z) is however not unique, because it depends on the choice of z_0 and of G(z_0).", "We will choose z_0 on the real axis such that z_0 = \\lim _{t \\rightarrow -\\infty } (t + 0 i), and we will take for G(z_0) the principal value of \\log [E(z_0)/\\bar{E}], namely\\begin{equation}G_\\infty = \\lim _{t \\rightarrow -\\infty } \\log \\left[1+\\frac{E_s(t)}{\\bar{E}}\\right] = 0.", "\\end{equation}We have then\\begin{equation}G(z) = \\lim _{t \\rightarrow -\\infty } \\int _{t+0i}^z \\frac{\\dot{E}(z^{\\prime })}{E(z^{\\prime })} {\\rm d}z^{\\prime }.", "\\end{equation}Let us now consider the integral in the complex plane z = t + i \\tau for \\omega = -\|\\omega \| <0,\\begin{equation}J_\\Gamma = \\oint _{\\Gamma } G(z) \\exp (-i \|\\omega \| z) {\\rm d}z, \\end{equation}where \\Gamma is again a contour that includes the real axis and returns to t = -\\infty from the semicircle in the lower half plane z = R \\exp (i \\phi ) with \\phi \\in [\\pi , 2 \\pi ].", "The integral of Eq.", "(\\ref {JGm}) can be decomposed into two terms\\begin{equation}J_\\Gamma = \\tilde{G}(\\omega ) + J_C \\end{equation}the first,\\begin{equation}\\tilde{G}(\\omega ) = \\int _{-\\infty }^\\infty G(t) \\exp \\left(i \\omega t\\right) {\\rm d}t , \\end{equation}being the Fourier transform of G(t), and the second, J_C, being the contribution of the infinite semicircle.", "The second contribution is equal to J_C = \\lim _{R \\rightarrow \\infty } J_R, where\\begin{equation}J_R = i \\int _{\\pi }^{2 \\pi } G[R \\exp (i \\phi )] \\exp \\lbrace i [-\|\\omega \| R \\exp (i \\phi )+ \\phi ]\\rbrace R {\\rm d}\\phi .", "\\end{equation}In the absence of poles in the lower half-plane we have J_\\Gamma = 0 and hence\\begin{equation}\\tilde{G}(\\omega ) = - J_C \\quad \\forall \\omega <0.", "\\end{equation}We have\\begin{eqnarray}\|J_R\| &\\le & \\int _{\\pi }^{2 \\pi } \\left\|G[R \\exp (i \\phi )] \\right\| \\exp [\|\\omega \| R \\sin (\\phi )] R {\\rm d}\\phi \\nonumber \\\\&\\le & M(R) \\int _{\\pi }^{2 \\pi } \\exp [\|\\omega \| R \\sin (\\phi )] R {\\rm d}\\phi , \\end{eqnarray}where M(R) = \\max _{\\phi \\in [\\pi , 2 \\pi ]} \\left\\lbrace \\left\|G[R \\exp (i \\phi )] \\right\|\\right\\rbrace .", "We have then\\begin{equation}\|J_R\| \\le 2 M(R) \\int _{0}^{\\pi /2} \\exp [-\|\\omega \| R \\sin (\\phi )] R {\\rm d}\\phi , \\end{equation}where the integral at right-hand side tends to a constant\\begin{equation}\\lim _{R \\rightarrow \\infty } \\int _{0}^{\\pi /2} \\exp [ - \|\\omega \| R \\sin (\\phi )] R {\\rm d}\\phi = \\frac{1}{\|\\omega \|}.", "\\end{equation}Using now Eq.", "(\\ref {Gz}) on the semicircle of radius R we obtain for \\phi \\in [\\pi , 2 \\pi ]\\begin{eqnarray}G_\\phi &=& \\lim _{R \\rightarrow \\infty } G[R \\exp (i \\phi )] \\nonumber \\\\&=& \\lim _{R \\rightarrow \\infty } \\log \\left\\lbrace 1 + \\frac{E_s[R\\exp (i \\phi )]}{\\bar{E}} \\right\\rbrace = 0, \\end{eqnarray}because the principal value of the logarithm is used in Eq.", "(\\ref {Gz0}) and, being \\lim _{t \\rightarrow -\\infty } G(t+0i) = G_{\\phi = \\pi } = 0, the principal value should be used, for continuity, in G_\\phi for all \\phi \\in (\\pi , 2 \\pi ].", "Consequently we also have \\lim _{R \\rightarrow \\infty } M(R) = \\max _{\\phi \\in [\\pi , 2 \\pi ]} G_\\phi = 0, and hence J_C = \\lim _{R \\rightarrow \\infty } J_R = 0.\\end{proof}The above results implies that also \\lim _{t \\rightarrow \\infty } G(t+0i) = G_{\\phi = 2 \\pi } = 0, so that asymptotically, on the real axis\\begin{equation}G(t) = \\log \\left[1+\\frac{E_s(t)}{\\bar{E}} \\right] \\rightarrow \\frac{E_s(t)}{\\bar{E}}, \\quad \|t\| \\rightarrow \\infty .", "\\end{equation}Being E(t) > \\bar{E}, \\forall t \\in \\mathbb {R}, and E(t) finite, the integrals of both \|G(t)\| and \|G(t)\|^2 over a finite interval are finite.", "At infinity, being E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B) for \\beta >0, Eqs.", "(\\ref {mod1}) and (\\ref {Hb}) show that \|E_s(t)\| \\simeq 1/\|t\|^2 for \|t\| \\rightarrow \\infty and hence the integrals of \|G(t)\| and \|G(t)\|^2 also converge at infinity.", "Consequently, G(t) \\in L^1 \\bigcap L^2.", "}}The above theorem shows that the ambiguity in the determination of the logarithm to be used in the restriction of G(z) to the real axis, G(t) = \\log [E(t)/\\bar{E}], is removed by the prescriptions that both \\lim _{t \\rightarrow -\\infty } G(t) = 0 and G(t) is a continuos function of t \\in \\mathbb {R}.", "}\\begin{Th} Let us assume E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B), and a constant \\bar{E} \\ne 0, and define E(t) = E_s(t) + \\bar{E} with \\bar{E} \\ne 0, such that E(t) \\ne 0, \\forall t \\in \\mathbb {R} and the trajectory of E(t) never encircles the origin for t \\in (-\\infty , \\infty ).", "Then, the phase of E(t) can be reconstructed by a logarithmic Hilbert transform\\begin{equation}\\phi (t) = \\overline{\\phi } + \\frac{1}{2 \\pi } \\mathrm {p.v.}", "\\int _{-\\infty }^{\\infty } {\\rm d}t^{\\prime } \\frac{\\log \\left[ \|E(t^{\\prime })\|^2 \\right]}{t^{\\prime }-t}, \\end{equation}where E(t) = \|E(t)\| \\exp [i \\phi (t)] and \\overline{E} = \|\\overline{E}\| \\exp (i \\overline{\\phi }).", "}\\end{Th}\\begin{proof} Under the hypotheses stated, theorem \\ref {Co10} ensures that G(t) \\in L^1 \\bigcap L^2 and \\tilde{G}(\\omega ) = 0 for \\omega < 0 and, hence, the hypotheses of lemma \\ref {Le1} are verified, so that\\begin{equation}G(t) = \\frac{i}{\\pi }\\, \\mathrm {p.v.", "}\\int _{-\\infty }^\\infty {\\rm d}t^{\\prime } \\frac{G(t^{\\prime })}{t^{\\prime }-t}.", "\\end{equation}Equation (\\ref {phi}) is then readily obtained by equating the imaginary parts of both sides of Eq.", "(\\ref {kk1}), using G(t) = \\log \|E(t)\| - \\log \\overline{E} + i [\\phi (t)-\\overline{\\phi }] and that\\begin{equation}\\frac{1}{\\pi } \\mathrm {p.v.}", "\\int _{-\\infty }^{\\infty } {\\rm d}t^{\\prime } \\frac{\\log \|\\overline{E}\|}{t^{\\prime }-t} = 0.", "\\end{equation}\\end{proof}}Notice that, being\\begin{equation}\\frac{1}{2 \\pi } \\mathrm {p.v.}", "\\int _{-\\infty }^{\\infty } {\\rm d}t \\int _{-\\infty }^{\\infty } {\\rm d}t^{\\prime } \\frac{\\log \\left[ \|E(t^{\\prime })\|^2 \\right]}{t^{\\prime }-t} = 0, \\end{equation}the phase bias \\overline{\\phi } is also the time average of \\phi (t)\\begin{equation}\\overline{\\phi }= \\lim _{T \\rightarrow \\infty } \\frac{1}{T} \\int _{-T/2}^{T/2} {\\rm d}t \\phi (t).", "\\end{equation}}}From the necessary and sufficient condition of theorem \\ref {Th3} more restrictive sufficient conditions can be derived.", "One is the following$ Corollary 1 Let us assume $E_s(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B), and define E(t) = E_s(t) + \\bar{E}, with \\bar{E} \\ne 0 a complex constant.", "Then, E(t) is of minimum phase, and hence its phase can be reconstructed by the logarithmic Hilbert transform (\\ref {phi}) when \\exists \\phi _0 \\in [0, 2 \\pi ) such that \\mathrm {real}[E(t) \\exp (i \\phi _0)] > 0, \\forall t \\in \\mathbb {R}.", "}$ Under the hypotheses of the theorem, the curve $E(t)$ never encircles the origin, hence the hypotheses of theorem are satisfied.", "This corollary can also be proven independently, under slightly less restrictive conditions.", "Theorem 1 Let $E_s(t) \\in L^1 \\bigcap L^2$ such that $\\tilde{E}_s(\\omega ) = 0$ , $\\forall \\omega < 0$ , and define $E(t) = E_s(t) + \\bar{E}$ , with $\\bar{E} \\ne 0$ a complex constant.", "Then, the phase of $E(t)$ can be reconstructed by the logarithmic Hilbert transform () when $\\exists \\phi _0 \\in [0, 2 \\pi )$ such that $\\mathrm {real}[E(t) \\exp (i \\phi _0)] > 0$ , $\\forall t \\in \\mathbb {R}$ .", "Let us define $F(t) = [E(t)/\\overline{E}-Z_0]/Z_0$ with $Z_0 = R \\exp (i \\phi _0)$ , and use this definition into the $G(t)$ given by Eq.", "(), to obtain $G(t) = \\log (Z_0) + \\log \\left[1 + F(t) \\right]$ , where for the logarithm we assume its principal value.", "We then note that the expansion $G(t) = \\log (Z_0) + \\sum _{n=1}^\\infty \\frac{(-1)^{n+1}}{n} F^n(t), $ is legitimate for $\|F(t)\|<1$ , that is inside the circle $\|E(t)/\\overline{E}-Z_0\|< \|Z_0\|$ centered in $Z_0$ and of radius $\|Z_0\|=R$ .", "In the limit $R \\rightarrow \\infty $ , the expansion is then legitimate for $F(t)$ inside the half plane delimited by a straight line passing through the origin and orthogonal to $Z_0$ and containing $Z_0$ , which is the region $F(t)$ belongs to by virtue of the condition $\\mathrm {real}[E(t) \\exp (i \\phi _0)] > 0$ , $\\forall t \\in \\mathbb {R}$ .", "Being the spectrum of $E(t)$ zero for $\\omega < 0$ , the spectrum of $F(t)$ and of all its powers $F^n(t)$ is also zero for $\\omega < 0$ .", "The series expansion (REF ) then shows that the spectrum of $G(t)$ exists and it is zero for $\\omega < 0$ .", "If we chose the determination of the logarithm such that $G(t) = \\log [E(t)/\\bar{E}] \\rightarrow 0$ as $\|t\| \\rightarrow \\infty $ , then being $E(t) \\ne 0, \\forall t \\in \\mathbb {R}$ and being $G(t) \\simeq E_s(t)/\\bar{E}$ for $\|t\| \\rightarrow \\infty $ with $E_s(t) \\in L^1 \\bigcap L^2$ , also $G(t) \\in L^1 \\bigcap L^2$ .", "The function $G(t)$ then fulfills once again the conditions of lemma REF so that its real and imaginary parts are the Hilbert transform of one another, and from this the thesis is deduced.", "Another sufficient condition, more restrictive than that given by corollary REF is that a signal $E(t) = \\bar{E} + E_s(t)$ is of minimum phase if $\|E_s(t)\|^2 < \|\\overline{E}\|^2$ for every $t$ .", "This condition was explicitly stated in [2] and [3] and, later, independently rediscovered by others, see for instance [5] where the condition was given in the context of wireless channel characterizations, and [6] where the condition was applied to optical measurements.", "It is immediate to show that, if this condition is satisfied, $E(t)$ never encircles the origin and it is consequently of minimum phase.", "In addition, being $\|E_s(t)\|^2 \\le \\int \\frac{{\\rm d}\\omega }{2 \\pi } \|\\tilde{E}_s(\\omega )\|^2.", "$ condition $\|\\overline{E}\|^2 > \|E_s(t)\|^2$ also includes the even more restrictive one $\|\\overline{E}\|^2 > \\int \\frac{{\\rm d}\\omega }{2 \\pi } \|\\tilde{E}_s(\\omega )\|^2 $ given in [14].", "Although we assumed $\\overline{E} \\ne 0$ , the results obtained are valid for $\\overline{E}$ arbitrarily small.", "The case $\\overline{E} = 0$ is a delicate one, because in this case, $G(t) \\notin L^2$ , so that $\\tilde{G}(\\omega )$ is not defined.", "In this case, however, the role of $\\overline{E}$ is not essential for convergence of the logarithmic Hilbert transform ().", "Indeed, the phase of every $E(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B) is well defined for every \\overline{E}, and it does not diverge for \\overline{E} = 0.", "For \\overline{E} \\ne 0, if E(t) does not encircle the origin, \\phi (t) given by Eq.", "(\\ref {phi}) gives the phase of E(t).", "Consequently, if E(t) does not encircle the origin the phase of E(t) can be still calculated for \\overline{E} = 0 using Eq.", "(\\ref {phi}) in the limit \\overline{E} \\rightarrow 0, which for what said does not diverge.", "We may therefore remove in the following the condition \\overline{E} = 0 assuming that this case is included as the limit for \\overline{E} \\rightarrow 0.", "}\\begin{Co} Let the two fields E_0(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B) and E_0^{\\prime }(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B) having the same intensity \|E_0(t)\|^2 = \|E_0^{\\prime }(t)\|^2.", "If E_0(t) and E_0^{\\prime }(t) do not encircle the origin, then they are both minimum phase signals and hence E_0(t) = E_0^{\\prime }(t) \\exp (i \\overline{\\phi }), \\forall t, where \\overline{\\phi } is an arbitrary constant phase.", "}}\\end{Co}$ An interesting and not immediately obvious property is now the following: Theorem 2 Given an arbitrary field $E(t) \\in \\tilde{\\mathcal {}{C}_\\beta (0,B) then\\begin{equation}E_0(t) = \|E(t)\| \\exp [i \\phi _{0}(t)], \\end{equation}where\\begin{equation}\\phi _{0}(t)= \\overline{\\phi } + \\frac{1}{2 \\pi } \\mathrm {p.v.}", "\\int _{-\\infty }^{\\infty } {\\rm d}t^{\\prime } \\frac{\\log \\left[ \|E(t^{\\prime })\|^2 \\right]}{t^{\\prime }-t}, \\end{equation}with \\overline{\\phi } an arbitrary phase, is band-limited to the same interval of E(t), namely (0, 2 \\pi B).", "}$ This theorem is parallel to an analogous one in [15] that gives the condition for a function to be the auto-convolution of a time limited function.", "If the analytic continuation of $E(t)$ , namely $E(t+i\\tau )$ , has no zeros for $\\tau <0$ , then we have $E(t) = E_0(t)$ for a suitable value of $\\overline{\\phi }$ .", "If this is not the case, any zero of $E(z)$ in the lower complex half plane, say $z_0 = t_0 - i \|\\tau _0\|$ , can be removed by multiplication by the pure phase modulation $H(t) = \\frac{t-t_0 - i \|\\tau _0\|}{ t-t_0 + i \|\\tau _0\|} = 1 - \\frac{ 2 }{1-i(t-t_0)/\|\\tau _0\|}, $ which adds a zero in the upper half plane symmetrically placed with respect to the real axis.", "The spectrum of the field $E^{\\prime }(t)$ after the phase modulation is $\\tilde{E}^{\\prime }(\\omega ) &=& \\tilde{E}(\\omega ) -2 \|\\tau _0\| \\int \\frac{{\\rm d}\\omega ^{\\prime }}{2 \\pi } \\tilde{E}(\\omega ^{\\prime }) u(\\omega -\\omega ^{\\prime }) \\nonumber \\\\&& \\times \\exp \\left[ i (\\omega -\\omega ^{\\prime }) (t_0-i \|\\tau _0\|) \\right], $ where $u(\\cdot )$ is the unit step function.", "The integral at right hand side of (REF ) is zero for $\\omega < 0$ because the non-zero regions of $E(\\omega ^{\\prime })$ and $u(\\omega -\\omega ^{\\prime })$ do not overlap, and it is also zero for $\\omega >2 \\pi B$ because in this case $u(\\omega -\\omega ^{\\prime })$ can be replaced by 1 and $\\int \\frac{{\\rm d}\\omega ^{\\prime }}{2 \\pi } \\tilde{E}(\\omega ^{\\prime }) \\exp \\left[ -i \\omega ^{\\prime } (t_0-i \|\\tau _0\|) \\right] = E(t_0-i \|\\tau _0\|) = 0.", "$ After the phase modulation, the spectrum is still zero for $\\omega < 0$ and $\\omega > 2 \\pi B$ , and the zero at $t_0 - i \|\\tau _0\|$ is replaced by a zero at $t_0 + i \|\\tau _0\|$ .", "This procedure can be repeated until all the zeros in the lower complex half plane are removed.", "At the end, the resulting field is a minimum phase signal still bandwidth limited to the same bandwidth of the original signal.", "Being the minimum phase signal unique, this signal is equal to $E_0(t)$ given by Eq.", "().", "Let us now consider the class of functions $\\tilde{\\mathcal {}{C}(0,B) band-limited in the interval (0,B), obtained from \\tilde{\\mathcal {}{C}_\\beta (0,B) in the limit of \\beta \\rightarrow 0^+.", "Theorem \\ref {Th0} insures that the class of band-limited signals with a common intensity profile always includes the minimum phase signal.", "To be more precise, theorem \\ref {Th0} insures that if one groups the functions E(t) \\in \\tilde{\\mathcal {}{C} (0,B) into functions with the same intensity profile I(t) = \|E(t)\|^2, any of these classes always includes the minimum phase function E_0(t) \\in \\tilde{\\mathcal {}{C} (0,B).", "Corollary \\ref {Cou} than shows that if the minimum phase condition is met, then the intensity profile uniquely determine the function E_0(t) with the exception of an immaterial rotation of the complex plane.", "From now on, we will refer for convenience to E(t) as the equivalence class of functions that differ from one another by an arbitrary constant phase factor, and with this caveat corollary \\ref {Cou} states that the minimum phase signal is unique.", "Consequently, all possible intensity profiles I(t) = \|E(t)\|^2 with E(t) \\in \\tilde{\\mathcal {}{C} (0,B) can be set into a one-to-one correspondence with the class of minimum phase functions E_0(t).", "}From theorem \\ref {Th0} we may also draw the important consequence that the ratio between a signal E(t) \\in \\tilde{\\mathcal {}{C}(0,B) that encircles N times the origin and the minimum phase signal E_0(t) with the same intensity profile \|E_0(t)\|^2 = \|E(t)\|^2, which we define as\\begin{equation}H_N(t) = \\frac{E(t)}{E_0(t)} , \\end{equation}has the form\\begin{equation}H_N(t) = \\prod _{k=1}^N \\left[1 - \\frac{ 2 }{1-i(t-t_k)/\|\\tau _k\|} \\right], \\end{equation}where t_k - i \|\\tau _k\| are the N zeros of E(t+i \\tau ) in the lower complex half plane.", "The function H_N(t) is the product of terms which are 1 minus a Lorentzian line-shape, centered on the real parts t_k and of width equal to the modulus of the imaginary part \|\\tau _k\| of each zero in the lower complex half plane.", "When these Lorentzian line-shapes are well separated, which may occur when the number N is small, we have\\begin{equation}H_N(t) \\simeq 1 - \\sum _{k=1}^N \\frac{2}{1-i(t-t_k)/\|\\tau _k\|}, \\end{equation}so that H_N(t) is mostly 1 with the exception of small regions of amplitude \|\\tau _k\| around the time t_k where the deviation from unit has a Lorentzian shape.", "The magnitude of the imaginary parts of the zeros in the lower complex half plane \|\\tau _k\| is proportional to the amplitude of the region around t_k where the reconstruction of the phase with the logarithmic Hilbert transform is inaccurate.", "}}}}}$" ], [ "Encoding information on the intensity of an electromagnetic field", "Let us now consider the following problem: What is the most efficient transmission over a bandwidth $B=1/T$ for the complex field $E(t)$ if we have the capability of modulating the optical field in modulus and phase and we perform square-law intensity detection of the transmitted signal at the receiver?", "The study of transmission systems where only direct detection is applied at the receiver became lately an intense area of research, targeting high capacity applications in the short reach range [16], [17], [18], [19], [20], [21].", "Although the information is not contained in the phase of the optical field, a suitable phase shaping is required to confine the signal within a limited bandwidth.", "One could in principle think of generating an arbitrary amplitude modulation at bandwidth $2B$ and then use a phase modulation equal to the logarithmic Hilbert transform of the intensity that makes this signal single sideband, as proposed in [20], but in general the spectrum of the signal after modulation is not zero for $\\omega > 2 \\pi B$ , and is in principle unbounded for positive frequencies.", "The analysis of the present paper gives a general solution to this problem.", "We have shown that all possible field patterns of a given optical bandwidth can be grouped into classes of fields with the same intensity pattern, and theorem REF ensured that to each of these classes belongs the minimum phase signal $E_0$ .", "Restricting the set of symbols to minimum phase signals therefore does not reduce the set of symbols available to transmission.", "The recently proposed Kramers Kronig (KK) transmission scheme [22] is capable of receiving without errors any minimum phase signal $E_0(t)$ and hence it maximizes the set of symbols that can be received over a given optical bandwidth.", "The principle of operation of the receiver is very simple: the receiver detects the intensity profile only, and the phase of the field is calculated from the intensity profile using the logarithmic Hilbert transform () over an integration time window much longer than the inverse of the optical field bandwidth.", "Assuming that chromatic dispersion is optically compensated or pre-compensated at the transmitter, we will show in the section that follows that a signal constellation like () with real$(a_n) > a$ and imag$(a_n) > a$ with $a$ a suitably chosen positive constant produces a minimum phase signal.", "It is enough, in general, to choose $a$ much smaller than the range of possible values of $a_n$ , so that the available symbols are almost all those belonging to one quadrant, say the first, of the complex plane, one quarter of all possible symbols available on the two quadratures.", "If some additive noise impairs the signal field, and if the noise is small enough that the perturbed field does not encircle the origin, then the perturbed field is still of minimum phase and faithfully detected by the receiver.", "Therefore, in spite of the fact that the quadratic receiver detects the field intensity only, and that the phase is calculated from the intensity profile and not independently measured, the receiver acts as a linear receiver (in the field) that is capable of detecting signals in the first quadrant only.", "Consequently, for high signal-to noise ratio, we may conjecture that the capacity of such system is of the order of the capacity (per unit bandwidth) of a coherent system in which the symbols are constrained in the first quadrant only, approximately two bit less than the capacity of a full, unconstrained, coherent system [22], [23]." ], [ "Numerical validation", "We numerically validated the results of the previous sections by testing on a signal made as the sequence of 512 waveforms of the form $(\\ref {mod})$ with $a_n = [(b + k_1) + i(b + k_2)] \\sqrt{T}$ , where $n$ runs from 1 to 512 and $k_1$ and $k_2$ are randomly chosen between 0 and 7.", "This signal corresponds to a shifted 16 quadrature-amplitude modulation (16QAM) constellation.", "The bias $b$ was chosen equal to two values, one $b_3 = 0.5$ insufficient to make the waveform of minimum phase, and the second $b_0 = 1.1$ , that makes the waveform a minimum phase one.", "Figure: Complex field of a shifted 16QAM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 3 =0.5b_3 = 0.5.The numerical analysis was performed using a Marlab program, performing the Hilbert transform over a finite time window $T_w$ that in the examples given was chosen as $T_w = 512 \\, T$ , by multiplication of the spectrum of the signal by $-i \\, \\mathrm {sgn}(\\omega )$ and using a fast Fourier transform (FFT) and inverse FFT routines, i.e.", "implicitly assuming a periodic signal instead of an $L^2$ one.", "The analysis of the periodic case implied by the use of the FFT can be done by replacing the kernel of the Hilbert transform $1/t$ by its $T_w$ -periodic counterpart $\\sum _{k=-\\infty }^\\infty \\frac{1}{t - k T_w} = \\frac{\\pi }{T_w} \\mathrm {cot}\\left(\\frac{\\pi t}{T_w} \\right), \\quad t \\ne \\tau , $ and limiting the integration over $\\tau $ to a single period $[-T_w/2, T_w/2)$ .", "This corresponds to replace Fourier integrals with Fourier series.", "All the theorems that we have discussed retain their validity.", "Theorem for instance can be shown without closing the curve $\\Gamma $ with a semicircle $C$ at infinity, because when $t$ spans on the real axis an entire period the function $E(t)$ describes in the complex plane a closed curve.", "While the rigorous analysis of the periodic case is beyond the scope of this paper, the correspondence between the periodic and the $L^2$ case can be obtained considering the case $T_w \\rightarrow \\infty $ .", "In the $L^2$ case, the field $\\overline{E}$ is the time average of the field $E(t)$ $\\overline{E} = \\lim _{T_w \\rightarrow \\infty } \\frac{1}{T_w} \\int _{-T_w/2}^{T_w/2} E(t) {\\rm d}t, $ so that in the periodic case the role of the field bias $\\overline{E}$ is played by the average of the signal over the length of the symbol sequence $T_w$ $\\overline{E}_\\mathrm {av} = \\frac{1}{T_w} \\int _{-T_w/2}^{T_w/2} {\\rm d}t E(t).", "$ An intuitive understanding of this correspondence can be obtained by looking at the spectrum of $E(t) = E_s(t) + \\bar{E}$ , namely $\\tilde{E}(\\omega ) = \\tilde{E}_s(\\omega ) + 2 \\pi \\bar{E} \\delta (\\omega )$ , and the spectrum of the periodic sequence.", "Both $\\overline{E}$ and $\\overline{E}_\\mathrm {av}$ are the amplitude of the spectral component at zero frequency, the Dirac delta function in $\\tilde{E}(\\omega )$ being replaced by the amplitude of the discrete spectral component at zero frequency in the periodic case.", "Figure REF shows by a blue dashed line a field $E(t)$ obtained with the lowest value of the bias, and by a red solid line the curve $E_0(t)$ reconstructed by the logarithmic Hilbert transform.", "Big blue dots represent the values of the $E(t)$ at $t = nT_w$ , and red smaller dots the values of the reconstructed field at the same times.", "The curve $E(t)$ has a winding number around the origin of 3.", "The red solid curve overlaps with the blue dashed one almost everywhere, with the exception of the vicinity of the points where the windings occur.", "The accuracy of the reconstruction is more evident if we plot the phase reconstructed by the logarithmic Hilbert transform on top of the phase of $E(t)$ .", "This comparison is shown in Figs.", "REF and REF , where we show by a red solid line the reconstructed phase and by a blue dashed line the phase of $E(t)$ .", "In Fig.", "REF three phase jumps corresponding to the three windings of $E(t)$ around the origin are clearly visible.", "Figure REF is the zoom of the plot of Fig.", "REF in the vicinity of the second phase jump, showing that the deviation of the phase reconstruction from the phase of $E(t)$ is confined to a small region around the jump.", "The numerical results confirm the conjecture proposed the previous section that the constraint of square-law intensity detection reduces the number of available symbols by approximately one quarter, implying a capacity reduction of two bits over the full coherent detection for the same optical bandwidth.", "Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =0.5b_3 = 0.5.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =0.5b_3 = 0.5.", "This is the zoom in the region of the second phase jump of Fig.", "In Figs.", "REF we show the ratio $\|H_3(t) -1\| = \|E(t)/E_0(t) - 1\|$ for the field $E(t)$ of Fig.", "REF .", "Figure REF clearly shows the three Lorentzian of amplitude 2 corresponding to the three encircling of the origin of $E(t)$ .", "Figure REF shows by a blue solid thin line the same curve in a semilogarithmic scale, and by a red dashed thick line an interpolation with the curve $H_3(t) - 1 = \\prod _{k=1}^3 \\left[1 - \\sum _{h=-\\infty }^{\\infty } \\frac{ 2 }{1 - i (t-t_k - h T_w)/\|\\tau _k\|}\\right], $ which, using Eq.", "(REF ), becomes $H_3(t) - 1 = \\prod _{k=1}^3 \\left[1 - \\frac{2 \\pi i \|\\tau _k\|}{T_w} \\mathrm {cot} \\left(\\pi \\frac{t-t_k+i \|\\tau _k\|}{T_w} \\right) \\right].", "$ Equation (REF ) was obtained adapting Eq.", "() to account for the temporal periodicity induced by the use of the FFT algorithm, which introduces an infinite number of replicas of the Lorentzian line-shapes spaced by the time window $T_w$ .", "The parameters were obtained by interpolation of the main peaks only and were for the real parts $t_1/T_w=108.5755$ , $t_2/T_w = 272.4868$ , and $t_3/T_w = 384.5205$ , and for the imaginary parts $\|\\tau _1\|/T_w = 0.0070$ , $\|\\tau _2\|/T_w = 0.027$ and $\|\\tau _3\|/T_w = 0.041$ .", "As shown in Fig REF , the expression given in Eq.", "(REF ) was accurate more than six orders of magnitude down to the main peaks.", "However, a single Lorentzian, i.e.", "only the dominant term with $h =0$ in Eq.", "(REF ), is accurate 2 orders of magnitudes down to the main peak, and is sufficient to exactly reproduce the plot in linear scale shown in Fig.", "REF .", "From the average time and the temporal width of each Lorentzian line-shape we were able to compute, with high accuracy, the position of the zeros of $E(t+i\\tau )$ in the lower complex half-plane without the need of numerical analytic continuation of $E(t)$ .", "Figure: \|H 3 (t)-1\|\|H_3(t)-1\| vs. t/Tt/T for a bias b 3 =1.1b_3 = 1.1.Figure: \|H 3 (t)-1\|\|H_3(t)-1\| vs. t/Tt/T for a bias b 3 =1.1b_3 = 1.1 shown in Fig.", ", in a semilogarithmic scale.", "The blue solid thin line is the numerical result reported in Fig.", ", the red dashed thick line is the interpolation using Eq.", "().Figures REF –REF shows the curves obtained with the same $a_n$ sequence used for Figs.", "REF –REF but with a larger value of the bias $b_0 = 1.1$ .", "In this case, there are no windings of $E(t)$ around the origin, and the reconstructed field coincides with the original one, $E_0(t) \\equiv E(t)$ .", "Figure REF shows the perfect reconstruction of the field at the sampling point by the overlap of the red and the big blue dots.", "Figure: Complex field of a shifted 16QAM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 3 =1.1b_3 = 1.1.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =1.1b_3 = 1.1.", "The smaller excursion of the phase shown here compared with that of Fig.", "is caused by the larger distance from the origin due to the larger bias.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =1.1b_3 = 1.1.", "This is the zoom in the same region of the phase jump of Fig.", ".The example given in Figs.", "REF –REF corresponds to a case in which beside the condition of no windings around the origin of the trajectories of $E(t)$ also the condition $\|E_s(t)\|^2 < \\overline{E}^2$ was fulfilled.", "Figures REF and REF show instead a case where the condition $\|E_s(t)\|^2 < \\overline{E}^2$ fails but the reconstruction of the phase and consequently of the field from the intensity profile is perfect because no windings around the origin occur.", "These curves were obtained using $a_n = (b + k) \\sqrt{T}$ , with $n$ running from 1 to 512 and $k$ randomly chosen between 0 and 7, corresponding to an amplitude modulation with 8 levels.", "The value of $b$ was $b_0 = 0.1$ .", "Figure: Complex field of a shifted 8AM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 0 =0.1b_0 = 0.1.Figure: Phase of a shifted 8AM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 0 =0.1b_0 = 0.1." ], [ "Time-frequency duality", "In this paper, we have considered functions $E(t)$ whose Fourier transform $\\tilde{E}(\\omega )$ was zero for $\\omega < 0$ , and our goal was to calculate the phase of $E(t)$ once its intensity $\|E(t)\|^2$ was measured.", "Once the necessary changes are made, this case has a one to one correspondence with the case studied in [1], [3], [4], [5], [6] of causal functions $E(t)$ , i.e.", "such that $E(t) = 0$ for $t \\le 0$ , where the goal was to find the phase of $\\tilde{E}(\\omega )$ once the power spectrum $\|E(\\omega )\|^2$ was measured over a range of frequency.", "The necessary changes include, for instance, the fact that the property $\\tilde{E}(\\omega ) =0$ for all $\\omega < 0$ implied, as we have seen, that the analytic continuation of $E(t)$ in the complex plane does not have poles in the lower complex half plane, whereas the causality of $E(t)$ considered in [1], [3], [4], [5], [6], namely $E(t) = 0$ for all $t \\le 0$ , implied that the analytic continuation of $\\tilde{E}(\\omega )$ in the complex $\\omega $ plane had no poles in the upper complex half plane.", "The necessary and sufficient condition becomes that $\\tilde{E}(\\omega )$ does not encircle the origin when $\\omega $ goes from $-\\infty $ to $\\infty $" ], [ "Conclusions", "We have given a necessary and sufficient condition for a function $E(t)$ to be of minimum phase, and hence for its phase to be univocally determined by its intensity $\|E(t)\|^2$ .", "The check of this condition requires only the plot of the function $E(t)$ for $t$ belonging to the real axis, and does not require the analytic continuation of $E(t)$ in the complex plane.", "We have shown that sufficient conditions previously proposed can be simply derived from this more general one.", "As an application to communication systems, we find that the recently proposed KK transmission scheme gives a practical way to decode all distinguishable band-limited fields when the detector is sensitive only to the intensity of the field and insensitive to its phase." ], [ "Encoding information on the intensity of an electromagnetic field", "Let us now consider the following problem: What is the most efficient transmission over a bandwidth $B=1/T$ for the complex field $E(t)$ if we have the capability of modulating the optical field in modulus and phase and we perform square-law intensity detection of the transmitted signal at the receiver?", "The study of transmission systems where only direct detection is applied at the receiver became lately an intense area of research, targeting high capacity applications in the short reach range [16], [17], [18], [19], [20], [21].", "Although the information is not contained in the phase of the optical field, a suitable phase shaping is required to confine the signal within a limited bandwidth.", "One could in principle think of generating an arbitrary amplitude modulation at bandwidth $2B$ and then use a phase modulation equal to the logarithmic Hilbert transform of the intensity that makes this signal single sideband, as proposed in [20], but in general the spectrum of the signal after modulation is not zero for $\\omega > 2 \\pi B$ , and is in principle unbounded for positive frequencies.", "The analysis of the present paper gives a general solution to this problem.", "We have shown that all possible field patterns of a given optical bandwidth can be grouped into classes of fields with the same intensity pattern, and theorem REF ensured that to each of these classes belongs the minimum phase signal $E_0$ .", "Restricting the set of symbols to minimum phase signals therefore does not reduce the set of symbols available to transmission.", "The recently proposed Kramers Kronig (KK) transmission scheme [22] is capable of receiving without errors any minimum phase signal $E_0(t)$ and hence it maximizes the set of symbols that can be received over a given optical bandwidth.", "The principle of operation of the receiver is very simple: the receiver detects the intensity profile only, and the phase of the field is calculated from the intensity profile using the logarithmic Hilbert transform () over an integration time window much longer than the inverse of the optical field bandwidth.", "Assuming that chromatic dispersion is optically compensated or pre-compensated at the transmitter, we will show in the section that follows that a signal constellation like () with real$(a_n) > a$ and imag$(a_n) > a$ with $a$ a suitably chosen positive constant produces a minimum phase signal.", "It is enough, in general, to choose $a$ much smaller than the range of possible values of $a_n$ , so that the available symbols are almost all those belonging to one quadrant, say the first, of the complex plane, one quarter of all possible symbols available on the two quadratures.", "If some additive noise impairs the signal field, and if the noise is small enough that the perturbed field does not encircle the origin, then the perturbed field is still of minimum phase and faithfully detected by the receiver.", "Therefore, in spite of the fact that the quadratic receiver detects the field intensity only, and that the phase is calculated from the intensity profile and not independently measured, the receiver acts as a linear receiver (in the field) that is capable of detecting signals in the first quadrant only.", "Consequently, for high signal-to noise ratio, we may conjecture that the capacity of such system is of the order of the capacity (per unit bandwidth) of a coherent system in which the symbols are constrained in the first quadrant only, approximately two bit less than the capacity of a full, unconstrained, coherent system [22], [23]." ], [ "Numerical validation", "We numerically validated the results of the previous sections by testing on a signal made as the sequence of 512 waveforms of the form $(\\ref {mod})$ with $a_n = [(b + k_1) + i(b + k_2)] \\sqrt{T}$ , where $n$ runs from 1 to 512 and $k_1$ and $k_2$ are randomly chosen between 0 and 7.", "This signal corresponds to a shifted 16 quadrature-amplitude modulation (16QAM) constellation.", "The bias $b$ was chosen equal to two values, one $b_3 = 0.5$ insufficient to make the waveform of minimum phase, and the second $b_0 = 1.1$ , that makes the waveform a minimum phase one.", "Figure: Complex field of a shifted 16QAM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 3 =0.5b_3 = 0.5.The numerical analysis was performed using a Marlab program, performing the Hilbert transform over a finite time window $T_w$ that in the examples given was chosen as $T_w = 512 \\, T$ , by multiplication of the spectrum of the signal by $-i \\, \\mathrm {sgn}(\\omega )$ and using a fast Fourier transform (FFT) and inverse FFT routines, i.e.", "implicitly assuming a periodic signal instead of an $L^2$ one.", "The analysis of the periodic case implied by the use of the FFT can be done by replacing the kernel of the Hilbert transform $1/t$ by its $T_w$ -periodic counterpart $\\sum _{k=-\\infty }^\\infty \\frac{1}{t - k T_w} = \\frac{\\pi }{T_w} \\mathrm {cot}\\left(\\frac{\\pi t}{T_w} \\right), \\quad t \\ne \\tau , $ and limiting the integration over $\\tau $ to a single period $[-T_w/2, T_w/2)$ .", "This corresponds to replace Fourier integrals with Fourier series.", "All the theorems that we have discussed retain their validity.", "Theorem for instance can be shown without closing the curve $\\Gamma $ with a semicircle $C$ at infinity, because when $t$ spans on the real axis an entire period the function $E(t)$ describes in the complex plane a closed curve.", "While the rigorous analysis of the periodic case is beyond the scope of this paper, the correspondence between the periodic and the $L^2$ case can be obtained considering the case $T_w \\rightarrow \\infty $ .", "In the $L^2$ case, the field $\\overline{E}$ is the time average of the field $E(t)$ $\\overline{E} = \\lim _{T_w \\rightarrow \\infty } \\frac{1}{T_w} \\int _{-T_w/2}^{T_w/2} E(t) {\\rm d}t, $ so that in the periodic case the role of the field bias $\\overline{E}$ is played by the average of the signal over the length of the symbol sequence $T_w$ $\\overline{E}_\\mathrm {av} = \\frac{1}{T_w} \\int _{-T_w/2}^{T_w/2} {\\rm d}t E(t).", "$ An intuitive understanding of this correspondence can be obtained by looking at the spectrum of $E(t) = E_s(t) + \\bar{E}$ , namely $\\tilde{E}(\\omega ) = \\tilde{E}_s(\\omega ) + 2 \\pi \\bar{E} \\delta (\\omega )$ , and the spectrum of the periodic sequence.", "Both $\\overline{E}$ and $\\overline{E}_\\mathrm {av}$ are the amplitude of the spectral component at zero frequency, the Dirac delta function in $\\tilde{E}(\\omega )$ being replaced by the amplitude of the discrete spectral component at zero frequency in the periodic case.", "Figure REF shows by a blue dashed line a field $E(t)$ obtained with the lowest value of the bias, and by a red solid line the curve $E_0(t)$ reconstructed by the logarithmic Hilbert transform.", "Big blue dots represent the values of the $E(t)$ at $t = nT_w$ , and red smaller dots the values of the reconstructed field at the same times.", "The curve $E(t)$ has a winding number around the origin of 3.", "The red solid curve overlaps with the blue dashed one almost everywhere, with the exception of the vicinity of the points where the windings occur.", "The accuracy of the reconstruction is more evident if we plot the phase reconstructed by the logarithmic Hilbert transform on top of the phase of $E(t)$ .", "This comparison is shown in Figs.", "REF and REF , where we show by a red solid line the reconstructed phase and by a blue dashed line the phase of $E(t)$ .", "In Fig.", "REF three phase jumps corresponding to the three windings of $E(t)$ around the origin are clearly visible.", "Figure REF is the zoom of the plot of Fig.", "REF in the vicinity of the second phase jump, showing that the deviation of the phase reconstruction from the phase of $E(t)$ is confined to a small region around the jump.", "The numerical results confirm the conjecture proposed the previous section that the constraint of square-law intensity detection reduces the number of available symbols by approximately one quarter, implying a capacity reduction of two bits over the full coherent detection for the same optical bandwidth.", "Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =0.5b_3 = 0.5.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =0.5b_3 = 0.5.", "This is the zoom in the region of the second phase jump of Fig.", "In Figs.", "REF we show the ratio $\|H_3(t) -1\| = \|E(t)/E_0(t) - 1\|$ for the field $E(t)$ of Fig.", "REF .", "Figure REF clearly shows the three Lorentzian of amplitude 2 corresponding to the three encircling of the origin of $E(t)$ .", "Figure REF shows by a blue solid thin line the same curve in a semilogarithmic scale, and by a red dashed thick line an interpolation with the curve $H_3(t) - 1 = \\prod _{k=1}^3 \\left[1 - \\sum _{h=-\\infty }^{\\infty } \\frac{ 2 }{1 - i (t-t_k - h T_w)/\|\\tau _k\|}\\right], $ which, using Eq.", "(REF ), becomes $H_3(t) - 1 = \\prod _{k=1}^3 \\left[1 - \\frac{2 \\pi i \|\\tau _k\|}{T_w} \\mathrm {cot} \\left(\\pi \\frac{t-t_k+i \|\\tau _k\|}{T_w} \\right) \\right].", "$ Equation (REF ) was obtained adapting Eq.", "() to account for the temporal periodicity induced by the use of the FFT algorithm, which introduces an infinite number of replicas of the Lorentzian line-shapes spaced by the time window $T_w$ .", "The parameters were obtained by interpolation of the main peaks only and were for the real parts $t_1/T_w=108.5755$ , $t_2/T_w = 272.4868$ , and $t_3/T_w = 384.5205$ , and for the imaginary parts $\|\\tau _1\|/T_w = 0.0070$ , $\|\\tau _2\|/T_w = 0.027$ and $\|\\tau _3\|/T_w = 0.041$ .", "As shown in Fig REF , the expression given in Eq.", "(REF ) was accurate more than six orders of magnitude down to the main peaks.", "However, a single Lorentzian, i.e.", "only the dominant term with $h =0$ in Eq.", "(REF ), is accurate 2 orders of magnitudes down to the main peak, and is sufficient to exactly reproduce the plot in linear scale shown in Fig.", "REF .", "From the average time and the temporal width of each Lorentzian line-shape we were able to compute, with high accuracy, the position of the zeros of $E(t+i\\tau )$ in the lower complex half-plane without the need of numerical analytic continuation of $E(t)$ .", "Figure: \|H 3 (t)-1\|\|H_3(t)-1\| vs. t/Tt/T for a bias b 3 =1.1b_3 = 1.1.Figure: \|H 3 (t)-1\|\|H_3(t)-1\| vs. t/Tt/T for a bias b 3 =1.1b_3 = 1.1 shown in Fig.", ", in a semilogarithmic scale.", "The blue solid thin line is the numerical result reported in Fig.", ", the red dashed thick line is the interpolation using Eq.", "().Figures REF –REF shows the curves obtained with the same $a_n$ sequence used for Figs.", "REF –REF but with a larger value of the bias $b_0 = 1.1$ .", "In this case, there are no windings of $E(t)$ around the origin, and the reconstructed field coincides with the original one, $E_0(t) \\equiv E(t)$ .", "Figure REF shows the perfect reconstruction of the field at the sampling point by the overlap of the red and the big blue dots.", "Figure: Complex field of a shifted 16QAM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 3 =1.1b_3 = 1.1.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =1.1b_3 = 1.1.", "The smaller excursion of the phase shown here compared with that of Fig.", "is caused by the larger distance from the origin due to the larger bias.Figure: Phase of a shifted 16QAM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 3 =1.1b_3 = 1.1.", "This is the zoom in the same region of the phase jump of Fig.", ".The example given in Figs.", "REF –REF corresponds to a case in which beside the condition of no windings around the origin of the trajectories of $E(t)$ also the condition $\|E_s(t)\|^2 < \\overline{E}^2$ was fulfilled.", "Figures REF and REF show instead a case where the condition $\|E_s(t)\|^2 < \\overline{E}^2$ fails but the reconstruction of the phase and consequently of the field from the intensity profile is perfect because no windings around the origin occur.", "These curves were obtained using $a_n = (b + k) \\sqrt{T}$ , with $n$ running from 1 to 512 and $k$ randomly chosen between 0 and 7, corresponding to an amplitude modulation with 8 levels.", "The value of $b$ was $b_0 = 0.1$ .", "Figure: Complex field of a shifted 8AM.", "Red solid line: reconstructed field E 0 (t)E_0(t), blue dashed line, detected field E(t)E(t), for a bias b 0 =0.1b_0 = 0.1.Figure: Phase of a shifted 8AM.", "Red solid line: reconstructed phase profile, blue dashed line, phase of E(t)E(t), vs. t/Tt/T, for a bias b 0 =0.1b_0 = 0.1." ], [ "Time-frequency duality", "In this paper, we have considered functions $E(t)$ whose Fourier transform $\\tilde{E}(\\omega )$ was zero for $\\omega < 0$ , and our goal was to calculate the phase of $E(t)$ once its intensity $\|E(t)\|^2$ was measured.", "Once the necessary changes are made, this case has a one to one correspondence with the case studied in [1], [3], [4], [5], [6] of causal functions $E(t)$ , i.e.", "such that $E(t) = 0$ for $t \\le 0$ , where the goal was to find the phase of $\\tilde{E}(\\omega )$ once the power spectrum $\|E(\\omega )\|^2$ was measured over a range of frequency.", "The necessary changes include, for instance, the fact that the property $\\tilde{E}(\\omega ) =0$ for all $\\omega < 0$ implied, as we have seen, that the analytic continuation of $E(t)$ in the complex plane does not have poles in the lower complex half plane, whereas the causality of $E(t)$ considered in [1], [3], [4], [5], [6], namely $E(t) = 0$ for all $t \\le 0$ , implied that the analytic continuation of $\\tilde{E}(\\omega )$ in the complex $\\omega $ plane had no poles in the upper complex half plane.", "The necessary and sufficient condition becomes that $\\tilde{E}(\\omega )$ does not encircle the origin when $\\omega $ goes from $-\\infty $ to $\\infty $" ], [ "Conclusions", "We have given a necessary and sufficient condition for a function $E(t)$ to be of minimum phase, and hence for its phase to be univocally determined by its intensity $\|E(t)\|^2$ .", "The check of this condition requires only the plot of the function $E(t)$ for $t$ belonging to the real axis, and does not require the analytic continuation of $E(t)$ in the complex plane.", "We have shown that sufficient conditions previously proposed can be simply derived from this more general one.", "As an application to communication systems, we find that the recently proposed KK transmission scheme gives a practical way to decode all distinguishable band-limited fields when the detector is sensitive only to the intensity of the field and insensitive to its phase." ] ]	1606.04861
[ [ "Theorem: A Static Magnetic N-pole Becomes an Oscillating Electric N-pole\n in a Cosmic Axion Field" ], [ "Abstract We show for the classical Maxwell equations, including the axion electromagnetic anomaly source term, that a cosmic axion field induces an oscillating electric N-moment for any static magnetic N-moment.", "This is a straightforward result, accessible to anyone who has taken a first year graduate course in electrodynamics." ], [ "FERMILAB-PUB-16-222-T Theorem: A Static Magnetic $N$ -pole Becomes an Oscillating Electric $N$ -pole in a Cosmic Axion Field Based upon invited talks given at Oxford Universty, “The Axion-like Particles Workshop,\" IPPP, Durham University, U. of Minnesota, and U. of Chicago, (spring of 2016), and “The Invisibles Webinar,\" Valencia, Spain, (spring 2015).", "Based upon invited talks given at Oxford Universty, “The Axion-like Particles Workshop,\" IPPP, Durham University, U. of Minnesota, and U. of Chicago, (spring of 2016), and “The Invisibles Webinar,\" Valencia, Spain, (spring 2015).", "Christopher T. Hill [email protected] Fermi National Accelerator Laboratory P.O.", "Box 500, Batavia, Illinois 60510, USA $ $ We show for the classical Maxwell equations, including the axion electromagnetic anomaly source term, that a cosmic axion field induces an oscillating electric $N$ -moment for any static magnetic $N$ -moment.", "This is a straightforward result, accessible to anyone who has taken a first year graduate course in electrodynamics.", "14.80.Bn,14.80.-j,14.80.-j,14.80.Da The action for electrodynamics in the presence of an axion field, $\\theta = a(x)/f_a$ , is [1]: $S =\\int d^4 x\\; \\left(-\\frac{1}{4}F_{\\mu \\nu }F^{\\mu \\nu } - \\frac{1}{4} g\\theta (x) F_{\\mu \\nu }\\widetilde{F}^{\\mu \\nu } \\right)$ where $\\widetilde{F}^{\\mu \\nu }=(1/2) \\epsilon ^{\\mu \\nu \\rho \\sigma }F_{\\rho \\sigma }$ .", "$g$ is the anomaly coefficient, which is model dependent but typically of order $\\sim 10^{-3}$ .", "Eq.", "(REF ) leads to Maxwell's equations, $\\partial _\\mu F^{\\mu \\nu } =g\\partial _\\mu \\theta \\widetilde{F}^{\\mu \\nu }$ .", "We specialize to a cosmic axion field in its rest-frame, where it is a pure oscillator plus a constant (or slowly varying function), $\\overline{\\theta }(x,t)$ , and $m_a$ is the axion mass: $\\theta (t) =\\theta _0\\sin (m_a (t-t_0)) +\\overline{\\theta }.", "\\qquad $ In this frame we assume electromagnetic fields of the form: $\\overrightarrow{E} = \\overrightarrow{E}_{r} $ and $\\overrightarrow{B} = \\overrightarrow{B}_{0}+\\overrightarrow{B}_{r} $ .", "$\\overrightarrow{B}_{0}$ is a large static, applied magnetic field, and $\\overrightarrow{E}_{r}$ and $\\overrightarrow{B}_{r}$ are small oscillating “response fields.”Note that $ \\overrightarrow{E}_{r} $ and $ \\overrightarrow{B}_{r} $ are of order $g\\theta _0 << 1$ where $\\theta _0\\sim 3\\times 10^{-19}$ when one matches to the local galactic halo density of $\\sim 300$ MeV/cm$^3$ [2].. Maxwell's Equations in these fields, to first order in $g\\theta _0 $ , become: $\\overrightarrow{\\nabla }\\times \\overrightarrow{B}_{r}-\\partial _{t}\\overrightarrow{E}_{r}& = & -g\\overrightarrow{B}_{0} \\partial _{t}\\theta \\nonumber \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{E}_{r}+\\partial _{t}\\overrightarrow{B}_{r}& =& 0$ and $\\ \\ \\overrightarrow{\\nabla }\\cdot \\overrightarrow{B}_{r}=\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}_{r}=0.$ These are standard and lead to, e.g., the RF-cavity solutions [2] Consider an applied magnetic field, $ \\overrightarrow{B}_0$ , arising from a static, classical, magnetic dipole, $\\overrightarrow{m} $ , with a pointike magnetization $\\overrightarrow{m}\\delta ^3(\\overrightarrow{r})$ .", "This produces the familiar result, $\\overrightarrow{B}_0= - \\frac{1}{4\\pi } \\left( \\frac{1}{r^3} \\right) \\left(\\overrightarrow{m}-\\frac{3\\overrightarrow{r}\\left( \\overrightarrow{r}\\cdot \\overrightarrow{m}\\right) }{r^2}\\right) +\\frac{2}{3}\\overrightarrow{m}\\delta ^3(\\vec{r})$ and $\\overrightarrow{\\nabla }\\cdot \\overrightarrow{B}_0 =0$ .", "This expression is well-known, such as in eq.", "(5.64) of Jackson [3].", "Observe, however, that eq.", "(REF ) can be rewritten in an equivalent form: $\\overrightarrow{B}_0= \\overrightarrow{m}\\delta ^3(\\vec{r})+\\frac{1}{4\\pi } \\overrightarrow{\\nabla }\\left( \\overrightarrow{m}\\cdot \\overrightarrow{\\nabla }\\frac{1}{r} \\right)$ Eq.", "(REF ) yields eq.", "(REF ) upon computing the gradient term by using the identity: ${\\nabla }_i\\frac{r_j}{r^3} = \\delta _{ij}\\frac{1}{r^3} -3 \\frac{r_i r_j}{r^5}+ \\frac{4\\pi }{3}\\delta _{ij}\\delta ^3(\\vec{r})$ Note that, if we contract $(ij)$ , then eq.", "(REF ) becomes $\\overrightarrow{\\nabla }\\cdot \\frac{\\vec{r}}{r^3} ={4\\pi }\\delta ^3(\\vec{r})$ , which is just Gauss' law (we emphasize that the singularities can be replaced by smooth, localized Gaussians).", "The Maxwell equations thus become: $\\overrightarrow{\\nabla }\\times \\overrightarrow{B}_{r}-\\partial _{t}\\overrightarrow{E}_{r}& = & -g\\partial _t \\theta \\left(\\overrightarrow{m}\\delta ^3(\\vec{r})+\\frac{1}{4\\pi } \\overrightarrow{\\nabla }\\left( \\vec{m}\\cdot \\overrightarrow{\\nabla }\\frac{1}{r} \\right)\\right)\\nonumber \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{E}_{r}+\\partial _{t}\\overrightarrow{B}_{r}& = &0$ We now make a redefinition of the electric field by shifting away the gradient term, where we assume $\\partial _t\\overline{\\theta }(x,t)\\approx 0$ : $\\overrightarrow{E}_{r}=\\overrightarrow{E}^{\\prime }_{r}+\\frac{1}{4\\pi }g \\widetilde{\\theta } \\;\\overrightarrow{\\nabla }\\left( \\overrightarrow{m}\\cdot \\overrightarrow{\\nabla }\\frac{1}{r} \\right)$ Here $\\widetilde{\\theta }(t) =\\int ^t_{t_0} d\\tau \\partial _t \\theta (t) = \\theta _0\\sin (m_a (t-t_0))$ and has the property that $\\widetilde{\\theta }(t)\\rightarrow 0$ as $m_a\\rightarrow 0$ .", "Note that as $m_a\\rightarrow 0$ then $\\partial _t\\theta \\rightarrow 0$ and $\\widetilde{\\theta }(t)\\rightarrow 0$ .", "The constant $\\overline{\\theta }$ has disappeared from the physics and the theory has the “shift symmetry” $\\overline{\\theta } \\rightarrow \\overline{\\theta } + k$ for any constant $k$ .", "This is called “axion decoupling,” and it is somewhat more involved with general $\\theta (x,t)$ [2].", "We note that in an RF-cavity, such as ADMX, an oscillating electric field develops along the cavity axis driven by the axion and the same eqs.", "(REF ) with $\\overrightarrow{E}_{r}\\propto \\widetilde{\\theta }(t)$ [2].", "Maxwell's equations act as a “high-pass filter” for $\\theta (t)\\rightarrow \\widetilde{\\theta }(t) $ .", "The shift is a purely longitudinal (gradient) term and cannot affect the radiation field.", "Since $\\overrightarrow{\\nabla }\\times \\overrightarrow{\\nabla }(X) = 0$ , we have $\\overrightarrow{\\nabla }\\times \\overrightarrow{E}_{r}= \\overrightarrow{\\nabla }\\times \\overrightarrow{E}^{\\prime }_{r}$ , and the second Maxwell equation, (REF ), is unaffected by the shift.", "Hence: $\\overrightarrow{\\nabla }\\times \\overrightarrow{B}_{r}-\\partial _{t}\\overrightarrow{E}^{\\prime }_{r}& = & -g\\partial _t \\theta \\overrightarrow{m}\\delta ^3(\\vec{r})\\nonumber \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{E}^{\\prime }_{r}+\\partial _{t}\\overrightarrow{B}_{r}& = &0$ The Maxwell equations, in terms of $\\overrightarrow{B}_r$ and $\\overrightarrow{E}^{\\prime }_{r}$ , describe the radiation field produced by the anomaly and magnetic dipole source.", "Note that the extended magnetic dipole field has completely disappeared, leaving only the pointlike source.", "Equations (REF ) are identical to those of an oscillating electric dipole moment $\\overrightarrow{p}(t)$ (the “Hertzian” dipole): $\\overrightarrow{\\nabla }\\times \\overrightarrow{B}_{r}-\\partial _{t}\\overrightarrow{E}^{\\prime }_{r}& = & -\\partial _t \\overrightarrow{p}(t)\\nonumber \\\\\\overrightarrow{\\nabla }\\times \\overrightarrow{E}^{\\prime }_{r}+\\partial _{t}\\overrightarrow{B}_{r}& = &0$ where we have the correspondence: $\\overrightarrow{p}(t)=g \\widetilde{\\theta }(t)\\overrightarrow{m}\\delta ^3(\\vec{r})$ One may have noticed that, upon performing the shift, eq.", "(REF ), the $\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}^{\\prime }_{r}$ equation is now modified, $\\!\\!\\!\\!\\!\\!\\!\\!", "\\!\\!\\!\\!\\!\\!\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}^{\\prime }_{r}& =& -\\frac{1}{4\\pi }g\\widetilde{\\theta }\\overrightarrow{\\nabla }^2 \\overrightarrow{m}\\cdot \\overrightarrow{\\nabla }\\frac{1}{r}= g\\widetilde{\\theta }(t)\\vec{m}\\cdot \\overrightarrow{\\nabla }\\delta ^3(\\overrightarrow{r}).$ In the second term, using eq.", "(REF ) we have: $\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}^{\\prime }_{r}=\\overrightarrow{\\nabla }\\cdot \\overrightarrow{p}$ However, eq.", "(REF ) together with eq.", "(REF ) are precisely the Maxwell equations obtained from an action containing an electric dipole term, $S = \\int d^4x \\; \\left(-\\frac{1}{4}F_{\\mu \\nu }F^{\\mu \\nu } -\\overrightarrow{p}(t)\\cdot \\overrightarrow{E}\\right)$ The axion causes the magnetic dipole to radiate.", "The radiation from a static magnetic moment in an axion field is identical to the Hertzian electric dipole radiation (see e.g., Jackson, [3] eqs.", "(9.18, 9.19)).", "In the near-zone the two terms on the rhs of eq.", "(REF ) cancel as $m_a\\rightarrow 0$ , causing $\\overrightarrow{E}_{r}$ to vanish.", "Hence there is no persistent electric field in the $\\widetilde{\\theta }\\rightarrow $ constant limit!", "In the far-zone we have: $\\overrightarrow{E}_{r}(x,t) & = & gm_{a}^{2}\\widetilde{\\theta }(t-r)\\left( \\frac{\\overrightarrow{m}}{r}-\\frac{\\overrightarrow{r}}{r^{2}}\\frac{\\overrightarrow{m}\\cdot \\overrightarrow{r}}{r}\\right)\\nonumber \\\\\\overrightarrow{B}_{r}(x,t) & = & -gm_{a}^{2}\\widetilde{\\theta }(t-r)\\left( \\ \\overrightarrow{m}\\ \\times \\frac{\\overrightarrow{r}}{r^{2}}\\right)$ Note the CP-violation in the alignment of the polarization of the vector $\\overrightarrow{E}_{r}$ with the axial vector $\\overrightarrow{m} $ , which arises from the background axion field.", "From eq.", "(REF ) we obtain the total emitted power, $P= g^{2}m_{a}^{4}\\theta _{0}^{2}\|\\overrightarrow{m}\|^{2}/12\\pi .$ This is equivalent to the quantum result for a spin-up to spin-up electron with $\|\\overrightarrow{m}\|^{2}\\rightarrow \\mu _{Bohr}^2 $ .", "Coherent assemblages of many electrons can produce potentially observable effects.", "The physical applications of this are discussed in [2].", "In the near-zone there is a cancellation of $\\overrightarrow{E}^{\\prime }_{r}$ with the shifted piece of eq.", "(REF ), so there is no persistent constant electric dipole field in $\\overrightarrow{E}_{r}$ in the $\\widetilde{\\theta }\\rightarrow $ constant limit (see eq.", "(56) in [2]).", "Alternatively, we can see this result directly from the action, eq.", "(REF ), without performing the shift.", "We decompose $\\overrightarrow{E}_{r}$ into transverse and longitudinal (gradient) components: $\\overrightarrow{E}_{r}=\\overrightarrow{E}_{rT} + \\overrightarrow{E}_{rL}$ where (i): $\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}_{rT}=0$ , and (ii): $\\overrightarrow{E}_{rL}= \\overrightarrow{\\nabla }[(1/\\overrightarrow{\\nabla }^2)\\overrightarrow{\\nabla }\\cdot \\overrightarrow{E}_{r}]$ .", "The anomaly term in the action of eq.", "(REF ), $\\int g\\theta \\overrightarrow{E}_r\\cdot \\overrightarrow{B}_0$ , becomes, upon integrating by parts and using (i), (ii): $\\int g\\theta \\overrightarrow{E}_{rT}\\cdot \\overrightarrow{m}\\delta ^3(\\overrightarrow{r})$ .", "Thus, only the transverse electric field couples to the axion-induced OEDM.", "The axionic OEDM radiates transverse radiation, and it will interact in the usual way, as in eq.", "(REF ) with an applied transverse $\\overrightarrow{E}_T$ (t), e.g., such as a cavity mode or light.", "Thus, we can write a complete action by replacing $\\overrightarrow{p}(t)$ by $\\overrightarrow{p}^{\\prime }(t)$ in eq.", "(REF ) where $\\overrightarrow{p}^{\\prime }(t)=\\overrightarrow{p}(t)-\\overrightarrow{\\nabla }(1/\\overrightarrow{\\nabla }^2)\\overrightarrow{\\nabla }\\cdot \\overrightarrow{p}(t)$ , as in [4], [2].", "We can build up a quadrupole source from a pair of dipoles, and an octupole source from a pair of quadrupoles, etc.", "Thus, by superposition, the axion causes any static magnetic $N$ -pole to become an oscillating electric $N$ -pole.", "We've seen that the extended magnetic dipole field is irrelevant to the radiation, only the point-like singularity matters, a result that translates to the $N$ -pole case.", "We thus call this an effective OEDM, in the sense of e.g., Fermi's effective weak interaction.", "It is “effective” as a leading order result in a tiny $g\\theta _0$ coupling constant.", "I thank W. Bardeen and A. Vainshtein for helpful discussions.", "This work was done at Fermilab, operated by Fermi Research Alliance, LLC under Contract No.", "DE-AC02-07CH11359 with the United States Department of Energy." ] ]	1606.04957
[ [ "Formation of the G-ring arc" ], [ "Abstract Since 2004, the images obtained by Cassini spacecraft's on-board cameras have revealed the existence of several small satellites in the Saturn system.", "Some of these small satellites are embedded in arcs of particles.", "While these satellites and their arcs are known to be in corotation resonances with Mimas, their origin remains unknown.", "This work investigates one possible process for capturing bodies into a corotation resonance, which involves raising the eccentricity of a perturbing body.", "Therefore, through numerical simulations and analytical studies, we show a scenario that the excitation of Mimas' eccentricity could capture particles in a corotation resonance and given a possible explanation for the origin for the arcs." ], [ "Introduction", "Since 2004, the Cassini spacecraft has returned to Earth a copious amount of data about the Saturnian system.", "In particular, its imaging system revealed the existence of several small satellites.", "There are papers [19], [5], [10] showing that Mimas perturbs the orbits of some of these satellites by resonant interactions.", "In 2006 data returned by Cassini revealed an arc of dust inside the G-ring.", "Since then, this region has been explored by [9] precisely calculated the G arc's mean motion using the Cassini's data.", "It revealed that this arc is in a 7:6 corotation resonance with Mimas.", "Through the analysis of images obtained between 2008 and 2009, it was found that the satellite Aegaeon was located inside this arc [10].", "And since this satellite is immersed in this arc, it is also trapped in this corotation resonance with Mimas.", "[10] confirm this corotation resonance through numerical simulations of the full equations of motion.", "They also identify that this satellite is in corotation resonance with characteristic angle $\\varphi _{{}_{CER}}=7\\lambda _{{}_{Mimas}}-6\\lambda _{{}_{Aegaeon}}-\\varpi _{{}_{Mimas}}.$ Their simulations also show that a nearby Lindblad resonance also influences the moon's motion [10].", "Prior this study and also from Cassini images two others satellites were discovered, Methone in 2004 [19] and Anthe in 2007 [5].", "The numerical analysis of the full equations of motion [19], [5] indicates that these satellites are in corotation resonances whose characteristic angles are, respectively, $&\\varphi _{{}_{CER}}&=15\\lambda _{{}_{Methone}}&-14\\lambda _{{}_{Mimas}}&-\\varpi _{{}_{Mimas}},\\\\&\\varphi _{{}_{CER}}&=11\\lambda _{{}_{Anthe}} &-10\\lambda _{{}_{Mimas}}&-\\varpi _{{}_{Mimas}}.&$ The Cassini images also show that Aegaeon, Anthe and Methone are immersed in tenuous arcs of particles [9], [11] and [10] said that these particles probably have been originated from the material knocked off, at low speeds, from the surface of these satellites.", "Therefore, these particles don't have enough energy to escape the corotation resonance and then remain close to the satellite filling the nearby space.", "With these evidences [10] concluded that the study of these satellites and their arcs may improve the understanding of the connection between satellites and rings.", "Because these objects and their arcs are in corotation resonances, these satellites may be seen as a distinct class of objects in the Saturn system.", "The existence of these satellites immersed in arcs of dust allows us to infer two possibilities for their origin.", "(i) The satellites were build-up by particles previously caught in resonance and their arcs are the vestiges of this formation.", "Alternatively, (ii) Mimas had captured satellites already formed, and consequently the arcs had originated from particles that had broken off from these satellites.", "Although we are not able to judge immediately which of the above statements are the correct one, we realize that both statements depend on the possibility of bodies been captured in corotation resonance with Mimas, either small particles or a full satellite.", "Thus, in this work we investigate the mechanism of capturing particles in corotation resonance.", "The corotation resonance exists only when the perturbing satellite has eccentricity different from zero [15].", "Then, in our problem, corotation depends on the eccentricity of Mimas.", "And it is known that, due to tidal effects, the eccentricity of Mimas should be lower than the current value [14].", "But Mimas has a higher eccentricity than most of Saturn's regular satellites, it is likely that his eccentricity was increased through a resonant interaction with another satellite.", "In this work we will consider the presence of Enceladus playing an important role in this scenario.", "The aim of this paper is to investigate the mechanisms which would make Mimas capture particles in a corotation resonance in the past of the Saturn system.", "The study will be made through numerical simulations and analytical studies which will be developed using the dynamics of three and four bodies problems considering the effects of non-spherical shape of Saturn.", "In our problem the main bodies are Saturn, Mimas, Enceladus and particles of G-ring.", "We will study the scenario where particles will be captured in corotation resonance with Mimas when Mimas passes through a resonance with Enceladus.", "As a result of this study, we will have a better understanding of the dynamics involved in the origin and stability of small satellites.", "In this study, we will focus on 7:6 corotation resonance In section we make a brief discussion on the corotation resonance.", "Section introduces the mechanism we develop to obtain the eccentricity variation of Mimas.", "Section shows the effects of this mechanism on Mimas orbit.", "Section presents the results of this mechanism when a ring of particles is immersed in the 7:6 corotation resonance region.", "In Section we analyze the process of capture due the migration.", "Finally, the concluding remarks for this study will be found in the section ." ], [ "Resonances in oblate planets", "When an object orbits an oblate planet, its orbit experiences effects of a potential which depends on the planet's zonal harmonic coefficients $J_2$ , $J_4$ , $\\cdots $ .", "The perturbations of that potential cause the rotation of the orbit in space, the precession of the unperturbed orbit.", "The rotation of the orbit generates three frequencies: $n$ , the mean motion; $\\kappa $ and $\\nu $ , radial/vertical epicyclic frequency, respectively [15].", "Thereby, considering the additional gravitational effects of a perturbing satellite on a particle, when this system is around an oblate planet, the orbit of the particle can be analysed through those frequencies.", "Those frequencies are associated with the precession of the node and pericenter of the orbit by $\\kappa = n - \\dot{\\varpi }$ and $\\nu = n - \\dot{\\Omega }$ and then the definition of the corotation resonance [15] occurs when: $\\varphi _{{}_{CR}} = j\\lambda ^{\\prime } + (k+p-j)\\lambda - k\\varpi ^{\\prime } - p\\Omega ^{\\prime },$ where the primed orbital elements belong to the perturbing satellite while the non-primed orbital elements belong to the particle, $j, k, p$ are integer values.", "When a particle is in corotation resonance with a perturbing satellite, some orbital parameters are modified.", "There are analytical models able to estimate the extent of these variations, for example, the Pendulum Model or the Hamiltonian Approach [15].", "The orbital parameters that is most strongly affected by a corotation resonance is the semi-major axis.", "Using the Pendulum Model, we can calculate the maximum width libration of the semi-major axis for corotation resonance.", "The maximum width for a corotation resonance is ([15], Eq.", "(10.10)) $W_{CR} = 8 \\left( \\frac{a\|R\|}{3Gm_p} \\right)^{1/2} a,$ where $a$ is the semi-major axis of the perturbed body, $m_p$ is the central body's mass and $R$ is the relevant term of the perturbing function, whose equation is $R = \\frac{Gm^{\\prime }}{a^{\\prime }} f_d(\\alpha ) \\, e^{\\prime }{}^{\|k\|} \\,s^{\\prime }{}^{\|p\|} \\, \\cos {\\varphi _{{}_{CER}}},$ where the primed parameters are of the perturbing satellite, with $m^{\\prime }$ , $a^{\\prime }$ , $e^{\\prime }$ and $s^{\\prime }$ as the mass, semi-major axis, eccentricity and a value associated to the inclination $I^{\\prime }$ , i.e.", "$s^{\\prime }=\\sin (I^{\\prime }/2)$ , $f_d(\\alpha )$ is a function in Laplace's coefficients for the directs terms of the perturbing function, $\\varphi _{{}_{CER}}$ is the corotation resonant angle, $k$ and $p$ are integers.", "Therefore, from the above equations, we expect that to existence of the corotation resonance, the perturbing satellite's eccentricity must be different from zero." ], [ "Model", "The current eccentricity of Mimas is approximately 0.02, and [14] pointed out that this value is relatively high and would imply a much higher value in the past, or it was recently excited.", "It is known that the eccentricity of a satellite can be excited due to resonances [15], but the recent value of Mimas' eccentricity cannot be explained by present resonances such as Mimas-Tethys 4:2 mean motion resonance [3], [2].", "Thus we suppose that Mimas experienced some event in the past that increased its eccentricity.", "[14] suggested that Mimas was captured by Enceladus or by Dione into a resonance while the eccentricity of Mimas was less than the current value.", "They verified that when Mimas came into some eccentricity-type resonance with one of these satellites, Mimas' eccentricity increased and even exceed its current value.", "After these satellites escaped from this resonance interaction, the tidal orbital evolution decreased their eccentricity to the current values, as we can notice in Figures 6, 7 and 9 from the paper of [14].", "Therefore, these kind of resonant encounters could have temporarily increased Mimas' eccentricity, which have flavoured the capture of particles in Mimas' corotation resonances.", "In our study we adopt a scenario where we have Mimas-Enceladus 3:2 e-Mimas resonance e-Mimas resonance is the notation of [14] for Mimas' eccentricity-type first order resonance.", "as discussed in [14].", "That resonance would induce an increase in Mimas' eccentricity even larger than the current one, and after a certain time those satellites could go out of that resonance.", "After the escape, due to the tidal effects, the eccentricity of Mimas would decay to the current value [14].", "In our scenario Mimas and Enceladus were closer to Saturn than they are today.", "Thus, we had to calculated a consistent position for them based on a satellite tidal evolution.", "For this task, we perform the procedures discussed below.", "First, we evaluate the ratio of the semi-major axes $\\alpha $ when Mimas and Enceladus were trapped in the 3:2 eccentricity mean motion resonance.", "When two satellites are in mean motion resonance, $\\alpha $ remains approximately constant and can be calculated with [3] $\\alpha = \\left( \\frac{p+q}{p}\\right)^{\\!-2/3} \\!\\!\\left( 1 \\!", "+ \\!", "\\frac{ q_{{}_1} \\dot{\\varpi }_{{}_M} \\!", "+ \\!q_{{}_2} \\dot{\\varpi }_{{}_E} \\!", "+ \\!q_{{}_3} \\dot{\\Omega }_{{}_M} \\!", "+ \\!q_{{}_4} \\dot{\\Omega }_{{}_E}}{n_{{}_E}(p+q)}\\right)^{\\!-2/3}\\hspace{-17.5pt},$ where $p$ , $q$ , $q_{{}_1}$ , $q_{{}_2}$ , $q_{{}_3}$ and $q_{{}_4}$ are integers; while $\\dot{\\varpi }_{{}_M}$ , $\\dot{\\varpi }_{{}_E}$ , $\\dot{\\Omega }_{{}_M}$ and $\\dot{\\Omega }_{{}_E}$ are precession rates of longitude of pericenter, longitude of ascending node for Mimas and Enceladus, respectively, and $n_{{}_E}$ is the mean motion of Enceladus.", "Using the angles values for the 3:2 eccentricity mean motion resonance in equation (REF ), we get the ratio between the semi-major axes of Mimas and Enceladus when they were trapped in that resonance [3].", "In this paper, the values of $\\dot{\\varpi }_{{}_M}$ , $\\dot{\\varpi }_{{}_E}$ , $\\dot{\\Omega }_{{}_M}$ and $\\dot{\\Omega }_{{}_E}$ are consistent with the geometric orbital elements [18].", "For the case of our resonance, we find the values of $q_{{}_1}$ , $q_{{}_2}$ , $q_{{}_3}$ and $q_{{}_4}$ from the comparison between general resonant angle, $\\varphi = p \\lambda - (p+q) \\lambda ^{\\prime }+ q_{{}_1} \\varpi + q_{{}_2} \\varpi ^{\\prime }+ q_{{}_3} \\Omega + q_{{}_4} \\Omega ^{\\prime },$ where $\\lambda $ , $\\varpi $ , and $\\Omega $ are the mean longitude, longitude of pericenter and longitude of the ascending node, respectively for the inner satellite, while those longitudes with prime represent the angles for the outer satellite, with resonant angle as, $\\varphi _{e} = 2\\lambda _{{}_M} - 3\\lambda _{{}_E} + \\varpi _{{}_M},$ where $\\lambda _{M}$ is the mean longitude of Mimas, $\\lambda _{E}$ is the mean longitude of Enceladus.", "After our evaluations, we obtain $\\alpha $ equal to $0.7637895$ .", "The next step was to find the semi-major axis of the satellites corresponding to our $\\alpha $ .", "We will call those semi-major axis ”ancient semi-major axes“ and it can be evaluated through the equation (the development of the following equation can be seen at appendix A) $a_{{}_{0M}} = \\left[ \\frac{a_{{}_M}^{13/2}\\,\\left(\\dfrac{m_{{}_E}}{m_{{}_M}}\\right)-a_{{}_E}^{13/2}}{\\left(\\dfrac{m_{{}_E}}{m_{{}_M}}\\right)-\\dfrac{1}{\\alpha ^{13/2}}}\\right]^{2/13},$ where $a_{{}_{0M}}$ is the Mimas' ancient semi-major axis, $a_{{}_M}$ , $a_{{}_E}$ , $m_{{}_M}$ and $m_{{}_E}$ are the current semi-major axes and mass of the satellites Mimas and Enceladus, respectively.", "Equation (REF ) is a function of $\\alpha $ , the Mimas and Enceladus' semi-major axes ratio.", "As we had used a value of $\\alpha $ when both satellites were in resonance with each other, we can locate the ancient semi-major axis of Enceladus using $a_{{}_{0E}}={a_{{}_{0M}}/\\alpha }$ .", "The results of $a_{0_{E}}$ and $a_{{}_{0M}}$ indicate that those satellites were more distant from each other in the past compared with their current positions.", "As expected, since the inner satellite migrate faster than the outer one.", "The values of Mimas and Enceladus ancient semi-major axis, calculated through equation (REF ), are approximated values.", "To obtain values which lead the system to enter in the resonance we try some values near the ancient semi-major axes until we got them close to a position in the verge of resonance.", "The tidal effect can cause Saturn's satellites migration with velocity orders between of e-7 and e-5km per year at their current positions [14], [12].", "Therefore, we should migrate Mimas and Enceladus with velocities close to those velocities.", "However, if we did that, it would take a long computational time, as we have noted in migration tests using that order of magnitude for the migration rate.", "As we are trying to prove the concept of corotation eccentricity resonance capture caused by the eccentricity of Mimas when it was enlarged due to mean motion resonance with Enceladus, therefore we used values which make our simulations more quickly.", "Thus, we migrate Mimas with a rate for the semi-major axis close to 1km per year and Enceladus around 0.01km per year.", "The external satellite will migrate slower than the internal one (see equation (REF ) and also [1]).", "To generate those rates, we inserted into the dynamic of each satellite a drag force $\\vec{F}_p = - \\gamma v\\vec{v},$ where $\\vec{v}$ is the velocity of the satellite, and $\\gamma $ a constant which will give the mentioned velocities for the semi-major axis.", "The value of semi-major axis will increase due to a negative sign for $\\gamma $ .", "Figure: Critical angle for Mimas-Enceladus 3:2 e-Mimas resonance (equation()) during the migration process.", "It shows a librationaround zero.", "To the end of tandem migration the critical angle increases itssweep following the increase on the eccentricity of Mimas.In a scenario with a lower eccentricity value for Enceladus, using the three-bodies dynamics for Saturn, Mimas and Enceladus, we found no chaotic behaviour for the resonance angle when we have Mimas-Enceladus 3:2 e-Mimas resonance.", "That is, we have found the same results that [14] had encountered for Mimas-Enceladus 3:2 e-Enceladus resonance.", "It means that Mimas does not escape from the resonance capture during our time integration.", "This integration had run for 3000years of arbitrary time (equivalent approximately to 300 million years, if we have used the corrects rates for migration).", "Tidal effects could have reduced a higher eccentricity of Enceladus, and in our scenario we verified that this higher eccentricity is necessary for Mimas to escape from the resonance.", "Higher values for Enceladus's eccentricity are possible due to earlier captures into others resonances as can be seen in [13].", "We used respectively 0.005 and 0.02 for the initial eccentricities of Mimas and Enceladus.", "These values make our scenario works very well as we will show below.", "Probably they may exist other mechanisms which could take Mimas out of the resonance other than this value for the eccentricity of Enceladus, but we are most interested in proving that the corotation resonance is able to capture particles, and we will investigate those features in other works." ], [ "Migration Effects on Mimas Orbit", "In the first part of our hypotheses, we treat the excitation on Mimas eccentricity due to its passage through the Mimas-Enceladus 3:2 e-Mimas resonance during the migration process of these satellites.", "To perform this simulation we used the Gauss-Radau spacings described by [7] with initial time step of 0.1 day.", "The dynamics of this process included Mimas, Enceladus and an oblate Saturn, and also the drag force of equation (REF ).", "For the initial conditions of Mimas and Enceladus, we use the consideration commented in Section to form the set of initial condition shown in table REF .", "Table: Initial Conditions of Enceladus and Mimas in Orbital Elements, after changes commented in Section .In Figure (REF ) we observe the behaviour of the critical angle for Mimas-Enceladus 3:2 e-Mimas resonance (equation (REF )).", "This behaviour occurs due to the passage of Mimas and Enceladus through the resonance during the migration process.", "We can see that at the very beginning of the simulation the Mimas-Enceladus system was not in resonance (the resonant angle circulates) and, due to the system migration, those satellites enter in the Mimas-Enceladus 3:2 e-Mimas resonance [14] in which the resonant angle librates around 0deg.", "We can see along the migration process a decreasing amplitude for the resonant angle, it may turn the resonance between them more robust.", "Subsequently, this amplitude begins to increase until it reaches the value of 180deg.", "At this point, we can say that Mimas and Enceladus are out of the resonance.", "In Figure (REF a) we can see the behaviour of Mimas' and Enceladus' semi-major axes during the migration process.", "Before they enter in resonance, the variation rates for the semi-major are the ones stated in the last section, with Enceladus migrating slower than Mimas (see zoom in Figure (REF a)).", "After they enter in resonance, the rate of Enceladus' semi-major axis variation increases until reaching a value slightly larger than Mimas, while the semi-major axis rate variation for Mimas decreases.", "These can be seen in the semi-major axis inclination showed in Figure (REF a).", "When they come out of resonance, the variation rates for the semi-major axis returns to their previous values.", "This passage through the resonance affects Mimas' eccentricity significantly, as shown in Figure (REF b).", "When Mimas is in resonance, its eccentricity increases to high values while Enceladus eccentricity remains constant.", "Despite the migration velocity, we have adopted, the behaviour for Mimas' eccentricity is in agreement with the results of [14].", "The eccentricity only stops growing when it reaches a certain equilibrium eccentricity (here value is higher 0.052, that was also found by [14]) and the satellites escape of Mimas-Enceladus 3:2 e-Mimas resonance.", "Although we do not show in this work, the eccentricities of Mimas and Enceladus should decrease after they went out of the resonance due to the tidal effects reaching the current values.", "These results show that Mimas' eccentricity could be enlarged and hence affect the width of the corotation resonance.", "In the next section we will study this effect on the capture of particles by the corotation resonance." ], [ "Capture in Corotation Resonance", "In the previous section we saw that the eccentricity of Mimas increases when Mimas and Enceladus pass through a 3:2 resonance during the tidal migration process.", "In this section, we will check if this increase in eccentricity will enable Mimas to capture particles in corotation resonance.", "To perform this study, we created a ring with 10 000 particles in a region where the resonance should appear when the eccentricity of Mimas increase, as we can see in Figure (REF a).", "Thus the particles were uniformly distributed with semi-major axis between 162409.9km and 162604.8km and with mean longitude between 0deg and 360deg.", "All particles have its other orbital elements fixed with the value 0.01075838 for eccentricity, 0.004053321deg for inclination, 342.0739deg for ascending node, and 326.3412deg for longitude of the pericenter.", "These fixed orbital elements were based on the orbital elements of Aegaeon after it was migrated toward Saturn and close to a position where the resonance should appear.", "This choice was made to improve our chances of capture and we are not interested in finding the best orbital configuration for the capture, but in the migration process as the cause of the capture in corotation resonance.", "For this experiment we integrate the full equations of motion for the four body model (Saturn, Mimas, Enceladus and a particle) plus equation (REF ) acting only in Mimas and Enceladus and representing the tide interaction in these bodies.", "We argue that the homogeneous rings do not raise tidal bulges in the planet like moons do, since a homogeneous ring does not raise a tidal bulge.", "Also, ring's particles have no interaction with each other by collisions or gravity.", "We also considered an oblate Saturn for all particles involved in the integration.", "The integrations were made using Gauss-Radau spacings described by [7] with initial time step of 0.1 day.", "We used the same initial conditions for Mimas and Enceladus of table REF .", "It is important to say that, although all 10 000 particles were integrated at the same time they do not interact with each other.", "Thus it is a four body problem plus Saturn oblateness for each particle.", "In panels (d) and (h) of Figure (REF ) we can see the evolution of Mimas' eccentricity and in panels (a), (b), (c), (e), (f) and (g) the effects this evolution make in the particles of the ring.", "In the very beginning, despite of some small variation, Mimas' eccentricity is not increasing as we can see in Figure (REF d).", "After some time the satellites enter the 3:2 e-Mimas resonance and Mimas' eccentricity increases affecting the ring, as shown in Figure (REF b).", "In Figure (REF c) it is possible to see trapped particles in the resonance which had moved upward due to Mimas migration.", "In the panels (e), (f) and (g) of Figure (REF ) we see more explicitly this phenomena where we clearly see six structures moving outside our initial ring.", "Those six lobes are consistent with a 7:6 corotation resonance.", "It is also possible to see that some particles were dragged outwards.", "To explicitly show that the eccentricity variation is the mechanism responsible for the capture in corotation resonance, we made other experiment passing the corotation resonance through the ring of particles, but with Mimas not varying its eccentricity.", "For that we take off Enceladus of the simulation and initiate Mimas with its present eccentricity of 0.02 and its semi-major axis 100km below the value shown in the table (REF ).", "For all other orbital elements, it was maintained the values of table (REF ), and then it was applied the migration process to Mimas.", "The ring of particles was the same as the last experiment.", "We noted that the capture of particles doesn't occur due to overlap between the Lindblad and corotation resonances.", "These two resonances have a separation about 19km (Figure (REF b) and the particles in the ring trapped by the corotation resonance feel the Lindblad perturbation [6].", "These simulations show that, in our accelerated tidal scenario, the overlap of resonances isn't the dominant mechanism to capture of particles.", "Figure: In panel (a) we present the localization of Corotation and Lindlbladresonances.", "These locations were found following the technique of ,with equations of.", "The Corotation and Lindlblad location movebecause Mimas is migrating during this simulation.", "In (b) it shows thedistance between these resonances.We can see that holes appear while the corotation resonance passes through the ring of particles (see panels (a) to (f) of Figure (REF )).", "As there are no particles inside these holes, we can affirm that particles weren't captured.", "This effect shows that the corotation resonance cannot capture neither loose particles unless it changes its width.", "However, we can see that some particles stay close to the corotation edge (Figure (REF e)) and Figure (REF f)).", "These particles are temporary captured in a region known as resonance stickiness [4].", "These particles move in the border of the lobes of the corotation resonance for a time and then they escape.", "Figure: In these six snapshots shown we had taken off Enceladus of thesimulation.", "Without the eccentricity enhancement due to 3:2 resonance thereis no capture in the corotation resonance.", "In last snapshot, all corotationsites have passed through the ring without capture any particle.With these results, we show that our scenario could explain the formation of the arc of the G-ring.", "In this scenario, we used a migration rate much above of that generated by tidal effect, and with a realistic migration rate, it must work as the same way.", "Actually, we had made our first experiments with rates consistent with tidal migration and observed the robustness of the corotation resonance.", "In one of our first experiments, which we have not shown here, we created an arc of 1847 particles in the resonance of Mimas where G ring arc was in date January 1st, 2004 [10], and simulated Mimas migrating with a velocity equal to 9.0E-9km/year toward Saturn by 1000years.", "The final of simulation we found that 98% of the particles was in corotation with Mimas, corroborating the robustness of the corotation resonance.", "In this scenario, it was needed a high initial eccentricity for Enceladus to force the escape of Mimas from the 3:2 e-Mimas resonance, but it is not a problem for our scenario because several other processes could take Mimas out of this resonance, for instance, an eventual resonance between Mimas and Dione." ], [ "Evolution of the Capture Resonance", "In the last section, we observed that the particles were captured in corotation resonance and dragged out of the ring we created.", "Now we will analyse these captures through the history of the captures.", "In Figure (REF ) we have the ring of particles in time 0 of our simulation and we indicate the particles that were captured.", "In the Figure (REF b), we identified with grey points the particles which were not captured in the Mimas migration process and with black points the particles which were captured in this process and didn't escape the corotation sites during the simulation.", "The final state of these captured particles (black points in Figure (REF b)) can be seen in panel (g) of Figure (REF ) occupying the corotation sites above the ring of particles.", "In the bottom part of the Figure (REF b), we saw a draft of the 7:6 corotation sites in its initial condition produced by Mimas eccentricity.", "The amplitude of the curves obeys the width of the corotation for the initial Mimas eccentricity (equation (REF )).", "With the Mimas migration, the width of this curve will enlarge.", "This enlargement will produce the captures in the corotation resonances.", "Figure: Evolution of the capture in corotation resonance through the ring ofparticles.", "Panel (b) of this graph shows the initial conditions of thecaptured particles.", "In panel (a) it was shown a histogram of capturedparticles in ranges of 10km for the semi-major axis.Figure: Amplitude evolution of the corotation resonance width based onequation () for each value of the Mimas eccentricityof Figure ().We can see in the panel (b) of Figure (REF ) that the black points are spread in the ring and they show some structures.", "In panel (a) we show the number of captured particles in bins of 10km for the semi-major axis.", "The excess particles in the low part of the disc occur due to Mimas initial eccentricity, as the corotation resonance has an initial width showed by the embedded black points encircled by the draft representing the corotation curves.", "Observing the structures and the histogram of the graph, it suggests that we hadn't a homogeneous rate of captures.", "Thus, the capture is not continuous, but in steps.", "The explanation for that can be obtained observing the graphs of Figure REF as the eccentricity oscillates while increases.", "The capture by the corotation resonance is complex.", "When Mimas was captured by the 3:2 resonance with Enceladus its eccentricity began to increase but not in a uniform way.", "In Figure (REF ) we see the evolution of this width based on equation (REF ) and the history of Mimas eccentricity shown in Figure (REF ).", "The net result of the process double the initial width, however locally it increases and decreases in a very noise form.", "Thus, particles were captured while there are others that escape from the corotation resonance.", "The particles that were captured and escaped are the ones in the stickiness caused by the corotation resonance observed in Figure (REF ).", "This complex dynamic explains the structures observed in Figure (REF ).", "This feature can be understood when we consider the capture probability in the corotation resonance, which depends on perturbing eccentricity [17], in our case the perturber is Mimas.", "The Mimas' eccentricity increases non-uniformly (Figure REF ), so the capture probability oscillates during the simulation.", "When the eccentricity increases, it favours the capture, and when it decreases, the escape likelihood increases.", "The result is a relative capture probability, because sometimes it is more likely to capture particles, but sometimes, it is less likely.", "It was that which creates the structures observed in Figure (REF ).", "These results lead us to conclude that the migration sweeping over the particles, as shown in Figure (REF ), with the change in the corotation resonance width producing a change in Mimas' eccentricity, as shown in Figure (REF ), is the process which could explain the capture of the one particle, or a group of particles, which produced the arc of the G-ring." ], [ "Conclusions", "In this paper, we showed that it is possible to explain the formation of the arc of the G-ring through a plausible scenario where the Saturn tides play the main role.", "The Saturn's tide could vary the semi-major axis of the satellites, which make them cross several resonances.", "In this process a low Mimas' eccentricity could have increased and could have caused the enlarging of the corotation region.", "In our experiment, when Mimas created the region of corotation particles, we had populated all the six structures of the corotation.", "Then, if there was a ring of particles in the corotation resonance capture region, we should observe other groups today.", "However, there exists only one arc.", "Our hypothesis is still plausible, since other effects, such as Poynting-Robertson drag may have destroyed the other groups, or even the region was not as homogeneous as we created them in this scenario.", "We believe that further studies for the arc of the G-ring formation should solve this problem.", "Other studies could also answer if the arc was formed by agglomeration or erosion, and all these information together could clarify how and why we have only one group observed in 7:6 corotation resonance." ], [ "Acknowledgments", "The authors want to thank Bruno Sicardy for the prolific discussions and helpful suggestions, the anonymous referee for his helpful suggestions, and the Brazilian science funding agencies CAPES, CNPq and FAPESP (grant 2011/08171-3)." ], [ "[16] affirm that the attraction of the tidal bulge raised on a planet by a satellite outside the synchronous orbit results in a gain of angular momentum by the satellite.", "This causes the orbit of the satellite to expand and the rate of change of its semi-major axis is $\\frac{\\dot{a}}{a} = 3 \\left( \\frac{G}{M_p} \\right)^{1/2} \\, k_{{}_{2P}}\\frac{R_p^5}{Q_P}\\frac{m}{a^{13/2}},$ where the parameters $a$ , $\\dot{a}$ , $m$ , are the semi-major axis, its variation and the mass of the satellite, whereas $G$ is the gravitational constant, $M_p$ is the mass of the planet, $k_{{}_{2p}}$ is the Love number of the planet and $Q_p$ is the dissipation factor.", "We consider that the semi-major axis of Mimas and Enceladus suffered changes described by Equation (REF ).", "If we integrate this equation, keeping $M_p$ , $k_{{}_{2p}}$ and $Q_p$ constant over time and using parameters of Mimas and Enceladus, we can find the values of semi-major axis for Mimas which enables its trapping with Enceladus into a resonance in the remote past.", "The first consideration is that the physical properties of the planet are constant, so we can write that $\\xi = 3 \\left( \\frac{G}{M_p} \\right)^{1/2} \\, k_{{}_{2P}} \\, \\frac{R_p^5}{Q_P},$ where $\\xi $ is a constant depending on the planet parameters.", "Consequently, the equation (REF ) becomes $\\dot{a} = {a^{-11/2}} \\, \\xi \\, {m}.$ And integrating it we have $\\frac{2}{13} \\left( {a^{13/2}} - {a_{{}_0}}^{13/2} \\right) = \\Delta t \\, \\xi \\, m.$ where the integral constant $a_{{}_0}$ is the value of semi-major axis at $t_{{}_0}$ , and $\\Delta t=t-t_{{}_0}$ .", "Calculating the value of this integral with the parameters of Mimas, we have $\\frac{2}{13} \\left( {a_{{}_M}^{13/2}}-{a_{{}_{0M}}^{13/2}} \\right) =\\Delta t_{{}_M} \\, \\xi \\, m_{{}_M},$ where ${a_{{}_M}}$ and $a_{{}_{0M}}$ are the current and ancient semi-major axis of Mimas, respectively, ${m_{{}_M}}$ is Mimas' mass and $\\Delta t_{{}_M}$ is the time spent for Mimas to vary its position from $a_{{}_{0M}}$ to ${a_{{}_M}}$ .", "Using that integral with parameters of Enceladus, we get $\\frac{2}{13} \\left( a_{{}_E}^{13/2} - a_{{}_{0E}}^{13/2} \\right) =\\Delta t_{{}_E} \\, \\xi \\, m_{{}_E},$ where $M$ was replaced with $E$ .", "Now, if we consider that $t$ is the current time and $t_{{}_0}$ is an instant that these satellites would be in resonance we have $\\Delta t_{{}_M} = \\Delta t_{{}_E}$ , and also the semi-major axes rate of Mimas and Enceladus for the resonance is $\\alpha ={a_{{}_{0M}}/a_{{}_{0E}}}$ .", "Then, from equations (REF ) and (REF ), we find that $a_{{}_{0M}} = \\left[\\frac{\\left(\\dfrac{m_{{}_E}}{m_{{}_M}}\\right) \\, a_{{}_M}^{13/2}-a_{{}_E}^{13/2}}{\\left(\\dfrac{m_{{}_E}}{m_{{}_M}}\\right) - \\dfrac{1}{\\alpha ^{13/2}}}\\right]^{2/13}.$ Therefore, with equation (REF ) we calculate an approximate value for the ancient semi-major axis of Mimas, just when that satellite was trapped in resonance with Enceladus, given an appropriate $\\alpha $ ." ] ]	1606.05190
[ [ "Scale-free networks with exponent one" ], [ "Abstract A majority of studied models for scale-free networks have degree distributions with exponents greater than $2$.", "Real networks, however, can demonstrate essentially more heavy-tailed degree distributions.", "We explore two models of scale-free equilibrium networks that have the degree distribution exponent $\\gamma = 1$, $P(q) \\sim q^{-\\gamma}$.", "Such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high.", "Our models exploit a rewiring mechanism.", "They are local in the sense that no knowledge of the network structure, apart from the immediate neighbourhood of the vertices, is required.", "These models generate uncorrelated networks in the infinite size limit, where they are solved explicitly.", "We investigate finite size effects by the use of simulations.", "We find that both models exhibit disassortative degree-degree correlations for finite network sizes.", "In addition, we observe a markedly degree-dependent clustering in the finite networks.", "We indicate a real-world network with a similar degree distribution." ], [ "Introduction", "Scale-free networks have been in the forefront of networks research for almost two decades.", "Examples range from social networks [1], [2], biological networks [3], [4] to artificial networks like the internet [5] and the world-wide web [6], [7].", "The widely accepted mechanism for the evolution of growing scale-free networks is preferential attachment [8].", "This mechanism has been extensively explored and many generalizations and modifications of the original model have been suggested [9], [10], [11], [12], [13].", "The same principle has also been applied to equilibrium networks [14].", "The preferential attachment mechanism has relations with random multiplicative processes ([15], [16], [17], [18]) which have relevance in many fields of statistical physics, and are known to produce skewed distributions.", "Besides preferential attachment, some other methods of producing scale-free networks have been exploited, for example, fitness-based models [19], [20], [21], merging processes [22], [23], [24], optimization models [25], [26], urn-based statistical ensembles [27], [28], networks embedded into metric spaces [29], [30], and others.", "Most well-studied real networks appear to have degree distribution exponents larger than two.", "As a result, and also due to the popularity of the preferential attachment mechanism, networks with smaller exponents have received much less attention.", "In such networks the mean degree diverges, and consequently the “natural” cutoff of the degree distribution scales with the system size in a different way compared to networks with higher exponents [24].", "The case of $\\gamma = 1$ is even more peculiar: the normalization condition implies that the cutoff of the degree distribution must remain finite in an infinite system.", "This circumstance makes it rather difficult to clearly identify such distributions in empirical data obtained from sparse networks.", "On the other hand, if the mean vertex degree is sufficiently large, this kind of distribution can be observed (see below), which justifies our investigation.", "To our knowledge, the only studied model for networks with $\\gamma = 1$ , was that of trees growing based on “the power of choice” (local optimization of the new connections) [26].", "Note that variations of the models with hidden variables [31], [32], [33] can also generate such degree distributions if one chooses an appropriate distribution of hidden variables.", "In the present paper we consider two simple equilibrium network models producing $\\gamma = 1$ .", "These rewiring models are local in the sense that the vertices need to “know” only the structure of their immediate neighbourhoods.", "We show that the resulting degree distribution has a simple exact solution in the sparse network limit, where the network is uncorrelated, which is a power-law of $\\gamma = 1$ with an exponential cutoff that is determined by the mean degree.", "We perform extensive numerical simulations of these models for finite networks, and observe marked disassortative degree–degree correlations.", "Figure: Schematic representation of the evolution mechanism for models 1 and 2." ], [ "The models", "A well-studied class of mechanisms that is known to produce power-law distributions, is random multiplicative processes [15], [16], [17], [18].", "The essence of such models is that the fluctuations of random variables are proportional to their values (independent fluctuations would result in classical Brownian dynamics).", "A simple example of such processes is discussed in [34] and is shown to generate power-law distributions of exponent 1.", "We use a similar principle.", "In our case, the random variables are the degrees of nodes in the network.", "Instead of fluctuations which are proportional to the values of the random variables, we apply fluctuations of a fixed size (rewiring one link at a time) with a probability that is proportional to the values of the random variables (degrees of nodes).", "Here we consider a model of equilibrium networks that realizes this scheme.", "Model 1.", "Consider an arbitrary connected graph.", "At every step of the evolution do the following: Choose an edge uniformly at random (edge $e$ in Fig.", "REF ).", "Reattach a neighbour (node $A$ in Fig.", "REF ) of one of its end nodes to the other (from node $B$ to node $C$ in Fig.", "REF ).", "Repeat the above procedure until equilibrium is reached.", "In the second step node $A$ is chosen uniformly at random from the set of all neighbours of $B$ which are not neighbours of $C$ (and are not themselves $C$ ).", "If there are no such nodes, then nothing should be done in this iteration.", "We denote the degree distribution by $P(q)$ .", "The joint degree distribution, i.e., the probability that the end nodes of a uniformly randomly chosen link have degrees $q$ and $q^{\\prime }$ is denoted by $P(q,q^{\\prime })$ , and the conditional probability that an end node of a random link has degree $q$ given that the other end node has degree $q^{\\prime }$ , by $P(q\|q^{\\prime })$ .", "In a given step of the evolution the probability of a node of degree $q$ to be chosen as node $B$ is $P_B(q) = \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle }$ and, similarly, the probability of a node of degree $q$ to be chosen as node $C$ is $P_C(q) = \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle } = P_B(q).$ Assuming that in equilibrium, clustering is purely a result of degree–degree correlations, we introduce $R(q,q^{\\prime })$ , the probability that if a node of degree $q$ is chosen as $B$ and a node of degree $q^{\\prime }$ as $C$ , then a rewiring is possible, i.e., that $B$ has at least one neighbour that is not a neighbour of $C$ : $R(1,q^{\\prime }) &=& 0,\\nonumber \\\\[5pt]R(q>1,q^{\\prime }) &=& 1 - \\left\\lgroup \\sum _k P(k\|q) P(q^{\\prime }\|k) \\frac{(q^{\\prime }-1)(k-1)}{Nq^{\\prime }P(q^{\\prime })} \\right\\rgroup ^{q-1}\\nonumber \\\\[5pt]&=&1 - \\left\\lgroup \\frac{\\langle q \\rangle ^2 (q^{\\prime }-1)}{Nqq^{\\prime }P(q)P(q^{\\prime })} \\sum _k \\frac{P(k,q)P(q^{\\prime },k)(k-1)}{kP(k)} \\right\\rgroup ^{q-1},$ where $N$ is the number of nodes in this network.", "Now we can write the probability that a node of degree $q$ is chosen as $B$ and rewiring is possible: $P_{B,r}(1) &=& 0,\\nonumber \\\\[5pt]P_{B,r}(q>1) &=& \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle } \\sum _{q^{\\prime }} P(q^{\\prime }\|q) R(q,q^{\\prime }).$ Similarly, the probability that a node of degree $q$ is chosen as $C$ and rewiring is possible: $P_{C,r}(q) = \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle } \\sum _{q^{\\prime }>1} P(q^{\\prime }\|q) R(q^{\\prime },q).$ It is easy to see from Eq.", "(REF ) that in the limit $N \\rightarrow \\infty $ , $R(q,q^{\\prime })=1$ for any $q > 1,q^{\\prime }$ and $R(1,q^{\\prime }) = 0$ .", "In this case, Eq.", "(REF ) reduces to $P_{B,r}(1) &=& 0,\\nonumber \\\\[5pt]P_{B,r}(q>1) &=& \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle },$ and Eq.", "(REF ) becomes $P_{C,r}(q) = \\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle } b(q),$ where $b(q) = 1 - P(1\|q)$ .", "Noting that in the stationary state the probability of a node of degree $q+1$ losing an edge must match the probability of a node of degree $q$ gaining an edge, we can write the stationary equation for the degree distribution: $P_{C,r}(q) = P_{B,r}(q+1).$ Substituting Eqs.", "(REF ) and (REF ) into Eq.", "(REF ), we have: $\\frac{1}{2} \\frac{qP(q)}{\\langle q \\rangle }b(q) = \\frac{1}{2} \\frac{(q+1)P(q+1)}{\\langle q \\rangle }.$ We see that degree–degree correlations appear only in $b(q)$ .", "If we assume that $b(q)$ is constant (this is a weaker assumption than assuming that correlations are entirely absent), then $b(q) = c = 1 - 1P(1)/\\langle q \\rangle $ .", "This can be easily seen in the following way.", "For any network, regardless of degree-degree correlations, the degree distribution of end nodes of links is $qP(q) / \\langle q \\rangle $ (the probability, that a randomly chosen end node of a randomly chosen link has degree $q$ ).", "The probability that an end node of a randomly chosen link has degree 1, is therefore $1P(1) / \\langle q \\rangle $ .", "This probability can also be written as: $\\frac{1P(1)}{\\langle q \\rangle } = \\sum _q P(1\|q) \\frac{qP(q)}{\\langle q \\rangle }.$ Assuming that $P(1\|q)$ is constant ($P(1\|q) = h$ ), we have $\\frac{1P(1)}{\\langle q \\rangle } = \\sum _q P(1\|q) \\frac{qP(q)}{\\langle q \\rangle } = \\sum _q h \\frac{qP(q)}{\\langle q \\rangle } = h.$ Therefore $b(q) = 1 - P(1\|q) = 1 - 1P(1) / \\langle q \\rangle = c$ .", "Then Eq.", "(REF ) is simply $qP(q)c = (q+1)P(q+1).$ The solution of Eq.", "(REF ) is the following: $P(q) = A q^{-1} e^{- q/q_{\\text{cut}}}.$ where $q_{\\text{cut}} = -1 / \\ln c$ .", "The cutoff $q_{\\text{cut}}$ may also be expressed in terms of the mean degree $\\langle q \\rangle $ .", "The constant $A$ in Eq.", "(REF ), from the normalization condition, is $A = \\frac{1}{\\sum _{q=1}^{\\infty } q^{-1} e^{- q/q_{{cut}}} } = -\\,\\frac{1}{\\ln (1 - e^{- 1/q_{\\text{cut}}})} \\cong \\frac{1}{\\ln q_{\\text{cut}}},$ and the mean degree is $\\langle q \\rangle = A \\sum _{q=1}^{\\infty } e^{- q/q_{\\text{cut}}} = \\frac{A}{1 - e^{- 1/q_{\\text{cut}}}} \\cong \\frac{q_{\\text{cut}}}{\\ln q_{\\text{cut}}},$ so $q_{\\text{cut}}$ $q_{\\text{cut}} \\cong \\langle q \\rangle \\ln \\langle q \\rangle $ is independent of the system size.", "Using Eq.", "(REF ), it is possible in principle to write the complete master equation for the degree–degree distribution $P(q,q^{\\prime })$ in the general case and solve it numerically.", "However, this undertaking would be very cumbersome, therefore, instead, in Sec.", "we present simulation results.", "Let us first introduce an alternative formulation of the above model.", "Model 2.", "Consider an arbitrary connected graph.", "At every step of the evolution do the following: Choose a node uniformly at random (node $A$ in Fig.", "REF ).", "Choose a neighbour of $A$ uniformly at random (node $B$ in this figure).", "Choose a 2nd neighbour of $A$ through $B$ (node $C$ ) uniformly at random from all 2nd neighbours of $A$ through $B$ .", "If no such node exists, do nothing in this iteration.", "Rewire $A$ from $B$ to $C$ (as in the figure).", "Repeat the above procedure until equilibrium is reached.", "In other words, a node rewires its connection from a randomly chosen nearest neighbour to a “descendant” of this neighbour.", "We can write the probability $P_B(q)$ that in a given step of the evolution a node of degree $q$ is selected to be node $B$ : $P_B(q) = \\sum _{q^{\\prime }} P_A(q^{\\prime }) P(B{:}q \| A{:}q^{\\prime }) = \\langle q \\rangle \\sum _{q^{\\prime }} \\frac{P(q,q^{\\prime })}{q^{\\prime }},$ where $P(B{:}q \| A{:}q^{\\prime })$ is the conditional probability that a node of degree $q$ is chosen to be node $B$ , given that a node of degree $q^{\\prime }$ was chosen to be node $A$ .", "Assuming, again, that in equilibrium, clustering is purely a result of degree-degree correlations, and considering the limit $N \\rightarrow \\infty $ (i.e.", "assuming that clustering goes to zero), we can write the probability $P_{BC}(q,q^{\\prime })$ that at a given step a node of degree $q$ is chosen as $B$ and a node of degree $q^{\\prime }$ as $C$ : $\\!\\!\\!\\!\\!P_{BC}(1,q^{\\prime })&=&0,\\nonumber \\\\[5pt]\\!\\!\\!\\!\\!P_{BC}(q>1,q^{\\prime })&=&P_B(q) P(C{:}q^{\\prime } \| B{:}q) =\\nonumber \\\\[5pt]\\!\\!\\!\\!\\!&=& \\langle q \\rangle \\sum _{q^{\\prime \\prime }} \\frac{P(q,q^{\\prime \\prime })}{q^{\\prime \\prime }} \\frac{P(q,q^{\\prime })}{qP(q)} \\langle q \\rangle =\\nonumber \\\\[5pt]\\!\\!\\!\\!\\!&=& \\langle q \\rangle ^2 \\frac{P(q,q^{\\prime })}{qP(q)} \\sum _{q^{\\prime \\prime }} \\frac{P(q,q^{\\prime \\prime })}{q^{\\prime \\prime }}.$ Again, requiring that in the stationary state the probability of a node of degree $q+1$ losing an edge match the probability of a node of degree $q$ gaining an edge, we can write the equation: $\\sum _{q^{\\prime }} P_{BC}(q+1,q^{\\prime }) = \\sum _{q^{\\prime }} P_{BC}(q^{\\prime },q).$ We see that even in the limit of infinite size, the situation is much more complex than Eq.", "(REF ).", "If we further assume that if $N \\rightarrow \\infty $ , then the equilibrium network is uncorrelated, i.e., that the degree-degree distribution factors, we find that Eq.", "(REF ) reduces to $P_B(q) = \\frac{qP(q)}{\\langle q \\rangle }.$ Then Eq.", "(REF ) takes the simple form: $P_{BC}(1,q^{\\prime }) &=& 0,\\nonumber \\\\[5pt]P_{BC}(q>1,q^{\\prime }) &=& \\frac{qP(q)q^{\\prime }P(q^{\\prime })}{\\langle q \\rangle ^2},$ and the stationary equation (Eq.", "(REF )) is now simply $\\frac{(q+1)P(q+1)}{\\langle q \\rangle } = \\frac{qP(q)}{\\langle q \\rangle } c$ with $c = 1 - 1P(1)/\\langle q \\rangle $ , which is identical to Eq.", "(REF ).", "We see that in the limit $N \\rightarrow \\infty $ , assuming that the network is then uncorrelated, the two model formulations are equivalent.", "Models 1 and 2 are closely related, exploiting the same mechanism.", "While model 1 is in fact a null model, there is a rationale behind model 2.", "In this model, a node redirects one of its connections to get the farthest possible reach by using only local information from its nearest neighbours (the lists of their neighbours).", "The redirection of a link, instead of the addition of a new one, corresponds to evolution with limited resources.", "In the following we employ simulations to investigate the behaviour of the two models in a wide range of system sizes.", "The simulations indicate that in the infinite sparse network limit, both formulations lead to uncorrelated equilibrium networks, although the models are significantly different for finite networks." ], [ "Simulations", "We performed simulations of varying system sizes and mean degrees, averaging over at least 10 realizations for each combination of parameters.", "The starting graph in each simulation was a connected random graph generated in the following way: first all the nodes were linked in a chain to ensure connectedness, then all remaining links were assigned to the nodes randomly.", "$T=10^{10}$ time steps (rewiring attempts) were used in each simulation, this ensured that equilibrium was reached in each case.", "The success rate for rewiring was above $95\\%$ for all parameter settings.", "First we investigate the case of $N \\gg \\langle q \\rangle $ , approaching the limit of large sparse networks.", "Clustering and degree-degree correlations are found diminishing as this limit is approached.", "Secondly we analyse more dense networks to compare the behaviour of the two model formulations.", "Figure: (Color online) Degree distribution of sparse equilibrium networks of mean degree 〈q〉=20\\langle q \\rangle = 20 and different sizes for model 1 (a) and model 2 (b)." ], [ "Sparse networks", "Degree distributions at equilibrium are shown in Fig.", "REF for both models, for different system sizes, and fixed mean degree $\\langle q \\rangle = 20$ .", "In both cases, for large system sizes, the uncorrelated form of the degree distribution Eq.", "(REF ), is approached, but this convergence is much slower for the second model.", "The choice of mean degree was limited by the system size for which simulations run in reasonable time for the computationally more demanding model 2.", "To study correlations and clustering, we measured the degree dependence of the average degree of nearest neighbours and the clustering coefficient (Figs.", "REF and REF ).", "In the plots we normalized the measured values $\\overline{q}_{nn}(q)$ and $C(q)$ by the values expected in the uncorrelated case.", "These corresponding uncorrelated values, denoted by $(\\overline{q}_{nn})_c$ and $(C)_c$ , are just the values calculated in the configuration model using the same structural characteristics $N$ , $\\langle q \\rangle $ , and $\\langle q^2 \\rangle $ as those obtained in the simulations: $(\\overline{q}_{nn})_c &=& \\frac{\\langle q^2 \\rangle }{\\langle q \\rangle },\\\\[5pt](C)_c &=& \\frac{1}{N \\langle q \\rangle } \\left\\lgroup \\frac{\\langle q^2 \\rangle - \\langle q \\rangle }{\\langle q \\rangle } \\right\\rgroup ^2.$ Figure: (Color online) Degree dependence of the mean degree q ¯ nn \\overline{q}_{nn} of the nearest neighbours of a node of degree qq for sparse equilibrium networks of mean degree 〈q〉=20\\langle q \\rangle = 20and different sizes.", "(a) model 1, (b) model 2.", "The mean degree q ¯ nn \\overline{q}_{nn} is normalized by its value for the corresponding uncorrelated network.Figure: (Color online) Degree dependence of the local clustering coefficient of sparse equilibrium networks of mean degree 〈q〉=20\\langle q \\rangle = 20and different sizes.", "(a) model 1, (b) model 2.", "The local clustering coefficient is normalized by its value for the corresponding uncorrelated network.Figures REF and REF confirm a convergence to an uncorrelated equilibrium state for large networks.", "It is interesting to note that although correlations are smaller in the second model, Fig.", "REF (b), compared to the first one, Fig.", "REF (a), the degree distributions in the second model at the same sizes are still further away from the form of Eq.", "(REF ).", "Model 1 exhibits stronger correlations and stronger degree dependence of the local clustering coefficient (Fig.", "REF (a) compared with Fig.", "REF (b)), even though the degree distributions in model 1 (Fig.", "REF (a)), for the system sizes considered, already practically coincide with the uncorrelated form of Eq.", "(REF )." ], [ "Denser networks", "We performed simulations of networks with higher mean node degrees, 200 and 50 (models 1 and 2, respectively), than in the preceeding subsection.", "This enabled us to observe stronger size effects in the degree distributions, Fig.", "REF (a), (b), than in Fig.", "REF at the same network sizes.", "Simulations for model 2 are particularly time consuming, so the mean degree, 50, has to be chosen smaller than 200 for model 1.", "The system sizes were chosen in a way to capture a wide range of behaviors in both models, using the highest possible mean degree (limited by computational time).", "Figures REF (a), (b) demonstrate markedly different degree distributions for models 1 and 2 at low network sizes.", "Note that the difference is not only in a hump present in Fig.", "REF (b) but this difference is well observable even in the range of small degrees.", "The degree-degree correlations for these networks demonstrate a stronger disassortative mixing, Fig.", "REF (a), (b), than for their less dense counterparts in Fig.", "REF (a), (b).", "The degree dependence of the local clustering coefficient is also more pronounced in denser networks, Fig.", "REF (a), (b), than in their less dense counterparts, Fig.", "REF (a), (b).", "This is especially well seen for model 1, compare respective Fig.", "REF (a) ($\\langle q \\rangle =200$ ) and Fig.", "REF (a) ($\\langle q \\rangle =20$ ).", "Figure: (Color online) Degree distribution of denser (than in Fig. )", "equilibrium networks of different sizes.", "(a): 〈q〉=200\\langle q \\rangle = 200, model 1; (b): 〈q〉=50\\langle q \\rangle = 50, model 2.Figure: (Color online) Degree dependence of the mean degree of nearest neighbours of dense equilibrium networks of varying size.", "(a): 〈q〉=200\\langle q \\rangle = 200, model 1; (b): 〈q〉=50\\langle q \\rangle = 50, model 2.Figure: (Color online) Degree dependence of the clustering coefficient of dense equilibrium networks of varying size.", "(a): 〈q〉=200\\langle q \\rangle = 200, model 1; (b): 〈q〉=50\\langle q \\rangle = 50, model 2." ], [ "A real-world example", "To demonstrate that degree distributions of $\\gamma = 1$ do exist in reality, we explore data from Facebook.", "The analyzed sample consists of all of the user-to-user links from the Facebook New Orleans (2009) networks [35].", "This sample has size $N = 63731$ and mean degree $\\langle q \\rangle = 25.64$ .", "Figure REF shows the measured Facebook degree distribution and the degree distribution from our model 1 using the same system size and mean degree.", "Notice the closeness of the two curves although no fitting was done.", "In this parameter range, the degree distribution of the model is already very close to the analytical form given by Eq.", "(REF ).", "Figure: (Color online) Degree distribution taken from a Facebook sample, compared to the degree distribution of model 1, using thesame system size and mean degree: N=63731N = 63731 and 〈q〉=25.64\\langle q \\rangle = 25.64.We see that the curve from our model 1 provides a good approximation of the empirical distribution.", "We stress that the underlying structures of the two networks are different, and so our models cannot be applied directly.", "Facebook is growing, and like the majority of social networks, it exhibits assortative correlations, while our equilibrium models produce disassortative ones.", "The Pearson correlation coefficients are $0.175$ and $-0.004$ for the Facebook sample and our model 1, respectively; with these parameters model 1 is already close to the uncorrelated sparse limit.", "Also, social networks have strong clustering, whereas our models have very low clustering coefficients for large systems (and clustering actually vanishes in the infinite size limit).", "The corresponding clustering coefficients are $0.148$ and $0.006$ .", "Nevertheless, Fig.", "REF indicates that such low exponents of the degree distribution do appear in reality.", "Therefore it may be useful to think outside the realm of conventional preferential attachment models in order to come closer to a full explanation of real-world network structures." ], [ "Discussion", "Previously studied network models generating degree distributions with exponents $\\gamma $ smaller than 2 exploited a set of rather intricate mechanisms and non-trivial ideas.", "In particular, these included fitness models [19], [20], accelerated growth, where the network becomes denser with time [12], [13], aggregation processes [24], the power of choice [26], etc.", "At first sight, the two equilibrium network models that we have considered in this paper are simpler.", "In the infinite size limit, both these rewiring models generate uncorrelated networks with the degree distributions $P(q) \\sim q^{-1} e^{-q/(\\langle q \\rangle \\ln \\langle q \\rangle )}$ .", "In finite networks, however, these models become essentially non-trivial due to constraints for rewiring which occur in this situation.", "We have found that these constraints lead to strong disassortative degree–degree correlations and to degree-dependent local clustering.", "They also markedly change the form of the degree distributions.", "The structural constraint is particularly strong for model 2, so the results for these two closely related models in finite systems significantly differ from each other.", "Finally, we emphasize a strong difference of these rewiring models from well studied equilibrium networks based on the preferential attachment mechanism [14].", "While networks in the present work demonstrate a power-law degree distribution in a wide range of mean degrees, the networks from Ref.", "[14] are scale-free only at a critical mean degree value.", "The resulting degree distributions are observable only if the mean number of connections of nodes in a network is sufficiently large.", "This is the case for a number of real-world networks, including social and neural networks.", "(The mean number of friends of adult Facebook users was already 338 in 2014 [36] and the mean number of synapses in brain neuronal networks is generally of the order of $10^3$ [37], [38]).", "We suggest that our results may be useful for understanding the structural properties of networks of this kind." ], [ "Acknowledgement", "This work was supported by the FET proactive IP project MULTIPLEX 317532." ] ]	1606.04973
[ [ "Using instrumental variables to disentangle treatment and placebo\n effects in blinded and unblinded randomized clinical trials influenced by\n unmeasured confounders" ], [ "Abstract Clinical trials traditionally employ blinding as a design mechanism to reduce the influence of placebo effects.", "In practice, however, it can be difficult or impossible to blind study participants and unblinded trials are common in medical research.", "Here we show how instrumental variables can be used to quantify and disentangle treatment and placebo effects in randomized clinical trials comparing control and active treatments in the presence of confounders.", "The key idea is to use randomization to separately manipulate treatment assignment and psychological encouragement messages that increase the participants' desire for improved symptoms.", "The proposed approach is able to improve the estimation of treatment effects in blinded studies and, most importantly, opens the doors to account for placebo effects in unblinded trials." ], [ "Introduction", "Placebo effects have draw a lot of interest and debate in medicine[1].", "They can be viewed as a simulation of an active therapy within a psychosocial context[1].", "Research in neurobiology has shown that placebo responses are accompanied by actual alterations in neural activity within brain regions involved in emotional regulation[1], [2], [3], [4].", "Hence, rather than inducing a simple bias in response, placebos can induce actual biological effects and improve clinical outcomes.", "Among the cognitive and emotional factors that have been proposed to contribute to placebo effects, the interaction between the desire for symptom change and the expected symptom intensity has been proposed as a key component giving raise to placebo effects[1].", "In the psychology literature, this interaction is known as the desire-expectation model of emotions[1], [5], [6], [7], which postulates that ratings of positive and negative emotional feelings are predicted by multiplicative interactions between ratings of desire and expectation.", "A number of experimental studies of placebo analgesia[1], [8], [9] have corroborated the role of the desire-expectation model as a trigger of placebo effects.", "These findings have important implications for both clinical practice and clinical trials.", "On one hand, clinicians should harness the placebo effect to improve the clinical outcome of their patients (by managing expectations and desires through ethical use of suggestions and optimum caregiver-patient interactions)[1].", "On the other hand, assessment of expectation and desire levels is also important in clinical trials since placebo effects might strongly influence the results of a study.", "In unblinded trials, it is widely recognized that the overall effect attributed to a treatment might actually correspond to a combination of treatment and placebo effects.", "However, placebo effects might still play a role in blinded trials as well[1].", "For instance, blinded studies evaluating the effectiveness of acupuncture[10] and of implantation of human embryonic dopamine neurons into the brains of persons with severe Parkinson disease[11] have shown that perceived treatment (or the treatment the participants thought they had received) can have stronger effects than the treatment actually received by the participants.", "These findings illustrate the relevance of measuring expectation, desire, and emotional levels in order to assess the contribution of placebo effects, and suggest that it is important to adjust for these variables when estimating treatment effects and interpreting the results of clinical trials[1].", "However, because it is generally impossible to rule out the presence of unmeasured confounders, simply measuring and adjusting for variables associated with placebo effects might not be enough to ensure a reliable estimation of the treatment effect.", "For instance, estimation based on regression models adjusting for the placebo related measurements still leads to biased estimates of the treatment effect, unless all confounders influencing the outcome variable enter the regression model." ], [ "The statistical method", "Here we present a statistical approach to disentangle treatment and placebo effects using instrumental variables[12], [13], [14] in randomized experiments.", "An instrumental variable (IV) is statistically independent from any unmeasured confounders, but is associated with the treatment variable and with the outcome variable (via its influence on the treatment variable alone).", "Use of IVs in randomized experiments allows the consistent estimation of treatment effects without the need to explicitly model the confounders (the technique even accounts for confounders the researcher is unaware about).", "Our proposed method requires the ability to assess variables associated with placebo effects (e.g., levels of expectancy, desire, and emotion), and uses randomization to separately manipulate a pair of variables.", "The first, corresponds to a psychological encouragement variable aiming to increase the desire for improved symptoms.", "The study participants are randomized according to whether they receive the psychological encouragement or not.", "This “psychological treatment\" IV allows the consistent estimation of the placebo effect on the outcome in the presence of confounders.", "The second, corresponds to a treatment assignment variable representing the random assignment of participants to active treatment or control therapy groups.", "It allows the estimation of the treatment effect on the outcome, after adjustment for the placebo effect.", "Mechanistically, the approach corresponds to a two-step procedure, which first estimates the contribution of the placebo effect on the outcome, and then the effect of the treatment on the residuals of the outcome variable after the contribution of the placebo effect has been removed.", "A graphical representation of the causal model underlying our approach is given in Figure REF a.", "Circled and un-circled nodes represent observed and unobserved variables, respectively.", "Arrows represent the causal influence of a variable on another, with the influence of unmeasured confounders shown as dotted arrows.", "The binary variable $Z$ represents the randomized treatment assigned to the participant (1 if participant is assigned to the active treatment group, and 0 if assigned to the control group), while $X$ represents the treatment actually received by the study participant (1 if the participant receives the active treatment, and 0 otherwise).", "It is important to model both assigned and received treatment variables since participants won't necessarily subscribe to their assigned treatment, and the experiment might suffer from imperfect compliance.", "Figure: Direct acyclic graph representation of the causal model underlying the proposed IV approach for disentangling treatment and placebo effects in unblinded clinical trials.", "Circled and un-circled nodes represent observed and unobserved variables, respectively.", "Arrows represent the causal influence of a variable on another, with the influence of confounders on variables shown as dotted arrows.", "The ZZ and XX nodes represent, respectively, the participant's assigned and received treatment, whereas QQ stands for the psychological encouragement treatment.", "The SS and PP variables represent the (unobserved) somatic and psychosomatic states of the participant, respectively.", "The EE, DD, II, and MM nodes stand for the participant's expectation of symptom intensity, desire for improved symptoms, desire-expectation interaction, and emotional level, respectively.", "The sets of variables UU, C 1 C_1, C 2 C_2, C 3 C_3, L 1 L_1, L 2 L_2, L 3 L_3, V 1 V_1, V 2 V_2, V 3 V_3, and ℋ\\mathcal {H} stand for unmeasured confounder variables.", "The YY node represents the outcome variable.", "Panel a shows the full model.", "Panel b shows the reduced model where the unobserved somatic and psychosomatic states of a participant are not directly represented in the causal model.The variable $S$ represents the unmeasured biochemical/physiological (somatic) state of a participant and mediates the effect of the treatment on the outcome variable, $Y$ .", "For instance, if $X$ represents a drug treatment, then $S$ could represent the physiological state induced by the biochemical pathways targeted by the drug.", "The causal effects of $X$ on $S$ and of $S$ on $Y$ are quantified, respectively, by $\\eta $ and $\\lambda $ .", "The outcome variable is also influenced by the unmeasured psychosomatic state of the participant, represented by $P$ .", "We allow $P$ to influence $Y$ via a direct path, quantified by $\\tau $ , and by an indirect path, mediated by $S$ , and quantified by the product $\\delta \\, \\lambda $ .", "The combined effect of the direct and indirect paths represents the placebo effect.", "The direct path from $P$ to $Y$ represents the influence of the psychosomatic state on the outcome mediated by biochemical and physiological pathways distinct from the pathways influenced by the active treatment, while the influence of $P$ on $S$ allows for the possibility that $P$ also influences the same pathways targeted by the treatment $X$ .", "(Experimental evidence that placebo effects influence biochemical pathways is provided, for example, in studies of placebo analgesia involving endogenous opioid systems[1], [15], [16], [17], [18], [19], [20].", "See also figure 2 in reference[21], for empirical support about pathways influenced by both psychosocial context and drug treatments.)", "The role played by the expectation-desire model of emotions is made explicit by the observed variables $E$ , $D$ , $I$ and $M$ , representing, respectively, the expected symptom intensity, the desire for symptom improvement, the interaction between expectation and desire, and the emotional level (measured, for example, by the participant's mood).", "According to the expectation-desire model, $M$ is directly influenced by $E$ , $D$ , and their interaction $I = E \\times D$ .", "The causal influence of $M$ on $P$ is quantified by $\\phi $ .", "In unblinded trials it is reasonable to expect that the treatment actually received by the participant will affect its expected symptom intensity, since participants who know they are receiving the active treatment will more likely experience an increase in their expectation to feel better.", "Hence, we include an arrow from $X$ to $E$ .", "The implication is that the treatment can influence the outcome not only via the participant's somatic state, but also by its psychosomatic state via the paths $X \\rightarrow E \\rightarrow M \\rightarrow P$ and $X \\rightarrow E \\rightarrow I \\rightarrow M \\rightarrow P$ .", "The binary variable $Q$ represents the randomized psychological encouragement IV assuming the value 1 when a encouragement message (aiming to increase the desire for symptom improvement) is applied to the participant, and 0 otherwise.", "In addition to the key variables described so far, it is important to recognize the existence of unmeasured confounders.", "Except for the exogenous variables $Z$ and $Q$ , that by construction are not associated with any unmeasured confounders, the model includes confounders influencing all pairs of endogenous variables other than $I$ , namely, $X$ , $E$ , $D$ , $M$ , $P$ , $S$ , and $Y$ .", "(It is not necessary to include confounders between $I$ and the other endogenous variables, since $I$ is deterministically computed as the product of $E$ and $D$ ).", "For instance, $U$ represents a set of unmeasured confounder variables influencing $X$ and $Y$ .", "In order to avoid cluttering the figure, the confounder variables influencing $S$ and $P$ and all other endogenous variables are represented by the vector of variables $\\mathcal {H}= (H_1, \\ldots , H_{11})^T$ .", "(For the same reason the figure does not explicitly shows the error terms, which account for unmeasured variables influencing each particular variable in the model and are uncorrelated with each other).", "It would be unrealistic to assume, for example, that the emotion of a participant is determined by $E$ , $D$ , and $I$ alone.", "Hence, the model allows sets of unmeasured confounders, such as $L_1$ , $L_2$ and $L_3$ , to influence emotion and expectation, emotion and desire, and expectation and desire, respectively.", "Similarly, it would be unrealistic to assume that emotion alone influences the psychosomatic state of a participant, and the model accommodates unmeasured confounders influencing these variables as well.", "Although, in practice, not all endogenous variables (other than $I$ ) will necessarily be influenced by confounders, the model still includes confounders for all 21 pairwise combinations of endogenous variables, since we want to derive estimators for the placebo and treatment effects under the most general setting possible.", "In practice, however, it is impossible to accurately measure the unobserved somatic and psychosomatic states of a participant.", "Hence, Figure REF b presents a reduced version where $S$ and $P$ are not explicitly represented in the graph.", "Assuming linear relationships between $S$ and $X$ , $P$ and $M$ , and $Y$ , $S$ , and $P$ , the causal influence of $X$ on $Y$ is given by $\\beta = \\eta \\, \\lambda $ , while the influence of $M$ on $Y$ is given by $\\psi = \\phi \\, \\tau + \\phi \\, \\delta \\, \\lambda $ .", "Under this reduced model the instrumental variable $Q$ allows for the consistent estimation of the net placebo effect, $\\psi $ , using the IV estimator $\\widehat{\\psi } = \\widehat{\\mbox{Cov}}(Q, Y)/\\widehat{\\mbox{Cov}}(Q, M)$ .", "Once the net placebo effect is estimated, it is possible to estimate the causal effect of $X$ on $Y$ using the IV estimator of the causal effect of $X$ on the residuals of the outcome variable after the removal of the placebo effect, $\\widehat{\\beta } = \\widehat{\\mbox{Cov}}(Z, \\widehat{R})/\\widehat{\\mbox{Cov}}(Z, X)$ , where $\\widehat{R} = Y - \\hat{\\psi } \\, M$ (see Methods for details)." ], [ "Performance evaluation", "We assessed the statistical performance of the proposed method (and compare it to a naive regression approach) in 16 simulation experiments evaluating the empirical type I error rate and empirical power of randomization tests for the null hypotheses that the placebo effect is zero, $H_0: \\psi = 0$ , and that the treatment effect is zero, $H_0: \\beta = 0$ .", "Descriptions of the randomization tests and simulation experiments are provided in the Methods.", "We simulated data from blinded and unblinded trials, in the presence and absence of confounders, according to the models presented in Figure REF .", "For each setting, we ran 4 separate simulation experiments generating data: (i) under the null for treatment and placebo effects; (ii) under the alternative for treatment, and null for placebo effects; (iii) the other way around; and (iv) under the alternative for treatment and placebo effects.", "Each simulation experiment employed 10,000 distinct synthetic data sets with diverse characteristics (see Methods).", "Although the randomization tests are non-parametric procedures free of distributional assumptions, we still generated data using gaussian errors in order to met the distributional requirements of the regression based analytical tests used in our comparisons.", "Figure: Models used in the simulation study.", "Panels a and b represent, respectively, blinded and unblinded trials influenced by confounders.", "For simplicity we include a single confounder variable per pair of endogenous variables (other than II), but still simulate confounding across the 10 possible pairwise combinations of the endogenous variables XX, YY, EE, MM, and DD.", "Panels c and d represent, respectively, unconfounded blinded and unblinded trials.", "For simulations under the null H 0 :ψ=0H_0: \\psi = 0 there are no arrows from MM to YY.", "Similarly, for simulations under H 0 :β=0H_0: \\beta = 0, there are no arrows from XX to YY.Figure: Empirical type I error rates of the placebo effect null, H 0 :ψ=0H_0: \\psi = 0, in both blinded and unblinded settings.", "Panels a and b show that, in the presence of confounders, the type I error rate of the IV approach is controlled at the exact nominal level (red and blue), whereas the regression based test leads to highly inflated error rates (orange and brown).", "Panels c and d show that, in the absence of confounding, both IV and regression approaches show well controlled errors.", "The nominal significance level is represented by α\\alpha .Figure: Empirical type I error rates for the treatment effect null, H 0 :β=0H_0: \\beta = 0, in the blinded setting.", "Panels a and b show that, in the presence of confounders, the type I error rates of the IV approaches are controlled at the exact nominal level (red and blue), whereas the regression based test leads to highly inflated error rates (brown).", "Panels c and d show that, in the absence of confounding, both IV and regression approaches show well controlled errors.", "The nominal significance level is represented by α\\alpha .Figure: Empirical type I error rates for the treatment effect null, H 0 :β=0H_0: \\beta = 0, in the unblinded setting.", "The two-step IV approach (blue) shows slightly inflated errors in the presence (panels a and b) and absence (panels c and d) of confounders.", "Note that the larger errors in panels c and d, in comparison to a and b, are likely due to the effective stronger influence of XX on MM in the simulations unaffected by confounders (the presence of confounders can considerably increase the amount of noise), so that adjustment by ψ ^\\hat{\\psi } leaks more information about XX in the absence than in the presence of confounders.", "The estimator adjusted by the true placebo effect (dark-orange) leads, nonetheless, to well controlled errors.", "The non-adjusted IV approach (red) leads to well controlled errors in the absence of placebo effects (panels a and c), but to highly inflated errors in the presence of placebo effects (panels b and d).", "Regression (brown) leads to highly inflated errors in the presence of confounders (panels a and b), but to well controlled error rates in their absence (panels c and d).Figure REF presents the results for the placebo effect tests, and shows that the error rates of the IV approach (red and blue) are controlled at the exact nominal levels in both blinded and unblinded settings, in the presence and absence of confounders.", "The regression approach (brown and dark-orange), on the other hand, shows highly inflated errors in the presence of confounders (panels a and b), since the association between $M$ and $Y$ , caused by confounders, is mistaken by an influence of $M$ on $Y$ .", "Being able to control type I error rates at the exact nominal level is a desirable statistical property, as it means that the test is neither conservative nor liberal.", "Figure REF presents the results for the treatment effect tests in the blinded setting.", "In addition to the two-step estimator (blue), we also evaluated the simple IV estimator $\\widehat{\\beta } = \\widehat{\\mbox{Cov}}(Z, Y)/\\widehat{\\mbox{Cov}}(Z, X)$ , which does not account for the placebo effect (red).", "The results show, again, well controlled error rates for both IV approaches, but inflated errors for the regression test (brown) in the presence of confounders (panels a and b).", "Figure REF presents the results for the unblinded case.", "All panels show slightly inflated errors for the two-step IV estimator (blue).", "The likely reason is that the estimated placebo effects are noisy and unable to completely block the influence of $X$ on $Y$ through the paths mediated by $M$ .", "To test this supposition, we evaluated an additional IV estimator, where the true placebo effect was used in the computation of the residuals (i.e., we estimated $\\beta $ by $\\widehat{\\beta } = \\widehat{\\mbox{Cov}}(Z, R)/\\widehat{\\mbox{Cov}}(Z, X)$ , where $R = Y - \\psi \\, M$ , instead of $\\widehat{\\beta } = \\widehat{\\mbox{Cov}}(Z, \\widehat{R})/\\widehat{\\mbox{Cov}}(Z, X)$ , where $\\widehat{R} = Y - \\hat{\\psi } \\, M$ ).", "Results based on this estimator (dark-orange) show that, indeed, adjustment by the true placebo effect leads to error rates controlled at the nominal level.", "The regression approach (brown) shows again highly inflated errors in the presence of confounders (panels a and b).", "Panels a and c show well controlled errors for the non-adjusted IV estimator (red) in the absence of placebo effects as, in this case, there are no paths from $X$ to $Y$ , and the association between $X$ and $Y$ induced by confounders is accounted by the IV estimator.", "Panels b and d, on the other hand, show highly inflated error rates in the presence of placebo effects since, in this case, $X$ can influence $Y$ through the paths mediated by $M$ .", "These observations suggest that, in practice, when analyzing the results of unblinded trials, we should first test for the existence of placebo effect, and then use the two-step IV estimator if $H_0: \\psi = 0$ is rejected, and the non-adjusted one if $H_0: \\psi = 0$ is accepted.", "While this strategy can decrease the chance of the two-step approach making a type I error in the absence of placebo effects, the estimator is still unable to avoid slightly inflated errors produced in the presence of placebo effects.", "We point out, however, that the two-step procedure still represents a strong improvement over the alternative approach of not adjusting for placebo effects in the presence of confounders (compare the red and blue curves in panel b of Figure REF ).", "Figure: Empirical power to detect placebo effects in the blinded and unblinded settings.", "Panels a and b show the results in the presence of confounders, whereas panels c and d show the results in their absence.", "The regression approach (brown and dark-orange) were considerably better powered than the IV approaches (blue and red) in the presence of confounders (panels a and b), but only slightly better powered in the absence of confounders (panels c and d).", "Both regression and IV approaches showed similar power under the blinded and unblinded settings.Figure: Empirical power for detecting treatment effects in the blinded setting.", "The regression approach (brown) tends to be better powered than the IV approaches in the presence of confounders (panels a and b), but only slightly better in the absence of confounding (panels c and d).", "The two-step IV approach (blue) tends to be better powered than the non-adjusted one (red) in the presence of placebo effects (panels b and d), but both IV approaches tend to be comparable in absence of placebo effects (panels a and c).Figure: Empirical power for detecting treatment effects in the unblinded setting.", "The regression approach (brown) tended to be better powered than the IV approaches in the presence of confounders (panels a and b), but comparable in the absence of confounding (panels c and d).", "The two-step IV approach (blue) tended to be slightly better powered than the non-adjusted one (red) in the presence of placebo effect (panel b), but comparable in the other panels.For completeness, we also report an evaluation of the empirical power (Figures REF , REF , and REF ).", "We point out, however, that power results are more sensitive to the choice of parameter values employed in the generation of the simulated data (e.g., sample size, the strength of treatment, placebo and confounding effects, and etc), than the type I error rates.", "In any case, these empirical power results, still serve to illustrate some general patterns.", "For instance, the regression tests tended to show considerably stronger power than the IV approaches in the presence of confounders (compare the brown and blue curves in panels a and b of Figures REF and REF ).", "We point out, however, that this increased power is likely an artifact of the biased estimates of $\\beta $ outputted by the regression approach.", "Figure REF , illustrates how the regression estimates tended to show larger bias than the estimates generated by the IV estimators (note the heavier tails of the brown density, in both blinded and unblinded cases).", "In other words, the increased power is likely a consequence of the overestimation of the treatment effect by the regression approach, which mistakenly interprets the association between treatment and outcome caused by unmeasured confounders as a stronger influence of the treatment on the outcome.", "Figure: Comparison of the bias of the regression and IV estimators.", "Panels a and b show the densities of the difference between true and estimated treatment effects, β-β ^\\beta - \\hat{\\beta }, in the blinded and unblinded settings, respectively.", "In both settings we observed larger bias in the regression estimates, in comparison to the IV approaches, as illustrated by the heavier tails of the brown densities.At least for the parameter ranges adopted in our simulations, we observed good empirical power of the IV approach to detect placebo effects, even when the correlation between psychological encouragement and emotional level was relatively low (Figure REF a).", "This suggests that the psychological encouragement treatment does not need to be highly effective in manipulating the emotional levels, in order for the approach to work well in practice.", "Similarly, Figure REF b shows good empirical power of the two-step IV approach to detect treatment effects when the correlation between the assigned and received treatment is moderate, suggesting that the proposed approach does not require high levels of compliance in order to perform well.", "Figure: Empirical power curves stratified by strength of association with the IV variable.", "Panel a shows the power curves for the placebo effect IV estimator ψ ^\\hat{\\psi }, stratified according to the correlation between QQ and MM (panel c shows the distribution of the correlation between QQ and MM across all simulations used to construct the power curves in panel a).", "Panel b shows the power curves for the two-step treatment effect IV estimator β ^\\hat{\\beta }, stratified according to the correlation between ZZ and XX (panel d shows the distribution of the correlation between ZZ and XX over the simulations used to estimate the power curves in panel b).", "Results based on blinded and unblinded simulations influence by confounders.Figure: Consistency of the ψ ^\\hat{\\psi } and β ^\\hat{\\beta } estimators.", "Panels a and b present, respectively, the densities of ψ-ψ ^\\psi - \\hat{\\psi } and β-β ^\\beta - \\hat{\\beta } for 5 increasing sample size ranges, and illustrate the consistency of the ψ ^\\hat{\\psi } and β ^\\hat{\\beta } estimators (which tend to get closer to the true parameter values as the sample size increases).", "Panel c shows that, as expected, the statistical power to detect a treatment effect increases with the sample size.", "Panel d, on the other hand, shows that increasing sample sizes do not reduce type I error rates, even though we are able to better estimate the placebo effects.", "The likely reason is that while larger sample sizes lead to better ψ ^\\hat{\\psi } estimates, they also increase the statistical power to detect very small effects, so that the advantage of a more precise estimate of ψ ^\\hat{\\psi } is counterbalanced by the increased propensity to detect small and spurious treatment effects as true signals.", "Results were based on data simulated from unblinded trials influenced by placebo effects and counfounders, as described in the Methods section.Figure: Randomization confidence intervals for placebo and treatment effects.", "The brown, dark-green and blue curves show the one-sided p-value profiles derived from randomization tests for 3 simulated data sets of increasing sizes (300, 900, and 2,700, respectively), generated under the unblinded setting influenced by confounders (all simulation parameters, other than sample size, were set to 1).", "The 95% confidence intervals for the placebo (panel a) and treatment effects (panel b) are shown by the respective double-headed colored arrows.", "The red vertical line corresponds to the true parameter values, ψ=1\\psi = 1 and β=1\\beta = 1.A natural question, at this point, is whether larger sample sizes (and, hence, more precise estimates of $\\hat{\\psi }$ ) would be able to decrease the slightly inflated error rates produced by the two-step estimator in unblinded trials.", "Figure REF presents additional simulation experiments showing that, while the empirical power and the $\\hat{\\psi }$ and $\\hat{\\beta }$ estimates are greatly improved by larger sample sizes, the type I error rates stay roughly the same (likely because larger sample sizes increase the ability of a test to detect small effects, since the randomization null distributions tend to be more concentrated around 0, so that the improved $\\hat{\\psi }$ estimates are counterbalanced by the increased propensity to detect small and spurious treatment effects).", "These results suggest that special care must be taken while interpreting the results of hypothesis tests in the unblinded case, even for large sample sizes.", "In any case, when the goal is estimation rather than testing, the consistency of the two-step estimator guarantees that the treatment estimates will converge to the true value as the sample size increases.", "This observation is particularly important in view of the current trend in the biomedical field, where researchers are shifting from relying exclusively in p-values and are paying more attention to parameter estimates and confidence intervals.", "To meet this latter need, we also describe in the Methods how to generate confidence intervals (CIs) for placebo and treatment effects by inverting randomization tests.", "Figure REF shows 95% CIs for the placebo and treatment effects, from 3 simulated data sets of increasing sizes.", "The randomization CIs inherit the statistical properties of the randomization tests, hence, the placebo effect CIs (and treatment effect CIs from blinded trials) are exact in the sense that a $100 (1 - \\alpha )$ % interval will contain the true parameter value $100 (1 - \\alpha )$ % of the time.", "Note that while the treatment effect CIs from unblinded trails won't be exact, they are still going to be centered around the estimated treatment effect, which will, nevertheless, converge to the true value as the sample size increases." ], [ "Discussion", "Clinical trials traditionally employ blinding to control the influence of placebo effects.", "It has being pointed out, however, that even blinded studies might be influenced by placebo effects, as the patients perceptions and beliefs about the treatment they think they received are able to trigger strong placebo effects[1], [11], [10].", "Recently, a number of statistical approaches have been proposed to quantify the contributions of treatment and placebo effects to a clinical outcome[22], [23], [24].", "These approaches, nonetheless, are tailored to blinded trials, and leverage blinding assessment data to quantify the amount of unmasking taking place during the trial.", "Our IV approach, on the other hand, allows the quantification of treatment and placebo effects not only in blinded, but also in unblinded trials.", "The key idea underlying the IV approach (what actually allows the consistent estimation of both treatment and placebo effects in the presence of unmeasured confounders), is the use of randomization to separately manipulate the treatment assignment and encouragement messages.", "In this sense, the proposed approach is similar in spirit (but not exactly equivalent) to a randomized treatment-belief trial (RTB)[25], where the treatment assignment is manipulated by randomization, whereas the belief is manipulated by varying the allocation ratio of participants assigned to control and treatment groups in a, necessarily, blinded trial.", "Hence, our IV approach can be viewed as a more flexible type of RTB that is applicable to both blinded and unblinded studies, and might be easier to administer than a standard RTB, which requires the stratification of study participants over several arms with distinct treatment/control allocation ratios in order to be able to assess placebo effects.", "The proposed IV approach enjoys appealing statistical properties.", "The IV estimators are consistent, meaning that the estimates converge to the true values as sample size increases.", "The randomization tests for placebo effects are exact in both blinded and unblinded trails, whereas the treatment effect tests are exact in blinded trials, but slightly liberal in unblinded ones.", "Furthermore, the confidence intervals obtained by inverting randomization tests inherit these appealing properties.", "An implicit assumption of the model in Figure REF a is that the placebo effect is mediated exclusively by the interplay of desire, expectation, and emotion.", "While it is believed that the desire-expectation model plays a key role in the triggering of placebo effects, other mechanisms, such as conditioning, might also be at work[1], [21].", "Clearly, when this is the case, a treatment effect estimate, adjusted by the desire-expectation component alone, will still be biased (although less biased than the estimate computed without accounting for it).", "In any case, if we are also able to assess and measure these additional mechanisms, then the same statistical framework can be used to obtain consistent estimates of treatment effects in the presence of confounders (we only need additional IVs to manipulate the additional placebo related variables).", "Figure REF shows an example.", "Figure: A more complex example.", "Panel a presents a more complex model where the placebo effect is mediated by MM (according to the desire-expectation model) but also by an additional variable AA.", "Assuming that a randomized instrument, WW, is available to manipulate AA, we can estimate the treatment effect using the estimator β ^ * =Cov ^(Z,R ^ * )/Cov ^(Z,X)\\hat{\\beta }^\\ast = \\widehat{\\mbox{Cov}}(Z, \\hat{R}^\\ast )/\\widehat{\\mbox{Cov}}(Z, X) where R ^ * =Y-ψ ^M-κ ^A\\hat{R}^\\ast = Y - \\hat{\\psi } \\, M - \\hat{\\kappa } \\, A.", "Panel b shows the empirical type I error rates for a simulation experiment under the unblinded setting influenced by confounders.", "The IV estimator adjusted by the true ψ\\psi and κ\\kappa values is able to control error rates at the nominal levels (dark-orange).", "The IV estimator adjusted by ψ ^\\hat{\\psi } and κ ^\\hat{\\kappa } shows slightly inflated errors (dark-green).", "As expected, adjustment with ψ ^\\hat{\\psi } alone (blue) leads to higher error rates than adjustment with both ψ ^\\hat{\\psi } and κ ^\\hat{\\kappa }.", "Similarly, the IV estimator using no adjustment (red) has higher errors than adjustment by ψ ^\\hat{\\psi } alone.", "The regression based estimator (brown) is adjusted by both MM and AA covariates, but still leads to inflated errors due to the presence of confounders.", "Panel c shows the empirical power results.From a pragmatic perspective, the proposed method is easy to implement.", "It only requires the ability to assess expectation, desire, and emotion, as well as, the development of a psychological encouragement IV, capable of manipulating the level of desire of a study participant.", "For example, in trials run into a clinic, a simple encouragement conversation with a caregiver would work as the “active treatment\" of the psychological encouragement IV.", "The desire and emotional level could then be recorded by a questionnaire or interview after the encouragement treatment, but prior to the measurement of the outcome variable.", "Another application of the proposed method (the one that actually motivated this work) is in the personalized monitoring of treatment response in mobile health.", "The statistical validity of using treatment assignment as an IV, in the context of longitudinal data provided by a single patient, has been established in reference[26].", "However, as pointed out by the authors, it is impossible to disentangle treatment and placebo effects based on the treatment assignment IV alone, since it is impossible to blind the patient to a self administered treatment.", "Implementation of the proposed IV approach in mobile health applications is also strait-forward.", "For instance, the psychological treatment could be delivered by encouragement messages popping up in the screen of a smartphone (according to a randomized schedule, where every day the participant has an equal chance of receiving, or not, the encouragement message), and the measurement of the emotional and desire levels can be assessed by short electronic surveys/quentionnaires delivered by the participant's smartphone on a daily basis.", "We expect the proposed method to play an important role in these personalized medicine[27], [28] applications.", "Finally, for both (population-based) clinical trials and personalized monitoring of treatment response, the instrument $Q$ serves the double role of disentangling placebo from treatment effects, and increasing the desire for improved symptoms.", "This latter capacity can induce a placebo effect and ultimately lead to more positive clinical outcomes.", "While the manipulation of the expectation for symptom intensity could, in principle, be used to consistently estimate a placebo effect under the proposed approach (i.e., we could have an IV influencing $E$ instead of $D$ ), the manipulation of expectation levels needs to be accompanied by the honest disclosure of the expected benefits of a treatment (and, in some cases, might raise ethical issues)[21].", "Manipulation of the desire for improved symptoms, on the other hand, provides an ethically defensible practice in the design of clinical trials and in the personalized monitoring of patients." ], [ "Identification of causal effects using instrumental variables", "We subscribe to the mechanism-based account of causation championed by Pearl[29].", "In this framework, the qualitative description of the assumptions regarding the causal relations between the variables involved in our proposed method, is encoded by the directed acyclic graph presented in Figure REF a.", "Assuming a linear relation between the outcome, $Y$ , and the unobserved somatic and psychosomatic state variables, $S$ and $P$ , we have that, $Y = \\mu _Y + \\lambda \\, S + \\tau \\, P + f_Y(U, \\mathcal {V}, \\mathcal {H}) + \\epsilon _Y~,$ where $\\mathcal {V}= (V_1, V_2, V_3)^T$ , $\\mathcal {H}= (H_1, \\ldots , H_{11})^T$ , $\\epsilon _Y$ represents an error term accounting for the unmeasured variables influencing exclusively $Y$ , and $f_Y(U, \\mathcal {V}, \\mathcal {H})$ represents is a general scalar function of the variables in $(U, \\mathcal {V}, \\mathcal {H})$ influencing $Y$ .", "Since $S$ and $P$ are unobserved variables, we need to derive the reduced model for the outcome variable that is not a function of $S$ and $P$ .", "Assuming a linear relation between $P$ and $M$ , and between $S$ and $P$ and $X$ , we have that, $P &= \\mu _P + \\phi \\, M + f_P(\\mathcal {H}) + \\epsilon _P~, \\\\S &= \\mu _S + \\eta \\, X + \\delta \\, P + f_S(\\mathcal {H}) + \\epsilon _S~, $ where $f_P(\\mathcal {H})$ and $f_S(\\mathcal {H})$ are arbitrary scalar functions of $\\mathcal {H}$ , and $\\epsilon _P$ and $\\epsilon _S$ are the respective error terms influencing $P$ and $S$ , respectively (we also assume that all variable specific error terms are uncorrelated).", "Substituting equations (REF ) and () into equation (REF ), we obtain the reduced outcome model, $Y = \\mu _Y^\\ast + \\beta \\, X + \\psi \\, M + f^\\ast (U, \\mathcal {V}, \\mathcal {H}) + \\epsilon _Y^\\ast $ where $\\beta = \\eta \\, \\lambda $ , $\\psi = \\phi \\, \\tau + \\phi \\, \\delta \\, \\lambda $ , $\\mu _Y^\\ast = \\mu _Y + \\lambda \\, \\mu _S + (\\tau + \\delta \\, \\lambda ) \\, \\mu _P$ , $\\epsilon _Y^\\ast = \\epsilon _Y + \\lambda \\, \\epsilon _S + (\\tau + \\delta \\, \\lambda ) \\, \\epsilon _P$ , and $f^\\ast (U, \\mathcal {V}, \\mathcal {H}) = f_Y(U, \\mathcal {V}, \\mathcal {H}) + \\lambda \\, f_S(\\mathcal {H}) + (\\tau + \\delta \\, \\lambda ) \\, f_P(\\mathcal {H})$ .", "Equation (REF ) represents the outcome model in Figure REF b.", "Because the instrumental variable $Q$ is randomized, and hence statistically independent of any variables that are not directly or indirectly influenced by $Q$ , it follows from equation (REF ) and standard properties of the covariance operator that, $\\mbox{Cov}(Q, Y) &= \\mbox{Cov}(Q, \\mu _Y^\\ast ) + \\beta \\, \\mbox{Cov}(Q, X) + \\psi \\, \\mbox{Cov}(Q, M) + \\\\ \\nonumber & \\;\\;\\;\\; + \\mbox{Cov}(Q, f^\\ast (U, \\mathcal {V}, \\mathcal {H})) + \\mbox{Cov}(Q, \\epsilon _Y^\\ast ) \\\\ \\nonumber &= \\psi \\, \\mbox{Cov}(Q, M)~,$ since $Q \\perp \\!\\!\\!\\perp \\mu _Y^\\ast $ , $Q \\perp \\!\\!\\!\\perp X$ , $Q \\perp \\!\\!\\!\\perp f^\\ast (U, \\mathcal {V}, \\mathcal {H})$ , and $Q \\perp \\!\\!\\!\\perp \\epsilon _Y^\\ast $ , and the respective covariance terms are 0 (here, the symbol $\\perp \\!\\!\\!\\perp $ stands for statistical independence).", "Therefore, $\\psi $ can be identified as, $\\psi = \\frac{\\mbox{Cov}(Q, Y)}{\\mbox{Cov}(Q, M)}~,$ as long as $\\mbox{Cov}(Q, M) \\ne 0$ (in practice, this condition is met if the psychological encouragement treatment can effectively manipulate the desire for improved symptoms, which, by its turn influences the emotional state, $M$ ).", "Now, if we let $R = Y - \\psi \\, M$ represent the residual of the outcome variable, after removal of the placebo effect, then we can rewrite equation (REF ) as, $R = \\mu _Y^\\ast + \\beta \\, X + f^\\ast (U, \\mathcal {V}, \\mathcal {H}) + \\epsilon _Y^\\ast ~.$ Because $Z$ is also randomized, it follows from equation (REF ) and the properties of the covariance operator that, $\\mbox{Cov}(Z, R) &= \\mbox{Cov}(Z, \\mu _Y^\\ast ) + \\beta \\, \\mbox{Cov}(Z, X) + \\\\ \\nonumber & \\;\\;\\;\\; + \\mbox{Cov}(Z, f^\\ast (U, \\mathcal {V}, \\mathcal {H})) + \\mbox{Cov}(Z, \\epsilon _Y^\\ast ) \\\\ \\nonumber &= \\beta \\, \\mbox{Cov}(Z, X)~,$ since $Z \\perp \\!\\!\\!\\perp \\mu _Y^\\ast $ , $Z \\perp \\!\\!\\!\\perp f^\\ast (U, \\mathcal {V}, \\mathcal {H})$ , and $Z \\perp \\!\\!\\!\\perp \\epsilon _Y^\\ast $ .", "Hence, the treatment effect $\\beta $ can be identified as, $\\beta = \\frac{\\mbox{Cov}(Z, R)}{\\mbox{Cov}(Z, X)}~,$ as long as $\\mbox{Cov}(Z, X) \\ne 0$ (in practice, this condition is met whenever there is some degree of compliance between assigned and received treatments).", "Note that in addition to the three core assumptions required by an IV[14] (namely, that it is statistically independent of any unmeasured confounders; is marginally associated with the treatment variable; and is associated with the outcome variable exclusively through its influence on the treatment variable), the above derivations require that $X$ and $M$ are linearly related to $Y$ , but make no assumptions about the relationship between $Y$ and the unmeasured confounders.", "Estimators for the placebo and treatment effects in equations (REF ) and (REF ) are presented in the next subsection.", "An alternative statistical framework, based on Rubin's potential outcomes approach to causality[30], [31], has been proposed in the literature to address partial compliance in studies involving binary instrumental and treatment variables[32], [31].", "While the method in reference[32] is not directly applicable to the estimation of the placebo effects, it could be used to estimate treatment effects (after removal of the placebo effect).", "We point out, however, that the estimator obtained from the potential outcome approach is still identical to the estimator derived from the mechanism based approach, so that statistical inferences based on randomization tests are still the same, independent of the causality framework one is willing to adopt." ], [ "Two-step estimation procedure", "Adopting a method of moments approach, we have that a consistent estimator for $\\psi $ is given by, $\\hat{\\psi } = \\frac{\\widehat{\\mbox{Cov}}(Q, Y)}{\\widehat{\\mbox{Cov}}(Q, M)} = \\frac{\\frac{1}{n} \\sum _{k=1}^{n} Q_{k} Y_{k} - (\\frac{1}{n} \\sum _{k=1}^{n} Q_{k}) (\\frac{1}{n} \\sum _{k=1}^{n} Y_{k})}{\\frac{1}{n} \\sum _{k=1}^{n} Q_{k} M_{k} - (\\frac{1}{n} \\sum _{k=1}^{n} Q_{k}) (\\frac{1}{n} \\sum _{k=1}^{n} M_{k})}~.$ Note that the above placebo effect estimator requires measurements of $M$ , but not of $E$ or $D$ .", "We point out, however, that if expectation and desire measurements are also available, then we can evaluate the validity of the desire-expectation model for the data at hand by checking whether the $E$ , $D$ , and $I$ variables are able to predict the $M$ measurements.", "We can also assess the effectiveness of the psychological treatment in influencing desire for better symptoms by estimating $\\mbox{Cor}(Q, D)$ .", "Direct estimation of the treatment effect in equation (REF ) using an IV estimator is unfeasible, as it would involve the unmeasured quantities $R_k = Y - \\psi \\, M_k$ .", "Therefore, in order to obtain a consistent estimator of the treatment effect, we adopt a two-step approach where we first estimate $R_k$ as $\\widehat{R}_{k} = Y_k - \\hat{\\psi } \\, M_k$ , for $k = 1, \\ldots , n$ , and then estimate $\\beta $ using, $\\hat{\\beta } = \\frac{\\widehat{\\mbox{Cov}}(Z, \\widehat{R})}{\\widehat{\\mbox{Cov}}(Z, X)} = \\frac{\\frac{1}{n} \\sum _{k=1}^{n} Z_{k} \\widehat{R}_{k} - (\\frac{1}{n} \\sum _{k=1}^{n} Z_{k}) (\\frac{1}{n} \\sum _{k=1}^{n} \\widehat{R}_{k})}{\\frac{1}{n} \\sum _{k=1}^{n} Z_{k} X_{k} - (\\frac{1}{n} \\sum _{k=1}^{n} Z_{k}) (\\frac{1}{n} \\sum _{k=1}^{n} X_{k})}~.$ Note that the IV estimators in equations (REF ) and (REF ) can produce highly inflated estimates when $\\widehat{\\mbox{Cov}}(Q, M) \\approx 0$ and $\\widehat{\\mbox{Cov}}(Z, X) \\approx 0$ .", "Hence, in practice, it is important to check the assumptions that psychological encouragement influences the emotion levels, and that the compliance between assigned and received treatments is not negligible." ], [ "Randomization tests for $H_0: \\psi = 0$ and {{formula:71b04f3c-fcad-463d-9645-1ac7564379d0}}", "We implemented randomization tests[33] for testing the presence of a placebo effect ($H_0: \\psi = 0$ versus $H_1: \\psi \\ne 0$ ), and of a treatment effect ($H_0: \\beta = 0$ versus $H_1: \\beta \\ne 0$ ).", "The randomization null distribution for the placebo effect is generated by evaluating the statistic $\\hat{\\psi }$ in equation (REF ) on a large number of shuffled versions of the data, where the $Y_k$ measurements are shuffled relative to the $(Q_k, M_k)$ measurements (whose connection is kept intact in order to preserve the association between the $Q$ and $M$ variables).", "The randomization null for treatment effect is generated by first calculating the residuals, $\\widehat{R}_{k} = Y_k - \\hat{\\psi } \\, M_k$ , where $\\hat{\\psi }$ is computed in the observed (not permuted) data, and then evaluating the statistic $\\hat{\\beta }$ in equation (REF ) in shuffled data sets, where $R_k$ is shuffled relative to $(Z_k, X_k)$ data (whose connection is kept intact to preserve the association between $Z$ and $X$ ).", "These randomization tests are non-parametric procedures and don't make any distributional assumptions about the data.", "However, because the identification of the causal effects assumes a linear relation between $Y$ and $X$ and $M$ , the validity of the tests is still contingent on this assumption." ], [ "Randomization confidence intervals", "Here we describe how to build confidence intervals for placebo and treatment effects using the p-values from randomization tests[34], [33].", "Throughout this section we use $\\theta $ to represent either the placebo effect, $\\psi $ , or the treatment effect, $\\beta $ .", "The procedure is strait-forward but requires a considerable amount of computation (which, nonetheless, can be easily parallelized).", "Assume for a moment that randomization tests for testing $H_0: \\theta = \\theta _j$ against one-sided alternative hypotheses $H_1: \\theta < \\theta _j$ and $H_1: \\theta > \\theta _j$ are available.", "Exploring the correspondence between confidence intervals and hypothesis tests, we obtain a $100 (1 - 2 \\alpha )$ % confidence interval (CI) for $\\theta $ by searching for a lower bound value, $\\theta _L$ , such that $H_0: \\theta = \\theta _L$ is rejected in favor of $H_1: \\theta > \\theta _L$ at a significance $\\alpha $ , and by searching for an upper bound value, $\\theta _U$ , such that $H_0: \\theta = \\theta _U$ is rejected in favor of $H_1: \\theta < \\theta _U$ at the same significance level[34].", "While an efficient algorithm for finding CI bounds has been proposed[34], the approach requires the specification of the significant level before hand.", "In order to avoid this constraint, we generate a one-sided randomization p-value profile which can be used to determine the $100 (1 - 2 \\alpha )$ % CI for any desired $\\alpha $ level.", "This p-value profile is generated as follows: ($i$ ) compute the observed placebo or treatment effect estimate, $\\hat{\\theta }$ ; ($ii$ ) for each $\\theta _j < \\hat{\\theta }$ , in a grid of decreasing $\\theta _j$ values, compute the randomization p-value from the one-sided test $H_0: \\theta = \\theta _j$ vs $H_1: \\theta > \\theta _j$ ; ($iii$ ) repeat step $ii$ until a p-value equal to zero is reached; ($iv$ ) for each $\\theta _j > \\hat{\\theta }$ , in a grid of increasing $\\theta _j$ values, compute the p-value from the one-sided test $H_0: \\theta = \\theta _j$ vs $H_1: \\theta < \\theta _j$ ; ($v$ ) repeat step $iv$ until a randomization p-value equal to zero is found.", "Before we explain how to generate null distributions for placebo effects different from zero, consider first the intention-to-treat (ITT) estimator, $\\widehat{ITT}_\\psi = \\frac{\\sum _{k=1}^{n} Y_k \\, 1\\!\\!1\\lbrace Q_k = 1 \\rbrace }{\\sum _{k=1}^{n} 1\\!\\!1\\lbrace Q_k = 1 \\rbrace } - \\frac{\\sum _{k=1}^{n} Y_k \\, 1\\!\\!1\\lbrace Q_k = 0 \\rbrace }{\\sum _{k=1}^{n} 1\\!\\!1\\lbrace Q_k = 0\\rbrace } = \\frac{\\widehat{\\mbox{Cov}}(Q, Y)}{\\widehat{\\mbox{Var}}(Q)}~.$ Instead of directly generating a randomization distribution under the null $H_0: \\psi = \\psi _j$ , we generate a randomization distribution under the equivalent null hypothesis that the intention-to-treat effect is equal to $\\psi _j \\, K_1$ , where $K_1 = \\widehat{\\mbox{Cov}}(Q, M)/\\widehat{\\mbox{Var}}(Q)$ is constant across all permutations of the response data used in the construction of the randomization null.", "(Note that, because $\\widehat{ITT}_\\psi = K_1 \\, \\hat{\\psi }$ the randomization tests based on $\\hat{\\psi }$ and $\\widehat{ITT}_\\psi $ estimators produce exactly the same p-value if we use the same permutations of the response data in the construction of their null distributions.)", "The practical advantage of the test based on $\\mbox{ITT}_\\psi $ effects is that it amounts to a simple two sample location problem for testing whether the difference in average response between the assigned treatment (psychological encouragement) and assigned control (no encouragement) groups is equal to $\\psi _j \\, K_1$ .", "The implementation of randomization tests for this two sample location problem is strait-forward[34]: we only need to add $\\psi _j \\, K_1$ for each $Y_k$ data point in the assigned control group (i.e., $k$ for which $Q_k = 0$ ), while leaving the response data from the assigned treatment group, $Q_k = 1$ , unchanged, and then run a randomization test for testing the null hypothesis that the $\\mbox{ITT}_\\psi $ effect is equal to zero, against the alternative one-sided hypothesis that it is positive, and against the alternative that it is negative.", "Similarly, for the treatment effects we consider the two-step ITT estimator, $\\widehat{ITT}_\\beta = \\frac{\\sum _{k=1}^{n} \\hat{R}_k \\, 1\\!\\!1\\lbrace Z_k = 1 \\rbrace }{\\sum _{k=1}^{n} 1\\!\\!1\\lbrace Z_k = 1 \\rbrace } - \\frac{\\sum _{k=1}^{n} \\hat{R}_k \\, 1\\!\\!1\\lbrace Z_k = 0 \\rbrace }{\\sum _{k=1}^{n} 1\\!\\!1\\lbrace Z_k = 0\\rbrace } = \\frac{\\widehat{\\mbox{Cov}}(Z, \\hat{R})}{\\widehat{\\mbox{Var}}(Z)}~,$ and generate randomization distributions under the equivalent null hypotheses $H_0: \\mbox{ITT}_\\beta = \\beta _j \\, K_2$ , where $K_2 = \\widehat{\\mbox{Cov}}(Z, X)/\\widehat{\\mbox{Var}}(Z)$ , by simply adding $\\beta _j \\, K_2$ for each $\\hat{R}_k$ data point in the assigned control group, $Z_k = 0$ , while leaving the residual data from the assigned treatment group, $Z_k = 1$ , unchanged (and then testing for the null that the $\\mbox{ITT}_\\beta $ is equal to zero, against the alternative one-sided hypotheses that it is positive and the alternative that it is negative0." ], [ "Adjustment for observed confounders", "If measured confounders influencing both $X$ and $Y$ are available, it is possible to adjust for them by simply working with the residuals of $X$ and $Y$ (computed by separately regressing $X$ and $Y$ on the confounders).", "Similarly, if measured confounders influencing both $M$ and $Y$ are available, it is possible to adjust for them by working with the respective residuals." ], [ "Regression based estimators and tests", "We compare the proposed IV estimators, and their respective randomization tests, to standard estimators and analytical hypothesis tests based on the linear regression of the outcome variable, $Y$ , on both the received treatment, $X$ , and emotion level, $M$ , according to the model, $Y = \\mu _Y + \\beta \\, X + \\psi \\, M + \\epsilon _Y$ .", "Under this regression based approach, we estimate $\\beta $ and $\\psi $ using ordinary least squares, and test the null hypotheses $H_0: \\psi = 0$ and $H_0: \\beta = 0$ using standard t-tests.", "In our simulations (described next), we generate data using gaussian errors, so that the distributional assumptions underlying the analytical t-tests are met." ], [ "Simulation experiments details", "We simulated data from blinded and unblinded settings, in the presence or absence of confounding, according to the models presented in Figure REF .", "For each of these settings, we run 4 separate simulation studies generating data: (i) under the null hypothesis that both treatment and placebo effect are zero, $H_0: \\beta = 0$ and $H_0: \\psi = 0$ ; (ii) under the alternative for treatment effects, $H_1: \\beta \\ne 0$ , but null for placebo effects, $H_0: \\psi = 0$ ; (iii) the other way around, $H_0: \\beta = 0$ and $H_1: \\psi \\ne 0$ ; and (iv) under the alternative for both treatment and placebo effects, $H_1: \\beta \\ne 0$ and $H_1: \\psi \\ne 0$ .", "Each simulated data set was generated as follows.", "The IVs $Z$ and $Q$ were sampled from $\\mbox{Bernoulli}(1/2)$ distributions.", "All confounding variables were sampled from $\\mbox{Normal}(0, 1)$ distributions.", "The binary variables $X$ , $E$ , and $D$ were generated by the threshold models, $X &= 1\\!\\!1\\lbrace \\theta _{XZ} \\, Z + \\theta _{XU} \\, U + \\theta _{XC_1} \\, C_1 + \\theta _{XC_2} \\, C_2 + \\theta _{XC_3} \\, C_3 + \\epsilon _{X} > 0 \\rbrace ~, \\\\E &= 1\\!\\!1\\lbrace \\theta _{EX} \\, X + \\theta _{EC_1} \\, C_1 + \\theta _{EL_1} \\, L_1 + \\theta _{EV_2} \\, V_2 + \\theta _{EL_3} \\, L_3 + \\epsilon _{E} > 0 \\rbrace ~, \\\\D &= 1\\!\\!1\\lbrace \\theta _{DQ} \\, Q + \\theta _{DV_1} \\, V_1 + \\theta _{DC_2} \\, C_2 + \\theta _{DL_2} \\, L_2 + \\theta _{DL_3} \\, L_3 + \\epsilon _{D} > 0 \\rbrace ~,$ where $\\epsilon _{X}$ , $\\epsilon _{E}$ , and $\\epsilon _{D}$ were sampled from $\\mbox{Normal}(0, 1)$ distributions.", "The interaction $I$ was generated as the product of $E$ and $D$ .", "Finally, the emotion and outcome data were generated from the linear models, $M = \\theta _{ME} \\, E + \\theta _{MD} \\, D + \\theta _{MI} \\, I + \\theta _{ML_1} \\, L_1 + \\theta _{ML_2} \\, L_2 + \\theta _{MC_3} \\, C_3 + \\theta _{MV_3} \\, V_3 + \\epsilon _{M}~,\\vspace{-8.5359pt}$ $Y = \\theta _{YX} \\, X + \\theta _{YM} \\, M + \\theta _{YU} \\, U + \\theta _{YV_1} \\, V_1 + \\theta _{YV_2} \\, V_2 + \\theta _{YV_3} \\, V_3 + \\epsilon _{Y}~,$ where $\\epsilon _{M}$ and $\\epsilon _{Y}$ were sampled from $\\mbox{Normal}(0, 1)$ distributions.", "(Note that the explicit form of the desire-expectation model of emotions is unimportant, as the estimator for $\\psi $ depends on the observed values of $M$ , but not of $E$ , $D$ , and $I$ , and does not require an explicit description of the functional relationships between $M$ and $E$ , $D$ , and $I$ .", "Hence, for simplicity, we adopt a simple linear relation, even though more sophisticated relations could have been used.)", "Each simulation experiment comprised 10,000 distinct synthetic data sets.", "Each simulated data set was generated using a unique combination of simulation parameter values.", "In order to select parameter values spread as uniformly as possible over the entire parameter range we employed a Latin hypercube design[35], optimized according to the maximin distance criterium [36], in the determination of the parameter values used on each of the 10,000 simulated data sets for each simulation experiment.", "We selected wide ranges for all model parameters.", "Explicitly, the parameters representing the effect of confounders on the observed variables (namely, $\\theta _{XU}$ , $\\theta _{XC_1}$ , $\\theta _{XC_2}$ , $\\theta _{XC_3}$ , $\\theta _{EC_1}$ , $\\theta _{EL_1}$ , $\\theta _{EV_2}$ , $\\theta _{EL_3}$ , $\\theta _{DV_1}$ , $\\theta _{DC_2}$ , $\\theta _{DL_2}$ , $\\theta _{DL_3}$ , $\\theta _{ML_1}$ , $\\theta _{ML_2}$ , $\\theta _{MC_3}$ , $\\theta _{MV_3}$ , $\\theta _{YU}$ , $\\theta _{YV_1}$ , $\\theta _{YV_2}$ , and $\\theta _{YV_3}$ ) were selected in the range $[-2, 2]$ for the simulations under the influence of confounders, but were set to 0 in the simulations under unconfounded conditions.", "The effect of $Z$ on $X$ ($\\theta _{XZ}$ ), and of $Q$ on $D$ ($\\theta _{DQ}$ ), as well as, the effects of $E$ , $D$ , and $I$ on $M$ ($\\theta _{ME}$ , $\\theta _{MD}$ , and $\\theta _{MI}$ ) were selected in the range $[1, 2]$ .", "The effect of $X$ on $E$ ($\\theta _{EX}$ ) was set to 0 in the blinded setting simulations, and selected in the range $[1, 2]$ in the unblinded simulations.", "The treatment effect ($\\beta $ ) and the placebo effect ($\\psi $ ) parameters were set to 0 in the simulations under the null hypothesis, and were selected in the range $[-2, 2]$ for the simulations under the alternative hypothesis.", "The range of sample size parameter, $n$ , was set to realistic values we expect to see in practice, $\\lbrace 100, 101, \\ldots , 1000\\rbrace $ .", "For any fixed significance level $\\alpha $ , the empirical type I error rate was computed as the fraction of the p-values smaller than $\\alpha $ over the data sets simulated under the null hypothesis, whereas the empirical power was calculated as the fraction of p-value smaller than $\\alpha $ over data sets generated under the alternative hypothesis." ], [ "Acknowledgements", "This work was funded by the Robert Wood Johnson Foundation." ] ]	1606.04896
[ [ "Solution of the Kirchhoff-Plateau problem" ], [ "Abstract The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension.", "We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact.", "In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure.", "Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori.", "Our mathematical results substantiate the physical relevance of the chosen model.", "Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments." ], [ "Introduction", "Liquid films spanning rigid frames have been of longstanding interest to physicists and mathematicians, thanks to the sheer beauty of the countless observable shapes.", "After the experimental investigations of Plateau [29], anticipated by Lagrange's [26] theoretical treatment of the minimal surface problem, the first satisfactory proofs of the existence of a surface of least area bounded by a fixed contour were provided only in the twentieth century by Garnier [18], Radó [30], and Douglas [15].", "This formed a basis for a wealth of mathematical investigations regarding minimal surfaces, concerning various aspects and generalizations of the classical Plateau problem.", "The interested reader is referred to the treatises by Dierkes, Hildebrandt & Sauvigny [13] and Dierkes, Hildebrandt & Tromba [14] for a comprehensive review of the formative contributions.", "An important generalization to the situation in which the boundary of the minimal surface is not fixed but is constrained to lie on a prescribed manifold was initially treated by Courant [9] and Lewy [27], whose work spurred a number of important mathematical contributions, as reviewed by Li [28].", "An existence theorem for a complementary generalization, in which part of the boundary is fixed and the remaining part is an inextensible but flexible string, was later proved by Alt [3].", "In the present article, we introduce a problem which combines those generalizations.", "We consider situations in which the boundary of the minimal surface lies on a deformable manifold, namely the surface of an elastic loop.", "The filament forming the loop is assumed to be thin enough to be modeled faithfully by a Kirchhoff rod, an unshearable inextensible rod which can sustain bending of its midline and twisting of its cross-sections (see Antman [4]).", "This is a mathematically one-dimensional theory that describes a three-dimensional object, endowed with a nonvanishing volume, since the material cross-sections have nonvanishing area.", "The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system composed by a closed Kirchhoff rod spanned by an area-minimizing surface.", "In recent years, some attention has been drawn to the Kirchhoff–Plateau problem following a paper by Giomi & Mahadevan [19], and the stability properties of flat circular solutions have been investigated, under various conditions regarding the material properties of the rod, by Chen & Fried [8], Biria & Fried [6], [7], Giusteri, Franceschini & Fried [20], and Hoang & Fried [25].", "An existence result for a similar problem was given by Bernatzky & Ye [5], but the elastic energy used therein fails to satisfy the basic physical requirement of invariance under superposed rigid transformations and thereby implicitly entails the appearance of unphysical forces.", "Moreover, a strong hypothesis is used to avoid the issue of self-contact.", "Importantly, in all of these studies the boundary of the spanning surface is assumed to coincide with the rod midline and not to lie on the surface of the rod.", "This amounts to a slenderness assumption.", "Moreover, the surface is viewed as diffeomorphic to a disk, except by Bernatzky & Ye [5], who employ the theory of currents.", "We relax both of these assumptions.", "Regarding the filament, we retain its three-dimensional nature for two physically well-justified reasons.", "First, there is a significant separation of scales between the typical thickness of the liquid film and the cross-sectional thickness of the filaments used in experimental investigations: a minimum of two orders of magnitude.", "Consequently, while the liquid film is still represented as two-dimensional, it seems appropriate to treat the bounding loop as a three-dimensional object.", "Second, the possibility of generating nontrivial shapes due to the interaction between the film and the bounding loop relies on the presence of anisotropic material properties of the filament, which are often associated with how its cross-sections are shaped.", "Furthermore, the physical presence of the bounding loop requires a proper treatment of the constraint of non-interpenetration of matter, which is clearly at play in real experiments and even becomes essential, since the bounding loop can sustain large deflections but remains constrained when self-contact occurs.", "If, in particular, the relative strength of surface tension with respect to the elastic response of the filament becomes large, then the compliance of the mathematical solution with physical requirements can only be guaranteed by including the non-interpenetration constraint.", "To include all these properties in a variational framework, we base our treatment of rod elasticity on Schuricht's [32] elegant approach.", "We introduce a minor simplification in the presentation of the constraint of local non-intrepenetration of matter, obtaining equivalent results specialized to the case of Kirchhoff rods.", "In developing our variational approach, the most delicate point involves the treatment of the spanning surface.", "The physical phenomenon at play is the minimization of the liquid surface area due to the presence of a homogeneous surface tension.", "Various mathematical models have been proposed with different characteristics.", "Representing the surface via a mapping from a manifold into the ambient space, despite being the first successful approach, poses severe and completely unphysical limitations on the topology of the surface.", "To cope with this issue, the theories of integral currents and of varifolds were applied to the Plateau problem by Federer & Fleming [17] and Almgren [1], respectively.", "However, their approaches also fail either to cover all the physical soap film solutions to the Plateau problem or to furnish a sufficiently general existence result.", "An alternate route was initiated by Reifenberg [31], who treated the surface as a point set that minimizes the two-dimensional Hausdorff measure.", "This purely spatial point of view, adopted also by De Pauw [12] and David [10], deals nicely with the topology of solutions, but makes it difficult to handle a generic boundary condition.", "A more complete treatment, which covers all the soap film solutions to the Plateau problem, has been developed by Harrison [21] and Harrison & Pugh [22] using differential chains.", "In that setting, it is possible to consider generic bounding curves due to the presence of a well-defined boundary operator.", "In relation to the problem at hand, the approach to the Plateau problem due to De Lellis, Ghiraldin & Maggi [11] proves to be superior in many respects.", "First and foremost, De Lellis, Ghiraldin & Maggi [11] treat the surface as the support of a Radon measure, adopting a spatial point of view, and thereby obtain the optimal soap film regularity defined by Almgren [2] and Taylor [33].", "Moreover, their definition of the spanning conditions, built on ideas of Harrison [21] (further developed by Harrison & Pugh [22], [23]), allows for an apt treatment of the free-boundary problem, whereas all prior approaches become rather difficult to use when the boundary of the spanning surface is not prescribed.", "Finally, their strategy has the physically relevant property of being insensitive to changes in the topology of the spanning surface, which can easily be observed during the relaxation to equilibrium of the system with an elastic bounding loop.", "Another interesting approach that allows for treatment of the free-boundary problem is provided in a recent paper by Fang [16], but its generalization to the case of a deformable bounding loop seems to require a more sophisticated apparatus than that employed in our treatment.", "We present the formulation of the Kirchhoff–Plateau problem in Section , where we introduce the energy functionals pertaining to the parts of our system and a suitable expression of the spanning condition.", "Of particular importance is also the introduction of various physical and topological constraints regarding the non-interpenetration of matter and the link and knot type of the bounding loop.", "Indeed, those natural requirements necessitate a variational approach to the problem, since they entail an inherent lack of smoothness, leaving open the question of the validity of the Euler–Lagrange equations at energy-minimizing configurations.", "Furthermore, they hinder the convexity properties of a seemingly innocuous functional.", "This becomes even more evident when the surface energy is added to the picture, so that the presence of multiple stable and unstable equilibria cannot in general be precluded.", "We prove our main existence result in Section , subsequent to establishing some preliminary facts.", "The essential feature of our treatment is a dimensional reduction which is performed by expressing the total energy of the system (bounding loop plus spanning surface) as a functional of the geometric descriptors of the bounding loop only.", "This is done by introducing a strongly nonlocal term entailing the minimization of the surface energy for a fixed shape of the bounding loop.", "Clearly, this step is justified by the existence of a solution to the problem of finding an area-minimizing surface spanning a three-dimensional bounding loop.", "We prove this (Theorem REF ) as a direct application of a result established by De Lellis, Ghiraldin & Maggi [11].", "Subsequently, it is necessary to adapt the arguments of De Lellis, Ghiraldin & Maggi [11] to our situation, in which the set that is spanned by the surface changes along minimizing sequences for the Kirchhoff–Plateau energy.", "This is accomplished in Lemma REF and Lemma REF and, based on these results, we establish the lower semicontinuity property that is needed to establish, in Theorem REF , the existence of a solution to the Kirchhoff–Plateau problem." ], [ "Formulation of the problem", "We seek to study the existence of a stable configuration of a liquid film that spans a flexible loop.", "Such a configuration is in reality metastable, since the liquid film will eventually break after becoming sufficiently thin, but we confine our attention to what happens before any such catastrophic event.", "The flexibility of the bounding loop represents a major difference between our problem and the Plateau problem, in which the liquid film spans a fixed boundary.", "It is therefore essential to model the elastic behavior of the loop in response to deformations, requiring a physical description much more sophisticated than that sufficient for a fixed boundary.", "In particular, we consider a loop formed by a slender filament with a nonvanishing cross-sectional thickness and subject to the physical constraint of non-interpenetration of matter.", "As discussed below, it is still reasonable to approximate the liquid film by a surface, since its thickness is typically much smaller than the cross-sectional thickness of the bounding loop, but an appropriate definition of how the surface spans the bounding loop is necessary.", "We next introduce the precise mathematical definitions needed to formulate the Kirchhoff–Plateau problem in a way that takes into account the physical requirements mentioned above.", "In so doing, we follow a variational approach by defining the energy of the different components of our system." ], [ "Preliminary considerations and assumptions", "The main assumption we impose on the bounding loop is that its length is much larger than the characteristic thickness of its cross-sections, which allows us to employ the theory of rods in our description.", "Within that theory, as presented for example by Antman [4], a rod is fully described by a curve in the three-dimensional Euclidean space, called the midline, a family of two-dimensional sets, describing the material cross-section at each point of the midline, and a family of material frames, encoding how the cross-sections are “appended” to the midline.", "Such a family of material frames corresponds also to a curve in the group of rotations of the three-dimensional space.", "We also assume that the rod is unshearable (namely, that the material cross-section at any point of the midline lies in the plane orthogonal to the midline at that point) and that its midline is inextensible.", "Together, these assumptions amount to choosing a Kirchhoff rod as a model for the filament from which the bounding loop is made.", "Under these assumptions, the shape of the loop is uniquely determined by assigning the shape of the cross-sections and three scalar fields: two flexural densities $\\kappa _1$ and $\\kappa _2$ together with a twist density $\\omega $ .", "From these fields we can reconstruct the midline $\\mathbf {x}$ and a director field $\\mathbf {d}$ , orthogonal to the tangent field $\\mathbf {t}:=\\mathbf {x}^{\\prime }$ , that gives the material frame as $\\lbrace (\\mathbf {t}(s),\\mathbf {d}(s),\\mathbf {t}(s)\\times \\mathbf {d}(s)):s\\in [0,L]\\rbrace $ , where $s$ is the arc-length parameter and $L$ is the total length of the midline.", "Indeed, once suitable conditions at $s=0$ are assumed, the fields $\\mathbf {x}$ and $\\mathbf {d}$ are the unique solution of the system of ordinary differential equations $\\left\\lbrace \\begin{aligned}&\\mathbf {x}^{\\prime }(s)=\\mathbf {t}(s)\\,,\\\\&\\mathbf {t}^{\\prime }(s)=\\kappa _1(s)\\mathbf {d}(s)+\\kappa _2(s)\\mathbf {t}(s)\\times \\mathbf {d}(s)\\,,\\\\&\\mathbf {d}^{\\prime }(s)=\\omega (s)\\mathbf {t}(s)\\times \\mathbf {d}(s)-\\kappa _1(s)\\mathbf {t}(s)\\,,\\end{aligned}\\right.$ for $s$ in $[0,L]$ , supplemented by the initial conditions $\\left\\lbrace \\begin{aligned}&\\mathbf {x}(0)=\\mathbf {x}_0\\,,\\\\&\\mathbf {t}(0)=\\mathbf {t}_0\\,,\\\\&\\mathbf {d}(0)=\\mathbf {d}_0\\,.\\end{aligned}\\right.$ It is not a priori granted that the solution to (REF )–(REF ) describes a closed loop.", "This property, being essential to the treatment of the Kirchhoff–Plateau problem, is imposed later as a constraint on the variational problem.", "For the variational problem that we plan to investigate, it is convenient to assume that each of the densities $\\kappa _1$ , $\\kappa _2$ , and $\\omega $ belongs to the Lebesgue space ${L}^{p}({[0,L];\\mathbb {R}})$ for some $p$ in $(1,\\infty )$ .", "This, by a classical result of Carathéodory (see, for instance, Hartman [24]), ensures that (REF )–(REF ) has a unique solution, with $\\mathbf {x}$ in ${W}^{2,p}({[0,L];\\mathbb {R}^3})$ and $\\mathbf {d}$ in ${W}^{1,p}({[0,L];\\mathbb {R}^3})$ , where ${W}^{n,p}({[0,L];\\mathbb {R}^d})$ denotes the Sobolev space of measurable functions from $[0,L]$ to $\\mathbb {R}^d$ with $n$ distributional derivatives in ${L}^{p}({[0,L];\\mathbb {R}^d})$ .", "We further assume $\|\\mathbf {t}_0\|=\|\\mathbf {d}_0\|=1$ .", "On this basis, we can use the structure of (REF )$_{2,3}$ to prove that $\|\\mathbf {t}(s)\|=\|\\mathbf {d}(s)\|=1$ for every $s$ in $[0,L]$ .", "The material cross-section at each $s$ is given by a compact simply connected domain $\\mathcal {A}(s)$ of $\\mathbb {R}^2$ such that the origin $\\mathbf {0}_2$ of $\\mathbb {R}^2$ belongs to $\\mathrm {int}(\\mathcal {A}(s))$ .", "A rod of finite cross-sectional thickness can then be described as the image in the three-dimensional Euclidean space of the set $\\Omega :=\\big \\lbrace (s,\\zeta _1,\\zeta _2):s\\in [0,L]\\text{ and }(\\zeta _1,\\zeta _2)\\in \\mathcal {A}(s)\\big \\rbrace $ through the map $\\mathbf {p}(s,\\zeta _1,\\zeta _2):=\\mathbf {x}(s)+\\zeta _1\\mathbf {d}(s)+\\zeta _2\\mathbf {t}(s)\\times \\mathbf {d}(s)\\,.$ By our assumptions, there exists an $R>0$ such that $\|\\zeta _1\|<R$ and $\|\\zeta _2\|<R$ for any $(s,\\zeta _1,\\zeta _2)$ in $\\Omega $ .", "We remark that, for the rod model to be a faithful representation of the mechanics of the filament from which the bounding loop is made, it is necessary that the maximum thickness $R$ be small compared to the length $L$ of the loop.", "Once the family of material cross-sections is assigned, any configuration of the rod in the (three-dimensional) ambient space corresponds to an element $\\mathsf {w}:=((\\kappa _1,\\kappa _2,\\omega ),\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ belonging to the Banach space $V:={L}^{p}({[0,L];\\mathbb {R}^3})\\times \\mathbb {R}^3\\times \\mathbb {R}^3\\times \\mathbb {R}^3\\,.$ Whereas all of the information regarding the shape of the rod is encoded in the flexural and twist densities, namely in the component $\\mathsf {w}_1=(\\kappa _1,\\kappa _2,\\omega )$ of $\\mathsf {w}$ belonging to ${L}^{p}({[0,L];\\mathbb {R}^3})$ , the clamping parameters $\\mathbf {x}_0$ , $\\mathbf {t}_0$ , and $\\mathbf {d}_0$ determine how the rod is translated and rotated in the ambient space.", "We denote the midline, the director field, and the rod configuration computed from (REF )–(REF ) and (REF ) for a given $\\mathsf {w}$ in $V$ as $\\mathbf {x}[\\mathsf {w}]$ , $\\mathbf {d}[\\mathsf {w}]$ , and $\\mathbf {p}[\\mathsf {w}]$ , respectively, and the bounding loop occupies the subset ${\\Lambda [\\mathsf {w}]}:=\\mathbf {p}[\\mathsf {w}](\\Omega )$ of $\\mathbb {R}^3$ ." ], [ "Individual contributions to the energy of the bounding loop", "We are now positioned to define the energy of the rod as a functional on $V$ .", "We consider three contributions entering this functional in an additive way: (i) the stored elastic energy, related with shape modifications; (ii) the non-interpenetration constraint; and (iii) the potential energy of an external load, such as the weight of the rod.", "The first term, being related only to shape deformations, can be expressed as the integral of an elastic energy density which depends only on $s$ and $\\mathsf {w}_1$ .", "We then introduce $f:\\mathbb {R}^3\\times [0,L]\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ and define the shape energy of the bounding loop as $E_\\mathrm {sh}(\\mathsf {w}):=\\int _0^Lf(\\mathsf {w}_1(s),s)\\,ds\\,,$ and we assume that $f(\\cdot ,s)$ is continuous and convex for any $s$ in $[0,L]$ , that $f(\\mathsf {a},\\cdot )$ is measurable for any $\\mathsf {a}$ in $\\mathbb {R}^3$ , and that $f(\\mathsf {a},s)$ is uniformly bounded below by a constant.", "These assumptions guarantee that $E_\\mathrm {sh}$ is weakly lower semicontinuous on the reflexive Banach space $V$ —a key property in applying the direct method of the calculus of variations.", "To ensure the necessary coercivity, we also impose the natural growth condition $f(\\mathsf {a},s)\\ge C_1\|\\mathsf {a}\|^p+C_2\\,,$ with $C_1>0$ and $C_2$ in $\\mathbb {R}$ .", "To include the local non-interpenetration constraint, we use the characteristic function of the closed subset $N$ of $V$ containing those elements $\\mathsf {w}$ such that $\\max _{(\\zeta _1,\\zeta _2)\\in \\mathcal {A}(s)}\\big (\\zeta _1\\kappa _2(s)-\\zeta _2\\kappa _1(s)\\big ) \\le 1$ for almost every $s$ in $[0,L]$ .", "We thus add to the energy of the loop the term $E_{\\mathrm {ni}}(\\mathsf {w}):={\\left\\lbrace \\begin{array}{ll}0 & \\text{if }\\mathsf {w}\\in N\\,,\\\\+\\infty & \\text{if }\\mathsf {w}\\in V\\setminus N\\,.\\end{array}\\right.", "}$ Condition (REF ) can be derived by relaxing the standard requirements that $\\mathbf {p}[\\mathsf {w}]$ be orientation preserving and locally injective (see Antman [4] and Schuricht [32]).", "We will prove, in Theorem REF , that our penalization strategy (which differs slightly from that of Schuricht [32]) is sufficient to guarantee the local injectivity of $\\mathbf {p}[\\mathsf {w}]$ on $\\mathrm {int}(\\Omega )$ for configurations with finite energy.", "The concept of local interpenetration of matter is illustrated in Figure REF .", "Figure: Local injectivity fails when adjacent material cross-sections interpenetrate.", "This is illustrated in the encircled region (magnified on the right) where two material cross-sections, traced on the surface, overlap, due to the excessive curvature of the midline.", "Global injectivity fails also when interpenetration occurs between cross-sections that lie far apart along the midline.", "This situation, in which local injectivity is not hindered, is depicted in the framed region.Finally, we account for the effects of the weight of the rod by considering the potential energy $E_\\mathrm {g}(\\mathsf {w}):=-\\int _\\Omega \\rho (s,\\zeta _1,\\zeta _2)\\mathbf {g}\\cdot \\mathbf {p}[\\mathsf {w}](s,\\zeta _1,\\zeta _2)\\, d(s,\\zeta _1,\\zeta _2)\\,,$ where $\\rho >0$ represents the mass density at each point of the rod and the vector $\\mathbf {g}$ represents the constant gravitational acceleration.", "From the above definitions, we obtain a weakly lower semicontinuous functional $E_\\mathrm {loop}(\\mathsf {w}):=E_\\mathrm {sh}(\\mathsf {w})+E_{\\mathrm {ni}}(\\mathsf {w})+E_\\mathrm {g}(\\mathsf {w})$ representing the total energy of the bounding loop." ], [ "Energy of the liquid film", "A fundamental tenet in physical chemistry is that he process of building an interface between two immiscible substances is accompanied by an energetic cost.", "Roughly speaking, each portion of each given substance prefers to be surrounded by the same substance and some interfacial energy is developed whenever this is not the case.", "For this reason, a droplet of water surrounded by air tends to assume a spherical shape: for a fixed droplet volume, it minimizes the area of the interface.", "To produce a liquid film, it is necessary to counteract this tendency, since doing so entails stretching the droplet, necessarily increasing the area of the liquid/air interface.", "We operate in two ways.", "On one hand, the solid bounding loop provides a third substance to which the liquid is attracted, since the energy density per unit area of the liquid/solid interface is lower than that associated with the liquid/air interface.", "On the other hand, it is possible to tamper a bit with the liquid to further reduce the energetic cost of the liquid/air interface.", "The second objective is accomplished by adding a small amount of surfactant to the liquid.", "Since the energy density per unit area of the liquid/air interface is lower for a higher surfactant concentration, surfactant molecules migrate towards the interface, leaving water in the bulk.", "In the liquid film configuration this produces two leaflets of surfactant phase (that lower the interfacial energy) covering a thin water layer (that provides a significant cohesion to the structure).", "In this context, we can define the surface tension $\\sigma $ of the liquid as the energy density per unit area of the liquid/air interface.", "It is physically reasonable to assume that $\\sigma $ is a homogeneous positive quantity, representing the ratio between the total interfacial energy and the surface area of the interface.", "To arrive at a mathematical model for the liquid film, we now combine a geometric approximation with the notion of interfacial energy discussed above.", "We assume that the thickness of the film (two surfactant leaflets plus the water layer) is negligible and we represent it as a two-dimensional object $S$ , but we keep track of the fact that it is built with two surfactant leaflets.", "We thus define the energy of the liquid film as $E_\\mathrm {film}(S):=2\\sigma \\mathcal {H}^2(S)\\,,$ where $\\mathcal {H}^d$ represents the $d$ -dimensional Hausdorff measure.", "Two remarks are in order.", "First, there is no evidence in (REF ) of the energy associated with the liquid/solid interface along which the film is in contact with the bounding loop.", "This is, in principle, a significant contribution, but the energy barrier that must be overcome to detach the film from the bounding loop is so high that its effect can be replaced by the spanning condition, discussed in the next section, designed to prevent detachment, encoding in essence the infinite height of the aforementioned barrier.", "The small corrections to that energy due to the size and shape of the liquid/solid interface are negligible as far as detachment is concerned.", "The influence of those corrections on the shape energy of the film is also negligible if the film is assumed to be of vanishing thickness.", "Indeed, they would influence the shape of the liquid/solid interface at length scales that are smaller than its typical thickness and, hence, not captured by our model.", "Figure: The thickness of the liquid film is orders of magnitude smaller than the cross-sectional thickness of the bounding loop.", "The white arrow points to the bright region where the thickness of the soap film slightly increases just before touching the bounding loop, realized using commercial fishing line with cross-sectional diameter of approximately 0.20.2 mm.", "Since most of the filament surface is not covered by the liquid film, the thickness of the latter must be considerably less than 0.20.2 mm.Second, it is important to justify the vanishing-thickness approximation, especially in view of the nonvanishing thickness that we attribute to the rod modeling the bounding loop.", "The typical thickness of a soap film is on the sub-micron scale, while a bounding loop made of a strand of human hair would have a cross-section with characteristic thickness of some tens of microns: this indicates that in many practical examples of liquid films bounded by flexible loops the characteristic thickness of the loop is at least two orders of magnitude greater than the thickness of the film (see Figure REF )." ], [ "The spanning condition", "We now provide a precise mathematical formulation of the conditions stipulating that the liquid film spans the bounding loop.", "In so doing, we borrow an elegant idea, that exploits notions of algebraic topology, introduced by Harrison [21] (and further developed by Harrison & Pugh [22], [23]).", "For our application it is convenient to present that idea in the form provided by De Lellis, Ghiraldin & Maggi [11], specialized to the particular setting in which the ambient space is three-dimensional.", "Definition 2.1 Let $H$ be a closed set in $\\mathbb {R}^3$ and consider the family $\\mathcal {C}_H:=\\big \\lbrace \\gamma :S^1\\rightarrow \\mathbb {R}^3\\setminus H:\\gamma \\text{ is a smooth embedding of }S^1\\text{ into }\\mathbb {R}^3\\big \\rbrace \\,.$ Then, any subset $\\mathcal {C}$ of $\\mathcal {C}_H$ is closed by homotopy (with respect to $H$ ) if $\\mathcal {C}$ contains all the elements $\\hat{\\gamma }$ of $\\mathcal {C}_H$ belonging to the same homotopy class $[\\gamma ]$ of any $\\gamma $ in $\\mathcal {C}$ .", "Such homotopy classes are elements of the first fundamental group $\\pi _1$ of $\\mathbb {R}^3\\setminus H$ .", "Given a subset $\\mathcal {C}$ of $\\mathcal {C}_H$ closed by homotopy, a relatively closed subset $K$ of $\\mathbb {R}^3\\setminus H$ is a $\\mathcal {C}$ -spanning set of $H$ if $K\\cap \\gamma \\ne \\emptyset $ for every $\\gamma $ in $\\mathcal {C}$ .", "Figure: An appropriate choice of homotopy classes determines which holes of a bounding loop with points of self-contact are covered.", "For the particular loop depicted here, which is subject to self contact without interpenetration at the central crossing point (black cross), if we seek a spanning set relative to the homotopy class of the loop aa or bb, spanning surfaces that cover only the hole on the left or on the right will be allowed, respectively.", "If, instead, we consider the homotopy class of the loop cc, both holes must be covered by the spanning set.The previous definition offers several advantages in comparison to more classical definitions.", "Most importantly, the set that is spanned by the surface representing the liquid film need not be a one-dimensional structure, it can be any closed set in the ambient space.", "This allows for the possibility that the surface spans the bounding loop ${\\Lambda [\\mathsf {w}]}$ of finite cross-sectional thickness.", "Another useful feature of this definition is that it allows for choice regarding the number of “holes to be covered” by the spanning set.", "Namely, when the set $H$ has a somewhat complex topology, it is permissible to restrict attention to surfaces that span only significant subregions of $H$ .", "To give a trivial example, illustrated in Figure REF , where $H$ is a loop subject to self contact, it is possible to seek surfaces that span one, the other, or both holes.", "Notice that, as shown below, the non-interpenetration constraint still allows for points on the surface of the bounding loop to come into contact, since it entails only a superposition of points on the surface of the rod.", "In particular, the subset $\\mathcal {D}_{\\Lambda [\\mathsf {w}]}$ of $\\mathcal {C}_{\\Lambda [\\mathsf {w}]}$ containing all $\\gamma $ that are not homotopic to a constant ($[\\gamma ]\\ne 1_{\\pi _1}$ ) is closed by homotopy.", "We will then seek a surface that is a $\\mathcal {D}_{\\Lambda [\\mathsf {w}]}$ -spanning set of the bounding loop ${\\Lambda [\\mathsf {w}]}$ .", "This is a maximal choice in the sense that we cannot include paths homotopic to a constant in the spanning condition.", "Indeed, since ${\\Lambda [\\mathsf {w}]}$ is a compact set, it is easy to see that any $\\lbrace 1_{\\pi _1}\\rbrace $ -spanning set of ${\\Lambda [\\mathsf {w}]}$ required to intersect all constant paths would fill up all of $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ .", "This would certainly not represent the behavior of a liquid film." ], [ "The Kirchhoff–Plateau problem", "The basic step in connecting our mathematical model to the experimental observations consists in finding a minimizer $\\mathsf {w}$ for the functional $E_\\mathrm {KP}(\\mathsf {w}):=E_\\mathrm {loop}(\\mathsf {w})+\\inf \\big \\lbrace E_\\mathrm {film}(S):S\\text{ is a \\mbox{$\\mathcal {D}_{\\Lambda [\\mathsf {w}]}$-spanning} set of }{\\Lambda [\\mathsf {w}]}\\big \\rbrace \\,,$ where $\\mathsf {w}$ belongs to a suitable subset $U$ of $V$ , that encodes the additional physical and topological constraints on the rod modeling the bounding loop.", "These constraints are: (i) the closure of the midline; (ii) the global gluing conditions; (iii) global non-interpenetration of matter; (iv) the knot type of the midline.", "When we combine the closure constraint with the definition of the Kirchhoff–Plateau functional $E_\\mathrm {KP}$ , a strong competition between the action of the spanning film and the elastic response of the bounding loop can arise.", "In a typical situation, the curvature of the loop tends to be minimized, producing somewhat wider configurations which, in turn, require spanning films with larger surface areas.", "The equilibrium shape is strongly influenced by the relative strength of surface tension with respect to the properties of the filament, but the proper inclusion of the global non-interpenetration constraint guarantees the physical relevance of the solutions the existence of which we establish.", "We will now present, following Schuricht [32], the precise formulation of the aforementioned constraints and a lemma, proving that the set $U$ is weakly closed in $V$ , as this is the essential property needed to establish the tractability of those constraints within our variational approach.", "The fact that the midline is a closed curve can be readily expressed by $\\mathbf {x}[\\mathsf {w}](L)=\\mathbf {x}[\\mathsf {w}](0)=\\mathbf {x}_0\\,,$ which we supplement with $\\mathbf {t}[\\mathsf {w}](L)=\\mathbf {t}[\\mathsf {w}](0)=\\mathbf {t}_0\\qquad \\text{and}\\qquad \\mathbf {d}[\\mathsf {w}](0)=\\mathbf {d}_0\\,.$ These clamping conditions (REF )–(REF ) are important in view of the preferred direction associated with the gravitational acceleration $\\mathbf {g}$ .", "Indeed, the weight term breaks the invariance of our problem under rigid rotations.", "Moreover, (REF )–(REF ) effectively describe the physical operation of holding the flexible structure at one point with tweezers.", "The right-hand sides of (REF )–(REF ), namely $\\mathbf {x}_0$ , $\\mathbf {t}_0$ , and $\\mathbf {d}_0$ , are referred to as the clamping parameters.", "Note that we do not require that $\\mathbf {d}[\\mathsf {w}](L)=\\mathbf {d}[\\mathsf {w}](0)$ , since the rod can be glued fixing an arbitrary angle between $\\mathbf {d}[\\mathsf {w}](L)$ and $\\mathbf {d}[\\mathsf {w}](0)$ (local gluing condition) while simultaneously respecting the clamping conditions.", "To prescribe how many times the ends of the rod are twisted before being glued together, we define the integer link type of the closed rod as the linking number of the closed midline $\\mathbf {x}[\\mathsf {w}]$ and the curve $\\mathbf {x}[\\mathsf {w}]+\\epsilon \\mathbf {d}[\\mathsf {w}]$ .", "Although a sufficiently small $\\epsilon $ can always be chosen, the curve may need to be closed as indicated by Schuricht [32].", "The global gluing conditions are then fixed by prescribing the angle between $\\mathbf {d}[\\mathsf {w}](L)$ and $\\mathbf {d}[\\mathsf {w}](0)$ and the link type of the closed rod.", "The non-interpenetration of matter can be enforced through the global injectivity condition $\\int _\\Omega \\det \\frac{\\partial \\mathbf {p}[\\mathsf {w}](s,\\zeta _1,\\zeta _2)}{\\partial (s,\\zeta _1,\\zeta _2)}\\,d(s,\\zeta _1,\\zeta _2)\\le \\mathcal {H}^3(\\mathbf {p}[\\mathsf {w}](\\Omega ))\\,,$ which, by (REF ), is equivalent to $\\int _\\Omega (1-\\zeta _1\\kappa _2(s)+\\zeta _2\\kappa _1(s))\\,d(s,\\zeta _1,\\zeta _2)\\le \\mathcal {H}^3(\\mathbf {p}[\\mathsf {w}](\\Omega ))\\,.$ Note that although condition (REF ) implies the global injectivity of $\\mathbf {p}[\\mathsf {w}]$ only on $\\mathrm {int}(\\Omega )$ , it is sufficient for our purposes, since we wish to allow for self-contact of the bounding loop, preventing only interpenetration.", "Clearly, global injectivity implies local injectivity, but the converse is not true.", "The local property is only a necessary condition for the global injectivity to hold.", "Indeed, as depicted in Figure REF , global injectivity may fail due to the overlapping of cross-sections belonging to regions of the rod that lie far apart along the midline and in which local injectivity holds true.", "To encode the knot type of the midline, we invoke the notion of isotopy class for closed curves.", "Definition 2.2 Let $\\mathbf {x}_i:[0,L]\\rightarrow \\mathbb {R}^3$ , $i=1,2$ , be two continuous curves with $\\mathbf {x}_i(L)=\\mathbf {x}_i(0)$ .", "The curves $\\mathbf {x}_1$ and $\\mathbf {x}_2$ are called isotopic (denoted $\\mathbf {x}_1\\simeq \\mathbf {x}_2$ ) if there are open neighborhoods $N_1$ of $\\mathbf {x}_1([0,L])$ and $N_2$ of $\\mathbf {x}_2([0,L])$ and a continuous mapping $\\Phi :N_1\\times [0,1]\\rightarrow \\mathbb {R}^3$ such that $\\Phi (N_1,\\tau )$ is homeomorphic to $N_1$ for all $\\tau $ in $[0,1]$ and $\\Phi (\\cdot ,0)=\\mathrm {Identity}\\,,\\quad \\Phi (N_1,1)=N_2\\,,\\quad \\text{and}\\quad \\Phi (\\mathbf {x}_1([0,L]),1)=\\mathbf {x}_2([0,L])\\,.$ By means of isotopy classes we can encode the knot type of the bounding loop as follows.", "We fix a continuous mapping $\\mathbf {\\ell }:[0,L]\\rightarrow \\mathbb {R}^3$ such that $\\mathbf {\\ell }(L)=\\mathbf {\\ell }(0)$ and we say that an element $\\mathsf {w}$ of $V$ , for which the closure and clamping conditions (REF )–(REF ) hold, has the knot type of $\\mathbf {\\ell }$ if $\\mathbf {x}[\\mathsf {w}]\\simeq \\mathbf {\\ell }\\,.$ We can now prove a theorem about the non-interpenetration constraint and our basic lemma about the weak closure of the set $U$ of competitors for our minimization problem.", "Theorem 2.3 Let $\\mathsf {w}=((\\kappa _1,\\kappa _2,\\omega ),\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ in $V$ be such that $E_\\mathrm {loop}(\\mathsf {w})<+\\infty $ .", "Then the configuration map $\\mathbf {p}[\\mathsf {w}]$ is locally injective on $\\mathrm {int}(\\Omega )$ .", "Furthermore, this mapping is open on $\\mathrm {int}(\\Omega )$ .", "If, in addition, $\\mathsf {w}$ satisfies (REF ), then $\\mathbf {p}[\\mathsf {w}]$ is also globally injective on $\\mathrm {int}(\\Omega )$ .", "Let $\\mathsf {w}$ in $V$ be fixed and consider a point $(\\bar{s},\\bar{\\zeta }_1,\\bar{\\zeta }_2)$ in $\\mathrm {int}(\\Omega )$ .", "Having defined ${A}_{\\varepsilon ,\\delta }:=\\big \\lbrace (s,\\zeta _1,\\zeta _2)\\in \\Omega : \|s-\\bar{s}\|<\\varepsilon ,\\;\|\\zeta _1-\\bar{\\zeta }_1\|<\\delta ,\\;\|\\zeta _2-\\bar{\\zeta }_2\|<\\delta \\big \\rbrace \\,,$ there exist constants $\\varepsilon >0$ and $\\delta >0$ such that $\\overline{A}_{\\varepsilon ,3\\delta }\\subset \\mathrm {int}(\\Omega )\\,.$ By our assumptions on the material cross-sections, there exists $R>0$ such that $\|\\zeta _1\|<R$ and $\|\\zeta _2\|<R$ for any $(s,\\zeta _1,\\zeta _2)$ in $\\Omega $ .", "By condition (REF ), $\\delta <R$ and it is possible to choose $\\varepsilon $ such that $\\int _{\\bar{s}-\\varepsilon }^{\\bar{s}+\\varepsilon }\|\\omega (s)\|\\,ds<\\frac{\\delta }{3R}\\qquad \\text{and}\\qquad \\mathbf {t}[\\mathsf {w}](s_1)\\cdot \\mathbf {t}[\\mathsf {w}](s_2)>\\frac{1}{2}$ for every $s_1$ and $s_2$ in $(\\bar{s}-\\varepsilon ,\\bar{s}+\\varepsilon )$ .", "Assuming $E_\\mathrm {loop}(\\mathsf {w})<+\\infty $ and recalling condition (REF ) together with (REF ), we infer that $1+\\zeta _2\\kappa _1(s)-\\zeta _1\\kappa _2(s)\\ge 2\\delta (\|\\kappa _1(s)\|+\|\\kappa _2(s)\|)$ for almost every $s$ in $(\\bar{s}-\\varepsilon ,\\bar{s}+\\varepsilon )$ and all corresponding $(s,\\zeta _1,\\zeta _2)$ in ${A}_{\\varepsilon ,\\delta }$ .", "We now show that $\\mathbf {p}[\\mathsf {w}]$ is injective on $A_{\\varepsilon ,\\delta }$ .", "Given two points $(a,\\zeta ^a_1,\\zeta ^a_2)$ and $(b,\\zeta ^b_1,\\zeta ^b_2)$ in ${A}_{\\varepsilon ,\\delta }$ , we assume that $\\mathbf {p}[\\mathsf {w}](a,\\zeta ^a_1,\\zeta ^a_2)=\\mathbf {p}[\\mathsf {w}](b,\\zeta ^b_1,\\zeta ^b_2)\\,.$ If $a=b$ , we use the definition (REF ) of the configuration mapping to find that $\\zeta ^a_1=\\zeta ^b_1$ and $\\zeta ^a_2=\\zeta ^b_2$ .", "We then argue by contradiction, by assuming $a<b$ , and we set $\\Delta s:=b-a\\qquad \\text{and}\\qquad \\tilde{\\mathbf {p}}(s):=\\mathbf {p}[\\mathsf {w}](s,\\zeta _1^b,\\zeta _2^b)\\,.$ Then, using (REF ) and (REF ), we obtain $\\tilde{\\mathbf {p}}(b)-\\tilde{\\mathbf {p}}(a)&\\mbox{}=\\Delta s\\int _0^1\\tilde{\\mathbf {p}}^{\\prime }(a+t\\Delta s)\\,dt\\\\&\\mbox{}=\\Delta s\\int _0^1\\big [(1-\\zeta _1^b\\kappa _2+\\zeta _2^b\\kappa _1)\\mathbf {t}[\\mathsf {w}]-\\omega \\zeta _2^b\\mathbf {d}[\\mathsf {w}]+\\omega \\zeta _1^b(\\mathbf {t}[\\mathsf {w}]\\times \\mathbf {d}[\\mathsf {w}])\\big ]\\,dt\\,,$ where $a+t\\Delta s$ is the argument of all functions in the second line.", "Since $\\big (\\mathbf {p}[\\mathsf {w}](b,\\zeta ^b_1,\\zeta ^b_2)-\\mathbf {p}[\\mathsf {w}](a,\\zeta ^a_1,\\zeta ^a_2)\\big )\\cdot \\mathbf {t}[\\mathsf {w}](a)=\\big (\\tilde{\\mathbf {p}}(b)-\\tilde{\\mathbf {p}}(a)\\big )\\cdot \\mathbf {t}[\\mathsf {w}](a)\\,,$ to obtain a contradiction with (REF ) it suffices to show that $\\big (\\tilde{\\mathbf {p}}(b)-\\tilde{\\mathbf {p}}(a)\\big )\\cdot \\mathbf {t}[\\mathsf {w}](a)>0\\,,$ thereby proving that $a=b$ and that $\\mathbf {p}[\\mathsf {w}]$ is locally injective.", "On introducing $\\alpha _1=\\int _a^b\|\\kappa _1(s)\|\\,ds\\,,\\qquad \\alpha _2=\\int _a^b\|\\kappa _2(s)\|\\,ds\\,,\\qquad \\text{and}\\qquad \\alpha _3=\\int _a^b\|\\omega (s)\|\\,ds\\,,$ it follows from (REF ) that $\\sup _{s\\in (a,b)}\|\\mathbf {d}[\\mathsf {w}](s)\\cdot \\mathbf {t}[\\mathsf {w}](a)\|\\le \\frac{\\alpha _2+\\alpha _3\\alpha _1}{1-\\alpha _3^2}$ and, analogously, that $\\sup _{s\\in (a,b)}\|\\mathbf {t}[\\mathsf {w}](s)\\times \\mathbf {d}[\\mathsf {w}](s)\\cdot \\mathbf {t}[\\mathsf {w}](a)\|\\le \\frac{\\alpha _1+\\alpha _3\\alpha _2}{1-\\alpha _3^2}\\,.$ On multiplying (REF ) by $\\mathbf {t}[\\mathsf {w}](a)$ and, recalling (REF ), (REF ), and (REF ), we easily obtain $\\big (\\tilde{\\mathbf {p}}(b)-\\tilde{\\mathbf {p}}(a)\\big )\\cdot \\mathbf {t}[\\mathsf {w}](a)\\ge \\Delta s \\int _0^1\\bigg (\\delta (\|\\kappa _1\|+\|\\kappa _2\|)-R\|\\omega \|\\frac{\\alpha _1+\\alpha _2}{1-\\alpha _3}\\bigg )\\,dt\\ge \\frac{\\delta }{2}(\\alpha _1+\\alpha _2)\\,.$ Now, if $\\alpha _1+\\alpha _2>0$ , (REF ) implies (REF ).", "Otherwise, $\\kappa _1(s)$ and $\\kappa _2(s)$ must both vanish for almost every $s$ in $(a,b)$ .", "This means that $\\mathbf {t}[\\mathsf {w}](s)=\\mathbf {t}[\\mathsf {w}](a)$ for every $s$ in $(a,b)$ ; hence, $\\big (\\tilde{\\mathbf {p}}(b)-\\tilde{\\mathbf {p}}(a)\\big )\\cdot \\mathbf {t}[\\mathsf {w}](a)=\\Delta s>0\\,,$ which is (REF ).", "We have therefore established the local injectivity of $\\mathbf {p}[\\mathsf {w}]$ and moreover that, being continuous, $\\mathbf {p}[\\mathsf {w}]$ is an open mapping on $\\mathrm {int}(\\Omega )$ .", "Given the local injectivity, the global injectivity follows from condition (REF ), as proved by Schuricht [32].", "Lemma 2.4 Let a continuous mapping $\\mathbf {\\ell }:[0,L]\\rightarrow \\mathbb {R}^3$ with $\\mathbf {\\ell }(L)=\\mathbf {\\ell }(0)$ , the global gluing conditions, and a real constant $M$ be given.", "Let also the clamping parameters $\\mathbf {x}_0$ , $\\mathbf {t}_0$ , and $\\mathbf {d}_0$ belonging to $\\mathbb {R}^3$ be given.", "Then the set $U:=\\big \\lbrace \\mathsf {w}=((\\kappa _1,\\kappa _2,\\omega ),\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)\\in V:E_\\mathrm {loop}(\\mathsf {w})<M\\text{ and (\\ref {eq:closure})--(\\ref {eq:knot}) hold}\\big \\rbrace $ is weakly closed in $V$ .", "If $U=\\emptyset $ the assertion is true.", "If $U\\ne \\emptyset $ , then the proof is a mere restatement of those of Lemma 3.9, Lemma 4.5, and Lemma 4.6 of Schuricht [32].", "Figure: A nonvanishing cross-sectional thickness is essential for distinguishing knot types in the presence of self-contact.", "As a trivial example, the trefoil knot (left) and the unknot (right) are clearly distinct when the cross-sections have a nonvanishing thickness, even in the presence of self-contact.", "On the contrary, in the vanishing-thickness limit (center), reaching self-contact from each of the two knotted configurations produces topologically equivalent structures and the distinction between a knot and an unknot is lost.We stress that the global injectivity condition (REF ) is crucial in ensuring the closure proved in Lemma REF .", "Indeed, just as the physical non-interpenetration of matter makes it impossible to change the type of a knot without tearing the loop that forms it, global injectivity entails that the knot type is preserved when passing to the limit in a sequence within the constrained set $U$ .", "In particular, the fact that constrained sets defined by different knot types are well separated in $V$ stems from the nonvanishing thickness of the cross-sections and condition (REF ) and would be inevitably lost in the vanishing-thickness limit, as illustrated in Figure REF ." ], [ "Existence of global minimizers", "In this section, we prove the existence of a solution to the Kirchhoff–Plateau problem.", "Specifically, we establish the existence of a global minimizer in $V$ for the total energy $E_\\mathrm {KP}$ , under appropriate physical and topological conditions.", "As necessary steps towards this goal, we first prove the existence of an energy-minimizing configuration for the bounding loop in the absence of the liquid film and then we demonstrate the existence of an area-minimizing spanning surface for a rigid bounding loop.", "This constitutes a somewhat more physical version of the Plateau problem, since the bounding loop is treated as a three-dimensional object.", "Those two results are quite straightforward, given the results of the previous section and the general theorem proved by De Lellis, Ghiraldin & Maggi [11].", "On the other hand, the proof of our main existence result requires establishing two lemmas in which the arguments of De Lellis, Ghiraldin & Maggi [11] are adapted to our situation, wherein the set that is spanned by the surface changes along minimizing sequences for the Kirchhoff–Plateau energy.", "Based on these results, we are able to verify the lower semicontinuity property that is needed to establish the existence of a solution of the Kirchhoff–Plateau problem.", "Moreover, Lemma REF provides an interesting extension of the main compactness result of De Lellis, Ghiraldin & Maggi [11], which could be used also in other contexts when studying the convergence of minimal surfaces induced by the convergence of the structure spanned by the surfaces.", "Theorem 3.1 (Minimization of the loop energy) Let a continuous mapping $\\mathbf {\\ell }:[0,L]\\rightarrow \\mathbb {R}^3$ satisfying $\\mathbf {\\ell }(L)=\\mathbf {\\ell }(0)$ , the global gluing conditions, and the clamping parameters $\\mathbf {x}_0$ , $\\mathbf {t}_0$ , and $\\mathbf {d}_0$ belonging to $\\mathbb {R}^3$ be given.", "If there exists $\\tilde{\\mathsf {w}}=(\\tilde{\\mathsf {w}}_1,\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ in $V$ such that $E_\\mathrm {loop}(\\tilde{\\mathsf {w}})<+\\infty $ and that complies with all of the constraints (REF )–(REF ), then there exists a minimizer $\\mathsf {w}=(\\mathsf {w}_1,\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ belonging to $V$ for the loop energy functional $E_\\mathrm {loop}$ and obeying the same constraints.", "Consider a minimizing sequence $\\lbrace \\mathsf {w}_k\\rbrace $ such that $E_\\mathrm {loop}(\\mathsf {w}_k)<M$ for some $M$ in $\\mathbb {R}$ , with $\\mathsf {w}_k$ belonging to the subset $U$ of the reflexive Banach space $V$ introduced in Lemma REF .", "The existence of a competitor $\\tilde{\\mathsf {w}}$ guarantees that $U$ is not empty.", "By the coercivity of $E_\\mathrm {loop}$ , $U$ is bounded in $V$ .", "Hence, it is possible to extract a weakly converging subsequence $\\mathsf {w}_{k_i}\\rightharpoonup \\mathsf {w}$ .", "Since, by Lemma REF , $U$ is weakly closed, $\\mathsf {w}$ must belong to $U$ and, by the weak lower semicontinuity of $E_\\mathrm {loop}$ , we conclude that $\\liminf _i E_\\mathrm {loop}(\\mathsf {w}_{k_i})\\ge E_\\mathrm {loop}(\\mathsf {w})$ .", "Since $\\lbrace \\mathsf {w}_{k_i}\\rbrace $ is a minimizing sequence, this proves that the weak limit $\\mathsf {w}$ is indeed a minimizer.", "From now on, for any $\\mathsf {w}$ in $V$ , we denote by ${\\Lambda [\\mathsf {w}]}$ the image of $\\Omega $ under the configuration map $\\mathbf {p}[\\mathsf {w}]$ and by $\\mathcal {S}[{\\mathsf {w}}]$ the set of all $\\mathcal {D}_{{\\Lambda [\\mathsf {w}]}}$ -spanning sets of $\\Lambda [{\\mathsf {w}}]$ , with $\\mathcal {D}_{\\Lambda [\\mathsf {w}]}$ defined as in Section REF .", "Moreover, for any $\\gamma \\colon S^1 \\rightarrow \\mathbb {R}^3$ and for any $r>0$ we denote by $U_r(\\gamma )$ the tubular neighborhood of radius $r$ around $\\gamma $ .", "Theorem 3.2 (Minimization of the surface energy) Fix $\\mathsf {w}$ in $V$ .", "If $\\alpha :=\\inf \\big \\lbrace E_\\mathrm {film}(S):S\\in \\mathcal {S}[{\\mathsf {w}}]\\big \\rbrace <+\\infty \\,,$ then there exists a relatively closed subset $M[\\mathsf {w}]$ of $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ that is a $\\mathcal {D}_{\\Lambda [\\mathsf {w}]}$ -spanning set of ${\\Lambda [\\mathsf {w}]}$ with $E_\\mathrm {film}(M[\\mathsf {w}])=\\alpha $ .", "Furthermore, $M[\\mathsf {w}]$ is an $(\\mathbf {M},0,\\infty )$ -minimal set in $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ in the sense of Almgren [2].", "In particular, $M[\\mathsf {w}]$ is countably $\\mathcal {H}^2$ -rectifiable.", "The proof is a direct application of the general theorem by De Lellis, Ghiraldin & Maggi [11].", "The countably $\\mathcal {H}^2$ -rectifiability of $M[\\mathsf {w}]$ follows from the regularity of $(\\mathbf {M},0,\\infty )$ -minimal sets established by Almgren [2].", "Definition 3.3 Let $A$ and $B$ be two closed nonempty subsets of a metric space $(M,d_M)$ .", "The Hausdorff distance between $A$ and $B$ is defined as $d_\\mathcal {H}(A,B):=\\max \\big \\lbrace \\sup _{a\\in A}\\inf _{b\\in B}d_M(a,b),\\sup _{b\\in B}\\inf _{a\\in A}d_M(a,b)\\big \\rbrace \\,.$ The topology induced by $d_\\mathcal {H}$ on the space of closed nonempty subsets of $M$ is the Hausdorff topology.", "Lemma 3.4 Consider a sequence of closed nonempty subsets $\\Lambda _k$ of $\\mathbb {R}^3$ converging in the Hausdorff topology to the closed set $\\Lambda \\ne \\emptyset $ .", "Assume that, for every $k$ in $\\mathbb {N}$ , we have countably $\\mathcal {H}^2$ -rectifiable sets $S_k$ belonging to $\\mathcal {P}(\\Lambda _k)$ , where $\\mathcal {P}(\\Lambda _k)$ is a good class in the sense of De Lellis, Ghiraldin & Maggi [11], and such that $\\mathcal {H}^2(S_k)=\\inf \\big \\lbrace \\mathcal {H}^2(S):S\\in \\mathcal {P}(\\Lambda _k)\\big \\rbrace <+\\infty \\,.$ Then the measures $\\mu _k:=\\mathcal {H}^2\\mathop {\\hbox{\\vrule height 7pt width .5pt depth 0pt\\vrule height .5pt width 6pt depth 0pt}}\\nolimits S_k$ constitute a bounded sequence, $\\mu _k\\stackrel{}{\\rightharpoonup }\\mu $ up to the extraction of a subsequence, and the limit measure satisfies $\\mu \\ge \\mathcal {H}^2\\mathop {\\hbox{\\vrule height 7pt width .5pt depth 0pt\\vrule height .5pt width 6pt depth 0pt}}\\nolimits S_\\infty \\,,$ where $S_{\\infty }:={\\rm spt}(\\mu )\\setminus \\Lambda $ is a countably $\\mathcal {H}^2$ -rectifiable set.", "The proof of this lemma requires minor modifications of the proof of Theorem 2 of De Lellis, Ghiraldin & Maggi [11].", "It is sufficient to observe that the convergence of $\\lbrace \\Lambda _k\\rbrace $ ensures that, whenever $\\mathbf {x}\\in S_\\infty $ , we have $d(\\mathbf {x},\\Lambda _k)>0$ for large enough $k$ .", "Then, all arguments in the proof of Theorem 2 of De Lellis, Ghiraldin & Maggi [11] are recovered asymptotically.", "Lemma 3.5 Let $\\lbrace {\\mathsf {w}}_k\\rbrace $ be a sequence weakly converging to ${\\mathsf {w}}$ in the subset $U$ of $V$ introduced in Lemma REF , let $S_k$ be an element of $\\mathcal {S}[{\\mathsf {w}}_k]$ , and fix $\\gamma $ in $\\mathcal {D}_{{\\Lambda [\\mathsf {w}]}}$ .", "Then, for any $\\varepsilon >0$ such that $U_{2\\varepsilon }(\\gamma )$ is contained in $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ , there exists $M=M(\\varepsilon )>0$ such that, for any $k$ large enough, $\\mathcal {H}^2(S_k \\cap U_\\varepsilon (\\gamma ))\\ge M\\,.$ Let us fix $\\varepsilon >0$ such that $U_{2\\varepsilon }(\\gamma )$ is contained in $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ , denote by $B_\\varepsilon (\\mathbf {0}_2)$ the open disk of $\\mathbb {R}^2$ with radius $\\varepsilon $ and centered at the origin of $\\mathbb {R}^2$ , and consider a diffeomorphism $\\Phi \\colon S^1 \\times B_\\varepsilon (\\mathbf {0}_2) \\rightarrow U_{\\varepsilon }(\\gamma )$ such that $\\Phi _{\|_{S^1\\times \\lbrace \\mathbf {0}_2\\rbrace }}=\\gamma $ .", "Let $\\mathbf {y}$ belong to $B_\\varepsilon (\\mathbf {0}_2)$ and set $\\gamma _{\\mathbf {y}}:=\\Phi _{\|_{S^1\\times \\lbrace \\mathbf {y}\\rbrace }}$ .", "Then $\\gamma _{\\mathbf {y}}$ in $[\\gamma ]$ represents an element of $\\pi _1(\\mathbb {R}^3 \\setminus {\\Lambda [\\mathsf {w}]})$ .", "Let $\\mathbf {x}_k$ and $\\mathbf {x}$ denote the midlines corresponding respectively to ${\\mathsf {w}}_k$ and ${\\mathsf {w}}$ .", "Since $\\lbrace {\\mathsf {w}}_k\\rbrace $ converges weakly to ${\\mathsf {w}}$ in $U$ , $\\lbrace \\mathbf {x}_k\\rbrace $ converges to $\\mathbf {x}$ strongly in $W^{1,p}([0,L];\\mathbb {R}^3)$ .", "In particular, $\\lbrace \\mathbf {x}_k\\rbrace $ converges to $\\mathbf {x}$ uniformly on $[0,L]$ , which implies that, for $k$ sufficiently large, $\\Lambda [{\\mathsf {w}}_k]$ is contained in a neighborhood $W$ of $\\Lambda [{\\mathsf {w}}]$ with $W\\cap U_\\varepsilon (\\gamma )=\\emptyset $ .", "Hence, for such $k$ and $\\varepsilon $ it follows that, for any $\\mathbf {y}$ in $B_\\varepsilon (\\mathbf {0}_2)$ , $\\gamma _{\\mathbf {y}}$ belongs to $\\mathbb {R}^3\\setminus \\Lambda [{\\mathsf {w}}_k]$ , which yields $S_k\\cap \\gamma _{\\mathbf {y}}\\ne \\emptyset $ because $S_k$ is in $\\mathcal {S}[{\\mathsf {w}}_k]$ .", "Take $\\mathbf {\\pi }: S^1 \\times B_\\varepsilon (\\mathbf {0}_2) \\rightarrow B_\\varepsilon (\\mathbf {0}_2)$ as the projection on the second factor and let $\\hat{\\mathbf {\\pi }}:=\\mathbf {\\pi }\\circ \\Phi ^{-1}$ .", "Then, $\\hat{\\mathbf {\\pi }}$ is Lipschitz-continuous and $B_\\varepsilon (\\mathbf {0}_2)$ is contained in $\\hat{\\mathbf {\\pi }}(S_k \\cap U_\\varepsilon (\\gamma ))$ , which entails that $\\pi \\varepsilon ^2=\\mathcal {H}^2(B_\\varepsilon (\\mathbf {0}_2))\\le \\mathcal {H}^2(\\hat{\\mathbf {\\pi }}(S_k \\cap U_\\varepsilon (\\gamma ))\\le ({\\rm Lip}\\,\\hat{\\mathbf {\\pi }})^2\\mathcal {H}^2(S_k \\cap U_\\varepsilon (\\gamma ))\\,.$ We thus conclude that $\\mathcal {H}^2(S_k \\cap U_\\varepsilon (\\gamma ))\\ge \\frac{\\pi \\varepsilon ^2}{({\\rm Lip}\\,\\hat{\\mathbf {\\pi }})^2}\\,,$ which establishes the inequality (REF ).", "Theorem 3.6 (Main existence result) Let a continuous mapping $\\mathbf {\\ell }:[0,L]\\rightarrow \\mathbb {R}^3$ satisfying $\\mathbf {\\ell }(L)=\\mathbf {\\ell }(0)$ , the global gluing conditions, and the clamping parameters $\\mathbf {x}_0$ , $\\mathbf {t}_0$ , and $\\mathbf {d}_0$ belonging to $\\mathbb {R}^3$ be given.", "If there exists $\\tilde{\\mathsf {w}}=(\\tilde{\\mathsf {w}}_1,\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ in $V$ such that $E_\\mathrm {KP}(\\tilde{\\mathsf {w}})<+\\infty $ and which complies with the constraints (REF )–(REF ), then there exists a solution $\\mathsf {w}=(\\mathsf {w}_1,\\mathbf {x}_0,\\mathbf {t}_0,\\mathbf {d}_0)$ to the Kirchhoff–Plateau problem belonging to $V$ , namely a minimizer of the total energy $E_\\mathrm {KP}$ satisfying (REF )–(REF ).", "Furthermore, the spanning surface $M[\\mathsf {w}]$ associated with the energy minimizing configuration by Theorem REF is an $(\\mathbf {M},0,\\infty )$ -minimal set in $\\mathbb {R}^3\\setminus {\\Lambda [\\mathsf {w}]}$ in the sense of Almgren [2].", "In particular, $M[\\mathsf {w}]$ is countably $\\mathcal {H}^2$ -rectifiable.", "Consider a minimizing sequence $\\lbrace {\\mathsf {w}}_k\\rbrace $ for $E_{\\rm KP}$ such that $E_{\\rm KP}({\\mathsf {w}}_k) < M$ for some $M$ in $\\mathbb {R}$ .", "In particular, we then have that $E_{\\rm loop}({\\mathsf {w}}_k) < M$ and we can choose ${\\mathsf {w}}_k$ in the subset $U$ of $V$ introduced in Lemma REF .", "As in the proof of Theorem REF , we can extract a weakly convergent subsequence ${\\mathsf {w}}_{k_i} \\rightharpoonup \\overline{\\mathsf {w}}$ with $\\overline{\\mathsf {w}} \\in U$ .", "To complete the proof, it remains to establish that $E_{\\rm KP}$ is weakly lower semicontinuous on $V$ .", "Given the weak lower semicontinuity of $E_{\\rm loop}$ , this is tantamount to proving that the functional ${\\mathsf {w}} \\mapsto \\inf \\big \\lbrace \\mathcal {H}^2(S) : \\textrm {S \\in \\mathcal {S}[{\\mathsf {w}}]}\\big \\rbrace $ is weakly lower semicontinuous.", "Fix a weakly convergent sequence ${\\mathsf {w}}_k \\rightharpoonup {\\mathsf {w}}$ in $U$ .", "Let $S_k$ belonging to $\\mathcal {S}[{\\mathsf {w}}_k]$ be given by Theorem REF such that $\\mathcal {H}^2(S_k) =\\inf \\big \\lbrace \\mathcal {H}^2(S) : \\textrm {$ S[wk]$}\\big \\rbrace \\,.$$Without loss of generality, we can assume that $ H2(Sk)C$ for some $ C>0$.", "For any $ k$ in $ N$, let $ k:=H2 Sk$.", "Then, up to the extraction of a subsequence, we have $ k $ on $ R3$ and we can set $ S0:=spt()[w]$.On applying Lemma \\ref {lem:diagonal} with $ k=[wk]$, we deduce that\\begin{equation}\\mu \\ge \\mathcal {H}^2\\mathop {\\hbox{\\vrule height 7pt width .5pt depth 0pt\\vrule height .5pt width 6pt depth 0pt}}\\nolimits S_0 \\textrm { on subsets of }\\mathbb {R}^3 \\setminus \\Lambda [{\\mathsf {w}}]\\,.\\end{equation}We next show that $ S0$ belongs to $ S[w]$.", "Assume by contradiction that there exists $$ in $ D[w]$ with $ S0= $ and pick $$ as given by Lemma \\ref {lem:intersection}.We then find that $ (U())=0$ and, therefore, that$$\\lim _k\\mathcal {H}^2(S_k \\cap U_{\\varepsilon }(\\gamma ))=0\\,,$$which contradicts the thesis (\\ref {key}) of Lemma \\ref {lem:intersection}.Hence, we obtain the chain of inequalities{\\begin{@align}{1}{-1}\\liminf _k\\inf \\lbrace \\mathcal {H}^2(S) : \\textrm {S \\in \\mathcal {S}[{\\mathsf {w}}_k]}\\rbrace &\\mbox{}\\ge \\liminf _k\\mathcal {H}^2(S_k)\\\\&\\mbox{}=\\liminf _k\\mathcal {\\mu }_k(\\mathbb {R}^3)\\ge \\mu (\\mathbb {R}^3)\\ge \\mathcal {H}^2(S_0)\\ge \\inf \\lbrace \\mathcal {H}^2(S) : \\textrm {S \\in \\mathcal {S}[{\\mathsf {w}}]}\\rbrace \\,,\\end{@align}}which establishes the lower semicontinuity of the functional (\\ref {funct}).$" ], [ "Concluding remarks", "We introduced a mathematical model for experiments in which a thin filament in the form of a closed loop is spanned by a liquid film.", "In a variational setting, the model is defined by the sum of the energies of the different components of this system.", "While the loop is modeled as a nonlinearly elastic rod which is inextensible and unshearable, namely a Kirchhoff rod, the elasticity of the liquid film is described by a homogeneous surface tension.", "Following Schuricht [32], we required that the loop satisfy the physical constraints of non-interpenetration of matter, fixed global gluing conditions, and fixed knot type.", "A crucial point in this treatment is that the cross-sectional thickness of the loop is nonzero, implying that it occupies a nonvanishing volume.", "This led us to consider a somewhat more physical version of the Plateau problem where the liquid film is represented by a surface with a boundary that is not prescribed, but is free to move along the lateral surface of the three-dimensional bounding loop.", "Our choice to retain a nonvanishing cross-sectional thickness of the loop, while attributing a vanishing thickness to the liquid film, is justified by the typically large separation between those two length-scales.", "By exploiting the framework recently proposed by De Lellis, Ghiraldin & Maggi [11] for the Plateau problem, we established the existence of a global minimizer, namely a stable solution, for the coupled Kirchhoff–Plateau problem, in which the boundary of the liquid film lies on the lateral surface of the deformable bounding loop.", "This was achieved by means of a dimensional reduction, performed in expressing the total energy of the system as a functional of the geometric descriptors of the rod only.", "To this end, a strongly nonlocal term, entailing the minimization of the surface energy for a fixed shape of the loop, was introduced and the effectiveness of this strategy guaranteed by the auxiliary proof of existence of a surface realizing such a minimization.", "A key step towards the final result is the adaptation, presented in Lemma REF , of the main compactness argument of De Lellis, Ghiraldin & Maggi [11] needed to deal with the deformability of the bounding loop.", "Combining the approaches of Schuricht [32], for the bounding loop, and of De Lellis, Ghiraldin & Maggi [11], concerning the liquid film, we established the existence of a solution that complies with important physical constraints on the topology of the bounding loop and of the surface.", "Indeed, the latter enjoys the soap film regularity identified by Almgren [2] and Taylor [33].", "An important outcome of our analysis is that the existence of a physically relevant solution is obtained irrespective of the relative strength of surface tension compared to the elastic response of the filament from which the bounding loop is made, which instead influences the multiplicity and the qualitative properties of solutions, as discussed below.", "Based on the framework introduced above, further investigations can be addressed in two main directions.", "First, it would be interesting to study the existence of unstable solutions for the Kirchhoff–Plateau problem.", "We expect that classical techniques used to establish similar results in the context of the Plateau problem will not be easily applicable here, while suitably adapted tools from nonsmooth critical point theory may prove to be required and effective.", "It would be also important to investigate the unstable solutions generated by bifurcation phenomena, which are expected to be a feature of our model.", "Indeed, the competition between the action of the spanning film and the elastic response of the filament can trigger the transition between different stability regimes, meanwhile affecting the multiplicity of solutions to the Kirchhoff–Plateau problem.", "Secondly, studying a dissipative dynamics of the system considered above represents a challenging task both from the analytical point of view, again due to the lack of smoothness inherent to our setting, and from the mechanical point of view, since a physically consistent representation of the dissipative phenomena at play might not be straightforward.", "Nevertheless, a dynamical strategy of the proposed kind would likely provide an excellent basis for the implementation of numerical schemes aimed at finding local minima of the Kirchhoff–Plateau functional.", "Acknowledgments.", "E. F. and G. G. G. gratefully acknowledge support from the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.", "L. L. is grateful for the kind hospitality of the Okinawa Institute of Science and Technology Graduate University during the inception of the present work.", "The authors thank Jenny Harrison and the anonymous reviewers for their insightful comments.", "The authors would also like to thank Stoffel Janssens and David Vazquez Cortes for their help with the preparation of Figure 2." ] ]	1606.05078
[ [ "Dynamical Contents of Unconventional Supersymmetry" ], [ "Abstract The Dirac Hamiltonian formalism is applied to a system in $(2+1)$-dimensions consisting of a Dirac field $\\psi$ minimally coupled to Chern-Simons $U(1)$ and $SO(2,1)$ connections, $A$ and $\\omega$, respectively.", "This theory is connected to a supersymmetric Chern-Simons form in which the gravitino has been projected out (unconventional supersymmetry) and, in the case of a flat background, corresponds to the low energy limit of graphene.", "The separation between first-class and second-class constraints is performed explicitly, and both the field equations and gauge symmetries of the Lagrangian formalism are fully recovered.", "The degrees of freedom of the theory in generic sectors shows that the propagating states correspond to fermionic modes in the background determined by the geometry of the graphene sheet and the nondynamical electromagnetic field.", "This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector $\\psi=0$, which is of interest for perturbation theory.", "For the latter the analysis is adapted to the case of manifolds with boundary, and the corresponding Dirac brackets together with the centrally extended charge algebra are found.", "Finally, the $SU(2)$ generalization of the gauge group is briefly treated, yielding analogous conclusions for the degrees of freedom." ], [ "Introduction", "Supersymmetry (SUSY) is a natural –and perhaps unique– way to unify internal and spacetime symmetries in the description of fundamental particles and interactions.", "In spite of its elegant appeal, it is puzzling that no evidence of supersymmetry has been seen in the current phenomenology.", "In the seminal work of Wess and Zumino, an action principle based on the idea of a supergauge symmetry was examined in a Lagrangian consisting of spin-0 and spin-1/2 fields, the Wess-Zumino (WZ) model.", "The conclusion there was that in order for this symmetry to close, its parameter had to be a Killing spinor of the background spacetime [1].", "This indicates that the existence of supersymmetry requires spacetime itself to possess some symmetry that allows for the existence of some sort of Killing spinors.", "Indeed, the superalgebra behind the WZ model is an extension of the Poincaré algebra, whose geometric interpretation calls for a maximally symmetric spacetime, namely Minkowski space.", "In the WZ model and in most supersymmetric particle models, the fields form an irreducible vector representation of the super-Poincaré algebra, a supermultiplet.", "This implies that bosons and fermions come in pairs with equal mass and other quantum numbers, but with spins differing by $\\hbar /2$ (superpartners).", "Since no such duplication of the particle spectrum has been observed, it is argued that SUSY must be broken at the energy scales that we have been able to explore, but it is supposedly restored at a sufficiently high energies.", "An alternative to this picture, where the fields do not form a vector multiplet but rather enter as parts of a connection can also be considered, and in that case degenerate superpartners are not necessarily present [2], [3].", "This is a generic feature, for example, of supersymmetric Chern-Simons (CS) theories, where bosonic and fermionic fields combine to form a connection for the supersymmetric graded Lie algebra [4], [5], [6].", "It is well known that CS theories in three dimensions for any Lie algebra have no local degrees of freedom [7].", "This is true also for CS theories based on graded Lie algebras [8], like in the case of the CS supergravity for the $\\mathfrak {osp}(2\|2)$ algebra.", "By contrast, a massive spin-1/2 field in a fixed three-dimensional background of has $2n$ propagating degrees of freedom, where $n=1$ for Majorana and $n=2$ for Dirac spinors [9], [10].", "Now, if in the $\\mathfrak {osp}(2\|2)$ CS theory the gravitino field $\\chi ^\\alpha _\\mu $ is split into a spin-$1/2$ Dirac spinor $\\psi ^\\alpha $ and the vielbein $e^a_\\mu $ , the fermionic sector of the reduced theory describes a Dirac fermion in a curved background, minimally coupled to $\\mathfrak {u}(1)$ and $\\mathfrak {so}(2,1)$ gauge connection one-forms $A=A_\\mu dx^\\mu $ and $\\omega ^a{}_b=\\omega ^a{}_{b\\mu } dx^\\mu $ , respectively [2].", "It is therefore only natural to inquire whether this reduced theory has zero local degrees of freedom (DOF) as the original CS system, or has four local degrees of freedom of a spin-$1/2$ Dirac fermion.", "The question is further complicated by the fact that in the reduced Lagrangian the dreibein are not Lagrange multipliers (their time derivatives $\\dot{e}^a$ appear explicitly in the action) and therefore $e^a_\\mu $ are in principle dynamical fields as well.", "The identification of the physical degrees of freedom can be addressed by direct application of Dirac's analysis of constrained Hamiltonian systems [11], which systematically separates the dynamical fields from the gauge degrees of freedom.", "In the case of CS theories, however, the separation between first and second-class constraints is a delicate issue, and the system considered here is not an exemption.", "The reduced action in [2] reads $I[\\psi ,e,A,\\omega ]= \\int \\frac{1}{2}\\Bigl [ \\overline{\\psi }{e} (\\overleftarrow{D} - \\overrightarrow{D}){e} \\psi + AdA + \\frac{1}{2} \\omega ^a{}_b d \\omega ^b{}_a + \\frac{1}{3} \\omega ^a{}_b \\omega ^b_c \\omega ^c_a \\Bigr ]\\, ,$ where ${e} \\equiv \\Gamma _a e^a = \\Gamma _a e^a_\\mu dx^\\mu $ and $e^a_\\mu $ are the dreibein (see Appendix for notation).", "In addition to the local $U(1)\\times SO(2,1)$ symmetry and spacetime diffeomorphisms, this action is invariant under local Weyl rescalings, $e^a_\\mu \\rightarrow \\lambda e^a_\\mu \\,, \\quad \\psi \\rightarrow \\lambda ^{-1} \\psi \\,, \\quad \\bar{\\psi } \\rightarrow \\lambda ^{-1}\\bar{\\psi } \\,,$ where $\\lambda (x)$ is a non-vanishing, real and differentiable function.", "All of these symmetries are in principle associated with first-class constraints that reduce the physical phase space.", "Varying the action with respect to $\\psi $ yields the Dirac equation with a mass term $m=\\frac{1}{2\|e\|} \\epsilon ^{\\mu \\nu \\rho } \\eta _{ab}e^{a}_{\\mu } D_\\nu e^b_\\rho $ (including hermiticity corrections), while varying with respect to $e^a_{\\mu }$ implies the vanishing of the energy-momentum tensor, $\\mathcal {T}^{\\mu \\nu }=\\frac{1}{2\|e\|}\\eta ^{ab} E_a^{\\mu }\\frac{\\delta L}{\\delta e_{\\nu }^{b}}+(\\mu \\leftrightarrow \\nu )$ , with $E_a^\\mu $ the inverse dreibein.", "In particular, the vanishing of the trace $\\mathcal {T}^{\\mu }{}_\\mu =0$ is consistent with the local scale invariance of the action.", "For a fixed background the local degrees of freedom should correspond to the $2n$ independent components of the Dirac field in flat spacetime.", "A quick analysis suggests that six out of the nine components of the dreibein can be eliminated by the conditions $\\mathcal {T}^{\\mu \\nu }=0$ , while the remaining three can be gauged away via two spatial diffeomorphisms and a Weyl scaling.", "In CS theories, time diffeomorphisms are not independent, which means their phase space generators are linear combinations of the remaining first-class constraints [7].", "As noted in [2], the closure the supersymmetry for (REF ) requires the parameter of the SUSY transformation to satisfy a subsidiary condition to ensure the variation $\\delta \\psi $ to have spin-1/2, like $\\psi $ itself.", "This subsidiary condition is satisfied if the SUSY parameter is required to be a Killing spinor of the background and, like in the original WZ system, this means that supersymmetry is a global (rigid) symmetry [3].", "Since this is not a gauge symmetry, it is not generated by a first-class constraint that would further reduce the number of physical degrees of freedom." ], [ "Hamiltonian analysis", "Splitting the fields and their derivatives into time ($t$ ) and spatial components ($i,j=1,2$ ), the Lagrangian (REF ) can be written, up to a boundary term, as $L = \\epsilon ^{ij}\\Bigl [-\\eta _{ab} \\dot{e}_i ^a e_j^b \\bar{\\psi }\\psi -\\dot{\\bar{\\psi }}\\Gamma _{ij}\\psi +\\bar{\\psi }\\Gamma _{ij}\\dot{\\psi } +\\frac{1}{2}\\eta _{ab}\\dot{\\omega }_i^a \\omega _j ^b + \\frac{1}{2}\\dot{A}_i A_j \\Bigr ] -e_t ^a K_a + \\omega _t ^a J_a +A_t K\\,,$ where we defined $\\Gamma _{ij}:=e_{i}^{a}e_{j}^{b}\\Gamma _{ab}$ , $\\omega ^a := \\frac{1}{2} \\epsilon ^{abc}\\omega _{bc}$ , and $K_a & := & 2\\epsilon ^{ij}\\Bigl [ \\eta _{ab}T_{ij}^b \\bar{\\psi }\\psi -e_i^b (\\bar{\\psi }\\Gamma _a \\Gamma _b \\overrightarrow{D}_j \\psi + \\bar{\\psi } \\overleftarrow{D}_j \\Gamma _b \\Gamma _a \\psi )\\Bigr ]\\,, \\\\J_{a} & := & \\epsilon ^{ij}\\eta _{ab}(\\frac{1}{2}R_{ij}^{b}-\\epsilon _{\\:cd}^{b}e_{i}^{c}e_{j}^{d}\\bar{\\psi }\\psi )\\,, \\\\K & := & \\epsilon ^{ij}(\\partial _{i}A_{j}-i\\bar{\\psi }\\Gamma _{ij}\\psi )\\,, $ where $R^a= d\\omega ^ a+ \\epsilon ^a_{\\,\\,\\, bc}\\omega ^b \\omega ^c$ is the Lorentz curvature and $T^{a}=de^{a}+\\epsilon ^a{}_{bc}\\omega ^b e^c$ is the torsion tensor of the background.", "Lagrangian (REF ) describes the evolution of $(21+4n)$ coordinate fields: $e^a_\\mu $ (nine), $\\omega ^a_\\mu $ (nine), $A_\\mu $ (three), $\\psi $ (2n) and $\\bar{\\psi }$ (2n); among them there are seven ($e^a_t$ , $\\omega ^a_t$ and $A_t$ ), whose time derivatives do not appear in the Lagrangian and are therefore Lagrange multipliers with vanishing canonical momenta.", "For the remaining components the Lagrangian contains only first time derivatives and therefore each momentum is a function of the coordinate fields.", "Thus, the following $(14+4n)$ primary constraints are obtained $\\varphi _{a}^{i} & = & p_{a}^{i}+2\\epsilon ^{ij}\\eta _{ab}e_{j}^{b}\\bar{\\psi }\\psi \\approx 0\\,,\\nonumber \\\\\\Omega & = & \\chi +\\epsilon ^{ij}\\Gamma _{ij}\\psi \\approx 0\\,,\\nonumber \\\\\\bar{\\Omega } & = & \\bar{\\chi }-\\epsilon ^{ij}\\bar{\\psi }\\Gamma _{ij}\\approx 0\\,,\\\\\\phi _{a}^{i} & = & \\pi _{a}^{i}-\\frac{1}{2}\\epsilon ^{ij}\\eta _{ab}\\omega _{j}^{b}\\approx 0\\,, \\nonumber \\\\\\phi ^{i} & = & \\pi ^{i}-\\frac{1}{2}\\epsilon ^{ij}A_{j}\\approx 0\\, .\\nonumber $ The seven combinations $K_a$ , $J_a$ , $K$ in (REF ) are then secondary constraints associated to the Lagrange multipliers.", "Moreover, the canonical Hamiltonian weakly vanishes and the total Hamiltonian can be taken as an arbitrary linear combination of all the constraintsHereafter we perform the integrations over the spatial slices $\\Sigma $ given by $t=constant$ , for which we do not consider a boundary.", "The cases where $ \\partial \\Sigma \\ne \\emptyset $ , which yield asymptotic charges, are discussed in Section REF ., $ H_T = \\int d^2 x\\left[ e^a_t K_a -\\omega ^a_t J_a - A_t K + \\varphi ^i_a \\lambda ^a_i +\\phi ^i_a \\Lambda ^a_i +\\bar{\\Lambda }\\Omega + \\bar{\\Omega }\\Lambda +\\lambda _i \\phi ^i \\right] \\,.$ It can be proved that the following seven linear combinations are first-class constraints (see Appendix for details) $\\tilde{J}_a & := & J_a +\\epsilon ^{}_{ac} \\varphi ^j_b e^c_j +\\frac{1}{2}(\\bar{\\Omega }\\Gamma _a \\psi - \\bar{\\psi }\\Gamma _a \\Omega ) + D_j \\phi ^j_a \\,, \\nonumber \\\\\\tilde{K} & := & K-\\frac{i}{2}(\\bar{\\Omega }\\psi -\\bar{\\psi }\\Omega )+\\partial _j \\phi ^j \\,,\\nonumber \\\\\\Upsilon & := & -e^b_j \\varphi ^j_b +\\bar{\\Omega }\\psi +\\bar{\\psi }\\Omega \\,, \\\\\\mathcal {H}_i & := & e^a_i K_a -e^a_i D_j \\varphi ^j_a + T^a_{ij} \\varphi ^j_a + \\bar{\\psi } \\overleftarrow{D}_i \\Omega +\\bar{\\Omega }\\overrightarrow{D}_{i}\\psi -\\omega _{i}^{a}\\tilde{J}_{a}-A_{i}\\tilde{K}+\\phi ^{j}F_{ij} +\\phi _{a}^{j}R^a_{ij} \\, .", "\\nonumber $ Here the (spatial) covariant derivative $D_i$ acts on each field according to its transformation properties, as in (REF ).", "Using (REF )-(REF ), the generators $\\mathcal {H}_i$ can be expressed as $\\mathcal {H}_i & = & \\left(\\partial _i A_j -\\partial _j A_i \\right)\\pi ^j -A_i \\partial _j \\pi ^j + \\left(\\partial _i \\omega ^a_j -\\partial _j\\omega ^a_i \\right)\\pi ^j_a -\\omega ^a_i \\partial _j \\pi ^j_a \\\\& & +\\left(\\partial _i e^a_j -\\partial _j e^a_i \\right)p^j_a -e^a_i \\partial _j p^j_a + \\partial _i \\overline{\\psi } \\chi + \\overline{\\chi }\\partial _i \\psi \\,,$ which can be readily seen to generate spatial diffeomorphisms on phase space functions $F$ as $\\lbrace F, \\int \\xi ^i \\mathcal {H}_i \\rbrace = \\mathcal {L}_{\\xi }F$ .", "This in turn means that $\\left\\lbrace \\mathcal {H}_{i}(x), \\mathcal {H}_{j}(y)\\right\\rbrace = \\mathcal {H}_{i}(y)\\partial _{j}^{(y)}\\delta ^{(2)}(x-y)-\\mathcal {H}_{j}(x)\\partial _{i}^{(x)}\\delta ^{(2)}(x-y)\\,,$ as expected from generators of spatial diffeomorphisms [12].", "On the other hand, it can be directly checked that $\\tilde{J}_a$ , $\\tilde{K}$ and $\\Upsilon $ generate $SO(2,1) \\times U(1) \\times Weyl$ transformations over all the fields and momenta.", "Indeed, they satisfy the (weakly vanishing) Poisson relations (REF ) with all the constraints, and one finds $ \\lbrace \\tilde{J}_{a},\\tilde{J}_{b}\\rbrace & = & \\epsilon _{ab}^{\\quad c}\\tilde{J}_{c}\\,,\\nonumber \\\\\\lbrace \\tilde{K},\\tilde{K}\\rbrace & = &\\lbrace \\Upsilon ,\\Upsilon \\rbrace =\\lbrace \\tilde{K},\\Upsilon \\rbrace = 0\\,, \\\\\\lbrace \\tilde{J}_{b},\\tilde{K} \\rbrace &=& \\lbrace \\tilde{J}_{b},\\Upsilon \\rbrace = 0\\,.", "\\nonumber $ Together with the generators of spatial diffeomorphisms these then form a first-class Poisson algebra.", "Note that performing a shift in the Lagrange multipliers of the form $\\lambda ^a_i \\rightarrow \\lambda ^{\\prime a}_i &=& -ve^a_i + \\lambda ^a_i \\,,\\nonumber \\\\\\Lambda ^{\\prime \\alpha }\\rightarrow \\Lambda ^{\\prime \\alpha }&=& v\\psi ^\\alpha + \\Lambda ^\\alpha \\,, \\\\\\overline{\\Lambda }_\\alpha \\rightarrow \\overline{\\Lambda }^{\\prime }_\\alpha &=& v\\overline{\\psi }_\\alpha + \\overline{\\Lambda }_\\alpha \\, , \\nonumber $ produces a shift in the total Hamiltonian (REF ), $H_T \\rightarrow H^{\\prime }_T = H_T + \\int v \\Upsilon d^2 x\\, .", "$ This accounts for the Weyl invariance (REF ) of the system.", "However, the absence of spatial derivatives in $\\Upsilon $ implies that such symmetry is generated by a purely local constraint with no associated asymptotic charges, as explained in detail below (recent examples of this fact can be found in [13] and references therein, see [14] for a thorough discussion).", "Weyl symmetry is thus a local redefinition of the fields without any observable effects.", "The corresponding symmetry breaking, however, leads to physical consequences as we will discuss." ], [ "Generic conformally invariant sector", "We now assume that in a genericFollowing [7] we understand by generic sectors those with a maximum number of degrees of freedom or, equivalently, a minimum number of independent first-class constraints.", "background the $(14+4n)$ time preservation equations of the primary constraints fix an equal number of Lagrange multipliers (see Appendix for details).", "The other seven parameters remain free in the total Hamiltonian (REF ), to form a linear combination of the first-class constraints.", "Choosing {$\\mathcal {H}_i$ , $\\tilde{J}_a$ , $\\tilde{K}$ , $\\Upsilon $ } as the basis of these generators, the total Hamiltonian can be written as $H_T = \\int d^{2}x \\left[\\xi ^{i}\\mathcal {H}_{i} + v \\Upsilon - \\omega _t^{a}\\tilde{J}_{a}-A_t \\tilde{K} \\right] \\,,$ Here the Lagrange multipliers $\\xi ^i$ , $v$ , $\\omega _t^a$ and $A_t$ are real and arbitrary functions on equal footing.", "As the Hamiltonian is a combination of first-class constraints, the time preservation relations are fulfilled by construction, and no additional (tertiary) constraints are produced in the Dirac algorithm.", "Note further that for any phase space function $F$ the Poisson bracket $\\lbrace F,H_T\\rbrace $ coincides with the corresponding Dirac bracket.", "Now, the expression (REF ) was obtained from (REF ) by choosing $e^a_t = \\xi ^i e^a_i \\, .$ This means that the three components $e^a_t$ are functions of the two free parameters $\\xi ^i $ , while it also implies a degenerate dreibein, $\|e\|=0$ .", "Although this may seem puzzling for a metric interpretation, it is dynamically consistent and allows to do the correct counting of the local degrees of freedom (see e.g., [7] and Appendix ).", "The choice (REF ) is equivalent to the gauge $N^{\\perp }=0$ in gravitation, which is perfectly acceptable as well as generic choices in ordinary gauge systems, i.e.", "Yang-Mills [16], [15].", "Furthermore, it also allows to write the generator of temporal diffeomorphisms as a linear combination of generators of local spatial diffeomorphisms, rescalings, Lorentz and $U(1)$ transformationsIt can be explicitly shown that $\\lbrace \\cdots ,\\int N\\mathcal {H} \\rbrace \\approx \\mathcal {L}_{N\\frac{\\partial }{\\partial t}}(\\cdots ) $ , which is a general property of coordinate invariant systems [9]., $\\mathcal {H}=\\xi ^i \\mathcal {H}_i +v\\Upsilon -\\omega ^a_t \\tilde{J}_a -A_t \\tilde{K} \\, .$ Note that the degenerate condition $\|e\|=0$ remains invariant under local Weyl symmetry.", "Next, we consider a choice in which the Weyl symmetry is broken and the $e^a_t$ remains arbitrary so that the vielbein need not to be degenerate." ], [ "Pure spin-1/2 generic sector", "We now examine a specific sector of the theory in which (REF ) is not imposed but the Weyl invariance is fixed instead.", "We consider a generic sector for the fields $e$ and $\\psi $ that restricts the fermionic excitations to have spin-1/2 only.", "A fermionic field $\\chi ^\\alpha _a$ transforms as a vector in the index $a$ and as a spinor in the index $\\alpha $ and therefore belongs to a representation $1 \\otimes 1/2=3/2 \\oplus 1/2$ of the Lorentz group.", "There is a unique decomposition of this field into irreducible representations $\\chi _a=\\chi ^{(3/2)}_a + \\chi ^{(1/2)}_a$ , where $[ \\delta ^b_a - \\frac{1}{3}\\Gamma _a \\Gamma ^b ] \\chi ^{(1/2)}_b =0\\, , \\\\\\Gamma ^a \\chi ^{(3/2)}_a =0\\, .", "$ In the case of the field $\\psi $ , the condition that it only carries spin-1/2 requires that $D_{\\mu }\\psi $ also belongs to the spin-$1/2$ representation and should therefore be in the kernel of the spin-$3/2$ projector, namely,Formally, if the scale has not been fixed the sector should be defined as the equivalence class of configurations satisfying (REF ) up to Weyl transformations.", "A manifestly covariant condition can be attained by introducing a gauge field for scale invariance $D_\\mu \\rightarrow D_\\mu +W_{\\mu }$ , as originally proposed by Weyl [14].", "$ [\\delta ^\\mu _\\nu -\\frac{1}{3}\\Gamma _\\nu \\Gamma ^\\mu ] D_\\mu \\psi =0 \\, ,$ where $\\Gamma _\\mu \\equiv e^a_\\mu \\Gamma _a$ .", "This implies that the system does not generate local spin-$3/2$ excitations –no gravitini– through parallel transport of the fermion.", "It may be regarded as a consistency condition for the system (REF ) if it is meant to describe a Dirac spinor.", "The general solution of (REF ) is $D_\\mu \\psi =\\Gamma _\\mu \\xi \\, , $ where $\\xi $ is an arbitrary Dirac spinor.", "Next, in order to study the dynamical content of the sector, we perform a partial gauge fixing.", "As shown in [2], the field equations for the action (REF ) require the torsion to be covariantly constant, $DT^a=0$ , where $D$ is the Lorentz covariant exterior derivative (see Appendix ).", "The general solution of this equation, with an appropriate local rescaling of the dreiben –using the freedom due to Weyl symmetry– is of the form $T^a = \\alpha \\epsilon ^a{}_{bc}e^b e^c \\,, $ where $\\alpha $ is an arbitrary (dimensionful) constant.", "Now, inserting (REF ), (REF ) in (REF ) we obtain $K_a = 2\\epsilon ^{ij}e^b_i e^c_j \\Bigl [2\\alpha \\epsilon _{abc} \\bar{\\psi }\\psi - (\\bar{\\xi } \\Gamma _a \\Gamma _b \\Gamma _c \\psi - \\bar{\\psi }\\Gamma _c \\Gamma _b \\Gamma _a \\xi ) \\Bigr ]\\, .", "$ In order for the constraint condition $K_a \\approx 0$ not to introduce additional restrictions on the fields, the right-hand-side of (REF ) must identically vanish.", "This demands $\\xi =\\alpha \\psi $ and therefore, this selects the sector The projector (REF ) is a generalization of the so-called `twistor operator', which defines conformal Killing spinors (REF ) in the absence of torsion [1], [17].", "Equation (REF ) can be regarded as the Killing spinor equation for a curved background [18].", "Remarkably, (REF ) and (REF ) are completely equivalent by virtue of the Dirac equation.", "$D_{\\mu }\\psi = \\alpha \\Gamma _{\\mu }\\psi \\,.", "$ Multiplying both sides by $\\Gamma ^\\mu $ , this reduces to the Dirac equation where the mass $m=3\\alpha $ is an integration constant related to the torsion of the background, in complete agreement with [2].", "Both (REF ) and (REF ) break local scale invariance, leaving only a global symmetry under $e^a\\rightarrow \\lambda e^a$ , $\\psi \\rightarrow \\lambda ^{-1}\\psi $ , $m \\rightarrow \\lambda ^{-1}m$ for constant $\\lambda $ .", "In analogy with SUSY, this rigid symmetry does not interfere with the counting of local DOF.", "As pointed out in [2], [18], the introduction of a dimensionful mass constant $m$ enables us to finally determine the scale for the theory.", "In Appendix , we show that the sector equations (REF ) and (REF ) can be used to consistently solve and preserve the remaining constraints.", "In fact, in this case one is enabled to explicitly determine the time evolution of $e$ and $\\psi $ , which is equivalent to the fact that Lagrange multipliers in the total Hamiltonian are also found in closed form (without using the degenerate gauge (REF )).", "We now show how the first-class generators arise to recover the residual symmetries of (REF ,REF ).", "In principle we will only assume the spatial components of these equations to hold, while the temporal parts will be recovered from Hamilton equations.", "Thus, note first that in this sector the combinations $\\tilde{K}_a & = & K_a -D_i \\varphi ^i_a + 2\\alpha \\epsilon ^b{}_{ac} e^c_i \\varphi ^i_b +\\alpha (\\bar{\\Omega }\\Gamma _a \\psi - \\bar{\\psi }\\Gamma _a \\Omega ) \\nonumber \\\\& & +2ie^b_i \\bar{\\psi } \\Gamma _{ab}\\psi \\phi ^{i} + 2\\epsilon ^b{}_{ac} e^c_i \\bar{\\psi } \\psi \\phi ^i_b \\, , $ are first-class constraints, as can be directly checked computing the Poisson brackets: $\\lbrace \\tilde{K}_{a},\\Omega \\rbrace \\approx \\lbrace \\tilde{K}_{a},\\bar{\\Omega }\\rbrace \\approx \\lbrace \\tilde{K}_{a},\\varphi _{b}^{i}\\rbrace & \\approx & 0 \\,, \\\\\\lbrace \\tilde{K}_{a},\\phi _{b}^{i}\\rbrace \\approx \\lbrace \\tilde{K}_{a},\\phi ^{i}\\rbrace & \\approx & 0 \\,, \\\\\\lbrace \\tilde{K}_{a}, \\tilde{K}_{b} \\rbrace &\\approx & 0 \\,.$ These three constraints are the generators of spacetime diffeomorphisms supplemented by gauge transformations and projected on the tangent space.", "This is seen from the identity $\\lbrace \\cdots ,e^a_i \\tilde{K}_a \\rbrace \\approx \\lbrace \\cdots , \\mathcal {H}_i + A_i \\tilde{K} + \\omega ^a_i \\tilde{J}_a \\rbrace \\,.$ We now set the Lagrange multipliers associated to the primary constraints in order to accommodate the seven first-class generators.", "The total Hamiltonian reads $H_T & = & \\int d^{2}x \\left[e^a_t K_a -\\omega ^a_t J_a - A_t K + \\varphi ^i_a \\lambda ^a_i + \\phi ^i_a \\Lambda ^a_i + \\bar{\\Lambda }\\Omega +\\bar{\\Omega } \\Lambda + \\lambda _i \\phi ^i\\right] \\nonumber \\\\& = & \\int d^{2}x \\left[-\\omega ^a_t \\tilde{J}_a - A_t \\tilde{K} + e_t^a \\tilde{K}_a \\right] =: \\int d^2 x\\:\\mathcal {H} \\, .", "$ Note that here we are implicitly fixing the Weyl freedom, i.e.", "we have assumed $v=0$ in the shift (REF ).", "This is required to preserve the sector.", "Indeed, the time evolution for the fields $(e^a_i,\\psi ,A_i,\\omega ^a_i)$ , by virtue of the Hamilton equations, leads to $D_t \\psi =\\dot{\\psi } -\\frac{i}{2}A_t \\psi + \\frac{1}{2}\\epsilon ^{abc}\\omega _{bt} \\Gamma _c \\psi & = & \\alpha \\Gamma _t \\psi \\,, \\\\T^a_{it}=\\partial _i e^a_t -\\dot{e}^a_i + \\omega ^a_{bi}e^b_t -\\omega ^a_{bt}e^a_i & = & 2\\alpha \\epsilon ^a_{\\:bc} e_i^b e_t^c\\,, \\\\F_{it} = \\partial _i A_t -\\dot{A}_i & = & 2ie_{i}^{a}e_t^{b}\\bar{\\psi }\\Gamma _{ab}\\psi \\,, \\\\R^a_{it} = \\partial _i \\omega ^a_t -\\dot{\\omega }_i + \\epsilon ^{abc}\\omega _{bi} \\omega _{ct} & = & 2\\epsilon _{\\:bc}^a e_i^b e_t ^c \\bar{\\psi }\\psi \\, , $ These are readily seen to recover the temporal parts of equations (REF ,REF ) and the constrains (,), thus agreeing with the Euler-Lagrange equations.", "As stated, an interesting feature of this gauge is that $e^a_t$ is not restricted at all, which is equivalent to the statement that the three constraints $\\tilde{K}_{a}$ are first-class.", "For regular configurations with $\|e\|\\ne 0$ , it is clear that $(\\mathcal {H},\\mathcal {H}_i)$ are then three independent constraints generating temporal and spatial diffeomorphisms, respectively.", "Nevertheless, even for a degenerate vielbein it is possible to define $\\mathcal {H}_{\\perp }:=\\epsilon _{\\,bc}^{a}e_{1}^{b}e_{2}^{c}\\tilde{K}_{a}\\,,$ which corresponds (up to normalization) to the generator of diffeomorphisms normal to the surfaces $t= constant$ , modulo gauge transformations.", "Defining the Lagrange multipliers $e^a_t$ , $A_t$ and $\\omega ^a_t$ as $e^a_t & = & N^{\\perp }\\epsilon ^a{}_{bc} e^b_1 e^c_2 + e^a_i N^i \\,,\\\\A_t & = & \\lambda - A_{i}N^i \\,,\\\\\\omega ^a_t & = & \\lambda ^a - \\omega ^a{}_i N^i\\,,$ the generator of time evolution takes the more familiar form [12] $\\mathcal {H}= N^{\\perp }\\mathcal {H}_{\\perp } + N^{i}\\mathcal {H}_i - \\lambda \\tilde{K}-\\lambda ^{a}\\tilde{J}_{a}.$ We thus find the expected $SO(2,1)\\times U(1)\\times \\text{Diff} $ residual symmetries and their corresponding generators.", "We anticipate that even though in this gauge choice there exist a different set of first-class contraints associated to diffeomorphisms, the number of DOF is the same and this is therefore a generic sector.", "This will be discussed in Section ." ], [ "Bosonic Vacuum", "The purely bosonic vacuum $\\bar{\\psi }=0=\\psi $ corresponds to a very particular configuration.", "In principle, it should not be regarded as a subsector of the previous case because it acquires additional degeneracies in the Dirac matrixThe Dirac matrix is defined as $\\Omega _{AB}:=\\lbrace \\phi _A,\\phi _B\\rbrace $ , where the indexes $A,B$ range over all the constraints [11].", "which lead to new first-class constraints.", "This is a direct consequence of the whole energy-momentum tensor of the Lagrangian formalism vanishes identically and there are no field equations to determine $e^a_\\mu $ , so the dreibein is a non-propagating gauge field in this case.", "Nevertheless, some of the first-class constraints found in the previous section turn out to be not functionally independent and therefore compensate the situation.", "As we will show, the whole picture results into an equal number of DOF, thus we can think of the vacuum as a generic sector.", "First note if the fermions vanish, (REF ) and (REF )-() imply $\\lbrace \\varphi _{a}^{i},\\varphi _{b}^{j}\\rbrace & = & \\lbrace \\varphi _{a}^{i},\\Omega \\rbrace =\\lbrace \\varphi _{a}^{i},\\bar{\\Omega }\\rbrace =\\lbrace \\varphi _{a}^{i},\\phi ^{j}\\rbrace =\\lbrace \\varphi _{a}^{i},\\phi _{b}^{j}\\rbrace =0 \\,,\\\\\\lbrace \\varphi _{a}^{i},K\\rbrace & = & \\lbrace \\varphi _{a}^{i},J_{a}\\rbrace =\\lbrace \\varphi _{a}^{i},K_{a}\\rbrace =0 \\, ,$ (where we have set $\\bar{\\psi }=0=\\psi $ after computing the Poisson brackets).", "Thus, we find six additional first-class constraints $\\varphi ^i_a \\approx 0$ , which generate arbitrary changes in the spatial components of the dreibein, $\\delta e_{i}^{a}=\\lbrace e_{i}^{a},\\int d^{2}x\\:\\lambda _{j}^{b}\\varphi _{b}^{j}\\rbrace =\\lambda _{i}^{a}\\,.$ As the time component $e^a_t$ is already a Lagrange multiplier, this in turn means that the dreibein is completely arbitrary (in particular it can be chosen to be invertible).", "In this sector, the first-class constraints (REF ) read $\\tilde{J}_{a} & = & J_{a}+\\epsilon _{\\:ac}^{b}\\varphi _{b}^{j}e_{j}^{c}+D_{j}\\phi _{a}^{j} \\,, \\\\\\tilde{K} & = & K+\\partial _{j}\\phi ^{j} \\,, \\\\\\Upsilon & = & -e_{j}^{b}\\varphi _{b}^{j} \\,, \\\\\\mathcal {H}_{i} & = & e_{i}^{a}K_{a}-e_{i}^{a}D_{j}\\varphi _{a}^{j}+T_{ij}^{a}\\varphi _{a}^{j} -\\omega _{i}^{a}\\tilde{J}_{a}-A_{i}\\tilde{K} \\,.", "$ Note that the Weyl invariance has not been fixed so the torsion components $T^a_{ij}$ remain undetermined.", "In this sector one can also identify $K_a \\approx 0$ as a first-class constraint (which is identically fulfilled).", "However, since (REF ) is quadratic in the fermionic variables, it can be shown that it does not act on the phase space, $\\lbrace K_{a},F\\rbrace =0\\,,$ for any function of the physical fields.", "As $K_a := \\frac{\\partial \\mathcal {L}}{\\partial e^a_t}$ can be regarded as the ($t-a$ ) components of the energy-momentum tensor, (REF ) is a consequence of the fact that the linearized version of $\\mathcal {T}^\\mu _{\\,\\,\\nu }=0$ is fulfilled identically.", "Considering this functional degeneracy of $K_a$ , we see that diffeomorphisms () are composed only of gauge transformations plus certain particular displacements of the vielbein.", "Moreover, it is clear that the $SO(2,1)\\times U(1)\\times \\text{Diff}\\times Weyl$ transformations are generated by a linear combination of the first-class constraints $\\tilde{J}_a ,\\tilde{K}$ and $\\varphi _{a}^{i}$ only.", "The remaining constraints, corresponding to $\\Omega $ , $\\bar{\\Omega }$ , $\\phi ^j_b$ and $\\phi ^j$ , are second-class as can be checked from their Poisson brackets (REF )." ], [ "Dirac brackets and charge algebra", "As the rank of the Dirac matrix is constant for a neighborhood of the vacuum in the phase space (the generic property is defined for open regions [7]), the above classification of constraints can in fact be applied for a small perturbation with $\\psi \\ne 0$ .", "We now illustrate this by computing the Dirac brackets.", "According to (REF ) and the definition of Dirac brackets [11], one finds $\\lbrace A_i,A_j \\rbrace _D & = & \\epsilon _{ij} ,\\quad \\lbrace \\omega ^a_i,\\omega ^b_j\\rbrace _D = \\epsilon _{ij}\\eta ^{ab} , \\quad \\lbrace \\bar{\\psi }_\\beta ,\\psi ^\\alpha \\rbrace _D = \\hat{\\Gamma }^\\alpha _{\\,\\, \\, \\beta } ,\\nonumber \\\\\\lbrace e^a_i,p^j_b \\rbrace _D & = & \\delta ^j_i\\delta ^a_b , \\quad \\lbrace p^j_b,\\psi \\rbrace _D = 2\\epsilon ^{ij}e^a_i \\hat{\\Gamma }\\Gamma _{ab} \\psi , \\quad \\lbrace p^{j}_{b},\\bar{\\psi }\\rbrace _D = 2\\epsilon ^{ij}e^a_i \\bar{\\psi }\\hat{\\Gamma }\\Gamma _{ab} \\, ,$ where $\\epsilon _{ij}\\epsilon ^{ik}=\\delta ^k_j$ and the matrix $\\hat{\\Gamma }$ is defined such that $2\\epsilon ^{ij}(\\Gamma _{ij})^\\alpha _{\\,\\, \\, \\beta }(\\hat{\\Gamma })^\\beta _{\\,\\, \\, \\gamma }=\\delta ^\\alpha _\\gamma \\, ,$ (explicitly, $\\hat{\\Gamma }=-\\frac{1}{4\|e\|g^{tt}}E^t_a\\Gamma ^a$ ).", "Note that the phase space reduces to the fields $(A_i, \\omega ^a_i, \\psi , \\bar{\\psi }, e^a_i, p^j_b)$ after the second-class constraints are strongly implemented.", "With this simplification the first-class generators (REF ) read $\\tilde{J}_{a} & = & J_{a}+\\epsilon _{\\:ac}^{b}\\varphi _{b}^{j}e_{j}^{c} \\,, \\\\\\tilde{K} & = & K \\,, \\\\\\Upsilon & = & -e_{j}^{b}\\varphi _{b}^{j} = -e_{j}^{b}p_{b}^{j} \\,, \\\\G_i:= \\mathcal {H}_{i} + \\omega _{i}^{a}\\tilde{J}_{a} + A_{i}\\tilde{K} & = & e_{i}^{a}K_{a}-e_{i}^{a}D_{j}\\varphi _{a}^{j}+T_{ij}^{a}\\varphi _{a}^{j} \\,.", "$ where $G_i$ is the generator of improved diffeomorphisms [7] (under the Weyl fixing of Section REF it simply reads $G_i=e^a_i \\tilde{K}_a$ ).", "As we will show, under appropriate boundary conditions $G_i$ does not contribute to asymptotic charges, i.e.", "it corresponds to proper gauge transformations.", "Following the Regge-Teitelboim approach [19], the smeared gauge generator must be supplemented by a boundary term $Q$ depending on the asymptotic gauge parameters, so it reads $S[\\xi ^i, \\lambda , \\lambda ^a,v]=\\int d^2x (\\xi ^i G_i + \\lambda \\tilde{K} + \\lambda ^a \\tilde{J}_a + v\\Upsilon ) + Q_G[\\xi ^i]+Q_{\\tilde{K}}[\\lambda ]+Q_{\\tilde{J}}[\\lambda ^a] \\, .$ These boundary terms correspond to the asymptotic charges associated to (global) gauge symmetriesThese conserved charges are determined by the boundary terms that must be added to the action (REF ) in order to have well defined functional derivatives with respect to the fields.", "In [20], for instance, the normalization factor is chosen to be $\\frac{k}{4\\pi }$ , where $k$ is the CS level.. As stated above, the Weyl scaling does not have an associated charge.", "In Appendix we give the form of the variation of the charges and integrate them.", "It is shown that setting the fermionic fields $(\\psi , \\bar{\\psi })$ to vanish asymptoticallyThis condition is preserved under all the gauge symmetries considered here (see Appendix for details).", "However, regarding supersymmetry, one needs to check the stability under such transformation by solving the Killing spinor equation for a certain background, as shown in [2] for the BTZ case.", "yields no boundary term $ Q_G[\\xi ^i]$ , so the conserved charge due to diffeomorphisms $\\mathcal {H}_i$ is solely due to $SO(2,1) \\times U(1)$ gauge transformations, as usual for CS systems in $2+1$ dimensions [20].", "Once the boundary terms have been determined one is able to recover the gauge transformations globally generated by $S$ under the Dirac bracket.", "For instance, direct computation yields explicit relations for the improved diffeomorphisms (,).", "As can be readily checked, the smeared constraints $\\tilde{K}$ , $\\tilde{J}_a$ and $\\Upsilon $ also yield the corresponding transformations.", "Finally, as shown in Appendix , the asymptotic algebra induced by these symmetries splits into a (local) direct product $SO(2,1) \\times U(1) $ with the corresponding central extensions: $\\lbrace Q_{\\tilde{K}}[\\lambda ], Q_{\\tilde{K}}[\\zeta ] \\rbrace _D &=& C_{\\tilde{K}}[\\lambda ,\\zeta ] \\, , \\\\\\lbrace Q_{\\tilde{J}}[\\lambda ^a], Q_{\\tilde{J}}[\\zeta ^a] \\rbrace _D &=& Q_{\\tilde{J}}[\\epsilon ^a_{\\,\\, bc}\\lambda ^b \\zeta ^c]+C_{\\tilde{J}}[\\lambda ^a,\\zeta ^a] \\, ,$ where the central terms $C_{\\tilde{K}}$ and $C_{\\tilde{J}}$ given in (REF ,REF ), do not depend on the dynamical fields but only on the gauge parameters.", "A further refinement of this algebra can be obtained by the well known procedure of imposing asymptotic conditions for the bosonic sector $(\\omega _\\mu , A_\\mu )$ [21]." ], [ "Degree of freedom count", "In a theory with $N$ dynamical field components (that is, excluding Lagrange multipliers), $F$ first-class and $S$ second-class constraints, the number of DOF is given by [22] $ g= \\dfrac{2N-2F-S}{2}\\, .$ In the system discussed here there are $N=14+4n$ dynamical field components, $A_i, \\omega ^a{}_{i}, e^a_i, \\psi , \\bar{\\psi }$ .", "The following table gives the values of $F$ and $S$ in different cases: Table: NO_CAPTIONIn all cases, formula (REF ) gives $g=2n$ , in complete agreement with the naive counting of Section .", "Note that the first two sectors share the same number of independent first-class constraints.", "For the second, one finds an additional diffeomorphism generator instead of the Weyl scaling.", "As the possibility of finding another first-class combination cannot be ruled out in general, one could in principle find a sector where all the three diffeomorphism generators and the Weyl scaling (in addition to $\\tilde{J}_a$ and $\\tilde{K}$ ) are independent, even though such a configuration would certainly be non-generic by definition.", "However, this would lead to an odd number of second-class constraints and a non-integer result for $g$ , according to (REF )." ], [ "Discussion and summary", "We have carried out the Dirac analysis for constrained Hamiltonian systems for the action composed of a spin-$1/2$ Dirac field minimally coupled to an electromagnetic potential and to the Lorentz connection in $(2+1)$ -dimensions.", "The action of the entire system (REF ) is obtained from a CS form for an $\\mathfrak {osp}(2\|2)$ connection, in which the spinorial component of the connection was split as $\\chi ^\\alpha _\\mu := e^a_\\mu (\\Gamma _a)^\\alpha _\\beta \\psi ^\\beta $ .", "This splitting has a number of nontrivial consequences for the dynamical contents of the theory: i) Instead of zero degrees of freedom of a generic CS action, this system has the four propagating DOF of a Dirac spinor; ii) The system acquires a proper Weyl rescaling symmetry, i.e., it has no associated Noether charge and can be directly fixed; iii) The metric structures –the dreibein and the induced metric– are invariant under SUSY, and therefore there is no need to include spin-3/2 fields (gravitini); iv) Supersymmetry is reduced from a gauge symmetry to a rigid/global invariance that is contingent on the features of the background geometry and the gauge fields; v) For the vacuum sector the dreibein becomes pure gauge and diffeomorphisms degenerate into $SO(2,1) \\times U(1)$ transformations.", "The Dirac formalism completely recovers the Lagrangian equations.", "The equations for the gauge fields $(\\omega , A)$ follow from the constraints and the Hamilton equations for these fields.", "Furthermore, it can be shown that the Dirac equation and equation $\\mathcal {T}^\\mu {}_\\nu =0$ are respectively equivalent to (REF ,) for an invertible dreibein.", "In fact, after Weyl fixing and computing the temporal evolution one gets $D_t \\psi = e^a_t \\zeta _a$ and $T^a_{ti}= e^b_t e^c_i T^a_{\\,\\,\\, bc}$ .", "Then, equations (REF ,) together with the constraint (REF ) can be covariantized to give $T^a_{\\mu \\nu } \\bar{\\psi } \\psi &=& \\bar{\\psi }\\Gamma ^a \\Gamma _{[\\mu } \\overrightarrow{D}_{\\nu ]} \\psi + \\bar{\\psi } \\overleftarrow{D}_{[\\nu } \\Gamma _{\\mu ]} \\Gamma ^a \\psi \\, , \\\\\\Gamma ^{\\mu } D_{\\mu }\\psi &=& \\frac{1}{4} T^a_{\\mu \\nu }\\Gamma ^{\\nu \\mu } \\Gamma _a \\psi \\,.$ The degeneracy of these equations follows from the fact that $\\mathcal {T}^\\mu _{\\,\\,\\,\\mu }$ is proportional to () and is a combination of the Dirac equation -plus its conjugate-, and therefore identically vanishes for this theory, which is in turn equivalent to Weyl invariance.", "It should be stressed that $g=2n$ is an upper bound for the number of local DOF, since in non-generic sectors there might be additional accidental first-class constraints and therefore fewer degrees of freedom, as it happens in some sectors of higher-dimensional CS systems [7].", "The general counting performed in Section REF proceeds under the assumption that this is not the case.", "The argument given there, using the degenerate gauge, even holds for the spin-$1/2$ sector of Section REF , but for that configuration it is illustrative to explicitly use the Weyl fixing instead (see the end of Appendix ).", "In that sense, the purpose of choosing a specific sector such as the spin-$1/2$ is twofold: On the one hand the Lagrange multipliers can be readily solved, allowing for an explicit solution of (REF ,) leading to a full realization of the first-class constraints.", "On the other, the Weyl symmetry is “gauged away\" in this case, providing a symmetry breaking mechanism.", "One is left with a global version of the scale invariance which is broken by fixing the fermion mass or the normalization of the dreibein.", "In this system SUSY seems to play a marginal role.", "It starts out as part of the gauge invariance of the action (REF ), then it is seen as a global (rigid) symmetry without first-class constraints associated to it, contingent on the existence of some spacetime symmetry, which need not occur in every spacetime background.", "The action and the equations are invariant under $\\delta \\psi & = & \\frac{1}{3}{D} \\epsilon \\, , \\quad \\delta \\overline{\\psi } = \\frac{1}{3}\\overline{\\epsilon }\\overleftarrow{{D} }\\nonumber \\\\\\delta A & = & -\\frac{i}{2}\\left( \\overline{\\psi }{e} \\epsilon +\\overline{\\epsilon }{e} \\psi \\right) \\nonumber \\\\\\delta \\omega ^a & = &-\\overline{\\psi }\\left( e^a+ \\epsilon ^a{}_{bc}e^b \\Gamma ^c \\right) \\epsilon - \\overline{\\epsilon } \\left( e^a - \\epsilon ^a{}_{bc}e^b \\Gamma ^c \\right)\\psi \\, , \\\\\\delta e^a & = & 0\\, \\nonumber $ where $\\epsilon $ satisfies the no-spin-3/2 condition, $[\\delta _\\nu ^\\mu - (1/3) \\Gamma _\\nu \\Gamma ^\\mu ]D_\\mu \\epsilon =0$ .", "This condition can be fulfilled provided the spacetime and the connection fields admit a Killing spinor of a certain kind [3].", "This is the case for the vacuum: AdS or Minkowski space without fermions or electromagnetic fields.", "This background is a full-BPS state preserving full supersymmetry, but there are configurations preserving 1/2 or 1/4 of SUSY, just like in 2+1 supergravity [23], [18].", "A bosonic vacuum $\\psi =0$ remains invariant under (REF ) provided ${D} \\epsilon =0$ , which is also a requirement that the background admit a Killing spinor.", "This unconventional SUSY can by extended to describe fermions in the fundamental representation of a non-Abelian internal group like $SU(2)$ [18].", "As we have seen above, the constraints associated to internal $U(1)$ and Lorentz $SO(2,1)$ symmetries decouple from the diffeomorphism and Weyl ones.", "By the same token, in a generic supersymmetric extension of an internal non-Abelian gauge symmetry, the fermion excitations turn out to be the only contribution to the local DOF.", "In order to illustrate this, let us consider the split Lagrangian for the $SU(2)$ theory, which, up to a global factor, reads [18]The CS form also contains an abelian form $b$ associated to the central charge in $su(2,1\|2)$ .", "However, $b$ decouples from the action and therefore does not enter in the dynamical analysis.", "$L_{SU(2) }& = & \\epsilon ^{ij}\\Bigl [-\\eta _{ab} \\dot{e}_i ^a e_j^b \\bar{\\psi }_A \\psi ^A -\\dot{\\bar{\\psi }}_A\\Gamma _{ij}\\psi ^A+\\bar{\\psi }_A\\Gamma _{ij}\\dot{\\psi }^A +\\frac{1}{2}\\eta _{ab}\\dot{\\omega }_i^a \\omega _j ^b + \\frac{1}{2}\\delta _{IJ}\\dot{A}^I_i A^J_j \\Bigr ] \\nonumber \\\\& & -e_t^a K_a + \\omega _t ^a J_a +A^I_t K_I\\,,$ Here the indexes $A=1,2$ transform under the $2\\times 2$ vector representation of $SU(2)$ (Pauli matrices), while $I=1,2,3$ refers to the adjoint representation (we follow the conventions of [18]).", "The primary constraints $(\\varphi ^i_a, \\phi ^i_a, \\phi ^i_I, \\Omega ^A, \\bar{\\Omega }_A)$ are defined in an analogous fashion to their $U(1)$ counterparts.", "If one omits the contraction in the $A$ index, i.e.", "$\\bar{\\psi }_A \\psi ^A = \\bar{\\psi } \\psi $ , the secondary constraints $K_a$ and $J_a$ adopt exactly the same form as (REF ,) where the covariant derivatives are now gauged by $SO(2,1) \\times SU(2)$ .", "The remaining constraint reads $K_{I} =\\epsilon ^{ij}\\delta _{IJ}(\\frac{1}{2}F^{J}_{ij}-i\\bar{\\psi }\\Gamma _{ij}\\sigma ^{J}\\psi ) = \\epsilon ^{ij}\\delta _{IJ}(\\partial _{i}A_{j}^{J}+\\frac{1}{2}\\epsilon _{\\,KL}^{J}A_{i}^{K}A_{j}^{L}-i\\bar{\\psi }\\Gamma _{ij}\\sigma ^{J}\\psi ) \\, .$ Then, one can show that $\\tilde{J}_a & := & J_a +\\epsilon ^{}_{ac} \\varphi ^j_b e^c_j +\\frac{1}{2}(\\bar{\\Omega }\\Gamma _a \\psi - \\bar{\\psi }\\Gamma _a \\Omega ) + D_j \\phi ^j_a \\,, \\nonumber \\\\\\tilde{K}_I & := & K_I-\\frac{i}{2}(\\bar{\\Omega }\\sigma _I \\psi -\\bar{\\psi }\\sigma _I\\Omega )+D_j \\phi ^j_I \\,,\\nonumber \\\\\\Upsilon & := & -e^b_j \\varphi ^j_b +\\bar{\\Omega }\\psi +\\bar{\\psi }\\Omega \\,, \\\\\\mathcal {H}_i & := & e^a_i K_a -e^a_i D_j \\varphi ^j_a + T^a_{ij} \\varphi ^j_a + \\bar{\\psi } \\overleftarrow{D}_i \\Omega +\\bar{\\Omega }\\overrightarrow{D}_{i}\\psi -\\omega _{i}^{a}\\tilde{J}_{a}-A^I_{i}\\tilde{K}_I+\\phi ^{j}_I F^I_{ij} +\\phi _{a}^{j}R^a_{ij} \\, .", "\\nonumber $ correspond to $F=9$ first-class combinations generating $SO(2,1) \\times SU(2) \\times Weyl \\times \\text{Diff}$ transformations, respectively.", "In account of these and the remaining $S=18+8n$ second class constraints, the original phase space of $N=18+8n$ variables only contains $g_{SU(2)}= \\dfrac{2N-2F-S}{2} = 4n \\,$ degrees of freedom for a generic sector, exactly matching the double of the $U(1)$ case due to the doubling of the fermion fields.", "SUSY is again not realized as a first-class constraint, but is a rigid transformation for certain backgrounds.", "Such matters, together with the computation of the asymptotic charges, were already treated in the original work.", "Unconventional supersymmetries can also be constructed in higher dimensions based on a gauge superalgebra containing $\\mathfrak {so}(2n,2)$ or $\\mathfrak {so}(2n-1,1)$ as a proper subalgebras.", "In odd dimensions $D=2n+1\\ge 5$ , a similar CS construction can be set up, while for $D=2n \\ge 4$ , since the CS forms are not defined, the construction requires a metric and the action can be of a Yang-Mills type.", "In both cases the fermionic part of the connection can be construed as a composite of a vielbein and a spin-1/2 Dirac field [3].", "For all $D\\ge 4$ , it can be expected that, as in the three-dimensional case discussed here, the vielbein would not contribute to the dynamic contents unless it possesses an independent kinetic term of its own; the effective gauge symmetry would correspond to the bosonic part of the superalgebra, and supersymmetry would be reduced to a rigid invariance conditioned by the existence of globally defined Killing spinors of the background.", "In other words, supersymmetry would be at most an approximate feature in some vacuum spacetime geometries, and the main footprint of its presence in the theory would be in the field content, the type of couplings and the parameters in the action.", "The conduction properties of graphene [24], [25], [26] can be very well described by the $\\pi $ -electrons in the two sublattices of the honeycomb structure as massless fermions in the long-wavelength limit [27].", "It was already conjectured that the system studied here could reproduce the behavior of these $\\pi $ -electrons [2], [18], while the very strong $\\sigma $ -bond of the remaining available electrons of the carbon atoms keep the geometry of the graphene layer fixed.", "Therefore it is expected that in the low energy (long wavelength) regime, the dynamical contents are essentially in the fermion sector, as pointed out here.", "Nevertheless, note that we have introduced a torsional mass term, which is required in principle by hermiticity.", "Such construction not only leads to a symmetry breaking mechanism but, it also allows the massive fermion to trigger a backreaction into the background, provided we use the contorsion as an effective cosmological constant.", "This implies a constant curvature background as illustrated in [2].", "Following that line, an idea to be experimentally explored is whether specific graphene layers (or graphene-like material) can be manufactured which admit Killing-spinors in order to measure some induced supersymmetric effects.", "This would provide low-energy graphene models to test high-energy physics theories, whose observable effects are beyond reach in current particle accelerators [28].", "Besides providing a rigorous tool for identifying the dynamical DOF, the Hamiltonian formalism could be the preliminary warm-up towards a quantization procedure [10], [11], eventually leading to a quantum theory of graphene.", "In the system described here, the only dynamical degrees of freedom are those of the Dirac fermion; the bosonic connections $A$ and $\\omega $ are described by Chern-Simons actions and therefore have no local degrees of freedom, while the dreibein is an artifact that can be gauged away.", "This means that the bosonic fields do not contribute to the quantum field theory other than as classical external fields; their quantum excitations are produced by nontrivial global holonomies of a topological nature.", "Such fields do not propagate and hence do not generate perturbative corrections.", "In particular, there should be no perturbative corrections generated by quantum fluctuations of the bosonic fields in graphene, the system should behave like a free electron field propagating in a curved classical background and would therefore be renormalizable.", "Further insight comes from the AdS$_3$ /CFT$_2$ duality and its generalizations in $2+1$ gravity [29], [20], [30], which are realized through a centrally extended canonical algebra in the asymptotic region associated to a quantum theory at the boundary.", "In the broader gauge/gravity context, the holographic description of graphene in the IR regime has been recently studied by means of a 3+1 D-brane embedding, exhibiting also conformal symmetry breaking due to the introduction of a mass gap scale as an integration constant [31]." ], [ "Acknowledgments", "We are grateful to P. D. Alvarez, W. Clemens, O. Fuentealba, J. Helayel-Neto, A. Iorio, F. Toppan and M. Valenzuela for many enlightening discussions.", "We wish to specially thank P. Salgado-Rebolledo for taking active part in the initial stages of this project.", "P. P. wishes to express his thanks to Prof. M. Henneaux for support and encouragement.", "A. G. also thanks CONICYT for financial support.", "This work has been partially funded through Fondecyt grant 1140155.", "The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt." ], [ "General definitions and useful properties", "Through this work we extensively use the Clifford algebra in $D=3$ .", "Some basic properties and definitions areWe adopt the convention $\\epsilon _{012}= -\\epsilon ^{012}=1$ and the definition $T_{[a_1...a_p]}=\\frac{1}{p!", "}\\delta _{a_1...a_p}^{b_1...b_p}T_{b_1...b_p}$ .", "In the coordinate basis, $\\epsilon _{ij}:=\\epsilon _{tij}$ .", "$\\lbrace \\Gamma _a,\\Gamma _b\\rbrace = 2\\eta _{ab}\\,, \\quad \\Gamma _{ab}:=\\Gamma _{[a}\\Gamma _{b]} = \\frac{1}{2}[\\Gamma _{a},\\Gamma _{b}]\\, , \\quad \\Gamma _{ab} = \\epsilon _{abc}\\Gamma ^{c} \\,, \\quad \\\\\\quad \\frac{1}{2} [ \\Gamma _{ab},\\Gamma _c ] =\\eta _{bc}\\Gamma _a -\\eta _{ac} \\Gamma _b \\,, \\quad \\Gamma _{abc} = \\epsilon _{abc} =\\frac{1}{2}\\lbrace \\Gamma _{ab},\\Gamma _{c}\\rbrace =\\Gamma _{[a\|}\\Gamma _b \\Gamma _{\|c]}$ Let $\\psi $ be a two-component Dirac spinor with Grassman parity odd.", "We define its Dirac conjugate by $\\overline{\\psi }=i\\psi ^{\\dagger }\\Gamma _0.$ or explicitly as $\\bar{\\psi }_\\beta = i \\psi ^{\\alpha * }(\\Gamma _0)_{ \\alpha \\beta }$ , $\\alpha , \\beta = 1,2$ .", "With this prescription, we have the conjugacy properties $(\\overline{\\chi }\\psi )^{\\ast }&=&\\overline{\\psi }\\chi , \\nonumber \\\\(\\overline{\\chi }\\Gamma _{a}\\psi )^{\\ast }&=&-(\\overline{\\psi }\\Gamma _{a}\\chi ).", "\\\\\\overline{\\left(\\Gamma _{a}\\psi \\right)} & = & -\\overline{\\psi }\\Gamma _{a} \\nonumber $ The starting point for the this model is to take the connection for the full $osp(2\|2)$ algebra [2] $\\mathcal {A}=A{Z}+\\omega ^{a}{J}_{a}+{\\overline{Q}}{e}\\psi +\\overline{\\psi }{e}{Q},$ where ${Z}$ is the $U(1)$ generator, $\\lbrace {J}_{a}\\rbrace $ is the set of generators of the Lorentz algebra $SO(2,1)$ , and $\\lbrace {\\overline{Q}}_{\\alpha },{Q}^{\\alpha }\\rbrace $ is the set of SUSY generators.", "The symbol ${e}$ is a compressed notation for ${e}=e^{a}\\Gamma _{a}$ .", "The algebra reads $\\left[{J}_a,{J}_{b}\\right] &=& \\epsilon _{abc}{J}^{c}, \\quad \\left[{J}_{a},{Q}^{\\alpha }\\right] = -\\frac{1}{2}(\\Gamma _a)^{\\alpha }_{\\beta }{Q}^{\\beta }, \\quad \\left[{J}_{a},{\\overline{Q}}_{\\alpha }\\right] = \\frac{1}{2}(\\Gamma _{a})^{\\beta }_{\\alpha }{\\overline{Q}}_{\\beta }, \\nonumber \\\\\\left[{Z},{Q}^{\\alpha }\\right] &=& \\frac{i}{2}{Q}^{\\alpha }, \\quad \\left[{Z},{\\overline{Q}}_\\alpha \\right] = -\\frac{i}{2}{\\overline{Q}}_{\\alpha }, \\quad \\left\\lbrace {Q}^{\\alpha }, {\\overline{Q}}_\\beta \\right\\rbrace =(\\Gamma ^{a})^{\\alpha }_{\\beta }{J}_{a}-i\\delta ^{\\alpha }_{\\beta }{Z},$ where the other (anti-)commutators are zero.", "The action (REF ) is given by $I[\\psi ,e,A,\\omega ]=\\int \\langle \\mathcal {A},d\\mathcal {A} \\rangle + \\frac{2}{3} \\langle \\mathcal {A}, \\mathcal {A}^2 \\rangle $ where the (super-)invariants traces are $\\langle {J}_a , {J}_b \\rangle =\\frac{1}{2}\\eta _{ab}, \\quad \\langle {Z}, {Z}\\rangle =\\frac{1}{2}, \\quad \\langle {\\overline{Q}}_\\alpha , {Q}^\\beta \\rangle = \\delta ^{\\beta }_{\\alpha }, \\quad \\langle {Q}^\\alpha , {\\overline{Q}}_\\beta \\rangle = -\\delta ^{\\alpha }_{\\beta }.$ The covariant derivative $D_{\\mu }$ induced by (REF ) appears naturally in (REF ).", "Acting on a Lorentz vector $\\Sigma _{a}$ and $1/2$ -spinors ($\\psi ^{\\alpha }$ and $\\overline{\\psi }_{\\alpha }$ ) this reads $D_{\\mu }\\Sigma _{a}&=&\\partial _{\\mu }\\Sigma _{a}+{\\epsilon }{_{ab}^{c}}\\omega ^{b}_{\\mu }\\Sigma _{c}, \\nonumber \\\\\\overrightarrow{D}_{\\mu }\\psi ^{\\alpha }&=&\\partial _{\\mu }\\psi ^{\\alpha }-\\frac{i}{2}A_{\\mu }\\psi ^{\\alpha }+\\frac{1}{2}\\omega ^{a}_{\\mu }(\\Gamma _{a})^{\\alpha }_{\\beta }\\psi ^{\\beta }, \\\\\\overline{\\psi }_{\\alpha }\\overleftarrow{D}_{\\mu }&=&\\partial _{\\mu }\\overline{\\psi }_{\\alpha }+\\frac{i}{2}A_{\\mu }\\overline{\\psi }_{\\alpha }-\\frac{1}{2}\\overline{\\psi }_{\\beta }(\\Gamma _{a})^{\\beta }_{\\alpha }\\omega ^{a}_{\\mu }= \\overline{(\\overrightarrow{D}_{\\mu }\\psi )}_{\\alpha }.", "\\nonumber $" ], [ "Momenta, Constraints and Poisson brackets", "For the starting action (REF ), the canonical momenta associated to the dynamical fields are given by $\\pi ^{i}\\approx \\frac{\\partial \\mathcal {L}}{\\partial \\dot{A}_{i}}=\\frac{1}{2}\\epsilon ^{ij}A_{j}, \\quad \\pi ^{i}_{a}\\approx \\frac{\\partial \\mathcal {L}}{\\partial \\dot{\\omega }^{a}_{i}} &=& \\frac{1}{2}\\epsilon ^{ij}\\eta _{ab}\\omega ^{b}_{j}, \\quad p^{i}_{a}\\approx \\frac{\\partial \\mathcal {L}}{\\partial \\dot{e}^{a}_{i}} =-2\\epsilon ^{ij}\\eta _{ab}e^{b}_{j}\\overline{\\psi }\\psi \\nonumber \\\\\\chi ^{\\alpha }\\approx \\frac{\\partial ^{L}\\mathcal {L}}{\\partial \\dot{\\overline{\\psi }}_{\\alpha }}=-\\epsilon ^{ij}(\\Gamma _{ij})^{\\alpha }_{\\,\\,\\,\\, \\beta }\\psi ^{\\beta } &,& \\overline{\\chi }_{\\alpha }\\approx \\frac{\\partial ^{R}\\mathcal {L}}{\\partial \\dot{\\psi }^{\\alpha }}=\\epsilon ^{ij}\\overline{\\psi }_{\\beta }(\\Gamma _{ij})^{\\beta }_{\\,\\,\\,\\, \\alpha }.$ The non-vanishing Poisson brackets between the fields and their respective momenta are defined as in [10] Hereafter we will omit the $\\delta ^2 (x-y)$ factors when computing the brackets.", "Spinor indexes may also be omitted for simplicity.", "$\\left\\lbrace A_i,\\pi ^j \\right\\rbrace &=& -\\left\\lbrace \\pi ^j, A_i \\right\\rbrace = \\delta ^{j}_{i}, \\quad \\left\\lbrace \\omega ^a_i, \\pi ^j_b \\right\\rbrace = -\\left\\lbrace \\pi ^j_b, \\omega ^a_i \\right\\rbrace = \\left\\lbrace e^{a}_i,p^j_{b}\\right\\rbrace =-\\left\\lbrace p^j_{b},e^{a}_i\\right\\rbrace =\\delta ^{j}_{i}\\delta ^{a}_{b}, \\nonumber \\\\\\left\\lbrace \\psi ^\\alpha , \\overline{\\chi }_\\beta \\right\\rbrace &=& \\left\\lbrace \\overline{\\chi }_\\beta , \\psi ^\\alpha \\right\\rbrace = \\delta ^\\alpha _\\beta , \\quad \\left\\lbrace \\overline{\\psi }_{\\alpha },\\chi ^{\\beta }\\right\\rbrace =\\left\\lbrace \\chi ^{\\beta },\\overline{\\psi }_{\\alpha }\\right\\rbrace =-\\delta ^{\\beta }_{\\alpha }.$ It is worth to note the relative sign between the two brackets on the last line: This choice is consistent with $\\overline{\\psi }_{\\alpha } = i\\psi ^ {\\beta }(\\Gamma _0)_{\\beta \\alpha } $ and $\\overline{\\chi }_{\\alpha } = i\\chi ^ {\\beta }(\\Gamma _0)_{\\beta \\alpha } $ .", "Now, the primary constraints (REF ) satisfy $\\lbrace \\varphi _{a}^{i},\\varphi _{b}^{j}\\rbrace = 4\\epsilon ^{ij}\\eta _{ab}\\bar{\\psi }\\psi ,\\quad \\lbrace \\Omega ,\\varphi _b^j\\rbrace &=&2\\epsilon ^{ij}e_i^a \\Gamma _a \\Gamma _b \\psi , \\quad \\lbrace \\bar{\\Omega },\\varphi _{b}^{j}\\rbrace = 2\\epsilon ^{ij}e_{i}^{a}\\bar{\\psi }\\Gamma _{b}\\Gamma _{a} ,\\nonumber \\\\\\lbrace \\bar{\\Omega },\\Omega \\rbrace = 2\\epsilon ^{ij}\\Gamma _{ij} , \\quad \\lbrace \\phi ^{i},\\phi ^{j}\\rbrace &=& -\\epsilon ^{ij} , \\quad \\lbrace \\phi _{a}^{i},\\phi _{b}^{j}\\rbrace = -\\epsilon ^{ij}\\eta _{ab}\\,.$ Using (REF ) and the definitions (REF ) one can show that the generators $\\tilde{J}_{a}$ , $\\tilde{K}$ and $\\Upsilon $ satisfy the following: $\\lbrace \\tilde{J}_{a}, \\phi _{b}^{i} \\rbrace &=&\\epsilon _{ab}^{\\quad c}\\phi _c^i , \\quad \\lbrace \\tilde{J}_{a},\\Omega \\rbrace =-\\dfrac{1}{2}\\Gamma _{a} \\Omega , \\quad \\lbrace \\tilde{J}_{a},\\bar{\\Omega }\\rbrace =\\frac{1}{2}\\bar{\\Omega }\\Gamma _{a} , \\quad \\lbrace \\tilde{J}_{a}, \\varphi _{b}^{i}\\rbrace = \\epsilon _{ab}^{\\quad c}\\varphi _{c}^{i}\\,, \\nonumber \\\\\\lbrace \\tilde{J}_{a},J_{b}\\rbrace &=&\\epsilon _{ab}^{\\quad c}J_{c}\\,, \\quad \\lbrace \\tilde{J}_{a},K_{b}\\rbrace =\\epsilon _{ab}^{\\quad c}K_{c}, \\quad \\lbrace \\tilde{K},\\Omega \\rbrace =\\frac{i}{2}\\Omega , \\quad \\lbrace \\tilde{K},\\bar{\\Omega }\\rbrace =-\\frac{i}{2}\\bar{\\Omega } , \\\\\\lbrace \\Upsilon ,\\varphi _{a}^{j}\\rbrace &=& -\\varphi _{a}^{j} , \\quad \\lbrace \\Upsilon ,K_{a}\\rbrace =-K_{a} , \\quad \\lbrace \\Upsilon ,\\Omega \\rbrace =\\Omega , \\quad \\lbrace \\Upsilon ,\\bar{\\Omega }\\rbrace = \\bar{\\Omega } , \\nonumber $ where the remaining brackets with constraints (REF )-(REF ) vanish strongly.", "We then conclude that the constraints $\\tilde{J}_{a}$ , $\\tilde{K}$ ,$\\Upsilon $ , together with the generator $\\mathcal {H}_i$ , are first-class.", "The consistency of the primary constraints (REF ) with respect to the extended Hamiltonian (REF ) yields the following set of equations $0 = \\left\\lbrace \\phi ^{i},H_{T}\\right\\rbrace &=&\\epsilon ^{ij}\\left(\\partial _{j}A_t +2ie^{a}_t e^{b}_{i}\\overline{\\psi }\\Gamma _{ab}\\psi -\\lambda _{j}\\right)\\,, \\nonumber \\\\0 = \\left\\lbrace \\phi ^i_a ,H_T \\right\\rbrace &=&\\epsilon ^{ij}\\left(\\eta _{ab}D_j \\omega ^b_t + 2\\epsilon _{abc}e^b_t e^c_j \\overline{\\psi }\\psi - \\eta _{ab}\\Lambda ^b_j \\right) \\,, \\nonumber \\\\0 = \\left\\lbrace \\varphi ^i_a, H_T \\right\\rbrace &=&2\\epsilon ^{ij}\\left(\\epsilon _{abc}\\omega ^b_t e^c_j \\overline{\\psi }\\psi -i A_t e^b_j \\overline{\\psi }\\Gamma _{ab}\\psi -2 \\eta _{ab}\\partial _j \\left(e^b_t \\overline{\\psi }\\psi \\right) -\\epsilon _{abc}\\omega ^b_j e^c_t \\overline{\\psi }\\psi \\right.", "\\\\&&\\left.", "+ e^b_t (\\overline{\\psi }\\overleftarrow{D}_j \\Gamma _a \\Gamma _b \\psi +\\overline{\\psi }\\Gamma _b \\Gamma _a\\overrightarrow{D}_j \\psi ) +2\\eta _{ab}\\lambda ^b_j \\overline{\\psi } \\psi + e^b_j \\left(\\overline{\\Lambda }\\Gamma _b \\Gamma _a \\psi + \\overline{\\psi }\\Gamma _a \\Gamma _b \\Lambda \\right) \\right) \\nonumber \\,, \\\\0 = \\left\\lbrace \\Omega ,H_T \\right\\rbrace &=&-\\epsilon ^{ij}\\left(iA_t e^a_i e^b_j \\Gamma _{ab}\\psi + \\epsilon _{abc}\\omega ^a_t e^b_i e^c_j \\psi + 2\\eta _{ab}e^a_t T^{b}_{ij}\\psi -2e^{a}_{t}e^{b}_{i}\\Gamma _{a}\\Gamma _{b}D_{j}\\psi \\right.", "\\nonumber \\\\&& \\left.", "+ 2 D_{j}\\left(e^a_t e^b_i \\Gamma _b \\Gamma _{a}\\psi \\right) +2\\lambda ^{a}_{i}e^{b}_{j}\\Gamma _{b}\\Gamma _{a}\\psi -2e^{a}_{i}e^{b}_{j}\\Gamma _{ab}\\Lambda \\right) \\,, \\nonumber \\\\0 = \\left\\lbrace \\overline{\\Omega } ,H_{T}\\right\\rbrace &=&\\epsilon ^{ij}\\left(iA_t e^{a}_{i}e^{b}_{j}\\overline{\\psi } \\Gamma _{ab}- \\epsilon _{abc}\\omega ^{a}_t e^{b}_{i}e^{c}_{j}\\overline{\\psi } - 2\\eta _{ab}e^a_t T^{b}_{ij}\\overline{\\psi } +2e^{a}_t e^{b}_{i}(\\overline{\\psi }\\overleftarrow{D}_{j})\\Gamma _{b}\\Gamma _{a} \\right.", "\\nonumber \\\\&&\\left.", "- 2 \\left(e^a_t e^b_i\\overline{\\psi }\\Gamma _a \\Gamma _b \\right) \\overleftarrow{D}_j -2\\lambda ^a_i e^b_j \\overline{\\psi }\\Gamma _a \\Gamma _b -2e^a_i e^b_j \\overline{\\Lambda }\\Gamma _{ab} \\right) \\,.", "\\nonumber $ This system of $(14+4n)$ equations determines up to an equal number of Lagrange multipliers, leaving seven free parameters.", "This means that in a generic sector (maximum rank), there are $S=14+4n$ second-class and $F=7$ first-class constraints.", "Also, if one choose $(e^a_t,\\omega ^a_t,A_t)$ as the free parameters, the consistency of the secondary constraints $K_a, J_a$ and $K$ can be readily shown to follow.", "In Appendix we exhibit a solution for (REF )." ], [ "Solving the consistency equations", "Let us now choose tensors $\\zeta _a$ and $T^a_{\\,\\,\\, bc}=T^a_{\\,\\,\\, [bc]}$ , depending on the dynamical fields, such that $T^a_{\\,\\,\\, bc} e^b_i e^c_j &=&T^a_{ij} \\\\&=& D_i e^b_j - D_j e^b_i \\,, \\nonumber \\\\e^a_i\\zeta _a &=& D_i\\psi \\,.", "$ Equation (REF ) relates the 9 Lorentz covariant components $T^a_{\\,\\,\\, bc}$ to the 3 field dependent quantities on the RHS.", "Similarly, equation () expresses the vector-spinor $\\zeta _a$ as functions of 2 components on the RHS.", "This means there are six real and one spinorial indeterminate components respectively For $\|e\|\\ne 0$ , one can put for instance $\\zeta _a= E^i_a D_i\\psi + E^t_a\\xi $ and $T^a_{\\,\\,\\, bc}=E^i_b E^j_c T^a_{ij}+ E^t_{[b} E^i_{c]}\\xi ^a_i$ for arbitrary $\\xi $ and $\\xi ^a_i$ ., which will be fixed by the consistency equations.", "Now, let us take the Lagrange multipliers in equation (REF ) as $\\lambda _{j}^{b} & = & -ve_{j}^{b}-\\epsilon _{\\:cd}^{b}\\omega _t ^{c}e_{j}^{d}+D_{j}e_t ^{b}+T_{\\,ac}^{b}e_t ^{a}e_{j}^{c}\\,, \\nonumber \\\\\\Lambda & = & v\\psi +\\frac{i}{2}A_t \\psi -\\frac{1}{2}\\omega _t ^{c}\\Gamma _{c}\\psi +e_t ^{a}\\zeta _a \\,, \\nonumber \\\\\\bar{\\Lambda } & = & v\\bar{\\psi }-\\frac{i}{2}A_t \\bar{\\psi }+\\frac{1}{2}\\omega _t ^{c}\\bar{\\psi }\\Gamma _{c}+\\bar{\\zeta _a}e_t ^{a} \\, , \\\\\\Lambda _{j}^{b} & = & D_{j}\\omega _t ^{b}+2e_t ^{c}e_{j}^{a}\\epsilon _{\\:ca}^{b}\\bar{\\psi }\\psi \\,, \\nonumber \\\\\\lambda _{j} & = & \\partial _{j}A_t +2ie_t ^{a}e_{j}^{b}\\epsilon _{abc}\\bar{\\psi }\\Gamma ^{c}\\psi \\,.", "\\nonumber $ After inserting (REF ) into the consistency conditions (REF ), these reduce to $0 &= & e_t ^{b}e_{j}^{c}(\\eta _{ad}T_{\\,cb}^{d}\\bar{\\psi }\\psi -\\bar{\\psi }\\Gamma _{a}\\Gamma _{[c}\\zeta _{b]}-\\bar{\\zeta }_{[b}\\Gamma _{c]}\\Gamma _{a}\\psi ) \\,, \\\\0 & = & \|e\| \\epsilon ^{abc}(\\Gamma _{ab}\\zeta _c - \\frac{1}{2}\\Gamma _{a}\\Gamma _{d}T_{\\,bc}^{d}\\psi ) \\,.", "$ together with the conjugate of the last equation.", "For an arbitrary dreibein, equation () can be used to fix the remaining free component of $\\zeta _a$ as a function of the dynamical fields and $T^a_{\\,\\,\\,bc}$ .", "On the other hand, using the constraint $K_a\\approx 0$ , one can show that (REF ) correspond to 6 independent equations for an equal number of free components in $T^a_{\\,\\,\\,bc}$ , once $\\zeta _a$ is replaced.", "Note now that the parameter $v$ does not show up in (REF ,), this indicates that the complete set of equations is not independent.", "In fact, one can readily check that the following shift $T^a_{\\,\\,\\,bc} \\rightarrow T^a_{\\,\\,\\,bc} + 2\\beta \\delta ^a_{[b} E_{c]}^t &, \\quad & \\zeta _c \\rightarrow \\zeta _c - \\beta E_c^t \\psi \\,,$ leaves (REF ,) and (REF ,) invariant.", "This is related to the Weyl invariance, shifting the multiplier $v \\rightarrow v - \\beta $ in (REF ).", "We thus have the following picture: If the three components $e^a_t$ remain arbitrary, then one can solve the $(14+4n)$ multipliers as in (REF ), but this leaves a degeneracy in $v$ to be fixed afterwards.", "Otherwise one may restrict one of the components $e^a_t$ while leaving the scaling parameter $v$ completely free, as we explain below.", "In view of the counting argument of Section , we expect in general that one combination among the 8 parameters $(A_t, \\omega ^a_t,e^a_t, v)$ will be found fixed in a generic sector, so that the number of functionally independent first-class constraints is reduced to $F=7$ .", "Note that the degeneracy in $v$ also suggests there could be certain configurations of the dynamical fields such that (REF ,) have no solution: The consistency equations would lead to secondary constraints in this sectors.", "The above reasoning is illustrated with the spin-$1/2$ sector described in Section REF .", "In that case one chooses the gauge $v=0$ a priori, and then proceed to count the DOF considering the residual symmetries.", "This gauge fixing is equivalent to choose the solution $T^a_{\\,\\,\\,bc} &=& 2\\alpha \\epsilon ^a_{\\,\\,\\,bc} \\, , \\\\\\zeta _a &=& \\Gamma _a \\psi \\, ,$ for (REF ,) and (REF ,), provided (REF ,REF ).", "Inserting this into the multipliers (REF ) and then into the total Hamiltonian (REF ), one directly gets the form (REF ).", "Note that this Hamiltonian preserves the gauge, and possesses only 7 free parameters $(A_t, \\omega ^a_t, e^a_t)$ corresponding to the generators of residual symmetries.", "On the other hand, in a generic sector one can always use the \"degenerate gauge\" (REF ) for counting purposes.", "Using $K_a \\approx 0$ , this election is readily seen to close the consistencies (REF ,) and puts the total Hamiltonian in the form (REF ).", "However, by doing so one needs to assume there is in fact a solution for $\\zeta _a$ and $T^a_{\\,\\,\\,bc}$ , in order to extend the sector for non-degenerate choices with $\|e\|\\ne 0$ .", "Thus, in any generic sector a realization of the first-class constraints can be easily obtained by means of the degenerate gauge, leaving also $F=7$ free parameters $(A_t, \\omega ^a_t,\\xi ^i,v)$ ." ], [ "Asymptotic charges on the bosonic vacuum", "In order to compute the charges in the smeared generator (REF ), we demand that the boundary terms induced by its variations with respect to the dynamical fields be well defined.", "That is, the variations of $Q$ should compensate the boundary contributions coming from the integration by parts on the bulk term.", "Varying the action (REF ) with (REF -), and upon integrating by parts one finds $\\delta Q_{G}[\\xi ^i] &=& - \\int _{\\partial \\Sigma }dx^l \\xi ^{i}[2(\\bar{\\psi }\\Gamma _{il}\\delta \\psi +\\delta \\bar{\\psi }\\Gamma _{li}\\psi )+\\epsilon _{jl}\\delta (e_{i}^{a}p_{a}^{j})-\\epsilon _{il}\\delta e_{k}^{a}p_{a}^{k}] \\, , \\\\\\delta Q_{\\tilde{J}} [\\lambda ^a]&=& -2\\int _{\\partial \\Sigma }dx^l\\eta _{ab}\\lambda ^{a}\\delta \\omega _{l}^{b} \\, , \\\\\\delta Q_{\\tilde{K}} [\\lambda ]&=&- 2\\int _{\\partial \\Sigma }dx^l \\lambda \\delta A_{l} \\, ,$ where $\\partial \\Sigma $ the boundary of the spatial slice $t=constant$ .", "It is clear that $\\delta Q_{\\tilde{J}}$ and $\\delta Q_{\\tilde{K}}$ can be readily integrated, i.e.", "the $\\delta $ can be removed, but for $\\delta Q_{G}[\\xi ^i]$ we need to give certain boundary conditions.", "If we impose $\\psi $ and $\\bar{\\psi }$ to vanish at the boundaryMore generally, one could consider for instance $\\psi \\sim \\bar{\\psi }\\sim O(\\frac{1}{r^2})$ , where the asymptotic region is defined by $r \\rightarrow \\infty $ .", "Then the leading order in $p^i_j \\approx 2\\epsilon ^{ij}\\eta _{ab}e^b_j \\bar{\\psi }\\psi $ depends on the fall-off of the dreibein, and if $e^a_i \\sim O(r)$ , all the asymptotic contributions in () still vanish., then we are led to also fix $p^k_a = 0$ by consistency with $\\varphi ^k_a \\approx 0$ .", "These (gauge-consistent) conditions then annihilate the charge associated to $G_i$ , while leaving the variation of the dreibein completely undetermined, as was discussed for the vacuum configuration in the bulk region.", "The conditions can thus be regarded as the natural asymptotic extension of such sector.", "With these global charges, the smeared generator $S$ has well defined functional derivatives and consistently acts on the fields $(A_i, \\omega ^a_i, \\psi , \\bar{\\psi }, e^a_i, p^j_b)$ through the Dirac bracket.", "Upon gauge fixing the bulk term in $S$ is identically dropped, leading to the so-called asymptotic charge algebra $\\lbrace Q_{\\tilde{K}}[\\lambda ], Q_{\\tilde{K}}[\\zeta ] \\rbrace _D &=&\\lbrace Q_{\\tilde{K}}[\\lambda ], S[0,\\zeta ;0] \\rbrace _D \\nonumber \\\\&=& \\delta _{\\tilde{K}[\\zeta ]} Q_{\\tilde{K}}[\\lambda ] \\nonumber \\\\&=& 2\\int _{d\\Sigma }dx^l \\lambda \\partial _l \\zeta \\nonumber \\\\&=& C_{\\tilde{K}}[\\lambda ,\\zeta ] \\, ,$ and similarly, $\\lbrace Q_{\\tilde{J}}[\\lambda ^a], Q_{\\tilde{J}}[\\zeta ^a] \\rbrace _D &=& Q_{\\tilde{J}}[\\epsilon ^a_{\\,\\, bc}\\lambda ^b \\zeta ^c]+ 2\\int _{d\\Sigma }dx^l \\eta _{ab} \\lambda ^a \\partial _l \\zeta ^b \\, , \\\\\\lbrace Q_{\\tilde{J}}[\\lambda ^a], Q_{\\tilde{K}}[\\lambda ] \\rbrace _D &=& 0 \\,.$ We see that the asymptotic algebra precisely corresponds to that of the CS theory for the direct product $SO(2,1) \\times U(1)$ , including the central extensions.", "A short remark regarding the gauge consistency of the boundary conditions is now appropriate.", "Obviously, the fermion cannot be excited in the asymptotic region by an $SO(2,1) \\times U(1)$ transformation (nor by diffeomorphisms or Weyl scaling, in contrast with SUSY).", "Also, as $\\tilde{J}_a$ and $\\tilde{K}$ are first class they preserve the constraint $\\varphi ^i_a = 0$ and thus our boundary conditions are gauge invariant.", "As a consequence, we see that $\\delta _{\\tilde{K}} Q_G = \\delta _{\\tilde{J}} Q_G= 0$ is consistent, i.e., valid in any gauge.", "Now, one could ask if $\\delta _G Q_{\\tilde{K}} = \\lbrace Q_{\\tilde{K}},Q_{G}\\rbrace =0$ also holds, and the same for $\\delta _G Q_{\\tilde{J}}$ .", "In order to see this, let us compute explicitly the transformations generated by the improved diffeomorphisms.", "They readNote that the improved diffeomorphism corresponds to a covariant generalization of the Lie derivative.", "$\\lbrace e^a_i, S[\\xi ^i;0] \\rbrace _D = D_i (\\xi ^j e^a_j) + \\xi ^j T_{ji} \\,, \\quad && \\lbrace \\psi , S[\\xi ^i;0] \\rbrace _D = \\xi ^i D_i\\psi \\,, \\quad \\lbrace \\bar{\\psi } , S[\\xi ^i;0] \\rbrace _D = \\xi ^i \\bar{\\psi } \\overleftarrow{D}_i \\, , \\\\\\lbrace A_i , S[\\xi ^i;0] \\rbrace _D = \\xi ^j F_{ji} \\,, && \\quad \\lbrace \\omega ^a_i, S[\\xi ^i;0] \\rbrace _D = \\xi ^j R^a_{ji} \\, .$ By virtue of () and the constraints (,), we conclude the $SO(2,1) \\times U(1)$ gauge fields are invariant under improved diffeomorphisms in the asymptotic region.", "This in turn yields $\\delta _G Q_{\\tilde{K}} = \\delta _G Q_{\\tilde{J}}=0$ , as expected.", "A far simpler argument can be repeated for the Weyl generator $\\Upsilon $ , which is consistent with the algebra (REF )." ] ]	1606.05239
[ [ "Veiled symmetry of disordered Parity-Time lattices: protected\n PT-threshold and the fate of localization" ], [ "Abstract Open, non-equilibrium systems with balanced gain and loss, known as parity-time ($\\mathcal{PT}$)-symmetric systems, exhibit properties that are absent in closed, isolated systems.", "A key property is the $\\mathcal{PT}$-symmetry breaking transition, which occurs when the gain-loss strength, a measure of the openness of the system, exceeds the intrinsic energy-scale of the system.", "We analyze the fate of this transition in disordered lattices with non-Hermitian gain and loss potentials $\\pm i\\gamma$ at reflection-symmetric sites.", "Contrary to the popular belief, we show that the $\\mathcal{PT}$-symmetric phase is protected in the presence of a correlated (periodic) disorder which leads to a positive $\\mathcal{PT}$-symmetry breaking threshold.", "We uncover a veiled symmetry of such disordered systems that is instrumental for the said protection, and show that this symmetry leads to new localization behavior across the $\\mathcal{PT}$-symmetry breaking transition.", "We elucidate the interplay between such localization and the $\\mathcal{PT}$-symmetry breaking phenomena in disordered $\\mathcal{PT}$-symmetric lattices, and support our conclusions with a beam-propagation-method analysis.", "Our theoretical predictions provide avenues for experimental realizations of $\\mathcal{PT}$-symmetric systems with engineered disorder." ], [ "Introduction", "Over the past decade, classical and quantum open systems in two categories have been intensely investigated for their non-equilibrium properties.", "The first category consists of systems that are in quasi-equilibrium and can be studied using linear response theory [1].", "The second category has systems that are far removed from equilibrium [2], making perturbative methods inapplicable.", "Open systems with balanced gain and loss, called parity-time ($\\mathcal {PT}$ )-symmetric systems, straddle the two categories.", "In the quantum context, $\\mathcal {PT}$ -symmetric systems refer to those described by a non-Hermitian Hamiltonian $H_{PT}\\ne H_{PT}^\\dagger $ that is invariant under combined parity ($\\mathcal {P}$ ) and time-reversal ($\\mathcal {T}$ ) operations and leads to a non-unitary time evolution.", "The spectrum of $H_{PT}$ is purely real when the non-Hermiticity is small and becomes complex-conjugate pairs when it exceeds a threshold set by the Hermitian part of the Hamiltonian.", "This transition is called the $\\mathcal {PT}$ -symmetry breaking transition [3].", "In the $\\mathcal {PT}$ -symmetric phase (real spectrum), the system is in a quasi-equilibrium state characterized by bounded, periodic oscillations in the system particle number.", "In the $\\mathcal {PT}$ -broken phase (complex spectrum), the system is far removed from equilibrium, and the particle number increases exponentially with time [4].", "Two decades ago, $\\mathcal {PT}$ -symmetric Hamiltonians were first studied for continuum models on an infinite line [5], [6], [7].", "The past five years, however, have made it clear that the experimentally relevant [8], [9], [10], [11], [12] ones are discrete lattice models [13], [14], [15], [16] or continuum models on a finite line [17], [18], [19].", "For a one dimensional lattice with $N$ sites, the parity operator represents reflection about the lattice center, i.e., $\\mathcal {P}_{mn}=\\delta _{m\\bar{n}}$ where $\\bar{n}=N+1-m$ is the reflection-counterpart of site $n$ .", "The time-reversal operator is given by complex conjugation, $\\mathcal {T}=*$ .", "A typical $\\mathcal {PT}$ -symmetric Hamiltonian consists of a Hermitian part $H_0$ that represents kinetic energy and a non-Hermitian part $\\Gamma $ that represents balanced gain and loss.", "The $\\mathcal {PT}$ -symmetric nature of $H_0$ itself implies that its eigenfunctions are either symmetric or antisymmetric, ensures that the odd-order perturbative corrections from the gain-loss potential $\\Gamma $ to the eigenenergies of $H_0$ vanish [20], and thus leads to a positive $\\mathcal {PT}$ -symmetry breaking threshold.", "Discrete $\\mathcal {PT}$ -symmetric lattice Hamiltonians have been realized in coupled resonators [10], [11], [12] and coupled optical waveguides with balanced gain and loss [9].", "Evanescently coupled optical waveguides are also an exceptional platform for simulating key quantum phenomena [21] including Bloch oscillations [22] and Anderson localization in one dimension due to arbitrarily weak disorder [23].", "Although initially predicted in the condensed-matter context [24], [25], [26], [27], these phenomena have been thoroughly investigated in waveguide lattices because the Maxwell wave equation, under paraxial approximation, is isomorphic to the Schrödinger equation for the wave-envelope function $\|\\psi (t)\\rangle $ [21].", "In a sharp contrast with the nature-given lattices in condensed matter systems, waveguide lattices can be fabricated with a wide range of site-to-site tunneling amplitudes and on-site potentials; local or long-ranged \"impurity\" potentials; and on-site or tunneling disorder.", "This versatility has permitted the observation of disorder-induced localization, its insensitivity to the source of the disorder, as well as the signatures of the disorder-source in Hanbury-Brown Twiss correlations in disordered waveguide lattices [28] and fibers [29].", "What is the fate of a disordered $\\mathcal {PT}$ -symmetric system?", "In general, the $\\mathcal {PT}$ -symmetric phase is fragile in the sense that an arbitrarily weak disorder reduces the symmetry-breaking threshold to zero [13], [20].", "It does so because a random disorder does not preserve the symmetries of the underlying Hamiltonian.", "A straightforward way to salvage the fragile $\\mathcal {PT}$ -symmetric phase is to require a $\\mathcal {PT}$ -symmetric disorder [30].", "However, this approach imposes highly non-local correlations on the randomness and is therefore difficult to implement, even with an engineered disorder.", "Thus questions about localization and $\\mathcal {PT}$ -symmetry breaking in a disordered $\\mathcal {PT}$ -symmetric system appear moot [31], [32].", "In this report, we show that the $\\mathcal {PT}$ -symmetric phase in a disordered system is not always fragile, and that it is protected against random tunneling or on-site potential disorder if the disorder has specific periodicities.", "We elucidate an underlying symmetry that is critical for the said protection.", "We investigate the distribution of $\\mathcal {PT}$ -breaking threshold in such disordered systems and its dependence on the nature (tunneling or on-site potential) and the distribution (Gaussian, uniform, etc.)", "of disorder.", "In Hermitian disordered systems, disorder-averaged single particle properties, such as density of states and the localization profile, do not depend upon these details.", "Here, we show that the distribution of $\\mathcal {PT}$ -symmetry breaking threshold is sensitive to those disorder attributes.", "Our results demonstrate that a disordered $\\mathcal {PT}$ -symmetric system exhibits novel properties absent in its Hermitian counterpart.", "Consider an $N$ -site tight-binding lattice with gain and loss potentials $\\pm i\\gamma $ located at parity symmetric sites $m_0\\le N/2$ and $\\bar{m}_0>N/2$ respectively; the lattice has open boundary conditions, meaning the first and the $N$ th site has only one neighbor each.", "The distance between the gain and the loss sites, $d=\\bar{m}_0-m_0$ , ranges from $N-1$ to one (two) when $N$ is even (odd).", "The non-Hermitian, $\\mathcal {PT}$ -symmetric Hamiltonian for this lattice is given by $H_{PT}=H_0+\\Gamma $ where $H_0 = & -J\\sum _{k=1}^{N-1}\\left(\|k\\rangle \\langle k+1\| +\|k+1\\rangle \\langle k\|\\right) & =H_0^\\dagger ,\\\\\\Gamma = & i\\gamma \\left( \|m_0\\rangle \\langle m_0\| -\|\\bar{m}_0\\rangle \\langle \\bar{m}_0\|\\right) &=-\\Gamma ^\\dagger .$ $J>0$ is the constant tunneling amplitude that sets the energy-scale for the Hermitian Hamiltonian and $\|k\\rangle $ is a single-particle state localized at lattice site $k$ .", "Since the Hamiltonian $H_{PT}$ commutes with the antilinear operator $\\mathcal {PT}$ , it follows that its spectrum is either purely real or consists of complex conjugate pairs [33], [34].", "The spectrum is real when $\\gamma \\le \\gamma _{PT}(m_0)$ where the $\\gamma _{PT}(m_0)$ denotes the gain-location dependent $\\mathcal {PT}$ -symmetry breaking threshold.", "When $N$ is even, the threshold is maximum when the gain and loss potentials are nearest to each other or farthest away from each other, i.e., $\\gamma _{PT}=J$ when $d=1$ and $d=N-1$ .", "When $N$ is odd, $\\gamma _{PT}\\rightarrow J/2$ when $d=2$ and $\\gamma _{PT}\\rightarrow J$ when $d=N-1$ .", "This unexpected robustness of the $\\mathcal {PT}$ -symmetry breaking threshold at the largest gain-loss distance is due to open boundary conditions [35], [36].", "In the presence of a random, uncorrelated disorder, the threshold is suppressed to zero, i.e., $\\gamma _{PT}=0$ .", "In the following subsection, we show that introducing a periodic disorder alleviates this problem.", "We consider two classes of Hermitian disorders, one in the tunneling amplitude and the second in the on-site potential, each with lattice period $p$ , $V_T & = & J\\lambda \\sum _{k=1}^{N-1} r_k \\left( \|k\\rangle \\langle k+1\| +\|k+1\\rangle \\langle k\|\\right),\\\\V_O & = & J\\Delta \\sum _{k=1}^N r_k \|k\\rangle \\langle k\|.$ The dimensionless numbers $\\lambda \\ge 0$ and $\\Delta \\ge 0$ represent the strength of tunneling and on-site disorder respectively, $\\lbrace r_1,\\ldots ,r_p\\rbrace $ are independent, identically distributed (i.i.d.)", "random numbers with zero mean and unit variance, and the periodic nature of disorder implies that $r_{k^{\\prime }}=r_k$ if $k^{\\prime }-k=0\\mod {p}$ .", "Figure REF (a)-(b) show the schematic of a disordered lattice with $N=11$ sites and gain potential $i\\gamma $ at site $m_0=3$ .", "The tunneling disorder $V_T$ has period $p=3$ , and the three independent, random tunnelings are given by $J_k=J(1+\\lambda r_k)$ .", "Figure REF (c)-(d) show an on-site-potential disordered lattice with $N=15$ sites, gain potential at site $m_0=4$ , and disorder period $p=4$ ; the four independent, random potentials are given by $V_k=J\\Delta r_k$ .", "Note that the periodic disorder potential in each case is not $\\mathcal {PT}$ -symmetric, i.e., $[\\mathcal {PT},V_{T,O}]\\ne 0$ .", "Therefore, conventional wisdom suggests that the $\\mathcal {PT}$ -symmetry breaking threshold in each case will be zero.", "Figure REF (e) shows the numerically obtained threshold $\\gamma _{PT}(m_0,p)$ for an $N=17$ lattice with tunneling disorder strength $\\lambda =1$ .", "The key features of the threshold phase diagram are as follows.", "It is nonzero only when $N+1=0\\mod {p}$ and $m_0=0\\mod {p}$ .", "Thus, when $p=2$ the threshold is nonzero only when $m_0$ is even, for $p=3$ it is nonzero for $m_0=\\lbrace 3,6\\rbrace $ , and for $p=6$ , it is nonzero only when $m_0=6$ .", "It is identically zero for periods $p=\\lbrace 4,5,7,8\\rbrace $ for any gain-site location $m_0$ .", "These results, obtained for a particular realization of the tunneling disorder, are generic.", "They show that a tunneling disorder with appropriately chosen period $p$ and gain locations $m_0$ leads to a positive $\\mathcal {PT}$ -symmetry breaking threshold with values comparable to that of a clean system, $\\gamma _{PT}\\sim J$ .", "Figure REF (f) shows the corresponding threshold results for an on-site disorder with strength $\\Delta =1$ .", "The salient features of the phase diagram are the same: $\\gamma _{PT}>0$ when $N+1=0\\mod {p}$ and $m_0=0\\mod {p}$ .", "Thus, periodicities $p=\\lbrace 2,3,6\\rbrace $ have a positive threshold for appropriate gain locations, while $\\gamma _{PT}=0$ for all other disorder periods.", "In addition, when $p=2$ (on-site, dimer disorder), the symmetry breaking threshold is nonzero for odd values of gain location as well.", "This is the only qualitative difference between the threshold results for tunneling vs. on-site disorders.", "It arises because for an odd $N$ and $p=2$ , the on-site disorder is always $\\mathcal {PT}$ -symmetric, i.e., $[\\mathcal {PT},V_O]=0$ .", "For an even lattice, both tunneling and on-site dimer disorders have $\\gamma _{PT}>0$ only when the gain potential site is even.", "Results in Figure REF (e)-(f) are surprising because they show that the symmetry breaking threshold is robust against disorders that are not parity symmetric [37].", "They hint at the existence of another (antilinear) symmetry [33], [34] that protects the threshold.", "In the next subsection, we uncover this veiled symmetry and discuss its consequences.", "The tunneling Hamiltonian of a uniform lattice can be expressed as $H_0= U D U^{\\dagger }$ where $D_{\\alpha \\beta }=\\epsilon _\\alpha \\delta _{\\alpha \\beta }$ is a diagonal matrix with eigenvalues $\\epsilon _\\alpha =-2J\\cos p_\\alpha $ , the unitary matrix with corresponding eigenfunctions has entries $U_{m\\alpha }=\\sqrt{2/(N+1)}\\sin (p_\\alpha m)$ , and $p_\\alpha =\\pi \\alpha /(N+1)$ are the quasimomenta consistent with open boundary conditions.", "The spectrum of $H_0$ is particle-hole symmetric, i.e., its eigenvalues satisfy $\\epsilon _{\\bar{\\alpha }}=-\\epsilon _\\alpha $ where $\\bar{\\alpha }=N+1-\\alpha $ .", "The eigenfunctions of $H_0$ satisfy $U_{\\bar{m}\\alpha }=(-1)^{\\alpha -1}U_{m\\alpha }$ and $U_{m\\bar{\\alpha }}=(-1)^{m-1}U_{m\\alpha }$ .", "The first equation implies that the eigenfunctions are either symmetric or antisymmetric; the second equation implies that the particle and hole eigenfunctions, i.e.", "eigenfunctions with opposite energies, are related by a staggered $\\pi $ -phase.", "For a given $H_0$ , we can generate a family of \"parity\" operators $P= USU^\\dagger $ where $S=\\mathrm {diag}(\\pm ,\\ldots ,\\pm 1)$ is a diagonal matrix with randomly chosen entries $\\pm 1$ ; there are $2^{N-1}$ such distinct operators.", "When $S=1_N$ the resultant operator is the identity and when $S_{kk^{\\prime }}=(-1)^{k-1}\\delta _{kk^{\\prime }}={\\mathcal {S}}$ , the resultant parity operator is $\\Pi =U{\\mathcal {S}}U^\\dagger =\\mathcal {P}$ .", "In the site-space representation, both matrices are sparse.", "For a random string in $S$ , the resultant \"parity\" operator is not a sparse matrix.", "This procedure is generalized to the case of a disordered Hermitian Hamiltonian $H=H_0+V(\\lambda ,\\Delta )$ and leads to a family of $2^{N-1}$ disorder-dependent operators $P(\\lambda ,\\Delta )$ .", "It is easy to show that $P(\\lambda ,\\Delta )$ is Hermitian, $P^2=1$ , $P\\mathcal {T}=\\mathcal {T}P$ , and $P(\\lambda ,\\Delta )$ commutes with the disordered Hamiltonian $H(\\lambda ,\\Delta )$ .", "Note that the special parity operator $\\Pi (\\lambda ,\\Delta )=U(\\lambda ,\\Delta ){\\tt S}U(\\lambda ,\\Delta )^\\dagger $ depends on the disorder and is not equal to the parity (reflection) operator on the lattice, i.e., $\\Pi (\\lambda ,\\Delta )\\ne \\mathcal {P}$ .", "Figure: Veiled symmetry of a disordered lattice.", "(a) For a uniform lattice, the operator Π=𝒫\\Pi =\\mathcal {P} is given by Π kk ' =δ kk ¯ \\Pi _{kk^{\\prime }}=\\delta _{k\\bar{k}}.", "Typical parity operators Π\\Pi for an N=11N=11 site lattice with different disorder strengths λ,Δ\\lambda ,\\Delta and periods pp are shown in (b)-(d).", "In each case Π kk ¯ =1\\Pi _{k\\bar{k}}=1 if and only if k=0modpk=0\\mod {p} and N+1=0modpN+1=0\\mod {p}.", "(e) The site-dependent asymmetry functions A(k)A(k) for an N=23N=23 site lattice with tunneling disorder.", "The disorder strength is λ=1\\lambda =1 and the number of disorder realizations is M=100M=100.", "The asymmetry vanishes only if the disorder period pp satisfies N+1=0modpN+1=0\\mod {p} and only on sites kk that are integer multiples of the disorder period pp.", "(f) Results for an on-site disorder with strength Δ=1\\Delta =1 show the same quantitative trend.", "This veiled symmetry of the eigenfunctions of a disordered lattice is instrumental to the positive 𝒫𝒯\\mathcal {PT}-symmetry breaking thresholds in Figure.", ".Figure REF (a)-(d) show typical features of the $\\Pi $ operator in the site basis.", "For a uniform lattice, panel (a), the $\\Pi $ operator is the same as the lattice reflection operator.", "In the presence of disorder with period $p$ , $\\Pi $ is not a sparse matrix and satisfies $\\Pi _{k\\bar{k}}=1$ if and only if both $N+1=0\\mod {p}$ and $k=0\\mod {p}$ hold true.", "These results are generic and apply for on-site potential disorder, panel (b); tunneling disorder, panel (c); or a combination of the two, panel (d).", "In all cases, $[H,\\Pi \\mathcal {T}]=0$ .", "A positive symmetry-breaking threshold, then, is possible if and only if the antilinear operator $\\Pi \\mathcal {T}$ also commutes with the gain-loss potential $\\Gamma $ , eq.().", "It is straightforward, albeit tedious, to verify that it is so only when $N+1=0\\mod {p}$ and $m_0=0\\mod {p}$ .", "An insight into the vanishing commutator, $[\\Gamma (m_0),\\Pi \\mathcal {T}]=0$ , is offered by the effect of periodic disorder on the eigenfunctions of the uniform lattice.", "When the disorder is zero, the eigenfunctions $U_{m\\alpha }$ are symmetric or antisymmetric, i.e., $U_{\\bar{m}\\alpha }=(-1)^{\\alpha -1}U_{m\\alpha }$ .", "They have equal weights on reflection-symmetric sites $m$ and $\\bar{m}$ for all $m$ .", "This property ensures that odd-order perturbative corrections to the eigenvalues $\\epsilon _\\alpha $ due to the gain-loss potential $\\pm i\\gamma $ vanish, and leads to a positive $\\mathcal {PT}$ breaking threshold [20].", "Are the eigenfunctions of the disordered Hamiltonian also reflection symmetric?", "To quantify this property, we define a site-dependent asymmetry function $A(k)=\\sum _{\\alpha =1}^N \\left\|U_{\\bar{k}\\alpha }(\\lambda ,\\Delta )+(-1)^\\alpha U_{k\\alpha }(\\lambda ,\\Delta )\\right\|.$ It follows that $A\\ge 0$ in general and for a uniform lattice, $A(k)=0$ for all $k$ .", "The asymmetry functions $A(k)$ for $M=100$ tunneling disorder realizations on an $N=23$ site lattice are shown in Figure REF (e).", "When the disorder period is $p=8$ (top panel), $A(k)=0$ only at sites $k=\\lbrace 8,16\\rbrace $ , whereas when $p=3$ (bottom panel) the function vanishes exactly when $k=0\\mod {3}$ .", "Figure REF (f) has the corresponding results for an on-site disorder.", "Once again, we see that $A(k)=0$ if and only if $k=0\\mod {p}$ .", "The asymmetry function is nonzero everywhere if either of the two constraints, $N+1=0\\mod {p}$ and $k=0\\mod {p}$ , is not satisfied.", "Results in Figure REF (e)-(f) show that the disordered eigenfunctions $U_{m\\alpha }(\\lambda ,\\Delta )$ are neither symmetric nor antisymmetric, but, when restricted to a specific set of sites, they show these symmetries [37].", "It follows that $[\\Gamma (m_0),\\Pi \\mathcal {T}]=0$ if and only if $m_0=0\\mod {p}$ and $N+1=0\\mod {p}$ .", "Thus, although the Hamiltonian $H_{PT}(\\lambda ,\\Delta )=H(\\lambda ,\\Delta )+\\Gamma $ is not $\\mathcal {PT}$ -symmetric, it is $\\Pi \\mathcal {T}$ -symmetric under these constraints.", "This veiled antilinear symmetry of the eigenfunctions of $H$ gives rise to the positive $\\mathcal {PT}$ breaking thresholds seen in Figure REF .", "Disordered models with positive $\\mathcal {PT}$ -symmetry breaking thresholds prompt a number of questions.", "How does the probability distribution function of the $\\mathcal {PT}$ -breaking threshold $PDF(\\gamma _{PT})$ depend on the strength of the disorder?", "Does it depend on the distribution of the disorder?", "Is it different for on-site and tunneling disorders?", "What is the fate of localization in $\\mathcal {PT}$ -symmetric systems?", "These questions are addressed in the following paragraphs.", "Figure REF shows the threshold distributions $PDF(\\gamma _{PT})$ in the presence of on-site potential disorder, panel (a), and tunneling disorder, panel (b).", "The results are for the $\\mathcal {PT}$ -symmetry breaking threshold at gain site $m_0=3$ in an $N=17$ lattice, obtained by using $M=5\\mathrm {x}10^4$ realizations of disorder with period $p=3$ .", "The horizontal axis in each panel is the dimensionless threshold $\\gamma _{PT}/J$ .", "Panel (a) shows that as the on-site disorder strength $\\Delta $ increases, the threshold distribution $PDF(\\gamma _{PT})$ becomes broader, and skewed towards values smaller than its clean-limit value.", "In addition, $PDF(\\gamma _{PT})$ is independent of the disorder distribution, i.e., it is the same whether the random, periodic potential is drawn from a Guassian distribution with zero mean and variance $\\Delta $ (blue open circles, yellow crosses) or a uniform distribution with the same mean and variance (green filled circles, red crosses).", "Qualitatively similar results are obtained for other lattice sizes $N$ , disorder periods $p$ , and gain potential locations $m_0$ as long as they satisfy the criteria $N+1=0\\mod {p}$ and $m_0=0\\mod {p}$ .", "These results are consistent with what we would expect.", "Introducing disorder suppresses the $\\mathcal {PT}$ -breaking threshold and the threshold distribution $PDF(\\gamma _{PT})$ - a single particle property - is independent of the underlying disorder distribution [38].", "Figure REF (b) shows that these expectations are rather simplistic.", "For a Gaussian tunneling disorder (blue open circles, yellow crosses), $PDF(\\gamma _{PT})$ is a bell shaped distribution centered about its clean-limit value.", "It becomes broader when the tunneling disorder strength $\\lambda $ is increased, and its center shifts towards the origin.", "For a uniform disorder (green filled circles, red crosses), we find that $PDF(\\gamma _{PT})$ is now a flat-top distribution approximately centered about its clean-limit value.", "We obtain qualitatively similar results for other lattice sizes, gain locations, and disorder periodicities that lead to a positive clean-limit threshold.", "These results are remarkable because for a tunneling disorder, the threshold distribution $PDF(\\gamma _{PT})$ mimics the disorder distribution and is not universal.", "Figure: 𝒫𝒯\\mathcal {PT}-symmetry breaking threshold distribution PDF(γ PT )PDF(\\gamma _{PT}) for the gain potential at site m 0 =3m_0=3, in an N=17N=17 site lattice with disorder period p=3p=3.", "(a) For an on-site potential disorder, the threshold distribution PDF(γ PT )PDF(\\gamma _{PT}) broadens as disorder strength Δ\\Delta increases and it is independent of the disorder distribution, Gaussian or uniform.", "(b) For the tunneling disorder, the threshold distribution PDF(γ PT )PDF(\\gamma _{PT}) mimics the disorder distribution, giving different results for a Gaussian disorder and the uniform disorder.", "(c) Localization in an N=39N=39 site lattice, with on-site disorder period p=10p=10, the initial state at the center of the lattice, and M=1000M=1000 disorder realizations.", "When γ=0\\gamma =0, the disorder-averaged intensity I d (k,t)I_d(k,t) shows satellite peaks at k=k 0 modpk=k_0\\mod {p} in addition to the usual peak at the initial site k 0 =20k_0=20.", "(d) when the gain-potential is turned on, γ/J=0.05\\gamma /J=0.05, intensity weight at the gain site m 0 =10m_0=10 increases with time.", "(e) Intensity profile I d (k,t)I_d(k,t) at time Jt=100Jt=100 shows that the increase in the intensity at the gain-site when γ>0\\gamma >0 (red open triangles) does not come at the expense of the intensity at other sites, but instead from the non-unitary time evolution.In one-dimensional Hermitian systems, a random disorder exponentially localizes all states.", "In transport experiments, this localization is inferred from a scaling analysis of the resistivity in the presence of disorder [26], [27].", "In optical-waveguide realizations of a Hermitian disordered lattice, it is manifest by a disorder-averaged intensity profile that, after an initial ballistic expansion, develops a steady-state value [21], [23], [28].", "For an initial state on site $k_0$ , the disorder-averaged intensity profile $I_d(k,t)=\|\\langle k\|\\psi (t)\\rangle \|^2_d$ is symmetrically and exponentially localized around that site.", "Here the subscript $d$ denotes averaging over different disorder realizations, $\|\\psi (t)\\rangle =G(t)\|\\psi (0)\\rangle $ , and the time-evolution operator is $G(t)=\\exp (-iHt)$ where we have used $\\hbar =1$ .", "In the Hermitian case, $\\gamma =0$ , the time-evolution operator is unitary, i.e., $G(t)^\\dagger G(t)=1$ .", "Therefore, the total disorder-averaged intensity is constant at each time, $\\sum _k I_d(k,t)=1$ .", "In the $\\mathcal {PT}$ -symmetric disordered case, there are two distinct scenarios.", "If the gain potential strength is smaller than the minimum threshold value, i.e., $\\gamma <\\gamma _{\\min }=\\min _{\\gamma }\\lbrace PDF(\\gamma _{PT})>0\\rbrace $ , the system is in the $\\mathcal {PT}$ -symmetric phase for each disorder realization.", "Therefore, its non-unitary time evolution has bounded intensity oscillations and at long times $Jt\\gg 1$ , it leads to a quasi steady-state intensity profile $I_d(k)$ with constant total intensity $\\sum _k I_d(k)>1$ [4], [30].", "When $\\gamma >\\gamma _{\\min }$ , for a fraction of the $M\\gg 1$ disorder realizations, the system is in the $\\mathcal {PT}$ -broken phase where the total intensity increases exponentially with time as does the intensity in the neighborhood of the gain site $m_0$ .", "As a result, the disorder-averaged intensity $I_d(k,t)$ develops a peak at the gain site $m_0$ whose weight increases with time.", "We note that in this regime, the intensity $I_d(k,t)$ does not reach a steady state value [4], [31], [32].", "Figure REF (c)-(e) encapsulates the effects of correlated disorder on the disorder-averaged site- and time-dependent intensity $I_d(k,t)$ .", "The results are for an $N=39$ site lattice with on-site potential disorder with periodicity $p=10$ , number of disorder realizations $M=10^3$ , and an initial state localized at the center of the lattice, $k_0=20$ .", "Panel (c) shows the disorder-averaged intensity $I_d(k,t)$ for the Hermitian case, $\\gamma =0$ .", "A periodic disorder leads to a steady-state profile $I_d(k)$ that is exponentially localized about the initial site $k_0=20$ , along with satellite peaks at sites $k=k_0\\mod {p}=\\lbrace 10,30\\rbrace $ .", "These satellite peaks are signatures of extended states that exist in one-dimensional systems with periodic disorder [39], [40].", "As the disorder strength $\\Delta $ is increased, the peak intensity of the satellites decreases.", "We remind the reader that when the disorder is purely random, and not periodic, these satellite peaks are absent.", "Panel (d) shows corresponding results for a disordered $\\mathcal {PT}$ -symmetric system with gain potential of strength $\\gamma /J=0.05$ at site $m_0=p=10$ (red filled circle); the corresponding loss potential $-i\\gamma $ at site $\\bar{m}_0=30$ is also shown (blue filled circle).", "We see that in addition to the hermitian localization peaks at sites $k=k_0\\mod {p}$ , a new peak emerges at the gain location.", "It arises because a disordered system with $\\gamma /J=0.05$ is, sometimes, in the broken $\\mathcal {PT}$ -symmetric phase.", "Panel (e) shows the disorder-averaged site-intensity profile $I_d(k,t)$ at time $Jt=100$ .", "In the Hermitian case, it shows localization peaks at the initial site $k_0=20$ and satellite peaks at sites $k=k_0\\mod {p}=\\lbrace 10,30\\rbrace $ (blue filled circles).", "In the $\\mathcal {PT}$ -symmetric case, the intensity values are essentially unchanged except in the vicinity of the gain site, where the it has increased by a factor of five (red open triangles).", "This interplay between the disorder-induced localization and the broken $\\mathcal {PT}$ -symmetry induced localization occurs even if there is no disorder-induced peak at $m_0$ in the Hermitian limit.", "The results for a finite $\\mathcal {PT}$ -symmetry breaking threshold in disordered lattices presented in Figure REF are based on a tight-binding approximation.", "In the experimental realizations of such lattices, however, a \"site\" has a transverse spatial extent, and the tunneling Hamiltonian in Eq.", "(REF ) represents a site-discretized version of the spatial second derivative in the continuum Schrödinger (or Maxwell) equation.", "Therefore, to test that our predictions are not artifacts of the lattice approximation, we obtain the time-evolution of the wave function $\\psi (x,t)$ in a waveguide array with realistic parameters [41] via the beam propagation method (BPM) [42], [43].", "The continuum Schödinger equation is given by $i\\partial _t\\psi =-\\partial ^2_x\\psi /2m +V(x)\\psi $ .", "Here, the effective mass is $m=k_0n_0^2/c$ , the potential is given by $V(x)=ck_0\\left[ 1- n(x)^2/n_0^2\\right]$ , $n_0$ is the cladding index of refraction, $c$ is the speed of light in vacuum, $n(x)$ is the position-dependent index of refraction in the waveguide array, and $k_0=2\\pi /\\lambda $ is the wave number of the rapidly varying part of the electric field $E(x,z,t)=\\exp \\left[ik_0z-(ck_0/n_0)t\\right]\\psi (x,t)$ which satisfies the Maxwell equation.", "Figure: BPM simulations of wave packet propagation in an N=8N=8 waveguide lattice in the presence of on-site disorder with period p=3p=3.", "The system parameters are λ\\lambda =633 nm, cladding index n 0 =1.45n_0=1.45, waveguide width W=10W=10 μ\\mu m, and uniform waveguide separation d=16.9586d=16.9586 μ\\mu m. The bar-chart at the top of each panel shows a random, periodic index-contrast distribution.", "The vertical scale in each bar-chart denotes the index-contrast Δn\\Delta n and ranges from 4.8x10 -4 4.8\\mathrm {x}10^{-4} to 5.2x10 -4 5.2\\mathrm {x}10^{-4}.", "(a) For a gain potential with strength γ=0.7\\gamma =0.7 cm -1 ^{-1} on the first site, the intensity I(x,z)I(x,z) shows a 𝒫𝒯\\mathcal {PT}-symmetry broken state.", "(b) With the same gain potential on the second site, the system is again in the 𝒫𝒯\\mathcal {PT}-broken phase.", "(c) With the same gain-potential on site m 0 =3=pm_0=3=p, the system is in the 𝒫𝒯\\mathcal {PT}-symmetric phase, as is also shown by site intensities that are one to two orders of magnitude smaller.", "(d) When the gain potential at site m 0 =3m_0=3 is doubled to γ=1.4\\gamma =1.4 cm -1 ^{-1}, the 𝒫𝒯\\mathcal {PT}-symmetry is broken.", "Note that the index-contrast profiles in (c) and (d) are the same.", "The BPM analysis confirms the predictions for a nonzero 𝒫𝒯\\mathcal {PT}-threshold in the presence of random, periodic disorder.The index of refraction $n(x)$ is different from that of the cladding only within each waveguide.", "In the limit of small contrast, $n(x)=n_0+\\Delta n$ with $\\Delta n/n_0\\sim 10^{-4}\\ll 1$ , the potential term becomes linearly proportional to the index contrast, i.e., $V(x)=2c k_0\\Delta n/n_0$ , and we implement the gain and loss potentials by adding appropriate imaginary parts to the index contrast.", "Thus, in the continuum model, a correlated on-site disorder means a random, periodic index-contrast, whereas the tunneling disorder is implemented by random, periodic waveguide separations [28], [41].", "Figure REF shows representative results of such simulations for an $N=8$ waveguide-lattice in the presence of an on-site disorder with period $p=3$ .", "The initial state, marked by a white semicircle, is a normalized Gaussian with width $\\sigma =W/2$ in the center waveguide, where $W$ is the width of each waveguide.", "Each panel shows the time- and space-dependent intensity $I(x,z=ct/n_0)$ where we have switched to the distance along the waveguide $z=ct/n_0$ as a measure of time for an easier comparison with experiments.", "The bar-chart at the top of each panel shows a randomly generated index contrast $\\Delta n(x)$ with period $p=3$ .", "The gain-potential waveguide is shown by a red bar, the reflection-symmetric lossy waveguide is shown by a blue bar, and the linear scale on the vertical axis in each bar-chart ranges from $\\Delta n=4.8\\mathrm {x}10^{-4}$ to $\\Delta n=5.2\\mathrm {x}10^{-4}$ .", "The intensity plot $I(x,z)$ in Figure REF (a) is for a gain potential $\\gamma =0.7$ cm$^{-1}$ in the first waveguide.", "It shows that at long times, $z\\ge 10$ cm, the intensity is largely confined to the gain waveguide and the system is in the broken $\\mathcal {PT}$ -symmetry phase.", "Panel (b) has the intensity plot with the same gain potential, $\\gamma =0.7$ cm$^{-1}$ , in the second waveguide; it also shows intensity localized in the gain waveguide and thus indicates that the system is in the $\\mathcal {PT}$ -broken phase.", "In each case, we note that the maximum intensity $I(x,z)$ is larger than the average intensity $I\\sim 1/N=0.125$ expected in each waveguide in the Hermitian limit.", "Panel (c) shows $I(x,z)$ with the same gain potential strength, $\\gamma =0.7$ cm$^{-1}$ , in the third waveguide.", "It is clear from the intensity plot that the system is in the $\\mathcal {PT}$ -symmetric phase.", "Note that the gain location $m_0=p=3$ satisfies $N+1=0\\mod {p}$ .", "Panel (d) shows that when the gain potential is doubled, $\\gamma =1.4$ cm$^{-1}$ , the system enters a $\\mathcal {PT}$ -broken phase and the resultant intensity is localized largely in the gain waveguide.", "The results presented in Figure REF are generic and demonstrate that our findings of a nonzero $\\mathcal {PT}$ -threshold in disordered lattices are robust (Figure REF ), as are our predictions for the veiled symmetry of their eigenfunctions (Figure REF ).", "In this paper we have introduced non-Hermitian lattice models with balanced gain and loss that are robust against random, periodic disorder.", "We have uncovered a veiled symmetry that is exhibited by eigenfunctions of such disordered, Hermitian lattices.", "Since this symmetry is phase-sensitive, it ensures equal weights at specific reflections-symmetric sites, but not equal wave functions [37].", "Therefore, any phase-insensitive observable will reflect the signatures of this symmetry.", "Experimentally, the models studied here can be realized in coupled waveguide arrays with one gain waveguide and one lossy waveguide.", "Ideally, if the on-site potentials or tunneling amplitudes are tunable - for example, via voltage-controlled top-gate heaters - it will permit experimental investigations of interplay between localization due to a periodic disorder and the $\\mathcal {PT}$ -symmetry breaking transition.", "Mathematically, the lattice models considered here correspond to tridiagonal matrices with Hermitian, random, periodic entries, in addition to non-Hermitian, fixed, gain-loss potential entries along the main diagonal.", "The statistical properties of eigenvalues of such matrices are essentially unexplored.", "In particular, the dependence of the threshold distribution $PDF(\\gamma _{PT})$ on the source and the distribution of disorder is, at this point, poorly understood.", "A generalization of these models to non-sparse matrices with a positive $\\mathcal {PT}$ -symmetry breaking threshold, if one were possible, will provide an approach to investigate the spectral properties random, $\\mathcal {PT}$ -symmetric matrices with real spectra.", "We thank Tony Lee and Ricardo Decca for insightful comments.", "This work was supported by NSF Grant no.", "DMR-1054020.", "Y.J.", "conceived the project.", "A.H. and F.O.", "carried out numerical calculations.", "All authors reviewed the manuscript.", "Competing financial interests The authors declare no competing financial interests." ] ]	1606.04964
[ [ "A combinatorial construction of an M_{12}-invariant code" ], [ "Abstract In this work we summarized some recent results to be included in a forthcoming paper.", "A ternary [66,10,36]_3-code admitting the Mathieu group M_{12} as a group of automorphisms has recently been constructed by N. Pace.", "We give a construction of the Pace code in terms of $M_{12}$ as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5,6,12).", "We also present a proof that the Pace code does indeed have minimum distance 36." ], [ "Introduction", "A large number of important mathematical objects are related to the Mathieu groups.", "It came as a surprise when N. Pace found yet another such exceptional object, a $[66,10,36]_3$ -code whose group of automorphisms is $Z_2\\times M_{12}$ (see [3]).", "We present here two constructions for this code, an algebraic construction which starts from the group $M_{12}$ in its natural action as a group of permutations on 12 letters, and a combinatorial construction in terms of the Witt design $S(5,6,12).$ We also prove that the code has parameters as claimed.", "In the next section we start by recalling some of the basic properties of $M_{12}$ and the small Witt design $S(5,6,12).$" ], [ "The ternary Golay code, $M_{12}$ and {{formula:b67a05ca-54e5-42c9-9183-2c3bb0487ed9}}", "The Mathieu group $M_{12}$ is sharply 5-transitive on 12 letters and therefore has order $12\\times 11\\times 10\\times 9\\times 8.$ It is best understood in terms of the ternary Golay code $[12,6,6]_3.$ The ternary Golay code has a generator matrix $(I\\vert P)$ where $I$ is the $(6,6)$ -unit matrix and $P=\\left(\\begin{array}{cccccc}0 & 1 & 1 & 1 & 1 & 1 \\\\1 & 0 & 1 & 1 & 2 & 2 \\\\1 & 1 & 0 & 2 & 1 & 2 \\\\1 & 1 & 2 & 0 & 2 & 1 \\\\1 & 2 & 1 & 2 & 0 & 1 \\\\1 & 2 & 2 & 1 & 1 & 0\\end{array}\\right).$ The group $M_{12}$ acts in terms of monomial operations on the ternary Golay code.", "Here we identify the 12 letters with the columns of the generator matrix and consider the action of $M_{12}$ as a group of permutations on those 12 letters $\\lbrace 1,2,\\dots ,12\\rbrace .$ It is generated by $h_1,h_2,h_3,h_4$ and $g$ where $h_1=(2,3,5,6,4)(8,9,11,12,10), h_2=(2,3)(4,5)(8,9)(10,11),$ $ h_3=(3,5,4,6)(9,11,10,12), h_4=(1,2)(5,6)(7,8)(11,12),$ $ g=(5,12)(6,11)(7,8)(9,10).$ The group $H=\\langle h_1,h_2,h_3,h_4\\rangle $ of order 120 is the stabilizer of $\\lbrace 1,2,3,4,5,6\\rbrace .$ Call a 6-set an information set if the corresponding submatrix is invertible, call it a block if the submatrix has rank $5.$ The terminology derives from the fact that the blocks define a Steiner system $S(5,6,12),$ the small Witt design.", "There are 132 blocks and $12\\times 11\\times 6$ information sets.", "The complement of a block is a block as well.", "The stabilizer of each 5-set is $S_5,$ the stabilizer of a block has order $10\\times 9\\times 8=720=6!$ and the stabilizer of an information set has order $5!$ The stabilizer of a 2-set has order $1440.$ This stabilizer is the group $P\\Gamma L(2,9)\\cong Aut(A_6).$ In the sequel we identify the 12 letters with a basis $\\lbrace v_1,\\dots ,v_{12}\\rbrace $ of a vector space $V=V(12,3)$ over the field with three elements and consider the corresponding action of $M_{12}$ on $V.$" ], [ "The 10-dimensional module of $M_{12}$", "Clearly $M_{12}$ acts on an 11-dimensional submodule of $V,$ the augmentation ideal $I=\\lbrace \\sum _{i=1}^{12} a_iv_i\\vert \\sum a_i=0\\rbrace $ and on a 1-dimensional submodule generated by the diagonal $\\Delta =v_1+\\dots +v_{12}.$ As we are in characteristic 3, we have $\\Delta \\in I,$ and $M_{12}$ acts on the 10-dimensional factor space $Z=I/\\langle \\Delta \\rangle .$ The $u_i=v_i-v_{12}, i\\le 11$ are a basis of $I$ and $z_i=\\overline{u_i}=u_i+\\Delta \\mathbb {F}_3, i\\le 10$ are a basis of $Z.$ Here $\\sum _{i=1}^{11} u_i=\\Delta ,$ hence $z_{11}=-z_1-\\dots -z_{10}.$" ], [ "The Pace code", "We consider the action of $M_{12}$ on the 10-dimensional $\\mathbb {F}_3$ -vector space $Z$ with its basis $z_i=\\overline{u_i}=v_i-v_{12}+\\Delta \\mathbb {F}_3, i=1,\\dots ,10.$ Recall that it is induced by the permutation representation on $\\lbrace v_1,\\dots ,v_{12}\\rbrace .$ This action defines embeddings of $M_{12}$ in $GL(10,3)$ and in $PGL(10,3).$ For each orbit of $M_{12}$ we consider the projective ternary code whose generator matrix has as columns representatives of the projective points constituting the orbit.", "Definition 1 Let $X\\subset \\lbrace 1,2,\\dots ,12\\rbrace , \\vert X\\vert =6.$ Define $v_X=\\sum _{i\\in X}v_i, z_X=\\overline{v_X}.$ It is in fact clear that $v_X\\in I,$ and $z_X\\in Z$ is therefore defined.", "Proposition 2 The $z_X\\in Z$ where $X$ varies over the blocks of $S(5,6,12)$ form an orbit of length 132 in $Z.$ In the action on projective points (in $PG(9,3)$ ), this yields an orbit of length $66.$ Clearly $M_{12}$ permutes the $z_X$ in the same way as it permutes the blocks $X.$ This yields an orbit of length 132 in $Z=V(10,3).$ If $\\overline{X}$ is the complement of $X,$ then $v_{\\overline{X}}+v_X=\\Delta ,$ hence $z_{\\overline{X}}=-z_X.$ It follows that $M_{12}$ acts transitively on the 66 points in $PG(9,3)$ generated by the $z_X$ (block $X$ and its complement generating the same projective point).", "Definition 3 Let $C$ be the $[66,10]_3$ -code whose generator matrix has as columns representatives of the orbit of $M_{12}$ on the $z_X$ where $X$ is a block.", "This is one way of representing the Pace code.", "Observe that each complementary pair of blocks contributes one column of the generator matrix.", "We may use as representatives the vectors $z_X$ where $X$ varies over the 66 blocks $X$ not containing the letter $12.$ As the stabilizer of a block in $M_{12}$ is $S_6$ it follows that the stabilizer of a point in the orbit equals the stabilizer of a complementary pair of blocks and is twice as large as $S_6.$ The stabilizer is $P\\Gamma L(2,9),$ of order $2\\times 6!$" ], [ "A combinatorial description", "We introduce some notation.", "Definition 4 Let ${\\cal B}$ be a family of subsets (blocks) of a $v$ -element set $\\Omega .$ Let $A,B\\subset \\Omega $ be disjoint subsets, $\\vert A\\vert =a, \\vert B\\vert =b.$ Define a matrix $G$ with $k=v-a-b$ rows and $n$ columns where $n$ is the number of blocks disjoint from $A.$ Here we identify the rows of $G$ with the points $i\\in \\Omega \\setminus (A\\cup B)$ and the columns with the blocks $X$ disjoint from $A.$ The entry in row $i$ and column $X$ is $=1$ if $i\\in X,$ it is $=0$ otherwise.", "As the entries of $G$ are $0,1$ we can consider them as elements of an arbitrary finite field $K.$ Define ${\\cal C}=C_{A,B}({\\cal B},K)$ to be the code generated by $G$ over $K.$ In words: the column of $G$ indexed by $X\\in {\\cal B}$ is the characteristic function of the set $X\\setminus B.$ We write $C_{a,b}({\\cal B},K)$ instead if the choice of the subsets $A,B$ does not matter.", "This is the case in particular if the automorphism group of ${\\cal B}$ is $(a+b)$ -transitive.", "Code ${\\cal C}$ is a $K$ -linear code of length $n.$ Its designed dimension is $k$ but the true dimension may be smaller.", "We have no clue what the minimum distance is.", "Observe that $C_{A,B}({\\cal B},K)$ is a subcode of $C_{A,\\emptyset }({\\cal B},K):$ a generator matrix of the smaller code arises from the generator matrix of the larger code by omitting some $\\vert B\\vert $ rows.", "Proposition 5 The Pace code from Definition REF is monomially equivalent to $C_{1,1}(S(5,6,12),\\mathbb {F}_3).$ The generator matrix of Definition REF has rows indexed by $i\\in \\lbrace 1,\\dots ,10\\rbrace $ and columns indexed by blocks $X$ of $S(5,6,12)$ not containing the letter $12.$ If also $11\\notin X,$ then the corresponding column is the characteristic function of $X.$ Let $11\\in X.$ As $z_{11}=-z_1-\\dots -z_{10}$ the entries in this column are $=0$ if $i\\in X, =2$ if $i\\notin X.$ Taking the negative of this column, we obtain the characteristic function of $\\overline{X}\\setminus \\lbrace 12\\rbrace .$ We arrive at the generator matrix of $C_{A,B}(S(5,6,12),\\mathbb {F}_3)$ where $A=\\lbrace 11\\rbrace , B=\\lbrace 12\\rbrace .$" ], [ "Combinatorial properties of the small Witt design", "The following elementary properties of the Steiner system $S(5,6,12)$ will be used in the sequel.", "Lemma 6 Let $\\Omega =\\lbrace 1,2,\\dots ,12\\rbrace $ and $A,B\\subset \\Omega , \\vert A\\vert =a, \\vert B\\vert =b$ and such that $A\\cap B=\\emptyset ,a+b\\le 5.$ Let $i(a,b)$ be the number of blocks which contain $A$ and are disjoint from $B.$ Then $i(b,a)=i(a,b)$ and $i(5,0)=1, i(4,0)=4, i(3,0)=12, i(2,0)=30, i(1,0)=66,$ $i(1,1)=36, i(2,1)=18, i(3,1)=8, i(4,1)=3, i(2,2)=10, i(3,2)=5.$ $i(5,0)=1$ is the definition of a Steiner 5-design, $i(b,a)=i(a,b)$ follows from the fact that the complements of blocks are blocks.", "The rest follows from obvious counting arguments.", "The following combinatorial lemmas may be verified by direct calculations using coordinates.", "Lemma 7 A family of five 3-subsets of a 6-set contains at least two 3-subsets which meet in 2 points.", "Lemma 8 Let $U\\subset \\lbrace 1,2,\\dots , 11\\rbrace $ such that $\\vert U\\vert =6.$ The number of blocks $B\\in {\\cal B}$ such that $\\vert B\\cap U\\vert =3$ is 20 if $U$ is a block, it is 30 if $U$ is not a block.", "Lemma 9 Let $\\Omega =\\lbrace 1,2,\\dots ,12\\rbrace $ and $\\Omega =A\\cup B\\cup C$ where $\\vert A\\vert =\\vert B\\vert =\\vert C\\vert =4$ and $P\\in C.$ The number of blocks which meet each of $A,B,C$ in cardinality 2 and avoid $P$ is at most $18.$" ], [ "The parameters of the Pace code", "Theorem 10 The Pace code is a self-orthogonal $[66,10,36]_3$ -code.", "In the remainder of this section we prove Theorem REF .", "We use the Pace code in the form $C=C_{A,B}(S(5,6,12),\\mathbb {F}_3)$ where $A=\\lbrace 12\\rbrace , B=\\lbrace 11\\rbrace ,$ see Definition REF .", "The length is $n=i(0,1)=66,$ the designed dimension is $k=10.$ Let ${\\cal B}$ be the blocks of $S(5,6,12)$ not containing $12.$ Observe that the columns of $G$ are the characteristic functions of $X\\setminus \\lbrace 11\\rbrace $ where $X\\in {\\cal B}.$ Let $r_i, 1\\le i\\le 10$ be the rows of the generator matrix of Definition REF .", "The codewords of $C$ have the form $\\sum _{i\\in U}r_i-\\sum _{j\\in V}r_j,$ where $U,V$ are disjoint subsets of $\\lbrace 1,\\dots ,10\\rbrace .$ The number of zeroes of this codeword is the nullity $\\nu (U,V),$ the number of blocks $X\\in {\\cal B}$ satisfying the condition that $\\vert X\\cap U\\vert $ and $\\vert X\\cap V\\vert $ have the same congruence mod $3.$ Let $c\\in \\lbrace 0,1,2\\rbrace $ be this congruence.", "We need to show that $\\nu (U,V)\\le 30$ for all $(U,V)$ except when $U=V=\\emptyset .$ This will prove the claim that the nonzero weights are $\\ge 36$ and also that the dimension is $10.$ Let $u=\\vert U\\vert , v=\\vert V\\vert ,$ let $W$ be the complement of $U\\cup V$ in $\\lbrace 1,\\dots ,11\\rbrace , w=\\vert W\\vert .$ Observe $u+v+w=11, w>0.$ We have $\\nu (U,V)=\\sum _ck_c(u,v,w),$ where $k_c(u,v,w)$ is the number of $X\\in {\\cal B}$ meeting each of $U,V,W$ in a cardinality congruent to $c$ mod 3.", "Observe that $k_c(u,v,w)$ is symmetric in its arguments as long as the side condition $w>0$ is satisfied.", "The weight of $r_i$ is $i(1,1)=36$ (this is case $u=1,v=0$ ).", "In particular $r_i\\cdot r_i=0.$ Also $r_i\\cdot r_j=0$ for $i\\ne j$ as $i(2,1)=18$ is a multiple of $3.$ It follows that $C$ is self-orthogonal.", "All codeword weights and nullities are therefore multiples of $3.$ It may be verified that $\\nu (U,V)<33$ in a case by case analysis, starting from large values of $u.$ If $u=10$ then $v=0,w=1$ and $\\nu (10,0)=k_0(10,0,1)=i(0,2)=30.$ Cases $u\\in \\lbrace 6,7,8,9\\rbrace $ are similar.", "Let $u=5.$ By symmetry it can be assumed $3\\le v\\le 5.$ In case $v=5$ we have $k_1(5,5,1)\\le 10, k_0(5,5,1)\\le 20,$ and in case $v=4$ we have $k_2(5,4,2)\\le 12, k_1(5,4,2)\\le 2+10=12, k_0(5,4,2)\\le 8,$ hence $\\nu (5,4)<33.$ As $k_2(5,3,3)\\le 18, k_1(5,3,3)\\le 9$ and $k_0(5,3,3)\\le 1+2\\times 2$ we have $\\nu (5,3)<33.$ The final case to consider is $(u,v,w)=(4,4,3).$ In case $c=0$ we have that $X$ meets two of the subsets $U,V,W$ in cardinality $3.$ If $W\\subset X,$ there are at most two such blocks.", "There are at most four blocks meeting each of $U,V$ in cardinality $3.$ It follows $k_0(4,4,3)\\le 6.$ If $c=1,$ then either $U\\subset X$ or $V\\subset X.$ It follows $k_1(4,4,3)\\le 6.$ The most difficult case is $c=2.$ Lemma REF states $k_2(4,4,3)\\le 18.$ We are done." ] ]	1606.04857
[ [ "Realistic model for a fifth force explaining anomaly in ${^8Be^} \\to\n {^8Be} \\;{e^+e^-}$ Decay" ], [ "Abstract A $6.8\\,\\sigma$ anomaly has been reported in the opening angle and invariant mass distributions of $e^+e^-$ pairs produced in ${^8Be}$ nuclear transitions.", "It has been shown that a protophobic fifth force mediated by a $17\\,\\textrm{MeV}$ gauge boson $X$ with pure vector current interactions can explain the data through the decay of an excited state to the ground state, ${^8Be^} \\to {^8Be}\\, X$, and then the followed saturating decay $X \\to e^+e^-$.", "In this work we propose a renormalizable model to realize this fifth force.", "Although axial-vector current interactions also exist in our model, their contributions cancel out in the iso-scalar interaction for ${^8Be^} \\to {^8Be} \\,X$.", "Within the allowed parameter space, this model can alleviate the $(g-2)_\\mu$ anomaly problem and can be probed by the LHCb experiment.", "Several other implications are discussed." ], [ "Introduction", "Recently, studies of decays of an excited state of ${^8Be}$ to its ground state have found a $6.8\\,\\sigma $ anomaly in the opening angle and invariant mass distribution of $e^+e^-$ pairs produced in these transitions [1].", "The discrepancy from expectations may be explained by unknown nuclear reactions or unidentified experimental effects, the observed distribution fits well by postulating the existence of a fifth force mediated by a new boson $X$ that is produced on-shell in ${^8Be^} \\rightarrow {^8Be}\\; X$ and decays promptly via $X\\rightarrow e^+e^-$ .", "The authors of Ref.", "[1] have simulated this process, including the detector energy resolution, which broadens the $m_{ee}$ peak significantly.", "They find that the $X$ boson mass should be $m_X^{} = 16.7\\pm 0.35 (\\textrm {stat})\\pm 0.5 (\\textrm {sys})$ MeV.", "It has been argued that the $X$ boson is likely a vector boson which couples non-chirally to the SM fermions [2], $L &=& -{1\\over 4} X_{\\mu \\nu } X^{\\mu \\nu } + {1\\over 2} m^2_X X_\\mu X^\\mu -X_\\mu J^\\mu _X\\nonumber \\\\&&\\textrm {with}~~J_\\mu ^{}=\\sum _{f=u,d,e,\\nu _e^{},...}^{}e\\varepsilon _f^{v}J_\\mu ^f=\\sum _{f=u,d,e,\\nu _e^{},...}^{}e\\varepsilon _f^{v}\\bar{f}\\gamma _\\mu ^{} f\\,.$ Here the superscript “$v$ \" on $\\varepsilon _f^{v}$ indicates the vector current coupling nature.", "To explain the experimental data, the couplings $\\varepsilon _f^v$ are determined from the following considerations.", "Assuming ${^8Be^} \\rightarrow {^8B_e}\\;X$ followed by $X\\rightarrow e^+ e^-$ saturating $X$ decay, one obtains [2] $\|\\varepsilon ^v_p + \\varepsilon ^v_n \|\\approx 0.011\\,,~~\|\\varepsilon _e^{v}\|\\gtrsim 1.3\\times 10^{-5}\\,.$ Here the fact that the interaction matrix element of $X$ with ${^8Be}$ and ${^8Be^}$ is iso-scalar interaction has been taken into account which implies that the interaction is proportional to $\\varepsilon _p^v + \\varepsilon _n^v$ .", "Note that $\\varepsilon _p^v = 2 \\varepsilon _u^v + \\varepsilon _d^v$ and $\\varepsilon _n^v = \\varepsilon _u^v + 2\\varepsilon _d^v$ .", "The parameters are also constrained from other experimental data.", "An important one comes from $\\pi ^0 \\rightarrow X \\gamma $ where the decay width is proportional to $N_\\pi =(Q_u \\varepsilon ^v_u - Q_d \\varepsilon ^v_d)^2$ resulting from a calculation similar to anomaly induced $\\pi ^0 \\rightarrow \\gamma \\gamma $ .", "Saturating the experimental limit $N_\\pi = (2\\varepsilon ^v_u +\\varepsilon ^v_d)/9= \\varepsilon ^v_p/9 < \\varepsilon ^2_{max}/9$ with $\\varepsilon _{max} = 8\\times 10^{-4}$ [3], one gets [2] $-0.067 < {\\varepsilon ^v_p\\over \\varepsilon ^v_n} < 0.078\\;.$ Therefore the coupling $\\varepsilon _p^v$ is suppressed compared with $\\varepsilon ^v_n$ .", "It has been suggested in Ref.", "[2] that the interaction might be protophobic with $\\varepsilon _p^v = 0$ .", "In this case, $&&\\varepsilon ^v_u = \\pm 3.7\\times 10^{-3}\\;,\\;\\;\\;\\;\\varepsilon ^v_d = \\mp 7.4\\times 10^{-3}\\;.", "$ Requiring $\\varepsilon ^v_e$ to satisfy the lower bound from SLAC E141 experiment[4], the stringent constraint from electron anomalous magnetic dipole moment $(g-2)_e$ [5], and also the precision studies of $\\bar{\\nu }_e-e$ scattering from TEXONO [6], one yields [2] $&&2 \\times 10^{-4} < \|\\varepsilon ^v_e \| < 1.4\\times 10^{-3}\\;,\\;\\;\\;\\;\|\\varepsilon ^v_e \\varepsilon ^v_{\\nu _e}\|^{1/2}< 7\\times 10^{-5}\\;.", "$ In general the $X$ boson may also have axial-vector current couplings to the SM fermions, i.e.", "$e \\varepsilon ^a_f \\bar{f} \\gamma _\\mu \\gamma _5f\\,.$ However, the $X$ boson is only allowed to give an extremely tiny contribution to the decay width of $\\pi ^0_{} \\rightarrow e^{+}_{}e^{-}_{}$ .", "This sentences the case that the $X$ boson has a sizable axial-vector current interaction with both the electron and the first-generation quarks.", "Alternatively, the $X$ boson can be allowed to sizably couple to the axial-vector current of either the electron or the first-generation quarks.", "The $X$ boson interactions discussed above are based on an effective theory approach.", "It would be interesting to have the $X$ boson be part of a consistent theory respecting the standard model (SM) symmetry $SU(3)^{}_C \\times SU(2)^{}_L \\times U(1)^{}_Y$ .", "For this purpose, one must consider more constraints from both theoretical and experimental constraints which make the task non-trivial.", "In this letter we show the first successful realization of this goal.", "Specifically we consider new gauge symmetries $U(1)_{Y^{\\prime }}$ and $U(1)_X$ in addition to the SM gauge group.", "The $U(1)_{Y^{\\prime }}$ gauges certain variations of generation number difference without beyond the SM fermions.", "The $X$ boson is just the $U(1)_X$ gauge boson and couples to the SM fermions through the $U(1)_{Y^{\\prime }}$ and $U(1)_X$ kinetic mixing.", "In our model, the $X$ boson has both vector and axial-vector current couplings.", "The vector current interactions are protophobic, while the axial-vector currents are not protophobic but have no contributions to the iso-scalar interaction for ${^8Be^} \\rightarrow {^8Be}\\; X$ .", "Within the allowed parameter space, this model can alleviate the $(g-2)_\\mu $ anomaly problem and can be tested by the LHCb experiment." ], [ "A realistic model", "From theoretical side, the SM fermions appear in form of chiral fields, implying that introduction of new gauge boson interaction may generate gauge anomalies which is not allowed.", "We here consider to construct a model free of gauge anomaly by using the anomaly cancellation among different generations, similar to the gauge anomaly free model for $L_i - L_j$ in the literature [7].", "The appearance of chiral fields in general makes the $X$ boson interaction not purely vector current type which may lead to complications and need to be carefully treated.", "Also the $X$ boson may interact with different generations in general, there are more constraints from data.", "It is remarkable that our model can explain all of the data well.", "We provide the details in the following.", "The key to our construction is to have a protophobic vector current first and then accommodate the constraints from ${^8Be^}\\rightarrow {^8Be} X$ and $\\pi ^0 \\rightarrow X \\gamma $ .", "To achieve this we introduce a $U(1)_{Y^{\\prime }}$ gauge symmetry whose vector current is protophobic.", "The assignment of quantum numbers for the three generations of fermions, under the $SU(3)_C\\times SU(2)_L\\times U(1)_Y\\times U(1)_{Y^{\\prime }}$ , are as the following $&&Q^1_L: (3,\\;2,\\;1/6)(-1)\\;,\\;\\;\\;\\;\\;\\;u^1_R: (3,\\;1,\\;2/3)(5)\\;,\\;\\;\\;\\;\\;\\;\\;\\;d^1_R: (3,\\;1,\\;-1/3)(-7)\\;,\\nonumber \\\\&&L^1_L: (1,\\;2,\\;-1/2)(\\beta )\\;,\\;\\;\\;\\;\\;\\;\\;e^1_R: (1,\\;1,\\;-1)(\\beta )\\;,\\nonumber \\\\&&Q^2_L: (3,\\;2,\\;1/6)(1)\\;,\\;\\;\\;\\;\\;\\;\\;\\;\\;u^2_R: (3,\\;1,\\;2/3)(-5)\\;,\\;\\;\\;\\;\\;d^2_R: (3,\\;1,\\;-1/3)(7)\\;,\\nonumber \\\\&&L^2_L: (1,\\;2,\\;-1/2)(-\\beta )\\;,\\;\\;\\;\\;e^2_R: (1,\\;1,\\;-1)(-\\beta )\\;,$ and the third generation does not have any $U(1)_{Y^{\\prime }}$ charges.", "One can easily check that the model is free of gauge anomaly.", "Expanding the interactions between the $U(1)_{Y^{\\prime }}$ gauge boson $Y^{\\prime }$ and the SM fermions, $Y^{\\prime }_\\mu J^\\mu _{Y^{\\prime }}$ , we have the current coupling to the $Y^{\\prime }$ field, $J^\\mu _{Y^{\\prime }} &=& g_{Y^{\\prime }}^{} \\left[\\bar{u}\\gamma ^\\mu (4 + 6\\gamma _5) u - \\bar{d} \\gamma ^\\mu (8 +6 \\gamma _5) d +\\beta \\bar{e} \\gamma ^\\mu e +{\\beta \\over 2} \\bar{\\nu }_e \\gamma ^\\mu (1-\\gamma _5)\\nu _e\\right]\\nonumber \\\\&-&g_{Y^{\\prime }}^{} \\left[\\bar{c}\\gamma ^\\mu (4 + 6\\gamma _5) c - \\bar{s} \\gamma ^\\mu (8 +6 \\gamma _5) s +\\beta \\bar{\\mu }\\gamma ^\\mu \\mu +{\\beta \\over 2} \\bar{\\nu }_\\mu \\gamma ^\\mu (1-\\gamma _5)\\nu _\\mu \\right]\\;, $ with $g_{Y^{\\prime }}^{}$ being the $U(1)_{Y^{\\prime }}$ gauge coupling.", "We cannot identify the $Y^{\\prime }$ boson as the desired $X$ boson.", "The reason is that the Higgs scalars giving the SM fermion masses will have non-trivial quantum numbers for both the SM and $U(1)_{Y^{\\prime }}$ gauge groups and will contribute to the $W$ , $Z$ and $Y^{\\prime }$ gauge boson masses.", "If $Y^{\\prime }$ is $X$ , it must have a small mass $17\\,\\textrm {MeV}$ , and the involved vacuum expectation values (VEVs) should be much smaller than the electroweak scale for a $g_{Y^{\\prime }}^{}$ explaining the anomalous ${^8Be^} \\rightarrow {^8Be}\\; {e^+e^-}$ .", "Another problem is that the couplings of neutrinos to $Y^{\\prime }$ are too large to satisfy the constraints mentioned previously.", "These problems must be solved for a realistic model.", "We solve the light mass problem by introducing an additional gauge symmetry $U(1)_X$ , under which the SM fermions are trivial.", "But through a kinetic mixing of $U(1)$ gauge fields [8], $-(\\epsilon /2) Y^{\\prime }_{\\mu \\nu } X^{\\mu \\nu }$ , the $X$ boson does interact with the SM fermions.", "By diagonalizing and normalizing the gauge fields $Y^{\\prime }$ and $X$ properly, up to the leading order in $\\epsilon $ , we give the couplings of the $X$ boson to the $J^\\mu _{Y^{\\prime }}$ current, $\\epsilon X_\\mu J^\\mu _{Y^{\\prime }}\\;.$ Since the $X$ boson does not carry the SM gauge group quantum numbers, one can generate a small mass $m^2_X = (g_X x_\\rho v_{\\rho })^2$ by introducing an SM-singlet scalar $\\rho $ with a $U(1)_X$ charge $x_\\rho $ and a VEV $v_{\\rho }/\\sqrt{2}$ .", "Here $g_X^{}$ is the $U(1)_X$ gauge coupling.", "Clearly, this small mass $m_X^{}$ does not affect the usual electroweak scale.", "We will assume the $X$ boson to have a mass of $17\\,\\textrm {MeV}$ .", "At the same time, one can introduce another scalar singlet $\\sigma $ with a $U(1)_{Y^{\\prime }}$ charge $y^{\\prime }_{\\sigma }$ and a VEV $v_{\\sigma }/\\sqrt{2}$ to contribute to the $Y^{\\prime }$ mass with $m^2_{Y^{\\prime }} = (g_{Y^{\\prime }} y^{\\prime }_{\\sigma } v_{\\sigma })^2$ .", "Assuming $m_{Y^{\\prime }}$ of the order of $\\textrm {TeV}$ , the contributions from the Higgs scalars transforming as the SM iso-doublets can be neglected since their VEVs are at the electroweak scale.", "We now discuss how to suppress the couplings of the $X$ boson to the electron neutrino $\\nu _{e}^{}$ .", "This is achieved by mixing $\\nu _{e}^{}$ with a new vector-like fermion $S=S_L^{}+S_R^{}$ which is a singlet under the SM and $U(1)_{Y^{\\prime }}$ gauge groups but carry a $U(1)_X$ charge $x_{S}$ .", "One can also introduce three gauge-singlet fermions $N_{Ri}~(i=1,2,3)$ to facilitate a canonical seesaw mechanism for generating the small neutrino masses.", "Let us take the first generation into account for illustration.", "With three iso-doublet Higgs scalars $\\phi _e(0,0)$ , $\\phi _{\\nu _e}(\\beta ,0)$ , and $\\eta (\\beta , -x_{S})$ , where the brackets following the fields describe the transformations under the $U(1)^{\\prime }_Y\\times U(1)_X^{}$ gauge groups, the terms responsible for the first-generation lepton masses are $L = - y^{}_e \\bar{L}^1_L L\\tilde{\\phi _e} e^{}_R-y^{}_N \\bar{L}^1_L\\phi _{\\nu _e} N^{}_R-\\frac{1}{2}M_N \\bar{N}_R^c N_R^{}- f_S^{}\\bar{L}^1_L\\eta S_R^{} - m_S^{}\\bar{S}_L^{} S_R^{}+\\textrm {H.c.}\\,.$ We emphasize that the mixing between the vector-like fermion and the electron neutrino will not be stringently constrained by the neutrino masses, instead, it will affect the Dirac equations of the left-handed electron neutrino, i.e.", "$\\mathcal {L}\\supset i\\bar{\\nu }^{}_{Le}\\left(1+U_S^{}\\right)\\partial \\!\\!\\!/\\, \\nu _{Le}^{}-\\frac{1}{2}\\bar{\\nu }_{Le}^{}m_\\nu ^{}\\nu _{Le}^{c}+\\textrm {H.c.}\\,.$ Here $U_S^{}$ is a real number mediated by the vector-like fermion while $m_\\nu ^{}$ is the neutrino mass suppressed by the right-handed neutrinos, $U_S^{}=f_S^{}\\frac{v_\\eta ^2}{2m_S^{2}}f_S^{\\dagger }\\,,~~m_\\nu ^{}=-y_N^{}\\frac{v_{\\phi _e}^2}{2M_N^{}}y_N^{T}\\,.$ We then should normalize the left-handed electron neutrino and its mass by $&&(1+U_S^{})^{\\frac{1}{2}}_{}\\nu _{Le}^{} \\rightarrow \\nu _{Le}^{}\\,,\\;\\;\\;\\;(1+U_S^{})^{-\\frac{1}{2}}_{}m_\\nu ^{}(1+U_S^{T})^{-\\frac{1}{2}}_{}\\rightarrow m_\\nu ^{} \\,.$ In principle, the right-handed neutrinos will also modify the kinetic term of the left-handed electron neutrino.", "However, this contribution is of the order of $m_\\nu ^{}/M_N^{}$ and hence is negligible.", "By integrating out the vector-like fermion, a term of $-x_S^{}g_X^{}\\bar{\\nu }_{Le}^{}U_S^{}\\gamma _\\mu ^{}\\nu _{Le}^{}X^\\mu _{}$ will be generated.", "Including the normalization according to Eq.", "(REF ), one finds the effective coupling of the electron neutrino $\\nu _e^{}$ to the $X$ boson should be ${\\beta \\epsilon g_{Y^{\\prime }}^{} \\over 2} \\bar{\\nu }_{e}^{}\\gamma ^{}_\\mu (1-\\gamma _5) \\nu _{e}^{} X^\\mu _{}\\rightarrow {\\beta \\epsilon g_{Y^{\\prime }}^{}\\over 2} \\frac{1-\\frac{g_X^{}x_S^{}}{\\beta \\epsilon g_{Y^{\\prime }}}U_{S}^{}}{1+U_{S}^{}}\\bar{\\nu }_{e}^{}\\gamma ^{}_\\mu (1-\\gamma _5) \\nu _{e}^{} X^\\mu _{}\\,.$ With an appropriate choice of parameters, the coupling of $X$ to $\\nu _e$ can be supressed, even to zero if $g_X^{}x_S^{} U_{S}^{} = \\epsilon g_{Y^{\\prime }}^{} \\beta $ .", "The demonstrations in Eqs.", "(REF -REF ) can be generalized for all of the three generations by introducing more iso-doublet Higgs scalars and vector-like fermions with proper $U(1)^{\\prime }_Y\\times U(1)_X^{}$ charges.", "In this case, the numbers $U_S^{}$ and $m_\\nu ^{}$ should be understood as a hermitian matrix and a symmetric matrix, respectively." ], [ "The fifth force", "Combining Eqs.", "(REF ), (REF ) and (REF ), we derive the parameters $\\varepsilon _f^v$ in the effective theory (REF ) by $\\varepsilon ^v_u =-\\frac{4 \\epsilon g_{Y^{\\prime }}^{}}{e}\\,,~~\\varepsilon ^v_d = \\frac{8 \\epsilon g_{Y^{\\prime }}^{}}{e}\\,,~~\\varepsilon ^v_e = - \\frac{ \\beta \\epsilon g_{Y^{\\prime }}^{}}{e}\\,,~~ \\varepsilon ^v_{\\nu _e} = {\\beta \\epsilon g_{Y^{\\prime }}^{} \\over 2 e} {1-g_XU_{S}/(\\beta \\epsilon g_{Y^{\\prime }})\\over 1+ U_{S}}\\,.~~$ Obviously, $\\varepsilon _p^v = 2 \\varepsilon _u^v + \\varepsilon ^v_d = 0$ .", "So, the vector current interaction of the $X$ boson is protophobic type as proposed in Ref.", "[2].", "At this moment, one may have naively concluded that they can easily fit the required numbers for explaining the ${^8Be^} \\rightarrow {^8Be}\\; e^+e^-$ data as given in Eqs.", "(REF ).", "However, the model above also contains axial-vector current interactions (REF ) with $\\varepsilon ^a_u= - \\frac{6\\epsilon g_{Y^{\\prime }}^{}}{e}\\,,~~\\varepsilon ^a_d =\\frac{6 \\epsilon g_{Y^{\\prime }}^{}}{e}\\,,~~\\varepsilon _e^a = 0\\,,~~\\varepsilon ^a_{\\nu _e}=-{\\beta \\epsilon g_{Y^{\\prime }}^{}\\over 2 e} {1-g_XU_{S}/(\\beta \\epsilon g_{Y^{\\prime }})\\over 1+ U_{S}}\\,.$ Therefore the model is actually protophobic only in the vector current interactions.", "It is necessary to check if the axial-vector current interactions can satisfy the experimental data.", "Remarkably, the axial-vector current interactions between the $X$ boson and the $u$ and $d$ quarks are proportional to $\\varepsilon ^a_u \\bar{u} \\gamma ^\\mu \\gamma _5 u + \\varepsilon _d^a \\bar{d} \\gamma ^\\mu \\gamma _5 d$ which now is the sum of a zero iso-scalar current $[(\\varepsilon ^a_u + \\varepsilon _d^a)/2] (\\bar{u} \\gamma ^\\mu \\gamma _5 u +\\bar{d} \\gamma ^\\mu \\gamma _5 d)\\equiv 0$ and a nonzero iso-vector current $[(\\varepsilon ^a_u - \\varepsilon _d^a)/2] (\\bar{u} \\gamma ^\\mu \\gamma _5 u -\\bar{d} \\gamma ^\\mu \\gamma _5 d)\\equiv \\!\\!\\!\\!\\!\\!/ \\,\\,0$ [9].", "On the other hand, the observed ${^8Be^} \\rightarrow {^8Be}\\, X$ process is irrelevant to the iso-vector currents because the initial and final hadrons are both isospin singlets.", "The iso-vector interaction may induce some physical effects, such as $\\pi ^0 \\rightarrow e^+e^-$ .", "However, the electron only has a vector current interaction with the $X$ boson so that the contribution from our model to $\\pi ^0\\rightarrow e^+e^-$ can be identically zero.", "No constraint can be obtained from this consideration.", "The protophobic nature in the vector current interactions of the $X$ boson results in an interaction term $\\pi ^0 \\tilde{X}_{\\mu \\nu } F^{\\mu \\nu }$ through triangle anomaly diagram which generates $\\pi ^0 \\rightarrow \\gamma \\gamma $ decay.", "Here $F^{\\mu \\nu }$ is the photon field strength.", "Experimental limit on search of $\\pi ^0 \\rightarrow X \\gamma $ constrains the interaction to be protophobic as mentioned earlier.", "Our model contains the axial-vector current couplings of the $X$ boson to the $u$ and $d$ quarks.", "Naively, one would expect the emergence of an interaction term of the type of $\\pi ^0 X_{\\mu \\nu }F^{\\mu \\nu }$ which affects the result.", "However, this term does not appear since it violates CP and is therefore forbidden.", "The analysis of $\\pi ^0 \\rightarrow X \\gamma $ in Ref.", "[2] still hold in our model.", "To explain the observed anomalous $^{8}_{}\\!Be$ nuclear transitions and fulfill all of the other experimental limits, one needs $\|\\varepsilon _n^v\| = \|12\\epsilon g_{Y^{\\prime }}/e\| = 0.011$ and $\|\\varepsilon _e^v\| =\|\\beta \\epsilon g_{Y^{\\prime }}/e\| <1.4\\times 10^{-3}_{}$ which result in $\\epsilon g_{Y^{\\prime }} = 2.78\\times 10^{-4}\\,,~~\|\\beta \|<1.53\\,.$" ], [ "Other implications", "Since in our model, the first two generations of charged fermions couple to the $X$ boson with a same strength, in particular, $\\varepsilon _d^{v} = \\varepsilon _s^{v} =2\\varepsilon _n^{v}/3$ , there may be constraints from data on $X$ production from other quarks.", "The value for $\|\\varepsilon _s^{v}\| = 0.0073$ is at tension with the boundary of the $90\\%$ c.l.", "allowed region from KOEL data[10] on $\\phi \\rightarrow \\eta X$ .", "But allowed at 3$\\sigma $ c.l..", "Improved data can test the model further.", "Furthermore, we have $\|\\varepsilon _\\mu ^{v} \|= \|\\varepsilon _e^{v}\|$ , which has an effect on $(g-2)_\\mu ^{}$ .", "One can calculate the $X$ boson contribution to $\\Delta a_\\mu $ which has a $3\\,\\sigma $ deviation, $\\Delta a_\\mu ^{} = 288(80)\\times 10^{-11}_{}$ [11].", "Using the $3\\,\\sigma $ upper bound of $\\varepsilon _e^{v}=1.4\\times 10^{-3}_{}$ ($\\beta = 1.53$ ), we obtain $\\Delta a_\\mu = 152\\times 10^{-11}$ from the $X$ boson contribution which improves the deviation to $1.5\\,\\sigma $ .", "We now discuss possible ways to further test the model.", "Besides continuing similar experiments with higher sensitivity for those already provided constraints, it would be good to find new ways for testing the model.", "One may carry out $e^+_{} e^-_{}\\rightarrow \\gamma X$ followed by measuring $e^+_{}e^-_{}$ with a center of mass energy $\\sqrt{s}$ at BES III and also at BELLE II.", "Since in our model $\\varepsilon _e^{v}$ is constrained to be less than $1.4 \\times 10^{-3}_{}$ , the cross section is typically less than $ 10^{-2}_{}\\,\\textrm {fb}$ which may be too small to be measured experimentally in the near future.", "At hadron collider because $\|\\varepsilon _{d}^{v}\|$ is as large as $7\\times 10^{-3}_{}$ , the cross section for, $pp \\rightarrow \\gamma X+\\textrm {jets}$ , may be larger.", "However, in the hadronic back ground the measurement will be very challenging.", "Exclusive decay of a meson $A$ to $B X$ followed by measuring $e^+_{}e^-_{}$ from on-shell $X$ decay may be very hopeful.", "If the initial state $A$ is a state with two constituent quarks (a quark with an anti-quark have the same absolute electric charge $\|Q_q^{}\|$ ), one then obtains, for the vector part of the current interaction.", "$R(X/\\gamma , Q_q)={Br(A\\rightarrow B X)_{Q_q}/Br(A\\rightarrow B\\gamma )_{Q_q}} = (\\varepsilon _q^v/Q_q)^2 $ .", "Assuming $X\\rightarrow e^+e^-$ saturating the $X$ decay, for $Q_q = 2/3$ and $Q_q = -1/3$ , we have, respectively $R(X/\\gamma ,\\;2/3)= \\left\|\\frac{1}{2}\\varepsilon _n^{v}\\right\|^2_{}\\approx 3.0\\times 10^{-5}$ and $R(X/\\gamma ,\\;-1/3) = \\left\|2\\varepsilon _n^{v}\\right\|^2_{}\\approx 4.8\\times 10^{-4}$ .", "When the axial-vector current contributions are included which will add terms proportional to $\|\\varepsilon ^a_q\|^2$ , ratios become larger.", "So the numbers $3.0\\times 10^{-5}$ and $4.8\\times 10^{-4}$ represent lower bounds for the ratios.", "The above bounds can be used to study radiative ($X$ boson) decays of the vector mesons $J/\\psi $ or the flavored vector mesons $D^{0}$ into a spin zero meson.", "We find the most promising decay mode is $D^{0} \\rightarrow D^0 X\\rightarrow D^0 e^+e^-$ for the reasons that $D^{0}\\rightarrow D^0 \\gamma $ has a large branching ratio $(38.1\\pm 2.9)\\% $ [5] and a large number of this decay can be copiously produced and studied at the LHCb.", "At the LHC run III, the LHCb may have an integrated luminosity of $15\\,\\textrm {fb}^{-1}$ which means that the event number for $D^{0} \\rightarrow D^0 \\gamma $ can reach about $5\\times 10^{12}_{}$ .", "The analysis for constraining $\\varepsilon _c^{v}=-\\varepsilon _n^{v}/3$ is similar to that carried out for constraining the dark photon mixing parameter in Ref.", "[12] where it was shown that for $m_X^{}\\simeq 17\\,\\textrm {MeV}$ the LHCb sensitivity for the mixing parameter can reach about $2.4\\times 10^{-5}$ with an integrated luminosity of $15\\,\\textrm {fb}^{-1}_{}$ .", "Normalizing their notation to ours, the sensitivity for $\|\\varepsilon _c^{v}\|$ can be $1.6\\times 10^{-5}$ .", "Our model can be tested at the LHCb.", "In the above discussions, the third-generation fermions do not interact with the $X$ boson.", "However, it may turns out that the third-generation quarks interact with the $X$ boson, but the second-generation does not.", "In this case, the above formulae can be used to study radiative ($X$ boson) decays of the vector mesons $\\Upsilon $ into a spin zero meson, or radiative ($X$ boson) decays of the flavored vector mesons $B^{0}_d$ and $B^{0}_s$ into a spin zero meson." ], [ "Conclusions", "In summary, we have proposed a realistic gauge anomaly free model with a $17\\,\\textrm {MeV}$ $X$ gauge boson mediating a fifth force to explain the anomaly reported in ${^8Be^} \\rightarrow {^8Be}\\; e^+e^-$ .", "In our model, the $X$ boson has both of the vector and axial-vector current couplings to the SM fermions.", "The vector current interactions have a protophobic nature.", "Meanwhile, the contribution from the axial-vector currents cancels out in the iso-scalar interactions for ${^8Be^} \\rightarrow {^8Be}\\; X$ .", "Furthermore, the model allows us to suppress the unexpected couplings of the $X$ boson to the electron neutrino.", "Within the allowed parameter space, the model can alleviate the anomaly in $(g-2)_\\mu $ .", "The $X$ boson also couples to the second or third generation of quarks and hence may induce $D^{0} \\rightarrow D^0 X \\rightarrow D^0\\;{e^+e^-}$ or $B^{0}_{d,s} \\rightarrow B^0_{d,s} X \\rightarrow B^0_{d,s}\\;{e^+e^-}$ which can be studied at the LHCb to probe the parameter space for explaining ${^8Be^} \\rightarrow Be X \\rightarrow Be\\; {e^+e^-}$ .", "For generating the required fermion masses, we need introduce multi iso-doublet Higgs scalars carrying different $U(1)_{Y^{\\prime }}$ and/or $U(1)_X$ charges.", "This means rich flavor changing phenomena including the anomaly in $h \\rightarrow \\mu \\tau $ from the LHC and the anomalies in $b\\rightarrow s \\mu ^+\\mu ^-$ transitions shown in experimental data.", "We will present detailed studies elsewhere.", "PHG was supported by the Shanghai Jiao Tong University (Grant No.", "WF220407201) and the Recruitment Program for Young Professionals (Grant No.", "15Z127060004).", "XGH was supported in part by MOE Academic Excellent Program (Grant No.", "102R891505) and MOST of ROC (Grant No.", "MOST104-2112-M-002-015-MY3), and in part by NSFC (Grant Nos.", "11175115 and 11575111) of PRC.", "This work was also supported by the Shanghai Laboratory for Particle Physics and Cosmology (Grant No.", "11DZ2260700)." ] ]	1606.05171
[ [ "Nonlinear light-Higgs coupling in superconductors beyond BCS: Effects of\n the retarded phonon-mediated interaction" ], [ "Abstract We study the contribution of the Higgs amplitude mode on the nonlinear optical response of superconductors beyond the BCS approximation by taking into account the retardation effect in the phonon-mediated attractive interaction.", "To evaluate the vertex correction in nonlinear optical susceptibilities that contains the effect of collective modes, we propose an efficient scheme which we call the \"dotted DMFT\" based on the nonequilibrium dynamical mean-field theory (nonequilibrium DMFT) to go around the difficulty of solving the Bethe-Salpeter equation and analytical continuation.", "The vertex correction is represented by the derivative of the self-energy with respect to the external driving field, which is self-consistently determined by the differentiated (\"dotted\") DMFT equations.", "We apply the method to the Holstein model, a prototypical electron-phonon-coupled system, to calculate the susceptibility for the third-harmonic generation including the vertex correction.", "The results show that, in sharp contrast to the BCS theory, the Higgs mode can contribute to the third-harmonic generation for general polarization of the laser field with an order of magnitude comparable to the contribution from the pair breaking or charge density fluctuations.", "The physical origin is traced back to the nonlinear resonant light-Higgs coupling, which has been absent in the BCS approximation." ], [ "Introduction", "Nonequilibrium dynamics of superconductors induced by intense laser excitations opens various possibilities of controlling emergent states of matter without destroying quantum coherence.", "[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] Specifically, for relatively low frequencies of the laser such as terahertz (THz) and mid-infrared, we can expect to suppress the generation of quasiparticles having high energies that might be quickly transformed into heat through inelastic collisions causing a destruction of quantum coherence.", "Recent experiments indeed report that a superconducting-like state can be generated from the normal state by such low-energy excitations.", "[1], [5], [10], [11], [13] In superconductors, there exists a collective mode called the Higgs amplitude mode, which plays an important role in low-energy dynamics.", "The mode corresponds to the coherent amplitude oscillation of the superfluid density, which has a long history of theoretical studies.", "[15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51] Experimental observation of the Higgs mode in superconductors has been reported by Raman scattering, [52], [53] and THz pump-probe experiments.", "[9], [12] It has also been reported in a THz pump experiment[12] that there emerges a third-harmonic generation (THG) in the nonlinear optical response that is resonantly enhanced when the doubled frequency ($2\\Omega $ ) of the incident light equals the superconducting gap ($2\\Delta $ ), which coincides with the energy of the collective Higgs mode at long wavelength.", "On the other hand, there also exist individual excitations (Cooper pair breaking or charge density fluctuations), whose lower bound in the energy spectrum resides at the same energy of $2\\Delta $ with a diverging density of states.", "The question then is to what extent these two contribute to the nonlinear optical response in superconductors and how strongly the light is nonlinearly coupled to the Higgs mode.", "[44], [49] In the BCS mean-field theory (with the random phase approximation), the contribution of pair breaking or charge density fluctuation to the THG susceptibility is expressed in a gauge invariant form (including the screening effect) as[49] $\\chi _0^{\\rm BCS}(\\Omega )&=\\sum _{\\mathbf {k}} (\\ddot{\\epsilon }_{\\mathbf {k}})^2 \\chi _{33}(\\mathbf {k},\\Omega )-\\frac{\\left[\\sum _{\\mathbf {k}} \\ddot{\\epsilon }_{\\mathbf {k}}\\chi _{\\rm 33}(\\mathbf {k},\\Omega )\\right]^2}{\\sum _{\\mathbf {k}}\\chi _{\\rm 33}(\\mathbf {k},\\Omega )}.$ Here $\\epsilon _{\\mathbf {k}}$ is the band dispersion, $\\ddot{\\epsilon }_{\\mathbf {k}}=\\sum _{ij} (\\partial ^2\\epsilon _{\\mathbf {k}}/\\partial k_i \\partial k_j)e_i e_j$ , $\\mathbf {e}$ is the polarization vector of light, and $\\chi _{33}(\\mathbf {k},\\Omega )&=-\\frac{i}{2}\\int \\frac{d\\omega }{2\\pi } {\\rm Tr}[\\tau _3 \\hat{G}_{\\mathbf {k}}(\\omega +2\\Omega )\\tau _3 \\hat{G}_{\\mathbf {k}}(\\omega )]^<,$ where $\\tau _3$ is the third component of the Pauli matrix, $\\hat{G}_{\\mathbf {k}}(\\omega )$ is the Nambu-Gor'kov Green's function, and $<$ denotes the lesser component based on the Langreth rule[54] (with the notation defined in Appendix ).", "For $s$ -wave superconductors, $\\hat{G}_{\\mathbf {k}}(\\omega )$ and $\\chi _{33}(\\mathbf {k},\\omega )$ depend respectively on the momentum through $\\epsilon _{\\mathbf {k}}$ , which allows one to change the momentum sum into an energy integral by inserting $1=\\int d\\epsilon \\delta (\\epsilon -\\epsilon _{\\mathbf {k}})$ .", "Then we can define expansions around the Fermi energy,[44], [50] $\\sum _{\\mathbf {k}} \\delta (\\epsilon -\\epsilon _{\\mathbf {k}}) \\ddot{\\epsilon }_{\\mathbf {k}}&=D(\\epsilon _F)(c_0+c_1 \\epsilon +c_2 \\epsilon ^2+\\cdots ),\\\\\\sum _{\\mathbf {k}} \\delta (\\epsilon -\\epsilon _{\\mathbf {k}}) (\\ddot{\\epsilon }_{\\mathbf {k}})^2&=D(\\epsilon _F)(\\tilde{c}_0+\\tilde{c}_1 \\epsilon +\\tilde{c}_2 \\epsilon ^2+\\cdots ),$ where $D(\\epsilon _F)$ is the density of states at the Fermi energy.", "In Ref.", "Tsuji2015, it is assumed that the constant terms in the expansions (REF ) and () can be removed by gauge transformations, so that the pair breaking effect in THG is less dominant than the Higgs mode.", "As pointed out in Ref.", "Cea2016, however, this holds in rather restricted situations, such as one dimensional (1D) lattices, 2D square and 3D simple cubic lattices with polarization $\\mathbf {e}$ respectively parallel to $(1,1)$ and $(1,1,1)$ directions, 3D body-centered-cubic lattice with $\\mathbf {e}$ parallel to $(1,0,0)$ , and so on.", "For a general lattice with a general polarization, the constant terms may survive, and the Higgs-mode contribution may be left subleading.", "Figure: Feyman diagrams for the non-resonant (a), mixed (b), and resonant (c)contributions to the THG susceptibility containing the effect of collective modesas vertex corrections.", "The solid (wavy) lines represent the electron (photon) propagators,while the shaded boxes represent the reducible four-point vertex function.", "Among the four photon lines,one is outgoing with an energy 3Ω3\\Omega , and the other three are incoming with an energy Ω\\Omega .Then the next question is: what will happen if one goes beyond the BCS approximation.", "In fact, the superconductor NbN used in the experiments[9], [12] is known to have a strong electron-phonon coupling ($\\lambda \\sim 1$ ), [55], [56], [57] where it is important to capture corrections from the BCS analysis.", "Indeed the argument in the previous paragraph heavily relies on the speciality of BCS: the (nonlinear) coupling to the light occurs only in a non-resonant form $\\ddot{\\epsilon }_{\\mathbf {k}}A(t)^2$ rather than in a resonant form $\\dot{\\epsilon }_{\\mathbf {k}}A(t)\\dot{\\epsilon }_{\\mathbf {k}}A(t^{\\prime })$ , where $\\mathbf {A}(t)=A(t)\\mathbf {e}$ is the vector potential, and $\\dot{\\epsilon }_{\\mathbf {k}}=\\sum _i (\\partial \\epsilon _{\\mathbf {k}}/\\partial k_i) e_i$ .", "The terminology (“resonant” and “non-resonant”) is here borrowed from literatures on Raman scattering.", "[58] These forms can be expressed as diagrams for the THG susceptibility [59] in Fig.", "REF (which in fact very much resemble Raman-scattering diagrams[58]), where the effect of collective modes is incorporated in the vertex correction, with the Higgs mode represented by an infinite series of ring diagrams in the $\\tau _1$ channel.", "[22], [24], [41], [44] Two photon lines attached together to electron lines represent the non-resonant coupling, while two single-photon lines attached separately represent the resonant coupling.", "Within the BCS theory, there is only the non-resonant couplingThis can be understood in Anderson's pseudospin picture.", "[15], [12], [44] The time-dependent BCS theory is equivalent to a pseudospin dynamics described by $\\partial \\mathbf {\\sigma }_{\\mathbf {k}}/\\partial t=2\\mathbf {b}_{\\mathbf {k}}\\times \\mathbf {\\sigma }_{\\mathbf {k}}$ , where $\\mathbf {\\sigma }_{\\mathbf {k}}$ is the pseudospin, and $\\mathbf {b}_{\\mathbf {k}}=(-{\\rm Re}\\Delta ,-{\\rm Im}\\Delta ,(\\epsilon _{\\mathbf {k}+\\mathbf {A}(t)}+\\epsilon _{\\mathbf {k}-\\mathbf {A}(t)})/2)$ is the pseudomagnetic field.", "The coupling to the light is provided by the $z$ component of the pseudomagnetic field, $\\epsilon _{\\mathbf {k}}+\\ddot{\\epsilon }_{\\mathbf {k}}A(t)^2/2+O(A^4)$ , which is in a form of the non-resonant coupling., and the mixed [Fig.", "REF (b)] and resonant (c) contributions to THG exactly vanish.", "This is confirmed by explicitly calculating the convolution of relevant three electron propagators, $\\int \\frac{d\\omega }{2\\pi } {\\rm Tr}[\\tau _1 \\hat{G}_{\\mathbf {k}}(\\omega +2\\Omega ) \\hat{G}_{\\mathbf {k}}(\\omega +\\Omega ) \\hat{G}_{\\mathbf {k}}(\\omega )]^<&=0\\quad (\\mbox{BCS}).$ However, this does not guarantee that these contributions would remain small if one goes beyond the BCS approximation.", "For example, the real part of the optical conductivity $\\sigma (\\Omega )$ vanishes for $\\Omega \\ne 0$ within the BCS theory, since $\\int \\frac{d\\omega }{2\\pi } {\\rm Tr}[\\hat{G}_{\\mathbf {k}}(\\omega +\\Omega )\\hat{G}_{\\mathbf {k}}(\\omega )]^<&=0\\quad (\\mbox{BCS}),$ in much the same way as in Eq.", "(REF ).", "In reality, however, the real part of the optical conductivity is nonzero and not even small.", "[9], [60] They become nonzero when one takes account of dynamical correlations such as the electron-phonon coupling (producing retarded interactions) or impurity scattering.", "In those situations, we can expect that the resonant and mixed contributions to the THG response may also be nonzero.", "Indeed, it has been shown in the study of Raman scattering for correlated electron systems that the resonant contribution can significantly enhance the non-resonant Raman response.", "[61], [62] This has motivated us to study here the nonlinear optical response of superconductors for electron-phonon coupled systems beyond the BCS approximation.", "Theoretically, it is quite challenging to evaluate all of the non-resonant, mixed and resonant diagrams involving the four-point vertex on an equal footing, since the vertex carries three independent momenta and frequencies.", "Therefore, we employ the dynamical mean-field theory (DMFT),[63] which assumes the momentum-independent self-energy and vertex function.", "Still, the calculation is quite demanding if one tries to evaluate the nonlinear response function by solving the Bethe-Salpeter equation and performing multiple analytical continuations.", "In higher dimensions in the thermodynamic limit, an analysis including the vertex correction has so far been performed only in exceptional cases, such as the Raman response of the Falicov-Kimball model.", "[61], [62], [64], [65] For the Hubbard model, the nonlinear optical response has been analyzed by Hartree-Fock approximation,[66] by DMFT without considering vertex corrections,[67], [68] and by exact diagonalization for small finite-size systems.", "[69], [70] For the Holstein model, higher-harmonic generation has been studied by Migdal approximation without considering vertex corrections.", "[71] For 1D Hubbard-Holstein model, THG response has been studied by the density-matrix renormalization group.", "[72] In this paper, we propose an efficient way to calculate the vertex correction for nonlinear optical susceptibilities, which we call the “dotted DMFT”[73], without directly solving the Bethe-Salpeter equation and performing analytical continuation.", "The idea is to let the nonequilibrium DMFT equations[74] differentiated (“dotted”) with respect to the external field to deduce a self-consistent equation for the vertex function represented by the dotted self-energy.", "We then apply the method to the Holstein model, a prototypical model for electrons interacting with local phonons giving retarded interactions among electrons.", "The results indicate that the resonant contribution from the Higgs mode to the THG susceptibility can indeed be comparable to those from the pair breaking or density fluctuations.", "In particular, the resonance of THG at $2\\Omega =2\\Delta $ can be enhanced by the nonlinear resonant coupling between the light and Higgs mode.", "The paper is organized as follows.", "In Sec.", ", we describe the model set-up that we use throughout the paper for the analysis of the nonlinear optical response in superconductors.", "In Sec.", ", we propose an efficient method (dotted DMFT) to evaluate the vertex correction for dynamical susceptibilities based on the nonequilibrium DMFT.", "Sec.", "describes the results of the THG susceptibility obtained by the dotted DMFT for the electron-phonon-coupled system.", "In Sec.", "we summarize the paper." ], [ "Model", "We take the Holstein model as a typical model for electrons interacting with local phonons, $H&=\\sum _{ij,\\sigma }t_{ij} (c_{i\\sigma }^\\dagger c_{j\\sigma }+{\\rm h.c.})-\\mu \\sum _i n_i\\\\&\\quad +\\omega _0 \\sum _i b_i^\\dagger b_i+g\\sum _i (b_i+b_i^\\dagger )(n_i-1).$ Here $c_{i\\sigma }^\\dagger $ ($c_{i\\sigma }$ ) is the creation (annihilation) operator for an electron at site $i$ with spin $\\sigma =\\uparrow ,\\downarrow $ , $t_{ij}$ is the hopping amplitude, $n_i=\\sum _\\sigma c_{i\\sigma }^\\dagger c_{i\\sigma }$ , $\\mu $ is the chemical potential, $b_i^\\dagger $ ($b_i$ ) is the creation (annihilation) operator for phonons having a frequency $\\omega _0$ , and $g$ is the electron-phonon coupling constant.", "We then apply the dynamical mean-field theory (DMFT) to solve the model.", "Since DMFT becomes exact for large spatial dimensions ($d\\rightarrow \\infty $ ), [75], [63] we take the hypercubic lattice, whose energy dispersion is $\\epsilon _{\\mathbf {k}}&=-2t\\sum _{i=1}^d \\cos k_i.$ As usually done, we scale the hopping as $t=t^\\ast /\\sqrt{2d}$ with a fixed $t^\\ast $ to obtain a meaningful fixed point in the large $d$ limit, which results in a gaussian density of states $D(\\epsilon )=e^{-\\epsilon ^2/2t^\\ast {}^2}/\\sqrt{2\\pi }t^\\ast $ .", "We use $t^\\ast $ as the unit of energy (frequency) throughout the paper.", "We concentrate on the half-filled electron system ($\\mu =0$ ), in which the particle-hole symmetry is fully respected.", "In the particle-hole symmetric case, the Higgs amplitude mode is safely decoupled from the phase mode, and the screening effect is absent.", "Away from half filling, the amplitude mode can hybridize with the phase mode in principle.", "However, we expect that the damping of the amplitude mode into the phase mode is suppressed in superconductors, since the phase mode is pushed to high energies ($\\sim $ the plasma frequency) due to the Anderson-Higgs mechanism.", "[17], [18], [19], [20] If one integrates out the phonon degrees of freedom, the electrons acquire an effective retarded interaction, $U(\\omega )&=g^2 D_0^R(\\omega ),$ where $D_0^R(\\omega )$ is the noninteracting retarded phonon Green's function, $D_0^R(\\omega )&=\\frac{2\\omega _0}{(\\omega +i\\gamma )^2-\\omega _0^2}.$ We introduce a parameter $\\gamma $ to regularize the phonon Green's function.", "In the static limit ($\\omega \\rightarrow 0$ ), the effective interaction approaches $U(\\omega =0)=-2g^2\\omega _0/(\\omega _0^2+\\gamma ^2)<0$ , i.e., the attractive interaction.", "The strength of the attractive interaction can be measured (within the unrenormalized Migdal approximation as introduced later) by a dimensionless parameter, $\\lambda &\\equiv \|U(\\omega =0)\| D(\\epsilon _F)=\\frac{2g^2\\omega _0 }{\\omega _0^2+\\gamma ^2}D(\\epsilon _F).$ When the attractive interaction is large enough and the temperature is low enough, the model exhibits a phase transition from the normal to superconducting states.", "In this paper, instead of taking the infinitesimal limit of $\\gamma $ ($\\rightarrow +0$ ), we keep it nonzero and regard it as a phenomenological parameter that represents the finite lifetime ($\\tau \\sim \\gamma ^{-1}$ ) of phonon oscillations.", "This is physically natural, since the phonon oscillation should be damped to some extent in real solids by various possible ways of scattering and energy dissipation.", "The electron-phonon coupling itself can induce the damping of phonons.", "[76] A finite $\\gamma $ is not necessarily phenomenological, but can be actually modeled by phonons coupled to a heat bath comprising many harmonic oscillators (Caldeira-Leggett-type model[77]).", "In the application of the dotted DMFT, which we shall introduce in the next section, it turns out that it is important to take a nonzero $\\gamma $ (avoiding infinitely long-lived phonons) to stabilize the convergence of the dotted DMFT calculation.", "In a similar manner, we introduce a small imaginary part $\\delta $ (broadening factor) in the noninteracting retarded electron Green's function, $G_{0\\mathbf {k}}^R(\\omega )&=\\frac{1}{\\omega +i\\delta +\\mu -\\epsilon _{\\mathbf {k}}},$ where $\\delta $ can be considered as a decay rate of noninteracting electrons.", "It can be modeled by electrons coupled to a bath composed of free fermions.", "[74], [78] Compared to $\\gamma $ , the stability of the dotted DMFT is less sensitive to $\\delta $ , so that we can take a much smaller value for $\\delta $ than for $\\gamma $ .", "To study the third-harmonic generation, we apply an ac electric field to the Holstein model.", "We use the temporal gauge to represent the electric field with a vector potential $\\mathbf {A}(t)=\\mathbf {e} A e^{-i\\Omega t}$ , where $\\mathbf {e}$ is the polarization vector ($\|\|\\mathbf {e}\|\|=1$ ), $A$ and $\\Omega $ is the amplitude and frequency of the vector potential, respectively.", "$A$ is related to the amplitude of the electric field $E$ via $A=E/(i\\Omega )$ .", "The ac field is minimally coupled to the electrons through the Peierls phase.", "The resulting form of the coupling is $\\sum _{\\mathbf {k}\\sigma } \\epsilon _{\\mathbf {k}+\\mathbf {A}(t)} c_{\\mathbf {k}\\sigma }^\\dagger c_{\\mathbf {k}\\sigma }$ in the kinetic term of the Hamiltonian, where we have put the elementary charge $e=1$ .", "The electron current is defined by $\\mathbf {j}(t)&=-i\\sum _{\\mathbf {k}} \\mathbf {v}_{\\mathbf {k}+\\mathbf {A}(t)} G_{\\mathbf {k}}^<(t,t),$ where $\\mathbf {v}_{\\mathbf {k}}=\\partial \\epsilon _{\\mathbf {k}}/\\partial {\\mathbf {k}}$ is the group velocity.", "We measure the current along the electric field, $j(t)=\\mathbf {j}(t)\\cdot \\mathbf {e}.$ The susceptibility for the third-harmonic generation $\\chi (\\Omega )$ is defined by the nonlinear current oscillating with the frequency $3\\Omega $ , $j^{(3)}(t)&=\\chi (\\Omega ) A^3 e^{-3i\\Omega t}.$ To obtain the THG susceptibility, we take the third derivative of Eq.", "(REF ) with respect to $A$ and then set $A=0$ .", "This involves the derivatives of $G$ , which are evaluated by means of the dotted DMFT, as will be explained in the next section.", "Since our model is infinite dimensional, it is not obvious how to choose the polarization.", "One convenient way is to take the direction parallel to $(1,1,\\dots ,1)$ , for which every direction is equivalent.", "However, as mentioned in the introduction, this choice has a bias that the pair breaking effect is suppressed in the THG response.", "Another simple choice is $(1,0,\\dots ,0)$ .", "This, on the other hand, is the direction that maximally enhances the pair breaking contribution.", "To let the situation as fair as possible, we choose a general direction $\\mathbf {e}\\propto (\\overbrace{\\underbrace{1,1,\\dots ,1}_m,0,\\dots ,0}^d),$ where $m$ is the number of dimensions along which the polarization vector has nonzero components.", "It is a kind of generalization of $(1,1,0)$ direction for the three-dimensional cubic lattice.", "We fix the ratio, $\\alpha =\\frac{m}{d}\\quad (0\\le \\alpha \\le 1),$ and take the limit of $d, m\\rightarrow \\infty $ .", "The parameter $\\alpha $ continuously interpolates the two limits of $(1,1,\\dots ,1)$ and $(1,0,\\dots ,0)$ .", "The advantage of this setup is that it greatly simplifies the momentum integral without putting a bias on the pair breaking effect.", "We need a very fine grid for the momentum integral to eliminate the finite-size effect, which is particularly severe in the calculation of the THG spectrum, since one has to resolve the superconducting gap structure in the very vicinity of the Fermi energy.", "One might apply DMFT to the two-dimensional square lattice as an approximation (instead of applying to the hypercubic lattice), but our experience indicates that the number of $k$ points that has to be taken is so huge that it is practically intractable.", "Figure: A schematic picture for the dotted dynamical mean-field theory (dotted DMFT) formalismfor the third-harmonic generation.", "Each equation holds forthe retarded, advanced, lesser, and greater Green's functions and self-energies, respectively.An analogous treatment can be generally applied to arbitrary dynamical susceptibilities.To see how the momentum integral is simplified, we expand $\\epsilon _{\\mathbf {k}+\\mathbf {A}(t)}$ in $A$ as $\\epsilon _{\\mathbf {k}+\\mathbf {A}(t)}&=\\epsilon _{\\mathbf {k}}+\\dot{\\epsilon }_{\\mathbf {k}}A e^{-i\\Omega t}+\\frac{1}{2}\\ddot{\\epsilon }_{\\mathbf {k}}A^2 e^{-2i\\Omega t}+\\cdots ,$ where the dot denotes the derivative with respect to $A$ , that is, $\\dot{\\epsilon }_{\\mathbf {k}}&=\\sum _{i=1}^d \\frac{\\partial \\epsilon _{\\mathbf {k}}}{\\partial k_i} e_i,\\\\\\ddot{\\epsilon }_{\\mathbf {k}}&=\\sum _{i,j=1}^d \\frac{\\partial ^2\\epsilon _{\\mathbf {k}}}{\\partial k_i \\partial k_j} e_i e_j,$ and so on.", "The THG susceptibility is expressed as a momentum integral of a function of $\\epsilon _{\\mathbf {k}}$ multiplied by some of $\\dot{\\epsilon }_{\\mathbf {k}}, \\ddot{\\epsilon }_{\\mathbf {k}}, \\dddot{\\epsilon }_{\\mathbf {k}}$ , and $\\ddddot{\\epsilon }_{\\mathbf {k}}$ [with the total number of derivatives being always four since $\\mathbf {v}_{\\mathbf {k}}$ in the definition of the current (REF ) contains one derivative while the other three come from the external field].", "For instance, let us consider a momentum integral of the form $&\\quad \\sum _{\\mathbf {k}} \\dot{\\epsilon }_{\\mathbf {k}} \\dddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})\\\\&=\\frac{1}{m^2}\\sum _{\\mathbf {k}} \\sum _{i=1}^m 2t\\sin k_i \\sum _{j=1}^m (-2t)\\sin k_j f(\\epsilon _{\\mathbf {k}})\\\\&=-\\frac{4t^2}{m^2}\\sum _{\\mathbf {k}} \\left(\\sum _{i=1}^m \\sin ^2 k_i+\\sum _{\\begin{array}{c}i,j=1\\\\ i\\ne j\\end{array}}^m \\sin k_i \\sin k_j \\right) f(\\epsilon _{\\mathbf {k}})$ with a certain function $f(\\epsilon )$ .", "Since directions $i=1,\\dots ,m$ are equivalent and the second term in the parentheses vanishes due to a cancellation between $k_i$ and $-k_i$ , we can simplify Eq.", "(REF ) as $&=-\\frac{4t^2}{m^2}\\sum _{\\mathbf {k}} m\\sin ^2 k_x f(\\epsilon _{\\mathbf {k}}).$ We can symmetrize the momenta $\\lbrace k_i\\rbrace $ in the integrand due to the cubic symmetry to have $&=-\\frac{4t^2}{m^2}\\sum _{\\mathbf {k}} \\frac{m}{d}\\sum _{i=1}^d \\sin ^2 k_if(\\epsilon _{\\mathbf {k}})\\\\&=-\\frac{4t^2}{m^2}\\sum _{\\mathbf {k}} \\frac{m}{d}\\left(\\sum _{i=1}^d \\sin k_i\\right)^2f(\\epsilon _{\\mathbf {k}}),$ where $4t^2(\\sum _{i=1}^d \\sin k_i)^2$ can be replaced by $t^\\ast {}^2$ using the joint density of states[79] $D(\\epsilon ,\\bar{\\epsilon })=D(\\epsilon )D(\\bar{\\epsilon })$ (with $\\bar{\\epsilon }=2t\\sum _i \\sin k_i$ ) and $\\int d\\bar{\\epsilon }D(\\bar{\\epsilon })\\bar{\\epsilon }^2=t^\\ast {}^2$ .", "Equation (REF ) is finally reduced to $\\sum _{\\mathbf {k}} \\dot{\\epsilon }_{\\mathbf {k}} \\dddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}) =-\\frac{t^\\ast {}^2}{d^2 \\alpha } \\sum _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})$ in the infinite-dimensional limit.", "The resulting form can be expressed as a single integral of a function of $\\epsilon =\\epsilon _{\\mathbf {k}}$ , which can be evaluated analytically in terms of the local Green's function and self-energy.", "Similar simplifications apply to all the possible terms in the THG susceptibility.", "In Appendix we summarize some useful formulae to simplify various types of momentum integrals." ], [ "Dotted DMFT", "In this section, we propose an efficient way to calculate the vertex correction for nonlinear dynamical susceptibilities, which we call the “dotted DMFT” (Fig.", "REF ).", "This is because the method enables us to evaluate derivatives of Green's function and the self-energy with respect to the external field, which are required to obtain nonlinear response functions.", "We explain the formulation in the context of third-harmonic generation here, but it can be generalized to arbitrary dynamical susceptibilities.", "To start with, let us assume that the system reaches the (time-periodic) nonequilibrium steady state in the long-time limit in the presence of an ac electric field.", "The steady state emerges due to the balance between continuous excitations by the electric field and an energy dissipation to a heat bath, i.e., the system considered must be an open system.", "In the time-periodic nonequilibrium steady state, the time translational symmetry is partially recovered for Green's function, $G(t+\\mathcal {T},t^{\\prime }+\\mathcal {T})=G(t,t^{\\prime })$ (with $\\mathcal {T}=2\\pi /\\Omega $ being the period of the driving field).", "In principle, one can determine the interacting Green's function within the nonequilibrium steady-state DMFT, or Floquet DMFT, [74], [80] which is capable of treating an arbitrarily large amplitude of the electric field.", "For the present purpose, on the other hand, it is sufficient to calculate Green's function up to the third order in the driving field.", "This motivates us to expand Green's function and the self-energy with respect to the driving field, $G(t,t^{\\prime })&=G_{\\rm eq}(t,t^{\\prime })+\\dot{G}(t,t^{\\prime };\\Omega ) A e^{-i\\Omega t^{\\prime }}\\\\&\\quad +\\frac{1}{2}\\ddot{G}(t,t^{\\prime };\\Omega ) A^2 e^{-2i\\Omega t^{\\prime }}+\\cdots ,\\\\\\Sigma (t,t^{\\prime })&=\\Sigma _{\\rm eq}(t,t^{\\prime })+\\dot{\\Sigma }(t,t^{\\prime };\\Omega ) A e^{-i\\Omega t^{\\prime }}\\\\&\\quad +\\frac{1}{2}\\ddot{\\Sigma }(t,t^{\\prime };\\Omega ) A^2 e^{-2i\\Omega t^{\\prime }}+\\cdots .$ Here $G_{\\rm eq}(t,t^{\\prime })$ and $\\Sigma _{\\rm eq}(t,t^{\\prime })$ are the equilibrium Green's function and self-energy, respectively, and the external field is assumed to be in a form of $\\mathbf {A}(t)={\\mathbf {e}}Ae^{-i\\Omega t}$ .", "If one considers a real field such as $\\mathbf {A}(t)={\\mathbf {e}}A\\cos \\Omega t$ , one has to extend the expansion including cross terms between $e^{-i\\Omega t}$ and $e^{i\\Omega t}$ .", "There is an ambiguity in the definition of the expansion coefficients: the factor $e^{-in\\Omega t^{\\prime }}$ in the $n$ th order can be replaced by $e^{-in\\Omega [x t+(1-x)t^{\\prime }]}$ ($x\\in \\mathbb {R}$ ).", "This is possible as long as the condition $G(t+\\mathcal {T},t^{\\prime }+\\mathcal {T})=G(t,t^{\\prime })$ holds.", "In this paper, we adopt the convention with $x=0$ .", "The advantage of expanding Green's function with respect to $A$ rather than directly treating the nonequilibrium Green's function is that the full time-translation symmetry is available at each order in the expansion.", "To see this, let us write the expansion as $G(t,t^{\\prime })&=G_{\\rm eq}(t,t^{\\prime })+G^{(1)}(t,t^{\\prime };\\Omega )A+\\frac{1}{2}G^{(2)}(t,t^{\\prime };\\Omega )A^2+\\cdots .$ At the $n$ th order, the term contains multiple of $n$ ac fields, so that it acquires the phase $e^{-in\\Omega \\bar{t}}$ when time translation $t\\rightarrow t+\\bar{t}$ is operated.", "For example, for $n=2$ we have $G^{(2)}(t+\\bar{t},t^{\\prime }+\\bar{t};\\Omega )&=e^{-2i\\Omega \\bar{t}} G^{(2)}(t,t^{\\prime };\\Omega ).$ Since $G^{(2)}(t,t^{\\prime })=\\ddot{G}(t,t^{\\prime }) e^{-2i\\Omega t^{\\prime }}$ by definition, $\\ddot{G}$ becomes time-translation invariant: $\\ddot{G}(t+\\bar{t},t^{\\prime }+\\bar{t};\\Omega )&=\\ddot{G}(t,t^{\\prime };\\Omega ).$ The same applies to all orders.", "This allows us to write $\\ddot{G}(t,t^{\\prime };\\Omega )$ as a single-time function $\\ddot{G}(t-t^{\\prime };\\Omega )\\equiv \\ddot{G}(t,t^{\\prime };\\Omega )$ , which can be Fourier transformed as $\\ddot{G}(t-t^{\\prime };\\Omega )&=\\int \\frac{d\\omega }{2\\pi } e^{-i\\omega (t-t^{\\prime })}\\ddot{G}(\\omega ;\\Omega ).$ This is a great advantage because it is no longer necessary to treat the two-time Green's function $G(t,t^{\\prime })$ in favor of a single-frequency function.", "The following formulation can be implemented in the same way as in equilibrium which enjoys the full time-translation symmetry.", "Now the task is to evaluate the expansion coefficients order by order.", "When the system has an inversion symmetry, the local Green's function and self-energy must be parity even, while the electric field is parity odd.", "This implies that odd-order expansion coefficients identically vanish.", "The leading contribution then comes from the second order.", "In order to determine the expansion coefficients, we differentiate every DMFT self-consistency equation with respect to $A$ , and extract the second-order coefficients.", "Let us start with the lattice Dyson equation, $G=\\sum _{\\mathbf {k}} (G_{0\\mathbf {k}}^{-1}-\\Sigma )^{-1}$ .", "If we take a double derivative with respect to $A$ on both sides of the equation (and use $\\dot{\\Sigma }=0$ ), we end up with the “dotted lattice Dyson equation”, $\\ddot{G}^{R,A,<,>}(\\omega ;\\Omega )=\\sum _{\\mathbf {k}} \\big \\lbrace G_{\\mathbf {k}}(\\omega +2\\Omega )[\\ddot{\\epsilon }_{\\mathbf {k}}+\\ddot{\\Sigma }(\\omega ;\\Omega )]G_{\\mathbf {k}}(\\omega )\\\\\\quad +2G_{\\mathbf {k}}(\\omega +2\\Omega )\\dot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega +\\Omega )\\dot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega )\\big \\rbrace ^{R,A,<,>},$ where $R,A,<$ , and $>$ denote the retarded, advanced, lesser, and greater components of nonequilibrium Green's functions, respectively.", "For the detailed definition we refer to Ref. noneqDMFTreview.", "For the notation of $R, A, <$ , and $>$ for products of nonequilibrium Green's functions, see Appendix .", "Note that when Green's function has a matrix form (as in the superconducting state), $\\ddot{\\hat{G}}^A(\\omega ;\\Omega )\\ne [\\ddot{\\hat{G}}^R(\\omega ;\\Omega )]^\\dagger $ , so that the advanced component has to be calculated independently of the retarded one.", "Similarly, we differentiate the impurity Dyson equation, $G=(\\mathcal {G}_0^{-1}-\\Sigma )^{-1}$ (with $\\mathcal {G}_0$ being the Weiss Green's function) twice with respect to $A$ to obtain the “dotted impurity Dyson equation”, $\\ddot{G}^{R,A,<,>}(\\omega ;\\Omega )&=-\\lbrace G(\\omega +2\\Omega )[(\\ddot{\\mathcal {G}}_0^{-1})(\\omega ;\\Omega )\\\\&\\quad -\\ddot{\\Sigma }(\\omega ;\\Omega )]G(\\omega )\\rbrace ^{R,A,<,>}.$ Here the double-dotted inverse of the Weiss Green's functions reads $(\\ddot{\\mathcal {G}}_0^{-1})^{R,A,<,>}(\\omega ;\\Omega )&=-[\\mathcal {G}_0^{-1}(\\omega +2\\Omega )\\ddot{\\mathcal {G}}_0(\\omega ;\\Omega )\\mathcal {G}_0^{-1}(\\omega )]^{R,A,<,>}.$ Figure: Diagrammatic representations of the dotted lattice Dyson equation () (a), and the dotted Migdal approximation () (b).The solid, wavy, and double-wavy lines represent the electron, photon, and phonon propagators, respectively.To close the equation for the dotted functions, we need an explicit (diagrammatic) solution for the nonequilibrium impurity problem.", "This depends on the model and approximation.", "In the present case of the Holstein model, we employ the (unrenormalized) Migdal approximation,[42], [81], [48], [51] $\\Sigma ^{<,>}(t,t^{\\prime })&=ig^2 D_0^{<,>}(t,t^{\\prime })G^{<,>}(t,t^{\\prime }),$ which assumes that phonons stay in equilibrium.", "After expanding $\\Sigma $ and $G$ with respect to $A$ as in Eqs.", "(REF ) and (), and compare both sides of the equation at the order $A^2$ , we obtain $\\ddot{\\Sigma }^{<,>}(t;\\Omega )&=ig^2D_0^{<,>}(t)\\ddot{G}^{<,>}(t;\\Omega ).$ The corresponding retarded and advanced components are given by $\\ddot{\\Sigma }^{R}(t;\\Omega )&=\\theta (t)[\\ddot{\\Sigma }^>(t;\\Omega )-\\ddot{\\Sigma }^<(t;\\Omega )],\\\\\\ddot{\\Sigma }^{A}(t;\\Omega )&=\\theta (-t)[\\ddot{\\Sigma }^<(t;\\Omega )-\\ddot{\\Sigma }^>(t;\\Omega )],$ where $\\theta (t)=1$ for $t\\ge 0$ and $=0$ otherwise (step function).", "In this way, the dotted DMFT naturally generates the impurity solver that is consistent with the approximation used in the equilibrium DMFT.", "The diagrammatic representation of the dotted lattice Dyson equation (REF ) and dotted Migdal approximation (REF ) are shown in Fig.", "REF .", "In particular, if we expand the dotted self-energy only with $G$ and $D_0$ , it has a ladder structure, which represents the contribution from the amplitude mode.", "[48] The technique can be applied to any models in principle, as far as the diagrammatic expression for the impurity problem is given.", "To see how it works further, let us take another prototypical example, the Hubbard model, $H&=\\sum _{ij} t_{ij} c_{i\\sigma }^\\dagger c_{j\\sigma }+U\\sum _i n_{i\\uparrow } n_{i\\downarrow },$ where $U$ is the on-site Coulomb interaction.", "The Hubbard model is of particular interest in its own right from the point of view of large nonlinear optical responses of Mott insulators.", "[82], [83] The diagrammatic approximation often used for the nonequilibrium impurity problem is the iterative perturbation theory (IPT),[84], [63] $\\Sigma _{\\sigma }^{<,>}(t,t^{\\prime })&=U^2 \\mathcal {G}_{0,\\sigma }^{<,>}(t,t^{\\prime }) \\mathcal {G}_{0,-\\sigma }^{>,<}(t^{\\prime },t) \\mathcal {G}_{0,-\\sigma }^{<,>}(t,t^{\\prime }),$ which is nothing but the bare second-order weak-coupling perturbation theory.", "The dotted impurity solution derived from IPT is given as $\\ddot{\\Sigma }_{\\sigma }^{<,>}(t;\\Omega )&=U^2 \\ddot{\\mathcal {G}}_{0,\\sigma }^{<,>}(t;\\Omega ) \\mathcal {G}_{0,-\\sigma }^{>,<}(-t) \\mathcal {G}_{0,-\\sigma }^{<,>}(t)\\\\&\\quad +U^2 e^{-2i\\Omega t} \\mathcal {G}_{0,\\sigma }^{<,>}(t) \\ddot{\\mathcal {G}}_{0,-\\sigma }^{>,<}(-t;\\Omega ) \\mathcal {G}_{0,-\\sigma }^{<,>}(t)\\\\&\\quad +U^2 \\mathcal {G}_{0,\\sigma }^{<,>}(t) \\mathcal {G}_{0,-\\sigma }^{>,<}(-t) \\ddot{\\mathcal {G}}_{0,-\\sigma }^{<,>}(t;\\Omega ).$ Note that the second term in Eq.", "(REF ) acquires a phase factor $e^{-2i\\Omega t}$ , since the definition of the dotted function [Eqs.", "(REF ), ()] is asymmetric between $t$ and $t^{\\prime }$ .", "In the symmetric case (i.e., $x=1/2$ ), the phase factor does not appear.", "Combining the dotted impurity solution with the dotted lattice and impurity Dyson equations, we can determine $\\ddot{G}$ and $\\ddot{\\Sigma }$ self-consistently.", "We summarize the algorithm flow for the dotted DMFT in Fig.", "REF .", "First, we solve the equilibrium DMFT in the real-time (real-frequency) formalism (the upper part of Fig.", "REF ).", "Once the self-consistency loop is converged, we move on to the second step of calculating the dotted functions (the lower part of Fig.", "REF ).", "We fix the external frequency $\\Omega $ , and iteratively solve the dotted DMFT self-consistency loop.", "This process runs for every chosen $\\Omega $ .", "Thus the computational cost for the dotted DMFT is roughly $N_\\Omega $ times that for the equilibrium DMFT, where $N_\\Omega $ is the number of $\\Omega $ values taken.", "After collecting the results for the set of $\\Omega $ s, we can calculate the THG susceptibility (REF ), which is obtained from the third derivative of the current (REF ) with respect to $A$ .", "The THG susceptibility is classified into the bare susceptibility $\\chi _0$ and vertex correction $\\chi _{\\rm vc}$ , $\\chi _{\\rm THG}(\\Omega )&=\\chi _0(\\Omega )+\\chi _{\\rm vc}(\\Omega ),$ according as whether it contains the derivative of the self-energy ($\\ddot{\\Sigma }$ ).", "Each term is further decomposed respectively as $\\chi _0(\\Omega )=\\sum _{i=1}^5 \\chi _0^{(i)}(\\Omega )$ and $\\chi _{\\rm vc}(\\Omega )=\\sum _{i=1}^2 \\chi _{\\rm vc}^{(i)}(\\Omega )$ , following the topological classification of the corresponding Feynman diagrams as displayed in Fig.", "REF .", "Combining Figs.", "REF and REF , one can see that $\\chi _{\\rm vc}^{(1)}$ contains the non-resonant [Fig.", "REF (a)] and mixed [Fig.", "REF (b)] diagrams, while $\\chi _{\\rm vc}^{(2)}$ contains the mixed [Fig.", "REF (b)] and resonant [Fig.", "REF (c)] ones.", "In the BCS approximation, only $\\chi _0^{(1)}, \\chi _0^{(3)}$ , and $\\chi _{\\rm vc}^{(1)}$ , which have the non-resonant coupling to the light, are nonzero as explained in the introduction, and the rest vanish exactly.", "On the other hand, when one goes beyond the BCS approximation all the terms are generally non-vanishing and cannot be neglected, so that one has to evaluate all of them.", "The explicit form of the bare susceptibilities are the following: $\\chi _0^{(1)}(\\Omega )&=-\\frac{i}{6}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\ddddot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega )]^<,\\\\\\chi _0^{(2)}(\\Omega )&=-\\frac{i}{2}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\dddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<-\\frac{i}{6}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +3\\Omega )\\dddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<,\\\\\\chi _0^{(3)}(\\Omega )&=-\\frac{i}{2}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\ddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +2\\Omega )\\ddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<,\\\\\\chi _0^{(4)}(\\Omega )&=-i\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\ddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +2\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<\\\\&\\quad -\\frac{i}{2}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +3\\Omega )\\ddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )+\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +3\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +2\\Omega )\\ddot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<,\\\\\\chi _0^{(5)}(\\Omega )&=-i\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +3\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +2\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega )]^<.$ For the notation of $<$ for products of nonequilibrium Green's functions, see Appendix .", "Note that $\\chi _0^{(i)}$ ($i=1,\\dots ,5$ ) do not contain $\\ddot{\\Sigma }$ , so that they can be computed independently of the dotted DMFT.", "The vertex corrections are also explicitly derived as $\\chi _{\\rm vc}^{(1)}(\\Omega )&=-\\frac{i}{2}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\ddot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega +2\\Omega )\\ddot{\\Sigma }(\\omega ;\\Omega )G_{\\mathbf {k}}(\\omega )]^<,\\\\\\chi _{\\rm vc}^{(2)}(\\Omega )&=-\\frac{i}{2}\\sum _{\\mathbf {k}} \\int \\frac{d\\omega }{2\\pi }[\\dot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega +3\\Omega )\\ddot{\\Sigma }(\\omega +\\Omega ;\\Omega )G_{\\mathbf {k}}(\\omega +\\Omega ) \\dot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega )+\\dot{\\epsilon }_{\\mathbf {k}} G_{\\mathbf {k}}(\\omega +3\\Omega )\\dot{\\epsilon }_{\\mathbf {k}}G_{\\mathbf {k}}(\\omega +2\\Omega ) \\ddot{\\Sigma }(\\omega ;\\Omega ) G_{\\mathbf {k}}(\\omega )]^<.$ Using the momentum integral formulae listed in Appendix , one can show that $\\chi _0^{(4)}$ , $\\chi _0^{(5)}$ , $\\chi _{\\rm vc}^{(1)}$ , and $\\chi _{\\rm vc}^{(2)}$ do not depend on the polarization parameter $\\alpha $ , while $\\chi _0^{(1)}$ , $\\chi _0^{(2)}$ , and $\\chi _0^{(3)}$ do.", "Figure: Feynman diagrams for the susceptibility for the third-harmonic generation.There are five (two) topologically different diagramsfor the bare susceptibility χ 0 \\chi _0 (vertex correction χ vc \\chi _{\\rm vc}).Solid and wavy lines represent the electron and external photon propagators, respectively,while the shaded box represents the vertex correction Σ ¨\\ddot{\\Sigma }.Among the four photon lines,one is outgoing with an energy 3Ω3\\Omega , while the other three are incoming with an energy Ω\\Omega .The photon lines attached directly to the vertex are incoming.The algorithm can be generalized to arbitrary higher-order derivatives; they are determined in turn from lower orders [$(G,\\Sigma ) \\rightarrow (\\ddot{G}, \\ddot{\\Sigma }) \\rightarrow (\\ddddot{G}, \\ddddot{\\Sigma })$ ] due to the hierarchical structure of the dotted DMFT self-consistency defined at each derivative order.", "Essentially the same method has been used to evaluate the optical conductivity for periodically driven systems in Floquet DMFT,[78] where $\\dot{G}$ and $\\dot{\\Sigma }$ are needed.", "The first derivative can be nonzero, since the parity symmetry is broken by the presence of the external driving field.", "So far we have formulated the dotted DMFT for the normal phase, but it is straightforward to extend the approach to the superconducting phase.", "There, we have to impose the following modifications: the electron Green's functions and self-energy should be represented by $2\\times 2$ matrices (Nambu-Gor'kov formalism), $\\epsilon _{\\mathbf {k}}, \\ddot{\\epsilon }_{\\mathbf {k}}$ , and $\\ddddot{\\epsilon }_{\\mathbf {k}}$ appearing in the dotted DMFT should be multiplied by $\\tau _3$ (the third component of the Pauli matrix), and the dotted impurity solution (REF ) for the Holstein model should be replaced by $\\ddot{\\hat{\\Sigma }}^{<,>}(t;\\Omega )&=ig^2D_0^{<,>}(t)\\tau _3\\ddot{\\hat{G}}^{<,>}(t;\\Omega )\\tau _3.$ Let us finally comment on the generality of the present formulation.", "Although we describe the dotted DMFT formulation for the THG susceptibility, it is not restricted to THG but can be generalized to arbitrary dynamical response functions.", "One can introduce an infinitesimal external field (not necessarily an electric field), and take the derivative with respect to it for observables.", "It may contain a derivative of the self-energy, which can be evaluated by the corresponding dotted DMFT, where the DMFT self-consistency equations are differentiated with respect to the external field.", "For example, the dynamical pair susceptibility,[40], [48] $\\chi _{\\rm pair}^R(\\Omega )&=-i\\int _0^\\infty dt e^{-i\\Omega t} \\langle [B_{\\mathbf {0}}(t), B_{\\mathbf {0}}(0)]\\rangle ,$ is defined as the response of the pairing amplitude $\\langle B_{\\mathbf {0}}\\rangle $ against an external pair potential $H_{\\rm ex}(t)=\\varepsilon B_{\\mathbf {0}}e^{-i\\Omega t}$ , where $B_{\\mathbf {0}}=\\sum _i (c_{i\\uparrow }^\\dagger c_{i\\downarrow }^\\dagger +c_{i\\downarrow }c_{i\\uparrow })$ is the bosonic pairing operator with the center-of-mass momentum $\\mathbf {q}=\\mathbf {0}$ .", "This quantity detects collective amplitude oscillations of the superconducting order parameter.", "The dotted DMFT is then constructed by differentiating the DMFT self-consistency with respect to the pair field potential.", "The resulting dotted lattice Dyson equation reads $\\dot{\\hat{G}}^{R,A,<,>}(\\omega ;\\Omega )&=\\sum _{\\mathbf {k}}\\lbrace \\hat{G}_{\\mathbf {k}}(\\omega +\\Omega )[\\tau _1+\\dot{\\hat{\\Sigma }}(\\omega ;\\Omega )]\\hat{G}_{\\mathbf {k}}(\\omega )\\rbrace ^{R,A,<,>},$ where we have adopted an extended notation of $(\\tau _1)^{R,A}=\\tau _1$ and $(\\tau _1)^{<,>}=0$ .", "Once the dotted DMFT is solved, the dynamical pair susceptibility can be calculated as $\\chi _{\\rm pair}^R(\\Omega )&=-i\\int \\frac{d\\omega }{2\\pi } {\\rm Tr} [\\tau _1 \\dot{\\hat{G}}^<(\\omega ;\\Omega )].$ In the next section, we demonstrate the results obtained with the dotted DMFT for the THG susceptibility along with the dynamical pair susceptibility." ], [ "Results", "Let us now turn to the results of the dotted DMFT for the superconducting phase of the Holstein model.", "The parameters are taken to be $g=0.8$ , $\\omega _0=0.6$ , $\\gamma =0.2$ , and $\\delta =0.005$ .", "This corresponds to the effective interaction of $\\lambda =0.77$ [Eq.", "(REF )], which is in the moderately correlated regime.", "The temperature is set to be $T=0.02$ , which is low enough for the system to be in the superconducting state.", "The polarization (REF ) is set to a general direction $\\alpha =0.5$ without having a bias on the pair breaking effect.", "In Fig.", "REF , we show the single-particle spectrum $A(\\omega )=-{\\rm Im}\\,G_{11}^R(\\omega )/\\pi $ (red curve) along with the dynamical pair susceptibility $-{\\rm Im}\\, \\chi _{\\rm pair}^R(\\omega )$ (REF ) (blue with the dots), with the latter calculated by the dotted DMFT.", "Previously, the dynamical pair susceptibility has been evaluated from the real-time simulation of the nonequilibrium DMFT,[48] which is one way to avoid solving the complicated Bethe-Salpeter equation for the vertex correction.", "Here the dotted DMFT serves as an alternative efficient method.", "As one can see in Fig.", "REF , the single-particle spectrum shows the superconducting gap $2\\Delta \\approx 0.12$ [note that we plot $A(\\omega /2)$ in Fig.", "REF ] with the coherence peak at the edge of the band gap.", "The pair susceptibility also exhibits a clear gap structure with a resonance peak at $\\omega =2\\Delta $ .", "The result is in agreement with the one previously reported.", "[48], [51] The resonance peak is produced by the vertex correction, which is immediately confirmed by the comparison to the bare susceptibility.", "This suggests that the peak in $\\chi _{\\rm pair}^R(\\omega )$ represents the collective oscillation of the pairing amplitude with the frequency $2\\Delta $ , which can be identified as the Higgs amplitude mode.", "The coincidence of the single-particle and two-particle gaps (up to the factor of 2) holds beyond the BCS approximation, as observed in the previous study.", "[48], [51] Figure: The single-particle spectrum A(ω/2)A(\\omega /2) andthe dynamical pair susceptibility - Im χ pair R (ω)-{\\rm Im}\\,\\chi _{\\rm pair}^R(\\omega )calculated by the (dotted) DMFTfor the superconducting phase of the Holstein model with g=0.8,ω 0 =0.6,T=0.02,γ=0.2g=0.8, \\omega _0=0.6, T=0.02, \\gamma =0.2, and δ=0.005\\delta =0.005.The results of the THG susceptibility $\|\\chi \|^2$ (proportional to the THG intensity observed in experiments) calculated by the dotted DMFT are plotted in Fig.", "REF .", "We show the result for each term $\\chi _0^{(i)}$ ($i=1,\\dots ,5$ ) and $\\chi _{\\rm vc}^{(i)}$ ($i=1,2$ ).", "First of all, we can see that all the terms contribute to the THG response, which is in sharp contrast to the BCS approximation where $\\chi _0^{(i)}$ ($i=2,4,5$ ) and $\\chi _{\\rm vc}^{(2)}$ identically vanish.", "In particular, $\\chi _0^{(3)}$ , $\\chi _0^{(5)}$ and $\\chi _{\\rm vc}^{(2)}$ exhibit dominant contributions.", "The resonance peak exists at $\\Omega =\\Delta \\approx 0.06$ in the spectra of $\\chi _0^{(3)}$ , $\\chi _{\\rm vc}^{(1)}$ , and $\\chi _{\\rm vc}^{(2)}$ .", "The peak in $\\chi _0^{(3)}$ can be interpreted as individual excitations due to Cooper pair breaking, while the peaks in $\\chi _{\\rm vc}^{(1)}$ and $\\chi _{\\rm vc}^{(2)}$ can be interpreted as collective excitations resonating with the Higgs amplitude mode, since that is the only known collective mode at energy $2\\Delta $ .", "As expected from the BCS approximation,[49] the effect of $\\chi _{\\rm vc}^{(1)}$ is a few orders of magnitude smaller than that of $\\chi _0^{(3)}$ (see the inset of Fig.", "REF ) if one chooses a general polarization direction (here $\\alpha =0.5$ ).", "On the other hand, the contribution of $\\chi _{\\rm vc}^{(2)}$ , which has been absent in the BCS approximation, is quite significant, and can be even larger than that of $\\chi _0^{(3)}$ .", "This result implies that if one resumes the factors that are not taken into account in the BCS approximation, such as the retarded nature of the pairing interaction through the electron-phonon coupling, the Higgs mode can become a prominent component in the THG spectrum.", "The corrections from the BCS theory are not necessarily small but can be drastic (at least when the electron-phonon coupling is large enough).", "Let us again recall that NbN, which is experimentally used in Refs.", "Matsunaga2013,Matsunaga2014, has the strong electron-phonon coupling,[55], [56], [57] so that such corrections from the BCS analysis should be seriously taken into account.", "$\\chi _0^{(5)}$ is also not negligible, but this component does not show a resonance with the Higgs mode at $\\Omega =\\Delta $ .", "The increase of the spectral weight towards low frequencies (especially for $\\chi _0^{(5)}$ and $\\chi _{\\rm vc}^{(2)}$ ) is due to the presence of nonzero $\\delta $ , with which the system accommodates low-energy excitations.", "It can be suppressed when $\\delta $ is reduced, so that we can ignore the low-energy features, although we cannot take the limit of $\\delta \\rightarrow +0$ for the dotted DMFT to be numerically stable.", "Figure: The THG susceptibility decomposed into the bare susceptibilities χ 0 (i) \\chi _0^{(i)} (i=1,⋯,5i=1,\\dots ,5)and vertex corrections χ vc (i) \\chi _{\\rm vc}^{(i)} (i=1,2i=1,2) for the superconducting phase of the Holstein modelcalculated with the dotted DMFT.The parameters are the same as in Fig.", ".The polarization direction of the laser field is taken to be α=0.5\\alpha =0.5.The susceptibilities are normalized by χ * =t * /d 2 \\chi ^\\ast =t^\\ast /d^2.The inset is a blowup of χ vc (1) \\chi _{\\rm vc}^{(1)}.Figure REF plots the total THG susceptibility $\|\\chi \|^2=\|\\chi _0+\\chi _{\\rm vc}\|^2$ as compared with the total bare susceptibility $\|\\chi _0\|^2$ .", "Here we subtract the low-energy increase of the spectral weight of $\\chi _0^{(5)}$ at $\\Omega <0.055$ from $\\chi _0$ , which is out of our interest and could be removed by reducing $\\delta $ .", "We can see that both $\\chi $ and $\\chi _0$ exhibit conspicuous resonance peaks at $\\Omega =\\Delta $ .", "Although the position and shape of the peak do not differ so much between $\\chi _0$ and $\\chi $ , the peak height does.", "With the parameters taken here, the height for $\\chi $ is enhanced about four times that for $\\chi _0$ due to the resonance with the Higgs mode.", "The main contribution comes from $\\chi _{\\rm vc}^{(2)}$ , as can be seen in Fig.", "REF .", "The resonance width for $\\chi _0$ is broadened as compared to that for $\\chi _0^{(3)}$ due to the spectral weight of $\\chi _0^{(5)}$ distributed around the peak.", "The amplitude ratio between $\\chi _0$ and $\\chi _{\\rm vc}$ can depend on various model parameters (especially we will discuss the phonon-frequency dependence below), but at least there is such a possibility in a certain realistic parameter regime that the vertex correction has a non-negligible effect.", "Figure: The intensity of the third-harmonic generation for the superconducting phase of the Holstein modelcalculated by the dotted DMFT.The bubble contribution (χ 0 \\chi _0) and the total susceptibility including the vertex corrections (χ 0 +χ vc \\chi _0+\\chi _{\\rm vc}) are plotted.The parameters are taken to be the same as those of Fig.", ".The polarization direction of the laser field is taken to be α=0.5\\alpha =0.5.We are now in position to compare the BCS and DMFT results by calculating the THG susceptibility within the BCS approximation for the same parameter set as those for DMFT.", "The electron-phonon coupling is translated into a static attractive interaction via $U=2g^2\\omega _0/(\\omega _0^2+\\gamma ^2)$ [see Eqs.", "(REF ) and (REF )].", "In the gap equation, we perform the momentum integral in the range of $\|\\epsilon _{\\mathbf {k}}\|\\le \\omega _0$ .", "The result is displayed in Fig.", "REF , which indicates that the effect of the vertex correction in BCS is rather small for a general polarization direction ($\\alpha =0.5$ here).", "The resonance width is much sharper and the peak height is higher in BCS than in DMFT, since the THG susceptibility diverges at $\\Omega =\\Delta $ in the limit of $\\delta \\rightarrow 0$ in the BCS theory.", "While these are consistent with the previous studies,[44], [49] the BCS result is markedly different from the DMFT result (Fig.", "REF ) that takes account of dynamical correlation effects.", "This is simply because $\\chi _{\\rm vc}^{(2)}$ is absent in the BCS approximation, whereas it is generally non-negligible if one considers the retardation in the phonon-mediated interaction (or other effects that are not included in the BCS approximation such as impurity scattering, Coulomb interaction, etc.).", "Figure: The intensity of the third-harmonic generation for the superconducting phase of the Holstein modelcalculated within the BCS approximation.The bubble contribution (χ 0 \\chi _0) and the total susceptibility including the vertex corrections (χ 0 +χ vc \\chi _0+\\chi _{\\rm vc}) are plotted respectively.The parameters are taken to be the same as in Fig.", ".The polarization direction of the laser field is taken to be α=0.5\\alpha =0.5.Note the difference in the scales of the axes from Fig.", ".To confirm that the retardation effect is essential in enhancing the contribution of the Higgs mode to the THG resonance, we calculate $\\omega _0$ dependence of the THG susceptibility.", "Here we focus on $\\chi _0^{(3)}(\\Omega )$ and $\\chi _{\\rm vc}^{(2)}(\\Omega )$ that are in charge of the resonance structures at $\\Omega =\\Delta $ .", "A systematic comparison between the susceptibilities at different $\\omega _0$ is made by tuning the electron-phonon coupling $g$ such that the superconducting gap is fixed to a constant ($2\\Delta \\approx 0.12$ ).", "In Fig.", "REF , we plot the height of the resonance peak for $\|\\chi _0^{(3)}(\\Omega )\|^2$ and $\|\\chi _{\\rm vc}^{(2)}(\\Omega )\|^2$ as a function of the phonon frequency $\\omega _0$ (with the parameters other than $\\omega _0$ and $g$ the same as in Fig.", "REF ).", "We can see that, as $\\omega _0$ decreases and the effective interaction (REF ) becomes more retarded, the resonance for $\\chi _{\\rm vc}^{(2)}$ is enhanced while that for $\\chi _0^{(3)}$ is suppressed.", "This is consistent with the expectation that in the opposite antiadiabatic (non-retarded) limit ($\\omega _0\\rightarrow \\infty $ ) the model approaches the attractive Hubbard model, where the Migdal approximation is replaced by the BCS approximation, and $\\chi _{\\rm vc}^{(2)}$ vanishes as explained in Sec. .", "The result suggests that the retardation effect in the electron-phonon coupling indeed plays a crucial role in amplifying the vertex correction $\\chi _{\\rm vc}^{(2)}$ .", "We can elaborate the physical meaning of the result as follows.", "As we discussed previously, the dominant diagrams contained in $\\chi _{\\rm vc}^{(2)}$ is the one with the resonant coupling to the light [Fig.", "REF (c)].", "This represents a process in which a single photon is absorbed and then emitted by electrons at different times (with the time separation $\\sim (2\\Delta )^{-1}$ ).", "The retardation effect due to the scattering of phonons (in the time scale of $\\omega _0^{-1}$ ) can propagate between these times.", "If $2\\Delta $ ($\\approx 0.12$ in the present case) and $\\omega _0$ are in the same order, the scattering amplitude relevant for the THG resonance can be effectively enhanced, as confirmed from the result in Fig.", "REF .", "Note that the resonance between coherent phonons and the order parameter oscillation in the regime of $\\omega _0\\sim 2\\Delta $ has been discussed in Ref.", "Schnyder2011." ], [ "Summary", "To summarize, we have studied the nonlinear optical response, especially the third-harmonic generation, for electron-phonon coupled superconductors by means of the dotted DMFT framework proposed in the present paper.", "The results show that, for general polarization of the light, there is a possibility that the Higgs amplitude mode can contribute to the THG resonance at $2\\Omega =2\\Delta $ with an order of magnitude comparable to contributions from the Cooper pair breaking or charge density fluctuations, which is in sharp contrast to the BCS result.", "The interaction between the light and Higgs mode can be mediated by the resonant coupling, which is induced by the retarded interaction through the electron-phonon coupling.", "This is confirmed by the observation that the intensity of the THG resonance due to the Higgs mode does indeed increase as the phonon frequency is reduced.", "Let us note that the electron-phonon coupling is just one of many possibilities that could enhance the Higgs-mode effect.", "These may include phonon renormalization, impurity scattering, dynamical correlation effects from the Coulomb interaction, non-local correlations beyond DMFT, etc.", "Figure: Phonon-frequency dependence of \|χ 0 (3) \| 2 \|\\chi _0^{(3)}\|^2 and \|χ vc (2) \| 2 \|\\chi _{\\rm vc}^{(2)}\|^2 at the resonance (Ω=Δ\\Omega =\\Delta )for the superconducting phase of the Holstein model with T=0.02T=0.02, γ=0.2\\gamma =0.2, and δ=0.005\\delta =0.005.The polarization direction of the laser field is taken to be α=0.5\\alpha =0.5.We tune gg for each ω 0 \\omega _0 such that the superconducting gap is fixed to a constant (2Δ≈0.122\\Delta \\approx 0.12).To make a relevance to the experiment,[12] it is interesting to investigate temperature dependence of the THG susceptibility.", "However, we expect that this strongly depends on details of the model, since the model adopted here only includes a single optical phonon mode, while in realistic situations acoustic phonons may play an important role at low temperatures.", "The temperature dependence may also be affected by mechanisms of energy dissipation.", "In this paper, we have assumed a simple dissipation characterized by the broadening parameters $\\gamma $ and $\\delta $ , but it can be more complicated in real systems.", "Moreover, the present method has a numerical instability when $\\gamma $ or $\\delta $ approaches zero.", "These issues will be left as a future problem.", "When both the pair breaking (charge fluctuation) and Higgs mode contribute with comparable magnitudes, as indicated here to be possible, it is desirable to distinguish them in experiments.", "One possibility is to look at the polarization dependence.", "[49] To this end, one needs to accurately evaluate the polarization dependence of the pair breaking and Higgs-mode contributions in a realistic manner.", "We wish to thank R. Shimano, R. Matsunaga, and Y. Murotani for illuminating discussions.", "H.A.", "is supported by JSPS KAKENHI (Grant No.", "26247057) from MEXT and ImPACT project (No.", "2015-PM12-05-01) from JST.", "N.T.", "is supported by JSPS KAKENHI Grants No.", "25800192 and No.", "16K17729." ], [ "Notation for products of nonequilibrium Green's functions", "In this appendix, we explain the notation for products of nonequilibrium Green's functions used throughout the paper.", "We define $[A(\\omega )B(\\omega ^{\\prime })]^{R,A}&\\equiv A^{R,A}(\\omega ) B^{R,A}(\\omega ^{\\prime }),\\\\[A(\\omega )B(\\omega ^{\\prime })]^{<,>}&\\equiv A^{R}(\\omega ) B^{<,>}(\\omega ^{\\prime })+A^{<,>}(\\omega ) B^{A}(\\omega ^{\\prime }),$ following the Langreth rule.", "[54] Here $R, A, <$ , and $>$ respectively denote the retarded, advanced, lesser, and greater components of nonequilibrium Green's functions.", "For the detailed definition of nonequilibrium Green's function, we refer to Ref. noneqDMFTreview.", "The definition (REF ), () can be used repeatedly for products involving more than two Green's functions.", "For example, it follows from the above definition that $[A(\\omega )B(\\omega ^{\\prime })C(\\omega ^{\\prime \\prime })]^{R,A}&=A^{R,A}(\\omega ) B^{R,A}(\\omega ^{\\prime }) C^{R,A}(\\omega ^{\\prime \\prime }),\\\\[A(\\omega )B(\\omega ^{\\prime })C(\\omega ^{\\prime \\prime })]^{<,>}&=A^{R}(\\omega ) B^{R}(\\omega ^{\\prime }) C^{<,>}(\\omega ^{\\prime \\prime })\\\\&\\quad +A^{R}(\\omega ) B^{<,>}(\\omega ^{\\prime }) C^{A}(\\omega ^{\\prime \\prime })\\\\&\\quad +A^{<,>}(\\omega ) B^{A}(\\omega ^{\\prime }) C^{A}(\\omega ^{\\prime \\prime }).$ If products explicitly contain $\\epsilon _{\\mathbf {k}}$ , we can regard it as a component of Green's function with the definition, $[\\epsilon _{\\mathbf {k}}]^{R,A}&\\equiv \\epsilon _{\\mathbf {k}},\\\\[\\epsilon _{\\mathbf {k}}]^{<,>}&\\equiv 0.$ For example, we have $[\\epsilon _{\\mathbf {k}} A(\\omega )B(\\omega ^{\\prime })]^{<,>}&=\\epsilon _{\\mathbf {k}}[A^{R}(\\omega ) B^{<,>}(\\omega ^{\\prime })+A^{<,>}(\\omega ) B^{A}(\\omega ^{\\prime })].$ The same applies to derivatives of $\\epsilon _{\\mathbf {k}}$ such as $\\dot{\\epsilon }_{\\mathbf {k}}, \\ddot{\\epsilon }_{\\mathbf {k}}$ ." ], [ "A momentum integral formula for the calculation of THG", "In this appendix, we summarize some useful formulae for the momentum integral employed in the calculation of the THG susceptibility in the dotted DMFT.", "As we have seen in Sec.", ", one frequently encounters a momentum integral of a function of $\\epsilon _{\\mathbf {k}}$ [Eq.", "(REF )] multiplied by some of $\\dot{\\epsilon }_{\\mathbf {k}}, \\ddot{\\epsilon }_{\\mathbf {k}}, \\dddot{\\epsilon }_{\\mathbf {k}}$ , and $\\ddddot{\\epsilon }_{\\mathbf {k}}$ [see Eqs.", "(REF )-() for the definition].", "In the main text, we have taken the polarization vector as $\\mathbf {e}&=\\frac{1}{\\sqrt{m}}(\\overbrace{\\underbrace{1,1,\\dots ,1}_m,0,\\dots ,0}^d).$ In the limit $d,m\\rightarrow \\infty $ with a fixed ratio $\\alpha =m/d$ , the momentum integral is reduced to an integral over the single variable $\\epsilon =\\epsilon _{\\mathbf {k}}$ , as described in Sec. .", "Here we list the results.", "For momentum integrals containing two derivatives, we have $\\sum _{\\mathbf {k}} \\ddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})&=-\\frac{1}{d} \\sum _{\\mathbf {k}} \\epsilon _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),\\\\\\sum _{\\mathbf {k}} (\\dot{\\epsilon }_{\\mathbf {k}})^2 f(\\epsilon _{\\mathbf {k}})&=\\frac{t^\\ast {}^2}{d} \\sum _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),$ which can be used in the calculation of $\\chi _{\\rm vc}^{(i)}$ [Eqs.", "(REF )-()] and the dotted lattice Dyson equation (REF ).", "For momentum integrals containing four derivatives, we have $\\sum _{\\mathbf {k}} \\ddddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})&=\\frac{1}{d^2\\alpha }\\sum _{\\mathbf {k}} \\epsilon _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),\\\\\\sum _{\\mathbf {k}} \\dot{\\epsilon }_{\\mathbf {k}} \\dddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})&=-\\frac{t^\\ast {}^2}{d^2\\alpha }\\sum _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),\\\\\\sum _{\\mathbf {k}} (\\ddot{\\epsilon }_{\\mathbf {k}})^2 f(\\epsilon _{\\mathbf {k}})&=\\frac{t^\\ast {}^2(1-\\alpha )}{d^2\\alpha }\\sum _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})+\\frac{1}{d^2}\\sum _{\\mathbf {k}} \\epsilon _{\\mathbf {k}}^2 f(\\epsilon _{\\mathbf {k}}),\\\\\\sum _{\\mathbf {k}} (\\dot{\\epsilon }_{\\mathbf {k}})^2 \\ddot{\\epsilon }_{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}})&=-\\frac{t^\\ast {}^2}{d^2}\\sum _{\\mathbf {k}} \\epsilon _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),\\\\\\sum _{\\mathbf {k}} (\\dot{\\epsilon }_{\\mathbf {k}})^4 f(\\epsilon _{\\mathbf {k}})&=\\frac{3t^\\ast {}^4}{d^2}\\sum _{\\mathbf {k}} f(\\epsilon _{\\mathbf {k}}),$ which can be used in the calculation of $\\chi _0^{(i)}$ [Eqs.", "(REF )-()].", "Once the momentum integral is reduced to an integral over $\\epsilon =\\epsilon _{\\mathbf {k}}$ , it can be further evaluated analytically.", "To see this, let us parametrize the self-energy (at half filling with $\\mu =0$ ) as $\\hat{\\Sigma }^R(\\omega )&=[1-Z(\\omega )]\\omega +\\phi (\\omega )\\tau _1.$ We define functions $\\hat{S}^\\pm (\\omega )&\\equiv Z(\\omega )\\omega \\pm \\phi (\\omega )\\tau _1,\\\\S^2(\\omega )&\\equiv [Z(\\omega )\\omega ]^2-\\phi (\\omega )^2,$ with which the retarded Green's function is represented as $\\hat{G}_{\\mathbf {k}}^R(\\omega )&=[\\omega -\\epsilon _{\\mathbf {k}}\\tau _3-\\hat{\\Sigma }^R(\\omega )]^{-1}=\\frac{\\hat{S}^+(\\omega )+\\epsilon _{\\mathbf {k}}\\tau _3}{S^2(\\omega )-\\epsilon _{\\mathbf {k}}^2}.$ As an example, let us consider the first term in the dotted lattice Dyson equation (REF ), whose retarded component can be evaluated with Eq.", "(REF ) as $&\\sum _{\\mathbf {k}} \\hat{G}_{\\mathbf {k}}^R(\\omega +2\\Omega )\\ddot{\\epsilon }_{\\mathbf {k}}\\tau _3 \\hat{G}_{\\mathbf {k}}^R(\\omega )\\\\&=-\\frac{1}{d}\\sum _{\\mathbf {k}}\\frac{\\hat{S}^+(\\omega +2\\Omega )+\\epsilon _{\\mathbf {k}}\\tau _3}{S^2(\\omega +2\\Omega )-\\epsilon _{\\mathbf {k}}^2}\\epsilon _{\\mathbf {k}}\\tau _3\\frac{\\hat{S}^+(\\omega )+\\epsilon _{\\mathbf {k}}\\tau _3}{S^2(\\omega )-\\epsilon _{\\mathbf {k}}^2}\\\\&=\\frac{1}{d}\\frac{\\hat{G}^R(\\omega )\\hat{S}^-(\\omega )-\\hat{G}^R(\\omega +2\\Omega )\\hat{S}^-(\\omega +2\\Omega )}{S^2(\\omega )-S^2(\\omega +2\\Omega )}\\\\&\\quad \\times [\\hat{S}^+(\\omega )+\\hat{S}^+(\\omega +2\\Omega )].$ Note that $\\hat{G}^R(\\omega )$ and $\\hat{S}^\\pm (\\omega ^{\\prime })$ commute with each other.", "In this way, every momentum integral appearing in the calculation of the dotted DMFT and nonlinear optical susceptibilities can be written in terms of the local Green's function and self-energy.", "This greatly reduces the computational cost of the dotted DMFT algorithm." ] ]	1606.05024
[ [ "On the Astrid asteroid family" ], [ "Abstract Among asteroid families, the Astrid family is peculiar because of its unusual inclination distribution.", "Objects at $a\\simeq$~2.764 au are quite dispersed in this orbital element, giving the family a \"crab-like\" appearance.", "Recent works showed that this feature is caused by the interaction of the family with the $s-s_C$ nodal secular resonance with Ceres, that spreads the inclination of asteroids near its separatrix.", "As a consequence, the currently observed distribution of the $v_W$ component of terminal ejection velocities obtained from inverting Gauss equation is quite leptokurtic, since this parameter mostly depends on the asteroids inclination.", "The peculiar orbital configuration of the Astrid family can be used to set constraints on key parameters describing the strength of the Yarkovsky force, such as the bulk and surface density and the thermal conductivity of surface material.", "By simulating various fictitious families with different values of these parameters, and by demanding that the current value of the kurtosis of the distribution in $v_W$ be reached over the estimated lifetime of the family, we obtained that the thermal conductivity of Astrid family members should be $\\simeq$ 0.001 W/m/K, and that the surface and bulk density should be higher than 1000 kg/m$^{3}$.", "Monte Carlo methods simulating Yarkovsky and stochastic YORP evolution of the Astrid family show its age to be $T$ = 140$\\pm$30 Myr old, in good agreement with estimates from other groups.", "Its terminal ejection velocity parameter is in the range $V_{EJ}= 5^{+17}_{-5}$~m/s.]", "Values of $V_{EJ}$ larger than 25 m/s are excluded from constraints from the current inclination distribution." ], [ "Introduction", "The Astrid asteroid family is characterized by an unusual distribution in the $(a,\\sin {(i)})$ plane, with a dispersion in inclination of its members at $a \\simeq $ 2.764 au much larger than that of members at other semi-major axis.", "[23] recently showed that this feature of the Astrid family is caused by its interaction with the $s-s_C$ nodal secular resonance with Ceres.", "Asteroid crossing this resonance are significantly dispersed in inclination, causing the crab-like appearance of the family.", "The unusual distribution in inclination of the Astrid family also produces other consequences.", "[12] observed that the current distribution of the $v_W$ component of terminal ejection velocities field computed from inverting Gauss equation for this family is characterized by a leptokurtic distribution, i.e., a distribution with larger tails and more peaked than a Gaussian.", "If we define as kurtosis the ratio of the fourth momenta of a distribution with respect to the fourth power of its standard deviation, that for a distribution of n random variable $x_i$ is given by: $k=\\frac{\\frac{1}{n}\\sum _{i=1}^{n}{(x_i-<x>)^4}}{(\\frac{1}{n}\\sum _{i=1}^{n}{(x_i-<x>)^2)^2}},$ where $<x>=\\frac{1}{n}\\sum _{i=1}^{n}{x_i}$ is the mean value of the distribution, then Pearson ${\\gamma }_2$ kurtosis is equal to k-3.", "Gaussian distributions are characterized by values of ${\\gamma }_2$ equal to 0.", "The value of the Pearson ${\\gamma }_2$ parameter for the whole Astrid family is quite high, but is closer to mesokurtic values if asteroids in the resonant region are excluded.", "In this work we investigate what information on key parameters describing the Yarkovsky effect, such as the thermal conductivity of material on the surface and the mass density, can be obtained by studying the orbital diffusion of fictitious members of several simulated Astrid families.", "By checking on what time-scales the current value of ${\\gamma }_2(v_W)$ can be reached, and for what values of the parameters describing the Yarkovsky force, constraints on the allowed range of values of these parameters can be, in principle, obtained.", "The independent constraints provided by secular dynamics (and from the current inclination distribution of Astrid members) could then be used to estimate the age of the Astrid family with a higher precision than that available for other families." ], [ "Family identification and local dynamics", "As a first step in our analysis we selected the Astrid family, as identified in [21] using the Hierarchical Clustering Method (HCM, [2]) and a cutoff of 60 m/s.", "489 members of the Astrid dynamical group were identified in that work.", "We also selected asteroids in the background of the family, defined as a box in the $(a,e,\\sin {(i)})$ domain.", "We selected asteroids to within the minimum and maximum values of Astrid proper elements, plus or minus 0.02 au, 0.02, and 0.02 in proper $a$ ,$e$ , and $\\sin {(i)}$ , respectively, with the exception of the maximum values in $a$ that was given by the semi-major axis of the center of the 5J:-2A mean-motion resonance.", "588 asteroids, 99 of which not members of the Astrid group, were identified in the background of the family so defined.", "Figure: A (a,e)(a,e) (top panel) and (a,sin(i)(a,\\sin {(i)} (bottom panel) projectionof members of the HCM Astrid cluster (489 members, black full dots),and of the local background (588 members, black open dots).", "Verticallines display the location of the local mean-motion resonances.", "Theorbital location of 1128 Astrid is identified by a large black circleand it is labeled.Figure: An (a,sin(i)(a,\\sin {(i)} projection of the 20 asteroids with taxonomicinformation (panel A) and of the 207 bodies with WISE albedo (panel B).See figures legends for the meaning of the used symbols.Fig.", "REF displays the orbital location of family members (black full dots) and local background asteroids (black open dots) in the $(a,e)$ (top panel) and $(a,\\sin {(i)})$ (bottom panel) domains.", "The Astrid family numerically dominates the population in the local background: 83.1% of the asteroids in the region are members of the HCM family.", "One can also notice the spread in $\\sin {(i)}$ of Astrid members at $a \\simeq 2.765$ au, caused by the nodal linear secular resonance with Ceres $s-s_C$ , as shown in [23].", "We then turned our attention to the physical properties of objects in the Astrid region.", "We checked which asteroids have information in the three major photometric/spectroscopic surveys (ECAS (Eight-Color Asteroid Analysis, [27]), SMASS (Small Main Belt Spectroscopic Survey, [6], [7]), and S3OS2 (Small Solar System Objects Spectroscopic Survey, [16]), in the Sloan Digital Sky Survey-Moving Object Catalog data, fourth release (SDSS-MOC4 hereafter, [14]), and in the WISE survey [18].", "Taxonomic information was deduced for the SDSS-MOC4 objects using the method of [13].", "We obtained taxonomic information for 20 asteroids, while 207 bodies had values of geometric albedo in the WISE data-set.", "Fig.", "REF displays our results for these objects.", "The Astrid family is a C-complex family, and C-complex objects dominate the local background: out of 207 bodies with information on geometric albedo, only 5 (2.4% of the total) have $p_V > 0.12$ , and are possibly associated with a S-complex composition.", "No taxonomic or albedo interlopers were identified in the Astrid HCM group.", "How much the local dynamics is responsible for the current shape of the Astrid family?", "To answer this question, we obtained dynamical maps in the domain of proper $(a,\\sin {(i)})$ with the method described in [9], based on the theory developed by [15].", "We integrated 1550 particles over 20 Myr under the gravitation influence of i) all planets and ii) all planets plus Ceres as a massive bodyThe mass of Ceres was assumed to be equal to $9.39 \\cdot 10^{20}$ kg, as determined by the Dawn spacecraft [25].", "with $SWIFT\\_MVSF$ , the symplectic integrator based on $SWIFT\\_MVS$ from the Swift package of [17], and modified by [4] to include on line filtering of osculating elements.", "The initial osculating elements of the particles went from 2.730 to 2.828 au in $a$ and from $1.00^{\\circ }$ to $2.45^{\\circ }$ in $i$ .", "We used 50 intervals in $a$ and 31 in $i$ .", "The other orbital elements of the test particles were set equal to those of Ceres at the modified Julian date of 57200.", "Figure: Dynamical maps for the orbital region of Astrid obtained byintegrating test particles under the influence of all planets (panel A),and all planets and Ceres as a massive body (panel B).", "Unstableregions associated with mean-motion resonances appear as verticalstrips.", "Secular resonance appear as inclined bands ofaligned dots.", "Dynamically stable regions are shown as uniformlycovered by black dots.", "Vertical lines display the location of the mainmean-motion resonances in the area.", "Black filled dots in panel B showthe locations of “likely resonators” in the s-s C s-s_C secularresonance.", "Likely resonators in the s-s C -g 5 +g 7 s-s_C-g_5+g_7 and s-s C -2(g 5 +g 7 )s-s_C-2(g_5+g_7)resonances are shown as full squares and full hexagons,respectively.Fig.", "REF displays our results for the two maps.", "For the case without Ceres (panel A) the orbital region of the Astrid family is quite stable and regular, with most of the perturbations caused the 3J:-1S:-1A and 5J:-2A mean-motion resonances.", "More interesting is the case where Ceres was treated like a massive body (panel B).", "As observed by [23], the linear nodal secular resonance $s-s_C$ now appears in the region.", "Objects whose pericenter frequency is within $\\pm 0.3$ arc-sec/yr from $s_C = -59.17$ arc-sec/yr, likely resonators in the terminology of [8], are shown as black full dots in this figure.", "Two other secular resonances involving the nodal frequency $s_C$ of Ceres are also observed.", "Since the difference for the values of the $g_5$ and $g_7$ precession frequency of the pericenter of Jupiter and Uranus is small (4.257 and 3.093 arcsec/yr, respectively, which yield a difference of 1.164 arcsec/yr [15]), resonances of resonant argument involving $s-s_C$ and combinations of these two frequencies that satisfy the D'Alembert rules of permissible arguments are close in proper element space with respect to the main resonance $s-s_C$ .", "In this work we called such resonances “harmonics” of the main resonance.", "We identified the $s-s_C-g_5+g_7$ and $s-s_C-2(g_5+g_7)$ harmonics, whose likely resonators are shown in Fig.", "REF as full squares and full hexagons, respectively.", "Figure: An (a,sin(i))(a,\\sin {(i)}) projection ofthe 489 HCM Astrid asteroids, with the likely resonators shown in thesame symbols code as in Fig.", "(panel A).Panel B show a projection in the (sin(i/2)cos(Ω-Ω C ),sin(i/2)sin(Ω-Ω C )(\\sin {(i/2)} \\cos {(\\Omega -{\\Omega }_{C})},\\sin {(i/2)} \\sin {(\\Omega -{\\Omega }_{C})} of the 19 asteroidsobserved to be in librating states of the s-s C s-s_C resonance.To study the resonant dynamics of the Astrid family members, we integrated the 489 HCM Astrid asteroids with the same scheme used to obtain the dynamical map in Fig.", "REF , panel B.", "We then i) identify the likely resonators in the $s-s_C$ resonance, and studied the time evolution of the resonant argument $\\Omega -{\\Omega }_C$ .", "We identified 96 likely resonators, and 19 objects (19.8% of the total) whose resonant argument librated around $\\pm 90^{\\circ }$ for 20 Myr, the length of the integration.", "Unfortunately, the limited number of objects in librating states of the $s-s_C$ resonance does not allow to use conserved quantities of this resonance to obtain information on the initial ejection velocity field, as done by [30] for the Agnia family and the $z_1$ secular resonance, or, more recently, by [11] for the Erigone family and the $z_2$ resonance.", "No asteroid was identified in librating states of the $s-s_C-g_5+g_7$ , $s-s_C-2(g_5+g_7)$ , and $s-s_c + g_5-2g_6+g_c$ resonances.", "We then computed proper values of the resonant frequency $s$ , its amplitude $\\sin {(i/2)}$ , and its phase $\\Omega $ for the 19 resonant objects and Ceres itself.", "Fig.", "REF displays an $(a,\\sin {(i)})$ projection of the 489 HCM Astrid asteroids, with the likely resonators shown in the same symbol code as in Fig.", "REF (panel A).", "Panel B show a projection in the $(\\sin {(i/2)} \\cos {(\\Omega -{\\Omega }_{C})},\\sin {(i/2)} \\sin {(\\Omega -{\\Omega }_{C})}$ of the 19 asteroids observed to be in librating states of the $s-s_C$ resonance.", "One can notice that i), as observed from [23], the spread in $\\sin {(i)}$ of Astrid family members is indeed caused by the $s-s_C$ nodal resonance, and that, ii) resonant asteroids seems to oscillate around the stable point at $\\Omega -{\\Omega }_{C} = 0^{\\circ }$ .", "No other stable point was identified in this work, and the width of the $s-s_C$ resonance is equal to $0.8$ arcsec/yr.", "To check how fast an initially tight cluster in the $(\\sin {(i/2)} \\cos {(\\Omega -{\\Omega }_{C})},\\sin {(i/2)} \\sin {(\\Omega -{\\Omega }_{C})}$ would be dispersed beyond recognition, so losing information about its initial configuration, we followed the approach of [30].", "We generated 81 clones of 183405 2002 YE4, the lowest numbered object in a librating state of the $s-s_C$ resonance.", "The clones are in a 9 by 9 grid in eccentricity and inclination, with a step of 0.00001 in eccentricity and 0.0001 in inclination, and the elements of 183405 as central values of the grid.", "As observed for the $z_2$ resonant asteroids in the Erigone family ([11], Fig.", "9), the initially tight cluster becomes uniformly dispersed along the separatrix of the $s-s_C$ resonance.", "To quantify this effect, we used the polar angle $\\Phi $ in the $(\\sin {(i/2)} \\cos {(\\Omega -{\\Omega }_{C})},\\sin {(i/2)} \\sin {(\\Omega -{\\Omega }_{C})}$ plane, as defined in [30].", "At each step of the numerical simulation, we computed the dispersion $D²_{\\Phi }$ in the polar angle $\\Phi $ defined as: $D^2_{\\Phi }=\\frac{1}{N(N-1)}{\\displaystyle \\sum }_{i \\ne j}({\\Phi }_i-{\\Phi }_j)^2,$ Figure: Temporal evolution of D Φ 2 D^2_{\\Phi } ofEq.", "for the 81 clones of 183405.", "The horizontalblack line display the level corresponding to an uniform distributionof bodies along a circle .", "The dotted linedisplay the median value of D Φ 2 D^2_{\\Phi } during the simulation.where N = 81 is the number of integrated bodies and ${\\Phi }_i$ is the polar angle of the $i$ -th body (i = 1,...,N).", "Since we started with a compact cluster, $D^2_{\\Phi }$ is initially small ($\\simeq 6.61^{\\circ }$ ), but grows with time because of the differential libration of the bodies in the resonance (Fig.", "REF ).", "After only $\\simeq $ 12 Myr, i.e., about two libration cycles of the $s-s_c$ resonance for 183405, the value of $D^2_{\\Phi }$ saturates at $\\simeq 103^{\\circ }$ , which corresponds to an uniform distribution of bodies along a circle [30].", "This sets a lower limit on the timescale for dispersion of asteroids in the $(\\sin {(i/2)} \\cos {(\\Omega -{\\Omega }_{C})},\\sin {(i/2)} \\sin {(\\Omega -{\\Omega }_{C})}$ plane.", "Any family that reached this resonance more than $\\simeq $ 12 Myr ago, would have had its members completely dispersed along the separatrix of the $s-s_c$ resonance, which suggests that Astrid resonant members reached this resonance more than 12 Myr ago." ], [ "Constraints on terminal ejection velocities from the current\ninclination distribution", "The Astrid family is the product of a relatively recent collision: [21] estimate its age to be $140\\pm 10$ Myr, while [26], using a V-shape criteria, estimate the family to be $150\\pm 32$ Myr old.", "Monte Carlo methods [19], [29], [30], [31] that simulates the evolution of the family caused by the Yarkovsky and YORP effects, where YORP stands for Yarkovsky-O'Keefe-Radzievskii-Paddack effect, could also be used to obtain estimates of the age and terminal ejection velocities of the family members (these models will be referred as “Yarko-Yorp” models hereafter).", "However, the age estimates from these methods depend on key parameters describing the strength of the Yarkovsky force, such as the thermal conductivity $K$ and bulk and surface density ${\\rho }_{bulk}$ and ${\\rho }_{surf}$ , that are in many cases poorly known.", "Before attempting our own estimate of the family age and terminal ejection velocity field, here we analyze what constraints could be obtained on the possible values of terminal ejection velocities of the original Astrid family from its current inclination distribution.", "In the Yarko-Yorp models, fictitious families are generated considering an isotropic velocity fieldNot all ejection velocities field are isotropic.", "If the fragmentation was not completely catastrophic, terminal velocities could be rather anisotropic.", "This could actually be the case for the Astrid family, as also discussed later in this paper.", "However, since in this section we are just interested in setting constraints to the maximum magnitude of the possible ejection velocity field, we prefer for this purpose to use a simpler approach., and assuming that the fragments are dispersed with a Gaussian distribution whose standard deviation follows the relationship: $V_{SD}=V_{EJ}\\cdot (5km/D),$ where $V_{EJ}$ is the terminal ejection velocity parameter to be estimated, and $D$ is the asteroid diameter.", "[21] estimated that the parent body of the Astrid family was 42.0 km in diameter, which yields an escape velocity of 33.0 m/s.", "Assuming that the $V_{EJ}$ parameter of the terminal ejection velocity field would be in the range $0.2 < \\beta < 1.5$ , with $\\beta = V_{EJ}/V_{esc}$ , as observed for most families in the main belt [12], then, expected values of $V_{EJ}$ would be in the range from 5 to 50 m/s.", "If we only consider objects with $a > 2.77$ au, so as to eliminate the asteroids that interacted with the $s-s_C$ resonance, then the currently observed minimum and maximum values of $\\sin {(i)}$ of family members are 0.0086 and 0.0148, respectively.", "Neglecting possible changes in $\\sin {(i)}$ after the family formation, which is motivated by the fact that the local dynamics does not seems to particularly affect asteroids in this region (see Fig.", "REF ), and will be further investigated later on, these values set constraints on the possible terminal ejection velocity parameter $V_{EJ}$ with which the family was created.", "Currently, only 7 objects not members of the family are observed at sines of inclinations lower that 0.016, i.e., 1.5% of the current number of family members.", "We generated synthetic families for values of $V_{EJ}$ from 5 m/s up to 40 m/s.", "Fig.", "REF show an $(a,\\sin {(i)}$ projection of the initial orbital dispersion of the members of the family generated for $V_{EJ} = 25$ m/s (panel A) and $V_{EJ} = 40$ m/s.", "Figure: An (a,sin(i)(a,\\sin {(i)} projection of the initial orbital dispersionof a family generated with V EJ =25V_{EJ} = 25 m/s (panel A) and V EJ =40V_{EJ} = 40 m/s(panel B).", "The full black circle identifies the location of 1128 Astrid(that essentially corresponds with the family barycenter), while thedashed lines show the minimum and maximum values of sin(i)\\sin {(i)} currentlyobserved for members of the Astrid family with a>2.77a > 2.77 au, i.e., thosethat did not yet interacted with the s-s C s-s_C secular resonance.", "The othersymbols have the same meaning as in Fig.", ".For $V_{EJ} = 25$ m/s 7 particles (1.5% of the total) had values of $\\sin {(i)}$ outside the range of values currently observed, while for $V_{EJ} = 40$ m/s these number was 55 (11.5% of the total).", "Based on these considerations, it seems unlikely that the ejection velocity parameter $V_{EJ}$ was larger than 25 m/s, or a larger number of asteroids outside the Astrid family at $a > 2.77$ au would be visible today.", "This implies that $\\beta = \\frac{V_{EJ}}{V_{esc}}$ was at most 0.76, excluding larger values associated with more catastrophic events." ], [ "Ejection velocities evolution", "[12] recently investigated the shape of the current distribution of the $v_W$ component of terminal ejection velocity fields and argued that families that were produced with a $V_{EJ}$ parameter smaller than the escape velocity from the parent body, are relatively young, and are located in dynamically less active regions, as is the case of the Astrid family, should be characterized by a leptokurtic distribution of the $v_W$ component.", "This because, assuming that initial ejection velocities followed a Gaussian distribution, fragments with initial ejection velocities less than the escape velocity from the parent body would not be able to escape.", "This would produce a distribution of ejection velocities more peaked and with larger tails than a Gaussian one, i.e., leptokurtic.", "While the subsequent dynamical evolution would tend to cause the distribution of ejection velocities to be more mesokurtic, this effect would be less intense for families such Astrid, that are both relatively young and in dynamically less active regions.", "One would therefore expect Astrid to be a relatively leptokurtic family.", "However, as also noticed in [12], the effect of the $s-s_C$ secular resonance tend to increase the dispersion in inclination values of the family members, and therefore of $v_W$ .", "While the current value of ${\\gamma }_2$ , the parameter associated with the kurtosis of the $v_W$ distribution (equal to 0 for mesokurtic or Gaussian distributions) of the whole Astrid family is quite large, ( ${\\gamma }_2 = 4.43$ ), if we only consider objects with $a > 2.77$ au that did not interacted with the secular resonance, the value of ${\\gamma }_2$ is just 0.39, more compatible with a relatively somewhat leptokurtic family.", "This shows that most of the leptokurtic shape of the currently observed Astrid family is therefore caused by the interaction of its members with the $s-s_C$ secular resonance.", "To investigate what information the $v_W$ component of the terminal ejection velocities could provide on the initial values of the $V_{EJ}$ parameter, we simulated fictitious Astrid families with the currently observed size-frequency distribution, values of the parameters affecting the strength of the Yarkovsky force typical of C-type asteroids according to [5], i.e., bulk and surface density equal to ${\\rho }_{bulk}={\\rho }_{surf} = 1300$ kg/m$^3$ , thermal conductivity $K =0.01$ W/m/K, thermal capacity equal to $C_{th} = 680$ J/kg/K, Bond albedo $A_{Bond} =0.02$ and infrared emissivity $\\epsilon = 0.9$ .", "We also generated fictitious families with $V_{EJ} = 5, 10, 15, 20$ , and 25 m/s, the most likely values of this parameter, according to the analysis of the previous section.", "Particles were integrated with $SWIFT\\_RMVSY$ , the symplectic integrator developed by [4] that simulates the diurnal and seasonal versions of the Yarkovsky effect, over 300 Myr and the gravitational influence of all planets plus Ceres.", "Values of $v_W$ were then obtained by inverting the third Gauss equation [20]: $\\delta i = \\frac{(1-e^2)^{1/2}}{na} \\frac{cos(\\omega +f)}{1+e cos(f)} \\delta v_W.$ where $\\delta i= i-i_{ref}$ , with $i_{ref}$ the inclination of the barycenter of the family, and $f$ and $\\omega +f$ assumed equal to 30$^{\\circ }$ and 50.5$^{\\circ }$ , respectively.", "Results from [12] show that the shape of the $v_W$ distribution is not strongly dependent on the values of $f$ and $\\omega +f$ .", "Figure: Time evolution of the Kurtosis parameter γ 2 {\\gamma }_2 for membersof a fictitious family with V EJ =5V_{EJ} = 5 m/s (panel A) and 10 m/s (panel B).The horizontal black line displays the current value of γ 2 {\\gamma }_2 forthe real whole Astrid family.", "The vertical lines identify the range ofpossible ages for the Astrid family, according to .Fig.", "REF displays the time evolution of the ${\\gamma }_2$ parameter of the $v_W$ distribution for the fictitious family with $V_{EJ} = 5$ m/s (panel A) and 10 m/s (panel B).", "The peak in the ${\\gamma }_2$ value occurs when most particles interacted with the $s-s_C$ secular resonance and had their inclination value increased by this resonance.", "The current value of ${\\gamma }_2$ of the Astrid family is not reached for any time inside the range of possible ages, as estimated by [26] (vertical red lines, the largest range of uncertainty for the age of this family in the literature.", "This range of ages corresponds to a 1-standard deviation confidence level, obtained by computing a Yarkovsky calibration, with 20% relative uncertainty, and with an assumed density of 1410 kg/m$^3$ ), neither for the simulations with $V_{EJ} = 5$ m/s nor that with $V_{EJ} = 10$ m/s.", "The situation is even worse for families with larger values of the ejection parameter, for which the peak in ${\\gamma }_2$ is achieved earlier.", "This suggests that standard parameters describing the Yarkovsky force may not apply for the Astrid family.", "[18] analyzed the effect that changing the values of the Yarkovsky parameters had on estimate of the family age, and found that the largest effect was associated with changes in the values of the thermal conductivity and bulk and surface density of asteroids, in that order.", "Based on these results, we first considered two other possible values of $K$ , 0.001 and 0.100 W/m/k, and repeated our simulations for $V_{EJ} = 10$ m/s.", "Results are shown in Fig.", "REF .", "Figure: Time evolution of the Kurtosis parameter γ 2 {\\gamma }_2 for membersof a fictitious family with V EJ =10V_{EJ} = 10 m/s and thermal conductivityKK = 0.001 W/m/K (panel A) and 0.100 W/m/k (panel B).", "In panel C and Dwe display results for KK = 0.001 W/m/K andρ bulk =ρ surf =900{\\rho }_{bulk}={\\rho }_{surf} = 900 kg/m 3 ^3,and ρ bulk =ρ surf =1700{\\rho }_{bulk}={\\rho }_{surf} = 1700 kg/m 3 ^3, respectively.The symbols have the same meaning as in Fig.", ".In both cases, the current value of ${\\gamma }_2$ is indeed achieved in the interval covering the uncertainty associated with Astrid age.", "In the second case, however, the fraction of objects with semi-major axis lower than 2.7646 au, that crossed the $s-s_c$ resonance, was too small at $t=182$ Myr (the maximum possible age for Astrid), when compared with the current value (15.8%).", "This suggests that $K$ = 0.001 W/m/K could be closer to the actual value of thermal conductivity of the real Astrid asteroids.", "We then considered the effect of changing the bulk and surface density, assumed equal, for simplicity.", "We used for the two sets of simulations ${\\rho }_{bulk}={\\rho }_{surf} = 900$ kg/m$^3$ and ${\\rho }_{bulk}={\\rho }_{surf} = 1700$ kg/m$^3$ , that are at the extreme of the range of values for the density of C-type asteroids [13].", "The other parameters were equal to previous values, and $K$ = 0.001 W/m/K.", "Fig.", "REF , panels C and D, displays our results.", "While the values of ${\\gamma }_2$ for the first simulation, do not reach the current value in the time interval covering the uncertainty associated with Astrid age, larger values of the density could be still compatible with our ${\\gamma }_2$ test.", "Overall, our results suggest that the thermal conductivity $K$ of Astrid members should be of the order of $K$ = 0.001 W/m/K, while the mean density of Astrid fragments should be higher than 1000 kg/m$^3$ .", "Remarkably, results obtained with the ${\\gamma }_2(V_W)$ method are in good agreement with those obtained from independent methods [26]." ], [ "Chronology of the Astrid family", "Now that the analysis of the current inclination distribution and our ${\\gamma }_2$ test provided independent constraint on the values of the $V_{EJ}$ parameter and of the thermal conductivity and density of Astrid members, we can try to obtain an independent age estimate for this family.", "We use the approach described in [10] that employs a Monte Carlo method [19], [29], [30], [31] to estimate the age and terminal ejection velocities of the family members.", "More details on the method can be found in [10].", "Essentially, the semi-major axis distribution of simulated asteroid families is evolved under the influence of the Yarkovsky effect (both diurnal and seasonal version), the stochastic YORP force, and changes in values of the past solar luminosity.", "Distributions of a $C$ -target function are then obtained through the equation: $0.2H=log_{10}(\\Delta a/C),$ where $H$ is the asteroid absolute magnitude, and $\\Delta a = a -a_{center}$ is the distance of each asteroid from its family center, here defined as the family center of mass.", "For the Astrid family this is essentially equal to the semi-major axis of 1128 Astrid itself.", "We can then compare the simulated $C$ -distributions to the observed one by finding the minimum of a ${\\chi }^2$ -like function: ${\\psi }_{\\Delta C}=\\sum _{\\Delta C}\\frac{[N(C)-N_{obs}(C)]^2}{N_{obs}(C)},$ where $N(C)$ is the number of simulated objects in the $i-th$ $C$ interval, and $N_{obs}(C)$ is the observed number in the same interval.", "Good values of the ${\\psi }_{\\Delta C}$ function are close to the number of the degrees of freedom of the ${\\chi }^2$ -like variable.", "This is given by the number of intervals in the $C$ minus the number of parameters estimated from the distribution (in our case, the the family age and $V_{EJ}$ parameter).", "Using only intervals with more than 10 asteroids, to avoid the problems associated with small divisors in Eq.", "REF , we have in our case 7 intervals for $C < 0$ (see Fig.", "REF , panel A) and 2 estimated parameters, and, therefore, 5 degrees of freedom.", "If we assume that the ${\\psi }_{\\Delta C}$ probability distribution follows a law given by an incomplete gamma function of arguments ${\\psi }_{\\Delta C}$ and the number of degrees of freedom, the value of ${\\psi }_{\\Delta C}$ associated with a 1-sigma probability (or 68.23%) of the simulated and real distributions being compatible is equal ${\\psi }_{\\Delta C}=4.3$ [24].", "Figure: Panel A: Histogram of the distribution of CC values for theAstrid family (blue line).", "The dashed line displays thepositive part of the CC distribution.Panels B: target function ψ ΔC {\\psi }_{\\Delta C} values in (Age,V EJ Age,V_{EJ})plane for a symmetrical bimodal distribution based on theCC negative values.", "The horizontal full white line display the value ofthe estimated escape velocity from the parent body,while the horizontal dashed white line refers to the V EJ =25V_{EJ} = 25 m/s limitobtained from the current inclination distribution inSect. .", "The black thickline displays the contour level of ψ ΔC {\\psi }_{\\Delta C} associatedwith a 1-sigma probability that the simulated andreal distribution were compatible.The reason why we only considered negative values of $C$ for our analysis is that the semi-major axis distribution (and, therefore, the $C$ one) is quite asymmetric.", "72.4% of family members are encountered at lower semi-major axis than that of 1128 Astrid.", "This reflects into a bimodal distribution of the $C$ values as well, with a more pronounced peak at negative $C$ values (see Fig.", "REF , panel A).", "Among the causes that could have produced this situation, i) the original fraction of retrograde rotators produced in the collision could have been higher, ii) the ejection velocity field could have been asymmetrical, with a large fraction of members ejected at lower semi-major axis, and iii) some of the members of the family at higher semi-major axis could have been lost in the 5J:-2A mean-motion resonance.", "Rather than account for any of these mechanisms, or better an unknown combination of the three, we preferred in this work to use a different approach.", "Since the most interesting dynamics occurs for values of semi-major axis lower than the family center, we just fitted the distribution of $C$ negative values using Eq.", "REF .", "Results of our simulations are shown in Fig.", "REF , panel B, that displays target function ${\\psi }_{\\Delta C}$ values in the ($Age,V_{EJ}$ ) plane.", "As determined from the previous section, we used $K$ = 0.001 W/m/K and ${\\rho }_{bulk}={\\rho }_{surf} = 1300$ kg/m$^3$ .", "Values of other parameters of the model such as $C_{YORP}$ , ${\\delta }_{YORP}$ and $c_{reorient}$ and their description can be found in [3].", "At 1-sigma level, we obtain $T = 135^{+15}_{-20}$ Myr, and $V_{EJ}= 5^{+17}_{-5}$ m/s.", "Overall, to within the nominal errors, we confirmed the age estimates of [21] and [26].", "Independent constraints from Sect.", "imply that $V_{EJ} < 25$ m/s, in agreement with our results." ], [ "Conclusions", "Our results could be summarized as follows: We identify the Astrid family in the domain of proper elements, and eliminated albedo and photometric interlopers.", "The Astrid family is a C-complex family and C-complex objects dominate the local background.", "19 members of the family are in $s-s_C$ resonant librating states, and appear to oscillate around the stable point at $\\Omega -{\\Omega }_{C} = \\pm 90^{\\circ }$ .", "The width of the librating region of the $s-s_C$ resonance is equal to $0.8$ arcsec/yr, and any cluster of objects injected into the resonance would have its members completely dispersed along the separatrix of the $s-s_c$ resonance on timescales of the order of 10 Myr.", "Assuming that the original ejection velocity field of the Astrid family could be approximated as isotropic, the $V_{EJ}$ parameter describing the standard deviation of terminal ejection velocity should not have been higher than 25 m/s, or the family would have been more dispersed in inclination than what currently observed.", "Interaction with the $s-s_C$ increased the value of the kurtosis of the distribution of the $v_W$ component of currently observed ejection velocities to the large value currently observed (${\\gamma }_2 =4.43$ ).", "Simulations of fictitious Astrid families with standard values of key parameters describing the strength of the Yarkovsky force for C-type asteroids, such as the thermal conductivity $K=0.01$ W/m/K, fails to produce a distribution of asteroids with ${\\gamma }_2(v_W)$ equal to the current value over the possible lifetime of the family.", "Constraints from the currently observed number of objects that crossed the $s-s_C$ region, suggest that $K$ could be closer to 0.001 W/m/K for the Astrid members.", "The bulk and surface density should be higher than 1000 kg/m$^3$ .", "Using a Monte Carlo approach to asteroid family determination [3], [10], and values of thermal conductivity and asteroid mass density obtained from the ${\\gamma }_2(v_W)$ tests, we estimated the Astrid family to be $T = 135^{+15}_{-20}$ Myr old, and its ejection velocity parameter to be in the range $V_{EJ}= 5^{+17}_{-5}$ m/s.", "In agreement with what found from constraints from the current inclination distribution of family members, values of $V_{EJ}$ larger than 25 m/s were not likely to have occurred.", "Overall, the unique nature of the Astrid family, characterized by its interaction with the $s-s_C$ secular resonance and by high values of the ${\\gamma }_2$ parameter describing the kurtosis of the $v_W$ component of the currently estimated ejection velocity field allowed for the use of techniques that provided invaluable constraints on the range of permissible values of parameters describing the Yarkovsky force, such as the surface thermal conductivity and density, not available for other asteroid families." ], [ "Acknowledgments", "We are grateful to the reviewer of this paper, Prof. Andrea Milani, for comments and suggestions that significantly improved the quality of this paper.", "We would like to thank the São Paulo State Science Foundation (FAPESP) that supported this work via the grant 14/06762-2, and the Brazilian National Research Council (CNPq, grant 305453/2011-4).", "This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE) and NEOWISE, which are a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration." ] ]	1606.05189
[ [ "Combinatorial Optimization of Work Distribution on Heterogeneous Systems" ], [ "Abstract We describe an approach that uses combinatorial optimization and machine learning to share the work between the host and device of heterogeneous computing systems such that the overall application execution time is minimized.", "We propose to use combinatorial optimization to search for the optimal system configuration in the given parameter space (such as, the number of threads, thread affinity, work distribution for the host and device).", "For each system configuration that is suggested by combinatorial optimization, we use machine learning for evaluation of the system performance.", "We evaluate our approach experimentally using a heterogeneous platform that comprises two 12-core Intel Xeon E5 CPUs and an Intel Xeon Phi 7120P co-processor with 61 cores.", "Using our approach we are able to find a near-optimal system configuration by performing only about 5% of all possible experiments." ], [ "Introduction", "Heterogeneous computing systems that consist of CPUs and accelerators such as Nvidia GPU [1] or Intel Xeon Phi [2] are becoming prevalent.", "Some of the most powerful supercomputers in the TOP500 list (November 2015, [3]) are heterogeneous at their node level.", "For example, a node of Tianhe-2 (no.", "1 in TOP500) comprises two Intel IvyBrigde CPUs and three Intel Xeon Phi co-processors; a node of Titan (no.", "2 in TOP500) contains one AMD Opteron CPU and one Nvidia Tesla GPU.", "Utilizing the computational power of all the available resources (CPUs + accelerators) in heterogeneous systems is essential to achieve good performance.", "However, due to different performance characteristics of their processing elements, achieving a good workload distribution across multiple devices on heterogeneous systems is non-trivial [4], [5], [6].", "Furthermore, optimal workload distribution is most likely to change for different applications, input problem sizes and available resources.", "Determining the optimal system configuration (including the number of threads, thread affinity, workload partitioning ratio for multi-core processors of the host and the accelerating devices) using brute-force may be prohibitively time consuming.", "Various approaches for workload distribution have been proposed.", "For example Augonnet et al.", "[7] propose a task scheduling library to handle the load balancing and the memory transfer.", "Scogland et al.", "[8] propose an adaptive worksharing library to schedule computational load across devices.", "Ravi and Agrawal [9] propose a dynamic scheduling framework that splits tasks into smaller ones and distributes them across processing elements on heterogeneous systems.", "Grewe and O'Boyle [10] propose a static partitioning approach to distribute OpenCL programs on heterogeneous systems.", "However, so far not much research was focused on using meta-heuristics to optimize the workload distribution of data-parallel applications, which considers various parameters such as: the number of threads, the thread affinity, and the workload partitioning ratio for host CPUs and co-processing devices.", "In this paper we propose an optimization approach that combines the Combinatorial Optimization and Machine Learning to determine near-optimal system configuration parameters of a heterogeneous system.", "We use Simulated Annealing as a combinatorial optimization approach to search for the optimal system configuration in the given parameter space, whereas for performance evaluation of the proposed system configurations during space exploration we use the Boosted Decision Tree Regression.", "The objective function that we aim to minimize is the application's execution time.", "To evaluate our approach we use a parallel application for DNA Sequence Analysis on a platform that comprises two 12-core Intel Xeon E5 CPUs and an Intel Xeon Phi 7120P co-processor with 61 cores.", "Using our optimization approach to determine the near-optimal system configuration we achieve a speedup of 1.74$\\times $ compared to the case when only the available resources of the host are used, and up to 2.18$\\times $ speedup compared to the case when all the resources of the accelerating device are used.", "Contributions: The major contributions of this paper are: A Combinatorial Optimization approach to explore the large system configuration space; A supervised Machine Learning approach to evaluate the performance of parallel applications; An approach that combines the combinatorial optimization heuristic with machine learning to determine a near-optimal system configuration, such that the execution time is decreased; Experimental evaluation of our approach; Performance comparison of our approach that utilizes both CPUs and accelerators, compared to CPU-only and accelerator-only approaches.", "The rest of the paper is organized as follows.", "Section provides background and motivation.", "Section describes the design and implementation of our optimization approach.", "Section presents our evaluation.", "This paper is compared and contrasted to the state-of-the-art related work in Section .", "We provide conclusions and discuss the future work in Section ." ], [ "Background and Motivation", "In this section we will motivate the need for optimized workload distribution across heterogeneous devices.", "To illustrate and motivate the problem of workload distribution on heterogeneous platforms and to evaluate the proposed approach, we will measure the execution time of a DNA Sequence Analysis application [11], [12] in a heterogeneous platform that is accelerated using an Intel Xeon Phi co-processor.", "Details related to the heterogeneous platform and the application used for experimentation will follow in the next sections." ], [ "Heterogeneous Computing Platforms with Intel Xeon Phi", "A typical heterogeneous platform that is accelerated with the Intel Xeon Phi is diagrammed in Figure REF .", "Such platforms may consist of one or two CPUs on the host (left-hand side of the figure), and one to eight accelerators (right-hand side of the figure).", "The host CPUs are of type Intel Xeon E5, which consists of 12 cores, each of them supports two hardware threads that amounts to a total of 48 threads.", "The L3 cache is split in two parts, in total it features a 30MB L3 cache.", "The Xeon phi accelerator has 61 cores, where each core supports four hardware threads, in total 244 threads per co-processor [2].", "The Xeon Phi comes with a lightweight Linux Operating System ($\\mu $ OS) that allows us to either run applications natively or offload them.", "One of the cores is used by the OS, the remaining 60 cores are used for experimentation.", "The Xeon Phi has a unified L2 cache memory of 30.5MB.", "One of the key features of the Intel Xeon Phi is its vector processing units that are essential to fully utilize the co-processor [13].", "Through the 512-bit wide SIMD registers it can perform 16 (16 wide $\\times $ 32 bit) single-precision or 8 (8 wide $\\times $ 64 bit) double-precision operations per cycle.", "The performance capabilities of the Intel Xeon Phi have been investigated by different researches within different domains [14], [15], [16]." ], [ "DNA Sequence Analysis", "For motivation purposes, and later on for evaluation of our approach we have used a high performance data analytic application for DNA Sequence Analysis [11], [12] that is based on Finite Automata and finds patterns (so called motifs) in large-scale DNA sequences.", "It allows efficient use of the computational resources of the host and accelerating device.", "The DNA Sequence Analysis application targets heterogeneous systems that are accelerated with the Intel Xeon Phi co-processor, and is able to exploit both the thread- and SIMD-level parallelism." ], [ "Motivational Experiment", "We measured the execution time of a DNA Sequence Analysis application [11], [12] on a simple heterogeneous system that consists of two Intel Xeon CPUs and one Intel Xeon Phi co-processor.", "In reality, heterogeneous systems may consist of several different types of accelerators with different performance capabilities.", "We run these experiments with different input sizes and number of CPU threads.", "To highlight the work-distribution problem we vary the distribution ratio across host and device.", "Figure REF shows the results of our experiments.", "The x-axis indicates the work distribution ratio, for instance $60/40$ means that 60% of the work is mapped to the host CPUs and the remaining 40% is mapped to the co-processor.", "The y-axis indicates the execution time, note that the values are normalized in a range from 1-10.", "In the first experiment, depicted in Fig.", "REF , we may observe that the lowest execution time is achieved when running on the CPU only.", "That is due to the relatively small input size used, where any work distribution makes the execution time be biased by the represented overhead.", "In the second experiment, shown in Fig.", "REF , we used a larger input size, therefore running on the 48 threads of the CPU or on the co-processor only is not the most effective mapping.", "We may observe that a work distribution of $70/30$ or $60/40$ is much faster.", "Figure REF shows the results when using the same input size but the number of CPU threads is reduced to 4.", "We may observe that the optimal work distribution is when we assign 70% of the work to the co-processor.", "Please note that in these experiments we consider only 11 possible workload partition ratios ($0, 10, 20, ..., 100$ ).", "In real-world problems this ratio can be any number in the interval 0-100.", "From the above experiments we may see that the optimal workload distribution depends on the input size and the available resources.", "If we consider more features (example, thread affinity, number of threads per core) or multiple accelerators with different performance characteristics, the number of all possible system configurations increases dramatically.", "Determining the optimal system configuration using brute-force may be prohibitively time expensive.", "The number of all possible system configurations is a product of parameter value ranges, $ \\prod _{i=1}^{n} R_{c_i} = R_{c_1} \\times R_{c_2} \\times .. \\times R_{c_n}$ where $C=\\lbrace c_1, c_2, ..., c_n\\rbrace $ is a set of parameters and each $c_i$ has a value range $R_{c_i}$ .", "In the next section we are going to propose an intelligent work distribution approach that is able to determine an optimal system configuration using combinatorial optimization and machine learning.", "Figure: DNA Sequence Analysis with different input sizes and number of CPU threads used.", "The execution time values are normalized in the range of 1-10.One of the most compelling features of the Intel Xeon Phi co-processor is the double advantage of transforming-and-tuning, which means that tuning an application on the Intel Xeon Phi for scaling (more cores and threads), vectorization and memory usage, stands to benefit an application when running on the Intel Xeon processors.", "Therefore, with not much programming investment application tailored for many-core Intel Xeon Phi co-processors can benefit when running on multi-core Intel Xeon CPUs, and vice-versa.", "To distribute the workload across the heterogeneous devices we use the offload programming model.", "We overlap the parts offloaded to the co-processor with the ones that are running on the host CPUs, which mitigates the idle time for both CPUs and accelerators.", "We target applications with divisible workload, which means that the workloads division can be adjusted arbitrarily.", "However, as seen in Section REF , in heterogeneous systems that have processing units of different speed, finding an optimal partitioning ratio for a given workload is non-trivial.", "In this section, we describe our approach for determining the optimal system configuration parameters (including number of threads, thread affinities, workload fraction) of a heterogeneous systems.", "The goal of our approach is to propose a near-optimal system configuration such that the overall execution time is minimized.", "The system parameters and their possible values are listed in Table REF .", "Table: The set of considered parameters and their values for our target system.To determine the optimal system configuration in a large parameter space one could try to naively enumerate over all possible parameter values, a technique we refer to as enumeration (also known as brute-force).", "The use of enumeration for design-space exploration in a real-world context may be prohibitively time consuming [17], [18], [19], [20].", "Therefore, we propose to use Simulated Annealing as a combinatorial optimization method to search for an optimal system configuration in a given parameter space.", "We may use measurements or model-based prediction for evaluation of the system performance for each system configuration.", "In comparison to the measurement based evaluation, the prediction-based is much faster but less accurate.", "Furthermore it requires training of the prediction model.", "In this paper, we consider using various optimization approaches: a) Enumeration and Measurements (EM) - Certainly determines the optimal system configuration, however it involves a very large number of performance experiments.", "The expected optimization effort is very high.", "Since EM has no performance prediction capabilities, for each program input the whole optimization process needs to be repeated.", "b) Enumeration and Machine Learning (EML) - uses machine learning to infer about the system performance.", "Since it has to examine all of the possible system configurations, the effort needed for parameter space exploration is still high.", "c) Simulated Annealing and Measurements (SAM) - uses Simulated Annealing to guide the parameter space exploration and measurements for performance evaluation of the proposed system configurations.", "This method significantly reduces the effort for parameter space exploration.", "d) Simulated Annealing and Machine Learning (SAML) - Compared to SAM, SAML provides the possibility to predict the system performance for new unseen system configurations, because it uses machine learning for performance evaluation.", "The properties of each of the proposed approaches are listed on Table .", "In what follows in this section we describe our approach for parameter space exploration using Simulated Annealing and our approach for performance prediction using Machine Learning.", "[t] Properties of optimization methods.", "Table: NO_CAPTION" ], [ "Using Simulated Annealing for Parameter Space Exploration", "Press et al.", "[21] describe several heuristics for solving optimization problems, including: Genetic Algorithms, Ant Colony Optimization, Simulated Annealing, Local Search, Tabu Search.", "Factors such as the type of the optimization problem and search space, the computational time, and demanded solution quality need to be considered when choosing the most convenient heuristic for a specific problem [22], [23].", "We have decided to use Simulated Annealing because of its ability to cope with very large discrete configuration space, and the ability to avoid getting stuck at local minimums, which makes it much better on average at finding an approximate global minimum on a large space.", "The name and inspiration comes from the process of annealing in metallurgy, a technique that includes heating and controlled cooling of materials.", "At high temperatures particles of the material have more freedom of movement, and as the temperature decreases the movement of particles is restricted as well.", "When the material is cooled slowly, the particles are ordered in the form of a crystal that represents its minimal energy.", "In the same way, in Simulated Annealing there is a temperature variable $T$ that controls the cooling process.", "One of the fundamental properties of the Simulated Annealing meta-heuristic is its ability to accept worse solutions at a higher temperature, therefore there is a corresponding chance to get out of local minimum, which enables a more extensive search for the global optimal solution.", "The lower the temperature, less likely it accepts new solutions [21].", "The method of Simulated Annealing is a suitable technique for optimization of large scale problems, especially the ones where the global optimum is hidden among many local optima.", "Examples like the traveling salesman problem (TSP) or designing complex integrated circuits are just some of many problems that can be solved using the Simulated Annealing.", "The space over which the objective function is defined is discrete and very large (factorial) configuration space, for example, in the TSP the set of possible orders of cities.", "In the context of the load balancing problem in heterogeneous systems, we define the configuration space as follows: workload fraction is a discrete value from 0-100, which indicates the percentage of the workload that needs to be executed in a specific device.", "For instance in a heterogeneous system with one CPU and one accelerator, if 40% of the workload is mapped to the host CPU, the remaining 60% is assigned to the accelerator(s); number of threads for the host CPU and the accelerator(s); the thread allocation strategy for the host CPU and the accelerator(s); The objective function $E$ (analog of energy) of our approach is to minimize the total execution time of an application, which basically is determined by the maximum of the $T_{host}$ and $T_{device}$ : $E = max(T_{host}, T_{device})$ Figure: The Structure of the Simulated Annealing Algorithm.An overview of the Simulated Annealing algorithm is depicted in Figure REF .", "The algorithm start by setting an initial temperature and creating a random initial solution.", "Then we begin looping until the annealing process has sufficiently cooled.", "We define the annealing schedule as follows: $T = T * (1 - coolingRate);$ where $coolingRate$ determines the cooling rate.", "The temperature variable plays a decisive role in the acceptance probability function.", "When a new solution is proposed, we first check if its energy $E^{\\prime }$ is lower than the energy of the current solution $E$ .", "If it is, we accept it unconditionally, otherwise we consider how much worse is the time of the proposed solution compared to the current one, and what is the temperature of the system.", "If the temperature is high, the system is more likely to accept solutions that are worse than the current one.", "The acceptance probability function $p$ is determined as follows: $p = exp((E - E^{\\prime }) / T)$ where $E^{\\prime }$ determines the energy of the newly generated solution.", "This function allows the system to get out of local optima, and find a new better one." ], [ "Using Machine Learning for Performance Evaluation", "The evaluation of the newly generated solutions by the Simulated Annealing can be done using measurements of actual program execution, or using machine learning approaches to predict the execution time of an application on the host $T_{host}$ and accelerator $T_{device}$ .", "In our approach we use the predicted execution time to determine the near-optimal system configuration.", "The aim is to balance the workload between the host and device(s) such that the total execution time is reduced.", "During the development of our performance prediction model we have considered various supervised machine learning approaches, including Linear Regression, Poisson Regression, and the Boosted Decision Tree Regression.", "In our performance prediction experiments, we achieved more accurate prediction results with the Boosted Decision Tree Regression.", "The Boosted Decision Tree Regression is a supervised machine learning algorithm that uses boosting to generate a group of regression trees and determine the optimal tree based on a loss function.", "Figure: The Predictive Model using Boosted Decision Tree RegressionThe execution time for most of the applications is mainly influenced by the input size, the available computing resources, and the thread allocation strategies.", "Therefore, we use these features to train and evaluate our prediction model.", "We have generated training data for training our performance prediction model by executing the application used during evaluation of our approach with different number of threads, thread affinities and input sizes.", "The main features including their possible values used to train and evaluate our prediction model are listed in Table REF .", "We generated data by running our experiments on two different environments (host and device).", "On the host we used $2, 4, 6, 12, 24, 36$ and 48 threads.", "We varied the thread affinities between none, scatter, and compact.", "On the accelerator we used $2, 4, 8, 16, 30, 60, 120, 180$ and 240 threads, whereas we varied the thread affinity strategies between balanced, scatter, and compact.", "We trained our model with different input fractions, varying from 0-100, which represents the percentage fraction of the input that needs to be examined in a specific device.", "In total the data of about 7200 experiments were used to train and evaluate the performance prediction model using the Boosted Decision Tree Regression.", "Half of the experiments were used to train the prediction model, and the other half were used for evaluation.", "Figure REF illustrates the process of training and predicting an unseen system configuration.", "The left hand side of the figure shows the training model, which basically takes as input a structured data set, and trains a model using the Boosted Decision Tree Regression algorithm.", "The gray colored boxes are used for evaluation of our approach.", "The right-hand side of the figure shows the Predictive Model, which takes the proposed system configurations as input, uses the trained model and predicts the execution time." ], [ "Evaluation", "In this section we evaluate experimentally our proposed combinatorial optimization approach for workload distribution on heterogeneous platforms.", "We describe the following: the experimentation environment evaluation of our prediction model comparison of the SAML and EM achieved performance improvement" ], [ "Experimentation Environment", "In this section we describe the experimentation environment used for the evaluation of our approach for workload sharing on heterogeneous platforms.", "We describe the system configuration, the application used for testing, its input dataset, and the parameters that define the system configuration.", "In Section REF we described the architecture of the heterogeneous platform used for performance evaluation of our approach.", "The major features of our system are listed in Table REF .", "In Section REF we talked about the application used for evaluation of our approach, that is a DNA Sequence Analysis application.", "We used the code generated by our PaREM tool [24] as a basis for our DNA Sequence Analysis application.", "The DNA sequence is basically a long string of characters.", "Each character indicates one of the nucleotide bases Adenine (A), Cytosine (C), Guanine (G), and Thymine (T).", "The size of the DNA sequences of various organisms is typically of several gigabytes.", "For experimentation, we used real-world DNA sequences of human (3.17GB), mouse (2.77GB), cat (2.43GB) and dog (2.38GB).", "These DNA sequences are extracted from the GenBank sequence database of the National Center for Biological Information [25].", "The parameters that define the system configuration for our combinatorial optimization approach are shown in Table REF .", "All the parameters are discrete.", "The considered values for the number of threads for host are $\\lbrace 2, 6, 12, 24, 36, 48\\rbrace $ , whereas for device are $\\lbrace 2, 4, 8, 16, 30, 60, 120, 180, 240\\rbrace $ .", "The thread affinity can vary between $\\lbrace none, compact, scatter\\rbrace $ for the host, and $\\lbrace balanced, compact, scatter\\rbrace $ for the device.", "The DNA Sequence Fraction parameter can have any number in the range $\\lbrace 0,..,100\\rbrace $ , such that if $60\\%$ of the DNA sequence is assigned for processing to the host, the remaining $100 - 60 = 40\\%$ is assigned to the device.", "Table: Emil: hardware architecture" ], [ "Evaluation of our Performance Prediction Model", "We have trained our performance prediction model for different input sizes.", "A total of 7200 experiments (2880 on host and 4320 on the device) were performed.", "We employed a standard validation methodology by using half of the experiments for training and the other half for evaluation.", "The predicted values are then compared to the measured values to calculate the prediction accuracy.", "We use the absolute error and the percent error to express the prediction accuracy, $absolute\\_error = \|T_{measured} - T_{predicted}\|$ $percent\\_error = 100 \\cdot absolute\\_error / T_{measured}$ Result 1: The execution times evaluated by our performance prediction model match well the execution time evaluated with measurements.", "Figure REF shows the measured and predicted execution time of DNA sequence analysis on the host CPUs.", "We perform the experiments for various number of threads, thread affinities, and fractions of the selected DNA sequences.", "The fractions include $2.5 - 100$ percent of the DNA sequence size.", "We may observe that predicted values match well the measured values execution times for most configurations.", "We observe similar behavior for none and compact thread affinities, but we elide these figures for space and simplicity.", "Figure: Performance prediction accuracy for the host.", "A total of 2880 experiments with DNA sequences of human, mouse, cat and dog were needed.", "Half of the experiments are used to train the model, and the other half to evaluate it.Figure REF depicts the measurement and prediction results of the execution time on the Intel Xeon Phi device for different number of threads and fractions of the selected DNA sequences.", "For most of the test cases the predicted execution time values match well the measured values.", "We have observed similar behavior when using 2, 4, 8, and 16 threads and varying the thread affinity to scatter and compact, but we elide their results for space and simplicity.", "Figure: Performance prediction accuracy for the device.", "A total of 4320 experiments with DNA sequences of human, mouse, cat and dog were needed.", "Half of the experiments are used to train the model, and the other half to evaluate it.Table: Performance prediction accuracy expressed via the absolute error [s] and percent error [%] for the hostTable: Performance prediction accuracy expressed via the absolute error [s] and percent error [%] for the deviceResult 2: The performance prediction model is able to accurately predict the execution time for unseen system configurations.", "The absolute and percent error are very low.", "Figure REF depicts a histogram of the frequency of performance prediction absolute error for the experiments running on the host CPUs.", "It shows that most of the absolute error values are low.", "For instance, 756 predictions have an absolute error less than $0.01$ seconds, 609 predictions have an absolute error in the range $0.01 - 0.02$ seconds, and the rest of the predictions have an absolute error in the range of $0.02 - 0.2$ .", "Figure: Error histogram for execution time predictions on the hostFigure REF depicts a histogram of the frequency of performance prediction absolute errors for the experiments running on the co-processor.", "Most of the predictions have an absolute error less than $0.3$ seconds.", "The error differences between the host and device error histograms is due to the larger span of execution times (0.9 - 42 seconds) on the device compared to host (0.74 - 5.5 seconds).", "However, that does not necessarily mean that the prediction model for the device is less accurate than the one on for the host (see the percent errors in Table REF and REF ).", "Figure: Error histogram for execution time predictions on the deviceThe average percent and absolute error that considers all the tested system configurations for different number of threads on the host is shown in Table REF .", "Table REF shows the average percent and absolute error for the experiments running on the co-processor.", "The average percent error for the experiments on the host is 5.239%, whereas the average percent error on the device is 3.132 %.", "The average absolute error on the host is 0.027 seconds, and 0.074 on the device.", "In the following section, we will show that the average prediction error of 5.239% and 3.132% enables us to satisfactory infer about the execution time during the evaluation of a given system configuration." ], [ "Comparison of SAML with EM", "The enumeration approach finds the system parameter values that result with the best performance by trying out all of the possible parameter values of the system under study.", "While this approach determines certainly the best system configuration, for the large search space of real-world problems enumeration may be prohibitively expensive.", "For the experiments used in this paper, despite the fact that we tested only what we considered reasonable parameter values (listed on Table REF in Section ), 19926 experiments were required when we used enumeration.", "Our heuristic-guided approach SAML that is based on Simulated Annealing and Machine Learning leads to comparatively good performance results, which requires only a relatively small set of experiments to be performed.", "For performance comparison, we use the absolute difference and percent difference, which are determined using the following equations: $absolute\\_difference = \|T_{EM} - T_{SAML}\|$ $percent\\_difference = 100 \\cdot absolute\\_difference / T_{EM}$ where $T_{EM}$ indicates the best execution time determined using EM, and $T_{SAML}$ indicates the execution time of our algorithm with a system configuration suggested by the SAML approach.", "Result 3 Using SAML we can determine a near-optimal system configuration by evaluating only about 5% of the total required experiments by EM Figure REF depicts the execution time of the selected application when running using the system configuration suggested by the simulated annealing.", "The solid horizontal line indicates the execution time of the system configuration determined by EM, which is considered as the optimal solution.", "The dashed horizontal line indicates the execution time of the optimal solution determined using EML.", "Simulated Annealing suggests at each iteration parameter values for the system configuration.", "We can adjust the number of iterations required by Simulated Annealing by changing the initial temperature, or adjusting the cooling function.", "We may observe that after 1000 iterations (that is only about 5% of the total possible configurations) our approach is able to determine a system configuration that results with a performance that is close to the performance of the system configuration determined with 19926 experiments when using EM.", "Please note that Simulated Annealing is a global optimization approach, and to avoid ending at a local optima during the search sometimes it accepts a worse system configuration that results with a higher execution time compared to the previous one.", "Figure: Performance comparison between the best system configuration determined by the Enumeration and Measurements (EM) and the near to optimal one determined by the Simulated Annealing and Measurements (SAM) and Simulated Annealing and Machine Learning (SAML).The EML and SAML use the predicted execution times to evaluate the proposed system configurations during the search space, however for fair comparison we use the measured values.", "That explains the results depicted on Figure REF and REF , where the execution time for EML is worse than the SAM and SAML for 750 or more iterations.", "In these cases, based on the predicted values the optimal execution time would be the ones indicated by the dashed lines, however they might be the cases with lowest prediction accuracy.", "Result 4 The system configurations determined using the SAML approach have low absolute and percent differences compared to the optimal solution determined by EM Table REF shows the percent difference of the SAML approach compared to the EM.", "We may observe that for 250 iterations the average percent difference is very high (19.685%), but by increasing the number of iterations to 500, 750 and 1000, the percent difference decreases significantly, into 14.067%, 11.846% and 10.129% respectively.", "Further increase of the number of iterations (1250, 1500, 1750 and 2000) results with a modest decrease of the percent difference (9.557, 8.599, 7.601, 6.849).", "However, since SAML is based on performance predictions, once the model is trained one can easily increase the number of iterations even more in order to achieve a higher accuracy.", "With respect to the absolute difference shown in Table REF , the determined system configurations using SAML with 250 iterations is only 0.075 seconds slower than the EM approach.", "Increasing the number of iterations into 500, 750 and 1000, decreases the absolute difference between the execution time into 0.054, 0.046 and 0.039 seconds respectively.", "Doubling the number of iterations required by SAML, we may achieve even closer absolute difference between EM and SAML, only 0.026 seconds.", "Table: Percent difference [%].", "The performance of system configuration suggested by SAML after 250, 500, 750, 1000, 1250, 1500, 1750, 2000 iterations is compared with the best one determined by EM.Table: Absolute difference [s].", "The performance of system configuration suggested by SAML after 250, 500, 750, 1000, 1250, 1500, 1750, 2000 iterations is compared with the best one determined by EM." ], [ "Performance improvement", "In this section we present the performance improvement when all the available resources of the host and device are utilized using the system configuration determined by the SAML approach.", "Please note that in what follows we present only the speedups achieved when comparing our approach with CPU-only (48 threads) and accelerator-only (244 threads) execution times.", "Comparing our approach with sequential execution is not relevant for this paper.", "Result 5 Our approach is able to determine system configurations that allow the applications to efficiently share its workload among the available resources.", "The results in Table REF demonstrate the performance improvement achieved when the system configuration determined by the SAML and EM is used for DNA sequence analysis compared to the case when all the available cores on the host are used.", "We achieve a maximal speedup of 1.74 after 1000 system configurations have been tried with SAML, whereas the maximal speedup that can be achieved using EM is 1.95.", "Table: Speedup achieved when host and device are used for DNA sequence analysis compared with the host only.", "We consider system configurations determined by EM and SAML after 250, 500, 750, 1000, 1250, 1500, 1750, 2000 iterations.Table REF shows the performance improvement that is achieved when the system configuration determined by the SAML and EM is used for DNA sequence analysis compared to the case when all the available cores on the device are used.", "The maximal achieved speedup using EM is 2.36.", "We achieve a close to maximal speedup (2.18) using only 1000 iterations.", "Table: Speedup achieved when host and device are used for DNA sequence analysis compared with the device only.", "We consider system configurations determined by EM and SAML after 250, 500, 750, 1000, 1250, 1500, 1750, 2000 iterations." ], [ "Related Work", "Efficient utilization of the combined computation power of the various computing units in heterogeneous systems requires optimal workload distribution.", "Recent related work proposed various approaches for workload distribution across different devices in heterogeneous systems.", "CoreTsar [8] is an adaptive worksharing library for workload scheduling across different devices.", "It is a directive based library that extends the accelerated OpenMP by introducing a cross-device worksharing directive.", "Such directives enable the programmer to specify the association between the computation and data.", "The library evaluates the speed of each device statically, then use these indicators to split the workload across different devices.", "Similarly Ayguadé et al.", "[26] investigated the extension of OpenMP to allow workload distribution on future iterations based on the results of first static ones.", "These approaches tend to minimize the required source code changes.", "In comparison, StarPU [7] and OmpSs [27] (task block models) require manual workload distribution by the developer, which may include significant structural source code changes.", "These powerful models for scheduling on heterogeneous systems are queue-based that basically split the workload into smaller tasks and queuing these tasks across the available resources.", "A similar approach based on priority queues is proposed by Dokulili et al.", "[16].", "A dynamic scheduling framework that divides tasks into smaller ones is proposed by Ravi and Agrawal [9].", "These task are distributed across different processing elements in a task-farm way.", "While making scheduling decisions, architectural trade-offs, computation and communication patterns are considered.", "Our approach considers only system runtime configuration and the input size that makes it a more general approach, which can be used with different applications and architecture.", "Odajima et al.", "[28] combines the pragma-based XcalableMP (XMP) [29] programming language with StarPU runtime system to utilize resources on each heterogeneous node for work distribution of the loop executions.", "XMP is used for work distribution and synchronization, whereas StarPU is used for task scheduling.", "Qilin [30] is a programming system that is based on a regression model to predict the execution time of kernels.", "Similarly to our approach, it uses off-line learning that is thereafter used in compile time to predict the execution time for different input size and system configuration.", "Grewe and O'Boyle [10] focus on workload distribution of OpenCL programs on heterogeneous systems.", "Their static based partitioning uses static analysis for code features extraction, which are used to determine the best partitioning across the different devices.", "Their approach relies on the architectural characteristics of a system.", "In comparison to the aforementioned approaches, in addition to using machine learning for evaluation of applications performance, we use combinatorial optimization to determine the near-optimal system configuration." ], [ "Summary and Future Work", "In this paper we have proposed a combinatorial optimization approach that uses machine learning to determine the system configuration (that is, the number of threads, thread affinity, and the DNA sequence fraction for the host and device) such that the overall execution time is minimized.", "We have observed that searching for the best system configuration using enumeration is time consuming, since it required many experiments.", "Using Simulated Annealing to suggest at each iteration parameter values for the system configuration after 1000 iterations we determined a system configuration that results with a performance that is close to the performance of the system configuration determined with 19926 experiments of enumeration.", "By running only about 5% of experiments we were able to find a near-optimal system configuration.", "Furthermore, we have proposed a Machine Learning approach that is able to predict the execution time for a system configuration.", "We have observed in our experiments that the average percent error of 4.2% (5.239% on the host, and 3.132% on the device) of the performance prediction enables us to satisfactory suggest near to optimal system configurations.", "Using the near optimal system configuration determined by the Simulated Annealing and Machine Learning we achieved a maximal speedup of $1.74\\times $ compared to the case when all the cores of the host are used, and up to $2.18\\times $ faster compared to the fastest execution time on the device.", "Future work will study adaptive workload-aware approaches." ] ]	1606.05134
[ [ "Calculation of thermal conductivity coefficients of electrons in\n magnetized dense matter" ], [ "Abstract The solution of Boltzmann equation for plasma in magnetic field, with arbitrarily degenerate electrons and non-degenerate nuclei, is obtained by Chapman-Enskog method.", "Functions, generalizing Sonin polynomials are used for obtaining an approximate solution.", "Fully ionized plasma is considered.", "The tensor of the heat conductivity coefficients in non-quantized magnetic field is calculated.", "For non-degenerate and strongly degenerate plasma the asymptotic analytic formulas are obtained, which are compared with results of previous authors.", "The Lorentz approximation, with neglecting of electron-electron encounters, is asymptotically exact for strongly degenerate plasma.", "We obtain, for the first time, in three polynomial approximation, with account of electron-electron collisions, analytical expressions for the heat conductivity tensor for non-degenerate electrons, in presence of a magnetic field.", "Account of the third polynomial improved substantially the precision of results.", "In two polynomial approximation our solution coincides with the published results.", "For strongly degenerate electrons we obtain, for the first time, an asymptotically exact analytical solution for the heat conductivity tensor in presence of a magnetic field.", "This solution has considerably more complicated dependence on the magnetic field than those in previous publications, and gives several times smaller relative value of a thermal conductivity across the magnetic field at $\\omega\\tau\\gtrsim 0.8$." ], [ "Introduction", "Observations of thermal emission from neutron stars (NS) provides information about the magnetic field strength and configuration, temperature, chemical composition of the outer regions, and about the properties of matter at higher densities, deeper inside the star see [1],[2].", "To derive this information, we need to calculate the structure and evolution of the star, and compare theoretical models with observational data.", "X-ray observations of thermal emission show periodic variabilities in single neutron stars [3], indicating to the anisotropic temperature distribution.", "It is produced at the low and intermediate density regions, such as the solid crust, where a complicated magnetic field geometry could cause a coupled magneto-thermal evolution.", "In some extreme cases, with a very high magnetic field, this anisotropy may even be present in the poorly known interior, where neutrino processes are responsible for the energy removal [4].", "The spectrum of these NSs in broad range from optics to X-ray band, cannot be reproduced by a spectrum of the surface with a unique temperature.", "Fitting of the spectrum of the X-ray source $ RXJ 1856.53754$ in this broad region is explained [2] by a small hot emitting area $10 - 20$ $ {\\rm km}^2$ , and an extended cooler component.", "Another piece of evidence that strongly supports the nonuniform temperature distribution are pulsations in the X-ray signal of some objects of amplitudes $5 - 30\\,\\%$ , some of which have irregular light curves that point towards a non-dipolar temperature distribution [5].", "Heat transfer in the envelopes of NS plays crucial role in many aspects of evolution of these stars.", "Thermal conductivity is the basic quantity needed for calculating the relationship between the internal temperature of a neutron star and its effective surface temperature.", "This relationship affects thermal evolution of the neutron star and its radiation spectra.", "To calculate thermal conductivity we should know the transport properties of a dense matter where electrons are degenerate, and form a nearly ideal Fermi-gas [6].", "The ions are usually treated as non-degenerate.", "They may be in a gaseous state, may form a Coulomb liquid or a Coulomb crystal [7].", "Under such conditions, electrons are the most important heat carriers, and the thermal conductivity is determined by electron motion.", "The magnetic field limits the motion of electrons in directions perpendicular to the field lines and, since they are the main carriers of the heat transport, the thermal conductivity in these directions is suppressed, while remaining unaffected along the field lines.", "The conductivity of electrons in NS and white dwarfs in presence of a magnetic field was studied in [8], [9].", "The ratio between thermal conductivity along and across magnetic field lines considered in [8] was taken as $\\frac{\\lambda _{\\perp }}{\\lambda _{\\parallel }} = \\frac{1}{1+ (\\omega \\tau )^{2}}.$ and was used also in [9].", "Here $\\omega $ is electron cyclotron frequency, $\\tau $ is the time between collisions.", "The influence of the magnetic field on the electron heat conductivity in the form (REF ) was used in subsequent papers, see [10], [11].", "Here we find an analytic solution for the heat conductivity tensor of strongly degenerate electrons in a magnetic field, in the Lorentz approximation, which is asymptotically exact in this case, showing a more complicated dependence on the magnetic field strength than (REF ).", "Classical methods of kinetic gas theory were developed by Maxwell, Boltzmann, Gilbert, Enskog and Chapman.", "These methods are presented in the monograph of Chapman and Cowling [12].", "They are based on the solution of the Boltzmann equation by method of successive approximations.", "As a zeroth approximation, the thermodynamic equilibrium distribution function is taken, which is a Maxwell distribution for a non-degenerate gas, and a Fermi-Dirac distribution in cases when degeneracy is important.", "The equilibrium distribution function is not an exact solution of Boltzmann equation in presence of non-uniformity.", "Following [12] we look for a solution of the Boltzmann equation in the first approximation by Sonyne (Laguerre) polynomial expansion, valid for a non-degenerate gas.", "To take into account a degeneracy, we use a set of orthogonal functions, as a generalization of Sonyne polynomials, suggested in [13], [14], [15], see also [16].", "Only two first terms of this expansion are taken often for calculations of the heat conductivity coefficients.", "It was shown in [17] that this approximation gives substantial errors for the coefficient of a heat conductivity, which become much smaller when 3 polynomial expansion is used.", "Here we calculate the heat conductivity tensor in a magnetic field using 3 terms of the expansion.", "We show the improvement of the precision, when mowing from 2 to 3 polynomials, on the example of the non-magnetized Lorentz gas, where an exact solution is known.", "The first application of the Boltzmann equation to the gas of charged particles was made by Chapman [12].", "Due to the divergence of the collision integral at large impact parameters for particles with the Coulomb interaction, the upper limit of integration over the impact parameter was taken at the length of the average distance between particles.", "Thus the coefficients of viscosity, heat conductivity and diffusion of gases composed of charged particles were obtained.", "Divergence of the collision integral for Coulomb interaction at large impact parameter shows that the scattering with large impact parameters and small change of momentum in one collision play more important role than collisions with big change of the momentum.", "Landau used this fact to simplify the Boltzmann collision integral [18].", "He expanded distribution function after collision in small variations of momentum and left first two terms.", "The integral obtained in this way is called Landau collision integral.", "Another derivation of the Landau collision integral was done by Chandrasekhar [19], who used the analog with the Brownian motion, described by the Fokker-Planck equation.", "The identity between Landau and Fokker-Planck collision integrals was shown in [20], see also [21].", "The kinetic coefficients in the non-degenerate plasma, with and without magnetic field had been calculated in [22], [23], [24], [25] using Chapmen-Enskog expansion method .", "Braginskii [26] calculated the kinetic coefficients for a non-degenerate plasma in magnetic field consisting of electrons and one sort of positively charged ions, using kinetic equations normalized to average velocities, different for the ions and electrons.", "Landau collision integral was used, and two polynomials were taken into account in the expansion.", "The same approach, was used in [27], where calculations of kinetic coefficients for a fully ionized plasma of a complex composition have been performed.", "Coefficients of the heat conductivity tensor in a degenerate stellar cores were calculated in Lorentz approximation for a hydrogen plasma in [28].", "A nonrelativistic calculation, based on on the quantum Lenard-Balescu transport equation for the thermal and electrical conductivity of plasma of highly degenerate, weakly coupled electrons and nondegenerate, weakly coupled ions was performed in [29].", "Shear viscosities of non-relativistic, relativistic and ultra-relativistic hard sphere gas were calculated by Chapman-Enskog method in [30],[31] The present work is devoted to the solution of the Boltzmann equation by Chapman-Enskog method for electrons in an arbitrary degenerate plasma.", "We find a tensor of the heat conductivity using the expansion in two and three polynomials, and, on the example of the Lorentz gas, we show that the method has a good convergence to the exact solution.", "We obtain, for the first time, in three polynomial approximation, with account of electron-electron collisions, analytical expressions for the heat conductivity tensor for non-degenerate electrons, in presence of a magnetic field.", "Account of the third polynomial improved substantially the precision of results.", "In two polynomial approximation our solution coincides with the published results.", "For strongly degenerate electrons we obtain, for the first time, an asymptotically exact analytical solution for the heat conductivity tensor in presence of a magnetic field.", "This solution has considerably more complicated dependence on the magnetic field than those in previous publications, and gives several times smaller relative value of a thermal conductivity across the magnetic field at $\\omega \\tau \\gtrsim 0.8$ ." ], [ "Boltzmann equations and transfer equations", "We use a Boltzmann equation for electrons, in a magnetic field, with an allowance of arbitrary degeneracy, and assuming them as non-relativistic We consider the electron gas in a crystal lattice of heavy nuclei, and take into account the interaction of the electrons with a nondegenerate nuclei and with one another.", "The nuclear component of the matter in the crust is in a crystal state, and therefore the isotropic part of the distribution function $f_{N0}$ may differ from the Maxwellian distribution.", "If the mass of the nucleus $m_{N}$ is much greater, than the electron mass $m_{e}$ , then to the terms $\\sim m_{e}/m_{N}$ the details of the distribution function $f_{N0}$ are unimportant, and the calculations can be made for arbitrary $f_{N0}$ .", "Boltzmann equation, which describes the time variation of the electron distribution function $f$ in presence of the electric and magnetic fields is written as [24], [25] $\\frac{\\partial f}{\\partial t} + c_{i}\\frac{\\partial f}{\\partial r_{i}}- \\frac{e}{m_e}(E_{i}+ \\frac{1}{c}\\varepsilon _{ikl} c_{k} B_{l})\\frac{\\partial f}{\\partial c_{i}} + J = 0.$ Here $(-e) , m_e$ are the charge (negative) and the mass of the electron, $E_{i},B_{i}$ are the strength of the electric field, and magnetic induction, $ J$ is a collision integral, $\\varepsilon _{ikl}$ is the totally antisymmetric Levi-Civita tensor, $c$ is the speed of the light.", "$\\begin{aligned}J = J_{ee} + J_{eN} = R \\int [f^{^{\\prime }}f_{1}^{^{\\prime }}(1-f)(1-f_{1})- \\\\- ff_{1}(1-f^{^{\\prime }})(1-f^{^{\\prime }}_{1})]\\times g_{ee}b\\,db\\, d\\varepsilon dc_{1i}+\\\\+ \\int [f^{^{\\prime }}f^{^{\\prime }}_{N}(1-f) - ff_{N}(1-f^{^{\\prime }}) ] \\times g_{eN}b\\,db\\, d\\varepsilon dc_{Ni}.\\end{aligned}$ Here, the impact parameter $b$ , and $\\varepsilon $ are geometrical parameters of particle collisions with relative velocities $g_{ee},\\quad g_{eN};\\qquad R=\\frac{2m_e^3}{h^3}.$ The integration in electron part of the collision integral in (REF ) is performed over the phase space of the incoming particles ($dc_{1i}$ ), and the physical space of their arrival ($b\\,db\\,d\\varepsilon $ ) [12].", "The velocity functions after collision are marked with touches.", "The Boltzmann equation for electrons with a binary collision integral (REF ) may be applied in conditions, when the electron gas may be considered as almost ideal, i.e.", "the kinetic energy of the electrons is much larger than the energy of electrostatic interactions.", "It is valid for plasma at sufficiently small density.", "In the neutron stars and white dwarfs we have an opposite conditions of plasma at very large density, when it is important to take into account the electrons degeneracy.", "It is known from the statistical physics, that a gas of strongly degenerate electrons becomes ideal, because large Fermi energy substitutes here the thermal energy [32].", "Therefore the calculations in this paper are applied to the low density, and high density plasma with degenerate electrons.", "Detailed discussion of the applicability of a binary collision integral (REF ), and its modifications for high density non-degenerate gases may be found in [12].", "Lets introduce the thermal velocity of electrons, $v_{i} =c_{i}-c_{0i}$ , where $c_{0i}$ is the mass-average velocity.", "So we can write the Boltzmann equation with respect to the thermal velocity in the form [25] $\\begin{aligned}\\frac{df}{dt} + v_{i}\\frac{\\partial f}{\\partial r_{i}} - \\left[ \\frac{e}{m_e}(E_{i}+ \\frac{1}{c}\\varepsilon _{ikl} v_{k}B_{l}) + \\frac{dc_{0i}}{dt}\\right] \\frac{\\partial f}{\\partial v_{i}} \\\\- \\frac{ e}{m_e c} \\varepsilon _{ikl} v_{k} B_{l}\\frac{\\partial f}{\\partial v_{i}} -\\frac{\\partial f}{\\partial v_{i}}v_{k} \\frac{\\partial c_{0i}}{\\partial r_{k}}+ J =0,\\end{aligned}$ where $\\frac{d}{dt}=\\frac{\\partial }{\\partial t}+c_{0i}\\frac{\\partial }{\\partial r_i}.$ The transfer equations for the electron concentration, total momentum, and electron energy, in the two-component mixture of electrons and nuclei, can be obtained in a usual manner from the Boltzmann equation in a quasi-neutral plasma [12], [23], [24], [25] as $\\frac{dn_{e}}{dt}+n_{e}\\frac{\\partial c_{0i}}{\\partial r_i}+\\frac{\\partial }{\\partial r_{i}}(n_{e}\\langle v_{i} \\rangle )=0,$ $\\rho \\frac{dc_{0i}}{\\mathit {dt}}=\\frac{1}{c}\\varepsilon _{\\mathit {ikl}}j_kB_l-\\frac{\\partial \\Pi _{ik}}{\\partial r_k},$ $\\begin{aligned}\\frac{3}{2}{kn_{e}}\\frac{{dT}}{{dt}} - \\frac{3}{2}kT\\frac{\\partial }{\\partial r_{i}}(n_{e}\\langle v_{i}\\rangle )+\\frac{\\partial q_{ei}}{\\partial r_{i}}+ \\Pi _{ik}^e\\frac{\\partial c_{0i}}{\\partial r_k}=\\\\=j_{i}(E_{i}+\\frac{1}{c}\\varepsilon _{ikl}c_{0k}B_{l}) - \\rho _{e}\\langle v_{i}\\rangle \\frac{dc_{0i}}{dt},\\end{aligned}$ where: $\\begin{aligned}\\Pi _{ik} & =\\sum \\limits _\\alpha { n_\\alpha m_\\alpha \\langle v^\\alpha _{i} v^\\alpha _{k}\\rangle }, & \\Pi _{ik}^e &= n_{\\alpha } m_{\\alpha }\\langle v_{i} v_{k}\\rangle ,\\end{aligned}$ $\\begin{aligned}\\langle v_{ \\alpha i} \\rangle & = \\frac{R}{n_{\\alpha }} \\int f v_{\\alpha i} dc_{\\alpha i},& n_e& = R\\int f dc_{ei},\\end{aligned}$ $\\begin{aligned}c_{0i} & = \\frac{1}{\\rho } \\sum \\limits _\\alpha {\\rho _{\\alpha }\\langle c_{ai} \\rangle },&j_{i} & =- n_{e}e\\left\\langle v_{i} \\right\\rangle ,\\end{aligned}$ $q_{ \\alpha i}= \\frac{1}{2}n_{\\alpha } m_\\alpha \\langle v_{\\alpha }^{2}v_{\\alpha i} \\rangle .$ Here summation is taken over the electrons and nuclei, $\\Pi _{ik}^e=P_e\\delta _{ik}$ , $P_e=\\frac{1}{3}n_e m_e \\langle v^{2}\\rangle $ , when we neglect the electron viscosity, $P_{e}$ is the electron pressure, $\\langle v_{i} \\rangle $ is in average electron velocity in the comoving system, $q_{i}$ is the electron heat flux, and $j_{i}$ is the electron electric current.", "Here and in the subsequent consideration we identify the mass average velocity with the average velocity of ions $c_{0i}= \\langle c_Ni\\rangle $ , and we consider the electric current and heat flux produced only by electrons.", "In the quasi neutral plasma the electron concentration $n_{e}$ is uniquely connected with the density $\\rho $ , defined by $\\langle A,Z \\rangle $ nuclei, $m_{N} = Am_{p}$ $\\begin{aligned}\\rho & = m_{N}n_{N},& n_e& = \\frac{Z\\rho }{m_{N}}.\\end{aligned}$" ], [ "Derivation of general equations for the first approximation function", "The Boltzmann equation can be solved by Chapmen-Enskog method of successive approximation [12].", "This method is used here for conditions, when distribution functions are close to their values in thermodynamic equilibrium, and deviations are considered in a linear approximation.", "Equation for second order deviation from the equilibrium distribution function had been derived in [33] for a simple gas, see also [12].", "The complexity of this equation, and rather narrow region where second order corrections could be important, strongly restricted the application of this approach.", "The zeroth approximation to the electron distribution function is a Fermi-Dirac distribution, which is found by equating to zero of the collision integral $J_{ee}$ from (REF ) $f_0 =[1+\\exp \\left(\\frac{m_e v^2-2\\mu }{2\\mathit {kT}}\\right)]^{-1}, \\quad R\\int {f_0dv_i}=n_e.$ Here, $\\mu $ is a chemical potential of electrons, $k$ is Boltzmann's constant, $T$ is the temperature.", "The nuclear distribution function in the zeroth approximation $f_{N0}$ is assumed to be isotropic with respect to the velocities and to depend on the local thermodynamic parameters; otherwise it can be arbitrary with the normalization: $n_N=\\int f_{\\mathit {N0}}\\mathit {dc}_{Ni},$ where $n_N$ is the concentration of nuclei, $n_e=Z\\,n_N$ , $Z$ is the charge of the nucleus.", "Using (REF ) in ()-(REF ), we obtain the zeroth approximation for the transfer equations.", "In this approximation $\\langle v_{i} \\rangle =0,\\,\\,\\, q_{i} = 0,\\,\\,\\, \\Pi _{ik} = (P_{e}+ P_{N})\\delta _{ik}$ $\\begin{aligned}n_e & =2 \\left( \\frac{2kTm_{e}}{h^{2}}\\right) ^{3/2} G_{3/2}(x_0),\\\\P_e &= 2\\mathit {kT} \\left( \\frac{2kTm_{e}}{h^{2}}\\right)^{3/2}G_{5/2}(x_0),\\end{aligned}$ $G_n(x_0)=\\frac{1}{\\Gamma (n)}\\int _0^\\infty { \\frac{x^{n-1}dx}{1+exp(x-x_{0})} },\\,\\,\\,x_0=\\frac{\\mu }{\\mathit {kT}},$ where $G_n(x_0)$ are Fermi integrals.", "In what follows, instead of $G_{n}(x_{0})$ we will write $G_{n}$ cause the argument is the same.", "In the first approximation, we seek for the function $f$ in the form: $f=f_0[1+\\chi (1-f_0)].$ The distribution function $f_{N0}$ is assumed to satisfy the relation: $\\frac{1}{n_{N}} \\int v_{Ni} v_{Nk}f_{N0}dc_{Ni} = \\delta _{ik} \\frac{kT}{m_{N}}.$ The function $\\chi $ admits representation of the solution in the form: $\\begin{aligned}\\chi & = -A_{i}\\frac{\\partial \\ln T}{\\partial r_{i}}-n_{e}D_{i}d_{i}\\frac{G_{5/2}}{G_{3/2}}, \\\\\\end{aligned}$ $d_{i} = \\frac{\\rho _{N}}{\\rho } \\frac{ \\partial \\ln P_{e}}{\\partial r_{i}}- \\frac{\\rho _{e}}{P_{e}}\\frac{1}{\\rho }\\frac{\\partial P_{N}}{\\partial r_{i}}+\\frac{e}{kT}(E_i+\\frac{1}{c} \\varepsilon _{ikl} c_{0k} B_{l}).$ The plasma is supposed to be quasineutral with a zero charge density.", "The functions $A_{i}$ and $D_{i}$ determine the heat transfer and diffusion.", "Substituting (REF ) in the equation for $\\chi $ we obtain equations for $A_{i}$ , $D_{i}$ [12].", "It was shown in [24],[25], that in presence of a magnetic field $B_{i}$ , the polar vector $A_i$ (and similarly $D_i$ ) may be searched for in the form: $\\begin{aligned}A_{i} & = A^{(1)}v_{i}+A^{(2)} \\varepsilon _{ijk}v_{j}B_{k}+A^{(3)}B_{i}(v_{j}B_{j}), \\\\\\end{aligned}$ Introducing a function: $\\xi = A^{(1)}+iBA^{(2)},$ and dimensionless velocity: $u_{i} =\\sqrt{\\frac{m_{e}}{2kT}}v_{i}$ , we obtain the system for $\\xi $ as $\\begin{aligned}f_{0}(1-f_{0})(u^{2} - \\frac{5G_{5/2}}{2G_{3/2}})u_{i} =\\frac{i}{3} \\frac{em_{e} B}{ \\rho k T c}u_{i}f_{0}(1-f_{0})\\left[\\int \\xi _{N} f_{N0}v_{N}^{2}dc_{Ni}- R\\int \\xi f_{0}(1-f_{0})v_{e}^{2}dc_{i}\\right] \\\\-iBf_{0}(1-f_{0})\\frac{e \\xi }{m_{e} c}u_{i}+ I_{ee}(\\xi u_{i})+I_{eN}(\\xi _{Ni} u_{Ni}),\\end{aligned}$ where $I_{ee}(\\xi u_{i}) = R\\int f_{0}f_{01}(1-f^{^{\\prime }}_{0})(1-f_{01}^{^{\\prime }})\\times (\\xi u_{i}\\\\\\nonumber +\\xi _{1}u_{1i}-\\xi ^{^{\\prime }} u_{i}^{^{\\prime }}-\\xi _{1}^{^{\\prime }}u^{^{\\prime }}_{1i})g_{ee}bdbd\\varepsilon dc_{1i},$ $I_{eN}(\\xi u_{Ni}) = \\int f_{0}f_{N0}(1-f^{^{\\prime }}_{0})(\\xi u_{i} \\\\ \\nonumber -\\xi ^{^{\\prime }} u_{i}^{^{\\prime }})g_{eN}bdbd\\varepsilon dc_{Ni}.$ According to [12], a solution for the function $\\xi $ is searched for in the form of the raw of orthogonal polynomials.", "Sonyne polynomial that were used in the classical work [12] are coefficients of the expansion of the function $(1-s)^{- \\frac{3}{2} -1} e^{\\frac{xs}{1-s}}$ in powers of $s$ : $(1-s)^{- \\frac{3}{2} -1} e^{\\frac{xs}{1-s}} = \\Sigma S^{(p)}_{3/2}(x)s^{p}.$ Sonyne polynomials are orthogonal: $\\int _0^\\infty e^{-x}S^{(p)}_{3/2}(x)S^{(q)}_{3/2}(x)x^{3/2}dx = \\frac{\\Gamma (p+ \\frac{5}{2})}{p!", "}\\delta _{pq},$ and $\\begin{aligned}S^{(0)}_{3/2}(x) = 1, \\quad S^{(1)}_{3/2}(x) = \\frac{5}{2} - x,\\quad \\end{aligned}$ $S^{(2)}_{3/2}(x) = \\frac{35}{8} - \\frac{7}{2}x+\\frac{1}{2}x^{2}.$ For a degenerate case we have to seek a solution of (REF ) in the form of an expansion in polynomials $Q_{n}$ that are orthogonal with the weight $f_{0}(1-f_{0})x^{3/2}$ , analogous to Sonyne polynomials [16].", "$\\begin{aligned}Q_{0}(x) & = 1, \\quad Q_{1}(x) = \\frac{5G_{5/2}}{2G_{3/2}} - x,\\\\Q_{2}(x)& = \\frac{35}{8}\\frac{G_{7/2}}{G_{3/2}} - \\frac{7G_{7/2}}{2G_{5/2}}x+\\frac{1}{2}x^{2}, \\quad x = u^{2}.\\end{aligned}$ The nonzero integrals of products of these polynomials with the corresponding weight function are $\\int _0^\\infty f_0(1-f_0)\\,x^{3/2}Q_{0}^2(x)dx =\\frac{3}{2}\\Gamma (3/2)G_{3/2}(x_0),\\qquad \\nonumber \\qquad \\qquad \\qquad \\\\\\int _0^\\infty f_0(1-f_0)\\,x^{3/2}Q_{1}^2(x)dx=\\frac{15}{4}\\Gamma (3/2)G_{3/2}(x_0)\\left(\\frac{7}{2}\\frac{G_{7/2}}{G_{5/2}}-\\frac{5}{2}\\frac{G_{5/2}^2}{G_{3/2}^2}\\right),\\qquad \\qquad \\qquad \\\\\\int _0^\\infty f_0(1-f_0)\\,x^{3/2}Q_{2}^2(x)dx=\\frac{105}{16}\\Gamma (3/2)G_{3/2}(x_0)\\left(-\\frac{35}{8}\\frac{G_{7/2}^2}{G_{3/2}^2}+\\frac{49}{2}\\frac{G_{7/2}^3}{G_{5/2}^2\\,G_{3/2}}-\\frac{63}{2}\\frac{G_{9/2}G_{7/2}}{G_{5/2}G_{3/2}}+\\frac{99}{8}\\frac{G_{11/2}}{G_{3/2}^2}\\right).\\nonumber $ We seek $\\xi $ and $A_3$ in the form: $\\begin{aligned}\\xi = a_{0}Q_{0}+a_{1}Q_{1}+a_{2}Q_{2},\\\\A^{(3)} = c_{0}Q_{0}+c_{1}Q_{1}+c_{2}Q_{2}.\\end{aligned}$ It is easy to show, with account of (REF ), (REF ), that the first term in the right side of (REF ) is $\\sim \\frac{m_e}{m_N}$ times smaller than the second one.", "Neglecting this term, multiplying (REF ) by $RQ_{0}(x)u_{i}$ , $RQ_{1}(x)u_{i}$ and $RQ_{2}(x)u_{i}$ and integrating with respect to $dc_{i}$ , we obtain a system of equation for the heat conductivity coefficients in the form $\\left\\lbrace \\begin{aligned}0 =-\\frac{3}{2} i \\omega n_{e}a_{0}+ a_{0}(a_{00}+b_{00}) +a_{1}(a_{01}+b_{01})+a_{2}(a_{02}+b_{02})\\\\-\\frac{15}{4} n_{e}\\left( \\frac{7G_{7/2}}{2G_{3/2}} -\\frac{5G_{5/2}^{2}}{2G_{3/2}^{2}}\\right) =- \\frac{15}{4} \\left(\\frac{7G_{7/2}}{2G_{3/2}} - \\frac{5G_{5/2}^{2}}{2G_{3/2}^{2}}\\right)i \\omega n_{e}a_{1} + a_{0}(a_{10}+b_{10})+ a_{1}(a_{11} + b_{11})+a_{2}(a_{12}+b_{12})\\\\0 = -\\frac{105}{16}\\left(-\\frac{35}{8}\\frac{G_{7/2}^2}{G_{3/2}^2}+\\frac{49}{2}\\frac{G_{7/2}^2}{G_{5/2}^2}\\frac{G_{7/2}}{G_{3/2}}-\\frac{63}{2}\\frac{G_{9/2}G_{7/2}}{G_{5/2}G_{3/2}}+\\frac{99}{8}\\frac{G_{11/2}}{G_{3/2}}\\right)i \\omega n_{e}a_{2}\\qquad \\qquad \\end{aligned} \\right.$ $+a_{0}(a_{20}+b_{20})+a_{1}(a_{21}+b_{21})+a_{2}(a_{22}+b_{22})$ Here $a_{jk}$ , $b_{jk}$ are matrix elements for collision integrals, $\\omega = \\frac{eB}{m_{e}c}$ is a cyclotron frequency." ], [ "Matrix elements: $b_{jk}$", "The matrix elements, $b_{jk}$ , connected with electron-nuclei collisions, are determined as follows: $b_{jk} = R \\int f_{0} f_{N0} (1-f_{0}^{^{\\prime }})Q_{j}(u^{2})u_{i}[Q_{k}(u^{2})u_{i} \\nonumber \\\\-Q_{k}(u^{^{\\prime }2})u_{i}^{^{\\prime }}]g_{eN}bdb d\\varepsilon dc_{Ni} dc_{i}, \\\\ \\nonumber k\\ge 0 .$ Introduce functions $\\bar{\\Omega }^{(l)}_{eN}(r)$ , defined as (see [12]) $\\bar{\\Omega }^{(l)}_{eN}(r) = 2\\int _0^\\infty f_{0}(1-f_{0})z^{2r+2}\\int _0^\\infty (1-\\cos ^{l} \\theta _{12})g_{12} bdbdz,$ where $z=\\left[\\frac{m_1 m_2}{2kT(m_1+m_2)}\\right]^{1/2}g_{12},\\quad g_{12}=\|v_1-v_2\|,$ for colliding particles \"1\", \"2\", $\\theta _{12}$ is the scattering angle.", "At collisions of electrons (\"2\") and nuclei (\"1\") a mass of the nuclei is much greater than electron mass $m_{N} \\gg m_{e}$ , so we can neglect energy exchange in a collision $u^{^{\\prime }2} \\approx u^{2},\\quad u u^{^{\\prime }} \\approx u^{2}(1-cos \\theta _{12}),\\quad g_{eN}\\approx v,\\quad z\\approx u,$ Using the relation (REF ), we obtain from (REF ) $b_{jk} = 8\\pi ^{2}\\left( \\frac{2kTm_{e}}{h^{2}}\\right) ^{3/2}\\left( \\frac{2kT}{m_{e}}\\right) ^{1/2}n_{N} \\int _0^\\infty f_{0}(1-f_{0})Q_{j}(x)Q_{k}(x)x^{2}\\int _0^\\infty (1-\\cos \\theta _{12} ) bdb dx,$ Instead of functions $\\bar{\\Omega }^{(l)}_{eN}(r)$ from (REF ) we can use in this case functions $\\widehat{\\Omega }^{(l)}_{eN}(r)$ defined as $\\widehat{\\Omega }^{(l)}_{eN}(r) = \\int _0^\\infty f_{0}(1-f_{0})x^{r+1}\\int _0^\\infty (1-\\cos ^{l}\\theta _{12}) bdbdx .$ With account of (REF ), the elements of a symmetric matrix $b_{ij}$ are written as $b_{00} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N}\\widehat{\\Omega }^{(1)}_{eN}(1),$ $\\begin{aligned}b_{01} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N}\\left( \\frac{5}{2}\\frac{G_{5/2}}{G_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(1) - \\widehat{\\Omega }^{(1)}_{eN}(2) \\right),\\end{aligned}$ $\\begin{aligned}b_{11} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N} \\left( \\frac{25}{4}\\frac{G^{2}_{5/2}}{G^{2}_{3/2}} \\widehat{\\Omega }^{(1)}_{eN}(1) -5 \\frac{G_{5/2}}{G_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(2) + \\widehat{\\Omega }^{(1)}_{eN}(3)\\right),\\end{aligned}$ $\\begin{aligned}b_{02} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N}\\left[ \\frac{35}{8}\\frac{G_{7/2}}{G_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(1)-\\frac{7}{2} \\frac{G_{7/2}}{G_{5/2}} \\widehat{\\Omega }^{(1)}_{eN}(2) + \\frac{1}{2}\\widehat{\\Omega }^{(1)}_{eN}(3)\\right],\\end{aligned}$ $\\begin{aligned}b_{12} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N}\\left[ \\frac{175}{16} \\frac{G_{7/2}G_{5/2}}{G^2_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(1)- \\frac{105}{8}\\frac{G_{7/2}}{G_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(2) + \\left(\\frac{5}{4}\\frac{G_{5/2}}{G_{3/2}}+\\frac{7}{2}\\frac{G_{7/2}}{G_{5/2}}\\right)\\widehat{\\Omega }^{(1)}_{eN}(3) - \\frac{1}{2}\\widehat{\\Omega }^{(1)}_{eN}(4)\\right],\\,\\,\\,\\end{aligned}$ $\\begin{aligned}b_{22} = 8\\pi ^{2}\\frac{(2kT)^2}{h^{3}}m_{e} n_{N}\\qquad \\qquad \\qquad \\\\\\times \\left[\\frac{35^2}{8^2}\\frac{G^2_{7/2}}{G^2_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(1)- \\frac{245}{8}\\frac{G^{2}_{7/2}}{G_{5/2}G_{3/2}}\\widehat{\\Omega }^{(1)}_{eN}(2) +(\\frac{49}{4}\\frac{G_{7/2}^2}{G_{5/2}^2}+\\frac{35}{8} \\frac{G_{7/2}}{G_{3/2}})\\widehat{\\Omega }^{(1)}_{eN}(3)- \\frac{7}{2}\\frac{G_{7/2}}{G_{5/2}}\\widehat{\\Omega }^{(1)}_{eN}(4)+\\frac{1}{4}\\widehat{\\Omega }^{(1)}_{eN}(5)\\right].\\end{aligned}$" ], [ "Functions $\\phi ^{(l)}_{12}$ , and Coulomb logarithm", "The functions $\\bar{\\Omega }^{(l)}_{eN}(r)$ from (REF ) may be written in the form $\\widehat{\\Omega }^{(l)}_{eN}(r) = 2\\left[\\frac{m_1 m_2}{2kT(m_1+m_2)}\\right]^{1/2}$ $\\times \\int _0^\\infty f_{0}(1-f_{0})z^{2r+2}\\phi _{12}^{(l)}dz,$ with $\\phi _{12}^{(l)}=\\int _0^\\infty (1-\\cos ^{l} \\theta _{12})g_{12} b \\,db.$ During the integration in (REF ) over the impact parameter $db$ the integral has a logarithmic divergency at infinity.", "It is removed in a more precise treatment of Coulomb collisions in plasma with account of correlation functions [34], where the upper limit of the integration $b_{max}$ appeared.", "Introducing $v_0=bg_{12}^2(m_1 m_2/m_0 e_1 e_2)$ , where $m_0=m_1+m_2$ , $e_1$ , $e_2$ are the absolute values of charges, we obtain after integration [12] $\\phi _{12}^{(1)}=\\left(\\frac{m_0 e_1 e_2}{m_1 m_2}\\right)^2 g_{12}^{-3}\\ln (1+v_{0max}^2),$ $\\phi _{12}^{(2)}=2\\left(\\frac{m_0 e_1 e_2}{m_1 m_2}\\right)^2 g_{12}^{-3}\\left[\\ln (1+v_{0max}^2)-\\frac{v^2_{0max}}{1+v^2_{0max}}\\right],,$ $v_{0max}=b_{max} g_{12}^2(m_1 m_2/m_0 e_1 e_2).$ In farther integration the value under the logarithm is taken as constant when the average value $\\bar{g}_{12}$ is taken instead of the variable $g_{12}$ .", "For the electron-nuclei collisions with $g_{12}\\approx v_e$ the approximate expression of the Coulomb logarithm is written in the form [35] $\\Lambda =\\frac{1}{2}\\ln (1+v_{0max}^2)\\approx {\\bar{\\Lambda }}_v=\\ln \\left(b_{\\rm max}\\bar{v_e^2} m_e\\over Z{\\rm e}^2\\right),\\quad \\Lambda \\gg 1,$ where $\\bar{v_e^2}={3kT\\over m_e}\\frac{G_{5/2}}{G_{3/2}}={3kT\\over m_e} \\qquad (ND) \\\\ \\nonumber ={3\\over 5}{h^2\\over m_{\\rm e}^2}\\left(3n_e\\over 8\\pi \\right)^{2/3} \\qquad (D).$ The value of $b_{max}$ is represented by the radius of Debye screening by electrons $r_{{\\cal D}e}$ , and ions $r_{{\\cal D}i}$ , and may be written as [35] ${1\\over {b_{\\rm max}}^2}={1\\over r_{{\\cal D} i}^2}+{1\\over r_{{\\cal D} e}^2}={4\\pi {\\rm e}^2\\over kT}\\left(n_NZ^2+n_e \\frac{G_{1/2}}{G_{3/2}}\\right),$ where $\\frac{G_{1/2}}{G_{3/2}}=1 \\hspace{28.45274pt} (ND)$ $\\hspace{85.35826pt} = 4(3\\pi ^2)^{1/3} \\frac{m_e kT}{h^2 n_e^{2/3}} \\qquad (D).$ Influence of quantum effects on the Debye screening was discussed in [9].", "The average frequency of electron-ion collisions $\\nu _{ei}$ is written in [36] in the form ${\\nu _{ei}} = \\frac{4}{3 } \\sqrt{\\frac{2\\pi }{m_{e}}}\\frac{Z^2e^4n_{N}\\Lambda }{(kT)^{3/2}G_{3/2}} \\frac{1}{1+e^{-x_0}}.$ In the limiting cases it is expressed as $\\nu _{ei}={4\\over 3}\\sqrt{2\\pi \\over m_e}{Z^2{\\rm e}^4n_N\\Lambda \\over (kT)^{3/2}} \\qquad (ND)$ $={32\\pi ^2\\over 3}m_e{Z^2{\\rm e}^4 \\Lambda n_N\\over h^3 n_{\\rm e}}\\qquad (D).$ The average time $\\tau _{ei}$ between (ei) collisions is the inverse value of $\\nu _{ei}$ , and is written as $\\tau _{nd} =\\frac{1}{{\\nu _{nd}}}= {3\\over 4}\\sqrt{m_e\\over 2\\pi }{(kT)^{3/2} \\over Z^2{\\rm e}^4n_N\\Lambda },$ $\\tau _{d} = \\frac{1}{{\\nu _{d}}}=\\frac{3 h^3 n_{e}}{32 \\pi ^2 m_{e} Z^2 e^4 \\Lambda n_{N}}.$" ], [ "$b_{jk}$ for non-degenerate electrons", "For non-degenerate electrons we have $\\exp {(x-x_0)} \\ll 1$ , $G_n\\approx e^{x_0}$ , so $\\widehat{\\Omega }^{(l)}_{eN}(r) = e^{x_0}\\int _0^\\infty e^{-x}x^{r+1}\\int _0^\\infty (1-\\cos ^{l} \\theta ) bdbdx .$ In this case it is more convenient to use functions $\\Omega ^{(l)}_{eN}(r) $ defined as $\\Omega ^{(l)}_{eN}(r) = \\frac{\\sqrt{\\pi }}{2} \\int _{0}^{\\infty } e^{-x}x^{r+1}\\int _{0}^{\\infty } (1-\\cos ^{l} \\theta ) bdbdx.$ Using (REF ),(REF ) we find expressions for non-degenerate case as $\\Omega ^{(1)}_{eN}(r) = \\sqrt{\\pi }\\frac{e^{4}\\Lambda Z^2}{(2kT)^2}\\Gamma (r),\\quad \\Gamma (1)=1;$ $\\Gamma (2)=1; \\quad \\Gamma (3)=2; \\quad \\Gamma (4)=6; \\quad \\Gamma (5)=24.$ Substituting (REF ) into (REF )-(REF ), taking into account that $G_n=e^{x_0},\\qquad e^{x_{0}} = \\frac{n_{e}}{2\\pi ^{3/2}}\\left(\\frac{h^2}{2kTm_{e}}\\right)^{3/2},$ and using (REF ), (REF ), we write $b_{jk}$ for non-degenerate electrons as $b_{00} =8\\sqrt{\\pi }\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{3n_e}{2\\tau _{nd}},$ $b_{01}=12\\sqrt{\\pi }\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{9n_e}{4\\tau _{nd}},$ $b_{11} = 26\\sqrt{\\pi }\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{39n_e}{8\\tau _{nd}},$ $\\begin{aligned}b_{02} = 15\\sqrt{\\pi }\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{45n_e}{16\\tau _{nd}},\\end{aligned}$ $b_{12} = \\frac{69\\sqrt{\\pi }}{2}\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{207n_e}{32\\tau _{nd}},$ $b_{22} = \\frac{433\\sqrt{\\pi }}{8}\\frac{n_{e} n_{N} e^4 Z^2\\Lambda }{(2kT)^{3/2}\\sqrt{m_{e}}}=\\frac{1299n_e}{128\\tau _{nd}}.$" ], [ "$b_{jk}$ for partially degenerate electrons", "To calculate $b_{ij}$ for degenerate electrons we use expressions for $\\widehat{\\Omega }^{(1)}_{eN}(r)$ and $ G_{n}(x_0)$ .", "With account of (REF ),(REF ) we obtain $\\widehat{\\Omega }^{(1)}_{eN}(r) = \\int _0^\\infty f_{0}(1-f_{0})x^{r+1} \\int _0^{b_{max}} (1-\\cos \\theta ) b db dx =2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\int _0^\\infty f_{0}(1-f_{0})x^{r-1}dx=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\Gamma (r)G_{r-1}(x_0).$ The integral in (REF ) is calculated exactly for $r=1$ $\\int _0^\\infty f_{0}(1-f_{0})x^{r-1}dx=[1+\\exp (-x_0)]^{-1}$ .", "Between non-degenerate electrons with large negative non-dimensional chemical potential at $x_0 \\ll -1$ , and strongly degenerate electrons with $x_0 \\gg 1$ there is a level of degeneracy at which $x_0=0$ .", "Let us calculate matrix elements $b_{jk}$ for the level of degeneracy, corresponding to $x_0=0$ .", "The functions $G_n(0)$ have the following numerical values, according to [32] $G_{n}(0)=\\frac{1}{\\Gamma (n)}\\int _0^\\infty \\frac{x^{n-1}dx}{1+e^x} =(1-2^{1-n})\\zeta (n),$ where Riemann $\\zeta $ -function has the following values for the indexes used here [37] $\\zeta (3/2)=2.612,\\quad \\zeta (5/2)=1.341,\\quad \\zeta ((7/2)=1.127,$ $\\zeta (9/2)=1.0547,\\quad \\zeta (11/2)=1.0252,\\quad \\zeta (2)=1.645,$ $ \\zeta (3)=1.202,\\quad \\zeta (4)=1.0823, \\quad \\zeta (5)=1.0369.$ The functions $G_{n}(0)$ have the following values $G_{3/2}(0)=0.765,\\quad G_{5/2}(0)=0.867,\\quad G_{7/2}(0)=0.928,$ $G_{9/2}(0)=0.9615,\\quad G_{11/2}=0.980,\\quad G_{2}(0)=1.645,$ $G_{3}(0)=0.9015,\\quad G_{4}(0)=0.947,\\quad G_{5}(0)=0.972.$ The functions $\\widehat{\\Omega }^{(1)}_{eN}(r)$ at $x_0=0$ , defined as $\\widehat{\\Omega }^{(1)}_{eN0}(r)$ , have the following values, according to (REF ), using (REF )-(REF ) $\\widehat{\\Omega }^{(1)}_{eN0}(1)=\\frac{e^4Z^2\\Lambda }{(2kT)^2}\\equiv \\, I,\\quad \\widehat{\\Omega }^{(1)}_{eN0}(2)=2\\ln {2}\\,I=1.39\\,I,$ $\\widehat{\\Omega }^{(1)}_{eN0}(3)=3.29\\,I,\\quad \\widehat{\\Omega }^{(1)}_{eN0}(4)=10.82\\,I,\\quad $ $\\widehat{\\Omega }^{(1)}_{eN0}(5)=45.46\\,I.$ The level of degeneracy $DL(0)$ at $x_0=0$ is defined as a ratio of the Fermi energy $\\varepsilon _{fe}$ to $kT$ .", "With account of (REF ), (REF ), we obtain $DL(0)=\\frac{\\varepsilon _{fe}}{kT}=\\frac{(3\\pi ^2n_e)^{2/3}h^2}{8\\pi ^2 m_e kT}$ $=\\frac{\\pi }{4}\\left(\\frac{3}{\\pi }\\right)^{2/3}[2G_{3/2}(0)]^{2/3}=1.011.$ At $x_0=0$ the expression for the electron concentration $n_{e0}$ from (REF ), and the average time $\\tau _{ei}$ between (ei) collisions, which is the inverse value of $\\nu _{ei}$ , are written, using (REF ),(REF ),(REF ) as $n_{e0}=2G_{3/2}(0) \\left( \\frac{\\mathit {kTm}_e}{2\\pi \\hbar ^2}\\right) ^{3/2}=2\\times 0.765 \\left( \\frac{\\mathit {kTm}_e}{2\\pi \\hbar ^2}\\right) ^{3/2},$ ${\\tau _{d0}} = \\frac{3}{4 } \\sqrt{\\frac{m_{e}}{2\\pi }}\\frac{(kT)^{3/2}G_{3/2}}{Z^2e^4n_{N}\\Lambda } (1+e^{-x_0})$ $=0.765 \\frac{3}{2} \\sqrt{\\frac{m_{e}}{2\\pi }}\\frac{(kT)^{3/2}}{Z^2e^4n_{N}\\Lambda }.$ Using (REF ), (REF , (REF ) we find from (REF )-(REF ) $ b_{00} = \\frac{8\\pi ^2e^4 Z^2 m_e\\Lambda }{h^3}n_N=\\frac{3}{2}\\frac{n_{e0}}{\\tau _{d0}}$ $\\begin{aligned}b_{01} = 2.16\\frac{n_{e0}}{\\tau _{d0}},\\quad b_{11} = 5.162\\frac{n_{e0}}{\\tau _{d0}},\\quad b_{02} =2.588\\frac{n_{e0}}{\\tau _{d0}},\\end{aligned}$ $\\begin{aligned}b_{12} = 6.671\\frac{n_{e0}}{\\tau _{d0}},\\quad b_{22} = 11.038\\frac{n_{e0}}{\\tau _{d0}}.\\end{aligned}$ For arbitrary level of degeneracy at $x_0\\ne 0$ the functions $G_n(x_0)$ in (REF ) are not expressed analytically, and should be calculated numerically for each $x_0$ , at corresponding DL $DL(x_0)=\\frac{(3\\pi ^2 n_e)^{2/3}h^2}{8\\pi ^2 m_e kT}=\\frac{\\pi }{4}\\left(\\frac{3}{\\pi }\\right)^{2/3}[2G_{3/2}(x_0)]^{2/3}.$ After numerical calculation of $G_n(x_0)$ , the matrix elements $b_{jk}$ at arbitrary $x_0$ are found in the same way as it is done above at $x_0=0$ ." ], [ "$b_{jk}$ for strongly degenerate electrons", "For strongly degenerate case $x_0 \\gg 1$ we use [32] the following expansions $\\begin{aligned}G_{r}(x_0) = \\frac{1}{\\Gamma (r)}\\left[ \\frac{x_{0}^{r}}{r} +\\frac{\\pi ^{2}}{6}(r-1)x^{r-2}_{0}+\\frac{7\\pi ^4}{360}(r-1)(r-2)(r-3)x_0^{r-4}\\right]\\,\\,\\,{\\rm for}\\,\\, r\\ge 1.\\end{aligned}$ $\\begin{aligned}\\Gamma (r)G_{r-1}(x_0) = (r-1)\\left[ \\frac{x_{0}^{r-1}}{r-1} +\\frac{\\pi ^{2}}{6}(r-2)x^{r-3}_{0}+\\frac{7\\pi ^4}{360}(r-2)(r-3)(r-4)x_0^{r-5}\\right]\\,\\,\\,{\\rm for}\\,\\, r\\ge 2.\\end{aligned}$ For strongly degenerate electrons $x_0 =\\frac{(3\\pi ^2 n_e)^{2/3}h^2}{8\\pi ^2 m_e kT} \\gg 1.$ We obtain than from (REF ),(REF ), omitting exponentially small terms $\\sim e^{-x_0}$ $\\widehat{\\Omega }^{(1)}_{eN}(1) = 2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\int _0^\\infty f_{0}(1-f_{0})dx=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2},$ $\\widehat{\\Omega }^{(1)}_{eN}(2) = 2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\Gamma (2)G_1(x_0)=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2} x_0,$ $\\widehat{\\Omega }^{(1)}_{eN}(3) = 2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\Gamma (3)G_2(x_0)=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}(x_0^2+\\frac{\\pi ^2}{3}),$ $\\widehat{\\Omega }^{(1)}_{eN}(4) = 2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\Gamma (4)G_3(x_0)=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}(x_0^3+\\pi ^2 x_0),$ $\\widehat{\\Omega }^{(1)}_{eN}(5) = 2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}\\Gamma (5)G_4(x_0)=2\\frac{e^{4} Z^2 \\Lambda }{(2kT)^2}(x_o^4+2\\pi ^2x_0^2+\\frac{7\\pi ^2}{15}).$ Using (REF ),(REF )-(REF ), we find from (REF )-(REF ) $b_{00} = \\frac{16\\pi ^2e^4 Z^2 m_e\\Lambda }{h^3}n_N=\\frac{3n_e}{2\\tau _{d}},$ $\\begin{aligned}b_{01} = \\frac{3\\pi ^2n_e}{4x_0\\tau _{d}},\\quad b_{11} = \\frac{\\pi ^2n_e}{2\\tau _{d}},\\quad b_{02} = -\\frac{7\\pi ^4 n_e}{320x_0^2\\tau _{d}},\\end{aligned}$ $\\begin{aligned}b_{12}=\\frac{709\\pi ^4 n_e}{960x_0\\tau _{d}}, \\quad b_{22}=\\frac{2\\pi ^4n_e}{15\\tau _{d}}.\\end{aligned}$" ], [ "Matrix elements: $a_{jk}$", "The matrix elements, $ a_{jk}$ , related to electron-electron collisions, are determined as follows: $a_{jk} = R^{2} \\int f_{0}f_{01}(1 - f_{0}^{^{\\prime }})(1- f_{01}^{^{\\prime }})Q_{j}(u^{2})u_{i}[Q_{k}(u^{2})u_{i} + Q_{k}(u_{1}^{2})u_{1i}- Q_{k}(u^{^{\\prime }2})u_{i}^{^{\\prime }} - Q_{k}(u_{1}^{^{\\prime }2})u_{1i}^{^{\\prime }}]g_{ee}bdb d\\varepsilon dc_{1i}dc_{i},$ Let's introduce following variables [12] $\\begin{aligned}G_{li} = \\frac{1}{2}(c_{i}+c_{1i})=\\frac{1}{2}(c^{^{\\prime }}_{i}+c^{^{\\prime }}_{1i}),\\qquad \\\\g_{ee,i} = c_{1i}-c_{i},\\quad g_{ee,i}^{^{\\prime }} = c_{1i}^{^{\\prime }}-c_{i}^{^{\\prime }},\\qquad \\\\g_{ee}=\|g_{ee,i}\|=\|g_{ee,i}^{^{\\prime }}\|=g_{ee}^{^{\\prime }},\\quad G_{0i}=G_{li}-c_{0i},\\\\v_{i}=G_{0i}-\\frac{1}{2}g_{ee,i},\\quad v_{i1}=G_{0i}+\\frac{1}{2}g_{ee,i},\\quad \\\\v^2+v_1^2=2G_0^2+\\frac{1}{2}g_{ee}^2\\quad \\qquad .\\end{aligned}$ Here $G_{li}$ is a velocity of the center of mass of two colliding electrons in the laboratory frame, $G_{0i}$ is the same value in the comoving frame, $g_{ee,i}$ is a relative velocity of two colliding electrons before encounter, $g_{ee,i}^{^{\\prime }}$ is the same value after encounter; $v_i$ and $v_{1i}$ are velocities of colliding electrons in the comoving frame, defined above.", "Introduce non-dimensional variables $\\begin{aligned}g_{i} = \\frac{1}{2}\\left( \\frac{m_{e}}{kT}\\right)^{1/2} g_{ee,i},\\quad g_{i}^{^{\\prime }} = \\frac{1}{2}\\left( \\frac{m_{e}}{kT}\\right)^{1/2} g_{ee,i}^{^{\\prime }},\\qquad \\\\g = \|g_i\|=\|g^{^{\\prime }}_i\|=g^{^{\\prime }},\\qquad G_i=\\left(\\frac{m_{e}}{kT}\\right)^{1/2}G_{0i},\\qquad \\\\dc_{i}dc_{1i} = \\left( \\frac{2kT}{m_{e}}\\right) ^{3}dG_{i}dg_{i},\\quad \\qquad \\\\u^2+u_1^2=G^2+g^2,\\quad u^2=u_i^2, \\quad u_{1}^2=u_{1i}^2,\\quad G^2=G_i^2.\\end{aligned}$ Here $u_i$ , $u_{1i}$ are non-dimensional velocities of electrons, defined above.", "The matrix elements $\\begin{aligned}a_{j0} = 8 \\left( \\frac{2kTm_{e}}{h^{2}}\\right)^{3}\\left( \\frac{kT}{m_{e}}\\right)^{1/2}\\int f_{0}f_{01}(1-f^{^{\\prime }}_{01})(1-f^{^{\\prime }}_{0}) Q_{j}(u^2) u_{i} [u_{i}+ u_{1i} - u^{^{\\prime }}_{i}- u_{1i}^{^{\\prime }}]gbdbd\\varepsilon dg_idG_i=0.\\end{aligned}$ $a_{j0}$ are equal to zero because the momentum conservation during encounter define the zero value in the brackets of (REF ).", "The nonzero matrix elements $a_{jk}\\,\\,(j,k\\ge 1)$ are defined as $\\begin{aligned}a_{jk} = 8 \\left( \\frac{2kTm_{e}}{h^{2}}\\right)^{3}\\left( \\frac{kT}{m_{e}}\\right)^{1/2}\\int f_{0}f_{01}(1-f^{^{\\prime }}_{01})(1-f^{^{\\prime }}_{0}) Q_{j}u_{i}[Q_{k} u_{i}+Q_{k} u_{1i} -Q_{k}^{^{\\prime }} u^{^{\\prime }}_{i}-Q_{k}^{^{\\prime }} u_{1i}^{^{\\prime }}]gbdbd\\varepsilon dg_idG_i.\\end{aligned}$ Here $Q_i$ are function of $u^2$ or $u_1^2$ , and $Q_i^{^{\\prime }}$ are function of $u^{^{\\prime }2}$ or $u_1^{^{\\prime }2}$ respectively." ], [ "$a_{jk}$ for non-degenerate electrons", "For non-degenerate case, at $x_0\\gg 1$ , $f_0 \\ll 1$ , polynomials $Q_i$ are reduced to $S_{3/2}^{(i)}$ , and we have from (REF ) the following expression $(j,k\\ge 1)$ $\\begin{aligned}a_{jk} = 8 \\left( \\frac{2kTm_{e}}{h^{2}}\\right)^{3}\\left( \\frac{kT}{m_{e}}\\right)^{1/2} e^{2x_{0}}\\int e^{-u^2-u^{^{\\prime }2}} S_{3/2}^{(j)}u_{i}[S_{3/2}^{(k)} u_{i}+S_{3/2}^{(k)} u_{1i} -S_{3/2}^{(k)^{\\prime }} u^{^{\\prime }}_{i}-S_{3/2}^{(k)^{\\prime }} u_{1i}^{^{\\prime }}]gbdbd\\varepsilon dg_idG_i.\\end{aligned}$ The integrals $[S_{3/2}^{(j)},S_{3/2}^{(k)}] = \\frac{2}{\\pi ^3}\\left( \\frac{kT}{m_{e}}\\right)^{1/2} \\int e^{-u^2-u^{^{\\prime }2}} S_{3/2}^{(j)}u_{i}[S_{3/2}^{(k)} u_{i}+S_{3/2}^{(k)} u_{1i} -S_{3/2}^{(k)^{\\prime }} u^{^{\\prime }}_{i}-S_{3/2}^{(k)^{\\prime }} u_{1i}^{^{\\prime }}]gbdbd\\varepsilon dg_idG_i$ are calculated in [12], and are defined by formulae $\\begin{aligned}\\quad [S_{3/2}^{(1)},S_{3/2}^{(1)}]= 4\\Omega _{ee}^{(2)}(2),\\qquad \\quad \\\\[S_{3/2}^{(1)},S_{3/2}^{(2)}]= 7\\Omega _{ee}^{(2)}(2)-2\\Omega _{ee}^{(2)}(3),\\quad \\\\[S_{3/2}^{(2)},S_{3/2}^{(2)}]=\\frac{77}{4}\\Omega _{ee}^{(2)}(2)-7\\Omega _{ee}^{(2)}(3)+\\Omega _{ee}^{(2)}(4).\\end{aligned}$ The functions $\\Omega _{ee}^{(l)}(r)$ are similar to functions (REF ), and are defined in [12] as $\\begin{aligned}\\Omega _{ee}^{(l)}(r) = \\frac{\\sqrt{\\pi }}{2} \\int _{0}^{\\infty } e^{-x}x^{r+\\frac{1}{2}}\\phi _{ee}(l)dx,\\\\\\phi _{ee}(l)=\\int _{0}^{\\infty } (1-\\cos ^{l} \\theta ) bdb,\\quad x=g^2.\\end{aligned}$ Using (REF ),(REF ),(REF ) in (REF ), we have $\\begin{aligned}a_{jk} = n_e^2 [S_{3/2}^{(j)},S_{3/2}^{(k)}]\\end{aligned}$ For plasma with $\\Lambda \\gg 1$ from (REF ) we have from (REF ), (REF ) $\\begin{aligned}\\phi _{ee}(2)\\approx \\frac{16e^4}{m_e^2 g_{ee}^3},\\qquad \\\\\\Omega _{ee}^{(2)}(r)=\\sqrt{\\pi }\\frac{e^4 \\Lambda }{\\sqrt{m_e} (kT)^{3/2}}\\Gamma (r).\\end{aligned}$ Using (REF ), we have from (REF ),(REF ), with account of (REF ), with $n_e= Z n_N$ $a_{11} = 4 n^{2}_{e}\\frac{\\sqrt{\\pi } \\Lambda e^4}{\\sqrt{m_e} (kT)^{3/2}}=\\frac{3}{\\sqrt{2}}\\frac{n_e}{Z\\tau _{nd}},$ $a_{12} = 3 n^{2}_{e} \\frac{\\sqrt{\\pi } \\Lambda e^4}{\\sqrt{m_e}(kT)^{3/2}}= \\frac{9}{4 \\sqrt{2}}\\frac{n_e}{Z\\tau _{nd}},$ $a_{22} = \\frac{45}{4} n^{2}_{e}\\frac{\\sqrt{\\pi } \\Lambda e^4}{\\sqrt{m_e}(kT)^{3/2}}=\\frac{135}{16 \\sqrt{2}}\\frac{n_e}{Z\\tau _{nd}}.$" ], [ "$a_{jk}$ for degenerate electrons", "The matrix elements $a_{jk}$ for strongly degenerate case had been found analytically in [15], see also [16].", "They were calculated for strongly degenerate neutrons in a nuclear matter in [15], and for the neutrons in the inner crust of a neutron star, with many free neutrons [16].", "It was found in the last case that in presence of nondegenerate heavy nuclei, and strongly degenerate neutron, the input of collisions between them in the heat transfer and diffusion coefficients is negligibly small, in comparison with neutron-nuclei collisions.", "The same situation we have for the strongly degenerate electrons, for which, using results of [15], the estimations give $a_{jk}\\sim b_{jk}/x_0^2\\ll b_{jk}$ for $x_0\\gg 1$ .", "Therefore for strongly degenerate electrons the Lorentz approximation, with account of collisions between light and heavy particles only, is asymptotically exact.", "The heat transfer coefficients for strongly degenerate electrons in presence of a magnetic field are calculated in section VIII.", "The situation is more complicated for partially degenerate electrons.", "In this case there are no analytical expressions for the matrix elements $a_{jk}$ , which should be found numerically by integration of multi-dimensional integrals in (REF ).", "Another problem is more serious.", "As shown on section VII, the precision of polynomial approximation is decreasing with increasing of the level of degeneracy.", "For non-degenerate electrons the result of three-polynomial approximation in Lorentz gas at $B=0$ is less than the exact result in Lorentz approximation by only about 2.2%, see (REF ) and (REF ).", "Similar calculations for moderately degenerate electrons at $x_0=0$ in (REF ) and (REF ) show that the result of three polynomial approximation is about 87% of the exact result.", "Therefore for stronger degeneracy the result of 3-polynomial approximation will be even farther (less) from the exact result, and to obtain good results in the polynomial approximation the number of polynomials should increase with the level of degeneracy.", "That leads to very cumbersome analytical calculations.", "In looks out that it is better to solve this problem by numerical calculations, if a good precision is needed.", "In astrophysical problems it could be enough to use the interpolation formulae between sufficiently exact results obtained in 3-polynomial approximation for nondegenerate electrons with account of $e\\,e$ collisions, and asymptotically exact result for strongly degenerate electrons in Lorentz approximation.", "The discussion of this problem is given in section IX." ], [ "Tensor of a heat conductivity", "The heat flux is expressed via the heat conductivity tensor in the form [24], [25], [38]: $q_{i} = -\\lambda _{ik} \\frac{\\partial T}{\\partial r_{k}},$ where $\\lambda _{ik}$ : $\\lambda _{ik} = \\frac{5}{2}\\frac{k^{2}Tn_e}{m_e} \\frac{G_{5/2}}{G_{3/2}} \\left\\lbrace \\left[a_{0}^{1} - \\left( \\frac{7}{2} \\frac{G_{7/2}}{G_{5/2}} - \\frac{5}{2} \\frac{G_{5/2}}{G_{3/2}}\\right) a_{1}^{1} \\right]\\delta _{ik}\\right.$ $- \\left.\\varepsilon _{ikn}B_{n}\\left[b_{0}^{1} - \\left( \\frac{7}{2} \\frac{G_{7/2}}{G_{5/2}} - \\frac{5}{2} \\frac{G_{5/2}}{G_{3/2}}\\right) b_{1}^{1}\\right] \\right.$ $ \\left.", "+B_{i}B_{k}\\left[c_{0}^{1} - \\left( \\frac{7}{2} \\frac{G_{7/2}}{G_{5/2}} - \\frac{5}{2} \\frac{G_{5/2}}{G_{3/2}}\\right)c_{1}^{1}\\right] \\right\\rbrace $ Here $a_{0}^1$ , $a_{1}^1$ ; and $b_{0}^1$ , $b_{1}^1$ are the real and imaginary parts of the coefficients $a_{0}$ and $a_{1}$ , respectively: $\\begin{aligned}a_{0} & = a_{0}^{1}+iBb_{0}^{1}, &a_{1} & = a_{1}^{1}+iBb_{1}^{1} \\\\B^2\\,c_{0}^{1} & = (a_{0}^{1})_{B=0} - a_{0}^{1},& B^2\\,c_{1}^{1} & = (a_{1}^{1})_{B=0}-a_{1}^{1}\\end{aligned}$ To find the coefficients $a_{0}$ ,$a_{1}$ for arbitrary electron degeneracy it is necessary to solve the system of equations (REF ) with matrix elements $b_{jk}$ from (REF )-(REF ), and matrix elements $a_{jk}$ from section V. For arbitrary degeneracy of electrons the coefficients in the heat conductivity tensor, as well as well as in 3 other tensors defining the transport of a heat and electrical current in a dense plasma, may be evaluated only numerically.", "In two limiting cases of non-degenerate, and strongly degenerate electrons the results are found analytically." ], [ "Heat conductivity tensor for non-degenerate electrons", "For non-degenerate electrons tensor (REF ) can be written as follows: $\\lambda _{ik} = \\frac{5}{2}\\frac{k^{2}Tn_e}{m_e} \\left[(a_{0}^{1} - a_{1}^{1})\\delta _{ik} - \\varepsilon _{ikn}B_{n}(b_{0}^{1} - b_{1}^{1}) + B_{i}B_{k}(c_{0}^{1} -c_{1}^{1})\\right]$ The system for 3-polynomial solution for the electrons in presence of magnetic field, following from (REF ), with account of (REF )-(REF ), (REF )-(REF ), is written as $\\left\\lbrace \\begin{aligned}0 = -\\frac{3}{2} i\\omega \\tau _{nd} a_{0} +\\frac{3}{2}a_{0} +\\frac{9}{4}a_{1}+\\frac{45}{16}a_{2}\\\\-\\frac{15}{4} \\tau _{nd} =-\\frac{15}{4}i\\omega \\tau _{nd} a_{1} +\\frac{9}{4}a_{0}+ \\frac{3}{2}\\left(\\frac{13}{4}+\\frac{\\sqrt{2}}{Z}\\right) a_{1}+\\frac{9}{8}\\left(\\frac{23}{4}+\\frac{\\sqrt{2}}{Z}\\right) a_{2}\\\\0 = -\\frac{105}{16} i \\omega \\tau _{nd} a_{2} + \\frac{45}{16}a_{0}+\\frac{9}{8}\\left(\\frac{23}{4}+\\frac{\\sqrt{2}}{Z}\\right)a_{1}+\\frac{3}{32}\\left(\\frac{433}{4}+\\frac{45\\sqrt{2}}{Z}\\right)a_{2}\\end{aligned} \\right.$ Two first equations at $a_2=0$ determine the 2-polynomial approximation, giving with account of (REF ) the following results for the case $B=0$ $\\begin{aligned}a_0=\\frac{15}{4}\\frac{\\tau _{nd}}{1+\\frac{\\sqrt{2}}{Z}},\\quad a_1=-\\frac{5}{2}\\frac{\\tau _{nd}}{1+\\frac{\\sqrt{2}}{Z}},\\qquad \\qquad \\\\\\end{aligned}$ $\\lambda ^{(2)}_{nd}=\\frac{125}{8}\\frac{k^2 T n_e}{m_e}\\frac{\\tau _{nd}}{1+\\frac{\\sqrt{2}}{Z}}=15.63\\frac{k^2 T n_e}{m_e}\\frac{\\tau _{nd}}{1+\\frac{\\sqrt{2}}{Z}}.$ The above results coincide with the results obtained in [23],[24].", "In 3-polynomial approximation we obtain the solution of (REF ) for $a_{0},\\,a_{1}$ , and heat conductivity coefficient for the case $B=0$ , with account of (REF ), as $\\begin{aligned}a_0=\\frac{165}{32} \\frac{1+\\frac{15\\sqrt{2}}{11Z}}{1+\\frac{61\\sqrt{2}}{16Z}+\\frac{9}{2Z^2}}\\tau _{nd},\\qquad \\qquad \\\\a_1=-\\frac{65}{8}\\frac{1+\\frac{45\\sqrt{2}}{52\\,Z}}{1+\\frac{61\\sqrt{2}}{16Z}+\\frac{9}{2Z^2}}\\tau _{nd},\\qquad \\qquad \\\\\\end{aligned}$ $\\begin{aligned}\\lambda ^{(3)}_{nd}=\\frac{2125}{64}\\frac{k^2 T n_e}{m_e}\\frac{1+\\frac{18\\sqrt{2}}{17Z}}{1+\\frac{61\\sqrt{2}}{16Z}+\\frac{9}{2Z^2}}\\tau _{nd},\\quad \\\\=33.20\\frac{k^2 T n_e}{m_e}\\frac{1+\\frac{18\\sqrt{2}}{17Z}}{1+\\frac{61\\sqrt{2}}{16Z}+\\frac{9}{2Z^2}}\\tau _{nd},\\quad .\\end{aligned}$ The value $Q=\\frac{64 m_e \\lambda ^{(3)}_{nd}}{2125 k^2 T n_e \\tau _{nd}}=\\frac{1+\\frac{18\\sqrt{2}}{17Z}}{1+\\frac{61\\sqrt{2}}{16Z}+\\frac{9}{2Z^2}},$ showing how nondegenerate electron-electron collisions decrease the heat conductivity coefficient at $B=0$ is presented in Table 1 for different values of $Z$ .", "Table: The values of QQ for different elements: hydrogen (Z=1); helium (Z=2); carbon (Z=6); oxygen (Z=8); iron (Z=26), which may be expectedin the outer layers of white dwarfs and neutron stars.In two polynomial approximation, taking $a_2=0$ , we obtain the solution of the system (REF ) in the form $a_{0} = \\frac{15}{4}\\tau _{nd}\\frac{1}{1+\\frac{\\sqrt{2}}{Z}- \\frac{5}{2}\\omega ^2\\tau _{nd}^2-\\left(\\frac{23}{4}+\\frac{\\sqrt{2}}{Z}\\right)i\\omega \\tau _{nd}},$ $a_{1} = -\\frac{5}{2}\\tau _{nd}\\frac{1-i\\omega \\tau _{nd} }{1+\\frac{\\sqrt{2}}{Z}- \\frac{5}{2}\\omega ^2\\tau _{nd}^2-\\left(\\frac{23}{4}+\\frac{\\sqrt{2}}{Z}\\right)i\\omega \\tau _{nd}},$ $a_{0}^1 = \\frac{15}{4}\\tau _{nd}\\frac{1+\\frac{\\sqrt{2}}{Z} - \\frac{5}{2} \\omega ^2 \\tau _{nd}^2 }{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},$ $b_{0}^1 = \\frac{15}{4}\\frac{\\omega \\tau _{nd}^2}{B}\\frac{\\frac{23}{4}+\\frac{\\sqrt{2}}{Z}}{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},$ $a_{1}^1 = -\\frac{5}{2}\\tau _{nd}\\frac{1+ \\frac{\\sqrt{2}}{Z}+ \\left(\\frac{13}{4}+\\frac{\\sqrt{2}}{Z}\\right) \\omega ^2 \\tau _{nd}^2 }{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},$ $b_{1}^1 = -\\frac{5}{2}\\frac{\\omega \\tau _{nd}^2}{B}\\frac{\\frac{19}{4} + \\frac{5}{2} \\omega ^2 \\tau _{nd}^2 }{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},$ In 3-polynomial approximation the solution of the system (REF ) is written as $a_{0} =\\frac{165}{32}\\tau _{nd}\\frac{1+\\frac{15\\sqrt{2}}{11\\,Z}-\\frac{35}{11}i\\omega \\tau _{nd}}{1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2-i\\omega \\tau _{nd}\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)},$ $a_{1} =-\\frac{65}{8}\\tau _{nd}\\frac{1+\\frac{45\\sqrt{2}}{52\\,Z}-\\frac{35}{26}\\omega ^2\\tau _{nd}^2-\\left(\\frac{713}{208}+\\frac{45\\sqrt{2}}{52Z}\\right)i\\omega \\tau _{nd}}{1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2-i\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)},$ $a_{0}^1 =\\frac{165}{32}\\tau _{nd}\\frac{\\left(1+\\frac{15\\sqrt{2}}{11\\,Z}\\right)\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]+\\frac{35}{11}\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)}{\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]^2+\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)^2},$ $b_{0}^1=\\frac{165}{32}\\frac{\\omega \\tau _{nd}^2}{B} \\frac{-\\frac{35}{11} \\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]+\\left(1+\\frac{15\\sqrt{2}}{11\\,Z}\\right)\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)}{\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]^2+\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)^2},$ $a_{1}^1 =-\\frac{65}{8}\\tau _{nd}\\frac{\\left(1+\\frac{45\\sqrt{2}}{52\\,Z}-\\frac{35}{26}\\omega ^2\\tau _{nd}^2\\right)\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]+\\left(\\frac{713}{208}+\\frac{45\\sqrt{2}}{52Z}\\right)\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)}{\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]^2+\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)^2},$ $b_{1}^1=-\\frac{65}{8}\\frac{\\omega \\tau _{nd}^2}{B}\\frac{\\left(1+\\frac{45\\sqrt{2}}{52\\,Z}-\\frac{35}{26}\\omega ^2\\tau _{nd}^2\\right)\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)-\\left(\\frac{713}{208}+\\frac{45\\sqrt{2}}{52Z}\\right) \\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]}{\\left[1+\\frac{61\\sqrt{2}}{16\\,Z}+\\frac{9}{2\\,Z^2}-\\left(\\frac{5385}{128}+\\frac{365\\sqrt{2}}{32\\,Z}\\right)\\omega ^2\\tau _{nd}^2\\right]^2+\\omega ^2\\tau _{nd}^2\\left(\\frac{1017}{64}+\\frac{667\\sqrt{2}}{32\\,Z}+\\frac{9}{2Z^2}-\\frac{175}{16}\\omega ^2\\tau _{nd}^2\\right)^2}.$ The values $c_0^1$ and $c_1^1$ in 2 and 3 polynomial approximations are defined using (REF ).", "The heat flux $q_i$ from (REF ),(REF ) may be written in the form $q_i = -\\frac{5}{2}\\frac{k^{2}Tn_e}{m_e}\\left[(a_{0}^{1} - a_{1}^{1})\\delta _{ik} - \\varepsilon _{ikn}B_{n}(b_{0}^{1} - b_{1}^{1}) + B_{i}B_{k}(c_{0}^{1} -c_{1}^{1})\\right]\\frac{\\partial T}{\\partial r_k}=q_i^{(1)}+q_I^{(2)}+q_i^{(3)},$ $q_i^{(1)} = -\\frac{5}{2}\\frac{k^{2}T n_e}{m_e}(a_{0}^{1} - a_{1}^{1})\\frac{\\partial T}{\\partial r_i}=-\\lambda _{nd}^{(1)}\\frac{\\partial T}{\\partial r_i},\\\\q_i^{(2)} = \\frac{5}{2}\\frac{k^{2}Tn_e}{m_e}\\varepsilon _{ikn}B_{n}(b_{0}^{1} - b_{1}^{1})\\frac{\\partial T}{\\partial r_k}=-\\varepsilon _{ikn}B_{n}\\lambda _{nd}^{(2)}\\frac{\\partial T}{\\partial r_k},\\\\q_i^{(3)} = -\\frac{5}{2}\\frac{k^{2}Tn_e}{m_e} B_{i}B_{k}(c_{0}^{1} -c_{1}^{1})\\frac{\\partial T}{\\partial r_k}=- B_{i}B_{k}\\lambda _{nd}^{(3)}\\frac{\\partial T}{\\partial r_k}.$ For 2-polynomial approximation we obtain $\\lambda _{nd}^{(12)}=\\frac{5}{2}\\frac{k^{2}T n_e}{m_e}(a_{0}^{1} - a_{1}^{1})=\\frac{25}{4}\\frac{k^{2}T n_e}{m_e}\\tau _{nd}\\frac{\\frac{5}{2}\\left(1+ \\frac{\\sqrt{2}}{Z}\\right)+ \\left(-\\frac{1}{2}+\\frac{\\sqrt{2}}{Z}\\right) \\omega ^2 \\tau _{nd}^2 }{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},\\\\\\lambda _{nd}^{(22)}=-\\frac{5}{2}\\frac{k^{2}T n_e}{m_e}(b_{0}^{1} - b_{1}^{1})=-\\frac{25}{4}\\frac{k^{2}T n_e}{m_e}\\frac{\\omega \\tau _{nd}^2}{B}\\frac{\\frac{107}{8}+ \\frac{3\\sqrt{2}}{2Z}+ \\frac{5}{2}\\omega ^2 \\tau _{nd}^2 }{\\left(1+\\frac{\\sqrt{2}}{Z}\\right)^2+ \\left(\\frac{449}{16}+\\frac{13}{2}\\frac{\\sqrt{2}}{Z}+\\frac{2}{Z^2}\\right)\\omega ^2\\tau _{nd}^2+ \\frac{25}{4}\\omega ^4\\tau _{nd}^4},\\\\B^2\\lambda _{nd}^{(32)}=\\lambda _{nd}^{(12)}(B=0)-\\lambda _{nd}^{(12)}.\\qquad \\qquad \\qquad \\qquad $ The expressions for heat conductivity coefficients in 3-polynomial approximation are very cumbersome, and are not presented here.", "They could be written explicitly using (REF ) - ().", "Using (REF ) we present another form of the components of the heat conductivity tensor in the magnetic field.", "Three components of the heat flux: parallel $q_{\|\|}$ , perpendicular $q_{\\perp }$ to the magnetic field $\\vec{B}$ , and \"Hall\" component of the heat flux $q_{\\rm hall}$ , perpendicular to both vectors $\\nabla T$ and $\\vec{B}$ , with account of (REF ) or (REF ) are defined by relations $q_{\|\|}=-\\lambda _{\|\|}\\nabla T_{\|\|}, $ $\\lambda _{\|\|}=\\frac{5}{2}\\frac{k^2 T n_e}{m_e}[a_0^1-a_1^1+B^2(c_0^1-c_1^1)]=\\lambda _{nd},$ $q_{\\perp }=-\\lambda _{\\perp }\\nabla T_{\\perp },\\quad \\lambda _{\\perp } =\\frac{5}{2}\\frac{k^2 Tn_e}{m_e}(a_0^1-a_1^1),$ $ q_{\\rm hall}=-\\lambda _{\\rm hall}\\frac{\\nabla T\\times \\vec{B}}{B}, \\quad \\lambda _{\\rm hall} =\\frac{5}{2}\\frac{k^2 T n_e}{m_e}B(b_0^1-b_1^1).$ The 2-polynomial results coincide with corresponding derivations obtained in [23], [24].", "The difference between 2 and 3 polynomial approximation may be characterised by comparison of values $Q_\\perp ^{(2)}$ and $Q_\\perp ^{(3)}$ in Fig.1.", "$Q_\\perp ^{(2)}=\\frac{\\lambda _{nd}^{(12)}}{\\lambda _{nd}^{(3)}},\\quad Q_\\perp ^{(3)}=\\frac{\\lambda _{nd}^{(13)}}{\\lambda _{nd}^{(3)}},$ where $\\lambda _{nd}^{(12)}$ is defined in (REF ), $\\lambda _{nd}^{(3)}$ is defined in (REF ), and $\\lambda _{nd}^{(13)}$ is defined from (REF ),(REF ),(REF ) in the same way as $\\lambda _{nd}^{(12)}$ .", "The functions $Q_\\perp ^{(2)}(\\omega \\tau _{nd})$ , $Q_\\perp ^{(3)}(\\omega \\tau _{nd})$ are presented in Fig.1 for carbon, at Z=6.", "In this figure we have $Q_\\perp ^{(2)}=0.023$ and $Q_\\perp ^{(3)}=0.014$ , at $\\omega \\tau =1$ .", "Figure: Comparison of 2 and 3 polynomial approximation fornondegenerate carbon plasma at Z=6, at different ωτ\\omega \\tau .The Lorentz approximation for solving a kinetic equation is applied when the mass of light particles (electrons) is much smaller than the the mass of heavy particles (nuclei), and in addition electron-electron collisions are neglected.", "In this approximation the linearized Boltzmann equation has an exact solution.", "This approximation works good for metal transport coefficients, where a strong electron degeneracy permits to neglect electron-electron collisions.", "Lorenz approximation is useful for checking approximate polynomial solution, because it gives a possibility to follow a convergence of the approximate solution to the exact one, increasing the number of polynomials.", "In different approaches the solution in Lorentz approximation was considered in [12],[39],[28],[40], , see also [35].", "The explicit exact solution in Lorentz approximation is obtained for the case $B=0$ .", "If we consider only the heat flux connected with the temperature gradient, at zero value of the diffusive vector $d_i$ from (REF ),(REF ), than we obtain the expression for the heat flux from [35] as $q_i=-{640k\\over \\Lambda }{m_e (kT)^4\\over n_NZ^2e^4h^3}\\left(G_5-{1\\over 2}\\frac{G_{5/2}}{G_{3/2}} G_4\\right)\\frac{\\partial {T}}{\\partial r_{\\rm i}}.$ In the limiting cases the coefficient in (REF ) is reduces to $\\lambda _{e}^l= {40\\sqrt{2}\\over \\pi ^{3/2} \\Lambda }k{n_e\\over n_N}{(kT)^{5/2}\\over {\\rm e}^4 Z^2\\sqrt{m_e}}= \\frac{320}{3\\pi }\\frac{k^2 Tn_e}{m_e}\\tau _{nd} \\quad (ND)$ $\\quad ={5\\over 64\\Lambda }{k^2Tn_{\\rm e}^2h^3\\over m_{\\rm e}^2n_NZ^2e^4}=\\frac{5\\pi ^2}{6}\\frac{k^2 Tn_e}{m_e}\\tau _d \\quad (D).$ Note, that the heat conductivity coefficient in (REF ) determines the heat flux at zero value of the diffusion vector $d_i=0$ .", "Often the heat conductivity coefficient is written for the case of zero value of the diffusion velocity $\\langle v_{ \\alpha i} \\rangle =0$ [12], [35].", "When the thermal conductivity and diffusion are calculated in the same procedure, both heat and diffusion fluxes are calculated without any restrictions on the diffusion vector or diffusion velocity.", "Such consideration will be performed elsewhere.", "The exact formulae in the Lorentz model are used [12] for estimation of the precision of the polynomial approximation.", "The input of electron-electron collisions into the heat conductivity coefficients for different $Z$ may be estimated from the plot of normalized 3-polynomial heat conductivity coefficients in the direction perpendicular to the magnetic field, introducing the value $Q_\\perp ^{(3l)}$ , defined as $Q_\\perp ^{(3l)}=\\frac{\\lambda _{nd}^{(13)}}{\\lambda _{e,nd}^{l}}.$ Here $\\lambda _{e,nd}^{l}$ is taken from the upper line in (REF ).", "The curves of this value, for different Z, including Z=$\\infty $ , related to Lorentz approximation, are plotted in Fig.2.", "The intersection of the plots with the y-axis in Fig.2 occurs in the points given in the Table 1, multiplied by $\\frac{\\lambda ^{(3)}_{nd}}{\\lambda _{e,nd}^{l}}=0,978$ .", "At $\\omega \\tau =1$ we have $Q_\\perp ^{(3l)}=0.0053,\\,\\,0.0060,\\,\\, 0.0083,\\,\\, 0.014$ for $Z=\\infty ,\\, 26,\\,6,\\,2$ respectively.", "Figure: The plots of the value Q ⊥ (3l) Q_\\perp ^{(3l)} as a function of ωτ\\omega \\tau in 3 polynomial approximation are presentedfor nondegenerate plasma of helium (Z=2), carbon (Z=6), iron (Z=26), for comparison with theLorentz plasma, formally corresponding to Z=∞Z=\\infty .", "The deviations from the Lorentz plasma are connected with the input of electron-electron collisions.The intersection of Lorentz 3-polynomial curve (Z=∞\\infty ) with y-axis at 0.978 is connected with deviation from the exact solution inLorentz approximation." ], [ "Polynomial calculations without account of collisions between electrons", "In order to test the precision of polynomial approximation for the heat conductivity coefficients we compare them with ones, obtained by exact solution in Lorentz approximation.", "Omitting electron-electron collisions we obtain in 3 polynomial approximation the following system $\\left\\lbrace \\begin{aligned}0 =-\\frac{3}{2} i \\omega n_{e}a_{0}+ a_{0}b_{00} +a_{1}b_{01}+a_{2}b_{02}\\\\-\\frac{15}{4} n_{e}\\left( \\frac{7G_{7/2}}{2G_{3/2}} -\\frac{5G_{5/2}^{2}}{2G_{3/2}^{2}}\\right) =- \\frac{15}{4} \\left(\\frac{7G_{7/2}}{2G_{3/2}} - \\frac{5G_{5/2}^{2}}{2G_{3/2}^{2}}\\right)i \\omega n_{e}a_{1} + a_{0}b_{10}+ a_{1} b_{11}+a_{2}b_{12}\\\\0 = -\\frac{105}{16}\\left(-\\frac{35}{8}\\frac{G_{7/2}^2}{G_{3/2}^2}+\\frac{49}{2}\\frac{G_{7/2}^2}{G_{5/2}^2}\\frac{G_{7/2}}{G_{3/2}}-\\frac{63}{2}\\frac{G_{9/2}G_{7/2}}{G_{5/2}G_{3/2}}+\\frac{99}{8}\\frac{G_{11/2}}{G_{3/2}}\\right)i \\omega n_{e}a_{2}+a_{0}b_{20}+a_{1}b_{21}+a_{2}b_{22}\\end{aligned} \\right.$" ], [ "Results for non-degenerate electrons", "In absence of the magnetic field, in Lorenz approximation with $a_{jk}=0$ , the system (REF ) is reduced to $\\left\\lbrace \\begin{aligned}0 = a_{0}b_{00} +a_{1}b_{01}+a_{2}b_{02}\\\\-\\frac{15}{4} n_{e} =a_{0}b_{10}+ a_{1} b_{11}+a_{2}b_{12}\\\\0 = a_{0}b_{20}+a_{1}b_{21}+a_{2}b_{22}\\end{aligned} \\right.$ With account of (REF )-(REF )this system is written as $\\left\\lbrace \\begin{aligned}0 = \\frac{3}{2}a_{0} +\\frac{9}{4}a_{1}+\\frac{45}{16}a_{2}\\\\-\\frac{15}{4} \\tau _{nd} =\\frac{9}{4}a_{0}+ \\frac{39}{8}a_{1}+\\frac{207}{32}a_{2}\\\\0 = \\frac{45}{16}a_{0}+\\frac{207}{32}a_{1}+\\frac{1299}{128}a_{2}\\end{aligned} \\right.$ This system is written for 3-polynomial approximation to the solution.", "Two first equations at $a_2=0$ determine the 2-polynomial approximation, giving with account of (REF ) the following results $a_0=\\frac{15}{4}\\tau _{nd},\\quad a_1=-\\frac{5}{2}\\tau _{nd},\\quad $ $\\lambda ^{(2)}_{ndl}=\\frac{25}{4}\\frac{5}{2}\\frac{k^2 T n_e}{m_e}\\tau _{nd}=15.63\\frac{k^2 T n_e}{m_e}\\tau _{nd}.$ In 3-polynomial approximation we obtain the solution of (REF ) for $a_{0},\\,a_{1}$ , and heat conductivity coefficient, with account of (REF ), as $a_0=\\frac{165}{32}\\tau _{nd},\\quad a_1=-\\frac{65}{8}\\tau _{nd},\\quad $ $\\lambda ^{(3)}_{ndl}=\\frac{425}{32}\\frac{5}{2}\\frac{k^2 T n_e}{m_e}\\tau _{nd}$ $=\\frac{2125}{64}\\frac{k^2 T n_e}{m_e}\\tau _{nd}=33.20\\frac{k^2 T n_e}{m_e}\\tau _{nd}.$ The heat coefficients obtained by the method of successive polynomial approximations should be compared with the exact solution $\\lambda _{nd}^{l}$ , obtained by Lorentz method (REF ) for non-degenerate electrons $\\lambda ^{(l)}_{nd}=\\frac{320}{3\\pi }\\frac{k^2 Tn_e}{m_e}\\tau _{nd}=33.95\\,\\frac{k^2 Tn_e}{m_e}\\tau _{nd}.$ It is clear that the 2- polynomial solution underestimate the coefficient of the heat conductivity for more than 50%, and the 3-polynomial solution differs from the exact solution only for about 2.2%.", "Equations or 3 polynomial approximation in presence of a magnetic field are obtained from (REF ) with account of (REF )-(REF ),(REF ) in the form $\\left\\lbrace \\begin{aligned}0 = -\\frac{3}{2} i\\omega \\tau _{nd} a_{0} +\\frac{3}{2}a_{0} +\\frac{9}{4}a_{1}+\\frac{45}{16}a_{2}\\\\-\\frac{15}{4} \\tau _{nd} =-\\frac{15}{4}i\\omega \\tau _{nd} a_{1} +\\frac{9}{4}a_{0}+ \\frac{39}{8}a_{1}+\\frac{207}{32}a_{2}\\\\0 = -\\frac{105}{16} i \\omega \\tau _{nd} a_{2} + \\frac{45}{16}a_{0}+\\frac{207}{32}a_{1}+\\frac{1299}{128}a_{2}\\end{aligned} \\right.$ Explicit solution of equations (REF ) for 2 and 3 polynomial approximations is determined by formulae (REF )-(REF ) at formally infinite value of $Z$ ." ], [ "Partially degenerate electrons", "For partially degenerate electrons at $x_0=0$ , with the degeneracy level $DL=\\frac{\\varepsilon _{fe}}{kT}=1.011$ , the system (REF ) is written in the form $\\left\\lbrace \\begin{aligned}0 =-1.5 i \\omega n_{e}a_{0}+ a_{0}b_{00} +a_{1}b_{01}+a_{2}b_{02}\\\\-3.88 n_{e} =- 3.88 i \\omega n_{e}a_{1} + a_{0}b_{10}+ a_{1} b_{11}+a_{2}b_{12}\\\\0 = -7.138 i \\omega n_{e}a_{2}+a_{0}b_{20}+a_{1}b_{21}+a_{2}b_{22}\\end{aligned} \\right.,$ where matrix elements $b_{jk}$ are defined in (REF )-(REF ).", "In absence of a magnetic field this system is reduced to $\\left\\lbrace \\begin{aligned}0 = a_{0}b_{00} +a_{1}b_{01}+a_{2}b_{02}\\\\-3.88 n_{e} =a_{0}b_{10}+ a_{1} b_{11}+a_{2}b_{12}\\\\0 = a_{0}b_{20}+a_{1}b_{21}+a_{2}b_{22}\\end{aligned} \\right.$ With account of (REF )-(REF ) this system may be written in the form $\\left\\lbrace \\begin{aligned}0 = 1.5 a_{0} +2.16 a_{1}+2.588 a_{2}\\\\-3.88 \\tau _{d0} =2.16 a_{0}+ 5.162 a_{1} +6.671 a_{2}\\\\0 = 2.588 a_{0}+6.671 a_{1}+11.038 a_{2}\\end{aligned} \\right.$ This system is written for 3-polynomial approximation to the solution.", "Two first equations at $a_2=0$ determine the 2-polynomial approximation, which with account of (REF ), give the following result $a_0=2.723\\tau _{d0},\\quad a_1=-1.891\\tau _{d0},\\quad $ $\\lambda ^{(2)}_{d0l}=5.043\\frac{5}{2}\\frac{k^2 T n_e}{m_e}\\tau _{nd}=12.61\\frac{k^2 T n_e}{m_e}\\tau _{d0}.$ In 3-polynomial approximation we obtain the solution of (REF ) for $a_{0},\\,a_{1}$ , and heat conductivity coefficient, with account of (REF ), as $a_0=3.533\\tau _{d0},\\quad a_1=-5.295\\tau _{d0},\\quad $ $\\lambda ^{(3)}_{d0l}=8.278\\frac{5}{2}\\frac{k^2 T n_e}{m_e}\\tau _{d0}=22.07\\frac{k^2 T n_e}{m_e}\\tau _{d0}.$ The heat coefficients obtained by the method of successive polynomial approximations should be compared with an exact solution $\\lambda _{nd}^{l}$ obtained by Lorentz method (REF ) for non-degenerate electrons $\\lambda ^{(l)}_{d0}=0.744\\frac{320}{3\\pi }\\frac{k^2 Tn_e}{m_e}\\tau _{d0}=25.26\\,\\frac{k^2 Tn_e}{m_e}\\tau _{d0}.$ It is clear that the 2- polynomial solution underestimate the coefficient of the heat conductivity for more than 50%, and the 3-polynomial solution differs from the exact solution for about 13%.", "So, a convergence of the polynomial approximation to the exact value takes place slower than for non-degenerate electrons in previous subsection." ], [ "Results for strongly degenerate electrons", "The non-diagonal matrix elements $b_{ik}$ , $i\\ne k$ for strongly degenerate case are much smaller than the diagonal one $b_{ii}$ , according to (REF )- (REF .", "In this case at $x_0\\gg 1$ , and neglecting terms $\\sim 1/x_0$ we obtain a simplified system (REF ) for 3-polynomial expansion as $\\left\\lbrace \\begin{aligned}0 =-\\frac{3}{2} i \\omega n_{e}a_{0}+ a_{0}b_{00}\\qquad \\qquad \\\\-\\frac{15}{4} n_{e}\\frac{2\\pi ^2}{15}=- \\frac{15}{4} \\frac{2\\pi ^2}{15}i \\omega n_{e}a_{1}+ a_{1} b_{11} \\qquad \\qquad \\\\0 = -\\frac{105}{16}\\left(-\\frac{35}{8}\\frac{G_{7/2}^2}{G_{3/2}^2}+\\frac{49}{2}\\frac{G_{7/2}^2}{G_{5/2}^2}\\frac{G_{7/2}}{G_{3/2}}-\\frac{63}{2}\\frac{G_{9/2}G_{7/2}}{g_{5/2}G_{3/2}}+\\frac{99}{8}\\frac{G_{11/2}}{G_{3/2}}\\right)i \\omega n_{e}a_{2}+a_{2}b_{22}\\end{aligned} \\right.$ Solution of the system (REF ) is written in the form, with account of (REF ),(REF ) $a_0=0,\\quad a_2=0,$ $a_1=\\frac{\\frac{15}{4} n_{e} \\frac{2\\pi ^2}{15}}{\\frac{15}{4}\\frac{2\\pi ^2}{15} i n_{e}\\omega - b_{11}}=\\frac{\\frac{\\pi ^2 n_e}{2}}{i\\omega \\frac{\\pi ^2 n_e}{2}- \\frac{\\pi ^2 n_e}{2\\tau _d}}=-\\frac{\\tau _d}{1-i\\omega \\tau _d}=-\\tau _d\\frac{1+i\\omega \\tau _d}{1+\\omega ^2\\tau _d^2}=a_1^{1}+iB b_1^{1},$ $c_1^1=-\\frac{\\tau _d}{B^2} \\frac{\\omega ^2\\tau _d^2}{1+\\omega ^2\\tau _d^2}.$ Using (REF ) we obtain the components of the heat conductivity tensor in the magnetic field for strongly degenerate electrons in polynomial approximation.", "3 components of the heat flux: parallel $q_{\|\|}^{sd}$ , perpendicular $q_{\\perp }^{(sd)}$ to the magnetic field $\\vec{B}$ , and \"Hall\" component of the heat flux $q_{\\rm hall}^{(sd)}$ , perpendicular to both vectors $\\nabla T$ and $\\vec{B}$ , with account of (REF ),(REF )-(REF ) are defined by relations $q_{\|\|}^{(sd)}=-\\lambda _{\|\|}^{(sd)}\\nabla T_{\|\|}, $ $\\lambda _{\|\|}^{(sd)}=-\\frac{\\pi ^2}{3}\\frac{k^2 T n_e}{m_e}(a_1^1+B^2\\,c_1^1)=\\frac{\\pi ^2}{3}\\frac{k^2 T n_e}{m_e}\\tau _d.$ $q_{\\perp }^{(sd)}=-\\lambda _{\\perp }^{(2)}\\nabla T_{\\perp },\\quad \\lambda _{\\perp }^{(sd)} =-\\frac{\\pi ^2}{3}\\frac{k^2 T n_e}{m_e}a_1^1$ $=\\frac{\\pi ^2}{3}\\frac{k^2 T n_e}{m_e}\\frac{\\tau _d}{1+\\omega ^2\\tau _d^2}$ $q_{\\rm hall}^{(sd)}=-\\lambda _{\\rm hall}^{(2)}\\frac{\\nabla T\\times \\vec{B}}{B}, \\quad \\lambda _{\\rm hall}^{(sd)} =-\\frac{\\pi ^2}{3}\\frac{k^2 T n_e}{m_e}B\\,b_1^1.$ Comparing (REF ) with the exact value for strongly degenerate electrons from Lorentz approximation (REF ) we see, that polynomial approximation, where the terms $\\sim x_0^{-1}$ are neglected, gives is 2.5 times smaller value than the exact one.", "This value, as well as a simple dependence of the heat conductivity tensor on the magnetic field $q_{\|\|}^{(sd)}/q_{\\perp }^{(sd)}=(1+\\omega ^2\\tau _d^2)$ follows also from a rough theory of heat conductivity and diffusion in presence of a magnetic field, based on the mean free path, which is described in [12].", "The value of the heat conductivity coefficient, following from this approach, was considered in [8],[9], and many subsequent papers.", "As mentioned above the heat flux calculated here is connected only with a temperature gradient, when the diffusion vector $d_i=0$ .", "In laboratory conditions when the electrical conductivity is small and electrical current is damped rapidly, another limiting case is considered, where $j_i\\sim \\langle v_i\\rangle =0$ .", "This restriction leads to linear connection between $d_i$ and $\\nabla T$ , what permits [12] to exclude $d_i$ and to express the heat flux as directly proportional to $\\nabla T$ , with another heat conductivity coefficient $\\lambda _j$ .", "For strongly degenerate electrons we have $\\lambda _j=0.4\\lambda _e^l=\\lambda _{\|\|}^{(sd)}$ , see (REF ),(REF ) and [35]." ], [ "Heat conductivity of strongly degenerate electrons in presence of magnetic field: Lorentz approximation", "The equation for the function $\\xi $ from (REF ),(REF ) may be written in the form, using relations $f_0^{\\prime }=f_0$ , $\\xi ^{\\prime }=\\xi $ , $u^{\\prime }_i=u_i\\,cos\\theta $ , and making integration over $dc_{Ni}$ with account of (REF ), as $f_{0}(1-f_{0})(u^{2} - \\frac{5G_{5/2}}{2G_{3/2}}) =-iBf_{0}(1-f_{0})\\frac{e \\xi }{m_{e} c}u_{i}+ f_{0}(1-f_{0})n_N\\xi \\int (1-\\cos \\theta ) g_{eN}bdbd\\varepsilon .$ The function $\\xi $ is defined by expression $\\xi =\\frac{u^2-\\frac{5}{2}\\frac{G_{5/2}}{G_{3/2}}}{2\\pi n_N \\int _0^\\infty (1-\\cos \\theta )g b db -i\\omega }.$ Using (REF )-(REF ) we obtain in Lorenz approximation, with $g_{12}=v$ , $\\int _0^\\infty (1-\\cos \\theta )g b db =2\\frac{e^4Z^2}{m_e^2 v^3}\\Lambda $ $\\xi =\\frac{u^2-\\frac{5}{2}\\frac{G_{5/2}}{G_{3/2}}}{4\\pi n_N \\left(\\frac{m_e}{2kT}\\right)^{3/2}\\frac{e^4Z^2}{m_e^2 u^3}\\Lambda - i\\omega }.$ According to (REF ) we have $\\xi =A^{(1)}+iB A^{(2)},$ $A^{(1)}=\\frac{(u^2-\\frac{5}{2}\\frac{G_{5/2}}{G_{3/2}})4\\pi n_N \\left(\\frac{m_e}{2kT}\\right)^{3/2}\\frac{e^4Z^2}{m_e^2 u^3}\\Lambda }{\\left[4\\pi n_N \\left(\\frac{m_e}{2kT}\\right)^{3/2}\\frac{e^4Z^2}{m_e^2 u^3}\\Lambda \\right]^2 + \\omega ^2}.$ $A^{(2)}=\\frac{\\omega }{B}\\frac{u^2-\\frac{5}{2}\\frac{G_{5/2}}{G_{3/2}}}{\\left[4\\pi n_N \\left(\\frac{m_e}{2kT}\\right)^{3/2}\\frac{e^4Z^2}{m_e^2 u^3}\\Lambda \\right]^2 + \\omega ^2}.$ $A^{(3)}=A^{(1)}(B=0)-A^{(1)}.$ The expression for the heat flux, following from (REF ),(REF ),(REF ),(REF ), (REF )-(REF ) is written as $q_i=-\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\left[\\delta _{ij}\\int _0^\\infty f_0(1-f_0)A^{(1)}x^{5/2}dx -\\varepsilon _{ijk}B_k\\int _0^\\infty f_0(1-f_0)A^{(2)}x^{5/2}dx \\right.\\nonumber \\\\\\left.+ B_i B_j \\int _0^\\infty f_0(1-f_0)A^{(3)}x^{5/2}dx\\right]\\frac{\\partial T}{\\partial x_j}=q_i^{(1)}+q_i^{(2)}+q_i^{(3)}, \\qquad x=u^2, \\qquad \\nonumber \\\\q^{(1)}_i=-\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\int _0^\\infty f_0(1-f_0)A^{(1)}x^{5/2}dx \\frac{\\partial T}{\\partial x_i}=-\\lambda ^{(1)}_{sd} \\frac{\\partial T}{\\partial x_i},\\qquad \\\\q^{(2)}_i=\\varepsilon _{ijk}B_k\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\int _0^\\infty f_0(1-f_0)A^{(2)}x^{5/2}dx\\frac{\\partial T}{\\partial x_j}=-\\varepsilon _{ijk}B_k \\lambda ^{(2)}_{sd}\\frac{\\partial T}{\\partial x_j}, \\qquad \\nonumber \\\\q_i^{(3)}=-B_i B_j \\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\int _0^\\infty f_0(1-f_0)A^{(3)}x^{5/2}dx \\frac{\\partial T}{\\partial x_j}=-B_i B_j\\lambda ^{(3)}_{sd} \\frac{\\partial T}{\\partial x_j},\\qquad \\nonumber $ For strongly degenerate electrons at $x_0\\gg 1$ the integrals in (REF ) with $A^{1},\\,\\, A^{2}\\,\\,A^{(3)}$ from (REF )-(REF ), are expressed analytically , using expansion formula [32]) $\\int _0^\\infty \\frac{f(x)dx}{e^{x-x_0}+1}=\\int _0^{x_0}f(x)dx+\\frac{\\pi ^2}{6}f^{^{\\prime }}(x_0)+...$ After partial integration we obtain the expression, which are suitable for integration by formula (REF ) $\\lambda ^{(1)}=\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\int _0^\\infty f_0\\frac{d\\,(A^{(1)}x^{5/2})}{dx} dx$ $\\lambda ^{(2)}=-\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\int _0^\\infty f_0\\frac{d\\,(A^{(2)}x^{5/2})}{dx} dx$ $B^2 A^{(3)}=A^{(1)}(B=0)-A^{(1)}, \\nonumber \\\\B^2 \\lambda ^{(3)}=\\lambda ^{(1)}(B=0)-\\lambda ^{(1)}.$ Applying (REF ) to the integrals (REF ),(REF ), we obtain $\\lambda ^{(1)}=\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\left[A^{(1)}(x_0)x_0^{5/2}+\\frac{\\pi ^2}{6}\\frac{d^2\\,(A^{(1)}x^{5/2})}{dx^2}\|_{x=x_0}\\right],$ $\\lambda ^{(2)}=-\\frac{2\\pi }{3}\\frac{m_e^4}{h^3T}\\left(\\frac{2kT}{m_e}\\right)^{7/2}\\left[A^{(2)}(x_0)x_0^{5/2}+\\frac{\\pi ^2}{6}\\frac{d^2\\,(A^{(2)}x^{5/2})}{dx^2}\|_{x=x_0}\\right],$ Using (REF ),(REF ), and writing the formula using $\\tau _{d}$ from (REF , we write the heat conductivity coefficients in the form $\\lambda ^{(1)}=\\frac{5\\pi ^2}{6}\\frac{k^2 T n_e}{m_e }\\tau _d\\left\\lbrace \\frac{1}{1+\\omega ^2\\tau _d^2}-\\frac{6}{5}\\frac{\\omega ^2\\tau ^2_d}{(1+\\omega ^2\\tau _d^2)^2}-\\frac{\\pi ^2}{10}\\left[\\frac{1}{1+\\omega ^2\\tau _d^2\\left(\\frac{x^3}{x_0^3}\\right)}\\right]^{^{\\prime \\prime }}\|_{x=x_0}\\right\\rbrace ,$ $\\lambda ^{(2)}=-\\frac{4\\pi ^2}{3}\\frac{k^2 T n_e}{m_e }\\frac{\\tau _d^2\\omega }{B}\\left\\lbrace \\frac{1}{1+\\omega ^2\\tau _d^2}-\\frac{3}{4}\\frac{\\omega ^2\\tau ^2_d}{(1+\\omega ^2\\tau _d^2)^2}-\\frac{\\pi ^2}{16}\\left[\\frac{1}{1+\\omega ^2\\tau _d^2\\left(\\frac{x^3}{x_0^3}\\right)}\\right]^{^{\\prime \\prime }}\|_{x=x_0}\\right\\rbrace ,$ In the case of strongly degenerate electrons the equations (REF )-(REF ),(REF ,(REF ) give an asymptotically exact solution for the heat conductivity coefficients, because collisions between electrons can be neglected in this case.", "The difference between the exact $[\\lambda ^{(1)}]/[\\lambda ^{(1)}(B=0)]$ from (REF ), and phenomenological (REF ) account of the magnetic field influence on the heat conductivity coefficients is presented in Fig.3.", "Here the ratios between the values, which are perpendicular and parallel to magnetic field, are plotted for $kT=0.09E_{f}$ .", "At $\\omega \\tau =1.5$ the exact value of this ratio is 4 times smaller than the phenomenological one.", "The heat flux defined in (REF )-(REF ) corresponds to the situation when the diffusion vector $d_1$ from (REF ) is zero.", "In general case the heat and diffusion (electrical current) fluxes are connected with each other by diffusion vector $d_i$ and temperature gradient $\\partial T/\\partial x_i$ [12].", "Figure: The plots of the ratio λ ⊥ /λ ∥ \\lambda _{\\perp }/\\lambda _{\\parallel } as a function of ωτ\\omega \\tau are presented for phenomenologically obtained heat conductivity (dash-dot line) for comparison with heat conductivity obtained by the solution of Boltzmann equation in Lorentz appoximation (solid line) with kT=0.09E f kT=0.09E_{f}." ], [ "Discussion", "In this paper a thermal conductivity tensor is found for arbitrary degenerate non-relativistic electrons in presence of a non-quantizing magnetic field.", "For nondegenerate electrons the conductivity tensor is derived from the solution of a Boltzmann kinetic equation by classical Chapman - Enskog method using an expansion on the Sonyne polynomials, with remaining two and three terms.", "The electron-nucleus and electron-electron collisions are taken into consideration.", "The tensor of the thermal conductivity is written for arbitrary local direction of the magnetic field and the temperature gradient, in the Cartesian coordinate system, following [25].", "Our results exactly coincide with the results of previous authors [22], [23], [24] in two polynomial case.", "The analytic solution in three polynomial approximation was not obtained before.", "The value of the thermal conductivity coefficient obtained in the well-known work of Braginskii [26] in two-polynomial approximation is two times less than our correspondong value.", "It is connected with an approach used in the paper [26], which is different from the classical Chapman - Enskog method [12].", "In his consideration one half of the thermal flux is hidden inside the so-called \"thermal force\", so that the resulting heat flux in the co-moving frame is the same in both considerations.", "The heat conductivity coefficients for strongly degenerate electrons, in presence of magnetic field, are obtained asymptotically exactly in Lorentz approximation, when the electron-electron collision may be neglected in comparison with electron-nuclei collisions at nondegenerate nuclei.", "In most works considering the heat conductivity in astrophysical objects, in the neutron stars in particular, following Flowers and Itoh [8], the influence of the magnetic field on the heat flux was taken into account phenomenologically using the coefficient $1/(1+\\omega ^2\\tau ^2)$ , which decreases the heat flux in the direction perpendicular to the direction of a magnetic field.", "Our results, obtained by the solution of Boltzmann equation show, that the influence of the magnetic field on the coefficients of heat conductivity is stronger, and has a more complicated character Fig.3.", "On the example of the Lorentz approximation it was shown that the precision of approximation by the raw of the orthogonal functions, analogous to Sonyne polynomials, decreases with increasing of the degeneracy level.", "For strongly degenerate electrons the number of functions, needed for good precision is increasing $\\sim x_0$ , at $x_0 \\gg 1$ , and for small number of number of functions the resulting heat conductivity coefficient at $B=0$ is 2.5 times smaller than the exact value.", "For moderately degenerate electrons with $x_0=0$ the approximation by three orthogonal functions gives the value of the heat conductivity coefficient approximately 13% smaller than the exact value (in Lorentz approximation, at $B=0$ ).", "In the same approximation for nondegenerate electrons the value of the heat conductivity coefficient is only 2.2% smaller than the exact value.", "Note that the electron-electron collision even more decrease the the value of the heat conductivity coefficients.", "Therefore, in three functional approximation, the Lorentz approach may give a more exact value for heat conductivity coefficient of moderately degenerate electrons, than with account of electron-electron collisions.", "The simple linear interpolation, between \"exact\" results for nondegenerate, and strongly degenerate electrons, may be suggested for all heat conductivity coefficients $\\lambda _i(x_0)$ in presence of the magnetic field as $\\lambda _i(x_0)=\\lambda _i^{(nd)}\\frac{1-x_0}{2-x_0}+\\lambda _i^{(sd)}\\frac{1}{2-x_0},\\quad {\\rm at}\\quad x_0\\le 0,$ $\\lambda _i(x_0)=\\lambda _i^{(nd)}\\frac{1}{2+x_0}+\\lambda _i^{(sd)}\\frac{1+x_0}{2+x_0},\\quad {\\rm at}\\quad x_0\\ge 0.$ Here the indices (nd), (sd) are related to nondegenerate and strongly degenerate values, respectively.", "The Chapmen-Enskog method could be used for sufficiently dense gas (plasma), where the time between collisions of particles is the smallest among other characteristic times.", "In presence of a magnetic field we have, in addition to the time of the system flyover, and characteristic time of the parameter variations in plasma, the time of the flight over Larmor circle $\\tau _L=\\frac{2\\pi }{\\omega }$ .", "This time should be much less than $\\tau $ , equal to $\\tau _{nd}$ or $\\tau _d$ , what leads to inequality, at which the Chapmen-Enskog method could be used, in the form $\\omega \\tau \\ll 2\\pi .$ Therefore the consideration in this paper could be safely applied at $\\omega \\tau \\lesssim 1$ , and for larger $\\omega \\tau $ only qualitative estimations could be obtained.", "Our calculations have been done for non-relativistic electrons, while in deep layers of the neutron star crust the relativistic effects become important.", "The main relativistic effect of increasing the effective electron mass may be taken into account approximately, following [9], by writing in all expressions the relativistic electron mass $m_{e*}=(m_e^2+p_{Fe}^2/c^2)^{1/2}$ instead of the rest mass $m_e$ .", "The account of quantum effects is connected with consideration of discrete Landau levels in strong magnetic fields.", "This complicated problem is not yet solved.", "The transport coefficients calculated here determine a heat flux carried by electrons in the case of zero diffusion vector $d_i$ .", "In a general case of nonzero diffusion vector $d_i$ and temperature gradient $\\partial T/\\partial x_i$ , the heat and diffusion (electrical current) fluxes are connected with each other, and are defined by 4 kinetic coefficients [12], having a tensor structure in presence of a magnetic field.", "The general consideration of heat and electrical conductivity of degenerate electrons will be done elsewhere.", "The new coefficients can be used for calculation of temperature distribution in white dwarfs, on the surface and in the crust of magnetized neutron star, as well as in the magnetized matter accreting to the magnetized neutron star.", "The temperature distribution over the surface of NS is important for understanding of the geometry of magnetic field inside the neutron star and near its surface.", "The work of GSBK and MVG was supported by the Russian Science Foundation grant No.", "15-12-30016." ] ]	1606.05226
[ [ "Detecting continuous gravitational waves with superfluid $^4$He" ], [ "Abstract Direct detection of gravitational waves is opening a new window onto our universe.", "Here, we study the sensitivity to continuous-wave strain fields of a kg-scale optomechanical system formed by the acoustic motion of superfluid helium-4 parametrically coupled to a superconducting microwave cavity.", "This narrowband detection scheme can operate at very high $Q$-factors, while the resonant frequency is tunable through pressurization of the helium in the 0.1-1.5 kHz range.", "The detector can therefore be tuned to a variety of astrophysical sources and can remain sensitive to a particular source over a long period of time.", "For reasonable experimental parameters, we find that strain fields on the order of $h\\sim 10^{-23} /\\sqrt{\\rm Hz}$ are detectable.", "We show that the proposed system can significantly improve the limits on gravitational wave strain from nearby pulsars within a few months of integration time." ], [ "Introduction", "The recent detection of gravitational waves (GW) marks the beginning of gravitational wave astronomy [1].", "The first direct detection confirmed the existence of gravitational waves emitted from a relativistic inspiral and merger of two large black holes, at a distance of $400M $ parsecs (pc).", "Indirect evidence for gravitational radiation was previously attained by the careful observation since 1974 of the decay of the orbit of the neutron star binary system PSR B1913+16 at a distance of 6.4 kpc, which agrees with the predictions from general relativity to better than 1% [2].", "In this paper, we discuss the potential to use a novel superfluid-based optomechanical system as a tunable detector of narrow-band gravitational wave sources, which is well suited for probing nearby pulsars at a distance of less than 10kpc.", "As we discuss below, in the frequency range exceeding $\\sim $ 500 Hz, this novel scheme has the potential to reach sensitivities comparable to Advanced LIGO.", "Figure: Left: Schematic of the proposed gravity wave sensor based on acoustic modes of superfluid helium.", "Two cylindrical geometries considered here are Gen1 (radius a=11a= 11cm, length L=50L= 50cm, mass M=2.7M= 2.7kg) and Gen2 (a=11a= 11cm, L=3L=3m, M=16M= 16kg).", "Right: Prototype of the detector with a=1.8a= 1.8cm, L=4L=4cm, M=6M=6g and resonant frequency 10 kHz.The GW detector under consideration is formed by high-$Q$ acoustic modes of superfluid helium parametrically coupled to a microwave cavity mode in order to detect small elastic strains.", "This setup was initially studied in Ref.", "[3], and is shown in fig.", "REF .", "The helium detector effectively acts as a Weber bar antenna [4] for gravitational waves, but with two important differences.", "Firstly, the $Q/T$ -factor of the helium is expected to be much larger than that of metals, where $Q$ is the acoustic quality factor, and $T$ is the mode temperature.", "Secondly, the acoustic resonance frequency can be changed by up to 50% by pressurization of helium without affecting the damping rate, making the detector both narrowband and tunable.", "The power spectrum of gravitational waves is expected to be extremely broad and is estimated to range from $10^{-16}$ to $10^3$ Hz [5], [6], [7] for known sources.", "Ground-based optical interferometers (such as LIGO, Virgo, GEO, TAMA) allow for a broad-band search for gravitational waves in the frequency range 10 Hz - 1 kHz.", "These detectors are expected to be predominantly sensitive to the chirped, transient, GW impulse resulting from the last moments of coalescing binaries involving compact objects (black holes(BH) and/or neutron stars (NS)) [8].", "Space-based interferometric detectors can in principle be sensitive to lower frequency gravitational waves, as they are not limited by seismic noise [9].", "Unlike broadband impulse sources, rapidly rotating compact objects such as pulsars are expected to generate highly coherent, continuous wave gravitational wave signals due to the off-axis rotating mass, with frequencies spanning from $\\sim $ 1 kHz for millisecond pulsars (MSPs) in binaries, to 1 Hz for very old pulsars [10], [11], [12], [13], [5].", "Given the unknown mass distribution of the pulsar, one can only estimate the strain field here at earth.", "However, several mechanisms give upper bounds to the strength of gravitational waves on earth.", "One such limit is the “spin down limit\", which is given by the observed spin-down rate of the pulsar, and the assumption that all of the rotational kinetic energy which is lost is in the form of gravitational waves [14].", "Another limit is given by the yield strength of the material which makes up the neutron star, and how much strain the crust can sustain before breaking apart due to centripetal forces [15].", "The presence of strong magnetic fields indicate a potential mechanism for producing and sustaining such strains due to deformation of the neutron star [16].", "However, without knowing the strength and direction of the internal magnetic fields in a pulsar, it is difficult to estimate a lower limit on the size of gravitational wave signal.", "The measurement of gravitational waves from pulsars would therefore give us crucial information about the interior of neutron stars.", "Since pulsars should emit continuous and coherent gravity waves at specific and known frequencies, we can use a narrowband detector and integrate the signal for long times, averaging away the incoherent detector noise.", "We show that for reasonable parameters, the superfluid helium detector can approach strain sensitivities of $1-5\\times 10^{-23}/\\sqrt{\\rm Hz}$ at around 1 kHz, depending on the size and $Q$ factor of the detector.", "Pulsar frequencies are observed to vary slightly due to random glitches $\\Delta f / f \\sim 10^{-6}-10^{-11}$ (older, millisecond pulsars being more stable) [17], and due to the motion of the earth around the sun and resulting doppler frequency shifts.", "The tunability of the acoustic resonance will be essential to track these shifts during long detection integration times.", "Simultaneous monitoring of the targeted pulsar electromagnetically can facilitate the required precision frequency tracking.", "The frequency agility can also allow for using the same acoustic resonator to look for signals from multiple pulsars in a similar frequency range.", "Recent measurements with LIGO and Virgo have unsuccessfully searched for the signals from 179 pulsars and have limited the strain field $h\\lesssim 10^{-25}$ for most pulsars after nearly a year of integration time[18].", "In a parallel development, hundreds of new pulsars have been discovered in the last few years by analyzing Gamma-Ray sources observed by the Fermi Large Area Telescope (Fermi-LAT), some less than 0.5 kpc from earth [19], [20].", "Together, these developments are signaling a promising path towards gravitational wave astronomy of pulsars.", "This paper is organized as follows.", "We start with an overview of continuous gravitational waves from pulsars to get an estimate for the strains produced on earth in Section .", "We then describe the superfluid helium detector and show how it functions as a detector for gravitational waves in Section .", "In Section , we provide the detection system requirements.", "We then compare this detector with other functional gravity wave detectors, and show the key fundamental differences between these detectors and our proposed detector in Section .", "Finally, we conclude with a brief summary of the key features of this detector and outlook in Section .", "A review of the relevant concepts and derivations are relegated to the appendices for the interested reader." ], [ "Sources of continuous gravitational waves", "The generation of gravitational waves can be studied by considering the linearized Einstein equations in the presence of matter [21].", "The computations are similar to the analogous case in electromagnetism [22], see Appendix A for details.", "However, in the absence of gravitational dipoles, a quadrupole moment $Q_{ij}$ is necessary to source gravitational waves.", "The emitted power of gravitational waves is found to be [23] $P = \\frac{G}{5 c^5} \\langle \\dddot{Q}_{ij} \\dddot{Q}^{ij} \\rangle ,$ i.e.", "it depends on the third time derivative of the quadrupole moment of the system, where $Q_{ij}:=\\rho \\int _{\\rm body} x_i x_j dV$ for a body of density $\\rho $ .", "In the far-field limit where size of the source ($GM/c^2$ ) $\\ll $ wavelength of gravity wave ($c/\\omega $ ) $\\ll $ distance to detector ($d$ ), the gravitational metric perturbation becomes $h_{ij}=\\frac{2G}{c^4 d} \\ddot{Q}_{ij},$ where $h$ is the gravitational perturbation tensor in transverse-traceless gauge.", "Since $G/c^4 \\sim 10^{-44} {\\rm N\\ s^4/kg^2}$ , one needs events with relativistic changes in mass quadrupole moment to have a measurable source of gravitational radiation on earth.", "As an estimate, if all the observed slowdown of the Crab pulsar was converted into gravitational radiation, the power would correspond to $P \\sim 4.5 \\times 10^{31}$ W ($10^5$ times the electromagnetic radiation power from the sun) [24].", "However, at a distance 2 kpc away from the pulsar (distance to earth), the power flux is $10^{-9} {\\rm W/m}^2$ and the metric perturbation is $h\\sim 10^{-24}$ .", "Even though the power flux is macroscopic and easily detectable in other forms (acoustic, electromagnetic, etc.", "), the resulting strain is very small due to the remarkably high impedance of space-time.", "This is at the heart of the difficulty with laboratory detection of gravity waves.", "The several astrophysical candidates for gravitational waves considered so far can be broadly classified into three categories: stochastic background, broadband impulses, and continuous sources [25], [26].", "It is estimated that there will be a broadband background of gravitational waves from the expansion of the early universe.", "Furthermore, there is a low frequency stochastic background due to gravitational waves emitted by masses moving in the galaxies.", "Impulse sources could stem from supernovae or mergers of compact objects.", "The latter is the primary source being searched for by most ground based detectors, and was recently observed by the LIGO detectors [1].", "Lastly, continuous gravitational waves can be expected from stellar binaries (albeit at very low frequencies), or from pulsars.", "We now discuss the generation of gravitational waves from asymmetric pulsars and the limits on the signal set on earth.", "Estimates of gravitational radiation from pulsars is an active area of theoretical research that goes back to early observations of pulsars[13].", "The mechanism for gravitational wave generation is assumed to be an asymmetric mass distribution.", "Several mechanisms are proposed for the deviation from axial symmetry in mass distribution, for example magnetic deformations, star quakes or instabilities due to gravitational or viscous effects [27], [17].", "However, due to the unknown equation of state, there is significant variability in estimates of mass asymmetry and thus gravitational wave strain from pulsars.", "Null results from measurements of GW strain from pulsars have already put limits on the equation of state [15].", "Assuming that the emission of gravitational waves is a contributing mechanism towards the observed slowdown of pulsars enables us to set upper bounds on the GW metric strain here on earth.", "We now estimate the gravitational wave perturbation strain from measured spin-down rates and briefly discuss the validity of this limit.", "We then present relevant numbers for a few millisecond pulsars (MSPs) of interest for our detector.", "Details about these derivations and typical parameters for other pulsars of interest are presented in Appendix B.", "For an ellipsoidal pulsar rotating about the $z$ -axis with frequency $\\omega _p$ , the two polarizations of $h$ are given by $h_+&=&-\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega _p^2\\cos {2\\omega _p t},\\\\h_\\times &=&\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega _p^2\\sin {2\\omega _p t},$ where $I_{zz}$ is the moment of inertia along the $z$ -axis and $\\epsilon $ characterizes the mass quadrupole ellipticity ($\\epsilon = (Q_{xx}-Q_{yy})/I_{zz}$ ).", "Thus, the strain changes at twice the rotation frequency of the pulsar.", "The energy flux for a continuous GW of polarization $A$ from a pulsar source is given by $s^A = \\frac{c^3}{16\\pi G} {\\overline{\\dot{h}_A \\left( t \\right)^2} },$ where $A\\in \\lbrace +, \\times \\rbrace $ and the bar indicates time-averaging.", "Typical neutron stars have mass 1-1.5 $M_{\\odot }$ (where $M_{\\odot }=2 \\times 10^{30}$ kg is the solar mass) and have a radius of around 10 km.", "Using these values, the moment of inertia amounts to $10^{38}$ kg-m$^2$ , the estimate used in previous GW searches, c.f.", "Ref.", "[18].", "The ellipticity parameter is estimated by assuming that the observed slow-down rate ($\\dot{\\omega }_p$ ) of a pulsar is entirely due to emission of gravitational waves.", "This estimate is then used to compute the upper-limit estimate for gravitation perturbation strain known as the spin down strain, $h_{sd}=-\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega _p^2=\\sqrt{\\frac{5GI_{zz}\\dot{\\omega }_p}{2c^3d^2\\omega _p}}.$ From Eq.", "REF , it is clear that given the values of distance ($d$ ), rotational frequency ($\\omega _p$ ) and spin-down frequency ($\\dot{\\omega }_p$ ) from astronomical observations, we can put limits on gravitational wave strains due to pulsars.", "While $h_{sd}$ is a useful first-principles upper limit, it over-estimates the strength of gravitational waves, particularly from young pulsars that are highly active electromagnetically.", "This has already been confirmed by braking index measurements [28], [29] and the negative results from recent GW detector data [18].", "However, long-lived and stable millisecond pulsars ($\\dot{\\omega _p/2\\pi <10^{-14}} {\\rm Hz\\ s^{-1}}$ ) such as the ones considered here have been proposed as likely sources of continuous gravity waves [30].", "The frequency stability and relatively low magnetic field indicate that unlike young pulsars like Crab and Vega, the dominant spin down mechanism in MSPs is more likely to be quadrupolar gravitational radiation.", "We can also set limits on GW generation mechanism by considering specific models of the interior of neutron stars, as discussed in ref.", "[27], and reviewed briefly in Appendix B.", "Assuming standard nuclear matter and breaking strain for elastic forces, the ellipticity sustained can be limited to less than $6\\times 10^{-7}$ , irrespective of the physics leading to deformations [31], [15].", "This can also be used to evaluate the GW strain amplitude limit $h_{\\epsilon n}$ .", "Since the strain limits $h_{sd}$ and $h_{\\epsilon n}$ come from different physics (conservation of angular momentum and balancing forces in stellar interior), we use the lower of the two as the upper limit for metric strain.", "Table: Table of millisecond Pulsars with rotation frequency greater than 500 Hz: ω p \\omega _p is the rotational frequency, f GW =ω p /πf_{GW} =\\omega _p/\\pi is the frequency of gravitational waves, ω ˙ p \\dot{\\omega }_p is the measured spin down rate, and dd is the distance to the pulsar in kilo-parsecs, h sd h_{sd} and h en h_{en} are the spin-down and elastic strain limits.", "These values are compared to the strain limit set by recent continuous GW surveys by interferometric detectors h 0 95% h_0^{95\\%} , and the strain limit for two identical Gen1 helium resonant detectors (with an integration time of 250 days) as shown in Eq.", ".", "Pulsars indicated by superscript * ^* were discovered by the Fermi gamma ray telescope.", "Pulsars J1747-4036 and J1902+5105 were only discovered in 2012 , and were not included in the LIGO+VIRGO analysis presented in Ref.", "Table REF details parameters for pulsars of interest with rotational frequency higher than 500 Hz, along with current limitations on GW strain from LIGO+VIRGO collaboration.", "Theoretical estimates of metric strain assuming spin-down limit, and elastic crust breakdown limit on ellipticity ($\\epsilon = 6\\times 10^{-7}$ ) from Ref.", "[15] are also given.", "Table REF also gives the strain estimate set by the helium detector outlined in figure REF (Gen1) that we will discuss in detail in the following sections.", "Several of these pulsars were discovered recently by analyzing gamma-ray sources from Fermi-LAT.", "The number of known fast spinning pulsars is expected to grow significantly as more sources are discovered and analyzed.", "Furthermore, there is growing evidence that the GeV excess emission in our galactic center is in fact due to hundreds of unresolved MSPs and not from dark-matter annihilation [33], [34].", "Since the strain due to gravity waves from pulsars is expected to be very small but coherent, one needs to integrate the signal from these detectors for a long time (typically several days, the last result being a compilation of $\\sim 250$ days of integration over multiple detectors [18] ).", "Also, in order to rule out noise we need to detect a gravitational wave signal from at least two different detectors.", "Strain sensitivity also improves as $\\sqrt{N_d}$ , where $N_d$ is the number of detectors [35].", "There has been computationally intensive analysis of many days of data from broadband detectors like LIGO to search for such gravitational wave signals.", "However, all such searches have so far been unsuccessful, although they have improved the upper bound on the emitted wave amplitudes.", "These upper bounds in turn constrain the equation of state of exotic neutron stars.", "In the following, we outline the proposal for a simple, low-cost, narrowband detector for these gravitational waves based on a superfluid helium optomechanical system.", "Being relatively simple and economical, superfluid helium detectors can also be set up in multiple locations to improve overall detection sensitivity.", "As a precursor to subsequent discussions, we present the central result of our work in figure REF , showing the limits set by different detectors for ten MSPs of interest from Table REF .", "Along with the spin-down limit and the strain limit set up the previous LIGO measurement [18], we also show the limits set up two different geometries of helium detectors that we discuss in detail in Sections and .", "Since we are detecting a continuous GW signal, the strain sensitivity improves with integration time.", "Here, we have assumed that the resonance frequency of the same acoustic mode can be tuned by up to 200 Hz without changing the $Q$ -factor (of $10^{11}$ ), thereby resonantly targeting each of these pulsars with the same detector.", "Figure: Strain sensitivity of various detectors for 10 MSPs of interest for the helium detector versus measurement time for two helium detectors with same sensitivity operating simultaneously, assuming a bath temperature of 5mK, (l,m,n)=(0,2,0)(l,m,n)=(0,2,0), and Q-Q-factor of 10 11 10^{11} for both Gen1(mass=2.66 kg, in blue) and Gen2(mass= 15.9 kg, in red) detectors.", "We also show the limits set by three interferometric sensors operating at LIGO-S6 sensitivity for 250 days and at Advanced LIGO design sensitivity for one year.", "As seen in the figure, the current limit on pulsar GW strains can be surpassed within a few days of integration time for Gen1, and in under a day for Gen2.", "Also shown is the spin-down limit for these pulsars." ], [ "Superfluid helium gravity detector", "The gravitational wave strain detector we propose is a resonant mass detector formed by acoustic modes of superfluid helium in a cavity parametrically coupled to a microwaves in a superconducting resonator.", "For the purpose of our calculations, we will treat the superfluid as an elastic medium with zero dissipation.", "At the temperatures we expect to operate this detector, $T<10$ mK, the normal fluid fraction $\\rho _n$ is expected to be $\\rho _n/\\rho _0<10^{-8}$ , where $\\rho _0$ is the total density of the fluid [37].", "For temperatures below $T<100$ mK, the dissipation of audio frequency acoustic waves is expected and found to be dominated by a three-phonon process, falling off as $T^{-4}$ .", "An elastic body (with dimensions $\\ll \\lambda _{GW}$ ) in a gravitational field will undergo deformation due to changes in space-time as a gravity wave passes by.", "For distances far away from the source of radiation, the space-time perturbation acts like an external tidal force [38], as also discussed in Appendix A.", "The equation of motion for the displacement field ${\\bf u}\\left(r,t \\right)$ of an elastic body is given by [39] $\\rho \\frac{\\partial ^2 {\\bf u}}{\\partial t^2}-\\mu _L \\nabla ^2 {\\bf u} -(\\lambda _L+\\mu _L)\\nabla (\\nabla .", "{\\bf u})=\\frac{1}{2}\\rho \\ \\ddot{\\bf h} {\\bf x},$ where $\\rho $ is the density, $\\lambda _L,\\mu _L$ are the Lamé coefficients for the elastic body and $\\ddot{\\bf h} {\\bf x}$ is the effective amplitude of the wave for a particular orientation of the detector that exerts an effective tidal force on the detector.", "This acoustic deformation can be broken into its eigenmodes ${\\bf u}\\left({\\bf r},t \\right)=\\sum _n\\xi _n\\left( t\\right){\\bf w}_n\\left( {\\bf r} \\right)$ .", "For this analysis, we assume our acoustic antenna is in a single eigenmode of frequency $\\omega _m$ , thus dropping index $n$ .", "In this analysis, we have used the notation where ${\\bf w}_n\\left( {\\bf r} \\right)$ is a dimensionless spatial mode function with unit amplitude, and the actual amplitude of the displacement field is in $\\xi (t)$ .", "Rigid boundary walls and absence of viscosity enables us to describe the acoustic modes accurately via a simple wave equation as opposed to Navier-Stokes equations typically used to describe fluid flow.", "The spatial modes are obtained by solving the acoustic equations of motion [40].", "For elastic deformations in enclosed spaces, the change of pressure $p({\\bf r})$ is described by $\\nabla ^2 p - \\frac{1}{c_s^2} \\frac{\\partial ^2 p}{\\partial t^2} =0$ with the speed of sound in the material (here helium) being $c_s$ .", "The particle velocity $\\bf {v} = \\dot{\\bf {u}}$ is related to pressure via $ \\partial \\bf {v}/\\partial t = - \\nabla p/\\rho $ .", "Thus each vector component of the velocity $\\bf {v}$ also satisfies the same wave equation as the pressure, but the components are not independent of each other.", "The full solution can be equivalently expressed in terms of the Helmholtz potential for the velocity, ${\\bf v}=\\nabla \\Phi ({\\bf r})$ .", "In terms of the potential, the acoustic pressure becomes $p= -\\rho \\partial \\Phi / \\partial t $ , and the potential satisfies the same wave equation $\\nabla ^2 \\Phi - \\frac{1}{c_s^2} \\frac{\\partial ^2 \\Phi }{\\partial t^2} =0$ As before, the time dependence can be explicitly separated via $\\Phi \\rightarrow \\Phi ({\\bf r}) \\xi (t)$ .", "For cylindrical symmetry the solution for the spatial part of the potential is $\\Phi (r,\\theta ,z) = J_m(k_{m}(n) r) \\cos (m \\theta ) \\cos \\left( k_z(l) (z+ \\frac{L}{2})\\right),$ where the wavevectors are found from the rigid boundary conditions $\\partial \\Phi / \\partial z = 0$ at $z = \\pm L/2$ and $\\partial \\Phi / \\partial r = 0$ at r=a, such that $k_z(l) = l \\pi /L$ with $l=0,1,2 ...$ and $k_{m}(n)$ follows from the n roots of $J^{\\prime }_m(k_{m}(n) a)=0$ .", "Having the solution for the potential, one can obtain the velocity vector field, and thus the spatial modes, via ${\\bf w}(r,\\theta ,z)=\\nabla \\Phi (r,\\theta ,z)/\|{\\bf w}_{\\rm max}\|$ , where $\|{\\bf w}_{\\rm max}\|$ is the maximum value of $\\nabla \\Phi (r,\\theta ,z)$ .", "These acoustic modes of helium in a superconducting cavity were experimentally studied by some of the authors in Ref.", "[3].", "We found these modes to be well-modeled by this theory and to have extremely high $Q$ -factors ($Q>10^{8}$ ) at 45 mK.", "Figure: The first few pressure modes with non-zero quadrupolar tensors for the cylindrical cavity.", "While the form of quadrupolar tensor (shown on the right) is similar for many modes, the constant could be different for each acoustic mode.For the purposes of this paper, we will simply add the finite linear dissipation to the acoustic resonance, parameterized as a finite $Q$ .", "For a damped acoustic resonator, eq.", "REF can be simplified to show that the displacement field $\\xi (t)$ satisfies the equation of motion $\\mu \\left(\\ddot{\\xi }+\\frac{\\omega _m}{Q_{He}}\\dot{\\xi }+\\omega _m^2\\xi \\right) = \\frac{1}{4}\\sum _{ij} \\ddot{h}_{ij}q_{ij},$ where $Q_{He}$ is the $Q$ -factor associated with the acoustic mode, $\\mu $ is the reduced mass for the particular eigenmode, $\\mu = \\int \\rho {\\bf w}^2 dV,$ and $q_{ij}$ is the dynamic part of the quadrupole moment, $q_{ij} = \\int {\\rho \\left( w_i x_j+x_i w_j - \\frac{2}{3}\\delta _{ij}{\\bf w}\\cdot {\\bf r}\\right) dV}.$ Figure REF shows the first few pressure modes of the cylindrical cavity that have a non-zero quadrupolar tensor, along with the form of the tensor.", "As can be deduced from eq.", "REF , several modes have a zero quadrupole moment due to symmetry.", "In their analysis of various antenna geometries for gravitational radiation detection, Hirakawa and co workers introduced two quantities to compare GW antennas spanning different size and symmetry groups [41].", "These are the effective area of the antenna ($A_G$ ) characterizing the GW-active part of the vibrational mode, and the directivity function ($d^A$ ), which characterizes the directional and polarization dependence of the GW sensor.", "They are defined as $A_G = \\frac{2}{\\mu M}\\sum q_{ij}^2$ and $d^A\\left(\\theta ,\\phi \\right) = \\frac{5}{4}\\frac{\\left( \\sum q_{ij}e_{ij}^A\\left({\\bf k}\\right) \\right)^2 }{\\sum q_{ij}^2}$ where $M$ is the total mass of the antenna and $ e_{ij}^A$ is the unit vector for incoming GW signal polarization $A$ ($A\\in \\lbrace +, \\times \\rbrace $ ) in arbitrary direction ${\\bf k}( \\theta , \\phi , \\psi )$ .", "The Euler angles $( \\theta , \\phi , \\psi )$ transform from the pulsar coordinate system to the detector co-ordinate system and are discussed in Appendix A, along with the explicit form of $ e_{ij}^A$ .", "In sum, the angles $\\theta $ and $\\phi $ describe the direction of the incoming gravitational wave, and $\\psi $ defines the polarization of the detector (rotation of the $x-y$ plane of the source).", "An important distinction between the proposed detector and other gravitational wave sensors, particularly the interferometric ones is that the orientation of the detector can be adjusted to optimize the directivity function for the astrophysical source in consideration due to its small size.", "This acts as another tunable parameter that can give significant enhancement in sensitivity for a particular source, as shown in fig.", "REF for a specific acoustic mode $(l,m,n)=(0,2,0)$ .", "In terms of previously defined expressions, the mean squared signal force from a continuous gravity wave source of polarization $A$ is given by $\\overline{f_G^2} &=& \\frac{2\\pi G}{5c^3}M \\mu \\omega _G^2 A_G d^A \\left( \\theta , \\phi \\right) s^A,\\\\&=&\\frac{1}{40}M \\mu \\omega _G^4 A_G d^A {\\overline{h_A \\left( t \\right)^2} }$ where $\\omega _G=2\\omega _p$ is the frequency of gravity wave.", "Here, we have assumed a delta-function gravity wave spectrum.", "Figure: The directivity patterns for acoustic mode (l,m,n)=(0,2,0)(l,m,n)=(0,2,0) of the cylindrical cavity.", "The +,×+,\\times polarization and total directivity functions are given for two different polarizations of the detector (Euler angle ψ=0,π/2\\psi =0,\\pi /2).", "The orientation of the detector can be adjusted to optimize the directivity function for the astrophysical source in consideration.As an example, we choose a cylindrical cavity of radius $a=10.8$ cm, length $L=50$ cm (from now onwards referred to as Gen1 or with subscript $(\\rm He,1)$ ).", "We focus on acoustic mode $f_{(0,2,0)}=1071$ Hz, which has an effective mass $\\mu = 0.625 M$ , and a large GR-active area of $A_{G}=0.629 \\pi a^2$ due to its quadrupolar shape, shown in figure REF .", "Another geometry considered in this work is a cylindrical cavity of the same radius, but length $L=3$ m (from now onwards referred to as Gen2 or with subscript $\\rm ( He, 2)$ ).", "Since the resonance frequency of the $[020]$ mode is independent of length, it has the same frequency.", "However, increasing the mass gives us a larger effective mass for the same area.", "Figure REF shows the various directivity functions for this acoustic mode that capture the angular dependence of the sensitivity of the detector." ], [ "Noise mechanisms and minimum detectable strain", "The system we are proposing and have been exploring in the laboratory[3] is a parametric transducer[42] and essentially similar to other optomechanical systems[43]: the acoustic motion of the superfluid and resulting perturbation of the dielectric constant modulates the frequency of a high-$Q$ superconducting microwave resonator.", "The details of the coupled acoustic and microwave system, sources of dissipation (phonon scattering, effect of isotopic impurities, radiation loss,) microwave and signal detection limits, effects of electrical dissipation, requirements on thermal stability, etc.", "will be the subject of another manuscript[44].", "Here we take a few central results of this analysis.", "The noise sources relevant to this system are the Brownian motion of the fluid driven by thermal/dissipative forces, the additive noise of the amplifier which is used to detect the microwave field, the added noise of the stimulating microwave field (phase noise), and possible back-action forces due to fluctuations of the field inside the microwave cavity (due to phase noise and quantum noise).", "We will assume for the purpose of this discussion that the challenging job of seismically isolating the superfluid cell from external vibrations has been accomplished as has been done for other gravitational wave detectors.", "Due to the high frequency and narrow bandwidth of the astrophysical source of interest, the strain noise due to Newtonian gravity fluctuations are expected not to be relevant for this detector [26].", "The effect of vortices in superfluid helium due to earth's rotation on the $Q$ -factor is unclear.", "However, using an annular cylinder or an equatorial mount allows for long integration times without the possibly detrimental effects due to vortices.", "For a sufficiently intense microwave pump, with sufficiently low phase noise, the thermal Brownian motion of the helium will dominate the noise.", "Assuming the device is pumped on the red-sideband, $\\omega _{pp} = \\omega _c - \\omega _m$ , and that the system is the side-band resolved limit, $\\omega _m > \\kappa _c$ , the upconversion rate of microwave photons is given by: $\\Gamma _{opt}=4(\\Delta p_{SQL}\\cdot g_0)^2 n_p /\\kappa _c$ , where $\\omega _{pp}$ , $\\omega _c$ , and $\\omega _m$ are the pump, cavity, and acoustic mode frequency respectively, $\\kappa _c=\\omega _c/Q_{Nb}$ is the cavity damping rate, $\\Delta p_{SQL}$ is the amplitude of the zero-point fluctuation of the pressure of the acoustic field, $n_p$ is the amplitude of the pump inside the cavity measured in quanta, and $g_0$ is the coupling between the acoustic and microwave field.", "For the geometry we consider here, Gen1: $l=0.5$ m, $d=0.108$ m, $\\omega _m=1071 \\cdot 2 \\pi $ Hz, $\\omega _c= 1.6\\cdot 2\\pi $ GHz, and $g_0=-7.5 \\cdot 10^{-11} \\cdot 2 \\pi $ Hz.", "To achieve a readout with noise temperature of 1mK, which means that the added noise of the amplifier is equal to the thermal noise amplitude when the helium is thermalized at 1mK, requires $n_p=6\\cdot 10^{9}$ microwave pump photons and a phase noise of $-145$ $db_c$ /Hz.", "To begin to dampen and cool the acoustic resonance with cavity backaction force, would require $n_p=10^{12}$ , and a phase noise of $-145$ $db_c$ /Hz.", "Microwave sources have been realized using whispering gallery modes of sapphire with phase noise of $-180$ $db_c$ /Hz.", "Together with a tunable superconducting cavity, it is possible to realize a source with sufficient low noise to broaden and cool this mode with backaction.", "Furthermore, as we will detail in our future work[44], $^3$ He impurities diluted into the $^4$ He are expected to add acoustic loss, additional to the 3-phonon process.", "To achieve $Q_{He}=10^{11}$ , we estimate that an isotopic purity of $n_{3}/n_{4}=10^{-11}$ is required.", "Due to the very low dielectric constant of helium ($\\epsilon _{He}=1.05$ ), the bare optomechanical coupling constant is small compared to typical micro-scale optomechanical systems: $g_0=\\Delta p_{SQL}\\cdot \\partial \\omega _{c}/\\partial \\Delta p=-7.5 \\cdot 10^{-11} \\cdot 2 \\pi $ Hz: this is the frequency shift of the Nb cavity, $\\omega _c$ , due to the zero-point fluctuations of the acoustic field of the helium, $\\Delta p_{SQL}$ .", "However, the relevant quantity is cooperativity, $C=\\Gamma _{opt}/\\gamma _{He}$ , which compares the rate of signal photon up-conversion, $\\Gamma _{opt}$ , to the loss rate of acoustic quanta to the thermal bath, $\\gamma _{He}=\\omega _{He}/Q_{He}$ .", "With quantum limited microwave detection (now possible with a number of new amplifiers), detection at the SQL is achieved when $C=1$ , and is the onset of significant backaction effects such as optomechanical damping and cooling.", "The key point is that for this system we expect to be able to realize very large $n_p$ .", "This is due to the very high $Q$ possible in Nb, ($Q_{Nb}\\sim 10^{11}$ is now routine for accelerator cavities [45], [46], even when driven to very high internal fields of $10^{7}V/m$ corresponding to $n_p=10^{23}$ ,) and dielectric losses and resulting heating at microwave frequency in liquid helium are expected to be negligible up to very high pump powers.", "Assuming the dielectric loss angle in helium is less than $10^{-10}$ , our estimates suggest that $n_p=10^{16}$ should be achievable before dissipative effects lead to significant heating of the helium sample at 5mK, far beyond the internal pump intensity used with micro-optomechanical systems and far above the onset of backaction effects, $C=1$ for $n_p=8 \\cdot 10^{11}$ .", "As a result, we are optimistic that SQL limited detection and significant backaction cooling and linewidth broadening are possible.", "Since the frequency and phase of the pulsar's gravity wave signal should be known through observations of the electromagnetic signal, single quadrature back-action evading, quantum non-demolition measurement techniques could be implemented[47].", "This has the advantage of avoiding the back-action forces from the cavity field fluctuations and can lower the phase noise requirements of the microwave pump.", "For a damped harmonic oscillator with $\\gamma _{He} = \\omega _m/Q_{He}$ in equilibrium with a thermal bath at overall effective temperature $T$ , the position noise spectral density is given by $ \\nonumber S^{\\rm th}_{\\xi \\xi }[\\omega ]=\\frac{k_B T}{\\mu \\omega _m^2} \\left\\lbrace \\frac{\\gamma _{He}/2}{(\\omega +\\omega _m)^2+\\gamma _{He}^2/4 }\\right.", "\\\\+\\left.\\frac{\\gamma _{He}/2}{(\\omega -\\omega _m)^2 +\\gamma _{He}^2/4}\\right\\rbrace .$ Assuming that noise at the detection frequency is dominated by the thermal noise of the acoustic mode, the force noise spectral density $S_{FF}$ is given by the relation $S_{\\xi \\xi }[\\omega ]=\|\\chi (\\omega )\|^2 S_{FF}[\\omega ]$ , with the susceptibility $\\chi (\\omega ) = [\\mu ((\\omega _m^2 -\\omega ^2)+i\\gamma _{He}\\omega )]^{-1}$ .", "For gravitational strain, using eq.", "REF , we find $S_{hh}[\\omega ] = 40 S_{FF}[\\omega ]/(\\mu M \\omega _G^4 d^A A_G)$ for a continuous gravity wave source at frequency $\\omega _G$ .", "Combining these, we find that for a resonant mass detector at $\\omega _G = \\omega _m$ , $S_{hh}[\\omega ] = \\frac{80 k_B T}{M d^A A_G Q_{He} \\omega _m^3}.$ The strain sensitivity of our detector is simply $\\sqrt{S_{hh}[\\omega ]}$ , and the minimum noise is $\\sqrt{S_{hh}[\\omega ]/\\tau _{\\rm int}}$ after an integration time $\\tau _{\\rm int}$ .", "The minimum detectable strain field with $2\\sigma $ certainty is therefore given by [30] $h_{min}\\approx 2\\sqrt{\\frac{S_{hh}[\\omega ]}{\\tau _{\\rm int}}}=\\sqrt{\\frac{320 k_B T }{M\\omega _{G}^3 A_G d^A Q_m}\\frac{1}{\\tau _{\\rm int}}}.$ The $2\\sigma $ uncertainty limit is used to be consistent with previously reported limits on $h_{min}$ set by LIGO [18].", "As an example, both cylindrical cavities considered in section have acoustic mode $f_{(0,2,0)}\\sim 1071$ Hz.", "This mode of the detector can easily be tuned (by under $\\pm $ 15 Hz) to be in resonance with pulsars J0034-0534, J1301+0833, and J1843-1113.", "Similarly, another acoustic mode ($f_{[2,0,1]}= 1425$ Hz) is found to have resonant frequencies in the vicinity ($<$ 8 Hz) of the frequency of gravitational waves from pulsar J1748-2446ad.", "Taking into account the different quadrupole tensors, effective mass and directivity functions for the different acoustic modes, Table 2 lists the minimum detectible strain for several pulsars for cylindrical detector Gen1 after 250 days of integration time (same time as the current LIGO+VIRGO estimates in Ref.", "[18]).", "Here we have assumed an acoustic $Q$ -factor of $ 10^{11}$ and thermal $T_{th} = 5$ mK for both geometries.", "Since the detector is small enough to be rotated or moved geographically to optimize signal from a particular pulsar, we have assumed $\\psi =0$ and $(\\theta ,\\phi )$ that maximizes the directivity.", "Table: Table of millisecond Pulsars of interest for helium detector Geo1 and Geo2.", "Here, ω p \\omega _p is the oulsar rotational frequency, f GW =ω p /π f_{GW} = \\omega _p/\\pi is the frequency of gravitational waves, as given in Table 1.", "These values are compared to the strain limit set by recent continuous GW survey by interferometric detectors , along with the strain limit for the helium resonant detectors Gen1 with an integration time of 250 days and Gen 2 with one year integration time, as shown in eq.", ".", "Here, ψ=0\\psi = 0, QQ-factor is 10 11 10^{11}, and the acoustic mode is given in square brackets.In order to compare the sensitivity of our proposed detector with other GW sensors, we pick a specific expected astrophysical source: gravity waves from pulsar J1301+0833, with $\\omega _G = 2 \\pi \\times 1084.76$ Hz.", "Gen1 (mode [020]) gives us sensitivity of $h_{min} = 3.4 \\times 10^{-23}/\\sqrt{\\rm Hz}$ , which is significantly below the sensitivity of LIGO, and comparable (within a factor of 2) to current sensitivity of advanced LIGO.", "Such a detector can surpass the LIGO +VIRGO estimate on minimum strain $ h_0^{95\\%} = 1.1 \\times 10^{-25}$ in under a week of integration time (under a month if $Q$ -factor is $10^{10}$ instead).", "Increasing the mass by a factor of 6 (by choosing Gen2), while assuming the same $Q$ -factor and noise characteristics, we can get sensitivity of $1.4 \\times 10^{-23}/\\sqrt{\\rm Hz}$ , which is below the strain sensitivity of advanced LIGO for this frequency range.", "Figure REF shows the minimum detectable strain as a function of integration time for various $Q$ factors for two resonant detectors operating at the same sensitivity.", "Figure REF also shows the sensitivity estimates for three interferometeric detectors operating at LIGO-S6 sensitivity, and at advanced LIGO design sensitivity, as used in ref.", "[18].", "As figure REF and Table 2 demonstrate, Gen2 can come within a factor of 2 of the spin-down limit for pulsar J1301+0833 (and several other pulsars) in a year of integration time.", "Considering the conjecture that the primary spin-down mechanism for MSPs is the emission of gravitational radiation, our detector seems a promising candidate for searches of continuous gravitational waves from this and similar other pulsars.", "Figure: Strain sensitivity versus measurement time for two helium detectors with same sensitivity operating simultaneously, assuming a bath temperature of 5mK, (l,m,n)=(0,2,0)(l,m,n)=(0,2,0), for Geometry 1(blue) and 2(red).", "We also show the limits set by three interferometric detectors operating at LIGO-S6 sensitivity (solid black), and the design sensitivity of advanced LIGO (dashed black).", "The stars shows the current limit on minimum strain set by LIGO, and the projected limit by Advanced LIGO.", "As seen in the figure, the current limit can be surpassed within a few days of integration time for Gen1, and under a day for Gen2.", "Also shown is the spin-down limit for pulsar J1301+0833.We would like to note that several noise suppression mechanisms (such as squeezed light injection) currently used in LIGO can also be employed here.", "More importantly, there are ways to squeeze the mechanical motion of the detector [48], [49], [50].", "This can significantly relax the size, $Q$ -factor and microwave noise requirements, increasing the sensitivity of our proposed detector significantly.", "For example, exploring methods to squeeze mechanical motion by changing the speed of sound periodically, and exploring other effects arising from parametric coupling between the helium acoustic modes and the microwave resonator container is a straight forward extension of the current setup, since the helium is already being pressurized and parametrically coupled to microwaves for resonant force detection.", "A detailed analysis of implementing these protocols for improved gravity wave sensing will be the subject of future research." ], [ "Comparison with other detectors", "The basic principle of the superfluid helium detector is analogous to that of other resonant mass sensors, such as Weber bars.", "The use of resonant mass GW detectors has a 50 year history, dating back to early experiments by Weber [4].", "There have been several proposals of using resonant mass detectors to search for GW from pulsars, for example Ref.", "[30], and a few continuous GW searches targeting specific pulsars, the most notable one being the Tokyo group experiment looking for signal from Crab Pulsar[51].", "Here we highlight several key differences in the implementation using superfluid helium.", "Mass: We discuss a kg-scale sample of helium which is $10^3$ times smaller than the typical resonant bar detectors.", "The low mass limits the utility of the helium detector to CW sources, where as the massive detectors are useful for burst sources.", "Nonetheless, there is high sensitivity for CW sources and the low mass makes a helium detector economical and small scale.", "One could deploy a few such detectors to seek coincidence and further improve sensitivity.", "$T/Q_{He}$ – temperature and quality factor: It is possible to cool an isolated sample of helium to temperatures less than 10mK and we are anticipating very low loss.", "For instance, helium at 25mK with $Q_{He} = 10^9$ has a ratio $T/Q_{He}\\ 10^3$ times smaller than the best value found in the literature, and potentially $10^6$ times smaller at lower temperature [44].", "Optomechanical damping: It appears possible to substantially increase the acoustic resonance linewidth without decreasing the force sensitivity by parametrically coupling to microwaves [3].", "While parametric transducers are also used in other resonant mass detectors [52], [53], the particular geometry and mechanism used in helium detector is expected to have lower noise characteristics [44].", "Frequency tunability: It is possible to change the speed of sound in helium by 50% by pressurization.", "This allows the apparatus to be frequency agile; thus searching several pulsars with the same detector.", "It also allows for long term tracking the same pulsar in the presence of deleterious frequency shifts.", "For example, the estimated Doppler shift of the GW signal from Crab pulsar is $\\sim 30$ mHz/year due to earth's motion.", "Our detector can be tuned to track this shift, allowing for months of integration time.", "By resonantly tracking the pulsar we also reduce SNR, and thereby the detection threshold.", "A standard figure of merit used in literature to compare various bar detectors of different materials is $\\eta = Q \\rho c_{s}^3$ [25].", "Typical values of $\\eta $ range from $10^{21}-10^{24}$ kg s$^{-3}$ .", "According to this metric, helium may seem like a poor choice for a bar detector, ($\\eta \\sim Q_{He}\\times 10^9$ kg s$^{-3}$ ).", "This figure of merit is made of the material specific parameters in the minimum detectable strain, as given in eq.", "REF .", "However, adding the temperature dependence, and the significantly large $Q$ -factors make the helium sensor comparable to the resonant bar detector.", "In addition, due to it's smaller size, temperature stability, seismic and acoustic isolation are much easier to maintain.", "Unlike interferometric detectors like LIGO conducting a broadband search for gravitational waves, the helium detector is narrowband, and works best for detection of continuous waves such as pulsars.", "Nevertheless, as highlighted in fig.", "REF , around 1kHz the setup described above has strain sensitivity within a factor of 4 (Gen 1), or in principle even surpassing the sensitivity of advanced LIGO by considering a larger volume of superfluid helium (Gen 2).", "This allows us to surpass the limits from previous CW searches of VIRGO+LIGO experiments ($h_{min}\\sim 10^{-25}$ ) within a week, or less depending on the detector size and $Q$ -factor.", "There are several ongoing and proposed detectors for gravitational waves, for example space-based interferometric detector eLISA [9], [54], atom interferometry based detector AGIS-LEO [55], and Pulsar Timing Arrays [56].", "These detectors operate at different frequency ranges, typically much lower than the ones considered here.", "The astrophysical sources of interest are therefore different from those of the helium detector.", "Finally, an important advantage of considering superfluid helium as a resonant GW sensor is that by designing different geometries and exploring different types of resonances, one could build detectors for a range of astrophysical sources.", "For example, by considering smaller containers or Helmholtz resonances in micro or nano-fluidic channels [57], it may be possible to build a resonant detectors for high frequency sources of gravity waves as explored in other devices [58], [59].", "Alternatively, larger containers or low-frequency Helmholtz resonances may be used to detect continuous GWs from young pulsars or binary systems.", "Since the technology required for the proposed superfluid helium gravity detector is space-friendly, it may be possible to design low frequency detectors for space missions if seismic noise becomes a deterrent." ], [ "Conclusions and Outlook", "As discussed in Section , there are several stringent requirements for low-noise operation of our proposed helium detector: isotopically pure sample, sub-10mK cryogenic environment, very low phase-noise microwave source, and isolation from environmental vibrations.", "Furthermore, due to the low density and speed of sound, a reasonable size ($\\sim 1$ m) bar detector made of helium can only be used for detection of continuous gravity waves.", "Despite these extreme requirements, using superfluid $^4$ He does have several advantages.", "The low intrinsic dissipation and dielectric loss and wide acoustic tunability are direct manifestations of the inherent quantum nature of the acoustic medium.", "Furthermore, due to the mismatch between the speed of sound in helium and niobium, there is an inherent acoustic isolation from the container.", "Since the container itself is in a macroscopic quantum state (superconductor), it further contributes to the extremely low-noise, high sensitivity nature of the proposed device by making an extremely high $Q$ microwave resonator with very high power-handling.", "Several ideas for future work are outlined in the manuscript at various places.", "They include investigating more complex geometries for stronger coupling to gravitational strain, or investigating other high-$Q$ acoustic resonances (Helmholtz resonances) in helium to detect other sources of continuous gravity waves.", "Also, many ideas from quantum optics and quantum measurement theory can be implemented in this system to increase bandwidth or sensitivity.", "For example, by periodically modulating the acoustic resonance frequency, it will be possible to upconvert out of resonance signals into helium resonance signals, thereby increasing the frequency tunability of our detector.", "Several techniques from quantum measurements can be applied to our proposed transduction scheme to avoid measurement backaction or to squeeze acoustic noise, thereby increasing the sensitivity further.", "Even without these techniques, the extreme displacement sensitivity ($\\sim 10^{-23}/\\sqrt{Hz}$ ) of this meter-scale device corresponds to a measurement of the width of milky way to cm-scale precision!", "This is again made possible by combining two macroscopic quantum states in the measurement scheme (a superfluid coupled to a superconductor).", "The resulting hybrid quantum sensor is an extremely low noise detector at low temperatures due to the robustness of the quantum state involved.", "As these experiments develop, proposing a more broadly functioning gravity wave detector may be possible, as well as the detection of other extremely small laboratory forces.", "We would like to acknowledge helpful conversations with Rana Adhikari, Yanbei Chen, Dan Lathrop, Pierre Meystre, David Blair and Nergis Mavalvala.", "We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF IQIM-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-1250) NSF DMR-1052647, the NSF ITAMP grant, and DARPA-QUANTUM HR0011-10-1-0066." ], [ "Brief introduction to Gravitational waves", "Gravitational waves are solutions to the linearized Einstein equations, where the perturbed metric can be written as $g_{\\mu \\nu } = \\eta _{\\mu \\nu } + h_{\\mu \\nu }$ .", "Here, $\\eta _{\\mu \\nu }=\\textrm {diag}[-1,1,1,1]$ is the Minkowski metric (which is a good approximation for our solar system) and $\|h_{\\mu \\nu }\| \\ll 1$ is a small perturbation of the metric.", "In free space, Einstein's equations of motion, which describe the dynamics of space-time, reduce to $R_{\\mu \\nu } = 0$ , where $R_{\\mu \\nu }$ is the Ricci-tensor constructed from the metric.", "Since only the weak-field limit is considered, terms that are of higher order in $h_{\\mu \\nu }$ can be neglected.", "In addition, general relativity has an inherent gauge freedom related to the choice of coordinates.", "In the Lorentz-gauge the equations of motion reduce to a wave equation as in electromagnetism: $R_{\\mu \\nu } = \\Box h_{\\mu \\nu } = \\left( - \\partial _t^2 + c^2 \\nabla ^2\\right) h_{\\mu \\nu } = 0 \\, .$ This is the wave equation for gravitational waves, which are small perturbations of flat space-time that propagate at the speed of light.", "A general plane-wave solution has the form $h_{\\mu \\nu }(\\vec{x}, t) = A_{\\mu \\nu } \\cos (\\omega t - \\vec{k} \\cdot \\vec{x} +\\varphi )$ , with the dispersion relation $\\omega = c\|\\vec{k}\|$ .", "Choosing the specific transverse-traceless gauge, and a coordinate system in which the wave propagates only in the z-direction, the only non-vanishing components of the gravitational wave tensor are the spatial components $h_{i j} = h_+\\left(t-\\frac{z}{c}\\right) {\\bf e}^{+}_{ij}\\left(\\hat{z}\\right) + h_{\\times } \\left(t-\\frac{z}{c}\\right) {\\bf e}^{\\times }_{ij}\\left(\\hat{z}\\right)$ where $h_+$ and $h_{\\times }$ are the two polarization components with the polarization tensors given by $\\bf {e}^+\\left(\\hat{z}\\right)=\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & -1 & 0 \\\\0 & 0 & 0 \\\\\\end{array}\\right.$ and $\\bf {e}^x\\left(\\hat{z}\\right)=\\left(\\begin{array}{ccc}0 & 1 & 0 \\\\1 & 0 & 0 \\\\0 & 0 & 0 \\\\\\end{array}\\right.$ Gravitational waves carry energy and have observable effects on matter.", "For test particles at a distance much shorter than the wavelength of the gravitational wave, the wave induces an effective time-dependent tidal force.", "To see this, it is convenient to use gauge-invariant quantities, such as the Riemann tensor which is invariant to linear order.", "Its only non-vanishing component is $R_{\\mu 0 \\nu 0} = -\\frac{1}{2} \\ddot{h}_{\\mu \\nu }$ , where the dot denotes differentiation with respect to coordinate time $t$ .", "The Riemann tensor captures how neighboring geodesics (i.e.", "world lines of free particles) change with respect to each other: the vector $x^{\\mu }$ that connects two geodesics follows the geodesic deviation equation $\\ddot{x}^{\\mu } = R^{\\mu }_{0 \\nu 0} x^{\\nu } = -\\frac{1}{2} \\ddot{h}_{\\mu \\nu } x^{\\nu }$ .", "This equation holds for geodesics that are close to each other as compared to the wave length $\\lambda $ of the gravitational wave, i.e.", "$x << \\lambda $ .", "From this equation follows the equation of motion for the distance between two neighboring test particles: $\\begin{split}\\ddot{x} & = \\frac{1}{2} \\left( \\ddot{h}_+ x + \\ddot{h}_{\\times } y \\right) , \\\\\\ddot{y}& = \\frac{1}{2} \\left( \\ddot{h}_{\\times } x - \\ddot{h}_+ y \\right) .\\end{split}$ The equations of motion are equivalent to the presence of an effective tidal force $F_i = \\ddot{h}_{i j} x^j/2$ that acts on the particles.", "The corresponding effective force is conservative and can therefore be represented by force lines, shown in fig.", "REF for a purely plus-polarized wave.", "For a general polarization, the force line diagram is rotated counter-clockwise by the angle $\\Psi $ where $\\tan (2 \\Psi ) = \\ddot{h}_{\\times } / \\ddot{h}_+$ .", "For only a plus-polarized wave ($h_{\\times }=0$ ), the solution to lowest order in $h$ is $\\begin{split}x(t) & = x(0) \\left(1 + \\frac{h_+(t)}{2} \\right) \\\\y(t) & = y(0) \\left(1 - \\frac{h_+(t)}{2} \\right).\\end{split}$ The distances between nearby points oscillate in the $x-$ and $y-$ directions, i.e.", "perpendicular to the gravitational wave.", "A cross-polarized wave has the same effect but with the x-y-plane rotated by $\\pi /4$ (see also fig.", "REF ).", "Figure: Force lines for the effective tidal force produced by a plus-polarized gravitational wave ().", "The force acts perpendicular to the direction of propagation of the wave.", "The same force lines, rotated counter clock-wise by Ψ=π/4\\Psi = \\pi /4, represent the effect of a cross-polarized wave.Figure: Dynamics of mass distribution in time, as the gravitational wave passes.", "Row (a) shows a ring of test particles for a passing plus-polarized wave, while row (b) shows the effect of a cross-polarized wave.Figure: The co-ordinate transformation angles from the detector frame (symbolized by the cylinder) to the source frame (symbolized by the star).", "The angle Ψ\\Psi in the x'-y'-plane defines the polarization of the gravitational wave.The detector co-ordinate axis ($x,y,z$ ) is not necessarily aligned with the gravitational wavefront emitted from the source ($x^{\\prime },y^{\\prime },z^{\\prime }$ ).", "To account for the angular dependence, the strain at the detector can be written as $h(t)=F_+(\\theta ,\\phi ,\\psi )h_+(t)+F_\\times (\\theta ,\\phi ,\\psi )h_\\times (t)$ where $(\\theta ,\\phi ,\\psi )$ are the Euler angles that convert from the pulsar co-ordinate system to the the detector plane, as shown in fig.", "REF , and $F_{+/\\times }(\\theta ,\\phi ,\\psi )$ are known as the detector pattern functions [60].", "While the angles $\\theta $ and $\\phi $ describe the direction of the incoming gravitational wave ($\\phi $ being rotation of the old $x-y$ plane along the z-axis, and $\\theta $ being the angle between the source and detector z-axis), $\\psi $ defines the polarization of the detector (rotation of the $x-y$ plane along source line of sight) [61], as shown is fig.", "REF .", "Compared to large ground-based sensors where the angle $\\psi $ is fixed, it can be used as a parameter for the helium detector, to be optimized for the particular pulsar in consideration.", "$F_{+/\\times }(\\theta ,\\phi ,\\psi )$ is defined as $F_A(\\theta ,\\phi ,\\psi ) = q_{ij}\\hat{e}_A^{ij}(\\theta ,\\phi ,\\psi ),$ where ${\\bf q}$ is the dynamic mass quadrupole tensor of the detector that we will visit later, and $\\hat{e}_A^{ij}(\\theta ,\\phi ,\\psi )$ (with $A\\in \\lbrace +,\\times \\rbrace $ ) are the unit vectors for the two polarizations of the gravitational wave given in eqs.", "REF , REF in the rotated basis, $\\hat{e}_A(\\theta ,\\phi ,\\psi )=R^{-1}_{XYZ}\\hat{e}_AR_{XYZ}$ , where the rotation matrix is given by $ R_{XYZ}(\\theta ,\\phi ,\\psi )=\\begin{bmatrix}\\cos {\\theta } \\cos {\\phi }& \\cos {\\theta }\\sin {\\phi } & -\\sin {\\theta }\\\\-\\cos {\\psi }\\sin {\\phi }+\\cos {\\phi }\\sin {\\theta }\\sin {\\psi } &\\cos {\\phi }\\cos {\\psi }+\\sin {\\phi }\\sin {\\theta }\\sin {\\psi } & \\cos {\\theta }\\sin {\\psi }\\\\\\cos {\\phi }\\cos {\\psi }\\sin {\\theta } +\\sin {\\phi }\\sin {\\psi }& \\cos {\\psi }\\sin {\\theta }\\sin {\\phi }-\\cos {\\phi }\\sin {\\psi } &\\cos {\\theta }\\cos {\\psi }\\end{bmatrix} $ ." ], [ "Gravitational wave perturbation from an ellipsoidal pulsar", "As mentioned in the previous section, the generation of gravitational waves can be studied by considering the linearized Einstein equations in the presence of matter.", "In this section, we focus on the mechanism for generation of CW gravity waves from pulsars.", "Let us assume a non-spherical pulsar, rotating about the $z$ -axis.", "In order to emit gravitational waves, the star needs to have some asymmetry along the rotational axis (i.e.", "$r_x\\ne r_y$ ).", "Assuming an ellipsoidal star with the axes coinciding with the principal axes of the solid of revolution, the extent of the ellipsoidal star can be described by equation $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1,$ where $a,b,c$ are the semi-axes along $x,y,z$ directions respectively.", "We also assume a constant mass density, $\\rho $ .", "The quadrupolar mass tensor is given by [23] $Q_{ij}:=\\rho \\int _{\\rm body} x_i x_j dV.$ Due to the (chosen) co-ordinate system being along the principal axes, the quadrupolar tensor is given by $Q=(1/5)M_p{\\rm diag}[a^2,b^2,c^2]$ , where $M_p$ is the mass of the pulsar.", "Assuming at time t=0, $Q(t=0)={\\rm diag}[Q_1,Q_2,Q_3]$ , and given the star rotates about the $z$ -axis with angular frequency $\\omega _p$ , $Q(t)=R_z(\\omega t)I(0)R^{-1}_z(\\omega t)$ We now define two parameters, $Q=Q_1+Q_2$ , and ellipticity $\\epsilon =(Q_1-Q_2)/I_{zz}$ , where $I_{zz}=(1/5)M(a^2+b^2)$ is the moment of inertia about the $z$ -axis [62].", "The quadrupolar tensor can now be written as $Q=\\begin{bmatrix}\\frac{1}{2}Q+\\frac{1}{2}\\epsilon I_{zz}\\cos {2 \\omega _p t} & -\\frac{1}{2}\\epsilon I_{zz}\\sin {2 \\omega _p t} & 0\\\\-\\frac{1}{2}\\epsilon I_{zz}\\sin {2 \\omega _p t} & \\frac{1}{2}Q -\\frac{1}{2}\\epsilon I_{zz}\\cos {2 \\omega _p t} & 0\\\\0 & 0 & Q_{3}\\end{bmatrix}.$ In the far-field limit, where size of the star (or $GM/c^2$ ) $\\ll $ wavelength of the gravity wave ($c/\\omega _p$ ) $\\ll $ distance to detector ($d$ ), the gravitational wave perturbation becomes $h_{ij}(t,x)=\\frac{2G}{c^4 d} \\ddot{Q}(t_r),$ where $h$ is the gravitational perturbation tensor in transverse-traceless gauge, and $t_r=(t-d/c)$ is the retarded time, given the detector is distance $d$ away from the source.", "Since time retardation gives an extra overall phase here, we ignore it for our purposes.", "Thus, $h=\\frac{2G}{c^4 d} 2\\epsilon I_{zz}\\omega _p^2 \\begin{bmatrix}-\\cos {2 \\omega _p t} & \\sin {2 \\omega _p t} & 0\\\\\\sin {2 \\omega _p t} & \\cos {2 \\omega _p t} & 0\\\\0 & 0 & 0\\end{bmatrix},$ with the two polarization components being $h_+&=&-\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega _p^2\\cos {2\\omega _p t}\\\\h_\\times &=&\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega _p^2\\sin {2\\omega _p t}$ We now estimate the perturbation in the metric due to a gravitational wave from a pulsar, $h$ .", "There are two unknowns here, $I_{zz}$ and $\\epsilon $ .", "These parameters depend on the composition of the neutron star.", "Even though the equation of state (relation between density and pressure) of neutron star is unknown, certain properties of a neutron stars are remarkably well understood, and agree with astronomical observations [17].", "The main bulk of the star is assumed to be a neutron fluid, with a solid crystalline crust made of heavy nuclei.", "A neutron star can thus be modeled as a giant nucleus, akin to a degenerate fermi gas of neutrons.", "Since Fermi energy is very high, temperature variations have little effect on the properties of the star [17].", "Thus, the equation of state determining the size and radius of the star are fairly well constrained.", "Most models show that the radius lies between 10.5 and 11.2 km and mass ranges between 0.5$M_{\\odot }$ and 3$M_{\\odot }$ , with all measured values close to 1.35$M_{\\odot }$ (from Keplerian analysis of pulsars in binary systems- 5% of all observed pulsars) [17].", "Putting in these values, the moment of inertia amounts to $10^{38}$ kg-m$^2$ , surprisingly close to the moment of inertia of earth!", "While there may be uncertainties here, the changes would amount to less than an order of magnitude in the estimate.", "Thus $I=10^{38}$ kg-m$^2$ is the estimate used in previous GW searches, c.f.", "[18].", "For pulsars in binary systems of known mass, one could use a more accurate empirical expression given in Ref.", "[63], $\\nonumber I_{zz}=4.42\\times 10^{37} && {\\rm kg\\ m^2}\\left(\\frac{M_p}{M_{\\odot }}\\right)\\left(\\frac{R_p}{{10 \\rm km}}\\right)^2 \\\\&&\\times \\left[1+5\\left(\\frac{M_p}{M_{\\odot }}\\right)\\left(\\frac{{\\rm km}}{R_p}\\right)\\right], $ where $R_p$ is the average radius of the pulsar of interest.", "The biggest uncertainty in estimating $h$ therefore comes from $\\epsilon $ , the ellipticity parameter that characterizes the mass asymmetry of the pulsar.", "We now present estimates on $\\epsilon $ from two different mechanisms: the spin-down energy conservation ($\\epsilon _{sd}$ ) and elastic strain on the NS crust ($\\epsilon _{n}$ ) and their corresponding GW strain limits.", "Spin-down Limit: Since the pulsar is spinning down, its rotational frequency is changing at some observable rate $\\dot{\\omega }_p$ .", "This amounts to a torque of $I_{zz}\\dot{\\omega }_p$ .", "If we assume that all of this spin-down is due to gravitational radiation, $I_{zz}\\dot{\\omega }_p=dL_z/dt$ , where $L_z$ is the angular momentum of the body along $z$ axis.", "Recalling that change in energy, $\\Delta E= \\omega _p \\Delta L_z$ , we arrive at $dL_z/dt=(1/\\omega _p)dE/dt$ .", "One can derive a more rigorous expression for change in angular momentum due to gravitational radiation from first principles.", "But for the special case of rotation around a principal axis, the expression gets simplified to the one above [62].", "The emitted power of gravitational radiation is given by [23], [62] $\\frac{dE}{dt}=\\frac{G}{5c^5}\\langle \\dddot{Q}_{ij}\\dddot{Q}^{ij} \\rangle =\\frac{32G }{5c^5}\\epsilon ^2 I_{zz}^2 \\omega _p^6,$ giving us $I_{zz}\\dot{\\omega }_p=\\frac{32G }{5c^5}\\epsilon ^2 I_{zz}^2 \\omega _p^5.$ Substituting these values allows us to solve for the ellipticity parameter, $\\epsilon _{sd}=\\left(\\frac{5c^5\\dot{\\omega }_p}{32GI_{zz}\\omega _p^5}\\right)^{1/2}.$ This in turn is used to compute the upper-limit estimate for gravitation perturbation strain in terms of constants and observational data $h_{sd}=-\\frac{4G}{c^4 d} \\epsilon I_{zz}\\omega ^2=\\sqrt{\\frac{5GI_{zz}\\dot{\\omega }}{2c^3d^2\\omega }}$ The spin-down strain estimate is a significant over-estimate of the strength of gravitational waves, particularly from young pulsars.", "As an example, at the surface of the Crab pulsar the ratio of gravitational to magnetic forces on an electron, $\\frac{GMm}{r^2}/\\frac{e\\omega r B}{c}\\approx 10^{-12},$ suggesting that electromagnetic forces have a crucial role to play in most observable properties of the star.", "Thus, for young pulsars it is reasonable to assume that dipolar electromagnetic radiation is the dominant way to lose angular momentum, and not quadrupolar gravitational radiation, as is eluded to by braking index measurements [28], [29].", "This means that spin-down strain estimate is a significant over-estimate of the strength of gravitational waves from these pulsars, a fact already confirmed by the recent GW detector data [18].", "A second class of pulsars, known as millisecond pulsars (or MSPs) are much longer-lived, slowly decaying, even speeding up at times.", "These are pulsars in binary systems with the transfer of mass (and angular momentum) from the companion star, leading to X-ray emission and speeding up of the rotations.", "The discovery of new MSPs has accelerated since the operation of gamma-ray detectors like Fermi-LAT.", "For example, 5 new MSPs were detected by radio searches of unidentified Fermi LAT gamma-ray sources in 2012 (two of these are included in Table I) [32].", "MSPs are remarkably stable ($\\dot{\\omega }_p/2\\pi <10^{-14}$ ) and were once considered strong candidates for long-term time-standard.", "There has only been one observed random glitch in the thousands of years of accumulated observation time [64].", "Small slow down rates suggest that MSPs are not as magnetized as some younger pulsars [30].", "It is the reason why we have ignored magnetic deformations as the primary mass asymmetry mechanism in this work.", "This electromagnetic stability indicates that that gravitational radiation might dominate over magnetic dipole radiation as the dominant energy loss mechanism in MSPs.", "Crustal Strain Limit: There are several mechanisms that contribute to the mass asymmetry.", "Either due to its formation in a supernova, or due to the presence of an accretion disk, the neutron star's rotation and magnetic axis might not coincide.", "The enormous Lorentz forces on the crust might then contribute to the asymmetric mass distribution.", "Alternatively, the mass distribution could also be changed significantly due to star quakes, internal magnetic fields, instabilities induced by gravitational effects, or viscosity of the dense matter [17].", "There are also exotic theories involving superfluid turbulence in stellar cores.", "Estimating the maximum elastic deformation sustained by a neutron star is an active field of research, see ref.", "[65], [27], and references therein for details.", "Ushomirsky et.", "al set limits on the maximum quadrupole moment for a NS in the presence of elastic forces, irrespective of the nature of strain on the crust [31].", "For standard parameters for $I_{zz}$ and breaking strain of the crust, this quadrupole moment leads to a maximum ellipticity of $\\epsilon _{n}\\sim 6\\times 10^{-7}$ for a conventional neutron star[15].", "This in turn can be used to evaluate limit on the gravitational wave strain amplitude, $h_{\\epsilon n}=-\\frac{4G}{c^4 d} \\epsilon _n I_{zz}\\omega _p^2.$ Both the spin down and strain mechanisms put upper limits on the metric perturbation due to different physics.", "Therefore, we assume that the strain due to GW from pulsars is smaller than the lower of the two limits.", "Table REF shows the observational data, along with theoretical and measured estimates of $h$ for pulsars of interest from the LIGO+VIRGO data.", "Table: Table of Pulsars of interest from ref.", ".", "h ϵn h_{\\epsilon n} gives the strain (from Eq. )", "if the ellipticity was ϵ=6×10 -7 \\epsilon =6\\times 10^{-7}, the maximum for a neutron star made of neutrons .", "We have assumed I zz =10 38 I_{zz}=10^{38} kg-m 2 ^2, consistent with LIGO literature.It is interesting to note that for the younger pulsars with small rotation frequencies and large spin down rates, the upper limit on GW strain is set by the elastic deformation limit, while for MSPs smaller spin-down rates lead to a significantly lower limit set by $h_{sd}$ .", "In both cases, the GW signal limit is typically below $10^{-27}$ .", "These limits provide an upper limit on GW strength due to different physics, it is possible (in fact expected) that the actual signal would be even lower.", "However, it is worth remembering that the observation (or even absence) of GW signal is the only known way to gain information about the interior of these exotic objects." ], [ "Search for optimal detector geometry", "As eq.", "REF suggests, the minimum detectable strain by the helium detector depends on several parameters.", "Thus, it is difficult to determine the best geometry for gravitational wave detection.", "Here we analyze several acoustic modes for a cylindrical detector that have a non-zero quadrupole tensor.", "We have chosen each of these geometries/modes to have a resonance frequency around 1075 Hz, and assumed a $Q$ -factor of $10^{11}$ .", "In particular, $h_{\\rm min}$ is evaluated for pulsar J1843-1113.", "Before presenting a table analyzing 7 lower modes of interest, we present a summary of some general trends: For the same mode and frequency, it is always advantageous to use a bigger mass (assuming $Q$ -factor remains the same).", "Higher $n$ modes have significantly smaller effective area than lower $n$ modes.", "This then contributes to lower strain sensitivity.", "Thus, it is advantageous to have the lowest $n$ mode for a given frequency.", "The maximum of the directivity function ($d^A(\\theta ,\\phi )$ ) can vary by up to a factor of 4 in cylindrical geometry depending on the mode of interest.", "Below we summarize various properties of ten different cylindrical geometries with similar resonance frequencies.", "Geometry 7 and 8 are used in the main text.", "Table: Table of modes and geometries of interest.", "We have chosen each of these geometries/modes to have a resonance frequency of 1075±51075\\pm 5 Hz, and assumed a QQ-factor of 10 11 10^{11}.", "In particular, h min h_{\\rm min} is evaluated for pulsar J1843-1113.", "ψ=π/2\\psi =\\pi /2, unless otherwise noted.We find that for $l=0$ modes, one needs a cylinder with length in meters to beat the 1kHz sensitivity limit of advanced LIGO.", "The [020] mode has a particularly strong coupling to gravity waves due to its quadrupolar mode shape, and we choose this mode for the detector geometry discussed in the main text.", "Due to its large effective area and directivity, the $[110]$ mode also efficiently couples to gravitational metric strain.", "Unfortunately, the [110] mode does not couple to microwaves, so the gravitational wave signal in this acoustic mode cannot be detected using our proposed optomechanical technique.", "Finally, we would like to mention that while this paper deals exclusively with cylindrical geometry, there possibly are other geometries that couple more strongly to gravitational strain.", "Exploring different detector geometries is an interesting numerical problem that we hope to address in the future." ] ]	1606.04980
[ [ "Combining multiscale features for classification of hyperspectral\n images: a sequence based kernel approach" ], [ "Abstract Nowadays, hyperspectral image classification widely copes with spatial information to improve accuracy.", "One of the most popular way to integrate such information is to extract hierarchical features from a multiscale segmentation.", "In the classification context, the extracted features are commonly concatenated into a long vector (also called stacked vector), on which is applied a conventional vector-based machine learning technique (e.g.", "SVM with Gaussian kernel).", "In this paper, we rather propose to use a sequence structured kernel: the spectrum kernel.", "We show that the conventional stacked vector-based kernel is actually a special case of this kernel.", "Experiments conducted on various publicly available hyperspectral datasets illustrate the improvement of the proposed kernel w.r.t.", "conventional ones using the same hierarchical spatial features." ], [ "Introduction", "Integration of spatial information paves the way for improved accuracies in hyperspectral image classification [1], as the use of spatial features extracted from image regions produces spatially smoother classification maps [2].", "A common approach to extract such features is to rely on multiscale representations, e.g.", "through (extracted) attribute profiles [3] or hierarchical spatial features [4].", "In this framework, features from multiple scales are extracted to model the context information around the pixels through different scales.", "Features computed at each scale are then concatenated into a unique (long) stacked vector.", "Such a vector is then used as input into a conventional classifier like SVM.", "Representative examples of this framework include [4], [5], [6].", "While defining kernels on stacked vectors is a simple and standard way to cope with hierarchical spatial features, it does not take into account the specific nature (i.e.", "hierarchical) of the data.", "Indeed, hierarchical spatial features extracted from hyperspectral images can rather be viewed as a sequence of data, for which structured kernels are commonly applied in other fields.", "Among the existing sequence structured kernels, the spectrum kernel based on subsequences of various lengths has been successfully applied in various domains, e.g.", "biology for protein classification [7], [8] or nature language processing for text classification [9].", "Its relevance for hyperspectral image classification remains to be demonstrated and is the main objective of this paper.", "Indeed, by applying the spectrum kernel onto hierarchical spatial features, we can explicitly take into account the hierarchical relationships among regions from different scales.", "To do so, we construct kernels on various lengths of subsequences embedded in the whole set of hierarchical spatial features instead of modeling this set as a single stacked vector, the latter actually being a particular case of the sequence kernel.", "Furthermore, we also propose an efficient algorithm to compute the spectrum kernel with all possible lengths, thus making realistic to apply such a kernel on hyperspectral images.", "The paper is organized as follows.", "We first briefly recall some background on hierarchical image representation (Sec. ).", "We then detail the concept of spectrum kernel (Sec.", "), and introduce an efficient algorithm for its computation.", "Evaluation of proposed method is detailed in Sec.", ", before giving a conclusion and discussing future works." ], [ "Hierarchical image representation", "Hierarchical image representation describes the content of an image from fine to coarse level (as illustrated in Fig.", "REF ) through a tree structure, where the nodes represent the image regions at different levels and the edges model the hierarchical relationships among those regions.", "Such representation is commonly used in the GEOgraphic-Object-Based Image Analysis (GEOBIA) framework [10] and can be constructed with hierarchical segmentation algorithms, e.g.", "HSeg [11].", "Let $n_1$ be a pixel of the image.", "Through hierarchical image representation, we write $n_i$ the nested image regions at level $i= 2,...,p_\\text{max}$ , with region at lower levels always being included in higher levels i.e.", "$n_1 \\subseteq n_2 \\ldots \\subseteq n_{p_\\text{max}}$ .", "The context information of pixel $n_1$ can be then described by its ancestor regions $n_i$ at multiple levels $i= 2,...,p_\\text{max}$ .", "More specifically, one can define the context information as a sequence $S= \\lbrace n_1,...,n_{p_\\text{max}}\\rbrace $ that encodes the evolution of the pixel $n_1$ through the different levels of the hierarchy.", "Each $n_{i}$ is described by a $D$ -dimensional feature $x_{i}$ that encodes the region characteristics e.g.", "spectral information, size, shape, etc." ], [ "Definition", "The spectrum kernel is an instance of kernels for structured data that allows the computation of similarities between contiguous subsequences of different lengths [7], [8].", "Originally designed for symbolic data, we propose here an adaptation to deal with hierarchical representations equipped with numerical features.", "Contiguous subsequences can be defined as $ s_{p} = (n_t, n_{t+1}$ $..., n_{t+p})$ , with $ t \\ge 1, t+p \\le p_\\text{max}$ and $p$ being the subsequence length.", "Fig.", "REF gives an example of a sequence and enumerates all its subsequences $s_p$ .", "Figure: A sequence SS (left) and all its subsequences s p s_p (right).The spectrum kernel measures the similarity between two sequences $S, S^{\\prime }$ by summing up kernels computed on all their subsequences.", "Let $S_p = \\lbrace s_p \\in S \\mid \|s_p\| = p\\rbrace $ be the set of subsequences with a specific length $p$ , the spectrum kernel can be written as: $\\begin{split}K(S,S^{\\prime })& = \\sum _{p} \\; \\omega _{p} ~ K(S_p,S^{\\prime }_p) \\; \\\\& = \\sum _{p} \\; \\omega _{p} \\sum _{\\begin{array}{c}s_{p} \\in S_p, s^{\\prime }_{p} \\in S^{\\prime }_p \\end{array}} K(s_p,s^{\\prime }_p ) \\; ,\\end{split}$ where the $p$ -spectrum kernel $ K(S_p,S^{\\prime }_p)$ is computed between the set of subsequences of length $p$ , and is further weighted by parameter $\\omega _{p}$ .", "In other words, it only allows the matching of subsequences with same length.", "The kernel between two subsequences $K(s_p,s^{\\prime }_p )$ is defined as the product of atomic kernels computed on individual nodes $k(n_{t+i},n^{\\prime }_{t^{\\prime }+i})$ , with $i$ denoting the position of nodes in the subsequence, following an ascending order $ 0 \\le i \\le p - 1$ : $K(s_{p},s^{\\prime }_{p})= \\prod _{\\begin{array}{c}i = 0\\end{array}}^{p-1} k(n_{t+i},n^{\\prime }_{t^{\\prime }+i} )\\; .$ $K(S,S^{\\prime })$ in Eq.", "(REF ) suffers a common issue for structured kernels: the kernel value highly depends on the length of the sequences, as the number of compared substructures greatly increases with the length of sequence.", "One can mitigate this problem by normalizing the kernel as: $K^(S,S^{\\prime })= \\frac{K(S,S^{\\prime }) }{ \\sqrt{K(S,S)}\\sqrt{K(S^{\\prime },S^{\\prime })}}.", "\\;$ In the sequel, we only use the normalized version $K^$ of the kernel (written $K$ for the sake of simplicity)." ], [ "Weighting", "Several common weighting schemes [8] can be considered: $\\omega _{p}= 1$ if $p = q$ and $\\omega _{p}= 0$ otherwise, yielding to a $q$ -spectrum kernel considering only subsequence with a given length $q$ : $K(S,S^{\\prime })=\\sum \\limits _{{s_{q} \\in S_q, s^{\\prime }_{q} \\in S^{\\prime }_q }} K(s_q,s^{\\prime }_q )$ ; $\\omega _{p}= 1$ for all $p$ , leading to a constant weighting with all lengths of subsequences; $\\omega _{p}= \\lambda ^p$ with $\\lambda \\in (0,1)$ , an exponentially decaying weight w.r.t.", "the length of the subsequences.", "It should be noted here that when using Gaussian kernel for the atomic kernel $k(n_{i},n_{i}^{\\prime }) = \\exp (-\\gamma \\Vert x_{i}-x^{\\prime }_{i}\\Vert ^2)\\; ,$ the kernel computed on the stacked vector $z = (x_{1}, \\cdots , x_{p_\\text{max}})$ comes down to the $p_\\text{max}$ -spectrum kernel: $ \\begin{split}& K(s_{p_\\text{max}},s^{\\prime }_{p_\\text{max}}) = \\prod _{\\begin{array}{c}i = 1\\end{array}}^{p_\\text{max}} \\exp (-\\gamma \\Vert x_{i}-x^{\\prime }_{i}\\Vert ^2)\\; = \\\\& \\exp \\left(\\sum _{\\begin{array}{c}i = 1\\end{array}}^{p_\\text{max}} (-\\gamma \\Vert x_{i}-x^{\\prime }_{i}\\Vert ^2)\\right)\\; = \\exp (-\\gamma \\Vert z-z^{\\prime }\\Vert ^2)\\;.\\end{split}$" ], [ "Kernel computation", "We propose here an efficient computation scheme to iteratively compute all the $p$ -spectrum kernels in a single run, yielding a complexity of $O(p_\\text{max}p^{\\prime }_\\text{max})$ .", "The basic idea is to iteratively compute the kernel on subsequences $s_{p}$ and $s^{\\prime }_p$ using previously computed kernels on subsequences of length $(p-1)$ .", "The atomic kernel $k(n_{i}, n^{\\prime }_{i^{\\prime }})$ thus needs to be computed only once, avoiding redundant computing.", "We define a three-dimensional matrix $M$ of size $p_\\text{max} \\times p^{\\prime }_\\text{max} \\times \\min (p_\\text{max}, p^{\\prime }_\\text{max})$ , where each element $M_{i,i^{\\prime },p}$ is defined as: $M_{i,i^{\\prime },p} = k(n_i,n^{\\prime }_{i^{\\prime }}) ( M_{{i-1},{i^{\\prime }-1} ,p-1} ) \\; .$ where $M_{0,0,0} = M_{0,i^{\\prime },0} = M _{i,0,0} = 1$ by convention.", "The kernel value for the $p$ -spectrum kernel is then computed as the sum of all the matrix elements for a given $p$ : $K(S_p,S^{\\prime }_p) = \\sum _{i,i^{\\prime }=1}^{p_{\\text{max}},p^{\\prime }_{\\text{max}}}M_{i,i^{\\prime },p} \\; .$" ], [ "Datasets and design of experiments", "We conduct experiments on 6 standard hyperspectral image datasets: Indian Pines, Salinas, Pavia Centre and University, Kennedy space center (KSC) and Botswana, considering a one-against-one SVM classifier (using the Java implementation of LibSVM [12]).", "We use Gaussian kernel as the atomic kernel $ k(\\cdot ,\\cdot )$ .", "Free parameters are determined by 5-fold cross-validation over potential values: the bandwidth $\\gamma $ (Eq.", "(REF )) and the SVM regularization parameter $C$ .", "We also cross-validate the different weighting scheme parameters: $q \\in \\lbrace 1, \\ldots , p_\\text{max}\\rbrace $ for the $q$ -spectrum kernel and $\\lambda \\in (0,1)$ for the decaying factor." ], [ "Results and analysis", "We randomly pick $n=\\lbrace 10,25,50\\rbrace $ samples per class from available ground truth for training, and the rest for testing.", "In the case of small number of pixels per class in Indian Pines dataset (total sample size for a class less than $2n$ ), we use half of samples for training.", "Hierarchical image representations are generated with HSeg [11] by increasing the region dissimilarity criterion $\\alpha $ .", "Parameter $\\alpha $ is empirically chosen: $\\alpha =[2^{-2},2^{-1},...,2^{8}]$ , leading to a tree that covers the whole scales from fine to coarse (top levels of whole image are discarded as they do not provide any additional information).", "Hierarchical levels $\\alpha = \\lbrace 2^2,2^4,2^6\\rbrace $ of Indian Pines are shown in Fig.", "REF as the fine, intermediate, coarse level for illustration.", "Features $x_{i}$ that describe each region are set as the average spectral information of the pixels that compose the region." ], [ "Comparison with state-of-the-art algorithms", "We compare our sequence-based kernel with state-of-the-art algorithms that take into account the spatial information relying on multiscale representation of an image: i) spatial-spectral kernel [2] that uses area filtering to obtain the spatial features (the filtering size is fixed so as to lead to the best accuracy); ii) attribute profile [3], using 4 first principal components with automatic level selection for the area attribute and standard deviation attribute as detailed in [13]; iii) hierarchical features stored on a stacked vector [4], [5], [6].", "For comparison purposes, we also report the pixel-based classification overall accuracies.", "All results are obtained by averaging the performances over 10 runs of (identical among the algorithms) randomly chosen training and test sets.", "First of all, in Tab.", "REF , we can see that the overall accuracies are highly improved when spatial information is included.", "Using hierarchical features computed over a tree (stacked vector or any version of the spectrum kernel) yields competitive results compared with state-of-the-art methods.", "By applying the proposed spectrum kernel on the hierarchical features rather than a kernel on a stacked vector, the results are further improved: best results for Indian Pines, Salinas, Pavia Centre, KSC and Botswana datasets are obtained with a spectrum kernel.", "We can observe that attribute profiles perform better for Pavia University.", "This might be due to the kind of hierarchical representation used, i.e.", "min and max-trees in the case of attribute profiles instead of HSeg in our case.", "Besides, the popularity of these profiles as well as the Pavia dataset result in optimizations of the scale parameters for years.", "However, the proposed spectrum kernel is not limited at all to the HSeg representation, and it is thus possible to apply it to min- and max-trees and attribute features.", "This will be explored in future studies." ], [ "Impact of the weighting scheme", "We study the impact of the different weighting schemes introduced in Sec.", "REF .", "Fig.", "REF shows that the stacked vector ($q = p_\\text{max}$ ) does not lead to the best performances, and that the best scale $q$ can not be determined beforehand as it depends on the dataset.", "For most setups, combination of different scales (constant weighting or decaying factor) allows the improvement of the accuracies.", "However, the best weighting scheme again depends on the considered dataset, and this calls for a more extensive study of weighting strategies.", "Table: Mean (and standard deviation) of overall accuracies (OA) computed over 10 repetitions using nn training samples per class for 6 hyperspectral image datasets.", "cc stands for constant weighting, qq for the qq-spectrum kernel and λ\\lambda for the decaying weight.", "Best results are boldfaced.Figure: The overall accuracies of qq-spectrum kernel with different lengths qq using n=50n = 50 training samples per class.", "Results for the stacked vector with Gaussian kernel are shown in dashed line." ], [ "Conclusion", "In this paper, we propose to use the spectrum kernel for applying machine learning on hierarchical features for hyperspectral image classification.", "The proposed kernel considers the hierarchical features as a sequence of data and exploits the hierarchical relationship among regions at multiple scales by constructing kernels on various lengths of subsequences.", "The method exhibits better performances than state-of-the-art algorithms for all but one tested dataset.", "We also show that combining different scales allows the improvement of the accuracies, but the way to combine them should be further explored.", "The use of optimal weights thanks to the multiple kernel learning framework [14] is the next step of our work." ] ]	1606.04985
[ [ "Large effects of subtle electronic correlations on the energetics of\n vacancies in alpha-Fe" ], [ "Abstract We study the effect of electronic Coulomb correlations on the vacancy formation energy in paramagnetic alpha-Fe within ab initio dynamical mean-field theory.", "The calculated value for the formation energy is substantially lower than in standard density-functional calculations and in excellent agreement with experiment.", "The reduction is caused by an enhancement of electronic correlations at the nearest neighbors of the vacancy.", "This effect is explained by subtle changes in the corresponding spectral function of the d-electrons.", "The local lattice relaxations around the vacancy are substantially increased by many-body effects." ], [ "Supplementary material", "In this supplementary material, we provide additional details concerning the supercell DFT+DMFT scheme set up for the calculation of the vacancy formation energies, as well as additional support for the physical picture presented in the main text." ], [ "Details of the methodology", "Our procedure is to take as a starting point a converged DFT calculation, using the full potential linearized augmented plane-wave (FLAPW) Wien2k code [46], which is then used to perform charge self-consistent DFT+DMFT.", "We use a k-point mesh with 8x8x8 points for the 16 atom supercell, and 4x4x4 points for the 54 atom supercell.", "GGA calculations are done with the PBE96 functional.", "Atomic sphere radii (RMTs) are reduced to 2.12 from the default value of 2.37 to allow for atomic position relaxation.", "A first set of relaxed atomic positions was obtained in the large supercell by letting the atoms relax in spin-polarized GGA using atomic forces, at the equilibrium volume corresponding to a spin-polarized GGA calculation.", "The same atomic positions were then used in paramagnetic GGA and DFT+DMFT, at the experimental volume.", "Then, a second and final set of relaxed positions was obtained by manually relaxing the two first coordination shells, in DFT+DMFT, at the experimental volume.", "We show that while the GGA relaxed positions are a good starting point, corrections due to many-body effects included within DMFT still modify the nearest-neighbor positions quite significantly.", "In order to avoid systematic errors in the vacancy formation energy due to a different computational setup, we used the same cell geometries in calculations with and without the vacancy.", "Charge self-consistent DFT+DMFT calculations are performed on the same 2x2x2 and 3x3x3 supercells using the TRIQS library[44] and its DFT tools package[45] based on the DFT+DMFT implementation of Ref.", "[42] within the Wien2k package [46].", "We construct Wannier orbitals representing the Fe 3$d$ states from the Kohn-Sham eigenstates within a window [-6.8 eV, 5.4 eV] around the Fermi level using the projection approach of Ref. [42].", "From the local impurity problem we obtain the DMFT self-energy on the 7 (with vacancy) or 8 (without vacancy) inequivalent iron atoms of our supercell.", "This formidable computational hurdle – hitherto never overcome – has been dealt with by resorting to the fast segment representation version of the continuous-time quantum Monte-Carlo (CTQMC) hybridization expansion algorithm in order to solve the seven impurity problems.", "The DFT+DMFT Hamiltonian is $H=H^{LDA} + H^{U} - H^{DC}$ where $H^{U} = \\sum _{mm^\\prime ,\\sigma } U^{\\sigma \\bar{\\sigma }}_{mm^\\prime }n_{m\\sigma }n_{m^\\prime ,\\bar{\\sigma }} + \\sum _{m\\ne m^\\prime ,\\sigma } U^{\\prime \\sigma \\sigma }_{mm^\\prime }n_{m\\sigma }n_{m^\\prime ,\\sigma }$ with $\\begin{split}U^{\\sigma \\bar{\\sigma }}_{mm^\\prime } &= \\mathcal {U}_{mm^\\prime mm^\\prime } \\\\J_{mm^\\prime } &= \\mathcal {U}_{mm^\\prime m^\\prime m} \\\\U^{\\prime \\sigma \\sigma }_{mm^\\prime } &= U^{\\sigma \\bar{\\sigma }}_{mm^\\prime } - J_{mm^\\prime }\\end{split}$ Here, $\\mathcal {U}$ is the full Slater-parameterized interaction matrix, and $H^U$ is the density-density part of the full Slater Hamiltonian.", "$H^{DC}$ corrects the double counting of interactions, as both $H^{U}$ and $H^{LDA}$ contain a part of the on-site electron-electron interaction.", "We take the around mean-field approximation for $H^{DC}$[54], [55].", "In the calculations for the system with a vacancy, the DMFT self-consistency condition corresponds to a set of seven equations, one for each inequivalent atom $\\alpha $ .", "They require the impurity Green's functions $G_{\\mathrm {imp}}^\\alpha (i\\omega _{n})$ to equal the respective projections of the lattice Green's functions: $\\left[G_{\\mathrm {imp}}^\\alpha (i\\omega _{n})\\right]_{mm^{\\prime }}=\\sum _{\\mathbf {k},\\nu \\nu ^{\\prime }}P_{\\ell m, \\nu }^{\\alpha }(\\mathbf {k})~G_{\\nu \\nu ^{\\prime }}(\\mathbf {k},i\\omega _{n})~\\left[P_{\\ell m^{\\prime },\\nu ^{\\prime }}^{\\alpha }(\\mathbf {k})\\right]^$ where the projector $P_{\\ell m, \\nu }^{\\alpha }(\\mathbf {k})$ denotes the scalar product between the Kohn-Sham state $(\\mathbf {k}, \\nu )$ and the local orbital of character $(\\ell m)$ on atom $\\alpha $ .", "The (inverse) correlated Green's function in the Kohn-Sham basis is given by $[G^{-1}(\\mathbf {k},i\\omega _{n})]_{\\nu \\nu ^{\\prime }} = (i\\omega _{n} +\\mu -\\varepsilon _{\\mathbf {k}}^{\\nu }) \\delta _{\\nu \\nu ^{\\prime }} -\\Sigma _{\\nu \\nu ^{\\prime }}(\\mathbf {k},i\\omega _{n}),$ where $\\varepsilon _{\\mathbf {k}}^{\\nu }$ are the Kohn-Sham energies and $\\Sigma _{\\nu \\nu ^{\\prime }}(\\mathbf {k},i\\omega _{n})$ is the upfolded self-energy $\\Sigma _{\\nu \\nu ^{\\prime }}(\\mathbf {k},i\\omega _{n})=\\sum _{\\alpha ,mm^{\\prime }}\\left[P_{\\ell m,\\nu }^{\\alpha }(\\mathbf {k})\\right]^~\\left[\\Sigma ^{\\alpha }(i\\omega _{n})\\right]_{mm^{\\prime }}~P_{\\ell m^{\\prime },\\nu ^{\\prime }}^{\\alpha }(\\mathbf {k}).$ with $\\left[\\Sigma ^{\\alpha }(i\\omega _{n})\\right]_{mm^{\\prime }} = \\left[\\Sigma _\\mathrm {imp}(i\\omega _n)\\right]_{mm^{\\prime }}-\\left[\\Sigma _{DC}\\right]_{mm^{\\prime }}$ being the difference between the impurity self-energy $\\Sigma _\\mathrm {imp}(i\\omega _n)$ and the double-counting correction $\\Sigma _{DC}$ .", "Finally, the interaction energy is computed as $E_{int} = \\frac{1}{2}\\sum _{ij}U_{ij}\\left<n_in_j\\right> - H^{DC}$ .", "The temperature is $\\beta =10$ , corresponding to $T=1162\\text{K}>T_{curie}$ , where bcc $\\alpha $ -Fe is paramagnetic.", "About 15 iterations of DFT+DMFT are needed before reaching convergence in the DFT+DMFT cycle.", "A further averaging of the total energy values is required to obtain precise enough values.", "In practice, at least 50 more cycles are needed, and the statistical uncertainty shown in table REF and Fig.", "1 of the main text is the empirical standard deviation of the value over these iterations.", "This uncertainty increases with the supercell size.", "Our $2 \\times 2 \\times 2$ and $3 \\times 3 \\times 3$ supercells have an overall cubic symmetry.", "However, the presence of a vacancy still breaks the on-site cubic point group symmetries for all iron atoms apart from the central one.", "For those atoms the Wien2k code uses local coordinate frames chosen in such a way as to have the highest possible on-site symmetry.", "Subsequently, our impurity problems are also solved for those local coordinates.", "Hence, a corresponding inverse rotation should be applied to the resulting self-energies if one wishes to compare them with that of perfect bcc Fe.", "The later is, of course, obtained for the standard coordinate frame with $x$ , $y$ , and $z$ axises along the cube edges.", "However, because we are using a density-density Hamiltonian instead of the full rotationally-invariant one, the self-energies obtained for those local frames are still somewhat different from that of perfect bcc iron even after the inverse rotation.", "Hence, in our calculations of the ideal supercells we employed the same local coordinate frames as for the supercells with vacancy in order to avoid spurious contributions of those rotations to the vacancy formation energy.", "Furthermore, we verified that the off-diagonal elements in the Green's functions stay small, so that we could neglect them." ], [ "Vacancy formation energies", "In Table REF we list the vacancy formation energy in bcc Fe obtained by different theoretical approaches and experiments.", "Figure 2 of the main text corresponds to a graphical representation of these data.", "Table: Vacancy formation energies.", "The statistical uncertainty shown is the empirical standard deviation of the value over the last 50 iterations." ], [ "Density of states of iron $e_g$ and {{formula:91a33c69-2cac-4d35-befa-87f217333f3d}} orbitals", "The density of states of figure 5 in the main part is shown below for the full energy range containing the Fe 3$d$ bands.", "Figure: Density of states on the e g e_g (full line) and t 2g t_{2g} (dashed line) on the first nearest neighbor to the vacancy (blue) and on the central atom (red).", "(Color online)" ] ]	1606.05121
[ [ "Reciprocity Calibration for Massive MIMO: Proposal, Modeling and\n Validation" ], [ "Abstract This paper presents a mutual coupling based calibration method for time-division-duplex massive MIMO systems, which enables downlink precoding based on uplink channel estimates.", "The entire calibration procedure is carried out solely at the base station (BS) side by sounding all BS antenna pairs.", "An Expectation-Maximization (EM) algorithm is derived, which processes the measured channels in order to estimate calibration coefficients.", "The EM algorithm outperforms current state-of-the-art narrow-band calibration schemes in a mean squared error (MSE) and sum-rate capacity sense.", "Like its predecessors, the EM algorithm is general in the sense that it is not only suitable to calibrate a co-located massive MIMO BS, but also very suitable for calibrating multiple BSs in distributed MIMO systems.", "The proposed method is validated with experimental evidence obtained from a massive MIMO testbed.", "In addition, we address the estimated narrow-band calibration coefficients as a stochastic process across frequency, and study the subspace of this process based on measurement data.", "With the insights of this study, we propose an estimator which exploits the structure of the process in order to reduce the calibration error across frequency.", "A model for the calibration error is also proposed based on the asymptotic properties of the estimator, and is validated with measurement results." ], [ "Introduction ", "[lines=2]Massive Multiple-input Multiple-output (massive MIMO) is an emerging technology with the potential to be included in next generation wireless systems, such as fifth-generation (5G) cellular systems.", "Massive MIMO departs from traditional multi-user MIMO approaches by operating with a large number of base station (BS) antennas, typically in the order of hundreds or even thousands, to serve a relatively small number of mobile terminals [1].", "Such a system setup results in a multitude of BS antennas that can be used in an advantageous manner from multiple points of view [2].", "One major challenge of operating with a large number BS antennas is that it renders explicit channel estimation in the downlink impractical.", "Basically, the overhead of channel estimation in the downlink and feeding back the channel estimate to the BS, scales linearly with the number of BS antennas, and quickly becomes unsupportable in mobile time-varying channels [3].", "To deal with this challenge, the approach adopted is to operate in time-division-duplex (TDD) mode, rely on channel reciprocity, and use uplink channel state information (CSI) for downlink precoding purposes [4].", "However, the presence of the analog front-end circuitry in practical radio units complicates the situation and makes the baseband-to-baseband channel non-reciprocal.", "Explained briefly, the baseband representation of the received signals [5] experience channels that are not only determined by the propagation conditions, but also by the transceiver front-ends at both sides of the radio link.", "While it is generally agreed that the propagation channel is reciprocal [6], the transceiver radio frequency (RF) chains at both ends of the link are generally not [7].", "Hence, in order to make use of the reciprocity assumption and rely on the uplink CSI to compute precoding coefficients, the non-reciprocal transceiver responses need to be calibrated.", "Such a procedure is often termed reciprocity calibration, and contains two steps: (i) estimation of calibration coefficients, and (ii) compensation by applying those to the uplink channel estimates.However, with the term reciprocity calibration, we will interchangeably refer to the estimation step, compensation step, or both.", "The context will, hopefully, make clear which of the previous cases is being addressed.", "Reciprocity calibration of small scale TDD MIMO channels has been a matter of study in recent years.", "Depending on the system setup and requirements, the approach adopted can take many forms.", "For example, [7] proposed a methodology based on bi-directional measurements between the two ends of a MIMO link to estimate suitable reciprocity calibration coefficients.", "This calibration approach falls in the class of \"over-the-air\" calibration schemes where users are involved in the calibration process.", "A different approach is to rely on dedicated hardware circuitry for calibration purposes, see [8], [9].", "Despite the possibilities of extending both mentioned calibration approaches to a massive MIMO context, e.g., [10], [11], recent calibration works suggest this is more difficult than previously thought.", "For example, [12] questions the feasibility of having dedicated circuits for calibration when the number of transceivers to be calibrated grows large, and [13] argues that the calibration protocols should preferably not rely on mobile units.", "It thus appears that an increasing trend in massive MIMO systems is to carry out the calibration entirely at the BS side only through over-the-air measurements.", "The first proposal in this vein was presented in [14].", "The work proposes an estimator for the calibration coefficients, which only makes use of channel measurements between BS antennas.", "More specifically, bi-directional channel measurements between a given BS antenna, so-called reference antenna, and all other antennas.", "This estimator was later generalized in order to calibrate large-scale distributed MIMO networks [15], [13].", "The estimation problem is formulated as constrained least-squares (LS) problem where the objective function uses channel measurements from a set of arbitrary antenna pairs of the network.", "The generality of this approach spurred many publications dealing with particular cases [16], [17], [18].", "Parallel work in mutual coupling based calibration was also conducted in [12].", "An estimator for the calibration coefficients, which enables maximum ratio transmission (MRT), was proposed for BS antenna arrays with special properties.", "Although it appears that over-the-air reciprocity calibration only involving the BS side is feasible, some matters need further investigation.", "Firstly, the approaches available in the literature for co-located BSs are not of great practical convenience.", "They either rely on antenna elements that need to be (carefully) placed in front of the BS antenna array solely for calibration purposes [14], or are only available for a restrictive case of antenna arrays [12].", "Secondly, most estimators for calibration have been derived from empirical standpoints, e.g., [12], [14], and respective extensions [15], [17], [18].", "It is not clear how far from fundamental estimation performance bounds, or how close to Maximum likelihood (ML) performance, such estimators are.", "Thirdly, most available calibration approaches are proposed for narrow-band systems.", "Such systems bandwidths are usually defined by the frequency selectivity of the propagation channel, which is typically much smaller than the frequency selectivity of the transceiver responses.", "This results in similar calibration coefficients for adjacent narrowband channels.", "Thus, it is of interest to model the statistical dependency of such calibration coefficients, and provide means to exploit this dependency in order to reduce the calibration error across frequency.", "Lastly, there is little publicly available work on validation of massive MIMO calibration schemes.", "The need for validation is high, as it helps answering many questions of practical nature.", "For example, [19] raises the question whether the channel reciprocity assumption holds when strong coupling between BS antennas exist, and [20] questions if calibration assumptions similar to the ones used in this work, hold for massive MIMO arrays.", "Below, we summarize the main contributions of this work.", "We propose a convenient calibration method mainly relying on mutual coupling between BS antennas to calibrate its non-reciprocal analog front-ends.", "We make no assumptions other than channels due to mutual coupling being reciprocal.", "We show that the narrow-band calibration coefficients can be estimated by solving a joint penalized-ML estimation problem.", "We provide an asymptotically efficient algorithm to compute the joint solution, which is a particular case of the EM algorithm.", "We validate our calibration method experimentally using a software-defined radio massive MIMO testbed.", "More specifically, we verify how the measured Error-Vector-Magnitude (EVM) of the downlink equalized signals decreases as the calibration accuracy increases, in a setup where three closely spaced single-antenna users are spatially multiplexed by one hundred BS antennas.", "We propose a non-white Gaussian model for the narrow-band calibration error based on the properties of the proposed estimator, and partially validate this model with measurements." ], [ "Notation", "The operators $(\\cdot )^$ , $(\\cdot )^{ T}$ , $(\\cdot )^{ H}$ , and $(\\cdot )^{ \\dagger }$ denote element-wise complex conjugate, transpose, Hermitian transpose, and Moore-Penrose pseudo-inverse, respectively.", "The element in the $n$ th row and $m$ th column of matrix $\\bf A$ is denoted by $\\big [ {\\bf A} \\big ]_{n,m}$ .", "The operator $\\mathrm {E}\\left\\lbrace \\cdot \\right\\rbrace $ denotes the expected value.", "$\\operatorname{Re}\\left\\lbrace \\cdot \\right\\rbrace $ and $\\operatorname{Im}\\left\\lbrace \\cdot \\right\\rbrace $ return the real and imaginary part of their arguments.", "The matrix ${\\bf I}$ denotes the identity matrix, and ${\\rm diag}\\left\\lbrace a_1, a_2, \\hdots a_M \\right\\rbrace $ denotes an $M \\times M$ diagonal matrix with diagonal entries given by $a_1, a_2, \\hdots , a_M$ .", "The operator $\\ln $ denotes the natural logarithm.", "The set of the complex numbers and the set containing zero and the real positive numbers are denoted by $\\mathbb {C}$ and $\\mathbb {R}_{\\ge 0}$ , respectively.", "The operator $\\setminus $ denotes the relative set complement.", "Finally, $\|\|\\cdot \|\|$ denotes the Frobenius norm." ], [ "Paper Outline", "The remaining sections of the paper are as follows.", "Section presents the signal models.", "Section introduces the state-of-the-art estimator for the calibration coefficients, proposes a novel estimator, and provides a comparative analysis by means of MSE and downlink sum-rate capacities.", "Section validates the proposed calibration method experimentally.", "Using the estimated calibration coefficients obtained from the experiments, the purpose of Section is twofold: i) it studies several aspects of the calibration coefficients across $4.5$ MHz of transceiver bandwidth, ii) it proposes a model for the calibration error of a narrowband system.", "Lastly, Section summarizes the key takeaways from this work." ], [ "Signal Models ", "This section starts by introducing the uplink and downlink signal models, and shows how downlink precoding can be performed using calibrated uplink channel estimates.", "Finally, it models the channels between BS antennas which we use for calibration purposes." ], [ "Uplink and Downlink Signal models ", "Let $K$ single-antenna users simultaneously transmit a pilot symbol in the uplink of a narrow-band MIMO system (e.g., a particular sub-carrier of an OFDM-MIMO system).", "Collecting the pilot symbols in the vector ${\\bf p} = [p_1 \\cdots p_K]^{ T}$ , the received signal by an $M$ -antenna base station can be written as ${\\bf y}_{\\rm UP} & = {\\bf H}_{\\rm UP} \\, {\\bf p} + {\\bf w} \\nonumber \\\\& = {\\bf R}_{\\rm B} {\\bf H}_{\\rm P} {\\bf T}_{\\rm U} \\, {\\bf p}+ {\\bf w}.$ In (REF ), the matrix ${\\bf R}_{\\rm B}~=~{\\rm diag}\\left\\lbrace \\!", "r^{\\rm B}_{1},\\cdots , r^{\\rm B}_{M} \\!\\right\\rbrace $ models the hardware response of $M$ BS receive RF chains (one RF chain per antenna), and the matrix ${\\bf T}_{\\rm U}~=~{\\rm diag}\\left\\lbrace \\!", "t^{\\rm U}_{1},\\cdots ,t^{\\rm U}_{K} \\!\\right\\rbrace $ models the hardware response of $K$ transmit RF chains (one chain per user).", "${\\bf H}_{\\rm P}$ is the propagation channel matrix, ${\\bf H}_{\\rm UP}$ is the, so-called, uplink radio channel, and ${\\bf w}$ is a vector modeling uplink noise.", "Under the reciprocal assumption of the propagation channel, the received downlink signal can be written as ${\\bf y}_{\\rm DL} = & {\\bf H}_{\\rm DL} \\, {\\bf z}{\\rm ^{\\prime }} + {\\bf w}{\\rm ^{\\prime }} \\nonumber \\\\= & {\\bf R}_{\\rm U} {\\bf H}_{\\rm P}^{ T} {\\bf T}_{\\rm B} \\, {\\bf z}{\\rm ^{\\prime }}+ {\\bf w}{\\rm ^{\\prime }}.$ In (REF ), the matrix ${\\bf R}_{\\rm U}~=~{\\rm diag}\\left\\lbrace \\!", "r^{\\rm U}_{1},\\cdots , r^{\\rm U}_{K} \\!\\right\\rbrace $ models the hardware response of the receive RF chains of the $K$ users, and the matrix ${\\bf T}_{\\rm B}~=~{\\rm diag}\\left\\lbrace \\!", "t^{\\rm B}_{1},\\cdots ,t^{\\rm B}_{M} \\!\\right\\rbrace $ models the hardware response of $M$ BS transmit RF chains.", "The entries of $\\bf w{\\rm ^{\\prime }}$ model downlink noise, ${\\bf H}_{\\rm DL}$ is the downlink radio channel, and ${\\bf z}{\\rm ^{\\prime }}$ is a vector with linearly precoded QAM symbols.", "In particular, ${\\bf z}{\\rm ^{\\prime }}=\\bf Px$ , where $\\bf P$ is the precoding matrix, and the entries of $\\bf x$ contain QAM symbols." ], [ "Calibration Coefficients ", "Assume that an error free version of the uplink radio channel, ${\\bf H}_{\\rm UP}$ , is available at the BS.", "The transpose of the result of pre-multiplying ${\\bf H}_{\\rm UP}$ with the matrix $\\alpha {\\bf T}_{\\rm B}{\\bf R}_{\\rm B}^{-1}$ , where $\\alpha \\in \\mathbb {C}\\setminus 0$ and $r_m \\ne 0, \\forall \\; m$ , is a matrix ${\\bf G}$ that, if used for precoding purposes by means of a linear filtering, is sufficient for spatially multiplexing terminals in the downlink with reduced crosstalk.", "This can be visualized by expanding ${\\bf G}$ as ${\\bf G} = & \\left( \\left( \\alpha {\\bf T}_{\\rm B}{\\bf R}_{\\rm B}^{-1} \\right) { \\bf H}_{\\rm UP} \\right)^{ T} \\nonumber \\\\= & \\; \\alpha {\\bf T}_{\\rm U} {\\bf H}_{\\rm P}^{ T} {\\bf T}_{\\rm B} \\nonumber \\\\= & \\; \\alpha {\\bf T}_{\\rm U} {\\bf R}_{\\rm U}^{-1} {\\bf H}_{\\rm DL}.$ From (REF ) we have that ${\\bf G}$ is effectively the true downlink radio channel ${\\bf H}_{\\rm DL}$ pre-multiplied with a diagonal matrix with unknown entries accounting for the user terminals responses $ {\\bf T}_{\\rm U} {\\bf R}_{\\rm U}^{-1}$ , and $\\alpha $ .", "The row space of ${\\bf G}$ is thus the same as of the downlink radio channel ${\\bf H}_{\\rm DL}$ .", "This is a sufficient condition to cancel inter-user interference if, for example, ZF precoding is used (i.e., ${\\bf H}_{\\rm DL} {\\bf G}^\\dagger $ is a diagonal matrix).", "From (REF ), it can also be seen that any non-zero complex scalar $\\alpha $ provides equally good calibration.This follows since both magnitude and phase of $\\alpha $ are not relevant in this calibration setup.", "The former holds since any real scaled channel estimate provides the same precoder matrix $\\bf P$ , if the precoder has a fixed norm.", "The latter follows from (REF ), since the (uniform phases of the) diagonal entries of $ {\\bf T}_{\\rm U} {\\bf R}_{\\rm U}^{-1}$ are unknown to the precoder in this calibration setup.", "Thus, the matrix ${\\bf C} = & { \\rm diag\\lbrace c_1,\\cdots ,c_M \\rbrace } \\nonumber \\\\= & {\\bf T}_{\\rm B}{\\bf R}_{\\rm B}^{-1}$ is the, so-called, calibration matrix, and $\\left\\lbrace c_m\\right\\rbrace $ are the calibration coefficients which can be estimated up to a common complex scalar $\\alpha $ .", "We remark that, although not strictly necessary to build estimators, the concept of a reference transceiver [14] can be used to deal with the ambiguity of estimating $\\left\\lbrace c_m\\right\\rbrace $ up to $\\alpha $ .Explained briefly, assuming $c_\\textit {ref}=1$ and solving for $\\left\\lbrace c_m\\right\\rbrace \\setminus c_\\textit {ref}$ , where $c_\\textit {ref}$ is the calibration coefficient associated with a reference transceiver.", "The remainder of the paper deals with estimation aspects of $c_m= t^{\\rm B}_m/r^{\\rm B}_m$ .", "Thus, for notational simplicity, we write $t_m=t^{\\rm B}_m$ , $r_m=r^{\\rm B}_m$ , ${\\bf R} = {\\bf R}_{\\rm B}$ , and ${\\bf T} = {\\bf T}_{\\rm B}$ .", "Also, we stack $\\left\\lbrace c_m\\right\\rbrace $ in the vector ${\\bf c} = [c_1 \\cdots c_M]^{ T}$ , for later use." ], [ "Inter-BS Antennas Signal model ", "To estimate the calibration coefficients $c_m$ we sound the $M$ antennas one-by-one by transmitting a sounding signal from each one and receiving on the other $M-1$ silent antennas.", "Let the sounding signal transmitted by antenna $m$ be $s_m = 1, \\forall \\; m$ , unless explicitly said otherwise.", "Also, let $y_{n,m}$ denote the signal received at antenna $n$ when transmitting at antenna $m$ .", "It follows that the received signals between any pair of antennas can be written as $\\left[ \\begin{array}{cc} y_{n,m}\\\\ y_{m,n} \\end{array} \\right] = h_{n,m} \\left[ \\begin{array}{cc} r_{n} t_{m} & 0 \\\\ 0 & r_{m} t_{n} \\end{array} \\right] \\left[ \\begin{array}{cc} s_{m} \\\\ s_{n} \\end{array} \\right] + \\left[ \\begin{array}{cc} n_{n,m} \\\\ n_{m,n} \\end{array} \\right],$ where $h_{n,m} = & \\; \\bar{h}_{n,m} + \\tilde{h}_{n,m} \\\\= & \\; \|\\bar{h}_{n,m}\| \\exp ( j2\\pi \\phi _{n,m} ) + \\tilde{h}_{n,m}\\;$ models the (reciprocal) channels between BS antennas.", "The first term $\\bar{h}_{n,m}$ describes a channel component due to mutual coupling between antenna elements, often stronger for closely spaced antennas, which we lay down a model for in Sec.", "REF .", "The terms $\|\\bar{h}_{n,m}\|$ and $\\phi _{n,m}$ denote the magnitude and phase of $\\bar{h}_{n,m}$ , respectively.", "The term $\\tilde{h}_{n,m}$ , which absorbs all other channel multipath contributions except for the mutual coupling (e.g., reflections by scatterers in front of the BS) is modeled by an i.i.d.", "zero-mean circularly symmetric complex Gaussian random variable with variance $\\sigma ^2$ .", "Non-reciprocal channel components are modeled by $r_m$ and $t_m$ which materially map to the cascade of hardware components, mainly in the analog front-end stage of the receiver and transmitter, respectively.", "We assume i.i.d.", "circularly symmetric zero-mean complex Gaussian noise contributions $n_{m,n}$ with variance $N_0$ .", "Letting $\\big [ {\\bf Y} \\big ]_{m,n} = y_{m,n}$ , the received signals can be expressed more compactly as $\\bf Y = R H T + N.$ Note that ${\\bf H} = {\\bf H}^{ T}$ is assumed, and the diagonal entries in the $M \\times M$ matrix $\\bf Y$ are undefined." ], [ "Modeling Mutual Coupling", "The purpose of this section is to provide a model for the mutual coupling between antenna elements, i.e.", "$\\bar{h}_{m,n}$ , as a function of their distance.", "Instead of pursuing a circuit theory based approach to model the effect of mutual coupling [19], our modeling approach uses S-parameter measurements from a massive MIMO BS antenna array [21].", "We note that this model is used only for simulation purposes, and not to derive any of the upcoming estimators of $\\bf c$ ." ], [ "Test Array Description", "The antenna array considered for modeling is a 2-dimensional planar structure with dual-polarized patch elements spaced by half a wavelength.", "More information about the antenna array can be found in [22].", "The dimensional layout of the array adopted for this work corresponds to the $4 \\times 25$ rectangular grid in the upper part of the array shown in Fig.", "REF .", "Only one antenna port is used per antenna element.", "For a given antenna, the polarization port is chosen such that its adjacent antennas - the antennas spaced by half wavelength - are cross-polarized.", "This setting provides, so-called, polarization diversity, and reduces mutual coupling effects between adjacent antennas since co-polarized antennas couple stronger [21].", "Figure: The massive MIMO lab setup used throughout this work.", "The BS is on the left side where a \"T\" shaped antenna array can be seen.", "Three closely spaced user antennas stand the middle of the picture." ], [ "Modeling coupling gains between antennas", "The channel magnitude $\|\\bar{h}_{n,m}\|$ between several pairs of cross and co-polarized antennas were measured in an anechoic chamber using a Vector Network Analyzer, at 3.7 GHz - the center frequency of the array.", "Fig.", "REF shows the measured channel magnitudes.", "Different channel magnitudes for the very same measured distance and polarization cases, are due mostly to the relative orientation of the antenna pair with respect to their polarization setup.", "For example, vertically (co-)polarized antennas couple more strongly when they are oriented horizontally.", "A linear LS fit was performed to model the coupling gain $\|\\bar{h}_{n,m}\|$ as a function of antenna distance.", "The phase $\\phi _{m,n}=\\phi _{n,m}$ is modeled uniformly in $[0,1]$ , as a clear dependence with distance was not found.", "Figure: Measured coupling magnitudes \|h ¯ n,m \|\|\\bar{h}_{n,m}\| between different antenna pairs.", "The circles corresponds to measurements between co-polarized antenna elements, and the crosses between cross polarized antenna elements.", "The variable d\\rm d corresponds to the physical distance between antenna elements.", "The straight lines represent the corresponding linear LS fits." ], [ "Estimation of the Calibration Coefficients", "In this section we deal with estimation aspects of the calibration matrix ${\\bf C=TR}^{-1}$ .", "We introduce the state-of-art estimator of $\\bf C$ [15], [13], and propose a novel iterative penalized-ML estimator.We note that the only assumption used to derive the estimators is ${\\bf H} = {\\bf H}^{ T}$ .", "The generality of this assumption allows the estimators to be used in other calibration setups than those of co-located MIMO systems, as it will be pointed out later.", "A comparative numerical analysis is made by means of MSE and sum-rate capacity.", "We conclude the section with two interesting remarks." ], [ "The Generalized Method of Moments estimator ", "Calibration of large-scale distributed MIMO systems using a similar system model to (REF ) was performed in [13] and [15].In their work, $h_{m,n}$ denotes the propagation channel between antennas of different BSs.", "The reciprocal model adopted for $h_{m,n}$ accounts for large-scale and small-scale fading.", "Based on the structure of the system model, the authors identified that $\\mathrm {E} \\left\\lbrace y_{n,m}c_n - y_{m,n}c_m \\right\\rbrace = 0 .$ Define $ g_{m,n} \\triangleq y_{n,m}c_n - y_{m,n}c_m $ , and ${ g({\\bf c}) }= \\left[ g_{1,2} \\dots g_{1,M} \\, g_{2,3} \\dots g_{2,M} \\dots g_{M-1,M} \\right] ^T$ .The dependency of $ g({\\bf c})$ on $y_{n,m}$ is explicitly left out, for notational convenience.", "An estimator for $\\bf c$ was proposed by solving $\\hat{c}_{\\rm GMM} = \\arg \\min _{ \\begin{array}{c}c \\\\ s.t.\\; f_c(c)=1 \\end{array}} { g^{ H}({\\bf c}) W g({\\bf c}})$ with $\\bf W = I$ .", "Two constraints were suggested to avoid the all-zero solution, namely $f_c(c)=c_1$ or $f_c(c)=\|\|{\\bf c} \|\|^2$ .", "By setting the gradient with respect to ${\\bf c}$ to zero, an estimator in closed-form was given.", "Next, we provide a few remarks on this estimation approach.", "A fact not identified in [13] and [15], is that this estimator is an instance of a estimation framework widely used for statistical inference in econometrics, namely the generalized method of moments (GMM).", "The variable $g_{m,n}$ - whose expectation is zero - is termed a moment condition within GMM literature [23].", "With a proper setting of the weighting matrix $\\bf W$ , it can be shown that the solution to ($\\ref {eq:GMMeCost}$ ) provides an estimator that is asymptotically efficient[23].", "However, no such claim can be made in the low signal-to-noise (SNR) regime, where an optimal form of $\\bf W$ is not available in the literature.", "This typically leads to empirical settings of $\\bf W$ , e.g., $\\bf W = I$ .", "As a result, moment conditions comprising measurements with low SNR constrain the performance since they are weighted equally.", "It thus appears that an inherent problem of the GMM estimator is the selection of $\\bf W$ .", "Nevertheless, it provides a closed-form estimator based on a cost function where nuisance parameters for calibration, as $h_{m,n}$ , are conveniently left out." ], [ "Joint Maximum Penalized-Likelihood estimation", "Here we address joint maximum penalized-likelihood estimation for $\\bf c$ and for the equivalent channel $\\bf \\Psi \\triangleq RHR$ .", "Noting that (REF ) can be written as $\\bf Y = & \\; \\bf R H R C + N \\nonumber \\\\= & \\; \\bf \\Psi C + N,$ the optimization problem can be put as $\\bf [ \\hat{c}, \\; \\hat{\\Psi }] & = \\arg \\max _{ \\bf c, \\Psi } \\ln p( {\\bf Y \| C, \\Psi } ) + {\\rm Pen}({\\bf C,\\Psi },\\epsilon ^{\\prime }) \\nonumber \\\\& = \\arg \\min _{ {\\bf c}, \\Psi } J_{\\rm ML}( {\\bf Y, C, \\Psi },\\epsilon )$ with $J_{\\rm ML}( {\\bf Y, C, \\Psi },\\epsilon ) = \|\| {\\bf Y - \\Psi C } \|\|^2 + {\\rm Pen}({\\bf C,\\Psi },\\epsilon ).$ Here, $p( {\\bf Y \| C, \\Psi })$ denotes the probability density function (PDF) of $\\bf Y$ conditioned on $\\bf C$ and $\\bf \\Psi $ , and ${\\rm Pen}({\\bf C,\\Psi },\\epsilon )$ is a penalty term parametrized by $\\epsilon =\\epsilon ^{\\prime } N_0$ with $\\epsilon \\in \\mathbb {R}_{\\ge 0}$ .", "There are many uses for the penalty term in ML formulations [24].", "Here, we use it mainly to control the convergence rate of the algorithm (presented in Sec.", "REF ), and use $\\epsilon $ as a tuning parameter.", "With this in mind, we pursue Ridge Regression and set the penalty term asRidge Regression [25] is an empirical regression approach widely used in many practical fields, e.g., Machine Learning [24], as it provides estimation robustness when the model is subject to a number of degeneracies.", "This turns out to the case in this work, and we point out why this occurs later.", "However, we emphasize that the main reason of adding the penalty terms is to control the convergence of the algorithm, which we also point out later why this is the case.", "To finalize, we parametrize the penalty term (REF ) with a single parameter in order simplify the convergence analysis and be able to extract meaningful insights.", "${\\rm Pen}({\\bf C,\\Psi },\\epsilon ) = \\epsilon ( \|\| {\\bf C}\|\|^2 + \|\|{\\bf \\Psi }\|\|^2 ).$ After some re-modeling, a vectorized version of (REF ) can be written as $\\bf \\widetilde{Y} = \\bf \\Psi _{\\rm eq}({\\bf \\tilde{\\Psi }}) c + \\widetilde{N},$ or as $\\bf {\\bf Y^{\\prime }} = \\bf C_{\\rm eq}({\\bf c }) {\\bf \\tilde{\\Psi }} + N^{\\prime },$ where ${\\bf \\tilde{\\Psi }}$ stacks all $\\psi _{n,m}=[{\\bf \\Psi }]_{n,m}$ into an $ (M^2-M)/2 \\times 1$ vector, and $\\bf \\Psi _{\\rm eq}({\\bf \\tilde{\\Psi }})$ and $\\bf C_{\\rm eq}({\\bf c })$ are equivalent observation matrices which are constructed from ${\\bf \\tilde{\\Psi }}$ and $\\bf c$ , respectively.", "The structure of these matrices is shown in Appendix A, but it can be pointed out that $\\bf \\Psi _{\\rm eq}({\\bf \\tilde{\\Psi }})$ and $\\bf C_{\\rm eq}({\\bf c })$ are a block diagonal, where each block is a column vector.", "From (REF ), it is seen that for a given $\\bf \\bf C_{\\rm eq}({\\bf c })$ , the penalized-ML estimator of ${\\bf \\tilde{\\Psi }}$ is given byThe factor 2 in the regularization term of (REF ) appears since $\\psi _{m,n}=\\psi _{n,m}$ .", "Note that $\\epsilon $ is considered as a constant during the optimization, otherwise it is obvious that $\\epsilon =0$ minimizes (REF ).", "${\\bf \\tilde{\\Psi }}_{\\rm ML} = \\Big ( {\\bf C}_{\\rm eq}^{ H}({\\bf c}) {\\bf C_{\\rm eq}(\\bf c)} + 2 \\epsilon {\\bf I} \\Big )^{-1} {\\bf C}_{\\rm eq}^{ H}(\\bf c){\\bf Y^{\\prime }},$ If in (REF ), we replace ${\\bf \\tilde{\\Psi }}$ by its estimate ${\\bf \\tilde{\\Psi }}_{\\rm ML}$ , then the penalized ML solution for $\\bf c$ is 2Collumn ${\\hat{{\\bf c}}_{\\rm ML}} = \\arg \\min _{ {\\bf c} } \|\| { \\bf Y^{\\prime } - \\bf C_{\\rm eq}({\\bf c })}&\\Big ( {\\bf C}_{\\rm eq}^{ H}({\\bf c}) {\\bf C_{\\rm eq}(\\bf c)} + 2 \\epsilon {\\bf I} \\Big )^{-1} \\nonumber \\\\& \\times {\\bf C}_{\\rm eq}^{ H}({\\bf c}){\\bf Y^{\\prime }} \|\|^2,$ ${\\hat{{\\bf c}}_{\\rm ML}}=\\arg \\min _{ {\\bf c} } \|\|{ \\bf Y^{\\prime } - \\bf C_{\\rm eq}({\\bf c })}\\Big ( {\\bf C}_{\\rm eq}^{ H}({\\bf c}) {\\bf C_{\\rm eq}(\\bf c)} + 2 \\epsilon {\\bf I} \\Big )^{-1} {\\bf C}_{\\rm eq}^{ H}({\\bf c}){\\bf Y^{\\prime }} \|\|^2,$ It is possible to further simplify $(\\ref {eq:EqModel5})$ for the case of unpenalized ML estimation ($\\epsilon =0$ ) and attack the optimization problem with gradient-based methods [26].", "We have implemented the conjugate gradient method in a Fletcher-Reeves setting with an optimized step-size through a line-search.", "However, this turns out to be far less robust than, and computationally more expensive to, the method provided next.", "Therefore we omit to provide the gradient in closed form." ], [ "An EM Algorithm to find the joint Penalized-ML Estimate", "Here we provide a robust and computational efficient algorithm to find the joint penalized-ML estimate of $\\bf c$ and $\\bf \\Psi $ .", "Instead of pursuing an approach similar to the one used to reach (REF ), the algorithm has its roots in the joint solution found by setting the gradient of $ J_{\\rm ML}( {\\bf Y, C, \\Psi ,\\epsilon })$ to zero.", "Before presenting the algorithm, we therefore briefly address this gradient approach.", "Each entry of ($\\ref {eq:EqModel}$ ) is given by $y_{n,m} = \\psi _{n,m}c_m + n_{n,m}$ .", "The derivative of $J_{\\rm ML}( {\\bf Y, C, \\Psi ,\\epsilon })$ with respect to $c_m^$ is given by $\\frac{ \\partial J_{\\rm ML}( {\\bf Y, C, \\Psi ,\\epsilon }) }{ \\partial c_m^* } = \\epsilon c_m + \\sum \\limits _{\\begin{array}{c}n =1 \\\\ n\\ne m\\end{array}}^{M} \|\\psi _{n,m}\|^2 c_m - y_{n,m}\\psi ^_{n,m}.", "$ Setting (REF ) to zero and solving for $c_m$ yields $c_m = \\left( \\epsilon + \\sum \\limits _{\\begin{array}{c}n =1 \\\\ n\\ne m\\end{array}}^{M} \|\\psi _{n,m}\|^2 \\right)^{-1} \\sum \\limits _{\\begin{array}{c}n =1 \\\\ n\\ne m\\end{array}}^{M} \\psi ^_{n,m} y_{n,m} ,$ which can be expressed in a vector form as $\\hat{{\\bf c}}_{\\rm ML} = \\Big ( {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf \\tilde{\\Psi }} ) {\\bf \\Psi }_{\\rm eq}({\\bf \\tilde{\\Psi }} ) + \\epsilon {\\bf I} \\Big )^{-1} {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf \\tilde{\\Psi }} ) { \\bf \\widetilde{Y}}.$ In a similar fashion, setting the derivative of $J_{\\rm ML}( {\\bf Y, C, \\Psi ,\\epsilon })$ with respect to $\\psi _{n,m}^$ to zero and solving for $\\psi _{n,m}$ provides $\\psi _{n,m} = \\left( \|c_{n}\|^2 + \|c_{m}\|^2 +2 \\epsilon \\right)^{-1} \\left( y_{m,n} c^_{n} + y_{n,m} c^_{m} \\right),$ which can be expressed in a vector form as (REF ).", "Equations (REF ) and (REF ) show the analytical form for each entry of the penalized-ML vector estimates, which will prove to be useful during the complexity analysis.", "Combining the results from (REF ) and (REF ) yield the joint solution $\\begin{bmatrix}{ \\hat{ \\bf c }}_{\\rm ML} \\\\{\\bf \\tilde{\\Psi }}_{\\rm ML}\\end{bmatrix}=\\begin{bmatrix}\\Big ( {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf {\\bf \\tilde{\\Psi }}_{\\rm ML}} ) {\\bf \\Psi }_{\\rm eq}({\\bf {\\bf \\tilde{\\Psi }}_{\\rm ML}} ) + \\epsilon {\\bf I} \\Big )^{-1} {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf {\\bf \\tilde{\\Psi }}_{\\rm ML}} ) { \\bf \\widetilde{Y}} \\\\\\Big ( {\\bf C}_{\\rm eq}^{ H}({\\hat{{\\bf c}}_{\\rm ML} }) {\\bf C_{\\rm eq}({\\hat{{\\bf c}}_{\\rm ML} })} + 2 \\epsilon {\\bf I} \\Big )^{-1} {\\bf C}_{\\rm eq}^{ H}({\\hat{{\\bf c}}_{\\rm ML} }){\\bf Y^{\\prime }}\\end{bmatrix}$ The particular structure of (REF ) suggests that a pragmatic approach for solving can be pursued.", "More specifically, (REF ) can be separated into two sub-problems, i.e., solving for $ { \\hat{ {\\bf c}}}_{\\rm ML} $ and ${\\bf \\tilde{\\Psi }}_{\\rm ML}$ separately.", "Since each of the solutions depend on previous estimates, the joint solution can be computed iteratively, by sequentially solving two separate regularized LS problems, given an initial guess.", "Since each iteration estimates $\\bf c$ and ${\\bf \\tilde{\\Psi }}$ separately, this approach can be seen as an instance of the EM algorithm [27], where the - often challenging - Expectation step is performed by estimating only the first moment of the nuisance parameters $\\left\\lbrace \\psi _{m,n}\\right\\rbrace $ .", "The convergence of the algorithm can be analyzed using standard methods, such as a distance between consecutive point estimates.", "The GMM estimator can be used to compute a reliable initial guess for iteration - in contrast to a purely random initialization.", "This is often good practice to ensure convergence to a suitable local optimum since $J_{\\rm ML}( {\\bf Y, C, \\Psi },\\epsilon )$ is not a convex function of its joint parameter space.", "For sake of clarity, Algorithm 1 summarizes the proposed iterative procedure.", "Observe that $\\epsilon $ , i.e.", "the penalty term parameter in (REF ), ends up regularizing both matrix inversions.", "This is of notable importance from two points-of-view: i) from an estimation (robustness) point-of-view, since the matrices to be inverted are constructed from parameter estimates (and thus are subject to estimation errors) and no favorable guarantee exists on their condition number, e.g., see (REF ).", "ii) from a convergence point-of-view, as it is well-known that the convergence rate of regularized LS adaptive filters is inversely proportional to their eigenvalue spread [28]; This property combo justifies why Ridge Regression was pursued in the first place.", "[t] Expectation-Maximization [1] Measurement matrix $\\bf Y$ , convergence threshold $\\Delta _{\\rm ML}$ , penalty parameter $\\epsilon $ , initial guess $\\hat{\\bf c}$ Initialization: set $\\Delta = \\delta $ where $\\delta > \\Delta _{\\rm ML} $ $ \\Delta \\ge \\Delta _{\\rm ML} $ ${\\bf \\tilde{\\Psi }}_{\\rm ML} = \\Big ( {\\bf C}_{\\rm eq}^{ H}({\\bf \\hat{c}}) {\\bf C_{\\rm eq}(\\bf \\hat{c})} + 2 \\epsilon {\\bf I} \\Big )^{-1} {\\bf C}_{\\rm eq}^{ H}(\\bf \\hat{c}) { \\bf Y^{\\prime }}$ $\\hat{{\\bf c}}_{\\rm ML} = \\Big ( {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf \\tilde{\\Psi }}_{\\rm ML} ) {\\bf \\Psi }_{\\rm eq}({\\bf \\tilde{\\Psi }}_{\\rm ML} ) + \\epsilon {\\bf I} \\Big )^{-1} {\\bf \\Psi }^{ H}_{\\rm eq}({\\bf \\tilde{\\Psi }}_{\\rm ML} ) { \\bf \\widetilde{Y}}$ $\\Delta = \|\| \\hat{{\\bf c}}_{\\rm ML} - \\hat{{\\bf c}} \|\|^2$ $\\bf \\hat{c}=\\hat{{\\bf c}}_{\\rm ML}$ Calibration coefficients estimate $\\hat{{\\bf c}}_{\\rm ML}$ A side remark regarding an application of the EM algorithm follows.", "We highlight that the calibration coefficients $\\bf c$ and the equivalent channels $\\psi _{m,n}=r_m h_{m,n} r_n$ are jointly estimated.", "As previously mentioned, this a feature is not present in the GMM estimator.", "Noticeably, this feature makes the EM algorithm robust and hence very suitable to calibrate distributed MIMO systems since channel fading (i.e., high variations of $\|h_{m,n}\|$ ) often occurs [13].", "As mentioned in Sec.", "REF , the system model used can be also representative to that of distributed systems." ], [ "Complexity Analysis ", "The complexity of each iteration of Algorithm REF is dominated by steps 3 and 4.", "Fortunately the block diagonal structure of the equivalent matrices allows for the inversions to be of reduced complexity, as detailed next.", "From (REF ), each calculation of $\\psi _{m,n}$ requires a few multiplications and additions.", "Since $\\big (M^2 - M\\big )/2$ such calculations are needed to compute (REF ), the complexity order of step 3 is $\\mathcal {O}(M^2)$ .", "Similarly, the complexity of step 4 is $\\mathcal {O}(M^2)$ which can be seen directly from (REF ).", "The explanation of the $\\mathcal {O}(M^2)$ behavior is that the complexity of each calibration coefficient $c_m$ is $\\mathcal {O}(M)$ , and $M$ such calibration coefficients need to be computed.", "Overall, each iteration of the EM algorithm is of complexity $\\mathcal {O}(M^2)$ , and the algorithm's complexity is $\\mathcal {O}(N_{\\rm ite} \\, M^2)$ , with $N_{\\rm ite}$ being the number of iterations needed for convergence.", "The number of iterations needed for convergence is studied in Sec.", "REF .", "As for the GMM estimator, the closed-form solutions presented in [13] and [15] have complexity orders of $\\mathcal {O}(M^3)$ , as they consist of an inverse of a Hermitian matrix of size $M-1$ , and of the eigenvector associated with the smallest eigenvalue of a Hermitian matrix of size $M$ .", "On a practical note, we remark that the computational complexity of both approaches does not stand as a prohibitive factor for BS arrays using hundreds or even several thousands of antennas.", "This is because calibration typically needs to be performed on a hourly basis [14], [22]." ], [ "Simulation setup for the MSE analysis", "We simulate reciprocity calibration over a $4 \\times 25$ rectangular array as the one in Fig.", "REF .", "The linear regression parameters obtained in Sec.", "REF are used to model the coupling gains $\\bar{h}_{m,n}$ .", "The $m$ th transceiver maps to the antenna in row $a_{\\rm row}$ and column $a_{\\rm col}$ of the array as $m= 25 (a_{\\rm row}-1) + a_{\\rm col}$ .", "The reference transceiver index is set to $\\textit {ref}=38$ , as it is associated with one of the most central antenna elements of the 2-D array.", "The Cramér-Rao Lower Bound (CRLB) is computed to verify the asymptotical properties of the estimators' error [27].", "From (REF ) and (REF ), it can be seen that if $\\bar{h}_{m,n}$ is assumed to be known, the PDF of $\\bf Y$ conditioned on $\\bf R$ and $\\bf T$ is a multivariate Gaussian PDF.", "This makes the CRLB of $\\bf c$ to have a well known closed-form, which is computed in Appendix B.", "The transmitter $t_m$ and receiver $r_m$ gains are set to $ t_m = ( 0.9 + \\frac{0.2 m}{M} \\exp (-j2 \\pi m/M) ) / t_\\textit {ref}$ and $ r_m = ( 0.9 + \\frac{0.2 (M-m)}{M} \\exp (j2 \\pi m/M) ) / r_\\textit {ref}$ , respectively.", "We used this deterministic setting for the transceivers, as it allows for a direct comparison of the parameter estimates' MSE with the CRLB.", "Moreover, this setting incorporates eventual mismatches within the transceivers complex amplitude which are in line with the magnitude variations measured from the transmitters/receivers of our testbed, i.e., spread of around 10-percent around the mean magnitude (and uniform phase).", "This spread is in line with transceiver models adopted in other calibration works [13].", "The variance $\\sigma ^2$ of the multipath propagation contribution during calibration is set to $-60$ dB.", "Our motivation for this value is as follows.", "If the closest physical scatter to the BS is situated, say, 15 meters away, then by Friis' law [29] we have a path loss of around $ 10 \\log _{10}( \\frac{4 \\pi d }{\\lambda } ) = 10 \\log _{10}( \\frac{4 \\pi (2\\times 15m) }{ 3 \\times 10^8/(3.7 \\times 10^9) } ) = 73$ dB per path.", "This number does not account for further losses due to reflections and scattering.", "Based on this, we use $-60$ dB as the power (variance) of the resulting channel stemming from a large number of such uncorrelated paths.", "For consistency with the reference antenna concept used in the CRLB computations, the MSE of the EM algorithm output $\\hat{{\\bf c}}_{\\rm ML}$ , is defined as $\\mathrm {MSE}_m & = \\mathrm {E} \\left\\lbrace \| c_m - \\left[ \\hat{{\\bf c}}_{\\rm ML} \\right]_{ m,1} / \\left[ \\hat{{\\bf c}}_{\\rm ML}\\right]_{ \\textit {ref},1} \|^2 \\right\\rbrace ,$ since the estimated \"reference\" coefficient $\\left[ \\hat{{\\bf c}}_{\\rm ML}\\right]_{ \\textit {ref},1}$ is not necessarily equal to 1.", "This is because the concept of reference antenna is not used by the EM algorithm.", "As for the GMM estimator, the constraint provided in [15] is adopted, i.e., $c_\\textit {ref}=1$ in (REF ), which is already coherent with the computed CRLB.", "The results are averaged over 1000 Monte-Carlo simulations, and the threshold $\\Delta _{\\rm ML}$ is set to $10^{-6}$ which, based on our experience, ensures that convergence is reached in many parameter settings.", "The initial guess for the EM algorithm is produced by the GMM estimator." ], [ "Estimators' MSE vs CRLB", "Fig.", "REF compares the MSE of the estimators with the CRLB for two transceiver cases.", "Both estimators appear to be asymptotically efficient.", "Noticeably, the performance gains of the EM algorithm can be grossly superior to the GMM (up to 10 dB), as it approaches the CRLB at much smaller values of $N_0$ .", "As mentioned previously, this is mainly because the GMM estimator does not appropriately weight moment conditions with less quality.", "Two remarks about the CRLB itself are now in place.", "i) As mentioned in Appendix B, the assumptions used during the CRLB computations, could result in an underestimated CRLB.", "Indeed, the results in Fig.", "REF suggest that the assumptions used during the CRLB computations do not affect its final value since the estimators' MSE asymptotically converges to the computed CRLB.", "This is convenient since (asymptotically) efficient estimators can still be built with limited information.", "ii) It was assumed that $\\phi _{m,n}$ - the phase of $\\bar{h}_{m,n}$ - is known during the CRLB computations, although it is originally modeled as a random variable in Sec.REF .", "However, if $\\phi _{m,n}$ is assumed to be known, the CRLB is independent of the value of $\\phi _{m,n}$ .", "This is because a phase rotation in $\\mu _{n,m}$ , does not influence (REF ), due to the structure of $\\Sigma ^{-1}$ .", "Thus, any realization of $h_{m,n}$ - from the model proposed in Sec.", "REF - provides the same CRLB result.", "From the previous two remarks and standard estimation theory [27], it follows that the (narrowband) calibration error - in the high SNR regime - produced by the studied estimators can be well modeled as a multivariate zero-mean Gaussian distribution with covariance matrix given by the transformed inverse Fisher information matrix, found in (REF ).", "The Gaussianity of the calibration error is further verified (experimentally) in Sec.", "REF .", "Figure: MSE of the GMM estimator and the EM algorithm (with ϵ=0\\epsilon =0), versus their CRLB (solid line), for 2 extreme transceiver cases.", "Namely, a transceiver associated with an antenna at the edge of the array, and a transceiver associated with an antenna adjacent to the reference.", "The CRLB plotted by a dashed line is discussed in Sec.", "." ], [ "Convergence of the EM algorithm", "The convergence is analyzed for $N_0=-40$ dB, which from Fig.", "REF appears to be a region where EM-based estimation provides significant gains compared to GMM.", "Fig.", "REF illustrates the role played by the regularization constant $\\epsilon $ in terms of convergence rate and MSE.", "Noticeably, the higher $\\epsilon $ the faster the algorithm appears to converge.", "The number of iterations until convergence $N_{\\rm ite}$ is seen to be much smaller than $M$ with large enough $\\epsilon $ (i.e., around 5 iterations when $\\epsilon =0.1$ ).If, instead, the initial guess is chosen randomly (e.g., calibration coefficients with unit-norm and i.i.d.", "uniform phases) then our simulations indicate that the order of $N_{\\rm ite}$ is $\\mathcal {O}(M)$ .", "However, increasing $\\epsilon $ indefinitely is not an option as it degrades the performance.", "Moreover, the results also indicate that proper tuning of $\\epsilon $ can provide MSE gains compared to the unregularized case which is asymptotically efficient (notice that this does not conflict with the CRLB theorem, as an estimator built with $\\epsilon \\ne 0$ is not necessarily unbiased).", "This was - to some extent - expected due the benefits of Ridge Regression as discussed in Sec.REF .", "With that, we identify that a fine tuning of $\\epsilon $ can provide many-fold improvements.", "We note that in the literature there is a number of approaches available that deal with optimization of regularization constants in standard (non-iterative) LS problems [24].", "However, they are not directly applicable to this work as they typically optimize single error metrics, and are in general computationally expensive.", "Here, our main use for $\\epsilon $ is to accelerate the convergence and provide estimation robustness to the algorithm, all achieved at no complexity cost.", "For this matter, we treat $\\epsilon $ as a hyperparameter (an approach widely adopted in regularized LS adaptive filtering [28]).", "Further investigation on fully automatizing the EM algorithm is an interesting matter of future work.", "For the remainder of the paper, we set $\\epsilon = 0$ and proceed accordingly, for simplicity.", "Figure: MSE per iteration of the EM algorithm, for different regularization constants ϵ\\epsilon .", "The plots are for N 0 =-40N_0=-40 dB, and the remaining simulation settings are the same as Fig.", ".", "Note the different scales of the plots." ], [ "Simulation Setup for Sum-rate Capacity Analysis ", "The same parameter setting as in Sec.", "REF is kept in this setup, and the remaining simulation framework is defined next.", "We assume that the uplink channel ${\\bf H}_{\\rm UP}$ is perfectly know to the BS, and that there are two noise sources in the system.", "The first noise source is downlink additive noise modeled by $\\bf w^{\\prime }$ , see (REF ).", "Here, $\\bf w^{\\prime }$ have i.i.d.", "zero-mean circularly symmetric complex Gaussian distributed random entries with variance $N_w$ equal to 1.", "The same model is used for the entries of the downlink channel matrix ${\\bf H}_{\\rm DL}$ .", "The second noise source is the error during estimation of $\\bf c$ (i.e., calibration error).", "With that, the precoded signal ${\\bf z^{\\prime }} = {\\bf P x}$ is subject to calibration errors.", "The transmit power constraint $\\mathbb {E}{\\left\\lbrace \|\|{\\bf z^{\\prime }}\|\|^2\\right\\rbrace }= K$ is used.", "Also, we set $K=10$ single antenna users, and assume $t^U_k = t^B_k$ and $r^U_k = r^B_k$ for sake of simplicity.", "The sum-rate capacities [30] are evaluated for different calibration cases.", "More specifically, when no calibration is employed (i.e., $\\hat{c}_m =1$ ), when calibration is performed with the GMM or the EM algorithm, for the case of perfect calibration (i.e., $\\hat{c}_m = c_m$ ), and as a baseline, when precoding is performed using the true downlink channel ${\\bf H}_{\\rm DL}$ .", "The analysis is performed with $N_0=-40$ dB, for the reasons mentioned during the convergence analysis." ], [ "Sum-rate Capacity Results", "Fig.", "REF shows the obtained sum-rates cumulative distribution functions (CDFs) for different precoding schemes [2].", "Similarly to the MSE results, EM-based calibration provides significant gains compared to the GMM case.", "The magnitude of these gains obviously depend on both the calibration (and communication system) setup.", "For example, there are no sum-rate differences when $N_0 \\rightarrow 0$ or $N_0 \\rightarrow \\infty $ , as both GMM and EM approaches converge to that of perfect calibration, or to the uncalibrated case, respectively.", "Thus, it in only in a certain region of $N_0$ values that EM based calibration provides gains.Our analysis based on a wide range of parameter values also indicates that, in general, stricter calibration requirements need to be met in order to release the full potential of ZF compared to MRT precoding (i.e., no sum-rate difference compared to the perfect calibrated case).", "Noticeably, this observation is in line with previous calibration studies [31].", "It is interesting that - for this setup - there is no fundamental loss in capacity between this calibration approach (i.e., precoding with perfectly calibrated uplink CSI) and precoding with the true downlink CSI.", "Quantifying this loss is out of scope of this work, however, the interested reader is referred to [32] for an overview on the loss of different types of reciprocity calibration.", "We now finalize the section with two interesting remarks." ], [ "Remark 1: Calibration with Reduced Measurement Sets ", "There are several benefits of using a reduced measurement set for calibration (e.g., by only relying on high quality measurements).", "This is possible as long as (REF ) is not under-determined.", "As an illustrative example, the dashed line in Fig.", "REF shows the CRLB when a reduced measurements set - comprising the measurements between antenna pairs whose elements are distanced by at most $1/\\sqrt{2}$ wavelengths - is used.", "The number of measurement signals in this case drops from $M(M-1)$ to less than $8M$ , since one antenna signals to, at most, 8 other antennas.", "The performance loss turns out to be insignificant, i.e.", "2 dB for the neighbor case and 4 dB for the edge case, considering the number of signals discarded.", "This indicates that the channels between neighbor antennas, which are dominated by mutual coupling, are the most important for calibration.", "Thus, there is an interesting trade-off between the asymptotic performance of an estimator and its computational complexity (proportional to the number of measurements).", "Another benefit of using reduced measurement sets is a possible reduction of resource overhead dedicated for calibration.", "This can be very important from a system deployment point-of-view.", "To finalize, we remark that ML closed form estimators can be also reached when reduced measurement sets are used.", "This can be the case for the current (general) calibration setup when a reduced set of measurements is used, or for the case of working with a full set of measurements when the calibration setup is a special case.", "An example of the latter is given next." ], [ "Remark 2: Closed-form Unpenalized ML Calibration for Linear arrays ", "Consider an $M$ -antenna linear array, and let $m$ index the antennas in ascending order starting at one edge of the linear array.", "Assume that mutual coupling only exists between adjacent antenna elements, and that the channel between any other antenna pairs is weak enough so that it can be neglected without any noticeable impact on performance.", "We summarize our findings in Proposition 1.", "Proposition 1: Using a reference antenna as a starting point, say $c_{1}=1$ , the unpenalized ML solution for any $c_{\\ell +1}$ , with $1\\le \\ell \\le M-1$ , can be obtained sequentially by $\\hat{c}_{\\ell +1} = \\hat{c}_{\\ell }\\frac{ y^_{\\ell +1,\\ell } y_{\\ell ,\\ell +1} }{ \|y_{\\ell +1,\\ell }\|^2 }.$ 2Collumn Proof: See Appendix C. Proof: See Appendix C. We can also deduce the following interesting corollary.", "Corollary 1: For any of the two constraints considered in (REF ), the GMM (vector) estimator coincides with (REF ) up to a common complex scalar.", "2Collumn Proof: See Appendix C. Proof: See Appendix C." ], [ "Validation of the calibration method in a massive MIMO testbed", "In this section, we detail the experiment performed to validate the proposed mutual coupling based calibration method.", "More specifically, we implemented it in a software-defined radio testbed, and performed a TDD transmission from 100 BS antennas to 3 single antenna terminals.", "Note that the analysis conducted in this section and in Sec.", "is measurement based.", "As stationarity is assumed in the analysis, we monitored the system temperature throughout the measurements and verified no significant changes.", "We also made an effort to keep static propagation conditions, and performed the experiments at late hours in our lab with no people around." ], [ "Brief Description of the Testbed", "Here we briefly outline the relevant features of the testbed for this work.", "Further information can be found in [22].", "The BS operates with 100 antennas, each antenna connected to one distinct transceiver.", "For simplicity, the same transceiver settings (e.g., power amplifier gain and automatic gain control) are used in both calibration and data communication stages for all radio units.", "This ensures that the analog front-ends yield the same response during both stages, thus the estimated calibration coefficients are valid during the communication stage." ], [ "Synchronization of the radios", "Time and Frequency synchronization is achieved by distributing reference signals to all radio units.", "However, this does not guarantee phase alignment between all BS transceiver radio chains which motivates reciprocity calibration.", "Once the measurements to construct the observation matrix $\\bf Y $ are performed, $\\bf c$ is estimated using the unpenalized EM algorithm.", "The following sequence of events is then performed periodically:" ], [ "Uplink Channel Estimation and Calibration", "Users simultaneously transmit frequency orthogonal pilot symbols.", "The BS performs LS-based channel estimation, and interpolates the estimates between pilot symbols.", "Reciprocity calibration is then performed independently per subcarrier, i.e.", "as in (REF ), for coherence purposes with Sec.", ".", "This calibrated version of the downlink channel is then used to construct a ZF precoder." ], [ "Downlink channel estimation and data transmission", "Downlink pilot symbols are precoded in the downlink and each user performs LS-based channel estimation.", "Using the estimates, each user recovers the payload data using a one-tap equalizer.", "We note that 4-QAM signaling per OFDM sub-carrier is used for uplink channel estimation and data transmission.", "The main parameters are shown in Table REF .", "Further information on the signaling protocol (e.g., uplink/downlink frame structure or uplink pilot design) is found on [22]." ], [ "Measurement Description", "The setup used in our experiments is shown in Figure REF .", "Although not being a typical propagation scenario found in cellular systems, this extreme setup - closely located users under strong line-of-sight conditions - requires high calibration requirements to be met if spatial separation of users is to be achieved.", "In addition, we use ZF precoding as it is known to be very sensitive to calibration errors [32].", "The EVM [33] of the downlink equalized received samples at each mobile station was evaluated, and used as performance metric for validation purposes.", "The rationale is that, with multiple mobile terminals, calibration errors are translated into downlink inter-user interference (and loss of array gain), which increases the EVM.", "Letting $r$ be the downlink equalized received sample when symbol $s$ is transmitted, the EVM is defined as $\\mathrm {EVM} = \\mathrm {E}\\left\\lbrace \\frac{ \|r-s\|^2 }{ \|s\|^2}\\right\\rbrace ,$ where the expectation is taken over all system noise sources (e.g., hardware impairments and thermal noise).", "Our estimate of (REF ) was obtained by averaging realizations of $\|r-s\|^2 / \|s\|^2$ over all OFDM sub-carriers and over received OFDM symbols.", "We estimated the EVM for different energy values of the uplink pilots and calibration signals.", "We do so in order to be able to extract insightful remarks for the analysis of the results.", "In particular, letting $E_{\\rm Pilot}=\\mathrm {E} \\left\\lbrace p_k p_k^* \\right\\rbrace $ in (REF ) denote the energy of the uplink pilot, which, for simplicity, is the same for all users, and let $E_{\\rm Cal}$ denote the energy of the sounding signal $s_{m}$ in (REF ), we estimated the EVM for a 2-dimensional grid of $E_{\\rm Pilot}$ and $E_{\\rm Cal}$ .", "The results reported next are given with respect to the relative energies $Er_{\\rm Pilot}= E_{\\rm Pilot} / E^{\\rm max}_{\\rm Pilot} $ and $Er_{\\rm Cal}= E_{\\rm Cal} / E^{\\rm max}_{\\rm Cal}$ , where $E^{\\rm max}_{\\rm Pilot}$ and $ E^{\\rm max}_{\\rm Cal}$ are the maximum energies of the uplink pilot and calibration signal used in the experiments.", "Other systems parameters (e.g., transmit power in the downlink) were empirically set and kept constant throughout the experiment." ], [ "Validation Results", "Fig.", "REF shows the measured EVMs for the 3 user terminals in our experiment.", "Before discussing the results, we remark that analyzing the EVM when $Er_{\\rm Cal}$ is reduced beyond $-30$ dB is not of fundamental interest, as it approaches the uncalibrated case (where high EVMs are to be expected).", "Overall, a positive trend is observed with increasing $Er_{\\rm Cal}$ until $-10$ dB.", "This reflects the BS ability of spatially separating users which increases with increasing the calibration quality.", "The fact that downlink EVMs down to $-10$ dB are achieved, which are much smaller than the EVMs when $Er_{\\rm Cal}= -30$ dB, i.e.", "close to the uncalibrated case, motivates our validation claim.", "It is possible to observe a saturation of the EVMs at high enough $Er_{\\rm Cal}$ and $Er_{\\rm Pilot}$ for all user cases.", "This is an expected effect in practical systems.", "Explained briefly, system impairments other than the calibration or the uplink channel estimation error, become the dominant error sources that bound the EVM performanceMobile terminals error sources (e.g., in-phase and quadrature imbalance or thermal noise) qualify for such impairments.", "For a given downlink transmit power, it is straightforward to understand how such impairments bound the downlink EVMs regardless of the calibration and uplink estimation quality..", "Remarkably, this saturation effect implies that the calibration SNR - available in a practical array as ours - is sufficiently large not to be the main impairment to constrain the system performance.", "Mutual coupling channels are thus reliable (and reciprocal enough), so that they can be used for signaling in order to calibrate the system.We note there exists an interesting theoretical trade-off between the calibration quality and the capacity of downlink channels with respect to the strength of mutual coupling.", "In practice, the proposed calibration method can be used in compact antenna arrays with very low coupling (say $-30$ dB between adjacent elements) provided that the transmit power during calibration is sufficient to provide good enough estimation SNR.", "In such a setup, the impact of coupling in the capacity is negligible.", "Figure: Measured EVM at each of the three user terminals during a massive MIMO downlink transmission." ], [ "Aspects of Wideband Calibration and Error Modeling", "A short summary of this section follows.", "Using the measurements from the Sec.", ", we treat the estimated calibration coefficients across OFDM sub-carriers as realizations of a discrete stochastic process.", "Using low rank approximation theory, we propose a parametrized low dimensional basis that characterizes the subspace spanned by this process accurately.", "Based on the reduced basis, we propose a wideband estimator that averages out the calibration error across frequency.", "Using the wideband estimator results, we validate the narrowband calibration error model proposed in Sec.", "REF .", "We remark that our experiment makes use of a bandwidth of $ F_s N_{\\rm sub} / N_{\\rm FFT} = 4.5$ MHz." ], [ "Wideband Remarks for the Calibration Coefficients", "Denote the calibration coefficient of BS antenna $m$ at the $k$ th OFDM sub-carrier as $C_m[k] = t_m^k / r_m^k $ .", "The variable $\\hat{C}_m [k]$ is the estimate of $C_m[k]$ at sub-carrier $k$ - obtained, e.g., with the EM algorithm - and is modeled as $\\hat{C}_m [k] = & C_m[k] + E_m[k] \\nonumber \\\\=& \|C_m[k]\|\\exp (j2\\pi \\zeta _m[k]) + E_m[k]$ where $E_m[k]$ is an i.i.d.", "random process representing the calibration error which is assumed zero-mean and independent of $C_m[k]$ .", "Let the random phasor process $\\exp (j2\\pi \\zeta _m[k])$ in ($\\ref {eq:CCsMod}$ ) absorb the phase shift stemming from the arbitrary time that a local oscillator needs to lock to a reference signal.", "Such phase shift is often modeled as uniformly distributed, and thus $\\mathrm {E}\\left\\lbrace \\exp (j2\\pi \\zeta _m[k])\\right\\rbrace = 0.$ Moreover, since local oscillators associated with different transceivers lock at arbitrary times, it is safe to assume $\\mathrm {E}\\left\\lbrace \\exp (j2\\pi \\zeta _m[k_1]) \\exp (-j 2\\pi \\zeta _n[k_2]) \\right\\rbrace = 0, \\; m\\ne n.$ Not making further assumptions on the statistics of $\\hat{C}_m [k]$ , we now proceed with a series expansion, but before doing so we make one last remark.", "The series expansion conducted next is performed based on measurements from the 100 testbed transceivers, and serves as an example approach to obtain a suitable basis for $\\hat{C}_m [k]$ .", "This can well apply to mass-production transceiver manufactures that can reliably estimate the statistical properties of the hardware produced.", "However, as our testbed operates with relatively high-end transceivers - compared to the ones expected to integrate commercial massive MIMO BSs - the dimensionality of the subspace verified in our analysis might be underestimated.", "Intuitively, the higher transceiver quality, the less basis functions are needed to accurately describe $\\hat{C}_m [k]$ .", "Nevertheless, the upcoming remarks apply for smaller bandwidths - than $4.5$ MHz - depending on the properties of the transceivers." ], [ "Principal Component Analysis", "From the assumption (REF ), it follows that the element at the $v_1$ th row and $v_2$ th column of the covariance matrix ${\\bf K}_m$ of $\\hat{C}_m [k]$ is defined as $[{\\bf K}_m]_{[v_1,v_2]}= \\mathrm {E}\\left\\lbrace \\hat{C}_m[v_1] \\; \\hat{C}^_m[v_2] \\right\\rbrace .$ From the assumption (REF ), it follows that the principal components of $\\hat{C}_m[k]$ are obtained by singular value decomposition (SVD) of ${\\bf K}_m$ only [34].", "Let the SVD of ${\\bf K}_m$ be written as ${\\bf K}_m = \\sum _{i=1}^{\\rm N_{SUB}} {\\bf u}^{ m}_i \\lambda ^{ m}_i ({\\bf u}^{ m}_i)^{ H},$ where $\\left\\lbrace {\\bf u}^m_i \\right\\rbrace _{i=1}^{\\rm N_{SUB}}$ are the principal components, and $ { \\lambda }^m_i $ is the power (variance) of the coefficient obtained from projecting $\\hat{C}_m [k]$ into ${\\bf u}^m_i$ .", "We use the convention $ \\lambda ^m_1 \\ge \\lambda ^m_2 \\dots \\ge \\lambda ^m_{N_{\\rm SUB}}$ , and $ {\\bf u}^m_i = \\big [ [{u}^m_i[1], \\cdots , {u}^m_i[N_{\\rm SUB}] \\big ]^{\\rm T} $ .", "Figure: Principal component and coefficients of C ^ m [k]\\hat{C}_m [k].", "Left) The 10 strongest normalized singular values for 20 transceivers; Middle) Magnitude of the principal component for 3 transceivers; Right) Phase of the principal component for 3 transceivers.Fig.", "REF shows several coefficients and basis functions of the expansion, that were estimated based on 100 realizations of $\\hat{C}_m [k]$ , each measured with $Er_{\\rm Cal} = 5$ dB (which from Fig.", "REF provides a relatively high calibration SNR).", "Noticeably, it appears that all processes (one per transceiver) live mostly in a one-dimensional sub-space and thus can be well described by their first principal component ${\\bf u}^{ m}_1$ .", "This fact also indicates that the contribution of the calibration error in the expansion is small, and thus the first principal component of $\\hat{C}_m [k]$ is also representative for the true coefficients $C_m [k]$ .", "Visual inspection indicates that both magnitude and phase of the first principal component can be well approximated with a linear slope across frequency.", "The inherent error of this approximation is very small compared to the magnitude of the process itself.", "We note that this linear trend holds for any transceiver of the array (not only for the ones shown in Fig.", "REF )." ], [ "Wideband Modeling and Estimation", "The previous analysis indicates that any first principal component can be well described by a linear magnitude slope $\\gamma _m$ , and a linear phase $\\xi _m$ across frequency.", "Such properties are well captured by the Laplace kernel $\\exp ( (\\gamma _m + j2\\pi \\xi _m)k ),$ for small values of $\|\\gamma _m\|$ (since the range of $k$ is finite).", "The final parameter to model a realization of the process is the complex offset $A_m$ .", "With that, the general model (REF ) can thus be re-written as $\\hat{C}_m[k] = A_m\\exp ( (\\gamma _m + j2\\pi \\xi _m)k ) + w_m[k],$ where $w_m[k]$ is a random process that absorbs: the calibration error $E_m[k]$ , the error due to the low rank approximation, and the error due to the linear modeling of the first principal component ${\\bf u}^{ m}_1$ .", "Given an observation $\\lbrace \\hat{C}_m[k]\\rbrace _{k=1}^{ N_{\\rm SUB}}$ , the ML estimator of $A_m$ , $\\xi _m$ and $\\gamma _m$ , namely, $\\hat{A}_m$ , $\\hat{\\xi }_m$ and $\\hat{\\gamma }_m$ is straightforward to derive [27].", "Thus, we define the wideband estimator of $\\hat{C}_m[k]$ as $\\hat{C}_m[k]^{ \\rm WB} = \\hat{A}_m\\exp ( (\\hat{\\gamma }_m + j2\\pi \\hat{\\xi }_m)k ).$ For illustration purposes, a realization of the ML wideband estimator $\\hat{C}_m[k]^{\\rm WB}$ is contrasted with that of the narrow-band estimator $\\hat{C}_m[k]$ in Fig.", "REF .", "The obtained error reduction is evident.", "Figure: A realization of the narrow-band estimator C ^ m [k]\\hat{C}_m[k], and the proposed wideband estimator C ^ i [k] WB \\hat{C}_i[k]^{\\rm WB}." ], [ "A Model for the Calibration Error", "Here, we use the wideband estimator results to verify the Gaussianity of the narrow-band calibration error proposed in Sec.", "REF .", "This is done under the two following main assumptions.", "1) The residual process $ E_m[k] = \\hat{C}_m[k] - C_m[k]$ is well described by $ \\hat{E}_m[k] = \\hat{C}_m[k] - \\hat{C}_m[k]^{\\rm WB}$.", "This is reasonable if $ \\mathrm {E}\\left\\lbrace \| \\hat{C}_m[k]^{\\rm WB} - C_m[k] \|^2 \\right\\rbrace \\ll \\mathrm {E}\\left\\lbrace \| \\hat{C}_m[k] - C_m[k] \|^2 \\right\\rbrace $ .", "To justify, the estimation gains scale linearly in the number of realizations [27], which is $ N_{\\rm SUB}=1200$ in this case.", "Assuming that: the estimation error is independent across realizations, the underlying model (REF ) describes the first principal component well, and the low rank approximation error is minuscule, there are gains of $10\\log _{10}N_{\\rm SUB}\\approx 30$ dB which justify the first main assumption.", "2) The residual process $ E_m[k] $ is ergodic.Ergodicity is necessary since each (independent) measurement of $\\hat{C}_m[k]$ takes about ten minutes with our test system (due to the locking time of the local oscillator to the reference signal).", "As potential system temperature drifts during the measurements can result in varying statistical properties, it is safer to perform the analysis based on one solely realization of $E_m[k]$ .", "This is met if $E_m[k]$ is stationary and the ensemble of $N_{\\rm SUB}$ samples is representative for statistical modeling.", "The former holds for small OFDM bandwidths (e.g., $4.5$ MHz) as the hardware impairments do not vary significantly across the band.", "The latter is also met, as we have $N_{\\rm SUB}=1200$ narrow-band estimators whose estimated errors $\\lbrace \\hat{E}_m[k] \\rbrace _{k=1}^{N_{\\rm SUB}}$ were found to be mutually uncorrelated.", "Fig.", "REF shows the empirical CDF of both real and imaginary parts of $\\lbrace \\hat{E}_m[k] \\rbrace _{k=1}^{N_{\\rm SUB}}$ - which we found to the uncorrelated - for two transceiver cases.", "Each of the empirical CDFs is contrasted with a zero-mean Gaussian distribution of equal variance.", "Overall, the empirical CDFs for both transceivers resemble a Gaussian CDF extremely well.", "The Gaussianity of the calibration error was further verified by passing a Kolmogorov-Smirnov test with $0.05$ significance level [35].", "We note that these observations hold not only for the two transceivers in Fig.", "REF , but for all transceivers of the array.", "Noticeably, the empirical distribution of the calibration error is in line with the asymptotic properties of ML estimators, i.e.", "the error can be modeled by an additive zero-mean Gaussian multivariate.", "The final element for a full characterization is its covariance matrix, relating the errors across antennas.", "A good approximation (at high SNR) is the inverse of the transformed Fisher Information matrix in (REF ).", "Noticeably, future calibration works can benefit from the convenience of safely assuming a non-white Gaussian calibration error.", "Figure: Empirical CDFs for the real and imaginary parts of the calibration error, for a transceiver at the edge of the array, and for an adjacent transceiver to the reference antenna.", "A Gaussian CDF of equal variance is plotted for both cases for comparison." ], [ "Conclusions", "We have proposed and validated a convenient calibration method which rely on mutual coupling to enable the reciprocity assumption in TDD massive MIMO systems.", "We verified that in a practical antenna array, the channels due to mutual coupling are reliable and reciprocal enough, so that they can be used for signaling in order to calibrate the array.", "The iterative ML algorithm is asymptotically efficient and outperforms current state-of-the-art estimators in an MSE and sum-rate capacity sense.", "Further improvements - in terms of MSE and convergence rate - can be harvested by proper tuning of its regularization hyperparameter.", "The calibration error can be further reduced by proper averaging over the radio bandwidth.", "More importantly, it did not stand as the main impairment to constraint the performance of the system, from our experiments.", "Our measurements also verified that the narrow-band calibration error (at high SNR) is Gaussian distributed, which is coherent with the theory of the estimator proposed.", "The convenience of safely assuming a non-white Gaussian calibration error can, hopefully, open the door for future analytical studies of calibrated TDD massive MIMO systems." ], [ "Acknowledgments", "This work was funded by the Swedish foundation for strategic research SSF, VR, the strategic research area ELLIIT, and the E.U.", "Seventh Framework Programme (FP7/2007-2013) under grant agreement n 619086 (MAMMOET).", "We also thank the Comm.", "Systems group in Bristol University, for letting us replicate several of our results in their testbed.", "Here we show the structure of the equivalent models.", "Define the column vector ${\\bf \\Psi }_{m} = \\left[ \\psi _{1,m} \\dots \\psi _{m-1,m} \\; \\psi _{m+1,m} \\dots \\psi _{M,m} \\right]^T$ .", "The equivalent channel matrix in (REF ) is written as ${\\bf \\Psi _{\\rm eq}({\\bf \\tilde{\\Psi }})} ={\\rm diag}\\left\\lbrace {\\bf \\Psi }_{1}, {\\bf \\Psi }_{2}, \\hdots , {\\bf \\Psi }_{M} \\right\\rbrace .$ Now define ${\\bf \\bar{c}}_{n,m} = [ c_{n} \\; c_{m} ]^T.$ Noting that $\\psi _{m,n} = \\psi _{n,m}$ , the equivalent matrix and the parameter vector in (REF ) are written as ${\\bf C_{\\rm eq}({\\bf c })} = {\\rm diag}\\left\\lbrace {\\bf \\bar{c}}_{1,2},\\cdots , {\\bf \\bar{c}}_{1,M} , {\\bf \\bar{c}}_{2,3}, \\cdots , {\\bf \\bar{c}}_{2,M} , \\cdots \\right\\rbrace ,$ and ${\\bf \\tilde{\\Psi }} = \\begin{bmatrix}\\psi _{2,1} \\dots \\psi _{M ,1} \\; \\psi _{3,2} \\dots \\psi _{M ,2} \\dots \\psi _{M ,M-1}\\end{bmatrix}^T.$ Here we compute the CRLB for the calibration coefficients $\\left\\lbrace c_m\\right\\rbrace \\setminus c_\\textit {ref}$ .", "The exclusion of $ c_\\textit {ref}$ is justified in the end of the calculations.", "This is achieved by assuming $t_{ ref}=r_\\textit {ref}=1$ , and treating $c_\\textit {ref} = t_\\textit {ref}/r_\\textit {ref}$ as known for estimation purposes.", "Define the $(4M-4) \\times 1$ vector ${\\bf \\theta }\\!=\\!\\left[ \\operatorname{Re}\\!\\left\\lbrace t_{1} \\right\\rbrace \\; \\operatorname{Im}\\!\\left\\lbrace t_{1} \\right\\rbrace \\; \\operatorname{Re}\\!\\left\\lbrace r_{1} \\right\\rbrace \\;\\operatorname{Im}\\!\\left\\lbrace r_{1} \\right\\rbrace \\; \\operatorname{Re}\\!\\left\\lbrace t_{2} \\right\\rbrace \\dots \\operatorname{Im}\\!\\left\\lbrace r_{M} \\right\\rbrace \\right]^T,$ where $t_\\textit {ref}$ and $r_\\textit {ref}$ do not enter.", "The CRLB for $\\left\\lbrace c_m\\right\\rbrace \\setminus c_\\textit {ref}$ is given by the diagonal entries of the transformed inverse Fisher information matrix [27] $\\text{var} ( \\hat{c}_m ) \\ge \\left[ \\frac{ q( \\theta ) }{\\partial \\theta } \\textbf {I}^{-1}( \\theta ) \\frac{ q( \\theta )}{\\partial \\theta }^{\\rm H} \\right]_{m,m}, \\; m\\ne \\textit {ref},$ where $\\textbf {I}( \\theta )$ is the Fisher information matrix of $\\theta $ .", "The transformation of $\\theta $ into the calibration coefficients is given by $ q(\\theta ) = \\left[ \\frac{ \\operatorname{Re}\\!\\left\\lbrace t_{1} \\right\\rbrace + j\\operatorname{Im}\\!\\left\\lbrace t_{1} \\right\\rbrace }{ \\operatorname{Re}\\!\\left\\lbrace r_{1} \\right\\rbrace + j \\operatorname{Im}\\!\\left\\lbrace r_{1} \\right\\rbrace } \\dots \\frac{ \\operatorname{Re}\\!\\left\\lbrace t_{M} \\right\\rbrace + j \\operatorname{Im}\\!\\left\\lbrace t_{M} \\right\\rbrace }{ \\operatorname{Re}\\!\\left\\lbrace r_{M} \\right\\rbrace + j \\operatorname{Im}\\!\\left\\lbrace r_{M} \\right\\rbrace } \\right]^T.$ We now compute $\\textbf {I}( \\theta )$ .", "Assuming that $\\bar{h}_{m,n}$ , $\\sigma ^2$ and $N_0$ are at hand,These assumptions are only used for the CRLB calculations, and were not used to derive any of the estimators.", "A possible implication is that the CRLB can be underestimated, but we will see that this is not the case from the simulations' results.", "the mean $\\mu _{n,m}$ and the covariance matrix $\\Sigma _{n,m}$ of ${\\bf y}_{n,m} = [y_{n,m} \\; y_{m,n}]^T$ are given by $\\mu _{n,m} = \\mathrm {E}\\left\\lbrace {\\bf y}_{n,m} \\right\\rbrace = \\bar{h}_{n,m} \\left[ r_nt_m \\; r_mt_n \\right]^T,$ $\\Sigma _{n,m} & = \\mathrm {E}\\left\\lbrace ({\\bf y}_{n,m} - \\mu _{n,m}) ({\\bf y}_{n,m}-\\mu _{n,m})^H \\right\\rbrace \\nonumber \\\\& = \\begin{bmatrix}\|r_n\|^2 \|t_m\|^2 \\sigma ^2 + N_0 & r_n t_m r_m^ t_n^* \\sigma ^2 \\\\r_m t_n r_n^* t_m^* \\sigma ^2 & \|r_m\|^2 \|t_n\|^2 \\sigma ^2 + N_0 \\\\\\end{bmatrix}.$ We can observe that the PDF of ${\\bf Y}^{\\prime \\prime }$ , where ${\\bf Y}^{\\prime \\prime } = \\left[ {\\bf y}^T_{1,2} \\dots {\\bf y}^T_{1,M} \\; {\\bf y}^T_{2,3} \\dots {\\bf y}^T_{2,M} \\dots {\\bf y}^T_{M-1,M} \\right]^T,$ conditioned on $\\bf \\theta $ , follows a multivariate Gaussian distribution, i.e., $p({\\bf Y^{\\prime \\prime }} \\vert \\theta ) \\sim \\mathcal {CN}(\\mu ,\\Sigma )$ , with mean $\\mu = \\left[ \\mu ^T_{1,2} \\dots \\mu ^T_{1,M} \\mu ^T_{2,3} \\dots \\mu ^T_{2,M} \\dots \\mu ^T_{M-1,M} \\right]^T$ and block diagonal covariance $\\Sigma = {\\rm diag}\\left\\lbrace \\Sigma _{1,2}, \\cdots , \\Sigma _{1,M} , \\Sigma _{2,3}, \\cdots , \\Sigma _{2,M} , \\cdots , \\Sigma _{M-1,M} \\right\\rbrace .$ With that, we have $\\left[ \\textbf {I}( \\theta ) \\right]_{i,j} = \\; \\operatorname{Tr}\\left\\lbrace \\Sigma ^{-1} \\frac{ \\partial \\Sigma }{ \\partial \\theta _i } \\Sigma ^{-1} \\frac{ \\partial \\Sigma }{ \\partial \\theta _j } \\right\\rbrace + 2 \\operatorname{Re} \\left\\lbrace \\frac{ \\partial \\mu ^H }{\\partial \\theta _i} \\Sigma ^{-1} \\frac{ \\partial \\mu }{\\partial \\theta _j} \\right\\rbrace ,$ with $1 \\le i \\le (4M-4)$ and $1 \\le j \\le (4M-4)$ .", "The remaining computations of $\\left[ \\textbf {I}( \\theta ) \\right]_{i,j}$ are straightforward and thus omitted.", "We note that without the convention of $t_\\textit {ref}=r_\\textit {ref}=1$ - and thus $\\theta $ is a $4M \\times 1$ vector instead - it can be shown that the map $\\theta \\mapsto \\mu $ is not injective which renders $\\textbf {I}( \\theta )$ not invertible.", "Thus, the convention of reference antenna is needed to be able to compute the CRLB.", "Here we derive the closed-form unpenalized (i.e.", "$\\epsilon =0$ ) ML estimator for the linear array setup described in Sec.", "REF .", "By leaving out the terms that do not depend on $\\bf c$ , it follows that, after a few manipulations, the optimization problem of (REF ) can be written as 2Collumn $\\left\\lbrace \\hat{c}_m \\right\\rbrace = & \\arg \\max _{ {\\bf c} } { {\\bf Y^{\\prime }}^H \\bf C_{\\rm eq}({\\bf c })} {\\bf C}_{\\rm eq}^{ \\dagger }( {\\bf c}){\\bf Y^{\\prime }} \\nonumber \\\\= & \\arg \\max _{ \\left\\lbrace c_m \\right\\rbrace } \\sum _{\\ell =1}^{M-1‎} f_{\\rm L}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell }),$ with $f_{\\rm L}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell }) = {\\bf y}_{\\ell +1,\\ell }^{ H} {\\bf \\bar{c}}_{\\ell ,\\ell +1} {\\bf \\bar{c}}_{\\ell ,\\ell +1}^H {\\bf y}_{\\ell +1,\\ell } / {\\bf \\bar{c}}_{\\ell ,\\ell +1}^H {\\bf \\bar{c}}_{\\ell ,\\ell +1}.$ See (REF ) for structure of ${\\bar{c}}_{\\ell ,\\ell +1}$ , and (REF ) for structure of ${\\bf y}_{m,n}$ .", "$\\left\\lbrace \\hat{c}_m \\right\\rbrace = \\arg \\max _{ {\\bf c} } { {\\bf Y^{\\prime }}^H \\bf C_{\\rm eq}({\\bf c })} {\\bf C}_{\\rm eq}^{ \\dagger }( {\\bf c}){\\bf Y^{\\prime }} = \\arg \\max _{ \\left\\lbrace c_m \\right\\rbrace } \\sum _{\\ell =1}^{M-1‎} f_{\\rm L}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell }),$ with $f_{\\rm L}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell }) = {\\bf y}_{\\ell +1,\\ell }^{ H} {\\bf \\bar{c}}_{\\ell ,\\ell +1} {\\bf \\bar{c}}_{\\ell ,\\ell +1}^H {\\bf y}_{\\ell +1,\\ell } / {\\bf \\bar{c}}_{\\ell ,\\ell +1}^H {\\bf \\bar{c}}_{\\ell ,\\ell +1}.$ See (REF ) for structure of ${\\bar{c}}_{\\ell ,\\ell +1}$ , and (REF ) for structure of ${\\bf y}_{m,n}$ .", "Our ability to solve (REF ) is due to the following property.", "Property 1: For the function $f_{\\rm L}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell })$ , the maximum over $c_{\\ell +1}$ equals $\|\|{\\bf y}_{\\ell +1,\\ell }\|\|^2$ , and thus it does not depend on $c_\\ell $ .", "Hence, the ML estimate of $c_{\\ell +1}$ , i.e.", "$\\hat{c}_{\\ell +1}$ , can be found for a given $c_{\\ell }$ .", "With that, the joint maximization problem (REF ) can be split into $\\hat{c}_{\\ell +1} = \\arg \\max _{ x } f_{\\rm L}(\\hat{c}_\\ell ,x,{\\bf y}_{\\ell +1,\\ell }).$ This optimization is a particular case of the Rayleigh quotient problem, and the solution is given in (REF ) when the reference element (i.e., the starting point) is chosen to be $c_1$ .", "We now provide a short proof for Corollary 1.", "For the case of linear arrays with coupling solely between adjacent antennas, the optimization problem in (REF ) can be written - ignoring any constraint for now - as $\\hat{c}_{\\rm GMM}= \\arg \\min _{ \\begin{array}{c}c \\end{array}} \\sum _{\\ell =1}^{M-1‎} f_{\\rm G}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell })$ where $ f_{\\rm G}(c_\\ell ,c_{\\ell +1},{\\bf y}_{\\ell +1,\\ell })=\|y_{\\ell +1,\\ell } c_{\\ell +1} -y_{\\ell ,\\ell +1} c_{\\ell }\|^2$ .", "We solve (REF ) using the following property.", "Property 2: Letting $\\hat{c}_{\\ell }$ be the ML estimator from (REF ), it follows that $f_{\\rm G}(\\hat{c}_{\\ell } ,\\hat{c}_{\\ell +1},{\\bf y}_{\\ell +1,\\ell } )=0, \\; \\forall \\ell .$ Thus, the GMM solution (under any of the 2 constraints) coincides with that of the ML up to a common complex scalar.", "Uniqueness follows since the GMM cost function is quadratic." ] ]	1606.05156
[ [ "On two pencils of cubic curves" ], [ "Abstract We give two examples of plane curve arrangements of pencil type which are very close to line arrangements, though the action of the monodromy operator on the first cohomology group of the Milnor Fiber has eigenvalues of order 5 and 6, showing that surprising situations can occur for larger classes of curve arrangements than for line arrangements.", "Our computations rely on the algorithm given by A. Dimca and G. Sticlaru which detects the non trivial monodromy eigenspaces of free curves." ], [ "Introduction", "Let $\\mathcal {C} : f = 0$ be a reduced curve of degree $d$ in the complex projective plane $\\mathbb {P}^2_$ Consider the corresponding complement $U = \\mathbb {P}^2_{\\mathcal {C}},$ and the global Milnor fiber $F$ defined by $f(x,y,z) = 1$ in 3 with monodromy action $h: F \\rightarrow F,\\, h(x) = \\exp (2\\pi i/d)\\cdot (x,y,z).$ To determine the eigenvalues of the monodromy operator $h^1 : H^1(F, \\rightarrow H^1(F,$ is a rather difficult problem.", "Assume that ${\\mathcal {C}}$ is a curve arrangement of pencil type, i.e.", "the defining equation has the form $f = q_1q_2 \\cdots q_m,$ for some $m \\ge 3,$ where $\\deg q_1 = \\cdots = \\deg q_m = k \\ge 2$ and the curves ${\\mathcal {C}}_i : q_i = 0$ for $i = 1,\\hdots ,m$ are all members of the pencil ${\\mathcal {P}}: u{\\mathcal {C}}_1 + v{\\mathcal {C}}_2.$ The situation when the curves ${\\mathcal {C}}_i$ are line arrangements was systematically considered, see [10], [11], [14], particularly with a view in understanding whether the monodromy action (REF ) is combinatorially determined.", "We mention just three striking facts in this direction, assuming that ${\\mathcal {C}}_i$ are line arrangements.", "The number $m$ of members of the pencil ${\\mathcal {P}}$ is at most 4, and the Hessian arrangement is the only known such pencil type arrangement with $m = 4,$ see [14].", "For this arrangement the corresponding eigenvalues are the roots of unity of order 4.", "If the line arrangement ${\\mathcal {C}}$ has only double and triple points, then the monodromy operator (REF ) is combinatorially determined, and the corresponding eigenvalues are cubic roots of unity, see [11].", "It is not known whether the monodromy operator (REF ) can have eigenvalues which are not roots of unity of order 3 or 4, see [11].", "On the other hand, it is known that if we consider larger classes of curve arrangements, e.g.", "conic and line arrangements, many new properties can occur.", "For instance, Terao's conjecture about the combinatorial invariance of freeness is open for line arrangements, but false for conic and line arrangements, see [12].", "Our main result, to be stated next, says that a similar situation occurs when looking at the properties 1. and 3. listed above.", "Theorem 1.1 Consider the pencil of cubic curves given by ${\\mathcal {P}}: u{\\mathcal {C}}_1 + v{\\mathcal {C}}_2$ in the complex projective plane $\\mathbb {P}^2_$ where ${\\mathcal {C}}_1 : q_1(x,y,z) = 3xyz + y^3 + z^3 = 0$ and ${\\mathcal {C}}_2 : q_2(x,y,z) = 3xyz + x^3 + z^3 = 0.$ Then the following hold.", "The two curves ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ meet transversally in 9 points, hence the generic member of the pencil ${\\mathcal {P}}$ is smooth.", "The number $m$ of singular members in the pencil ${\\mathcal {P}}$ is 5 and they are the fibers of the points $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}_1$ that are listed as follows: for $(u^{\\prime } : v^{\\prime }) = (1 : 0)$ and $(u^{\\prime } : v^{\\prime }) = (0 : 1)$ we get two nodal irreducible cubics, for $(u^{\\prime } : v^{\\prime }) = (1 : 1)$ we get a union of three concurrent lines, and for $(u^{\\prime } : v^{\\prime }) = (1:-t_i),$ where $i = 1,2$ and $t_i$ 's are the roots of the equation $t^2 + 3t + 1 = 0,$ we get each time a triangle.", "The union of the singular members in the pencil ${\\mathcal {P}}$ is a free curve ${\\mathcal {C}}: f = q_1q_2(q_1- q_2)(q_1^2-3q_1q_2 + q_2^2) = 0$ of degree $d=15$ and with exponents $d_1 = 4$ and $d_2 = 10,$ having 9 lines and two nodal cubics as irreducible components.", "The monodromy operator (REF ) associated to the curve ${\\mathcal {C}}$ has eigenvalues which are roots of unity of order 5.", "More precisely, the Alexander polynomial of the curve ${\\mathcal {C}}$ is given by $\\Delta _{\\mathcal {C}}(t) = (t-1)^7(t^5 -1)^3.$ Remark 1.2 The curve arrangement ${\\mathcal {C}}$ in Theorem REF contains the line arrangement ${\\mathcal {A}}: (q_1 -q_2)(q_1^2 -3q_1q_2 + q_2^2) = 0,$ which is a (3,3)-net, has 9 lines, 10 triple points and 6 nodes.", "Such (3,3)-nets are discussed in [7].", "The next pencil is briefly discussed in [13], see the Pappus arrangement completed in the last section.", "Theorem 1.3 Consider the pencil of cubic curves given by ${\\mathcal {P}}: u{\\mathcal {C}}_1 + v{\\mathcal {C}}_2$ in the complex projective plane $\\mathbb {P}^2_$ where ${\\mathcal {C}}_1 : q_1(x,y,z) = y(x-y-z)(2x+y-z) = 0$ and ${\\mathcal {C}}_2 : q_2(x,y,z) = xz(2x-5y+z) = 0.$ Then the following hold.", "The two curves ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ meet transversally in 9 points, hence the generic member of the pencil ${\\mathcal {P}}$ is smooth.", "The number $m$ of singular members in the pencil ${\\mathcal {P}}$ is 6 and they correspond to the points $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}^1_ that are listed as follows: for $ (u' : v') = (1: 0), (u' : v') = (0 : 1)$ and $ (u' : v') = (1 : 1)$ we get each time a triangle, and for $ (u' : v') = (ti : -1)$ where $ i = 1,2,3$ and $ ti$^{\\prime }s are the roots of the equation $$125t^3 +399t^2 + 339t+1 = 0,$$ we get each time a nodal cubic.\\item The union of the singular members in the pencil $ P$ is a free curve $${\\mathcal {C}}: f = q_1q_2(q_1 -q_2)(125q_1^3 -399 q_1^2 q_2 + 339q_1q_2^2 -1) = 0$$ of degree 18 and with exponents $ d1 = 4$ and $ d2 = 13,$ having 9 lines and three nodal cubics as irreducible components.\\item The monodromy operator (\\ref {eq1}) associated to the curve $ C$ has eigenvalues which are roots of unity of order 6.", "More precisely, the Alexander polynomial of the curve $ C$ is given by$$\\Delta _{\\mathcal {C}}(t)=(t-1)^7(t^6-1)^4.$$$ Remark 1.4 The curve arrangement ${\\mathcal {C}}$ in Theorem REF contains the line arrangement ${\\mathcal {A}}: q_1q_2(q_1 -q_2) = 0,$ which is a (3,3)-net, has 9 lines, 9 triple points and 9 nodes.", "Such (3,3)-nets are discussed in [7].", "Remark 1.5 All the irreducible components of the curve ${\\mathcal {C}}$ in Theorem REF are rational curves, and all the singularities of ${\\mathcal {C}}$ are ordinary multiple points, namely eight of order 2, one of order 3 and nine of order 5.", "In other words, our curve ${\\mathcal {C}}$ is very close to a line arrangement, both globally and locally.", "Note that the sum of the total Milnor numbers of the singular members in ${\\mathcal {P}}$ is $1 + 1 + 4 + 3 + 3 = 12 = 3(3-1)^2,$ as predicted by the theory, see [4], [13].", "For the curve ${\\mathcal {C}}$ in Theorem REF , similar results apply, in particular in this case the sum of the total Milnor numbers of the singular members in ${\\mathcal {P}}$ is $1 + 1 + 1 + 3 + 3 + 3 = 3(3-1)^2.$ All the singularities are weighted homogeneous for obvious reasons, fact which allows us to use the results from [4] and [13].", "The Alexander polynomials of the free curves ${\\mathcal {C}}$ in Theorem REF and Theorem REF can be computed using the algorithm described in [8].", "Remark 1.6 The characteristic polynomials $\\Delta _{\\mathcal {C}}^q(t)$ of the operators $h^q, \\,0\\le q \\le 2,$ are related by the following formula (see [6]) $\\Delta _{\\mathcal {C}}^0(t)\\Delta ^1_{\\mathcal {C}}(t)^{-1}\\Delta ^2_{\\mathcal {C}}(t)=(t^d-1)^{\\chi (U)},$ where $\\chi (U)$ denotes the Euler characteristic of $U.$ Since the singularities are isolated, ${\\mathcal {X}}(U)$ can be easily computed (see for instance [6]) and since the curve ${\\mathcal {C}}$ is reduced, $\\Delta ^0_{\\mathcal {C}}(t)=t-1.$ It follows that the operator $h^2$ is completely determined by $h^1$ and we can reduce to study the Alexander polynomial $\\Delta _{\\mathcal {C}}(t):=\\Delta _{\\mathcal {C}}^1(t).$ Proof of Theorem REF Without loss of generality, one can assume that $x=1,$ and we have that $(1:y:z)$ is solution of the system $S:\\left\\lbrace \\begin{array}{ccc}3yz + y^3 +z^3 & =0 \\\\3yz+ 1 + z^3 & =0\\end{array}\\right.$ implies $y\\in \\lbrace 1,j,j^3\\rbrace ,\\,j=\\exp (2\\sqrt{i}\\pi /3),$ by subtracting the two equations.", "If $y=1$ then $z$ is solution of the equation $E_1: z^3+3z+1=0;$ if $y=j$ then $z$ is solution of the equation $E_2: z^3+3jz+1=0;$ and if $y=j^2$ then $z$ is solution of the equation $E_3: z^3+3j^2z+1=0.$ Denote by $\\mathcal {S}_i$ the solution set of $E_i,$ with $i=1,2,3.$ It is clear that $\\mathcal {S}_i \\cap \\mathcal {S}_j = \\emptyset ,$ for all $i \\ne j$ and the system (REF ) has nine solutions: $\\lbrace (1:1:z),\\,z\\in \\mathcal {S}_1\\rbrace \\cup \\lbrace (1:j:z),\\,z\\in \\mathcal {S}_2\\rbrace \\cup \\lbrace (1:j^2:z),\\,z\\in \\mathcal {S}_3\\rbrace .$ Hence $\|{\\mathcal {C}}_1 \\cap {\\mathcal {C}}_2\| = 9$ and the intersection points of the curves ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ have intersection multiplicity 1, i.e.", "the two curves meet transversally and the base locus of the pencil ${\\mathcal {P}}$ is smooth.", "First recall that the member in the pencil $\\mathcal {P}$ corresponding to the point $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}_{^1 is the curveu\\mathcal {C}_1+v\\mathcal {C}_2,where u=-v^{\\prime } and v=u^{\\prime }.", "If (u^{\\prime }:v^{\\prime })=(1:0), the corresponding curve \\mathcal {C}_2 is an irreductible nodal curve with node (0:1:0).", "Similarily, for (u^{\\prime }:v^{\\prime })=(0:1) we get an irreductible nodal curve {\\mathcal {C}}_1 with node (1:0:0).", "Suppose now that u^{\\prime } and v^{\\prime } are both non zero.", "Without loss of generality, one can assume u^{\\prime }=1 and we have that the corresponding member u{\\mathcal {C}}_1+ {\\mathcal {C}}_2 is singular if and only if the following system admits a solution:\\begin{equation}\\left\\lbrace \\begin{array}{ccc}3(u+1)xyz + uy^3 +(u+1)z^3 + x^3 & =0 & (i)\\\\(u+1)yz+ x^2 & =0 & (ii)\\\\(u+1)xz+uy^2 & =0 & (iii)\\\\(u+1)xy + (u+1)z^2& =0 &(iv)\\end{array}\\right.\\end{equation}}If $ (u':v')=(1:1),$ then $ -C1+C2: x3-y3$ is the union of three concurrent lines:\\begin{center}x=y,x=jy and x=j^2y.\\end{center}Assume now $ (u':v')=(1:v'), v' 1.$ The case $ y=0$ is an absurd, since then $ x=z=0$ with (ii) and (iv).", "If $ y = 1,$ we have $ x=0 z=0$ with (ii) and $ u=0$ with (i), while $ z=0 x=0$ with (ii) and $ u=0$ with (i), which are both in contradiction with our hypothesis $ v'0$.", "Finally, $ (x:1:z), x,z 0,$ is a solution of (\\ref {S1}) if and only if$$\\left\\lbrace \\begin{array}{ccc}3(u+1)xz + u +(u+1)z^3 + x^3 & =0 & (i^{\\prime })\\\\(u+1)z+ x^2 & =0 & (ii^{\\prime })\\\\(u+1)xz+u & =0 & (iii^{\\prime })\\\\x + z^2& =0 &(iv^{\\prime })\\end{array}\\right.$$and we have with $ (iv')$ and $ (iii')$ that $ z3=uu+1,$ which implies with (ii^{\\prime }) that $ u3+3u+1=0.$ Denote by $ ti, i=1,2,$ the solutions of the equation $$t^2+3t+1=0,$$ and by $ i$ a complex cubic root of $ titi+1.$ Then the singularities of the member $ tiC1 + C2$ associated to the point $ (1:-ti)$ are the following three points\\begin{center}(-\\alpha _i^2:1:\\alpha _i),(-j^2\\alpha _i^2:1:j \\alpha _i) and (-j\\alpha _i^2:1:j^2\\alpha _i),\\end{center}which are not colinear and hence form a triangle.\\item Since $ t1+t2=-3$ and $ t1t2=1,$ the union $ C$ of the singular members in the pencil $ P$ listed in 2. is the curve of degree $ d=15$ defined by the homogeneous polynomial $$f = q_1q_2(q_1- q_2)(q_1^2-3q_1q_2 + q_2^2).$$ Since the singularities are all weighted homogeneous, we can apply the results from \\cite {D1} (see Theorem 1.14) or \\cite {Val} (see Theorem 2.7) to show that $ C$ is free with exponents (4,10).\\item Recall that the $ 1-$eigenspace $ H1(F,1$ of the monodromy (\\ref {eq1}) is a pure Hodge structure of type $ (1,1),$ and that the sum of the non trivial eigenspaces $ d=1, 1H1(F,$ is pure of weight 1.", "For any $ k[1,d],$ it is known that there exists a spectral sequence $ E(f)k$ whose first term is constructed from the Koszul complex in $ x,y,z]$ of the partial derivatives of $ f,$ and whose limit $ E(f)k$ gives the action of the monodromy on the graded pieces $ H(F,, =(-2-1k/d),$ with respect to the pole order filtration $ P,$ which contains the Hodge filtration $ F$ and satisfies $ P2=0.$ In \\cite {D3}, A. Dimca and G. Sticlaru showed that the computation of the second terms given in Equation (\\ref {graduate}) is sufficient to detect all the non trivial eigenpaces of the operator (\\ref {eq1}), see \\cite [Theorem 1.2]{D3}.", "More in detail, for any $ k[1,d]$, these second terms are given by$ $E_2^{1,0}(f)_k= Gr_P^1 H^1(F,_{\\lambda },$ where $\\lambda = \\exp (-2\\sqrt{-1}\\pi k/d).$ Furthermore, when the curve is free, the authors also describe an algorithm which computes the dimensions $\\dim E_2^{1,0}(f)_k.$ By using this algorithm and the computer algebra software Singular [2], we get that the only non zero dimensions of second terms of the form (REF ) are listed as follows: $\\dim E_2^{1,0}(f)_6=1,\\,\\dim E_2^{1,0}(f)_9=2,\\,\\dim E_2^{1,0}(f)_{12}=3,$ and $\\dim E_2^{1,0}(f)_{15}=10.$ First, since $H^1(F,_1=H^1(U,$ and the curve $\\mathcal {C}$ is the union of 11 irreductible curves, it is well known that $\\dim H^1(F,_1=b_1(U)=10,$ see for instance [5].", "Using [8] we have that the multiplicities $m(\\lambda )$ of the eigenvalues $\\lambda \\ne 1$ of the monodromy operator (REF ) satisfy: $2 \\le m(\\lambda )\\le 3$ , for $k=6,9;$ $m(\\lambda )= 3$ , for $k=3,12;$ $m(\\lambda )= 0$ , for $k\\notin \\lbrace 3,6,9,12,15\\rbrace .$ The non trivial monodromies listed above are the roots of the unity of order 5.", "Since the monodromy operator $h^1$ is definied over $\\mathbb {Q},$ it is known that $\\Delta _{{\\mathcal {C}}}(t) \\in \\mathbb {Q}[t]$ is a product of cyclotomic polynomials $\\varphi _n$ with $n$ dividing $d=15$ , and it follows that $\\Delta _{\\mathcal {C}}(t)= (t-1)^7(t^5-1)^3.$ Proof of Theorem REF Let us compute explicitely the intersection $\\mathcal {C}_1\\cap \\mathcal {C}_2$ by solving the system of equations $q_1(x,y,z)=q_2(x,y,z)=0.$ If $x=0,$ then we get $(0:0:1),(0:1:1)$ and $(0:-1:1).$ Assume now $x=1.$ The first equation gives $y=0,y=1-z$ or $y=z-2.$ If $y=0,$ then $z=0$ or $-2$ and we get $(1:0:0)$ and $(1:0:-2);$ If $y=1-z,$ then $z=0$ or $\\frac{1}{2}$ gives $(1:1:0)$ and $(2:1:1);$ If $y=z-2,$ then with $z=0$ or $z=3$ we find $(1:-2:0)$ and $(1:1:3).$ Hence $\|{\\mathcal {C}}_1 \\cap {\\mathcal {C}}_2\| = 9$ and ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ meet transversally.", "Let us list all the singular members $u\\mathcal {{\\mathcal {C}}}_1+v \\mathcal {{\\mathcal {C}}}_2$ associated to points $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}_1,$ where $u^{\\prime }=v$ and $v^{\\prime }=-u.$ Let us study separately the cases $(u^{\\prime }:v^{\\prime })=(1:0)$ (a), $(u^{\\prime }:v^{\\prime })=(0:1)$ (b), $(u^{\\prime }:v^{\\prime })=(1:1)$ (c), and $(u^{\\prime }:v^{\\prime })=(u^{\\prime }:-1)$ with $u^{\\prime } \\ne 0,-1$ (d).", "If $(u^{\\prime }:v^{\\prime })=(1:0),$ then the singularities of the associated member $\\mathcal {C}_2$ are the solutions of the sytem $\\left\\lbrace \\begin{array}{ccc}xz(2x-5y+z) & =0 & (i)\\\\z(4x-5y+z) & =0 & (ii)\\\\-5xz & =0 & (iii)\\\\x(2x-5y+2z) & =0 & (iv) \\\\\\end{array}\\right.$ Without loss of generality, one can assume $y=1$ and we have $x=0$ or $z=0$ with $(iii).$ If $x=0,$ then $(ii)\\Rightarrow z=0$ or $z=5$ that is $(0:1:0)$ and $(0:1:5)$ are solutions of (REF ).", "The case $z=0$ and $x=5/2$ implied by $(iv)$ gives the third solution $(5:2:0)$ and we get a triangle.", "Similarily, for $(u^{\\prime }:v^{\\prime })=(0:1)$ the singularities of $\\mathcal {C}_1$ are the solutions of $\\left\\lbrace \\begin{array}{ccc}y(x-y-z)(2x+y-z)& =0 & (i)\\\\y(4x-y-3z) & =0 & (ii)\\\\2x^2-3y^2-2xy-3xz+z^2& =0 & (iii)\\\\-y(3x-2z)& =0 & (iv)\\\\\\end{array}\\right.$ Without loss of generality, one can assume $x=1$ and we have with $(iv)$ that $y=0$ or $z=\\frac{3}{2}.$ If $y=0,$ then $z$ is solution of $z^2-3z+2$ with $(iii)$ and $(1:0:1)$ and $(1:0:2)$ are solutions of (REF ).", "Assume $z=\\frac{3}{2}.$ Then $(i)$ and $(ii)$ both imply $y=0$ or $y=-\\frac{1}{2},$ while equation $(iii)$ implies $y=-\\frac{1}{2}$ or $-\\frac{1}{6}.$ Hence $y=-\\frac{1}{2}$ and $(2:-1:3)$ is the third solution of (REF ), i.e.", "we get once again a triangle.", "If $(u^{\\prime }:v^{\\prime })=(1:1),$ then the singularities of the associated member $\\mathcal {C}_1-\\mathcal {C}_2$ are the solutions of the sytem $\\left\\lbrace \\begin{array}{cc}y(x-y-z)(2x+y-z)-xz(2x-5y+z) & =0 \\\\y(4x-y-3z)-z(4x-5y+z) & =0 \\\\2x^2-3y^2-2xy+2xz+z^2 & =0 \\\\-y(3x-2z)-x(2x-5y+2z) & =0 \\\\\\end{array}\\right.$ Without loss of generality, one can assume $y=1$ and the previous system becomes: $\\left\\lbrace \\begin{array}{ccc}(x-1-z)(2x+1-z)-xz(2x-5+z) & =0 & (i)\\\\4x-1-3z-z(4x-5+z) & =0 & (ii)\\\\2x^2-3-2x+2xz+z^2 & =0 & (iii)\\\\-3x+2z-x(2x-5+2z) & =0 & (iv)\\\\\\end{array}\\right.$ Hence $z(1-x)=x(x-1)$ with $(iv).$ If $x=1,$ then with equations $(ii)$ and $(iii)$ we have that $z^2+2z-3=0,$ which gives the solutions $(1:1:1)$ and $(1:1:-3).$ If $x\\ne 1,$ then $(iv)\\Rightarrow z=-x$ and we get the system $\\left\\lbrace \\begin{array}{cc}(2x-1)(3x+1)+x^2(x-5) & =0 \\\\3x^2+2x-1 & =0 \\\\x^2-2x-3 & =0 \\\\z & =-x \\\\\\end{array}\\right.$ with $(-1:1:1)$ as unique solution.", "Assume $(u^{\\prime }:v^{\\prime })=(u^{\\prime }:-1)$ with $u^{\\prime }\\ne 0,-1.$ Then the associated curve $\\mathcal {C}_1+v\\mathcal {C}_2$ is singular whenever the following system has a solution: $\\left\\lbrace \\begin{array}{cc}y(x-y-z)(2x+y-z)+vxz(2x-5y+z) & =0 \\\\y(2x+y-z)+2y(x-y-z)+vz(2x-5y+z)+2vxz & =0 \\\\(x-2y-z)(2x+y-z)+y(x-y-z) - 5vxz & =0 \\\\-y(2x+y-z)-y(x-y-z)+vx(2x-5y+z) +vxz & =0 \\\\\\end{array}\\right.$ Without loss of generality, one can assume that $y=1,$ that is: $\\left\\lbrace \\begin{array}{ccc}(x-1-z)(2x+1-z)+vxz(2x-5+z) & =0 & (i)\\\\(2x+1-z)+2(x-1-z)+vz(2x-5+z)+2vxz & =0 & (ii)\\\\(x-2-z)(2x+1-z)+(x-1-z) - 5vxz & =0 & (iii)\\\\-(2x+1-z)-(x-1-z)+vx(2x-5+z) +vxz & =0 & (iv)\\\\\\end{array}\\right.$ First, we have with $(iv)$ that $2z(1+vx)=x(-2vx+5v+3).$ If $1+vx=0,$ then $x=\\frac{5v+3}{2v}$ with the previous equality.", "Since $x=-\\frac{1}{v}$ we get $v=-1,$ which contradicts our assumption.", "Hence one can assume $1+vx\\ne 0$ and by rewriting the equations of the previous system as polynomials in $(x])[z]$ we get: $\\mathcal {S}:\\left\\lbrace \\begin{array}{ccc}(1+vx)z^2 + x(-3-5v+2vx)z + 2x^2-x-1 & =0 & (i)\\\\vz^2 + (-3-5v+4vx)z +4x-1& =0 & (ii)\\\\z^2 + x(-3-5v)z +2x^2-2x-3 & =0 & (iii)\\\\\\frac{1}{2(1+vx)}x(-2vx+5v+3)& = z & (iv)\\\\\\end{array}\\right.$ Now, by injecting (REF ) in $(i),$ the latter gives $\\begin{array}{ccc}\\frac{1}{1+vx} (2x^2-x-1) & =z^2 & (i^{\\prime })\\\\\\end{array}$ Let us now replace $z$ and $z^2$ by $(iv)$ and $(i^{\\prime })$ in $(ii)$ and $(iii)$ of $\\mathcal {S},$ multiplicate $(ii)$ and $(iii)$ by $2(1+vx),$ and rewrite the obtained equations $(ii^{\\prime })$ and $(iii^{\\prime })$ as polynomials in $(u])[x].$ Then the system $\\mathcal {S}$ is equivalent to the following one: $\\mathcal {S^{\\prime }}:\\left\\lbrace \\begin{array}{ccc}\\frac{1}{1+vx} (2x^2-x-1) & =z^2 & (i^{\\prime })\\\\-8v^2x^3 + 30v(v+1)x^2 + (-25v^2-34v-1)x+ 2(-v-1)& =0 & (ii^{\\prime })\\\\10v(v+1)x^3 + (-25v^2-34v-1)x^2 + 6(-v-1)x -8& =0 & (iii^{\\prime })\\\\\\frac{1}{2(1+vx)}x(-2vx+5v+3)& = z & (iv)\\\\\\end{array}\\right.$ Remark that $(i^{\\prime })$ and $(iv)$ are equivalent whenever equations $(ii^{\\prime })$ and $(iii^{\\prime })$ are both satisfied.", "Indeed, from $(i^{\\prime })$ and $(iv)$ we have that $\\frac{1}{4(1+vx)^2}x^2(-2vx+5v+3)^2= \\frac{1}{1+vx}(2x^2-x-1),$ which is equivalent to the following equation: $4v^2x^4 -20v(1+v)x^3 + (25v^2+34v+1)x^2 + 4(1+v)x+4=0.$ Finally, by adding to equation $(iii^{\\prime })$ two times equation (REF ), we get exactly equation $(ii^{\\prime }),$ up to multiplication by $(-x).$ It follows that the solutions of $\\mathcal {S^{\\prime }}$ are given by the roots of the resultant of the two polynomials of $(ii^{\\prime })$ and $(iii^{\\prime })$ .", "One can compute that this resultant is $110592v^3(v+1)^3(125v^3+399v^2+339v+1)$ and if $t_i,\\,i=1,2,3,$ are the roots of the equation $125t^3+399t^2+339t+1,$ then the curve $\\mathcal {C}_1+t_i\\mathcal {C}_2$ is singular with singular points $(x:1:z),$ where $x$ is solution of $\\left\\lbrace \\begin{array}{ccc}-8t_i^2x^3 + 30t_i(t_i+1)x^2 + (-25t_i^2-34t_i-1)x+ 2(-t_i-1)& =0 & (ii^{\\prime })\\\\10t_i(t_i+1)x^3 + (-25t_i^2-34t_i-1)x^2 + 6(-t_i-1)x -8& =0 & (iii^{\\prime })\\\\\\end{array}\\right.$ and $z=\\frac{1}{2(1+t_ix)}x(-2t_ix+5t_i+3).$ By performing $4(ii^{\\prime }) + (t_i-1)(iii^{\\prime })$ we have that $x$ is solution of $2t_i(5t_i-1)(t_i-5)x^2-(t_i-1)(25t_i^2-154t_i+1)x -2(47t_i^2-62t_i-1)=0.$ Then it is possible to guess the number of singularities of the three curves $\\mathcal {C}_1+t_i\\mathcal {C}_2,\\,i=1,2,3,$ by using a cardinality argument and the total Milnor number of the curve $\\mathcal {C}.$ Indeed, since the pencil is generic from 1., we have with [4] that the sum of the Milnor numbers of all the singularities of the degree 3 members $u{\\mathcal {C}}_1+v{\\mathcal {C}}_2$ listed before is equal to $3(3-1)^2=12.$ It follows that the not yet known singularities (at least one for each curve ${\\mathcal {C}}_1+t_j{\\mathcal {C}}_2$ ) contribute to $12-(3\\times 1 + 3\\times 1+ 3\\times 1)=3.$ Hence each member ${\\mathcal {C}}_1+t_i{\\mathcal {C}}_2$ has exactly one singularity and is an irreductible nodal curve.", "Since $t_1+t_2+t_3=\\frac{-399}{125},\\,t_1t_2+t_1t_3+t_2t_3=\\frac{339}{125}$ and $t_1t_2t_3=\\frac{-1}{124}$ with Viete formula, the union ${\\mathcal {C}}$ of the singular members of the pencil $\\mathcal {P}$ listed above is defined by the homogeneous degree 18 polynomial $f= q_1q_2(q_1-q_2)(125q_1^3-399q_1^2q_2 + 339q_1q_2^2 -1).$ Since the singularities are all weighted homogeneous, we can deduce from the results in [4] and [13] that ${\\mathcal {C}}$ is free with exponents (4,13).", "By applying the algorithm described in [8] we get this time that the only non zero dimensions of the second terms of the form (REF ) are listed as follows: $\\dim E_2^{1,0}(f)_6=1,\\, \\dim E_2^{1,0}(f)_9=2,\\, \\dim E_2^{1,0}(f)_{12}=3,\\, \\dim E_2^{1,0}(f)_{15}=4,$ and $\\dim E_2^{1,0}(f)_{18}=11.$ Using [8] we have that the multiplicities $m(\\lambda )$ of the eigenvalues $\\lambda \\ne 1$ of the monodromy operator (REF ) satisfy: $m(\\lambda )=4,$ for $k=3,15$ ($\\lambda $ of order 6); $3 \\le m(\\lambda )\\le 4$ , for $k=6,12$ ($\\lambda $ of order 3); $2 \\le m(\\lambda )\\le 4$ , for $k=9$ ($\\lambda $ of order 2); $m(\\lambda )= 0$ , for $k\\notin \\lbrace 3,6,9,12,15,18\\rbrace .$ In particular, $H^1(F,_\\lambda \\ne 0 \\Rightarrow \\dim H^1(F,_\\lambda \\le 4,$ if $\\lambda \\ne 1.$ Let $S:= \\mathbb {P}^1_\\lbrace (u^{\\prime }:v^{\\prime }) \\in \\mathbb {P}^1_\\,\|\\,\\,(u^{\\prime }:v^{\\prime }) \\text{\\,is\\,a\\,singular\\,member\\,in\\,}{\\mathcal {P}}\\rbrace $ be the complement of the six points described in 2.", "Then by considering the surjective morphism $r: U \\rightarrow S,\\,r(x:y:z)=(q_1(x,y,z):q_2(x,y,z))$ , and applying [1], [9] or [3] we get that $\\dim H^1(F)_{\\lambda ^k} \\ge -\\chi (S)=4,$ for any $\\lambda ^k \\ne 1.$ Hence the Alexander polynomial has de following expression: $\\Delta _{\\mathcal {C}}(t)=(t-1)^7(t^6-1)^4.$ Acknowledgements.", "The author thanks Alexandru Dimca for his suggestions and motivating discussions which led her to obtain the results in this paper.", "She is supported by the BREMEN TRAC Postdoctoral Fellowships for Foreign Researchers." ], [ "Proof of Theorem ", " Without loss of generality, one can assume that $x=1,$ and we have that $(1:y:z)$ is solution of the system $S:\\left\\lbrace \\begin{array}{ccc}3yz + y^3 +z^3 & =0 \\\\3yz+ 1 + z^3 & =0\\end{array}\\right.$ implies $y\\in \\lbrace 1,j,j^3\\rbrace ,\\,j=\\exp (2\\sqrt{i}\\pi /3),$ by subtracting the two equations.", "If $y=1$ then $z$ is solution of the equation $E_1: z^3+3z+1=0;$ if $y=j$ then $z$ is solution of the equation $E_2: z^3+3jz+1=0;$ and if $y=j^2$ then $z$ is solution of the equation $E_3: z^3+3j^2z+1=0.$ Denote by $\\mathcal {S}_i$ the solution set of $E_i,$ with $i=1,2,3.$ It is clear that $\\mathcal {S}_i \\cap \\mathcal {S}_j = \\emptyset ,$ for all $i \\ne j$ and the system (REF ) has nine solutions: $\\lbrace (1:1:z),\\,z\\in \\mathcal {S}_1\\rbrace \\cup \\lbrace (1:j:z),\\,z\\in \\mathcal {S}_2\\rbrace \\cup \\lbrace (1:j^2:z),\\,z\\in \\mathcal {S}_3\\rbrace .$ Hence $\|{\\mathcal {C}}_1 \\cap {\\mathcal {C}}_2\| = 9$ and the intersection points of the curves ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ have intersection multiplicity 1, i.e.", "the two curves meet transversally and the base locus of the pencil ${\\mathcal {P}}$ is smooth.", "First recall that the member in the pencil $\\mathcal {P}$ corresponding to the point $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}_{^1 is the curveu\\mathcal {C}_1+v\\mathcal {C}_2,where u=-v^{\\prime } and v=u^{\\prime }.", "If (u^{\\prime }:v^{\\prime })=(1:0), the corresponding curve \\mathcal {C}_2 is an irreductible nodal curve with node (0:1:0).", "Similarily, for (u^{\\prime }:v^{\\prime })=(0:1) we get an irreductible nodal curve {\\mathcal {C}}_1 with node (1:0:0).", "Suppose now that u^{\\prime } and v^{\\prime } are both non zero.", "Without loss of generality, one can assume u^{\\prime }=1 and we have that the corresponding member u{\\mathcal {C}}_1+ {\\mathcal {C}}_2 is singular if and only if the following system admits a solution:\\begin{equation}\\left\\lbrace \\begin{array}{ccc}3(u+1)xyz + uy^3 +(u+1)z^3 + x^3 & =0 & (i)\\\\(u+1)yz+ x^2 & =0 & (ii)\\\\(u+1)xz+uy^2 & =0 & (iii)\\\\(u+1)xy + (u+1)z^2& =0 &(iv)\\end{array}\\right.\\end{equation}}If $ (u':v')=(1:1),$ then $ -C1+C2: x3-y3$ is the union of three concurrent lines:\\begin{center}x=y,x=jy and x=j^2y.\\end{center}Assume now $ (u':v')=(1:v'), v' 1.$ The case $ y=0$ is an absurd, since then $ x=z=0$ with (ii) and (iv).", "If $ y = 1,$ we have $ x=0 z=0$ with (ii) and $ u=0$ with (i), while $ z=0 x=0$ with (ii) and $ u=0$ with (i), which are both in contradiction with our hypothesis $ v'0$.", "Finally, $ (x:1:z), x,z 0,$ is a solution of (\\ref {S1}) if and only if$$\\left\\lbrace \\begin{array}{ccc}3(u+1)xz + u +(u+1)z^3 + x^3 & =0 & (i^{\\prime })\\\\(u+1)z+ x^2 & =0 & (ii^{\\prime })\\\\(u+1)xz+u & =0 & (iii^{\\prime })\\\\x + z^2& =0 &(iv^{\\prime })\\end{array}\\right.$$and we have with $ (iv')$ and $ (iii')$ that $ z3=uu+1,$ which implies with (ii^{\\prime }) that $ u3+3u+1=0.$ Denote by $ ti, i=1,2,$ the solutions of the equation $$t^2+3t+1=0,$$ and by $ i$ a complex cubic root of $ titi+1.$ Then the singularities of the member $ tiC1 + C2$ associated to the point $ (1:-ti)$ are the following three points\\begin{center}(-\\alpha _i^2:1:\\alpha _i),(-j^2\\alpha _i^2:1:j \\alpha _i) and (-j\\alpha _i^2:1:j^2\\alpha _i),\\end{center}which are not colinear and hence form a triangle.\\item Since $ t1+t2=-3$ and $ t1t2=1,$ the union $ C$ of the singular members in the pencil $ P$ listed in 2. is the curve of degree $ d=15$ defined by the homogeneous polynomial $$f = q_1q_2(q_1- q_2)(q_1^2-3q_1q_2 + q_2^2).$$ Since the singularities are all weighted homogeneous, we can apply the results from \\cite {D1} (see Theorem 1.14) or \\cite {Val} (see Theorem 2.7) to show that $ C$ is free with exponents (4,10).\\item Recall that the $ 1-$eigenspace $ H1(F,1$ of the monodromy (\\ref {eq1}) is a pure Hodge structure of type $ (1,1),$ and that the sum of the non trivial eigenspaces $ d=1, 1H1(F,$ is pure of weight 1.", "For any $ k[1,d],$ it is known that there exists a spectral sequence $ E(f)k$ whose first term is constructed from the Koszul complex in $ x,y,z]$ of the partial derivatives of $ f,$ and whose limit $ E(f)k$ gives the action of the monodromy on the graded pieces $ H(F,, =(-2-1k/d),$ with respect to the pole order filtration $ P,$ which contains the Hodge filtration $ F$ and satisfies $ P2=0.$ In \\cite {D3}, A. Dimca and G. Sticlaru showed that the computation of the second terms given in Equation (\\ref {graduate}) is sufficient to detect all the non trivial eigenpaces of the operator (\\ref {eq1}), see \\cite [Theorem 1.2]{D3}.", "More in detail, for any $ k[1,d]$, these second terms are given by$ $E_2^{1,0}(f)_k= Gr_P^1 H^1(F,_{\\lambda },$ where $\\lambda = \\exp (-2\\sqrt{-1}\\pi k/d).$ Furthermore, when the curve is free, the authors also describe an algorithm which computes the dimensions $\\dim E_2^{1,0}(f)_k.$ By using this algorithm and the computer algebra software Singular [2], we get that the only non zero dimensions of second terms of the form (REF ) are listed as follows: $\\dim E_2^{1,0}(f)_6=1,\\,\\dim E_2^{1,0}(f)_9=2,\\,\\dim E_2^{1,0}(f)_{12}=3,$ and $\\dim E_2^{1,0}(f)_{15}=10.$ First, since $H^1(F,_1=H^1(U,$ and the curve $\\mathcal {C}$ is the union of 11 irreductible curves, it is well known that $\\dim H^1(F,_1=b_1(U)=10,$ see for instance [5].", "Using [8] we have that the multiplicities $m(\\lambda )$ of the eigenvalues $\\lambda \\ne 1$ of the monodromy operator (REF ) satisfy: $2 \\le m(\\lambda )\\le 3$ , for $k=6,9;$ $m(\\lambda )= 3$ , for $k=3,12;$ $m(\\lambda )= 0$ , for $k\\notin \\lbrace 3,6,9,12,15\\rbrace .$ The non trivial monodromies listed above are the roots of the unity of order 5.", "Since the monodromy operator $h^1$ is definied over $\\mathbb {Q},$ it is known that $\\Delta _{{\\mathcal {C}}}(t) \\in \\mathbb {Q}[t]$ is a product of cyclotomic polynomials $\\varphi _n$ with $n$ dividing $d=15$ , and it follows that $\\Delta _{\\mathcal {C}}(t)= (t-1)^7(t^5-1)^3.$" ], [ "Proof of Theorem ", " Let us compute explicitely the intersection $\\mathcal {C}_1\\cap \\mathcal {C}_2$ by solving the system of equations $q_1(x,y,z)=q_2(x,y,z)=0.$ If $x=0,$ then we get $(0:0:1),(0:1:1)$ and $(0:-1:1).$ Assume now $x=1.$ The first equation gives $y=0,y=1-z$ or $y=z-2.$ If $y=0,$ then $z=0$ or $-2$ and we get $(1:0:0)$ and $(1:0:-2);$ If $y=1-z,$ then $z=0$ or $\\frac{1}{2}$ gives $(1:1:0)$ and $(2:1:1);$ If $y=z-2,$ then with $z=0$ or $z=3$ we find $(1:-2:0)$ and $(1:1:3).$ Hence $\|{\\mathcal {C}}_1 \\cap {\\mathcal {C}}_2\| = 9$ and ${\\mathcal {C}}_1$ and ${\\mathcal {C}}_2$ meet transversally.", "Let us list all the singular members $u\\mathcal {{\\mathcal {C}}}_1+v \\mathcal {{\\mathcal {C}}}_2$ associated to points $(u^{\\prime }:v^{\\prime })\\in \\mathbb {P}_1,$ where $u^{\\prime }=v$ and $v^{\\prime }=-u.$ Let us study separately the cases $(u^{\\prime }:v^{\\prime })=(1:0)$ (a), $(u^{\\prime }:v^{\\prime })=(0:1)$ (b), $(u^{\\prime }:v^{\\prime })=(1:1)$ (c), and $(u^{\\prime }:v^{\\prime })=(u^{\\prime }:-1)$ with $u^{\\prime } \\ne 0,-1$ (d).", "If $(u^{\\prime }:v^{\\prime })=(1:0),$ then the singularities of the associated member $\\mathcal {C}_2$ are the solutions of the sytem $\\left\\lbrace \\begin{array}{ccc}xz(2x-5y+z) & =0 & (i)\\\\z(4x-5y+z) & =0 & (ii)\\\\-5xz & =0 & (iii)\\\\x(2x-5y+2z) & =0 & (iv) \\\\\\end{array}\\right.$ Without loss of generality, one can assume $y=1$ and we have $x=0$ or $z=0$ with $(iii).$ If $x=0,$ then $(ii)\\Rightarrow z=0$ or $z=5$ that is $(0:1:0)$ and $(0:1:5)$ are solutions of (REF ).", "The case $z=0$ and $x=5/2$ implied by $(iv)$ gives the third solution $(5:2:0)$ and we get a triangle.", "Similarily, for $(u^{\\prime }:v^{\\prime })=(0:1)$ the singularities of $\\mathcal {C}_1$ are the solutions of $\\left\\lbrace \\begin{array}{ccc}y(x-y-z)(2x+y-z)& =0 & (i)\\\\y(4x-y-3z) & =0 & (ii)\\\\2x^2-3y^2-2xy-3xz+z^2& =0 & (iii)\\\\-y(3x-2z)& =0 & (iv)\\\\\\end{array}\\right.$ Without loss of generality, one can assume $x=1$ and we have with $(iv)$ that $y=0$ or $z=\\frac{3}{2}.$ If $y=0,$ then $z$ is solution of $z^2-3z+2$ with $(iii)$ and $(1:0:1)$ and $(1:0:2)$ are solutions of (REF ).", "Assume $z=\\frac{3}{2}.$ Then $(i)$ and $(ii)$ both imply $y=0$ or $y=-\\frac{1}{2},$ while equation $(iii)$ implies $y=-\\frac{1}{2}$ or $-\\frac{1}{6}.$ Hence $y=-\\frac{1}{2}$ and $(2:-1:3)$ is the third solution of (REF ), i.e.", "we get once again a triangle.", "If $(u^{\\prime }:v^{\\prime })=(1:1),$ then the singularities of the associated member $\\mathcal {C}_1-\\mathcal {C}_2$ are the solutions of the sytem $\\left\\lbrace \\begin{array}{cc}y(x-y-z)(2x+y-z)-xz(2x-5y+z) & =0 \\\\y(4x-y-3z)-z(4x-5y+z) & =0 \\\\2x^2-3y^2-2xy+2xz+z^2 & =0 \\\\-y(3x-2z)-x(2x-5y+2z) & =0 \\\\\\end{array}\\right.$ Without loss of generality, one can assume $y=1$ and the previous system becomes: $\\left\\lbrace \\begin{array}{ccc}(x-1-z)(2x+1-z)-xz(2x-5+z) & =0 & (i)\\\\4x-1-3z-z(4x-5+z) & =0 & (ii)\\\\2x^2-3-2x+2xz+z^2 & =0 & (iii)\\\\-3x+2z-x(2x-5+2z) & =0 & (iv)\\\\\\end{array}\\right.$ Hence $z(1-x)=x(x-1)$ with $(iv).$ If $x=1,$ then with equations $(ii)$ and $(iii)$ we have that $z^2+2z-3=0,$ which gives the solutions $(1:1:1)$ and $(1:1:-3).$ If $x\\ne 1,$ then $(iv)\\Rightarrow z=-x$ and we get the system $\\left\\lbrace \\begin{array}{cc}(2x-1)(3x+1)+x^2(x-5) & =0 \\\\3x^2+2x-1 & =0 \\\\x^2-2x-3 & =0 \\\\z & =-x \\\\\\end{array}\\right.$ with $(-1:1:1)$ as unique solution.", "Assume $(u^{\\prime }:v^{\\prime })=(u^{\\prime }:-1)$ with $u^{\\prime }\\ne 0,-1.$ Then the associated curve $\\mathcal {C}_1+v\\mathcal {C}_2$ is singular whenever the following system has a solution: $\\left\\lbrace \\begin{array}{cc}y(x-y-z)(2x+y-z)+vxz(2x-5y+z) & =0 \\\\y(2x+y-z)+2y(x-y-z)+vz(2x-5y+z)+2vxz & =0 \\\\(x-2y-z)(2x+y-z)+y(x-y-z) - 5vxz & =0 \\\\-y(2x+y-z)-y(x-y-z)+vx(2x-5y+z) +vxz & =0 \\\\\\end{array}\\right.$ Without loss of generality, one can assume that $y=1,$ that is: $\\left\\lbrace \\begin{array}{ccc}(x-1-z)(2x+1-z)+vxz(2x-5+z) & =0 & (i)\\\\(2x+1-z)+2(x-1-z)+vz(2x-5+z)+2vxz & =0 & (ii)\\\\(x-2-z)(2x+1-z)+(x-1-z) - 5vxz & =0 & (iii)\\\\-(2x+1-z)-(x-1-z)+vx(2x-5+z) +vxz & =0 & (iv)\\\\\\end{array}\\right.$ First, we have with $(iv)$ that $2z(1+vx)=x(-2vx+5v+3).$ If $1+vx=0,$ then $x=\\frac{5v+3}{2v}$ with the previous equality.", "Since $x=-\\frac{1}{v}$ we get $v=-1,$ which contradicts our assumption.", "Hence one can assume $1+vx\\ne 0$ and by rewriting the equations of the previous system as polynomials in $(x])[z]$ we get: $\\mathcal {S}:\\left\\lbrace \\begin{array}{ccc}(1+vx)z^2 + x(-3-5v+2vx)z + 2x^2-x-1 & =0 & (i)\\\\vz^2 + (-3-5v+4vx)z +4x-1& =0 & (ii)\\\\z^2 + x(-3-5v)z +2x^2-2x-3 & =0 & (iii)\\\\\\frac{1}{2(1+vx)}x(-2vx+5v+3)& = z & (iv)\\\\\\end{array}\\right.$ Now, by injecting (REF ) in $(i),$ the latter gives $\\begin{array}{ccc}\\frac{1}{1+vx} (2x^2-x-1) & =z^2 & (i^{\\prime })\\\\\\end{array}$ Let us now replace $z$ and $z^2$ by $(iv)$ and $(i^{\\prime })$ in $(ii)$ and $(iii)$ of $\\mathcal {S},$ multiplicate $(ii)$ and $(iii)$ by $2(1+vx),$ and rewrite the obtained equations $(ii^{\\prime })$ and $(iii^{\\prime })$ as polynomials in $(u])[x].$ Then the system $\\mathcal {S}$ is equivalent to the following one: $\\mathcal {S^{\\prime }}:\\left\\lbrace \\begin{array}{ccc}\\frac{1}{1+vx} (2x^2-x-1) & =z^2 & (i^{\\prime })\\\\-8v^2x^3 + 30v(v+1)x^2 + (-25v^2-34v-1)x+ 2(-v-1)& =0 & (ii^{\\prime })\\\\10v(v+1)x^3 + (-25v^2-34v-1)x^2 + 6(-v-1)x -8& =0 & (iii^{\\prime })\\\\\\frac{1}{2(1+vx)}x(-2vx+5v+3)& = z & (iv)\\\\\\end{array}\\right.$ Remark that $(i^{\\prime })$ and $(iv)$ are equivalent whenever equations $(ii^{\\prime })$ and $(iii^{\\prime })$ are both satisfied.", "Indeed, from $(i^{\\prime })$ and $(iv)$ we have that $\\frac{1}{4(1+vx)^2}x^2(-2vx+5v+3)^2= \\frac{1}{1+vx}(2x^2-x-1),$ which is equivalent to the following equation: $4v^2x^4 -20v(1+v)x^3 + (25v^2+34v+1)x^2 + 4(1+v)x+4=0.$ Finally, by adding to equation $(iii^{\\prime })$ two times equation (REF ), we get exactly equation $(ii^{\\prime }),$ up to multiplication by $(-x).$ It follows that the solutions of $\\mathcal {S^{\\prime }}$ are given by the roots of the resultant of the two polynomials of $(ii^{\\prime })$ and $(iii^{\\prime })$ .", "One can compute that this resultant is $110592v^3(v+1)^3(125v^3+399v^2+339v+1)$ and if $t_i,\\,i=1,2,3,$ are the roots of the equation $125t^3+399t^2+339t+1,$ then the curve $\\mathcal {C}_1+t_i\\mathcal {C}_2$ is singular with singular points $(x:1:z),$ where $x$ is solution of $\\left\\lbrace \\begin{array}{ccc}-8t_i^2x^3 + 30t_i(t_i+1)x^2 + (-25t_i^2-34t_i-1)x+ 2(-t_i-1)& =0 & (ii^{\\prime })\\\\10t_i(t_i+1)x^3 + (-25t_i^2-34t_i-1)x^2 + 6(-t_i-1)x -8& =0 & (iii^{\\prime })\\\\\\end{array}\\right.$ and $z=\\frac{1}{2(1+t_ix)}x(-2t_ix+5t_i+3).$ By performing $4(ii^{\\prime }) + (t_i-1)(iii^{\\prime })$ we have that $x$ is solution of $2t_i(5t_i-1)(t_i-5)x^2-(t_i-1)(25t_i^2-154t_i+1)x -2(47t_i^2-62t_i-1)=0.$ Then it is possible to guess the number of singularities of the three curves $\\mathcal {C}_1+t_i\\mathcal {C}_2,\\,i=1,2,3,$ by using a cardinality argument and the total Milnor number of the curve $\\mathcal {C}.$ Indeed, since the pencil is generic from 1., we have with [4] that the sum of the Milnor numbers of all the singularities of the degree 3 members $u{\\mathcal {C}}_1+v{\\mathcal {C}}_2$ listed before is equal to $3(3-1)^2=12.$ It follows that the not yet known singularities (at least one for each curve ${\\mathcal {C}}_1+t_j{\\mathcal {C}}_2$ ) contribute to $12-(3\\times 1 + 3\\times 1+ 3\\times 1)=3.$ Hence each member ${\\mathcal {C}}_1+t_i{\\mathcal {C}}_2$ has exactly one singularity and is an irreductible nodal curve.", "Since $t_1+t_2+t_3=\\frac{-399}{125},\\,t_1t_2+t_1t_3+t_2t_3=\\frac{339}{125}$ and $t_1t_2t_3=\\frac{-1}{124}$ with Viete formula, the union ${\\mathcal {C}}$ of the singular members of the pencil $\\mathcal {P}$ listed above is defined by the homogeneous degree 18 polynomial $f= q_1q_2(q_1-q_2)(125q_1^3-399q_1^2q_2 + 339q_1q_2^2 -1).$ Since the singularities are all weighted homogeneous, we can deduce from the results in [4] and [13] that ${\\mathcal {C}}$ is free with exponents (4,13).", "By applying the algorithm described in [8] we get this time that the only non zero dimensions of the second terms of the form (REF ) are listed as follows: $\\dim E_2^{1,0}(f)_6=1,\\, \\dim E_2^{1,0}(f)_9=2,\\, \\dim E_2^{1,0}(f)_{12}=3,\\, \\dim E_2^{1,0}(f)_{15}=4,$ and $\\dim E_2^{1,0}(f)_{18}=11.$ Using [8] we have that the multiplicities $m(\\lambda )$ of the eigenvalues $\\lambda \\ne 1$ of the monodromy operator (REF ) satisfy: $m(\\lambda )=4,$ for $k=3,15$ ($\\lambda $ of order 6); $3 \\le m(\\lambda )\\le 4$ , for $k=6,12$ ($\\lambda $ of order 3); $2 \\le m(\\lambda )\\le 4$ , for $k=9$ ($\\lambda $ of order 2); $m(\\lambda )= 0$ , for $k\\notin \\lbrace 3,6,9,12,15,18\\rbrace .$ In particular, $H^1(F,_\\lambda \\ne 0 \\Rightarrow \\dim H^1(F,_\\lambda \\le 4,$ if $\\lambda \\ne 1.$ Let $S:= \\mathbb {P}^1_\\lbrace (u^{\\prime }:v^{\\prime }) \\in \\mathbb {P}^1_\\,\|\\,\\,(u^{\\prime }:v^{\\prime }) \\text{\\,is\\,a\\,singular\\,member\\,in\\,}{\\mathcal {P}}\\rbrace $ be the complement of the six points described in 2.", "Then by considering the surjective morphism $r: U \\rightarrow S,\\,r(x:y:z)=(q_1(x,y,z):q_2(x,y,z))$ , and applying [1], [9] or [3] we get that $\\dim H^1(F)_{\\lambda ^k} \\ge -\\chi (S)=4,$ for any $\\lambda ^k \\ne 1.$ Hence the Alexander polynomial has de following expression: $\\Delta _{\\mathcal {C}}(t)=(t-1)^7(t^6-1)^4.$ Acknowledgements.", "The author thanks Alexandru Dimca for his suggestions and motivating discussions which led her to obtain the results in this paper.", "She is supported by the BREMEN TRAC Postdoctoral Fellowships for Foreign Researchers." ] ]	1606.04858
[ [ "A characterization of arithmetic functions satisfying\n $f(u^{2}+kv^{2})=f^{2}(u)+kf^{2}(v)$" ], [ "Abstract In this paper, we mainly discuss the characterization of a class of arithmetic functions $f: N \\rightarrow C$ such that $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v \\in N)$.", "We obtain a characterization with given condition, propose a conjecture and show the result holds for $k \\in \\{2, 3, 4, 5 \\}$." ], [ "Introduction", " Let $C$ denote the set of all complex numbers and $N$ be the set of all positive integers.", "An arithmetic function is defined as $f: N \\rightarrow C$ , which is an important research branch of number theory.", "A arithmetic function defined on $N$ is called multiplicative if $f(mn)=f(m)f(n)$ for all coprime $m,n \\in N$ , and is called completely multiplicative if $f(mn)=f(m)f(n)$ for all $m,n \\in N$ .", "In recent years, a lot of work on arithmetic function satisfying some Cauchy-like functional equation have been done by researchers.", "As was in [5], Spiro showed that if the multiplicative function $f: N \\rightarrow C$ satisfies $f(p+q)=f(p)+f(q)$ for all primes $p,q$ and there exists $n_0 \\in N$ such that $f(n_0) \\ne 0$ , then $f(n)=n$ for all $n$ .", "Later, Fang [6] extended the conclusion to the equation $f(p+q+r)=f(p)+f(q)+f(r)$ .", "Dubickas et al.", "[1] improved the conclusion to general case $f(p_{1}+p_{2}+ \\cdots +p_{k})=f(p_{1})+f(p_{2})+ \\cdots +f(p_{k})$ , where $k \\ge 2$ is fixed, and Chen et al.", "[10] studied the multiplicative function $f$ satisfies $f(p+q+n_{0})=f(p)+f(q)+f(n_{0})$ , where $p,q$ are prime and $n_{0}$ is a positive integer.", "Pong [3] considered that if the completely multiplicative function satisfies $f(p+q+pq)=f(p)+f(q)+f(pq)$ , $p, q$ are primes, and there exists some $p_{0}$ such that $f(p_{0}) \\ne 0$ , then $f(n)=n$ for all $n$ .", "De Koninck et al.", "[7] described that if the multiplicative function satisfies $f(1)=1$ and $f(p+m^{2})=f(p)+f(m^{2})$ when $p$ is a prime, then $f(n)=n$ for all $n$ .", "Chung [9] characterized the multiplicative and completely multiplicative functions satisfy $f(u^{2}+v^2)=f^{2}(u)+f^{2}(v)$ .", "Phong [4] proved that for any $ k \\in N$ , if the multiplicative function satisfies $f(u^{2}+v^2+k)=f(u^{2})+f(v^2+k)$ , $gcd(m, 2k)=1$ and $f(4)f(9) \\ne 0$ , then $f(n)=n$ for all $n$ .", "Indlekofer et al.", "[8] also discussed the multiplicative function satisfies $f(u^{2}+v^2+k+1)=f(u^{2}+k)+f(v^2+1)$ , $f(2) \\ne 0$ and $f(5) \\ne 1$ , then $f(n)=n$ for all $n$ , when $gcd(n,2)=1$ holds.", "Recently, Bašić [2] considered a modification of the functional equation treated by Chung: $f(u^{2}+v^2)=f^{2}(u)+f^{2}(v)$ , and proved that all such functions can be grouped into three families, namely $f(n) \\equiv 0, f(n)=\\pm n, f(n)=\\pm \\frac{1}{k+1}$ , and the sign is subjected to the precondition.", "In this paper, we mainly consider the general cases: $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ , we obtain a characterization with given condition, propose a conjecture and show the result holds for $k \\in \\lbrace 2, 3, 4, 5 \\rbrace $ ." ], [ "Main result and problem", " Let $k,u,v\\in N$ , in order to characterize the function satisfying $f(u^{2}+kv^{2})=f^{2}(u)+kf^{2}(v)$ , we prove the following theorem firstly.", "Theorem 2.1 Let $k\\ge 2$ be a positive integer and $A=\\left\\lbrace \\begin{array}{ll}6, & k=2; \\\\7, & k=3; \\\\2k, & k\\ge 4,\\end{array}\\right.$ $f: N \\rightarrow C$ satisfy $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ for all $u,v\\in N$ .", "If one of the following holds for all $n$ $(1\\le n\\le A)$ : (1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if there exist } u, v \\in N \\mbox{ such that } n=u^{2}+kv^{2}; \\\\\\pm n, & \\mbox{ otherwise},\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{k+1}, & \\mbox{ if there exist } u, v \\in N \\mbox{ such that } n=u^{2}+kv^{2}; \\\\\\pm \\frac{1}{k+1}, & \\mbox{ otherwise},\\end{array}\\right.$ Then the corresponding one of (1) , (2) , (3) holds for all $n\\in N$ .", "Proof.", "Let $a, b, c, d$ be integers with $ab \\pm kcd > 0$ and $ad \\pm bc > 0$ .", "Firstly, we have $(ab+kcd)^{2}+k(ad-bc)^{2}=(ab-kcd)^{2}+k(ad+bc)^{2}.$ Applying $f$ to both sides of (REF ) and using equation (REF ) we have $ f^{2}(ab+kcd)+kf^{2}(ad-bc)=f^{2}(ab-kcd)+kf^{2}(ad+bc).$ Now we only need to show one of (1), (2), (3) holds when $n>A$ .", "By induction on $n$ , we complete the proof by the following two cases.", "Case 1: $k$ is odd.", "Then $k \\ge 3$ .", "Subcase 1.1: $n$ is odd.", "Let $n=2l+1$ , then by $n > A$ we have $l\\ge \\left\\lbrace \\begin{array}{cc}4, & \\mbox{ if } k=3; \\\\k, & \\mbox{ if } k\\ge 5.\\end{array}\\right.$ We take $a=l-\\frac{k-1}{2}, b=2, c=d=1$ in (REF ), then $2l-2k+1>0,$ $l-\\frac{k-1}{2}-2>0$ and $ f^{2}(n)=f^{2}(2l-2k+1)+kf^{2}(l-\\frac{k-1}{2}+2)-kf^{2}(l-\\frac{k-1}{2}-2).$ Subcase 1.1.1: (1) holds for $ 1\\le n\\le A$ .", "By the inductive assumption and $f(n)\\equiv 0$ for $ 1\\le n\\le A $ , we have $f^{2}(2l-2k+1)=f^{2}(l-\\frac{k-1}{2}+2)=f^{2}(l-\\frac{k-1}{2}-2)=0$ .", "Then $f^{2}(n)=0$ by (REF ) and thus $f(n)=0$ .", "Subcase 1.1.2: (2) holds for $ 1\\le n\\le A$ .", "If there exist $u, v \\in N$ such that $n=u^{2}+kv^{2}$ , then by the inductive assumption and (2) holds for $ 1\\le n\\le A$ , we have $f^{2}(u)=f^{2}(v)=\\frac{1}{(k+1)^{2}}$ , and thus $f(n)=\\frac{1}{k+1}$ by (REF ).", "Otherwise, by the inductive assumption and (2) holds for $ 1\\le n\\le A$ , we have $f^{2}(2l-2k+1)=f^{2}(l-\\frac{k-1}{2}+2)=f^{2}(l-\\frac{k-1}{2}-2)=\\frac{1}{(k+1)^{2}}$ .", "Then $f(n)=\\pm \\frac{1}{k+1}$ by (REF ).", "Subcase 1.1.3: (3) holds for $ 1\\le n\\le A$ .", "If there exist $u, v \\in N$ such that $n=u^{2}+kv^{2}$ , then by the inductive assumption, we have $f^{2}(u)=u^2$ , $f^{2}(v)=v^2$ , thus $f(n)=n$ by (REF ).", "Otherwise, by the inductive assumption, we have $f^{2}(2l-2k+1)=(2l-2k+1)^{2}$ , $f^{2}(l-\\frac{k-1}{2}+2)=(l-\\frac{k-1}{2}+2)^{2}$ and $f^{2}(l-\\frac{k-1}{2}-2)=(l-\\frac{k-1}{2}-2)^{2}$ .", "Then we have $f^{2}(n)=(2l-2k+1)^{2}+k(l-\\frac{k-1}{2}+2)^{2}-k(l-\\frac{k-1}{2}-2)^{2}=4l^{2}+4l+1=(2l+1)^{2}=n^2$ by (REF ), and thus $f(n)=\\pm n$ .", "Combining the above three subcases, we complete the proof when $n$ is odd.", "Subcase 1.2: $n$ is even.", "Let $n=2l$ , then $l \\ge k+1$ by $n > A$ .", "We take $a=2l-k, b=c=d=1$ in (REF ), then $2l-2k>0$ , $2l-k-1>0$ and $ f^{2}(n)=f^{2}(2l-2k)+kf^{2}(2l-k+1)-kf^{2}(2l-k-1).$ Subcase 1.2.1: (1) holds for $ 1\\le n\\le A$ .", "By the inductive assumption and $f(n)\\equiv 0$ for $ 1\\le n\\le A $ , we have $f^{2}(2l-2k)=f^{2}(2l-k+1)=f^{2}(2l-k-1)=0$ .", "Then $f^{2}(n)=0$ by (REF ) and thus $f(n)=0$ .", "Subcase 1.2.2: (2) holds for $ 1\\le n\\le A$ .", "If there exist $u, v \\in N$ such that $n=u^{2}+kv^{2}$ , then by the inductive assumption and (2) holds for $ 1\\le n\\le A$ , we have $f^{2}(u)=f^{2}(v)=\\frac{1}{(k+1)^{2}}$ , and thus $f(n)=\\frac{1}{k+1}$ by (REF ).", "Otherwise, by the inductive assumption and (2) holds for $ 1\\le n\\le A$ , we have $f^{2}(2l-2k)=f^{2}(2l-k+1)=f^{2}(2l-k-1)=\\frac{1}{(k+1)^{2}}$ .", "Then $f(n)=\\pm \\frac{1}{k+1}$ by (REF ).", "Subcase 1.2.3: (3) holds for $ 1\\le n\\le A$ .", "If there exist $u, v \\in N$ such that $n=u^{2}+kv^{2}$ , then by the inductive assumption, we have $f^{2}(u)=u^2$ , $f^{2}(v)=v^2$ , thus $f(n)=n$ by (REF ).", "Otherwise, by the inductive assumption, we have $f^{2}(2l-2k)=(2l-2k)^{2}$ , $f^{2}(2l-k+1)=(2l-k+1)^{2}$ and $f^{2}(2l-k-1)=(2l-k-1)^{2}$ .", "Then we have $f^{2}(n)=(2l-2k)^{2}+k(2l-k+1)^{2}-k(2l-k-1)^{2}=4l^{2}=n^2$ by (REF ), and thus $f(n)=\\pm n$ .", "Combining the above three subcases, we complete the proof when $n$ is even.", "Case 2: $k$ is even.", "Subcase 2.1: $n$ is odd.", "Let $n=2l+1$ , then by $n > A$ we have $l\\ge \\left\\lbrace \\begin{array}{cc}3, & \\mbox{ if } k=2; \\\\k, & \\mbox{ if } k\\ge 4.\\end{array}\\right.$ We take $a=2l-k+1, b=c=d=1$ in equation (REF ), then $2l-2k+1>0$ , $2l-k>0$ and $ f^{2}(n)=f^{2}(2l-2k+1)+kf^{2}(2l-k+2)-kf^{2}(2l-k).", "$ Subcase 2.2: $n$ is even.", "Let $n=2l$ , then by $n > A$ we have $l\\ge \\left\\lbrace \\begin{array}{cc}4, & \\mbox{ if } k=2; \\\\k+1, & \\mbox{ if } k\\ge 4.\\end{array}\\right.$ We take $a=l-\\frac{k}{2}, b=c=d=1$ in equation (REF ), then $2l-2k>0$ , $l-\\frac{k}{2}-2>0$ and $ f^{2}(n)=f^{2}(2l-2k)+kf^{2}(l-\\frac{k}{2}+2)-kf^{2}(l-\\frac{k}{2}-2).", "$ Similar to the proof of Case 1, we can complete the proof by the assumption, (REF ), (REF ) or (REF ), so we omit it.", "$\\square $ Based on the result of Theorem REF , we propose the following conjecture for further research.", "Conjecture 2.2 Let $k\\in N$ , $f: N \\rightarrow C$ satisfy (REF ) for all $u,v\\in N$ .", "Then one of the following holds: (1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if there exist } u, v \\in N \\mbox{ such that } n=u^{2}+kv^{2}; \\\\\\pm n, & \\mbox{ otherwise}.\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{k+1}, & \\mbox{ if there exist } u, v \\in N \\mbox{ such that } n=u^{2}+kv^{2}; \\\\\\pm \\frac{1}{k+1}, & \\mbox{ otherwise}.\\end{array}\\right.$ In the following sections, we will show Conjecture REF holds for the cases $k=2,3,4,5$ ." ], [ "The proof of $k=2$", " In this section, we will prove Conjecture REF holds for $k=2$ .", "Lemma 3.1 Let $f: N \\rightarrow C$ satisfy $f(u^{2}+2v^2)=f^{2}(u)+2f^{2}(v)$ for all $u,v\\in N$ , $f(1)=a$ and $f(2)=b$ .", "Then we have $f^{2}(3)=9a^{4},$ $f^{2}(4)=\\frac{1}{2}(27a^{4}-3a^{2}+2b^{2}),$ $f^{2}(5)=27a^{4}-2a^{2},$ $f^{2}(6)=36a^{4}-4a^{2}+b^{2}.$ Proof.", "Since $f(1)=a, f(2)=b$ and $3=1^{2}+2\\times 1^{2}$ , then we have $f(3)=3a^{2}$ and thus (REF ) holds.", "Noting that $27=3^{2}+2\\times 3^{2}=5^{2}+2\\times 1^{2}, 33=1^{2}+2\\times 4^{2}=5^{2}+2\\times 2^{2}$ and $54=2^{2}+2\\times 5^{2}=6^{2}+2\\times 3^{2}$ , then we have $\\left\\lbrace \\begin{array}{ll}f^{2}(3)+2f^{2}(3)=f^{2}(5)+2f^{2}(1), \\\\f^{2}(1)+2f^{2}(4)=f^{2}(5)+2f^{2}(2), \\\\f^{2}(2)+2f^{2}(5)=f^{2}(6)+2f^{2}(3).\\end{array}\\right.$ Solving (REF ), we have (REF )-(REF ) hold, which complete the proof.", "$\\square $ Now we evaluate $f(1)$ .", "Lemma 3.2 $f(1)\\in \\lbrace 0,1,-1, \\frac{1}{3}, -\\frac{1}{3}\\rbrace $ .", "Proof.", "Firstly, by $f(1)=a, f(2)=b$ and $f(6)=f(2^{2}+2\\times 1^{2})=f^{2}(2)+2f^{2}(1)=b^{2}+2a^{2}$ , we have $ f^{2}(6)=(b^{2}+2a^{2})^{2}.$ Similarly, by $f(9)=f(1^{2}+2\\times 2^{2})=f^{2}(1)+2f^{2}(2)=a^{2}+2b^{2}$ , we have $ f^{2}(9)=(a^{2}+2b^{2})^{2}.$ Noting that $51=1^{2}+2\\times 5^{2}=7^{2}+2\\times 1^{2}$ and $99=1^{2}+2\\times 7^{2}=9^{2}+2\\times 3^{2}$ , then we have $f^{2}(1)+2f^{2}(5)=f^{2}(7)+2f^{2}(1),$ $f^{2}(1)+2f^{2}(7)=f^{2}(9)+2f^{2}(3)$ and thus $ f^{2}(9)=90a^{4}-9a^{2}.$ Combining (REF ), (REF ), (REF ) and (REF ), we have $\\left\\lbrace \\begin{array}{ll}(b^{2}+2a^{2})^{2}=36a^{4}-4a^{2}+b^{2}, \\\\(a^{2}+2b^{2})^{2}=90a^{4}-9a^{2}.\\end{array}\\right.$ Then $b^{2}=\\frac{39a^{4}-7a^{2}}{12a^{2}-4},$ and $(\\frac{39a^{4}-7a^{2}}{12a^{2}-4})^{2}+\\frac{4a^{2}(39a^{4}-7a^{2})}{12a^{2}-4}+4a^{4}=36a^{4}-4a^{2}+\\frac{39a^{4}-7a^{2}}{12a^{2}-4}$ .", "By simplifying this equation, we have $a^{2}(a-1)(a+1)(3a-1)(3a+1)(15a^{2}-4)=0.$ Thus $a_{1}=a_{2}=0, a_{3}=1, a_{4}=-1, a_{5}=\\frac{1}{3}, a_{6}=-\\frac{1}{3}, a_{7}=\\frac{2\\sqrt{15}}{15}, a_{8}=-\\frac{2\\sqrt{15}}{15}.$ Now we show that $f(1)=\\pm \\frac{2\\sqrt{15}}{15}$ is impossible.", "Otherwise, we have $a^{2}=\\frac{4}{15}$ and $b^{2}=-\\frac{17}{15}$ by (REF ).", "Since $f(12)=f(2^{2}+2\\times 2^{2})=f^{2}(2)+2f^{2}(2)=3b^{2}$ , then we have $ f^{2}(12)=9b^{4}.$ On the other hand, we note that $108=10^{2}+2\\times 2^{2}=6^{2}+2\\times 6^{2}$ and $216=4^{2}+2\\times 10^{2}=12^{2}+2\\times 6^{2}$ , then we have $ f^{2}(10)+2f^{2}(2)=f^{2}(6)+2f^{2}(6),$ $f^{2}(4)+2f^{2}(10)=f^{2}(12)+2f^{2}(6)$ and $ f^{2}(12)=f^{2}(4)+4f^{2}(6)-4b^{2}=\\frac{315a^{4}-35a^{2}+2b^{2}}{2}.$ By (REF )-(REF ), we have $\\frac{315a^{4}-35a^{2}+2b^{2}}{2}-9b^{4}=0.$ Unfortunately, when we put $a^{2}=\\frac{4}{15}$ and $b^{2}=-\\frac{17}{15}$ in $\\frac{315a^{4}-35a^{2}+2b^{2}}{2}-9b^{4}$ , we obtain $\\frac{315a^{4}-35a^{2}+2b^{2}}{2}-9b^{4}\\ne 0$ .", "Thus it implies a contradiction and we complete the proof.", "$\\square $ By Lemmas REF -REF and direct calculation, we obtain the following Coroll1ary.", "Corollary 3.3 Let $1\\le n \\le 6$ and $f$ satisfy the function equation (REF ).", "Then one of the following holds.", "(1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if } n\\in \\lbrace 3,6\\rbrace ; \\\\\\pm n, & \\mbox{ otherwise}.\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{3}, & \\mbox{ if } n\\in \\lbrace 3,6\\rbrace ; \\\\\\pm \\frac{1}{3}, & \\mbox{ otherwise}.\\end{array}\\right.$ Now, by Theorem REF and Corollary REF , we obtain Theorem REF directly.", "Theorem 3.4 Conjecture REF holds for $k=2$ ." ], [ "The proof of $k=3$", " In this section, we will prove Conjecture REF holds for $k=3$ .", "Lemma 4.1 Let $f: N \\rightarrow C$ satisfy $f(u^{2}+3v^2)=f^{2}(u)+3f^{2}(v)$ for all $u,v\\in N$ , $f(1)=a$ and $f(2)=b$ .", "Then we have $f^{2}(3)=\\frac{1}{3}(8b^{2}-5a^{2}), $ $f^{2}(4)=5b^{2}-4a^{2}, $ $f^{2}(5)=8b^{2}-7a^{2}, $ $f^{2}(6)=\\frac{1}{3}(35b^{2}-32a^{2}), $ $f^{2}(7)=16b^{2}-15a^{2}, $ Proof.", "Since $f(1)=a, f(2)=b$ and $28=5^{2}+3\\times 1^{2}=1^{2}+3\\times 3^{2}=4^{2}+3\\times 2^{2}$ and $52=2^{2}+3\\times 4^{2}=5^{2}+3\\times 3^{2}$ , we have $\\left\\lbrace \\begin{array}{ll}f^{2}(5)=3f^{2}(3)-2a^{2}, \\\\f^{2}(4)=3f^{2}(3)+a^{2}-3b^{2},\\\\b^{2}+3f^{2}(4)=f^{2}(5)+3f^{2}(3).\\end{array}\\right.$ Solving (REF ), we have (REF ), (REF ) and (REF ) hold.", "Similarly, it is easy to find that $84=6^{2}+3\\times 4^{2}=3^{2}+3\\times 5^{2}$ and $124=7^{2}+3\\times 5^{2}=4^{2}+3\\times 6^{2}$ , then we have $\\left\\lbrace \\begin{array}{ll}f^{2}(6)+3f^{2}(4)=f^{2}(3)+3f^{2}(5), \\\\f^{2}(7)+3f^{2}(5)=f^{2}(4)+3f^{2}(6).\\end{array}\\right.$ Solving (REF ), we have (REF ) and (REF ) hold.", "Then we complete the proof.", "$\\square $ Now we evaluate $f(1)$ .", "Lemma 4.2 $f(1)\\in \\lbrace 0,1,-1, \\frac{1}{4}, -\\frac{1}{4}\\rbrace $ .", "Proof.", "Firstly, by $f(1)=a$ and $f(4)=f(1^{2}+3\\times 1^{2})=f^{2}(1)+3f^{2}(1)=4f^{2}(1)=4a^{2}$ , we have $ f^{2}(4)=16a^{4}.$ Noting that $112=10^{2}+3\\times 2^{2}=2^{2}+3\\times 6^{2}$ , $208=14^{2}+3\\times 2^{2}=10^{2}+3\\times 6^{2}$ and $304=16^{2}+3\\times 4^{2}=14^{2}+3\\times 6^{2}$ , then we have $\\left\\lbrace \\begin{array}{ll}f^{2}(10)+3f^{2}(2)=f^{2}(2)+3f^{2}(6), \\\\f^{2}(14)+3f^{2}(2)=f^{2}(10)+3f^{2}(6), \\\\f^{2}(16)+3f^{2}(4)=f^{2}(14)+3f^{2}(6),\\end{array}\\right.$ and $ f^{2}(16)=9f^{2}(6)-5f^{2}(2)-3f^{2}(4)=85b^{2}-84a^{2}.$ On the other hand, by $f(2)=b$ and $f(16)=f(2^{2}+3\\times 2^{2})=f^{2}(2)+3f^{2}(2)=4f^{2}(2)=4b^{2}$ , we have $ f^{2}(16)=16b^{4}.$ Combining (REF ), (REF ), (REF ) and (REF ), we have $\\left\\lbrace \\begin{array}{ll}5b^{2}-4a^{2}=16a^{4}, \\\\85b^{2}-84a^{2}=16b^{4}.\\end{array}\\right.$ Then $b^{2}=\\frac{1}{5}(16a^{4}+4a^{2}),$ and $17(16a^{4}+4a^{2})-84a^{2}=\\frac{16}{25}(16a^{4}+4a^{2})^{2}$ .", "By simplifying this equation, we have $a^{2}(a-1)(a+1)(4a-1)(4a+1)(16a^{2}+25)=0.$ Thus $a_{1}=a_{2}=0, a_{3}=1, a_{4}=-1, a_{5}=\\frac{1}{4}, a_{6}=-\\frac{1}{4}, a_{7}=\\frac{5}{4}i, a_{8}=-\\frac{5}{4}i.$ Now we show that $f(1)=\\pm \\frac{5}{4}i$ is impossible.", "Otherwise, we have $a^{2}=-\\frac{25}{16}$ and $b^{2}=\\frac{105}{16}$ by (REF ).", "Noting that $f(7)=f(2^{2}+3\\times 1^{2})=f^{2}(2)+3f^{2}(1)=b^{2}+3a^{2}$ , by (REF ), we have $(b^{2}+3a^{2})^{2}-(16b^{2}-15a^{2})=0.$ But when we put $a^{2}=-\\frac{25}{16}$ and $b^{2}=\\frac{105}{16}$ in $(b^{2}+3a^{2})^{2}-(16b^{2}-15a^{2})$ , we obtain $(b^{2}+3a^{2})^{2}-(16b^{2}-15a^{2})\\ne 0$ which implies a contradiction and we complete the proof.", "$\\square $ By Lemmas REF -REF and direct calculation, we obtain the following Coroll1ary.", "Corollary 4.3 Let $1\\le n \\le 7$ and $f$ satisfy the function equation (REF ).", "Then one of the following holds.", "(1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if } n\\in \\lbrace 4,7\\rbrace ; \\\\\\pm n, & \\mbox{ otherwise}.\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{4}, & \\mbox{ if } n\\in \\lbrace 4,7\\rbrace ; \\\\\\pm \\frac{1}{4}, & \\mbox{ otherwise}.\\end{array}\\right.$ Now, by Theorem REF and Corollary REF , we obtain Theorem REF directly.", "Theorem 4.4 Conjecture REF holds for $k=3$ ." ], [ "The proof of $k=4$", " In this section, we will prove Conjecture REF holds for $k=4$ .", "Lemma 5.1 Let $f: N \\rightarrow C$ satisfy $f(u^{2}+4v^2)=f^{2}(u)+4f^{2}(v)$ for all $u,v\\in N$ , $f(1)=a$ and $f(2)=b$ .", "Then (REF )-(REF ) and (REF ) hold: $f^{2}(8)=21b^{2}-20a^{2}.", "$ Proof.", "Since $f(1)=a, f(2)=b$ , $20=4^{2}+4\\times 1^{2}=2^{2}+4\\times 2^{2}$ , $65=7^{2}+4\\times 2^{2}=1^{2}+4\\times 4^{2}$ and $68=8^{2}+4\\times 1^{2}=2^{2}+4\\times 4^{2}$ , we have $f^{2}(4)=5f^{2}(2)-4f^{2}(1)=5b^{2}-4a^{2},$ $f^{2}(7)=f^{2}(1)+4f^{2}(4)-4f^{2}(2)=16b^{2}-15a^{2},$ and $ f^{2}(8)=f^{2}(2)+4f^{2}(4)-4f^{2}(1)=21b^{2}-20a^{2}.$ Noting that $200=10^{2}+4\\times 5^{2}=2^{2}+4\\times 7^{2}$ and $104=10^{2}+4\\times 1^{2}=2^{2}+4\\times 5^{2}$ , we have $\\left\\lbrace \\begin{array}{ll}f^{2}(10)+4f^{2}(5)=f^{2}(2)+4f^{2}(7), \\\\f^{2}(10)+4f^{2}(1)=f^{2}(2)+4f^{2}(5).\\end{array}\\right.$ Solving (REF ), we obtain $f^{2}(10)=33b^{2}-32a^{2}$ and (REF ) holds.", "Similarly, it is easy to find that $100=6^{2}+4\\times 4^{2}=8^{2}+4\\times 3^{2}$ , $265=3^{2}+4\\times 8^{2}=11^{2}+4\\times 6^{2}$ and $125=11^{2}+4\\times 1^{2}=5^{2}+4\\times 5^{2}$ .", "Then we have $\\left\\lbrace \\begin{array}{ll}f^{2}(6)+4f^{2}(4)=f^{2}(8)+4f^{2}(3), \\\\f^{2}(3)+4f^{2}(8)=f^{2}(11)+4f^{2}(6), \\\\f^{2}(11)+4f^{2}(1)=f^{2}(5)+4f^{2}(5).\\end{array}\\right.$ Solving (REF ), (REF ) and (REF ) hold, and we complete the proof.", "$\\square $ Now we evaluate $f(1)$ .", "Lemma 5.2 $f(1)\\in \\lbrace 0,1,-1, \\frac{1}{5}, -\\frac{1}{5}\\rbrace $ .", "Proof.", "Firstly, by $f(1)=a$ and $f(5)=f(1^{2}+4\\times 1^{2})=f^{2}(1)+4f^{2}(1)=5f^{2}(1)=5a^{2}$ , we have $ f^{2}(5)=25a^{4}.$ Similarly, by $f(2)=b$ and $f(20)=f(2^{2}+4\\times 2^{2})=f^{2}(2)+4f^{2}(2)=5f^{2}(2)=5b^{2}$ , we have $ f^{2}(20)=25b^{4}.$ Noting that $404=20^{2}+4\\times 1^{2}=2^{2}+4\\times 10^{2}$ , by (REF ) we have $ f^{2}(20)=f^{2}(2)+4f^{2}(10)-4f^{2}(1)=133b^{2}-132a^{2}.$ Combining (REF ), (REF ), (REF ) and (REF ), we have $\\left\\lbrace \\begin{array}{ll}8b^{2}-7a^{2}=25a^{4}, \\\\133b^{2}-132a^{2}=25b^{4}.\\end{array}\\right.$ Then we have $b^{2}=\\frac{1}{8}(25a^{4}+7a^{2}),$ and $\\frac{133}{8}(25a^{4}+7a^{2})-132a^{2}=\\frac{25}{64}(25a^{4}+7a^{2})^{2}$ .", "By simplifying this equation, we have $a^{2}(a-1)(a+1)(5a-1)(5a+1)(5a^{2}+8)=0.$ Thus $a_{1}=a_{2}=0, a_{3}=1, a_{4}=-1, a_{5}=\\frac{1}{5}, a_{6}=-\\frac{1}{5}, a_{7}=\\frac{2\\sqrt{1}0}{5}i, a_{8}=-\\frac{2\\sqrt{1}0}{5}i.$ Now we show that $f(1)=\\pm \\frac{2\\sqrt{1}0}{5}i$ is impossible.", "Otherwise, we have $a^{2}=-\\frac{8}{5}$ and $b^{2}=\\frac{33}{5}$ by (REF ).", "Noting that $f(8)=f(2^{2}+4\\times 1^{2})=f^{2}(2)+4f^{2}(1)=b^{2}+4a^{2}$ , by (REF ), we have $(b^{2}+4a^{2})^{2}-(21b^{2}-20a^{2})=0.$ Unfortunately, when we put $a^{2}=-\\frac{8}{5}$ and $b^{2}=\\frac{33}{5}$ in $(b^{2}+4a^{2})^{2}-(21b^{2}-20a^{2})$ , we obtain $(b^{2}+4a^{2})^{2}-(21b^{2}-20a^{2})\\ne 0$ which implies a contradiction and we complete the proof.", "$\\square $ By Lemmas REF -REF and direct calculation, we obtain the following Coroll1ary.", "Corollary 5.3 Let $1\\le n \\le 8$ and $f$ satisfy the function equation (REF ).", "Then one of the following holds.", "(1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if } n \\in \\lbrace 5,8\\rbrace ; \\\\\\pm n, & \\mbox{ otherwise}.\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{5}, & \\mbox{ if } n \\in \\lbrace 5,8\\rbrace ; \\\\\\pm \\frac{1}{5}, & \\mbox{ otherwise}.\\end{array}\\right.$ Now, by Theorem REF and Corollary REF , we obtain Theorem REF directly.", "Theorem 5.4 Conjecture REF holds for $k=4$ ." ], [ "The proof of $k=5$", " In this section, we will prove Conjecture REF holds for $k=5$ .", "Lemma 6.1 Let $f: N \\rightarrow C$ satisfy $f(u^{2}+5v^2)=f^{2}(u)+5f^{2}(v)$ for all $u,v\\in N$ , $f(1)=a$ and $f(2)=b$ .", "Then (REF )-(REF ), (REF ), (REF ) and (REF ) hold: $f^{2}(9)=\\frac{1}{3}(80b^{2}-77a^{2}), $ $f^{2}(10)=33b^{2}-32a^{2}.$ Proof.", "Since $f(1)=a, f(2)=b$ and $21=4^{2}+5\\times 1^{2}=1^{2}+5\\times 2^{2}$ , $84=8^{2}+5\\times 2^{2}=2^{2}+5\\times 4^{2}$ , $69=7^{2}+5\\times 2^{2}=8^{2}+5\\times 1^{2}$ , $129=7^{2}+5\\times 4^{2}=2^{2}+5\\times 5^{2}$ , $345=10^{2}+5\\times 7^{2}=5^{2}+5\\times 8^{2}$ , $54=3^{2}+5\\times 3^{2}=7^{2}+5\\times 1^{2}$ , $81=6^{2}+5\\times 3^{2}=1^{2}+5\\times 4^{2}$ , $126=9^{2}+5\\times 3^{2}=1^{2}+5\\times 5^{2}$ , we have $\\left\\lbrace \\begin{array}{c}f^{2}(4)+5f^{2}(1)=f^{2}(1)+5f^{2}(2), \\\\f^{2}(8)+5f^{2}(2)=f^{2}(2)+5f^{2}(4), \\\\f^{2}(7)+5f^{2}(2)=f^{2}(8)+5f^{2}(1), \\\\f^{2}(7)+5f^{2}(4)=f^{2}(2)+5f^{2}(5), \\\\f^{2}(10)+5f^{2}(7)=f^{2}(5)+5f^{2}(8), \\\\f^{2}(3)+5f^{2}(3)=f^{2}(7)+5f^{2}(1), \\\\f^{2}(6)+5f^{2}(3)=f^{2}(1)+5f^{2}(4), \\\\f^{2}(9)+5f^{2}(3)=f^{2}(1)+5f^{2}(5).\\end{array}\\right.$ Solving (REF ), (REF )-(REF ), (REF ), (REF )-(REF ) hold, so we complete the proof.", "$\\square $ Now we evaluate $f(1)$ .", "Lemma 6.2 $f(1)\\in \\lbrace 0,1,-1, \\frac{1}{6}, -\\frac{1}{6}\\rbrace $ .", "Proof.", "Firstly, by $f(1)=a$ and $f(6)=f(1^{2}+5\\times 1^{2})=f^{2}(1)+5f^{2}(1)=6f^{2}(1)=6a^{2}$ , we have $ f^{2}(6)=36a^{4}.$ Similarly, by $f(2)=b$ and $f(9)=f(2^{2}+5\\times 1^{2})=f^{2}(2)+5f^{2}(1)=b^{2}+5a^{2}$ , we have $ f^{2}(9)=(b^{2}+5a^{2})^{2}.$ Combining (REF ), (REF ), (REF ) and (REF ), we have $\\left\\lbrace \\begin{array}{ll}\\frac{1}{3}(35b^{2}-32a^{2})=36a^{4} \\\\(b^{2}+5a^{2})^{2}=\\frac{1}{3}(80b^{2}-77a^{2}).\\end{array}\\right.$ Then we have $b^{2}=\\frac{1}{35}(108a^{4}+32a^{2}),$ and $(\\frac{108a^{4}+32a^{2}}{35}+5a^{2})^{2}=\\frac{1}{3}(\\frac{80(108a^{4}+32a^{2})}{35}-77a^{2})$ .", "By simplifying this equation, we have $a^{2}(a-1)(a+1)(6a-1)(6a+1)(36a^{2}+175)=0.$ Thus $a_{1}=a_{2}=0, a_{3}=1, a_{4}=-1, a_{5}=\\frac{1}{6}, a_{6}=-\\frac{1}{6}, a_{7}=\\frac{5\\sqrt{7}}{6}i, a_{8}=-\\frac{5\\sqrt{7}}{6}i.$ Now we show that $f(1)=\\pm \\frac{5\\sqrt{7}}{6}i$ is impossible.", "Otherwise, we have $a^{2}=-\\frac{175}{36}$ and $b^{2}=\\frac{2465}{36}$ by (REF ).", "Noting that $581=24^{2}+5\\times 1^{2}=9^{2}+5\\times 10^{2}$ , we have $f^{2}(24)=f^{2}(9)+5f^{2}(10)-5f^{2}(1)=\\frac{1}{3}(575a^{4}-572a^{2}).$ On the other hand, we know $f(24)=f(2^{2}+5\\times 2^{2})=6f^{2}(2)=6b^{2}$ , then by (REF ) we have $\\frac{1}{3}(575a^{4}-572a^{2})-36b^{4}=0.$ Unfortunately, when we put $a^{2}=-\\frac{175}{36}$ and $b^{2}=\\frac{2465}{36}$ in the left side of (REF ), we obtain $\\frac{1}{3}(575a^{4}-572a^{2})-36b^{4}\\ne 0.$ Thus it implies a contradiction and we complete the proof.", "$\\square $ By Lemmas REF -REF and direct calculation, we obtain the following Corollary.", "Corollary 6.3 Let $1\\le n \\le 10$ and $f$ satisfy the function equation (REF ).", "Then one of the following holds.", "(1) $f(n) \\equiv 0$ ; (2) $f(n) =\\left\\lbrace \\begin{array}{ll}n, & \\mbox{ if } n \\in \\lbrace 6,9\\rbrace ; \\\\\\pm n, & \\mbox{ otherwise}.\\end{array}\\right.$ (3) $f(n) =\\left\\lbrace \\begin{array}{ll}\\frac{1}{6}, & \\mbox{ if } n \\in \\lbrace 6,9\\rbrace ; \\\\\\pm \\frac{1}{6}, & \\mbox{ otherwise}.\\end{array}\\right.$ Now, by Theorem REF and Corollary REF , we complete the proof of Theorem REF .", "Theorem 6.4 Conjecture REF holds for $k=5$ ." ], [ "Some remarks", " So far, we know Conjecture REF holds for $k=1,2,3,4,5$ by [2] and the results of Sections 3-6.", "In fact, for $k=2,3,4,5$ , firstly, we find the formulae of $f(3), f(4), \\ldots , f(A)$ with $f(1)$ and $f(2)$ , then evaluate the value of $f(1)$ , and show the result holds for all $n$ with $1\\le n\\le A$ by direct calculation, finally, we show Conjecture REF holds for $k=2,3,4,5$ by Theorem REF .", "Similarly, we can show the cases $k=6, 7, \\ldots $ by the similar methods, so we omit it.", "Of course, we expect there exists better methods to simplify the proof.", "By the way, we find an interesting fact.", "The formulae (REF )-(REF ), (REF ) and (REF )-(REF ) all hold for $k=3,4,5$ .", "Are they also hold for $k=6, 7$ or more?", "We donot know, but it is worth expecting." ] ]	1606.05039
[ [ "An equation for the quench propagation velocity valid for high field\n magnet use of REBCO coated conductors" ], [ "Abstract Based on a study of the thermophysical properties, we derived a practical formula for the normal zone propagation velocity appropriate for REBa$_2$Cu$_3$O$_{7-x}$ coated conductors in high magnetic fields.", "An analytical expression to evaluate the current sharing temperature as a function of the operating conditions is also proposed.", "The presented study has allowed us to account for experimental results not fully understood in the framework of the models widely used in the literature.", "In particular, we provided a fundamental understanding of the experimental evidence that the normal zone propagation velocity in REBa$_2$Cu$_3$O$_{7-x}$ coated conductors can be mainly determined by the operating current, regardless of the applied field and temperature." ], [ "AIP/123-QED [Title]An equation for the quench propagation velocity valid for high field magnet use of REBCO coated conductors M. BonuraDepartment of Applied Physics (GAP) and Department of Quantum Matter Physics (DQMP), University of Geneva, quai Ernest Ansermet 24, CH-1211 Geneva, Switzerland.C.", "SenatoreDepartment of Applied Physics (GAP) and Department of Quantum Matter Physics (DQMP), University of Geneva, quai Ernest Ansermet 24, CH-1211 Geneva, Switzerland.", "Based on a study of the thermophysical properties, we derived a practical formula for the normal zone propagation velocity appropriate for REBa$_2$ Cu$_3$ O$_{7-x}$ coated conductors in high magnetic fields.", "An analytical expression to evaluate the current sharing temperature as a function of the operating conditions is also proposed.", "The presented study has allowed us to account for experimental results not fully understood in the framework of the models widely used in the literature.", "In particular, we provided a fundamental understanding of the experimental evidence that the normal zone propagation velocity in REBa$_2$ Cu$_3$ O$_{7-x}$ coated conductors can be mainly determined by the operating current, regardless of the applied field and temperature.", "Quench propagation velocity, coated conductors, HTS, REBCO, NZPV, CCs Second generation high temperature superconductors (2G HTS), i.e.", "REBa$_2$ Cu$_3$ O$_{7-x}$ (REBCO) coated conductors (CCs), are paving the way for the development of superconducting magnets that exceed the limits of LTS based technologies.", "[1], [2], [3], [4] HTS are not subjected to stability issues as LTS, and can in principle be operated without any protection.", "[5] However, this is usually not the case for HTS magnets, whose protection against quench-induced damages is motivated by their high cost.", "Different strategies are available.", "Active protection by distributed quench heaters is the one chosen for the 32 T magnet under construction at the NHMFL.", "[1], [6] Coils made using the no-insulation winding technique revealed to be self protecting, since the current can automatically bypass the normal spot.", "[7], [2] The normal zone propagation velocity ($NZPV$ ), i.e.", "the velocity of the superconducting/normal boundary during a quench, plays a key role in the quench process.", "The longitudinal component of the $NZPV$ assumes values two/three orders of magnitude smaller in 2G HTS than in LTS.", "[5] This makes the quench detection a critical issue for HTS magnets.", "[8] The propagation of the perturbation along the conductor depends on the properties of the conductor itself whilst the transverse component of the $NZPV$ is also determined by any other materials possibly present in the winding.", "Low values for the longitudinal $NZPV$ make the quench propagation in the transverse direction more relevant for HTS than for LTS.", "Properties of CCs are evolving very rapidly because of the intensive research carried out by industry and University research centers.", "Hence, reliable and practical tools to determine the $NZPV$ as a function of the conductor properties are highly desirable.", "This is the main motivation of this letter, which proposes a new analytical procedure to evaluate the longitudinal $NZPV$ in CCs with an accuracy higher than that associated with formulas commonly used in the literature.", "[5], [9] The results of our study have been compared with experimental data obtained in a commercial CC manufactured by SuperPower.", "[10] The $NZPV$ can be evaluated following different approaches.", "It can be measured experimentally in conditions as close as possible to those realized in a winding.", "However, measurements are complex and time consuming.", "[10] Numerical simulations are another way to assess the $NZPV$ , particularly useful in case of systems with complex design.", "[11] The most practical way to evaluate the $NZPV$ is to use the formulas that result from the solution of the heat equation describing the quench process.", "[5], [9], [12] A good agreement between experimental and calculated values has been demonstrated in case of LTS.", "Nevertheless, the validity of the approximations made when solving the equations may fail for HTS.", "[5], [9] In spite of this, scientists and magnet designers continue using confidently the analytical approach to reach an understanding of quenches in HTS, due to the difficulties encountered when evaluating the $NZPV$ experimentally or numerically.", "Based on the experimental study of the thermal conduction properties $\\kappa (T,B)$ , in the following we derive a practical equation for the $NZPV$ appropriate for 2G HTS in presence of high magnetic fields.", "The differential equation describing the adiabatic quench process in a superconductor is:[5] $C(T)\\frac{\\partial T}{\\partial t}=\\nabla \\cdot [\\kappa (T)\\nabla T]+g_j(T) .$ In Eq.", "(REF ), the left-hand side represents the variation in the internal energy density of the conductor, $C$ being the volumetric specific heat.", "On the right-hand side, the first term describes the thermal conduction in the conductor; the second term is the Joule heating, whose explicit form in case of composite superconductors is $g_j(T)=\\rho _m(T)J_m(T)J$ , where $\\rho _m$ is the matrix electrical resistivity, $J_m$ and $J$ the current density in the matrix and in the composite, respectively.", "[13] Whetstone and Roos derived an analytical expression for the $NZPV$ in adiabatic conditions, assuming that the normal-superconducting boundary during a quench can be represented by a translating coordinate system moving at $NZPV$ .", "[14] The formula was successively modified by Bellis and Iwasa in order to take into account the effects due to the current sharing between superconductor and metal matrix.", "[13] This adaptation is particularly important in the case of HTS because the temperature range over which the current sharing occurs is much larger than in LTS.", "[13] The deduced expression is: $NZPV\\approx J\\bigg [\\frac{1}{\\rho _n(T_t)\\kappa _n(T_t)}\\bigg (C_n(T_t)-\\frac{1}{\\kappa _n(T_t)}\\frac{d\\kappa _n}{dT}\\bigg \|_{T=T_t} \\nonumber \\\\\\times \\int _{T_{Op}}^{T_t}C_S(T)dT\\bigg )\\int _{T_{Op}}^{T_t}C_S(T)dT\\bigg ]^{-1/2} .$ The subscripts n and s refer to the normal and superconducting state, respectively.", "However, in REBCO CCs one can consider that $C_n\\approx C_s$ and $\\kappa _n \\approx \\kappa _s$ , because variations in the overall specific heat and thermal conductivity due to the transition from the superconducting to the normal state are negligible.", "[15], [16] The transition temperature, $T_t$ , has been introduced in place of $T_C$ in Eq.", "(REF ) in order to define an effective superconducting/normal boundary during a quench when current sharing effects are important.", "In LTS, $T_t$ is normally considered to be the average value between the temperature at which the current sharing starts ($T_{CS}$ ), and the critical temperature ($T_C$ ), i.e.", "$T_t\\equiv (T_{CS}+T_C)/2$ .", "In HTS, $T_t$ is more properly evaluated as the temperature at which the heat generation term in Eq.", "(REF ) assumes its average value in the current sharing temperature range, i.e.", "when $g_j(T_t)=\\bigg [\\int _{T_{CS}}^{T_C} g_j(T)dT\\bigg ]/(T_C-T_{CS}) .$ Neglecting the temperature dependence of the material properties, Eq.", "(REF ) may be simplified to $NZPV\\approx \\frac{J}{C}\\bigg [\\frac{\\rho \\kappa }{(T_t-T_{Op})}\\bigg ]^{1/2} .$ Different approaches have been proposed to evaluate the $NZPV$ by Eq.", "(REF ).", "Iwasa has shown that for $(T_t-T_{Op})/T_{Op}\\ll 1$ , $C$ , $\\rho $ , and $\\kappa $ can be conveniently evaluated at $\\tilde{T}\\equiv (T_{op}+T_t)/2$ , i.e.", "at the mean value between the operating and transition temperatures.", "[5] On the other hand, Wilson has proposed to evaluate $\\rho $ , and $\\kappa $ at $T=T_t$ , using for $C$ the average value in the range $T_{Op}-T_C$ .", "[9] The two procedures lead to similar values for the $NZPV$ in LTS and are in general not applicable to HTS.", "Nevertheless, their use has been extended in the practice also to HTS,[5], [16], [10] because of the complications encountered when solving the more general Eq.", "(REF ).", "In particular, the calculation of the $NZPV$ from Eq.", "(REF ) is hindered by the need of details about the $\\kappa (T)$ curve of the conductor at the conditions realized in a magnet.", "It is worth mentioning that, in order to bypass these difficulties, an approximation of Eq.", "(REF ) alternative to Eq.", "(REF ) has been derived under the less stringent assumption that only the $T$ dependence of the electrical resistivity is negligible.", "[17], [18], [19] However, this hypothesis is correct only for LTS because their transition temperature typically falls in the low-temperature region where $\\rho \\approx \\rho _{Res}$ , the residual electrical resistivity.", "Figure: Temperature dependence of the longitudinal thermal conductivity at different fields for the CC from SuperPower (product ID: SC S4050).", "The field has been applied parallel to the tape surface, perpendicularly to the thermal-flow direction.", "Inset: specific heat data as measured on the CC.Recently, we reported on the thermal conduction properties of REBCO CCs from different manufacturers in magnetic fields up to 19 T.[15] In Figure 1, the experimental $\\kappa (T)$ curves of the CC from SuperPower are shown as measured at $B=0$ , 7 and 19 T. The solid line is the best-fit curve obtained by fitting the experimental data at $B=0$ considering that $\\kappa \\approx \\kappa _{Cu}\\cdot s_{Cu}$ .", "[15] $\\kappa _{Cu}$ is the thermal conductivity of the copper, whose dependence on the $RRR$ is described in Ref.", "[20], and $s_{Cu}$ is the conductor's cross-section fraction occupied by the stabilizer.", "Dashed lines, associated with in-field data, have been calculated in the framework of the Wiedemann-Franz law supposing that $\\kappa (T,B)\\approx [\\rho (T,0)/\\rho (T,B)] \\kappa (T,0)$ and using magnetoresistance data measured on Cu specimens extracted from the CC, as described in Ref.[15].", "At low $T$ , electron-defect scattering processes dominate the heat propagation.", "Thus, the effect of the magnetic field is in some way analogous to that of disorder in the system: both reduce the electron mean free path and, consequently, $\\kappa $ .", "On increasing $T$ , the field-induced effects on $\\kappa $ become less important and the $\\kappa (T)$ curves associated with different $B$ values approach each other.", "This is a consequence of the fact that electron-phonon scattering events start to be more relevant than electron-defect ones in determining the heat conduction for $T\\gtrsim 50$ K. Data reported in Figure 1 exhibit typical features of $\\kappa (T,B)$ curves of REBCO CCs produced by different manufacturers,[15] and provide us the necessary understanding to formulate a practical expression for the $NZPV$ suitable for 2G HTS.", "$T_t$ assumes values higher than $\\approx 45$ K in HTS.", "In this range of temperatures, the derivative of $\\kappa (T)$ is strongly reduced on increasing the field, as implied by Figure 1.", "It follows that in Eq.", "(REF ) the term $\\frac{1}{\\kappa _n(T_t)}\\frac{d\\kappa _n}{dT}\|_{T=T_t}\\times \\int _{T_{Op}}^{T_t}C_S(T)dT$ becomes negligible with respect to $C_n(T_t)$ in case of operation at intense fields, and Eq.", "(REF ) can be approximated by: $NZPV\\approx J \\bigg [\\frac{\\rho (T_t)\\kappa (T_t)}{C(T_t)\\int _{T_{Op}}^{T_t}C(T)dT}\\bigg ]^{1/2}\\, .$ In Figure 2, the relative error made when using Eq.", "(REF ) in place of Eq.", "(REF ) is plotted as a function of $T_t$ , for different operating conditions ($T_{Op},B$ ), as determined for the SuperPower tape.", "The error decreases on augmenting $B$ and is always smaller than $7\\%$ at 19 T. Results very similar to those reported in Figure 2 have been obtained for the CCs from other manufacturers investigated in Ref. [15].", "Indeed, the validity of the approximation that leads to Eq.", "(REF ) relies on the $\\kappa (T,B)$ properties of copper, which gives the predominant contribution to the overall thermal conductivity of the tape.", "Thus, Eq.", "(REF ) can be generally used to study quench processes in Cu-stabilized 2G HTS in the presence of intense fields.", "For the sake of completeness, we want to mention that Dresner published in 1994 a study in which closed formulas for the $NZPV$ are derived considering specific dependencies of the specific heat on the temperature.", "In the general case of an arbitrary dependence of $C$ on $T$ , he proposed to solve Eq.", "(REF ) disregarding the entire term $\\nabla \\cdot [\\kappa (T)\\nabla T]$ when $T>T_C$ .", "This corresponds to neglect not only the term $\\kappa \\nabla ^2T$ as done (and justified) by Whetstone and Roos but also the term $\\nabla \\kappa \\nabla T$ .", "These assumptions lead to an expression for the $NZPV$ formally analogous to Eq.", "(REF ), with $T_C$ in the place of $T_t$ , since the author did not consider the current sharing effect.", "[21] However, Dresner did not justify the hypothesis of neglecting the $T$ dependence of $\\kappa $ .", "The validity of this assumption, which leads to Eq.", "(REF ), has been fully demonstrated in this letter in the case of Cu-stabilized REBCO CCs submitted to intense fields.", "Figure: Relative error when using Eq.", "() in the place of the more general Eq.", "() for different operating conditions (T Op ,BT_{Op},B), as determined for the CC from SuperPower.Eq.", "(REF ) can be further simplified considering that in the framework of the Wiedemann-Franz law $\\rho (T_t)\\kappa (T_t)\\approx LT_t$ .", "This reduces the parameters needed to perform the calculation to: $L$ , $C$ , and $T_t$ (apart from $J$ ).", "$L$ values of REBCO CCs are available in the literature[15].", "It has been shown that $L$ does not depend noticeably on $B$ .", "[15] The specific heat of CCs can be calculated from data of the component materials, considering that $C(T)=\\sum v_iC_i(T)$ , $v_i$ being the volume fraction occupied by the $i^{th}$ component.", "It is expected that the predominant contributions come from the substrate and the stabilizer, because of the large $v_i$ values.", "However, we have investigated experimentally the $C(T)$ curve using a Quantum Design PPMS, in order to get more precise results.", "Data relative to the tape from SuperPower are shown in the inset of Figure 1.", "Details on the critical current surface of the CC are needed to determine $T_t$ .", "Recently, it has been shown that the $T$ dependence of $J_C$ of CCs from different manufacturers can be described over a broad range of temperatures and fields by an exponential law, $J_{C}(T,B)=J_{C}(T=0,B)e^{-T/T^}$ .", "Deviations from this behavior are observed at temperatures $\\gtrsim 50$ K.[22] The exponential dependence of $J_C$ is connected with the presence of defects generating weak isotropic pinning and $T^$ is the characteristic pinning energy at these defects.", "[23], [24] Other dominant pinning mechanisms can lead to different $J_C(T,B)$ characteristics.", "[25] The $T^$ values associated with tapes from different manufacturers, for different orientations between the field and the tape surface, are reported in Ref.[22].", "From the expression for $J_{C}(T,B)$ one can easily deduce the following formula that relates $T_{CS}$ to parameters directly chosen by the magnet designer, namely the operating conditions and the current margin $T_{CS}\\approx T_{Op}-T^ln\\frac{I_{Op}}{I_C(B_{Op},T_{Op})} \\, .$ Figure: Dependence of the NZPVNZPV on the operating current at different T Op T_{Op} and BB, deduced from Eq.", "() (lines and symbols) and from Eq.", "() following the procedure described by Iwasa (lines without symbols).$T_t$ can be evaluated from Eq.", "(REF ), using $T_{CS}$ values from Eq.", "(REF ) and the expression for $J_C(T,B)$ reported in Ref. [22].", "We have also verified that the approximated formula $T_t\\equiv (T_{CS}+T_C)/2$ leads to a good estimation for $T_t$ , with differences within $\\approx 5\\%$ , when $T\\gtrsim 20$ K. At 4.5 K, the discrepancies increase up to $\\approx 10\\%$ .", "Table: Critical Current Densities of the investigated CCIn Figure 3, we report the $NZPV$ as determined from Eq.", "(REF ) considering that $\\rho (T_t)\\kappa (T_t)\\approx L T_t$ , using $L$ values reported in Ref.", "[15], $T_{t}$ calculated using the definition given in Eq.", "(REF ), and experimental $C(T)$ data shown in the inset of Figure 1.", "$I_{Op}$ has been varied in the range 0.2-0.9 $I_C$ , using for $I_C(T,B)$ the values reported in Table 1.", "These have been measured on CCs extracted from the same batch of the sample used for the thermal conduction studies, with the field applied parallel to the wide surface of the tape.", "Figure 3 shows that the $NZPV$ is mainly determined by the operating current.", "Points associated with different temperatures and fields approximatively reconstruct a single line in a log-log plot, defining a power-law dependence of the $NZPV$ on $I_{Op}$ .", "This behavior is unexpected if compared to what observed in LTS.", "Indeed, both in NbTi and Nb$_3$ Sn a clear dependence of the $NZPV$ on $B$ is found.", "[17] The contrast between the result shown in Figure 3 and what is observed in LTS is certainly related with the different $I_C(T,B)$ characteristics of the materials.", "$NZPV$ values calculated from Eq.", "(4) following the procedure described by Iwasa[5] are shown in Figure 3 as lines without symbols.", "Data associated with different operating conditions do not lie all on a same straight line in a log-log plot.", "Discrepancies between results from Eq.", "(REF ) and Eq.", "(REF ) become more evident, both qualitatively and quantitatively, on decreasing the operating temperature.", "This is worth to underline in view of applications of CCs in very high field magnets.", "Our results about the dependence of the $NZPV$ on $I_{Op}$ at different operating conditions are confirmed by the experimental $NZPV$ studies performed on a SuperPower tape extracted from another batch with respect to ours.", "[10] The experimental confirmation strengthens the validity of the analytical procedure proposed in this letter to determine the $NZPV$ .", "When comparing quantitatively experimental data with theoretical expectations, one has to take into account the so-called minimum propagation current ($I_{mp}$ ) i.e.", "the operating current below which there is no quench triggering even for pulses with an energy exceeding the stability margin.", "[16], [10] $I_{mp}$ values reported in the literature for CCs are in the range $10-30$ A.", "[16] The effect of $I_{mp}$ on the measured $NZPV$ can be neglected when $I_{Op}\\gg I_{mp}$ .", "Samples investigated by us and in Ref.", "[10] present slightly different $I_C$ characteristics.", "Nevertheless, we verified that for $I_{Op}\\gtrsim 100$ A, which is much larger than expected $I_{mp}$ , discrepancies between values shown in Figure 3 and data from Ref.", "[10] are below 25%.", "Results from Eq.", "(REF ), combined with longitudinal and transverse $\\kappa $ data, allow calculating the transverse $NZPV$ .", "[5], [9] In Ref.", "[26] we have reported experimental values for the square root of the ratio between the transverse and longitudinal components of $\\kappa $ for various CCs.", "Typical values are of the order of 0.1.", "These data provide lower limits for the anisotropy of the $NZPV$ in a winding, since the contact thermal resistance or the presence of other materials, which could reduce the overall transverse $\\kappa $ , have not been considered.", "In summary, an approximated equation has been derived for the longitudinal $NZPV$ , Eq.", "(REF ), particularly suitable for 2G HTS in intense magnetic fields.", "An analytical expression to evaluate the current sharing temperature as a function of the operating conditions, Eq.", "(REF ), has also been proposed.", "The presented study has allowed us to take into account experimental results not fully understood in the framework of models widely used in the literature.", "Financial support was provided by the SNSF (Grants No.", "PP00P2$\\_$ 144673 and No.51NF40-144613) and by FP7 EuCARD-2.", "EuCARD-2 is cofounded by the partners and the European Commission under Capacities 7th Framework Programme, Grant Agreement 312453.", "The authors thank Piotr Komorowski and Christian Barth for useful discussions." ] ]	1606.05073
[ [ "How many faces can be recognized? Performance extrapolation for\n multi-class classification" ], [ "Abstract The difficulty of multi-class classification generally increases with the number of classes.", "Using data from a subset of the classes, can we predict how well a classifier will scale with an increased number of classes?", "Under the assumption that the classes are sampled exchangeably, and under the assumption that the classifier is generative (e.g.", "QDA or Naive Bayes), we show that the expected accuracy when the classifier is trained on $k$ classes is the $k-1$st moment of a \\emph{conditional accuracy distribution}, which can be estimated from data.", "This provides the theoretical foundation for performance extrapolation based on pseudolikelihood, unbiased estimation, and high-dimensional asymptotics.", "We investigate the robustness of our methods to non-generative classifiers in simulations and one optical character recognition example." ], [ "Introduction", "In multi-class classification, one observes pairs $(z, y)$ where $y \\in \\mathcal {Y} \\subset \\mathbb {R}^p$ are feature vectors, and $z$ are unknown labels, which lie in a countable label set $\\mathcal {Z}$ .", "The goal is to construct a classification rule for predicting the label of a new data point; generally, the classification rule $h: \\mathcal {Y} \\rightarrow \\mathcal {Z}$ is learned from previously observed data points.", "In many applications of multi-class classification, such as face recognition or image recognition, the space of potential labels is practically infinite.", "In such a setting, one might consider a sequence of classification problems on finite label subsets $\\mathcal {Z}_1 \\subset \\cdots \\subset \\mathcal {Z}_K$ , where in the $i$ -th problem, one constructs the classification rule $h^{(i)}:\\mathcal {Y} \\rightarrow \\mathcal {Z}_i$ .", "Supposing that $(Z, Y)$ have a joint distribution, define the accuracy for the $i$ -th problem as $\\text{acc}^{(i)} = \\Pr [h^{(i)}(Y) = Z\|Z \\in \\mathcal {Z}_i].$ Using data from only $\\mathcal {Z}_k$ , can one predict the accuracy achieved on the larger label set $\\mathcal {Z}_K$ , with $K> k$ ?", "This is the problem of performance extrapolation.", "A practical instance of performance extrapolation occurs in neuroimaging studies, where the number of classes $k$ is limited by experimental considerations.", "Kay et al.", "[1] obtained fMRI brain scans which record how a single subject's visual cortex responds to natural images.", "The label set $\\mathcal {Z}$ corresponds to the space of all grayscale photographs of natural images, and the set $\\mathcal {Z}_1$ is a subset of 1750 photographs used in the experiment.", "They construct a classifier which achieves over 0.75 accuracy for classifying the 1750 photographs; based on exponential extrapolation, they estimate that it would take on the order of $10^{9.5}$ photographs before the accuracy of the model drops below 0.10!", "Directly validating this estimate would take immense resources, so it would be useful to develop the theory needed to understand how to compute such extrapolations in a principled way.", "However, in the fully general setting, it is impossible on construct non-trivial bounds on the accuracy achieved on the new classes $\\mathcal {Z}_K \\setminus \\mathcal {Z}_k$ based only on knowledge of $\\mathcal {Z}_k$ : after all, $\\mathcal {Z}_k$ could consist entirely of well-separated classes while the new classes $\\mathcal {Z}_K \\setminus \\mathcal {Z}_k$ consist entirely of highly inseparable classes, or vice-versa.", "Thus, the most important assumption for our theory is that of exchangeable sampling.", "The labels in $\\mathcal {Z}_i$ are assumed to be an exchangeable sample from $\\mathcal {Z}$ .", "The condition of exchangeability ensures that the separability of random subsets of $\\mathcal {Z}$ can be inferred by looking at the empirical distributions in $\\mathcal {Z}_k$ , and therefore that some estimate of the achievable accuracy on $\\mathcal {Z}_K$ can be obtained.", "The assumption of exchangeability greatly limits the scope of application for our methods.", "Many multi-class classification problems have a hierarchical structure [2], or have class labels distributed according to non-uniform discrete distributions, e.g.", "power laws [3]; in either case, exchangeability is violated.", "It would be interesting to extend our theory to the hierarchical setting, or to handle non-hierarchical settings with non-uniform prior class probabilities, but again we leave the subject for future work.", "In addition to the assumption of exchangeability, we consider a restricted set of classifiers.", "We focus on generative classifiers, which are classifiers that work by training a model separately on each class.", "This convenient property allows us to characterize the accuracy of the classifier by selectively conditioning on one class at a time.", "In section 3, we use this technique to reveal an equivalence between the expected accuracies of $\\mathcal {Z}_k$ to moments of a common distribution.", "This moment equivalence result allows standard approaches in statistics, such as U-statistics and nonparametric pseudolikelilood, to be directly applied to the extrapolation problem, as we discuss in section 4.", "In non-generative classifiers, the classification rule has a joint dependence on the entire set of classes, and cannot be analyzed by conditioning on individual classes.", "In section 5, we empirically study the performance of our classifiers.", "Since generative classifiers only comprise a minority of the classifiers used in practice, we applied our methods to a variety of generative and non-generative classifiers in simulations and in one OCR dataset.", "Our methods have varying success on generative and non-generative classifiers, but seem to work badly for neural networks.", "Our contribution.", "To our knowledge, we are the first to formalize the problem of prediction extrapolation.", "We introduce three methods for prediction extrapolation: the method of extended unbiased estimation and the constrained pseudolikelihood method are novel.", "The third method, based on asymptotics, is a new application of a recently proposed method for estimating mutual information [4]." ], [ "Setting", "Having motivated the problem of performance extrapolation, we now reformulate the problem for notational and theoretical convenience.", "Instead of requiring $\\mathcal {Z}_k$ to be a random subset of $\\mathcal {Z}$ as we did in section 1, take $\\mathcal {Z}=\\mathbb {N}$ and $\\mathcal {Z}_k = \\lbrace 1,\\hdots , k\\rbrace $ .", "We fix the size of $\\mathcal {Z}_k$ without losing generality, since any monotonic sequence of finite subsets can be embedded in a sequence with $\|\\mathcal {Z}_k\| = k$ .", "In addition, rather than randomizing the labels, we will randomize the marginal distribution $p(y\|z)$ of each label; Towards that end, let $\\mathcal {Y} \\subset \\mathbb {R}^p$ be a space of feature vectors, and let $\\mathcal {P}(\\mathcal {Y})$ be a measurable space of probability distributions on $\\mathcal {Y}$ .", "Let $\\mathbb {F}$ be a probability measure on $\\mathcal {P}$ , and let $F_1, F_2,\\hdots $ be an infinite sequence of i.i.d.", "draws from $\\mathbb {F}$ .", "We refer to $\\mathbb {F}$ , a probability measure on probability measures, as a meta-distribution.", "The distributions $F_1,\\hdots , F_k$ are the marginal distributions of the first $k$ classes.", "Further assuming that the labels are equiprobable, we rewrite the accuracy as $\\text{acc}^{(t)} = \\frac{1}{t}\\sum _{i=1}^t \\Pr _{F_i}[h^{(t)}(Y) = i].$ where the probabilities are taken over $Y \\sim F_i$ .", "In order to construct the classification rule $h^{(t)}$ , we need data from the classes $F_1,\\hdots , F_t$ .", "In most instances of multi-class classification, one observes independent observations from each $F_i$ which are used to construct the classifier.", "Since the order of the observations does not generally matter, a sufficient statistic for the training data for the $t$ -th classification problem is the collection of empirical distributions $\\hat{F}_1^{(t)},\\hdots ,\\hat{F}_t^{(t)}$ for each class.", "Henceforth, we make the simplifying assumption that the training data for the $i$ -th class remains fixed from $t =i, i+1,\\hdots $ , so we drop the superscript on $\\hat{F}_i^{(t)}$ .", "Write $\\hat{\\mathbb {F}}(F)$ for the conditional distribution of $\\hat{F}_i$ given $F_i = F$ ; also write $\\hat{\\mathbb {F}}$ for the marginal distribution of $\\hat{F}$ when $F \\sim \\mathbb {F}.$ As an example, suppose every class has the number of training examples $r \\in \\mathbb {N}$ ; then $\\hat{F}$ is the empirical distribution of $r$ i.i.d.", "observations from $F$ , and $\\hat{\\mathbb {F}}(F)$ is the empirical meta-distribution of $\\hat{F}$ .", "Meanwhile, $\\hat{\\mathbb {F}}$ is the true meta-distribution of the empirical distribution of $r$ i.i.d.", "draws from a random $F \\sim \\mathbb {F}$ ." ], [ "Multiclass classification", "Extending the formalism of Tewari and Bartlett [5]As in their framework, we define a classifier as a vector-valued function.", "However, we introduce the notion of a classifier as a multiple-argument functional on empirical distributions, which echoes the functional formulation of estimators common in the statistical literature., we define a classifier as a collection of mappings $\\mathcal {M}_i: \\mathcal {P}(\\mathcal {Y})^k \\times \\mathcal {Y} \\rightarrow \\mathbb {R}$ called classification functions.", "Intuitively speaking, each classification function learns a model from the first $k$ arguments, which are the empirical marginals of the $k$ classes, $\\hat{F}_1,\\hdots , \\hat{F}_k$ .", "For each class, the classifier assigns a real-valued classification score to the query point $y \\in \\mathcal {Y}$ .", "A higher score $\\mathcal {M}_i(\\hat{F}_1,\\hdots , \\hat{F}_k, y)$ indicates a higher estimated probability that $y$ belongs to the $k$ -th class.", "Therefore, the classification rule corresponding to a classifier $\\mathcal {M}_i$ assigns a class with maximum classification score to $y$ : $h(y) = \\text{argmax}_{i \\in \\lbrace 1,\\hdots , k\\rbrace } \\mathcal {M}_i(y).$ For some classifiers, the classification functions $\\mathcal {M}_i$ are especially simple in that $\\mathcal {M}_i$ is only a function of $\\hat{F}_i$ and $y$ .", "Furthermore, due to symmetry, in such cases one can write $\\mathcal {M}_i(\\hat{F}_1,\\hdots , \\hat{F}_k, y) = \\mathcal {Q}(\\hat{F}_i, y),$ where $\\mathcal {Q}$ is called a single-class classification function (or simply classification function), and we say that $\\mathcal {M}$ is a generative classifier.", "Quadratic discriminant analysis and Naive Bayes [6] are two examples of generative classifiersFor QDA, the classification function is given by $\\mathcal {Q}_{QDA}(\\hat{F}, y) = -(y - \\mu (\\hat{F}))^T \\Sigma (\\hat{F})^{-1} (y-\\mu (\\hat{F})) - \\log \\det (\\Sigma (\\hat{F})),$ where $\\mu (F) = \\int y dF(y)$ and $\\Sigma (F) = \\int (y-\\mu (F))(y-\\mu (F))^T dF(y)$ .", "In Naive Bayes, the classification function is $\\mathcal {Q}_{NB}(\\hat{F}, y) = \\sum _{i=1}^n \\log \\hat{f}_i(y_i),$ where $\\hat{f}_i$ is a density estimate for the $i$ -th component of $\\hat{F}$ .", ".", "The generative property allows us to prove strong results about the accuracy of the classifier under the exchangeable sampling assumption, as we see in Section 3." ], [ "Performance extrapolation for generative classifiers", "Let us specialize to the case of a generative classifier, with classification function $\\mathcal {Q}$ .", "Consider estimating the expected accuracy for the $k$ -th classification problem, $p_k \\stackrel{def}{=} \\textbf {E}[\\text{acc}^{(k)}].$ In the case of a generative classifier, we have $p_k = \\textbf {E}[acc^{(k)}] = \\textbf {E}\\left[\\frac{1}{k}\\sum _{i=1}^k \\Pr _{Y \\sim F_i}[\\mathcal {Q}(\\hat{F}_i, Y) > \\max _{j \\ne i}\\mathcal {Q}(\\hat{F}_j, Y)]\\right].$ Define the conditional accuracy function $u(\\hat{F}, y)$ which maps a distribution $\\hat{F}$ on $\\mathcal {Y}$ and a test observation $y$ to a real number in $[0,1]$ .", "The conditional accuracy gives the probability that for independently drawn $\\hat{F}^{\\prime }$ from $\\hat{\\mathbb {F}}$ , that $\\mathcal {Q}(\\hat{F}, y)$ will be greater than $\\mathcal {Q}(\\hat{F}^{\\prime }, y)$ : $u(\\hat{F}, y) = \\Pr _{\\hat{F}^{\\prime } \\sim \\hat{\\mathbb {F}}}[\\mathcal {Q}(\\hat{F}, y) > \\mathcal {Q}(\\hat{F}^{\\prime }, y)].$ Define the conditional accuracy distribution $\\nu $ as the law of $u(\\hat{F}, Y)$ where $\\hat{F}$ and $Y$ are generated as follows: (i) a true distribution $F$ is drawn from $\\mathbb {F}$ ; (ii) the empirical distribution $\\hat{F}$ is drawn from $\\hat{\\mathbb {F}}(F)$ (i.e., the training data for the class), (iii) the query $Y$ is drawn from $F$ , with $Y$ independent of $\\hat{F}$ (i.e.", "a single test data point from the same class.)", "The significance of the conditional accuracy distribution is that the expected accuracy $p_t$ can be written in terms of its moments.", "Theorem 3.1.", "Let $\\mathcal {Q}$ be a single-distribution classification function, and let $\\mathbb {F}$ , $\\hat{\\mathbb {F}}(F)$ be a distribution on $\\mathcal {P}(\\mathcal {Y}).$ Further assume that $\\hat{\\mathbb {F}}$ and $\\mathcal {Q}$ jointly satisfy the tie-breaking property: $\\Pr [\\mathcal {Q}(\\hat{F}, y) = \\mathcal {Q}(\\hat{F}^{\\prime }, y)] = 0$ for all $y \\in \\mathcal {Y}$ , where $\\hat{F}, \\hat{F}^{\\prime } \\stackrel{iid}{\\sim } \\hat{\\mathbb {F}}$ .", "Let $U$ be defined as the random variable $U = u(\\hat{F}, Y)$ for $F \\sim \\mathbb {F}$ , $Y \\sim F$ , and $\\hat{F} \\sim \\hat{\\mathbb {F}}(F)$ with $Y \\perp \\hat{F}$ .", "Then $p_k = \\textbf {E}[U^{k-1}],$ where $p_k$ is the expected accuracy as defined by (REF ).", "Proof.", "Write $q^{(i)}(y) = \\mathcal {Q}(\\hat{F}_i, y)$ .", "By using conditioning and conditional independence, $p_k$ can be written $p_k &= \\textbf {E}\\left[ \\frac{1}{k}\\sum _{i=1}^k \\Pr _{F_i}[q^{(i)}(Y) > \\max _{j\\ne i} q^{(j)}(Y)] \\right]\\\\&= \\textbf {E}\\left[ \\Pr _{F_1}[q^{(1)}(Y) > \\max _{j\\ne 1} q^{(j)}(Y)] \\right]\\\\&= \\textbf {E}_{F_1}[\\Pr [q^{(1)}(Y) > \\max _{j\\ne 1} q^{(j)}(Y)\|\\hat{F}_1, Y]]\\\\&= \\textbf {E}_{F_1}[\\Pr [\\cap _{j > 1} q^{(1)}(Y) > q^{(j)}(Y)\|\\hat{F}_1, Y]]\\\\&= \\textbf {E}_{F_1}[\\prod _{j > 1}\\Pr [q^{(1)}(Y) > q^{(j)}(Y)\|\\hat{F}_1, Y]]\\\\&= \\textbf {E}_{F_1}[\\Pr [q^{(1)}(Y) > q^{(2)}(Y)\|\\hat{F}_1, Y]^{k-1}]\\\\&= \\textbf {E}_{F_1}[u(\\hat{F}_1, Y)^{k-1}] = \\textbf {E}[U^{k-1}].$ $\\Box $ Theorem 3.1 tells us that the problem of extrapolation can be approached by attempting to estimate the conditional accuracy distribution.", "The $(t-1)$ -th moment of $U$ gives us $p_t$ , which will in turn be a good estimate of $\\text{acc}^{(t)}$ .", "While $U = u(\\hat{F}, Y)$ is not directly observed, we can obtain unbiased estimates of $u(\\hat{F}_i, y)$ by using test data.", "For any $\\hat{F}_1,\\hdots , \\hat{F}_k$ , and independent test point $Y \\sim F_i$ , define $\\hat{u}(\\hat{F}_i, Y) = \\frac{1}{k -1}\\sum _{j \\ne i} I(\\mathcal {Q}(\\hat{F}_i, Y) > \\mathcal {Q}(\\hat{F}_j, Y)).$ Then $\\hat{u}(\\hat{F}_i, Y)$ is an unbiased estimate of $u(\\hat{F}_i, Y)$ , as stated in the following theorem.", "Theorem 3.2.", "Assume the conditions of theorem 3.1.", "Then defining $V = (k-1)\\hat{u}(\\hat{F}_i, y),$ we have $V \\sim \\text{\\emph {Binomial}}(k-1, u(\\hat{F}_i, y)).$ Hence, $\\textbf {E}[\\hat{u}(\\hat{F}_i, y)] = u(\\hat{F}_i, y).$ In section 4, we will use this result to estimate the moments of $U$ .", "Meanwhile, since $U$ is a random variable on $[0, 1]$ , we also conclude that $p_t$ follows a mixed exponential decay.", "Let $\\alpha $ be the law of $-\\log (U)$ .", "Then from change-of-variables $\\kappa =-\\log (u)$ , we get $p_t = \\textbf {E}[U^{t-1}] =\\int _0^1 u^{t-1} d\\nu (u) = \\int _0^1 e^{t\\log (u)} \\frac{1}{u}d\\nu (u) =\\int _{\\mathbb {R}^{+}} e^{-\\kappa t} d\\alpha (\\kappa ).$ This fact immediately suggests the technique of fitting a mixture of exponentials to the test accuracy at $t =2,3,\\hdots , k$ : we explore this idea further in Section 4.1." ], [ "Properties of the conditional accuracy distribution", "The conditional accuracy distribution $\\nu $ is determined by $\\mathbb {F}$ and $\\mathcal {Q}$ .", "What can we say about the the conditional accuracy distribution without making any assumptions on either $\\mathbb {F}$ or $\\mathcal {Q}$ ?", "The answer is: not much.", "For an arbitrary probability measure $\\nu ^{\\prime }$ on $[0,1]$ , one can construct $\\mathbb {F}$ and $\\mathcal {Q}$ such that the conditional accuracy $U$ has the distribution $\\nu ^{\\prime }$ , even if one makes the perfect sampling assumption that $\\hat{F}=F.$ Theorem 3.3.", "Let $U$ be defined as in Theorem 3.1, and let $\\nu $ denote the law of $U$ .", "Then, for any probability distribution $\\nu ^{\\prime }$ on $[0,1]$ , one can construct a meta-distribution $\\mathbb {F}$ and a classification function $\\mathcal {Q}$ such that the conditional accuracy $U$ has distribution $\\nu ^{\\prime }$ under perfect sampling (that is, $\\hat{F} = F$ .)", "Proof.", "Let $G$ be the cdf of $\\nu $ , $G(x) = \\int _0^x d\\nu (x)$ , and let $H(u) = \\sup _x \\lbrace G(x) \\le u\\rbrace $ .", "Define $\\mathcal {Q}$ by $\\mathcal {Q}(\\hat{F}, y) = {\\left\\lbrace \\begin{array}{ll}0 &\\text{ if }\\mu (\\hat{F}) > y + H(y)\\\\0 & \\text{ if }y + H(y) > 1 \\text{ and }\\mu (\\hat{F}) \\in [H(y) - y, y]\\\\1 + \\mu (\\hat{F}) - y &\\text{ if } \\mu (\\hat{F}) \\in [y, y + H(y)]\\\\1 + y + \\mu (\\hat{F}) &\\text{ if }\\mu (\\hat{F}) + H(y) > 1 \\text{ and }\\mu (\\hat{F}) \\in [0, H(y) - y].\\end{array}\\right.", "}$ Let $\\theta \\sim \\text{Uniform}[0,1]$ , and define $F \\sim \\mathbb {F}$ by $F = \\delta _\\theta $ , and also $\\hat{F} = F.$ A straightforward calculation yields that $\\nu = \\nu ^{\\prime }$ .", "$\\Box $ On the other hand, we can obtain a positive result if we assume that the classifier approximates a Bayes classifier.", "Assuming that $F$ is absolutely continuous with respect to Lebesgue measure $\\Lambda $ with probability one, a Bayes classifier results from assuming perfect sampling ($\\hat{F} = F$ ) and taking $\\mathcal {Q}(\\hat{F}, y) = \\frac{dF}{d\\Lambda }(y)$ .", "Theorem 3.4. states that for a Bayes classifier, the measure $\\nu $ has a density $\\eta (u)$ which is monotonically increasing.", "Since a `good' classifier approximates the Bayes classifier, we intuitively expect that a monotonically increasing density $\\eta $ is a good model for the conditional accuracy distribution of a `good' classifier.", "Theorem 3.4.", "Assume the conditions of theorem 3.1, and further suppose that $\\hat{F} = F$ , $F$ is absolutely continuous with respect to $\\Lambda $ with probability one, that $\\mathcal {Q}(\\hat{F}, y) = \\frac{dF}{d\\Lambda }(y)$ , and that $F\|Y$ has a regular conditional probability distribution.", "Let $\\nu $ denote the law of $U$ .", "Then $\\nu $ has a density $\\eta (u)$ on $[0, 1]$ which is monotonic in $u$ .", "Proof.", "It suffices to prove that $\\nu ([u, u + \\delta ]) < \\nu ([v, v + \\delta ])$ for all $0 < u < v < 1$ and $0 < \\delta < 1-v$ .", "Let $\\mathcal {P}_{ac}(\\mathcal {Y})$ denote the space of distributions supported on $\\mathcal {Y}$ which are absolutely continuous with respect to $p$ -dimensional Lebesgue measure $\\Lambda $ .", "Let $\\mathbb {Y}$ denote the marginal distribution of $Y$ for $Y \\sim F$ with $F \\sim \\mathbb {F}$ .", "Define the set $J_y(A) =\\lbrace F \\in \\mathcal {P}_{ac}(\\mathcal {Y}): u(F, y) \\in A\\rbrace .$ for all $A \\subset [0, 1].$ One can verify that for all $y \\in \\mathcal {Y}$ , $\\Pr _\\mathbb {F}[J_y([u, u + \\delta ])\|Y=y] \\le \\Pr _\\mathbb {F}[J_y([v, v + \\delta ])\|Y=y],$ using the fact that $\\mathbb {F}$ has no atoms.", "Hence, we obtain $\\Pr [U \\in [u-\\delta , u + \\delta ]] = \\textbf {E}_{\\mathbb {Y}}[\\Pr _\\mathbb {F}[J_Y([u, u + \\delta ])\|Y]]\\le \\textbf {E}_{\\mathbb {Y}}[\\Pr _\\mathbb {F}[J_Y([v, v + \\delta ])\|Y]] = Pr[U \\in [v - \\delta , v + \\delta ]].$ Taking $\\delta \\rightarrow 0$ , we conclude the theorem.", "$\\Box $" ], [ "Estimation", "Suppose we have $m$ independent test repeats per class, $y^{(i),1}\\hdots , y^{(i), m}$ .", "Let us define $V_{i,j} = \\sum _{\\ell \\ne i} I(\\mathcal {M}_i(\\hat{F}_1,\\hdots , \\hat{F}_k, y^{(i, j)}) > \\mathcal {M}_\\ell (\\hat{F}_1,\\hdots , \\hat{F}_k, y^{(i, j)})),$ which coincides with the definition (REF ) in the special case that $\\mathcal {M}$ is generative.", "At a high level, we have a hierarchical model where $U$ is drawn from a distribution $\\nu $ on $[0, 1]$ and then $V_{i, j} \\sim \\text{Binomial}(k, U)$ .", "Let us assume that $U$ has a density $\\eta (u)$ : then the marginal distribution of $V_{i, j}$ can be written $\\Pr [V_{i,j} = \\ell ] = \\begin{pmatrix}k \\\\ \\ell \\end{pmatrix}\\int _0^1 u^\\ell (1-u)^{k-\\ell } \\eta (u) du.$ However, the observed $\\lbrace V_{i, j}\\rbrace $ do not comprise an i.i.d.", "sample.", "We discuss the following three approaches for estimating $p_t =\\textbf {E}[U^{t-1}]$ based on $V_{i, j}$ .", "The first is an extension of unbiased estimation based on binomial U-statistics, which is discussed in Section 4.1.", "The second is the pseudolikelihood approach.", "In problems where the marginal distributions are known, but the dependence structure between variables is unknown, the pseudolikelihood is defined as the product of the marginal distributions.", "For certain problems in time series analysis and spatial statistics, the maximum pseudolikelihood estimator (MPLE) is proved to be consistent [7].", "We discuss pseudolikelihood-based approaches in Section 4.2.", "Thirdly, we note that the high-dimensional theory of Anon 2016 [4] can be applied for prediction accuracy, which we discuss in Section 4.3." ], [ "Extensions of unbiased estimation", "If $V \\sim \\text{Binomial}(k, U)$ , then an unbiased estimator of $U^t$ exists if and only if $0 \\le t \\le k$ .", "The theory of U-statistics [8] provides the minimal variance unbiased estimator for $U^t$ : $U^t = \\textbf {E}\\left[\\begin{pmatrix}V \\\\ t\\end{pmatrix}\\begin{pmatrix}k \\\\ t\\end{pmatrix}^{-1}\\right].$ This result can be immediately applied to yield an unbiased estimator of $p_t$ , when $t \\le k$ : $\\hat{p}_t^{UN} = \\frac{1}{km}\\sum _{i=1}^k\\sum _{j=1}^{m} \\begin{pmatrix}V_{i, j} \\\\ t-1\\end{pmatrix}\\begin{pmatrix}k \\\\ t-1\\end{pmatrix}^{-1}.$ However, since $\\hat{p}_t^{UN}$ is undefined for $k \\ge t$ , we can use exponential extrapolation to define an extended estimator $\\hat{p}_t^{EXP}$ for $k > t$ .", "Let $\\hat{\\alpha }$ be a measure defined by solving the optimization problem $\\text{minimize}_{\\alpha } \\sum _{t=2}^{k} \\left(\\hat{p}_t^{UN} - \\int _0^\\infty \\exp [-t\\kappa ] d\\alpha (\\kappa )\\right)^2.$ After discretizing the measure $\\hat{\\alpha }$ , we obtain a convex optimization problem which can be solved using non-negative least squares [9].", "Then define $\\hat{p}_t^{EXP} = {\\left\\lbrace \\begin{array}{ll}\\hat{p}_t^{UN}&\\text{ for }t \\le k,\\\\\\int _0^\\infty \\exp [-t\\kappa ] d\\hat{\\alpha }(\\kappa ))&\\text{ for }t > k.\\end{array}\\right.", "}$" ], [ "Maximum pseudolikelihood", "The (log) pseudolikelihood is defined as $\\ell (\\eta ) = \\sum _{i=1}^k \\sum _{j=1}^{m} \\log \\left(\\int u^{V_{i, j}} (1-u)^{k - V_{i, j}} \\eta (u) du\\right),$ and a maximum pseudolikelihood estimator (MPLE) is defined as any density $\\hat{\\eta }$ such that $\\ell (\\hat{\\eta }_{MPLE}) = \\sup _{\\eta } \\ell (\\eta ).$ The motivation for $\\hat{\\eta }_{MPLE}$ is that it consistently estimates $\\eta $ in the limit where $k \\rightarrow \\infty $ .", "However, in finite samples, $\\hat{\\eta }_{MPLE}$ is not uniquely defined, and if we define the plug-in estimator $\\hat{p}_t^{MPLE} = \\int u^{t-1} \\hat{\\eta }_{MPLE}(u) du,$ $\\hat{p}_t^{MPLE}$ can vary over a large range, depending on which $\\hat{\\eta } \\in \\text{argmax}_{\\eta } \\ell _t(\\eta )$ is selected.", "These shortcomings motivate the adoption of additional constraints on the estimator $\\hat{\\eta }$ .", "Theorem 3.4. motivates the monotonicity constraint that $\\frac{d\\hat{\\eta }}{du} > 0$ .", "A second constraint is to restrict the $k$ -th moment of $\\hat{\\eta }$ to match the unbiased estimate.", "The addition of these constraints yields the constrained PMLE $\\hat{\\eta }_{CON}$ , which is obtained by solving $\\text{maximize }\\ell (\\eta ) \\text{ subject to }\\int u^{k-1} \\eta (u) du = \\hat{p}_k^{UN}\\text{ and }\\frac{d\\hat{\\eta }}{du} > 0.$ By discretizing $\\eta $ , all of the above maximization problems can be solved using a general-purpose convex solverWe found that the disciplined convex programming language CVX, using the ECOS second-order cone programming solver, succeeds in optimizing the problems where the dimension of the discretized $\\eta $ is as large as 10,000 [10, 11]..", "While the added constraints do not guarantee a unique solution, they improve estimation of $\\eta $ and thus improve moment estimation (Figure 1.)" ], [ "High-dimensional asymptotics", "Under a number of conditions on the distribution $\\mathbb {F}$ , including (but not limited to) having a large dimension $p$ , Anon [4] relate the accuracy $p_t$ of the Bayes classifier to the mutual information between the label $z$ and the response $y$ : $p_t = \\bar{\\pi }_t(\\sqrt{2I(Z; Y)}).$ where $\\bar{\\pi }_k(c) = \\int _{\\mathbb {R}} \\phi (z - c) \\Phi (z)^{k-1} dz.$ While our goal is not to estimate the mutual information, we note that the results of Anon 2016 imply a relationship between $p_k$ and $p_K$ for the Bayes accuracy under the high-dimensional regime: $p_K = \\bar{\\pi }_K\\left(\\bar{\\pi }_k^{-1}(p_k)\\right).$ Therefore, under the high-dimensional conditions of [4] and assuming that the classifier approximates the Bayes classifier, we naturally obtain the following estimator $\\hat{p}_t^{HD} = \\bar{\\pi }_K\\left(\\bar{\\pi }_k^{-1}(\\hat{p}_k^{UN})\\right).$" ], [ "Results", "We applied the methods described in Section 4 on a simulated gaussian mixture (Figure 2) and on a Telugu character classification task [12] (Table 1.)", "For the simulated gaussian mixture, we vary the size of the initial subset from $k=3$ classes to $k=K=50$ classes, and extrapolate the performance for gaussian mixture model, multinomial logistic, and one-layer neural network (with 10 sigmoidal units.)", "Figure 3 shows how the predicted $K$ -class accuracy changes as $k$ is varied.", "We see that the predicted accuracy curves for QDA and Logistic have similar behavior, even though QDA is generative and multinomial logistic is not.", "All three methods perform better on QDA and logistic classifiers than on the neural network: in fact, for the neural network, the test accuracy of the initial set, $\\text{acc}^{(k)}$ , becomes a better estimator of $\\text{acc}^{(K)}$ than the three proposed methods for most of the curve.", "We also see that the exponential extrapolation method, $\\hat{p}^{EXP}$ , is more variable than constrained pseudolikelihood $\\hat{p}^{CONS}$ and high-dimensional estimator $\\hat{p}^{HD}$ .", "Additional simulation results can be found in the supplement.", "In the character classification task, we predict the 400-class accuracy of naive Bayes, multinomial logistic regression, SVM [6], $\\epsilon $ -nearest neighbors$k$ -nearest neighbors with $k = \\epsilon n$ for fixed $\\epsilon > 0$, and deep neural networksThe network architecture is as follows: 48x48-4C3-MP2-6C3-8C3-MP2-32C3-50C3-MP2-200C3-SM.", "48x48 binary input image, $m$ C3 is a 3x3 convolutional layer with $m$ output maps, MP2 is a 2x2 max-pooling layer, and SM is a softmax output layer on 20 or 400 classes.", "using 20-class data with 103 training examples per class (Table 1).", "Taking the test accuracy on 400 classes (using 50 test examples per class) as a proxy for $\\text{acc}^{(400)}$ , we compare the performance of the three extrapolation methods; as a benchmark, also consider using the test accuracy on 20 classes as an estimate.", "The exponential extrapolation method performs well only for the deep neural network.", "Meanwhile, constrained PMLE achieves accurate extrapolation for two out of four classifiers: logistic and SVM but failed to converge for the the deep neural network (due to the high test accuracy).", "The high-dimensional estimator $\\hat{p}^{HD}$ performs well on the multinomial logistic, SVM, and deep neural network classifiers.", "All three methods beat the benchmark (taking the test accuracy at 20) for the first four classifiers; however, the benchmark is the best estimator for the deep neural network, similarly to what we observe in the simulation (albeit with a shallow network rather than a deep network.)", "Figure: Predictions for acc (50) \\text{acc}^{(50)} as kk, the size of the subset, is varied.", "Our methods work better for QDA and Logistic than Neural Net; overall, p ^ EXP \\hat{p}^{EXP} has higher variability than p ^ CONS \\hat{p}^{CONS} and p ^ HD \\hat{p}^{HD}.Table: Performance extrapolation: predicting the accuracy on 400 classes using data from 20 classes on a Telugu character dataset.", "(*) indicates failure to converge.ϵ=0.002\\epsilon = 0.002 for ϵ\\epsilon -nearest neighbors." ], [ "Discussion", "Empirical results indicate that our methods generalize beyond generative classifiers.", "A possible explanation is that since the Bayes classifier is generative, any classifier which approximates the Bayes classifier is also `approximately generative.'", "However, an important caveat is that the classifier must already attain close to the Bayes accuracy on the smaller subset of classes.", "If the classifier is initially far from the Bayes classifier, and then becomes more accurate as more classes are added, our theory could underestimate the accuracy on the larger subset.", "This is a non-issue for generative classifiers when the training data per class is fixed, since a generative classifier approximates the Bayes rule if and only if the single-class classification function approximates the Bayes optimal single-class classification function.", "On the other hand, for classifiers with built-in model selection or representation learning, it is expected that the classification functions become more accurate, in the sense that they better approximate a monotonic function of the Bayes classification functions, as data from more classes is added.", "Our results are still too inconclusive for us to recommend the use of any of these estimators in practice.", "Theoretically, it still remains to derive confidence bounds for the generative case; practically, additional experiments are needed to establish the reliability of these estimators in specific applications.", "There also remains plenty of room for new and improved estimators in this area: for instance, fixing the instability of the constrained pseudolikelihood estimator when the test accuracy is high." ], [ "Acknowledgments", "We thank John Duchi, Steve Mussmann, Qingyun Sun, Jonathan Taylor, Trevor Hastie, Robert Tibshirani for useful discussion.", "CZ is supported by an NSF graduate research fellowship.", "[1] Kay, K. N., Naselaris, T., Prenger, R. J., & Gallant, J. L. (2008).", "“Identifying natural images from human brain activity.” Nature, 452(March), 352-355.", "[2] Deng, J., Berg, A. C., Li, K., & Fei-Fei, L. (2010).", "“What does classifying more than 10,000 image categories tell us?” Lecture Notes in Computer Science, 6315 LNCS(PART 5), 71-84.", "[3] Garfield, S., Stefan W., & Devlin, S. (2005).", "“Spoken language classification using hybrid classifier combination.\"", "International Journal of Hybrid Intelligent Systems 2.1: 13-33.", "[4] Anonymous, A.", "(2016).", "“Estimating mutual information in high dimensions via classification error.” Submitted to NIPS 2016.", "[5] Tewari, A., & Bartlett, P. L. (2007).", "“On the Consistency of Multiclass Classification Methods.” Journal of Machine Learning Research, 8, 1007-1025.", "[6] Hastie, T., Tibshirani, R., & Friedman, J., (2008).", "The elements of statistical learning.", "Vol.", "1.", "Springer, Berlin: Springer series in statistics.", "[7] Arnold, Barry C., & Strauss, D. (1991).", "“Pseudolikelihood estimation: some examples.\"", "Sankhya: The Indian Journal of Statistics, Series B: 233-243.", "[8] Cox, D.R., & Hinkley, D.V.", "(1974).", "Theoretical statistics.", "Chapman and Hall.", "ISBN 0-412-12420-3 [9] Lawson, C. L., & Hanson, R. J.", "(1974).", "Solving least squares problems.", "Vol.", "161.", "Englewood Cliffs, NJ: Prentice-hall.", "[10] Hong, J., Mohan, K. & Zeng, D. (2014).", "“CVX.", "jl: A Convex Modeling Environment in Julia.\"", "[11] Domahidi, A., Chu, E., & Boyd, S. (2013).", "\"ECOS: An SOCP solver for embedded systems.\"", "Control Conference (ECC), 2013 European.", "IEEE.", "[12] Achanta, R., & Hastie, T. (2015) \"Telugu OCR Framework using Deep Learning.\"", "arXiv preprint arXiv:1509.05962 ." ] ]	1606.05228
[ [ "Zero kinematic viscosity-magnetic diffusion limit of the incompressible\n viscous magnetohydrodynamic equations with Navier boundary conditions" ], [ "Abstract We investigate the zero kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions in a smooth bounded domain $\\Omega\\subset\\mathbb{R}^3$.", "We obtain the uniform regularity of solutions with respect to the kinematic viscosity coefficient and the magnetic diffusivity coefficient.", "These solutions are uniformly bounded in a conormal Sobolev space and $W^{1,\\infty}(\\Omega)$ which allow us to take the zero kinematic viscosity-magnetic diffusion limit.", "Moreover, we also get the rates of convergence." ], [ "Introduction", "We consider the following incompressible viscous magnetohydrodynamic (MHD) equations ([8], [6]) $&\\partial _tv^\\epsilon -\\epsilon \\Delta v^\\epsilon +v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon -\\frac{1}{2}\\nabla (\|v^\\epsilon \|^2-\|H^\\epsilon \|^2)+\\nabla p^\\epsilon =0,\\\\&\\partial _tH^\\epsilon -\\epsilon \\Delta H^\\epsilon +v^\\epsilon \\cdot \\nabla H^\\epsilon -H^\\epsilon \\cdot \\nabla v^\\epsilon =0,\\\\&\\mathrm {div}\\,v^\\epsilon =\\mathrm {div}\\,H^\\epsilon =0$ in $(0,T)\\times \\Omega $ , where $\\Omega $ is a smooth bounded domain of $\\mathbb {R}^3$ .", "The unknowns $v^\\epsilon $ and $H^\\epsilon $ represent the fluid velocity and the magnetic field, respectively.", "The pressure $p^\\epsilon $ can be recovered from $v^\\epsilon $ and $H^\\epsilon $ via an explicit Calder¨®n-Zygmund singular integral operator ([7]).", "We add to $v^\\epsilon $ and $H^\\epsilon $ the following initial and boundary conditions $&v^\\epsilon \\cdot n=0,\\quad (Sv^\\epsilon \\cdot n)_\\tau =-\\zeta v^\\epsilon _\\tau \\quad \\text{on}~~~~\\partial \\Omega ,\\\\&H^\\epsilon \\cdot n=0,\\quad (SH^\\epsilon \\cdot n)_\\tau =-\\zeta H^\\epsilon _\\tau \\quad \\text{on}~~~~\\partial \\Omega ,\\\\&(v^\\epsilon ,H^\\epsilon )\|_{t=0}=(v_0,H_0)\\quad \\text{in}\\quad \\Omega ,$ where $n$ stands for the outward unit normal vector to $\\Omega $ , $\\zeta $ is a coefficient measuring the tendency of the fluid to slip on the boundary, $S$ is the strain tensor defined by $Su=\\frac{1}{2}(\\nabla u+\\nabla u^t),\\nonumber $ where $\\nabla u^t$ denotes the transpose of the matrix $\\nabla u$ , and $u_\\tau $ stands for the tangential part of $u$ on $\\partial \\Omega $ , i.e.", "$u_\\tau =u-(u\\cdot n)n.\\nonumber $ This kind boundary condition (REF ) was introduced by Navier in [17] to show that the velocity is propositional to the tangential part of the stress.", "It allows the fluid slip along the boundary and are often used to model rough boundaries.", "The Navier boundary condition (REF ) can be generalized to the following form ([10]) $u\\cdot n=0,~~(Su\\cdot n)_\\tau +Au=0,$ where $A$ is a $(1,1)$ -type tensor on the boundary $\\partial \\Omega $ .", "When $A=\\zeta \\, \\text{Id}$ (here Id denotes the identity matrix), (REF ) is reduced to the standard Navier boundary condition.", "For smooth functions, we can get the form of the vorticity $u\\cdot n=0,~~n\\times \\omega =[Bu]_\\tau \\quad \\text{on}\\quad \\partial \\Omega ,$ where $\\omega =\\nabla \\times u$ is the vorticity and $B=2(A-S(n))$ ([24]).", "In this paper we are interested in the existence of strong solution to the problem (REF )-() with uniform bounds on an interval of time independent of $\\epsilon $ and taking the limit $\\epsilon \\rightarrow 0$ to obtain the ideal incompressible MHD equations, i.e.", "$&\\partial _tv+v\\cdot \\nabla v-H\\cdot \\nabla H-\\frac{1}{2}\\nabla (\|v\|^2-\|H\|^2)+\\nabla p=0,\\\\&\\partial _tH+v\\cdot \\nabla H-H\\cdot \\nabla v=0,\\\\&\\mathrm {div}\\,v=\\mathrm {div}\\,H=0$ with the following slip boundary conditions: $v\\cdot n=H\\cdot n=0.$ When taking $H^\\epsilon =0$ in the system (REF )-(), it is reduced to the classical incompressible Navier-Stokes equations and there are many literature on the vanishing viscosity limit of it.", "In the case that there is no boundary, a uniform time of existence and the vanishing viscosity limit have been obtained, see [13], [15], [20].", "When the boundary appear, it is usually difficult to do higher order energy estimates near boundary because of the appearing of the boundary layer [18].", "In particular, for the incompressible Navier-Stokes equations with no-slip boundary condition, the vanishing viscosity limit of it is wildly open except when the initial data is analytic [21], [22] or the initial vorticity is located away from the boundary in the two-dimensional half plane [14].", "On the other hand, considering the incompressible Navier-Stokes system with Navier boundary conditions, more results are available, see, for example, [26], [2], [3], [4], [12], [5].", "Xiao and Xin [26] investigate the vanishing viscosity limit to incompressible Navier-Stokes equation with the boundary conditions $u\\cdot n=0,~~n\\times \\omega =0\\quad \\text{on} \\quad \\partial \\Omega .$ Because the main part in the boundary layer vanishes (i.e.", "$V=0$ in (REF ) below), they can obtain the local existence of strong solution with some uniform bounds in $H^3(\\Omega )$ and the vanishing viscosity limit.", "Their approaches overcame the compatibility issues of the nonlinear terms with (REF ).", "The authors in [2] got uniform estimates in $W^{k,p}(\\Omega )$ with $k\\ge 3$ and $p\\ge 2$ .", "The main reason is that the boundary integrals vanishes on flat portions of the boundary, see also [3], [4].", "Later, the results in [26], [2] was generalized by Berselli and Spirito [5] to a general bounded domain under certain restrictions on the initial data.", "In order to analysis the effect of the boundary layer in a general bounded domain, Iftimie and Sueur [12] constructed the boundary layer for the incompressible Navier-Stokes equations with the Navier boundary condition (REF ) in the form $u^\\epsilon (t,x)=u^\\epsilon (t,x)+\\sqrt{\\epsilon }V\\Big (t,x,\\frac{\\phi (x)}{\\sqrt{\\epsilon }}\\Big )+O(\\epsilon ),$ where the function $V$ vanishes for $x$ outside a small neighborhood of $\\partial \\Omega $ and $\\phi (x)$ is the distance between $x$ and $\\partial \\Omega $ for $x$ in a neighborhood of $\\partial \\Omega $ .", "The layers constructed in [12] are of width $O(\\sqrt{\\epsilon })$ like the Prandtl layer [18], but are of amplitude $O(\\sqrt{\\epsilon })$ (The Prandtl layer is of width $O(\\sqrt{\\epsilon })$ and of amplitude $O(1)$ ).", "So it is impossible to obtain the $H^3(\\Omega )$ or $W^{2,p}(\\Omega )$ ($p$ large enough) uniform estimates for the incompressible Navier-Stokes equations.", "Recently, Masmoudi and Rousset [16] considered the the vanishing viscosity limit for the incompressible Navier-Stokes equation with the boundary condition (REF ) in anisotropic conormal Sobolev spaces which can eliminate the effects of normal derivatives near the boundary.", "They obtained uniform regularity and the convergence of the viscous solutions to the inviscid ones by compactness argument.", "Recently, some results in [16] was extended to the compressible isentropic Navier-Stokes equations with Navier boundary conditions [23], [19].", "Moreover, based on the results in [16], the rates of convergence were obtained by Gie and Kelliher [10] and Xiao and Xin [24], respectively.", "In [27], Xiao, Xin and Wu studied the inviscid limit for the system (REF )-() with the boundary conditions $\\left\\lbrace \\begin{array}{l}v^\\epsilon \\cdot n=0,\\quad n\\times \\omega _v^\\epsilon =0\\quad \\text{on}\\quad \\partial \\Omega ,\\\\H^\\epsilon \\cdot n=0,\\quad n\\times \\omega _H^\\epsilon =0\\quad \\text{on}\\quad \\partial \\Omega ,\\end{array}\\right.$ where they used the approaches similar to that in [26] and formulated the boundary value in a suitable functional setting so that the stokes operator is well behaved and the nonlinear terms fall into the desired functional spaces.", "These facts allow them to get the uniform regularity for the viscous incompressible MHD system through the Galerkin approximation and a priori energy estimates.", "Here we investigate the inviscid limit for the system (REF )-() with the Navier boundary conditions (REF )-() in a $3D$ bounded domain in the framework of anisotropic conormal Sobolev spaces.", "Due to the strong coupling between $v^\\epsilon $ and $H^\\epsilon $ , a priori estimates become more complicated than that in [16] on the incompressible Navier-Stokes equations.", "We obtain uniform regularity of the solutions and, with this well-posedness theory, pursue the vanishing viscosity limit to the problem (REF )-().", "Moreover, we also obtain some rates of convergence for $v^\\epsilon $ and $H^\\epsilon $ .", "Hence our results can be regarded as generalizations of those in [10], [16], [24] to incompressible MHD eqautions.", "Our first result of this paper reads as follows.", "Theorem 1.1 Let $m$ be an integer satisfying $m>6$ and $\\Omega $ be a $C^{m+2}$ domain.", "Assume that the initial data $(v_0,H_0)$ satisfy $(v_0,H_0) \\in \\mathcal {E}^m, ~~(\\nabla v_0,\\nabla H_0)\\in W^{1,\\infty }_{co}(\\Omega ),$ $\\nabla \\cdot v_0=\\nabla \\cdot H_0=0,~~ v_0\\cdot n\|_{\\partial \\Omega }=H_0\\cdot n\|_{\\partial \\Omega }=0.$ Then, there exist $T_0>0$ and $\\widetilde{C}$ , independent of $\\epsilon \\in (0,1]$ and $\|\\zeta \|\\le 1$ , such that there exists a unique solution of the problem (REF )-() satisfying $(v^\\epsilon ,H^\\epsilon )\\in C([0,T_0],\\mathcal {E}^m)\\nonumber $ and $&\\sup _{t\\in [0,T_0]}\\big {\\lbrace }\\Vert (v^\\epsilon ,H^\\epsilon )(t)\\Vert _m+\\Vert (\\nabla v^\\epsilon ,\\nabla H^\\epsilon )(t)\\Vert _{m-1}+\\Vert (\\nabla v^\\epsilon ,\\nabla H^\\epsilon )(t)\\Vert _{1,\\infty }\\big {\\rbrace }\\nonumber \\\\&\\qquad \\quad +\\epsilon \\int ^{T_0}_0(\\Vert \\nabla ^2v^\\epsilon (t)\\Vert ^2_{m-1}+\\Vert \\nabla ^2H^\\epsilon (t)\\Vert ^2_{m-1})dt\\le \\widetilde{C},$ Here $ \\mathcal {E}^m:=\\lbrace u\\,\|\\,u\\in H^m_{co}(\\Omega ),\\nabla u\\in H^{m-1}_{co}(\\Omega ) \\rbrace $ and the meanings of $ W^{1,\\infty }_{co}(\\Omega )$ , $H^m_{co}(\\Omega )$ , $\\Vert \\cdot \\Vert _m$ and $\\Vert \\cdot \\Vert _{m,\\infty }$ will be explained in detail in next section.", "Remark 1.1 When the Navier boundary conditions (REF ) and () are replaced by the following $\\left\\lbrace \\begin{array}{l}v^\\epsilon \\cdot n=0,\\quad n\\times \\omega _v^\\epsilon =[Bv^\\epsilon ]_\\tau \\quad \\text{on}\\quad \\partial \\Omega ,\\\\H^\\epsilon \\cdot n=0,\\quad n\\times \\omega _H^\\epsilon =[BH^\\epsilon ]_\\tau \\quad \\text{on}\\quad \\partial \\Omega ,\\end{array}\\right.$ we can also obtain the same results as those in Theorem REF , where $B=2(A-S(n))$ and A is a $(1,1)$ -type tensor on the boundary $\\partial \\Omega $ .", "Remark 1.2 Theorem REF still holds if we replace the boundary conditions (REF ) and () by the slightly generalized one: $\\left\\lbrace \\begin{array}{l}v^\\epsilon \\cdot n=0,\\quad (Sv^\\epsilon \\cdot n)_\\tau =-\\zeta _1 v^\\epsilon \\quad \\text{on}\\quad \\partial \\Omega ,\\\\H^\\epsilon \\cdot n=0,\\quad (SH^\\epsilon \\cdot n)_\\tau =-\\zeta _1 H^\\epsilon \\quad \\text{on}\\quad \\partial \\Omega ,\\end{array}\\right.$ where $\\zeta _1$ and $\\zeta _2$ are two different constants.", "We now give some comments on the proof of Theorem REF .", "The main steps of the proof are similar to those in [16] in some sense.", "However, due to the strong coupling between $v^\\epsilon $ and $H^\\epsilon $ , we need to overcome some new difficulties and to face more complicated energy estimates.", "First, we get a conormal energy estimates in $H^m_{co}$ (see the definition in next section) for $(v^\\epsilon ,H^\\epsilon )$ .", "Here, we define $P^\\epsilon _1+P^\\epsilon _2:=p^\\epsilon -\\frac{1}{2}(\|v^\\epsilon \|^2-\|H^\\epsilon \|^2)$ , where $P^\\epsilon _1$ and $P^\\epsilon _2$ satisfy corresponding boundary value problems (see (REF ) and (REF ) below), respectively.", "By doing this decomposition, we can avoid higher order terms which are out of control.", "In the second step, we estimate $\\Vert (\\partial _n v^\\epsilon ,\\partial _nH^\\epsilon )\\Vert _{m-1}$ .", "Due to the incompressible conditions (), both $\\partial _nv^\\epsilon \\cdot n$ and $\\partial _nH^\\epsilon \\cdot n$ can be easily controlled by the $H^m_{co}$ norm of $(v^\\epsilon ,H^\\epsilon )$ .", "Thanks to the the Nvier-slip boundary conditions, it is convenient to study $\\eta ^\\epsilon _v=(Sv^\\epsilon n+\\zeta v^\\epsilon )_\\tau $ and $\\eta ^\\epsilon _H=(SH^\\epsilon n+\\zeta H^\\epsilon )_\\tau $ .", "We find that $\\eta ^\\epsilon _v$ and $\\eta ^\\epsilon _H$ satisfy equations with homogeneous Dirichlet boundary conditions, and we shall thus prove a control of $\\Vert (\\eta ^\\epsilon _v,\\eta ^\\epsilon _H)\\Vert _{m-1}$ by performing energy estimates on the equations solved by $(\\eta ^\\epsilon _v,\\eta ^\\epsilon _H)$ .", "The third step is to estimate $P^\\epsilon _1$ and $P^\\epsilon _2$ .", "Note that they satisfy nonhomogeneous elliptic equations with Neumann boundary conditions.", "By using the regularity theory of elliptic equations with Neumann boundary conditions, we get the estimates on the pressure terms.", "Finally, we need to estimate $\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }$ and $\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty }$ .", "Similar to the second step, we find equivalent quantities $\\overline{\\eta }^\\epsilon _v$ and $\\overline{\\eta }^\\epsilon _H$ .", "However, due to the strong coupling between $\\overline{\\eta }^\\epsilon _v$ and $\\overline{\\eta }^\\epsilon _H$ , we cannot deal with the system on $\\overline{\\eta }^\\epsilon _v$ and $\\overline{\\eta }^\\epsilon _H$ directly as that in [16].", "Instead, we need further to introduce another two quantities $\\eta _1:=\\overline{\\eta }^\\epsilon _v+\\overline{\\eta }^\\epsilon _H $ and $\\eta _2:=\\overline{\\eta }^\\epsilon _v-\\overline{\\eta }^\\epsilon _H$ .", "We then estimate $\\eta _1$ and $\\eta _2$ , respectively.", "Based on Theorem REF and Remark REF , we justify the vanishing viscosity limit as follows: Theorem 1.2 Assume that $(v_0,H_0)$ belong to $H^3(\\Omega )$ and satisfy the same assumptions as in Theorem REF .", "Let $(v,H)$ be the smooth solution of (REF )-(REF ) with the initial data $(v,H)\|_{t=0}=(v_0,H_0) $ on $[0,T_1]$ .", "Let $(v^\\epsilon ,H^\\epsilon )$ be the solution of (REF )-() with the boundary condition (REF ) and the initial data $(v^\\epsilon ,H^\\epsilon )\|_{t=0}=(v_0,H_0)$ .", "Then there exists a $T_2=\\min \\lbrace T_0,T_1\\rbrace >0$ such that $\\Vert v^{\\epsilon }&-v \\Vert ^2_{L^2}+\\Vert H^{\\epsilon }-H\\Vert ^2_{L^2}\\nonumber \\\\&+\\epsilon \\int _0^t(\\Vert (v^{\\epsilon }-v)(s)\\Vert _{H^1}^2+\\Vert ( H^{\\epsilon }-H)(s)\\Vert _{H^1}^2)\\,ds\\le C\\epsilon ^\\frac{3}{2}\\quad on \\quad [0,T_2],\\\\\\Vert v^{\\epsilon }&-v\\Vert ^2_{H^1}+\\Vert H^{\\epsilon }-H\\Vert ^2_{H^1}\\nonumber \\\\&+\\epsilon \\int _0^t(\\Vert (v^{\\epsilon }-v)(s)\\Vert ^2_{H^2}+\\Vert ( H^{\\epsilon }-H)(s)\\Vert _{H^2}^2)\\,ds\\le C\\epsilon ^\\frac{1}{2}\\quad on\\quad [0,T_2]$ for $\\epsilon $ small enough.", "Consequently, $\\begin{split}\\Vert v^{\\epsilon }&-v\\Vert ^p_{W^{1,p}}+\\Vert H^{\\epsilon }-H\\Vert ^p_{W^{1,p}}\\le C\\epsilon ^\\frac{1}{2}\\quad on\\quad [0,T_2]\\end{split}$ for $2\\le p<\\infty $ and $\\epsilon $ small enough, and $\\begin{split}\\Vert v^{\\epsilon }-v\\Vert _{L^\\infty ({[0,T_2]\\times \\Omega })}+\\Vert H^{\\epsilon }-H\\Vert _{L^\\infty ([0,T_2]\\times \\Omega )}\\le C \\epsilon ^\\frac{3}{10}.\\end{split}$ We now outline the proof of Theorem REF .", "Our approaches are similar to those in [24], but due to the strong coupling between magnetic field and velocity field, we meet some new difficulties.", "We first give the rates of the convergence in $L^\\infty (0,T_2;L^2(\\Omega ))$ and $L^\\infty ([0,T_2]\\times \\Omega )$ by using an elementary energy estimate for the difference of the solutions between the incompressible viscous MHD equations and the ideal incompressible MHD equations and the Gagliardo-Nirenberg interpolation inequality.", "Next, because we find that it is very difficult to estimate some boundary terms caused by multiplying (REF ) by $\\Delta (v^\\epsilon -v)$ and () by $\\Delta (H^\\epsilon -H)$ directly in the proof of the rate of the convergence in $L^\\infty (0,T_2;H^1(\\Omega ))$ , we turn to consider the Stokes problem (REF )-().", "Indeed, we can get $\\Vert u\\Vert _{H^2}\\le \\Vert P\\Delta u\\Vert +\\Vert u\\Vert \\,\\,\\text{for}\\,\\, u\\in W_{B},$ where $W_{B}$ is defined in Lemma REF and $P$ is Leray projector.", "Finally, we replace $\\Delta (v^\\epsilon -v)$ and $\\Delta (H^\\epsilon -H)$ by $P\\Delta (v^\\epsilon -v)$ and $P\\Delta (H^\\epsilon -H)$ to do prove the rates of the convergence in $L^\\infty (0,T_2;H^1(\\Omega ))$ and $L^\\infty (0,T_2; W^{1,p}(\\Omega ))$ .", "This paper is organized as follows.", "In the following section, we give some assumptions on the domain and the definitions on conormal Sobolev spaces, and present some inequalities.", "In Section , we prove a priori energy estimates and give the proof of Theorem REF .", "Finally, we prove Theorem REF in Section .", "Throughout the paper, we shall denote by $\\Vert \\cdot \\Vert _{H^m }$ and $\\Vert \\cdot \\Vert _{W^{1,\\infty } }$ the usual Sobolev norms in $\\Omega $ and $\\Vert \\cdot \\Vert $ for the standard $L^2$ norm.", "The letter $C$ is a positive number which may change from line to line, but independent of $\\epsilon \\in (0,1]$ and $\|\\zeta \|\\le 1$ ." ], [ "Preliminaries", "We first state the assumptions on the bounded domain $\\Omega \\subset \\mathbb {R}^3$ and then introduce some norms.", "We assume that $\\Omega $ has a covering such that $\\Omega \\subset \\Omega _0 \\cup _{k=1}^n\\Omega _k,$ where $\\overline{\\Omega _0}\\subset \\Omega $ and in each $\\Omega _k$ there exists a function $\\psi _k$ such that $\\Omega \\cup \\Omega _k=\\lbrace \\,x=(x_1,x_2,x_3)\\,\|\\,x_3>\\psi _k(x_1,x_2)\\,\\rbrace \\cup \\Omega _k,\\\\\\partial \\Omega \\cup \\Omega _k=\\lbrace \\,x=(x_1,x_2,x_3)\\,\|\\,x_3=\\psi _k(x_1,x_2)\\,\\rbrace \\cup \\Omega _k.$ We say that $\\Omega $ is $C^m$ if the functions $\\psi _k$ are $C^m$ functions.", "To define the conormal Sobolev spaces, we consider $(Z_k)_{1\\le k\\le N}$ , a finite set of generators of vector fields that are tangent to $\\partial \\Omega $ , and set $H^m_{co}(\\Omega ):=\\big {\\lbrace } f\\in L^2(\\Omega )\\,\\big {\|}\\, Z^If\\in L^2(\\Omega )~~~ \\text{for}~~\\, \|I\|\\le m,\\,\\,\\, m\\in \\mathbb {N}\\big {\\rbrace },$ where $I=(k_1,...,k_m)$ , $Z^I:=Z_{k_1}\\cdot \\cdot \\cdot Z_{k_m}$ .", "We define the norm of $H^m_{co}(\\Omega )$ : $ \\Vert f\\Vert ^2_m:=\\sum _{\|I\|\\le m}\\Vert Z^If\\Vert ^2_{L^2}.$ We say a vector field, $u$ , is in $H^m_{co}(\\Omega )$ if each of its components is in $H^m_{co}(\\Omega )$ and $ \\Vert u\\Vert ^2_m:=\\sum _{i=1}^3\\sum _{\|I\|\\le m}\\Vert Z^Iu_i\\Vert ^2_{L^2}$ is finite.", "In the same way, we set $ \\Vert f\\Vert _{m,\\infty }:=\\sum _{\|I\|\\le m}\\Vert Z^If\\Vert _{L^\\infty },$ $\\Vert \\nabla Z^mu\\Vert ^2:=\\sum _{\|I\|\\le m}\\Vert \\nabla Z^Iu\\Vert ^2_{L^2},$ and we say that $f\\in W^{m,\\infty }_{co}(\\Omega )$ if $\\Vert f\\Vert _{m,\\infty }$ is finite.", "By using above covering of $\\Omega $ , we can assume that each vector field is supported in one of $\\lbrace \\Omega _i\\rbrace _{i=0}^n$ .", "Also, we note that the $\\Vert \\cdot \\Vert _{m}$ norm yields a control of the standard $H^m$ norm in $\\Omega _0$ , whereas if $\\Omega _i \\cap \\partial \\Omega \\ne {\\emptyset }$ , there is no control of the normal derivatives.", "Since $\\partial \\Omega $ is given locally by $x_3=\\psi (x_1,x_2)$ (we omit the subscript $k$ for notational convenience), it is convenient to use the coordinates: $\\Psi :(y,z)\\mapsto (y,\\psi (y)+z)=x.$ A local basis is thus given by the vector fields $(\\partial _{y^1},\\partial _{y^1},\\partial _z)$ where $\\partial _{y^1}$ and $\\partial _{y^2}$ are tangent to $\\partial \\Omega $ on the boundary and in general $\\partial _z$ is usually not a normal vector field.", "We sometimes use the notation $\\partial _{y^3}$ for $\\partial _z$ .", "By using this parametrization, we can take suitable vector fields compactly supported in $\\Omega _i$ in the definition of the $\\Vert \\cdot \\Vert _m$ norms: $Z_i=\\partial _{y^i}=\\partial _i+\\partial _i\\psi \\partial _z,~~i=1,2,\\quad Z_3=\\varphi (z)\\partial _z,$ where $\\varphi (z)=\\frac{z}{1+z}$ is a smooth and supported function in $(0,+\\infty )$ and satisfies $\\varphi (0)=0,~~\\varphi ^{\\prime }(0)>0,~~\\varphi (z)>0\\,\\,\\,\\,\\text{for}\\,\\,\\,\\,z>0.$ In this paper, we shall still denote by $\\partial _i, i=1,~2,~3$ or $\\nabla $ the derivatives with respect to the standard coordinates of $\\mathbb {R}^3$ .", "The coordinates of a vector field $u$ in the basis $(\\partial _{y^1},\\partial _{y^1},\\partial _z)$ will be denote by $u^i$ , thus $u=u^1\\partial _{y^1}+u^2\\partial _{y^2}+u^3\\partial _z.$ We denote by $u_i$ the coordinates in the standard basis of $\\mathbb {R}^3$ , i.e.", "$u=u_1\\partial _1+u_2\\partial _2+u_3\\partial _3.$ Denote by $n$ the unit outward normal vector which is given locally by $n(x)=n(\\Psi (y,z))=\\frac{1}{\\sqrt{1+\|\\nabla \\psi (y)\|^2}}\\left(\\begin{array}{c}\\partial _1\\psi (y)\\\\\\partial _2\\psi (y)\\\\-1\\end{array}\\right)$ and by $\\Pi $ the orthogonal projection $\\Pi (x)=\\Pi (\\Psi (y,z))u=u-[u\\cdot n(\\Psi (y,z))]n(\\Psi (y,z) $ which gives the orthogonal projector onto the tangent space of the boundary.", "Note that $n$ and $\\Pi $ are defined in the whole $\\Omega _k$ and do not depend on $z$ .", "By using these notations, the Navier boundary conditions (REF ) and () read: $&v^\\epsilon \\cdot n=0, \\quad \\ \\Pi \\partial _nv^\\epsilon =\\theta (v^\\epsilon )-2\\zeta \\Pi v^\\epsilon ,\\\\&H^\\epsilon \\cdot n=0, \\quad \\Pi \\partial _nH^\\epsilon =\\theta (H^\\epsilon )-2\\zeta \\Pi H^\\epsilon ,$ where $\\theta $ is the shape operator (second fundamental form) of the boundary, $\\theta (v^\\epsilon ):=\\Pi ((\\nabla n)v^\\epsilon )$ and $\\theta (H^\\epsilon ):=\\Pi ((\\nabla n)H^\\epsilon )$ .", "First, we introduce a well-known inequality.", "Lemma 2.1 ([1], [26]) For $u\\in H^s(\\Omega ) \\,(s\\ge 1)$ , we have $\\Vert u\\Vert _{H^s(\\Omega )}\\le \\,C\\,(\\Vert \\nabla \\times u\\Vert _{H^{s-1}(\\Omega )}+\\Vert \\nabla \\cdot u\\Vert _{H^{s-1}(\\Omega )}+\\Vert u\\Vert _{H^{s-1}(\\Omega )}+\|u\\cdot n\|_{H^{s-\\frac{1}{2}}(\\partial \\Omega )}).\\nonumber $ Next, we introduce the Korn's inequlity which play an important role in energy estimates below.", "Lemma 2.2 (Korn's inequality[9]) Let $\\Omega $ be a bounded Lipschitz domain of $\\mathbb {R}^3$ .", "There exists a constant $C>0$ depending only on $\\Omega $ such that $\\nonumber \\Vert u\\Vert _{H^1(\\Omega )}\\le C\\,(\\Vert u\\Vert _{L^2(\\Omega )}+\\Vert S(u)\\Vert _{L^2(\\Omega )}),\\quad \\forall ~ u\\in (H^1(\\Omega ))^3.$ Third, we also need the following anistropic Sobolev embedding and trace estimates.", "Lemma 2.3 ([16], [23]) Let $m_1\\ge 0$ and $m_2\\ge 0$ be integers, $u\\in H^{m_1}_{co}(\\Omega )\\cap H^{m_2}_{co}(\\Omega )$ and $\\nabla u\\in H^{m_2}_{co}(\\Omega )$ .", "Then we have $&\\Vert u\\Vert ^2_{{L^\\infty (\\Omega )}}\\le C\\,(\\Vert \\nabla u\\Vert _{m_2}+\\Vert u\\Vert _{m_2})\\Vert u\\Vert _{m_1},\\quad m_1+m_2\\ge 3,\\\\&\|u\|^2_{H^s(\\partial \\Omega )}\\le C\\,(\\Vert \\nabla u\\Vert _{m_2}+\\Vert u\\Vert _{m_2})\\Vert u\\Vert _{m_1},\\quad m_1+m_2\\ge 2s\\ge 0.$ Fourth, we introduce the following Gagliardo-Nirenberg-Moser inequality which will be used frequently.", "Lemma 2.4 ([11]) Let $u, v\\in L^\\infty (\\Omega )\\cap H^k_{co}(\\Omega )$ , we have $\\nonumber \\Vert Z^{\\alpha _1}uZ^{\\alpha _2}v\\Vert \\le C\\, (\\Vert u\\Vert _{L^\\infty (\\Omega )}\\Vert v\\Vert _k+\\Vert v\\Vert _{L^\\infty (\\Omega )}\\Vert u\\Vert _k),\\quad \|\\alpha _1\|+\|\\alpha _2\|=k.$ Finally, the following decomposition on $H^s$ contributes to the proof of the convergence rate in $H^1$ .", "Lemma 2.5 ([25]) For $H^s(\\Omega ) \\,(s\\ge 0)$ , we have $H^s(\\Omega )=\\nabla \\times (FH\\cap H^{s+1}(\\Omega ))\\oplus (HG\\cap H^s(\\Omega ))\\oplus (GG\\cap H^s(\\Omega )),\\nonumber $ where $&FH=\\big {\\lbrace }u\\, {\|}\\,u=\\nabla \\times \\varphi ,\\,\\, \\varphi \\in H^1(\\Omega ),\\,\\, \\nabla \\cdot \\varphi =0,\\,\\, n\\times \\varphi =0\\,\\,\\, \\text{on} \\,\\,\\,\\partial \\Omega \\big {\\rbrace },\\\\&HG=\\Big {\\lbrace }u\\, {\|}\\,u=\\nabla \\varphi ,\\,\\,\\,\\Delta \\varphi =0,\\,\\,\\varphi =c_i\\,\\,\\, \\text{on} \\,\\,\\,\\Gamma _i,\\,\\,\\,\\bigcup _i\\Gamma _i=\\partial \\Omega \\Big {\\rbrace },\\\\&GG=\\big {\\lbrace }u\\, {\|}\\,u=\\nabla \\varphi ,\\,\\,\\varphi \\in H^1_0(\\Omega )\\big {\\rbrace }.$" ], [ "A priori estimates and proof of Theorem ", "The main aim of this section is to prove the following a priori estimates which is the crucial step in the proof of Theorem REF .", "Theorem 3.1 For $m>6$ and a $C^m$ domain $\\Omega $ , there exists a constant $C>0$ , independent of $\\epsilon \\in (0,1]$ and $\|\\zeta \|\\le 1$ , such that for any sufficiently smooth solution defined on $[0,T]$ of the problem (REF )-() in $\\Omega $ , we have $N_m(t)\\le C\\,\\Big {(} N_m(0)+(1+t+\\epsilon ^3t^2)\\int _0^t(N_m^2(s)+N_m(s))\\,ds\\Big {)},\\quad \\forall \\,\\, t\\in [0,T],$ where $\\!\\!\\!", "N_m(t):=\\Vert v^\\epsilon \\Vert _m^2+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}^2+\\Vert \\nabla v^\\epsilon \\Vert ^2_{1,\\infty }+\\Vert H^\\epsilon \\Vert _m^2+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}^2+\\Vert \\nabla H^\\epsilon \\Vert ^2_{1,\\infty }.$ Since the proof of Theorem REF is quite complicated and lengthy, we divided the proof into the following subsections." ], [ "Conormal Energy Estimates", "In this subsection, we first give the basic $L^2$ energy estimates.", "Lemma 3.1 For a smooth solution of the problem (REF )-(), we have $\\frac{1 }{2}\\frac{d }{dt}(\\Vert v^\\epsilon (t)\\Vert ^{2}+\\Vert H^\\epsilon (t)\\Vert ^{2})&+2\\epsilon (\\Vert Sv^\\epsilon \\Vert ^2+\\Vert SH^\\epsilon \\Vert ^2)\\nonumber \\\\&+2\\epsilon \\zeta \\int _{\\partial \\Omega }\\big {(}\\!\\mid v^\\epsilon _\\tau \\mid ^{2}+\\mid H^\\epsilon _\\tau \\mid ^{2}\\!", "\\big {)}=0$ for every $\\epsilon \\in (0,1]$ and $\|\\zeta \|\\le 1$ .", "Multiplying (REF ) and () by $v^\\epsilon $ and $H^\\epsilon $ respectively, using the boundary condition, and integrating by parts, we obtain $\\frac{1}{2}\\frac{d}{dt}(\\Vert v^\\epsilon \\Vert ^2+\\Vert H^\\epsilon \\Vert ^2)&-\\epsilon (\\Delta v^\\epsilon ,v^\\epsilon )-\\epsilon (\\Delta H^\\epsilon ,H^\\epsilon )\\nonumber \\\\&-(H^\\epsilon \\cdot \\nabla H^\\epsilon ,v^\\epsilon )-(H^\\epsilon \\cdot \\nabla v^\\epsilon ,H^\\epsilon )=0,$ where $(\\cdot ,\\cdot )$ stands for the $L^2$ scalar product.", "By integrating by parts and using the boundary conditions, we get $\\nonumber (H^\\epsilon \\cdot \\nabla H^\\epsilon ,v^\\epsilon )+(H^\\epsilon \\cdot \\nabla v^\\epsilon ,H^\\epsilon )=0.$ Now, let us treat the terms with the viscous coefficient $\\epsilon $ in (REF ).", "Thanks to integrations by parts and the boundary condition (REF ), we have $(\\epsilon \\Delta v^\\epsilon ,v^\\epsilon )=2\\epsilon (\\nabla \\cdot Sv^\\epsilon ,v^\\epsilon )&=-2\\epsilon \\Vert Sv^\\epsilon \\Vert ^2+2\\epsilon \\ \\int _{\\partial \\Omega }((Sv^\\epsilon )\\cdot n)\\cdot v^\\epsilon \\nonumber \\\\&=-2\\epsilon \\Vert Sv^\\epsilon \\Vert ^2-2\\epsilon \\zeta \\int _{\\partial \\Omega }\|v^\\epsilon _\\tau \|^2.$ Similarly, we have $(\\epsilon \\Delta H^\\epsilon ,H^\\epsilon )=-2\\epsilon \\Vert SH^\\epsilon \\Vert ^2-2\\epsilon \\zeta \\int _{\\partial \\Omega }\|H^\\epsilon _\\tau \|^2.$ Putting (REF ) and (REF ) into (REF ), we then obtain (REF ).", "Now, we turn to the higher order energy estimates.", "Lemma 3.2 For every $m\\ge 0$ , a smooth solution of the problem (REF )-() satisfies the estimate $\\frac{d }{dt}&(\\Vert v^\\epsilon (t)\\Vert _{m}^{2}+\\Vert H^\\epsilon (t)\\Vert _{m}^{2})+ \\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _{m}^{2}+\\Vert \\nabla H^\\epsilon \\Vert _{m}^{2})\\nonumber \\\\\\le & C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})( \\Vert v^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla v^\\epsilon \\Vert _{m-1}^{2}+\\Vert H^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla H^\\epsilon \\Vert _{m-1}^{2})\\nonumber \\\\&+C\\,\\Vert \\nabla ^2 P^\\epsilon _1\\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+C\\epsilon ^{-1}\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}^2,$ where the pressure $P^\\epsilon :=p^\\epsilon -\\frac{1}{2}(\|v^\\epsilon \|^2-\|H^\\epsilon \|^2):=P^\\epsilon _1+P^\\epsilon _2$ .", "Here, $P^\\epsilon _1$ is the“Euler\" part of the pressure which solves $\\left\\lbrace \\begin{array}{l}\\Delta P^\\epsilon _1=-\\nabla \\cdot (v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon )\\quad \\text{in}\\quad \\Omega ,\\\\\\partial _nP^\\epsilon _1=-(v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon )\\cdot n\\quad \\text{on}\\quad \\partial \\Omega \\end{array}\\right.$ and $P^\\epsilon _2$ is the “Navier-Stokes\" part of the pressure which solves $\\left\\lbrace \\begin{array}{l}\\Delta P^\\epsilon _2=0\\quad \\text{in}\\quad \\Omega ,\\\\\\partial _nP^\\epsilon _2=\\epsilon \\Delta v^\\epsilon \\cdot n\\quad \\text{on}\\quad \\partial \\Omega .\\end{array}\\right.$ The estimate for $m=0$ has been given in Lemma REF .", "Now we assume Lemma REF have been proved for $\|\\alpha \|\\le m-1$ and prove that it holds for $\|\\alpha \|=m$ .", "We apply $Z^\\alpha $ to (REF )-() for $\|\\alpha \|=m$ to obtain $&\\partial _tZ^\\alpha v^\\epsilon +v^\\epsilon \\cdot \\nabla Z^\\alpha v^\\epsilon -H^\\epsilon \\cdot \\nabla Z^\\alpha H^\\epsilon + Z^\\alpha \\nabla P^\\epsilon =\\epsilon Z^\\alpha \\Delta v^\\epsilon +\\mathcal {C}_1,\\\\&\\partial _tZ^\\alpha H^\\epsilon +v^\\epsilon \\cdot \\nabla Z^\\alpha H^\\epsilon -H^\\epsilon \\cdot \\nabla Z^\\alpha v^\\epsilon =\\epsilon Z^\\alpha \\Delta H^\\epsilon +\\mathcal {C}_2,$ where $&\\mathcal {C}_1:=-[Z^\\alpha ,v^\\epsilon \\cdot \\nabla ]v^\\epsilon +[Z^\\alpha ,H^\\epsilon \\cdot \\nabla ]H^\\epsilon ,\\\\&\\mathcal {C}_2:=-[Z^\\alpha ,v^\\epsilon \\cdot \\nabla ]H^\\epsilon +[Z^\\alpha ,H^\\epsilon \\cdot \\nabla ]v^\\epsilon .$ Consequently, we get from the standard energy estimate that $\\frac{1}{2}\\frac{d}{dt}(\\Vert Z^\\alpha v^\\epsilon \\Vert ^2+\\Vert Z^\\alpha H^\\epsilon \\Vert ^2)=&\\epsilon (Z^\\alpha \\Delta v^\\epsilon ,Z^\\alpha v^\\epsilon )+\\epsilon (Z^\\alpha \\Delta H^\\epsilon ,Z^\\alpha H^\\epsilon )\\nonumber \\\\&+(\\mathcal {C}_1,Z^\\alpha v^\\epsilon )+(\\mathcal {C}_2,Z^\\alpha H^\\epsilon )-(Z^\\alpha \\nabla P^\\epsilon ,Z^\\alpha v^\\epsilon ).$ First, by Lemma REF , we obtain $\|(\\mathcal {C}_1,Z^\\alpha v^\\epsilon )+(\\mathcal {C}_2,Z^\\alpha H^\\epsilon )\|\\le \\,&C\\,(\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})\\nonumber \\\\&(\\Vert v^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla v^\\epsilon \\Vert _{m-1}^{2}+\\Vert H^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla H^\\epsilon \\Vert _{m-1}^{2}).$ Next, we estimate the terms with the viscosity coefficient $\\epsilon $ .", "We have $\\epsilon \\int _\\Omega Z^\\alpha \\Delta v^\\epsilon \\cdot Z^\\alpha v^\\epsilon =2\\epsilon \\int _\\Omega (\\nabla \\cdot Z^\\alpha Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon +2\\epsilon \\int _\\Omega ([Z^\\alpha ,\\nabla \\cdot ]Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon .$ Now, by integrating by parts, we get from the first term on the right hand side of (REF ) that $\\epsilon \\int _\\Omega (\\nabla \\cdot Z^\\alpha Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon =&-\\epsilon \\int _\\Omega Z^\\alpha Sv^\\epsilon \\cdot \\nabla Z^\\alpha v^\\epsilon +\\epsilon \\int _{\\partial \\Omega }((Z^\\alpha Sv^\\epsilon )\\cdot n)\\cdot Z^\\alpha v^\\epsilon \\nonumber \\\\=&-\\epsilon \\Vert S(Z^\\alpha v^\\epsilon )\\Vert ^2-\\epsilon \\int _\\Omega [Z^\\alpha ,S]v^\\epsilon \\cdot \\nabla Z^\\alpha v^\\epsilon \\nonumber \\\\&+\\epsilon \\int _{\\partial \\Omega }((Z^\\alpha Sv^\\epsilon )\\cdot n)\\cdot Z^\\alpha v^\\epsilon .$ Thanks to Lemma REF , there exists a $c_0>0$ such that $\\epsilon \\int _\\Omega (\\nabla \\cdot Z^\\alpha Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon \\le &-c_0\\epsilon \\Vert \\nabla (Z^\\alpha v^\\epsilon )\\Vert ^2+C\\Vert v^\\epsilon \\Vert ^2_m+C\\epsilon \\Vert \\nabla Z^\\alpha v^\\epsilon \\Vert \\Vert \\nabla v^\\epsilon \\Vert _{m-1}\\nonumber \\\\&+\\epsilon \\int _{\\partial \\Omega }((Z^\\alpha Sv^\\epsilon )\\cdot n)\\cdot Z^\\alpha v^\\epsilon .$ It remains to estimate the boundary term of (REF ).", "Before we treat the boundary term, we have the following observations.", "Due to the Navier boundary condition (REF ), we get $\|\\Pi \\partial _nv^\\epsilon \|_{H^m(\\partial \\Omega )}\\le \|\\theta (v^\\epsilon )\|_{H^m(\\partial \\Omega )}+2\\zeta \|\\Pi v^\\epsilon \|_{H^m(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^m(\\partial \\Omega )}.$ To estimate the normal part of $\\partial _nv^\\epsilon $ , we can use the divergence free condition to write $\\nabla \\cdot v^\\epsilon =\\partial _nv^\\epsilon \\cdot n+(\\Pi \\partial _{y_1}v^\\epsilon )^1+(\\Pi \\partial _{y_2}v^\\epsilon )^2.$ Hence, we easily get $\|\\partial _nv^\\epsilon \\cdot n\|_{H^{m-1}(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}.$ From (REF ) and (REF ), we have $\|\\nabla v^\\epsilon \|_{H^{m-1}(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}.$ Thanks to $v^\\epsilon \\cdot n=0$ on the boundary, we immediately obtain that $\|(Z^\\alpha v^\\epsilon )\\cdot n\|_{H^1(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )},\\quad \|\\alpha \|=m.$ Now we return to deal with the boundary term of (REF ) as follows $\\int _{\\partial \\Omega }((Z^\\alpha Sv^\\epsilon )\\cdot n)\\cdot Z^\\alpha v^\\epsilon =&\\int _{\\partial \\Omega }Z^\\alpha (\\Pi (Sv^\\epsilon \\cdot n))\\cdot \\Pi Z^\\alpha v^\\epsilon \\nonumber \\\\&+\\int _{\\partial \\Omega }Z^\\alpha (\\partial _nv^\\epsilon \\cdot n)Z^\\alpha v^\\epsilon \\cdot n+\\mathcal {C}_b^v,$ where $\\mathcal {C}_b^v=\\int _{\\partial \\Omega }[\\Pi ,Z^\\alpha ](Sv^\\epsilon \\cdot n)\\cdot \\Pi Z^\\alpha v^\\epsilon +\\int _{\\partial \\Omega }[n,Z^\\alpha ](Sv^\\epsilon \\cdot n)Z^\\alpha v^\\epsilon \\cdot n.$ Due to (REF ) and (REF ), we can easily obtain that $&\|C_b^v\|\\le C\\,\|\\nabla v^\\epsilon \|_{H^{m-1}(\\partial \\Omega )}\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}^2,\\\\&\\Big {\|}\\int _{\\partial \\Omega }Z^\\alpha (\\Pi (Sv^\\epsilon \\cdot n))\\cdot \\Pi Z^\\alpha v^\\epsilon \\Big {\|}\\le C\\,\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}^2.$ By integrating by parts along the boundary, we have that $\\Big {\|}\\int _{\\partial \\Omega }Z^\\alpha (\\partial _nv^\\epsilon \\cdot n)Z^\\alpha v^\\epsilon \\cdot n\\Big {\|}\\le C\\,\|\\partial _nv^\\epsilon \\cdot n\|_{H^{m-1}(\\partial \\Omega )}\|Z^\\alpha v^\\epsilon \\cdot n\|_{H^1(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|^2_{H^m(\\partial \\Omega )}.$ Hence, we get from (REF ), (REF ), and (REF )-(REF ) that $\\epsilon \\int _\\Omega (\\nabla \\cdot Z^\\alpha Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon \\le \\, &C\\,(\\Vert v^\\epsilon \\Vert ^2_m+\\epsilon \\Vert \\nabla Z^m v^\\epsilon \\Vert \\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\epsilon \|v^\\epsilon \|^2_{H^m(\\partial \\Omega )})\\nonumber \\\\&-c_0\\epsilon \\Vert \\nabla (Z^\\alpha v^\\epsilon )\\Vert ^2.$ Next, we deal with the second term of the right hand side of (REF ), i.e.$\\epsilon \\int _\\Omega ([Z^\\alpha ,\\nabla \\cdot ]Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon $ .", "We can expand it as a sum of terms under the form $\\epsilon \\int _\\Omega \\beta _k\\partial _k(Z^{\\tilde{\\alpha }}Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon ,\\quad \|\\tilde{\\alpha }\|\\le m-1.\\nonumber $ By using integrations by parts and (REF ), we have $\\epsilon \\Big {\|}\\int _\\Omega \\beta _k\\partial _k(Z^{\\tilde{\\alpha }}Sv^\\epsilon )\\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\le \\, C\\,\\epsilon (\\Vert \\nabla Z^{m-1}v^\\epsilon \\Vert \\Vert \\nabla Z^mv^\\epsilon \\Vert +\\Vert v^\\epsilon \\Vert ^2_m+\|v^\\epsilon \|_{H^{m}(\\partial \\Omega )}^2).$ Consequently, from (REF ) and (REF ), we get $\\epsilon \\Big {\|}\\int _\\Omega Z^\\alpha \\Delta v^\\epsilon \\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\le &\\, C\\,\\big {\\lbrace }\\Vert v^\\epsilon \\Vert ^2_m+\\epsilon \\Vert \\nabla Z^m v^\\epsilon \\Vert \\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\epsilon \|v^\\epsilon \|^2_{H^m(\\partial \\Omega )}\\nonumber \\\\&+\\epsilon \\Vert \\nabla Z^m v^\\epsilon \\Vert \\Vert \\nabla Z^{m-1}v^\\epsilon \\Vert _{m-1}\\big {\\rbrace }-c_0\\epsilon \\Vert \\nabla (Z^\\alpha v^\\epsilon )\\Vert ^2.$ Similarly, for the term $\\epsilon (Z^\\alpha \\Delta H^\\epsilon \\cdot Z^\\alpha H^\\epsilon )$ in the right hand side of (REF ), we have $\\epsilon \\Big {\|}\\int _\\Omega Z^\\alpha \\Delta H^\\epsilon \\cdot Z^\\alpha H^\\epsilon \\Big {\|}\\le &\\,C\\,\\big {\\lbrace }\\Vert H^\\epsilon \\Vert ^2_m+\\epsilon \\Vert \\nabla Z^m H^\\epsilon \\Vert \\Vert \\nabla H^\\epsilon \\Vert _{m-1}+\\epsilon \|H^\\epsilon \|^2_{H^m(\\partial \\Omega )}\\nonumber \\\\&+\\epsilon \\Vert \\nabla Z^m H^\\epsilon \\Vert \\Vert \\nabla Z^{m-1}H^\\epsilon \\Vert _{m-1}\\big {\\rbrace }-c_0\\epsilon \\Vert \\nabla (Z^\\alpha H^\\epsilon )\\Vert ^2.$ Finally, we estimate the term involving the pressure $P^\\epsilon $ in (REF ).", "We have $\\Big {\|}\\int _{\\Omega }Z^\\alpha \\nabla P^\\epsilon \\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\le &\\, \\Vert \\nabla ^2P_1^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+\\Big {\|}\\int _{\\Omega }Z^\\alpha \\nabla P^\\epsilon _2\\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\nonumber \\\\\\le \\,&\\Vert \\nabla ^2P_1^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+C\\Vert \\nabla P_2^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m\\nonumber \\\\&+\\Big {\|}\\int _{\\Omega }\\nabla Z^\\alpha P^\\epsilon _2\\cdot Z^\\alpha v^\\epsilon \\Big {\|}.$ Now, we focus on the last term of (REF ).", "By integrating by parts, we obtain $\\Big {\|}\\int _{\\Omega }\\nabla Z^\\alpha P^\\epsilon _2\\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\le C\\,\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}\\Vert \\nabla Z^\\alpha v^\\epsilon \\Vert +\\Big {\|}\\int _{\\partial \\Omega }Z^\\alpha P^\\epsilon _2Z^\\alpha v^\\epsilon \\cdot n\\Big {\|}.\\nonumber $ To estimate the boundary term, we note that when $m=1$ , (REF ) can be obtained easily.", "Here, we assume that $m\\ge 2$ .", "By integrating by parts along the boundary, we get $\\nonumber \\Big {\|}\\int _{\\partial \\Omega }Z^\\alpha P^\\epsilon _2Z^\\alpha v^\\epsilon \\cdot n\\Big {\|}\\le C\\,\|Z^{\\widetilde{\\alpha }}P^\\epsilon _2\|_{L^2(\\partial \\Omega )}\|Z^\\alpha v^\\epsilon \\cdot n\|_{H^1(\\partial \\Omega )},$ where $\|\\widetilde{\\alpha }\|=m-1$ .", "By using (REF ) and Lemma REF , we have $\\Big {\|}\\int _{\\Omega }Z^\\alpha \\nabla P^\\epsilon \\cdot Z^\\alpha v^\\epsilon \\Big {\|}\\le &\\Vert \\nabla ^2P_1^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+C\\,\\Vert \\nabla P_2^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m\\nonumber \\\\&+C\\,\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}\\Vert \\nabla Z^\\alpha v^\\epsilon \\Vert +\\epsilon ^{-1}\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}^2\\nonumber \\\\&+\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _m\\Vert v^\\epsilon \\Vert _{m}+\\Vert v^\\epsilon \\Vert ^2_{m}).$ Consequently, from (REF ), (REF )-(REF ) and (REF ), we have $\\frac{1}{2}\\frac{d}{dt}&(\\Vert Z^\\alpha v^\\epsilon \\Vert ^2+\\Vert Z^\\alpha H^\\epsilon \\Vert ^2)+c_0\\epsilon \\Vert \\nabla (Z^\\alpha v^\\epsilon )\\Vert ^2+c_0\\epsilon \\Vert \\nabla (Z^\\alpha v^\\epsilon )\\Vert ^2\\nonumber \\\\\\le &\\, C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla v^\\epsilon \\Vert _{m-1}^{2}+\\Vert H^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla H^\\epsilon \\Vert _{m-1}^{2})\\nonumber \\\\&+C\\,\\big {\\lbrace }\\epsilon \\Vert \\nabla Z^m v^\\epsilon \\Vert \\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\epsilon \|v^\\epsilon \|^2_{H^m(\\partial \\Omega )}+\\epsilon \\Vert \\nabla Z^m v^\\epsilon \\Vert \\Vert \\nabla Z^{m-1}v^\\epsilon \\Vert _{m-1}\\nonumber \\\\&+\\epsilon \\Vert \\nabla Z^m H^\\epsilon \\Vert \\Vert \\nabla H^\\epsilon \\Vert _{m-1}+\\epsilon \|H^\\epsilon \|^2_{H^m(\\partial \\Omega )}+\\epsilon \\Vert \\nabla Z^m H^\\epsilon \\Vert \\Vert \\nabla Z^{m-1}H^\\epsilon \\Vert _{m-1}\\nonumber \\\\&+\\Vert \\nabla ^2P_1^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla P_2^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}\\Vert \\nabla Z^mv^\\epsilon \\Vert \\nonumber \\\\&+\\epsilon ^{-1}\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}^2+\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _m\\Vert v^\\epsilon \\Vert _{m}+\\Vert v^\\epsilon \\Vert ^2_{m})\\big {\\rbrace }.$ Next, by using Lemma REF , Young's inequality, the assumptions with respect to $\|\\alpha \|\\le m-1$ , we have $\\frac{1}{2}\\frac{d}{dt}&(\\Vert v^\\epsilon \\Vert ^2_m+\\Vert H^\\epsilon \\Vert ^2_m)+c_0\\epsilon \\Vert \\nabla v^\\epsilon \\Vert ^2_{m-1}+c_0\\epsilon \\Vert \\nabla v^\\epsilon \\Vert ^2_{m-1}\\nonumber \\\\\\le &\\, C(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla v^\\epsilon \\Vert _{m-1}^{2}+\\Vert H^\\epsilon \\Vert _{m}^{2}+ \\Vert \\nabla H^\\epsilon \\Vert _{m-1}^{2})\\nonumber \\\\&+ C(\\Vert \\nabla ^2P_1^\\epsilon \\Vert _{m-1}\\Vert v^\\epsilon \\Vert _m+\\epsilon ^{-1}\\Vert \\nabla P_2^\\epsilon \\Vert _{m-1}^2).$ This ends the proof of Lemma REF ." ], [ "Normal Derivative Estimates.", "In this subsection, we provide the estimates for $\\Vert \\nabla v^\\epsilon \\Vert _{m-1}$ and $\\Vert \\nabla H^\\epsilon \\Vert _{m-1}$ .", "Noticing that $\\Vert \\chi \\partial _{y^i}v^\\epsilon \\Vert _{m-1}\\le C\\,\\Vert v^\\epsilon \\Vert _{m},\\quad \\Vert \\chi \\partial _{y^i}H^\\epsilon \\Vert _{m-1}\\le C\\,\\Vert H^\\epsilon \\Vert _{m},\\quad i=1,2,$ it suffices to estimate $\\Vert \\chi \\partial _nv^\\epsilon \\Vert _{m-1}$ and $\\Vert \\chi \\partial _nH^\\epsilon \\Vert _{m-1}$ , where $\\chi $ is compactly supported in one of the $\\Omega _i$ and with value one in a vicinity of the boundary.", "We shall thus use the local coordinates (REF ).", "Due to (REF ), we immediately obtain that $\\Vert \\chi \\partial _nv^\\epsilon \\cdot n\\Vert _{m-1}\\le C\\,\\Vert v^\\epsilon \\Vert _m,\\quad \\Vert \\chi \\partial _nH^\\epsilon \\cdot n\\Vert _{m-1}\\le C\\,\\Vert H^\\epsilon \\Vert _m.$ Thus, it remains to estimate $\\Vert \\chi \\Pi \\partial _nv^\\epsilon \\Vert _{m-1}$ and $\\Vert \\chi \\Pi \\partial _nH^\\epsilon \\Vert _{m-1}$ .", "We define $&\\eta ^\\epsilon _v:=\\chi \\Pi ((\\nabla v^\\epsilon +(\\nabla v^\\epsilon )^t)n)+2\\zeta \\chi \\Pi v^\\epsilon ,\\\\&\\eta ^\\epsilon _H:=\\chi \\Pi ((\\nabla H^\\epsilon +(\\nabla H^\\epsilon )^t)n)+2\\zeta \\chi \\Pi H^\\epsilon .$ In view of the Navier boundary conditions (REF ) and (), we have $\\eta ^\\epsilon _v=0,\\quad \\eta ^\\epsilon _H=0\\quad \\text{on}\\quad \\partial \\Omega .$ Moreover, since $\\eta ^\\epsilon _v$ and $\\eta ^\\epsilon _H$ have another forms in the vicinity of the boundary $\\partial \\Omega $ : $&\\eta ^\\epsilon _v=\\chi \\Pi \\partial _nv^\\epsilon +\\chi \\Pi (\\nabla (v^\\epsilon \\cdot n)-\\nabla n\\cdot v^\\epsilon -v^\\epsilon \\times (\\nabla \\times n)+2\\zeta v^\\epsilon ),\\\\&\\eta ^\\epsilon _H=\\chi \\Pi \\partial _nH^\\epsilon +\\chi \\Pi (\\nabla (H^\\epsilon \\cdot n)-\\nabla n\\cdot H^\\epsilon -H^\\epsilon \\times (\\nabla \\times n)+2\\zeta H^\\epsilon ),$ we easily get that $\\Vert \\chi \\Pi \\partial _nv^\\epsilon \\Vert _{m-1}\\le &\\, C\\,(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _{m}+\\Vert \\partial _nv^\\epsilon \\cdot n\\Vert _{m-1})\\nonumber \\\\\\le &\\, C\\,(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _{m}),\\\\\\Vert \\chi \\Pi \\partial _nH^\\epsilon \\Vert _{m-1}\\le &\\, C\\,(\\,\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _{m}+\\Vert \\partial _nH^\\epsilon \\cdot n\\Vert _{m-1})\\nonumber \\\\\\le &\\, C\\,(\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _{m}).$ Hence, it remains to estimate $\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}$ and $\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}$ .", "We have the following conormal estimates for $\\eta ^\\epsilon _v$ and $\\eta ^\\epsilon _H$ .", "Lemma 3.3 For every $m\\ge 1$ , we have $\\frac{1}{2}\\frac{d}{dt}&(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}^2+\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}^2)+\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert ^2_{m-1}+\\Vert \\nabla \\eta ^\\epsilon _H\\Vert ^2_{m-1})\\nonumber \\\\\\le &\\,C\\,(1+\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty })\\nonumber \\\\&\\times (\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}^2+\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}^2+\\Vert v^\\epsilon \\Vert _m^2+\\Vert H^\\epsilon \\Vert _m^2+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}^2+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}^2)\\nonumber \\\\&+C\\,\\big {(}(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _m)(\\Vert \\nabla ^2P^\\epsilon _1\\Vert _{m-1}+\\Vert \\nabla P^\\epsilon \\Vert _{m-1})+\\epsilon ^{-1}\\Vert \\nabla P^\\epsilon _2\\Vert ^2_{m-1}\\big {)}.$ Setting $M_v=\\nabla v^\\epsilon $ and $M_H=\\nabla H^\\epsilon $ , we get from (REF )-() that $&\\partial _tM_v-\\epsilon \\Delta M_v+v^\\epsilon \\cdot \\nabla M_v-H^\\epsilon \\cdot \\nabla M_H=( M_H)^2-(M_v)^2-\\nabla ^2P^\\epsilon ,\\\\&\\partial _tM_H-\\epsilon \\Delta M_H+v^\\epsilon \\cdot \\nabla M_H-H^\\epsilon \\cdot \\nabla M_v= M_v M_H-M_H M_v.$ Hence, $\\eta ^\\epsilon _v$ and $\\eta ^\\epsilon _H$ solve the equations $&\\partial _t\\eta ^\\epsilon _v-\\epsilon \\Delta \\eta ^\\epsilon _v+v^\\epsilon \\cdot \\nabla \\eta ^\\epsilon _v-H^\\epsilon \\cdot \\nabla \\eta ^\\epsilon _H=F_v^b+F_v^\\chi +F_v^\\kappa -2\\chi \\Pi (\\nabla ^2P^\\epsilon n),\\\\&\\partial _t\\eta ^\\epsilon _H-\\epsilon \\Delta \\eta ^\\epsilon _H+v^\\epsilon \\cdot \\nabla \\eta ^\\epsilon _H-H^\\epsilon \\cdot \\nabla \\eta ^\\epsilon _v=F_H^b+F_H^\\chi +F_H^\\kappa ,$ where $F_v^b=&-\\chi \\Pi ((\\nabla v^\\epsilon )^2+((\\nabla v^\\epsilon )^t)^2-(\\nabla H^\\epsilon )^2-((\\nabla H^\\epsilon )^t)^2)n-2\\zeta \\chi \\Pi \\nabla P^\\epsilon ,\\\\F_v^\\chi =&-\\epsilon \\Delta \\chi (\\Pi 2 Sv^\\epsilon n+2\\zeta \\Pi v^\\epsilon )-2\\epsilon \\nabla \\chi \\cdot \\nabla (\\Pi 2 Sv^\\epsilon n+2\\zeta \\Pi v^\\epsilon )\\\\&+(v^\\epsilon \\cdot \\nabla \\chi )\\Pi (2Sv^\\epsilon n+2\\zeta v^\\epsilon )-(H^\\epsilon \\cdot \\nabla \\chi )\\Pi (2SH^\\epsilon n+2\\zeta H^\\epsilon ),\\\\F_v^\\kappa =&\\chi (v^\\epsilon \\cdot \\nabla \\Pi )(2Sv^\\epsilon n+2\\zeta v^\\epsilon )+\\chi \\Pi (2Sv^\\epsilon (v^\\epsilon \\cdot \\nabla )n)-\\epsilon \\chi (\\Delta \\Pi )(2Sv^\\epsilon n+2\\zeta v^\\epsilon )\\\\&-2\\epsilon \\chi \\nabla \\Pi \\cdot \\nabla (2Sv^\\epsilon n+2\\zeta v^\\epsilon )-\\epsilon \\chi \\Pi (2Sv^\\epsilon \\Delta n+2\\nabla Sv^\\epsilon \\cdot \\nabla n)\\\\&-\\chi (H\\cdot \\nabla \\Pi )(2SH^\\epsilon n+2\\zeta H^\\epsilon )-\\chi \\Pi (2SH^\\epsilon (H\\cdot \\nabla )n),\\\\F_H^b=&-\\chi \\Pi (M_H M_v+M_v^tM_H^t-M_v M_H-M_H^t M_v^t)n,\\\\F_H^\\chi =&-\\epsilon \\Delta \\chi (\\Pi 2 SH^\\epsilon n+2\\zeta H^\\epsilon )-2\\epsilon \\nabla \\chi \\cdot \\nabla (\\Pi 2 SH^\\epsilon n+2\\zeta H^\\epsilon )\\\\&+(v^\\epsilon \\cdot \\nabla \\chi )\\Pi (2SH^\\epsilon n+2\\zeta \\Pi H^\\epsilon )-(H^\\epsilon \\cdot \\nabla \\chi )\\Pi (2Sv^\\epsilon n+2\\zeta \\Pi v^\\epsilon ),\\\\F_H^\\kappa =&\\chi (v^\\epsilon \\cdot \\nabla \\Pi )(2SH^\\epsilon n+2\\zeta H^\\epsilon )+\\chi \\Pi (2SH^\\epsilon (v^\\epsilon \\cdot \\nabla )n)-\\epsilon \\chi (\\Delta \\Pi )(2SH^\\epsilon n\\\\&+2\\zeta H^\\epsilon )-2\\epsilon \\chi \\nabla \\Pi \\cdot \\nabla (2SH^\\epsilon n+2\\zeta H^\\epsilon )-\\epsilon \\chi \\Pi (2SH^\\epsilon \\Delta n+2\\nabla SH^\\epsilon \\cdot \\nabla n)\\\\&-\\chi (H\\cdot \\nabla \\Pi )(2Sv^\\epsilon n+2\\zeta v^\\epsilon )-\\chi \\Pi (2Sv^\\epsilon (H\\cdot \\nabla )n).$ Let us start with the case of $m=1$ .", "By using the standard $L^2$ energy estimate, we get $\\frac{1}{2}&\\frac{d}{dt}(\\Vert \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\eta ^\\epsilon _H\\Vert ^2)+\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\nabla \\eta ^\\epsilon _H\\Vert ^2)\\nonumber \\\\&=(F_v^b+F_v^\\chi +F_v^\\kappa ,\\eta ^\\epsilon _v)+(F_H^b+F_H^\\chi +F_H^\\kappa ,\\eta ^\\epsilon _H)-2(\\chi \\Pi (\\nabla ^2P^\\epsilon n),\\eta ^\\epsilon _v).$ Now we estimate the right-hand side terms of (REF ).", "We easily arrive at $\\Vert F_v^b\\Vert _{m-1}+\\Vert F_H^b\\Vert _{m-1}\\le \\,&C\\,(\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert \\nabla H^\\epsilon \\Vert _{m-1})\\nonumber \\\\&+C\\,\\Vert \\nabla P^\\epsilon \\Vert _{m-1},\\\\\\Vert F_v^\\kappa \\Vert _{m-1}+\\Vert F_H^\\kappa \\Vert _{m-1}\\le \\,&C\\,\\epsilon (\\Vert \\chi \\nabla ^2v^\\epsilon \\Vert _{m-1}+\\Vert \\chi \\nabla ^2H^\\epsilon \\Vert _{m-1}+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}\\nonumber \\\\&+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _{m}+\\Vert H^\\epsilon \\Vert _{m})\\nonumber \\\\&+C\\,(\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _{m-1}\\nonumber \\\\&+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}).$ Next, since $F_v^\\chi $ and $F_H^\\chi $ are supported away from the boundary, we can control any derivatives by the norm $\\Vert \\cdot \\Vert _m$ .", "We immediately get $\\Vert F_v^\\chi \\Vert _{m-1}+\\Vert F_H^\\chi \\Vert _{m-1}\\le &\\,C\\,\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _m)\\nonumber \\\\&+C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _m+\\Vert H^\\epsilon \\Vert _m).$ Finally, we estimate $(\\chi \\Pi (\\nabla ^2P^\\epsilon n),\\eta ^\\epsilon _v)$ .", "Noting that $P^\\epsilon =P^\\epsilon _1+P^\\epsilon _2$ , we get $\|(\\chi \\Pi (\\nabla ^2P^\\epsilon n),\\eta ^\\epsilon _v)\|\\le \\Vert \\nabla ^2P^\\epsilon _1\\Vert \\Vert \\eta ^\\epsilon _v\\Vert +\\Big {\|}\\int _{\\Omega }\\chi \\Pi (\\nabla ^2P^\\epsilon _2 n)\\cdot \\eta ^\\epsilon _v\\Big {\|}.$ Since $\\eta ^\\epsilon _v=0$ on the boundary, we can integrate by the parts the last term in (REF ) to obtain $\\Big {\|}\\int _{\\Omega }\\chi \\Pi (\\nabla ^2P^\\epsilon _2 n)\\cdot \\eta ^\\epsilon _v\\Big {\|}\\le C\\Vert \\nabla P^\\epsilon _2\\Vert (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert +\\Vert \\eta ^\\epsilon _v\\Vert ).$ Consequently, from (REF )-(REF ), (REF ), (REF ), we have $\\frac{1}{2}\\frac{d}{dt}&(\\Vert \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\eta ^\\epsilon _H\\Vert ^2)+\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\nabla \\eta ^\\epsilon _H\\Vert ^2)\\nonumber \\\\\\le &\\, C\\,\\epsilon (\\Vert \\chi \\nabla ^2v^\\epsilon \\Vert +\\Vert \\chi \\nabla ^2H^\\epsilon \\Vert +\\Vert \\nabla v^\\epsilon \\Vert _1+\\Vert \\nabla H^\\epsilon \\Vert _1)(\\Vert \\eta ^\\epsilon _v\\Vert +\\Vert \\eta ^\\epsilon _H\\Vert )\\nonumber \\\\&+C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert ^2_1+\\Vert H^\\epsilon \\Vert ^2_1+\\Vert \\nabla v^\\epsilon \\Vert ^2+\\Vert \\nabla H^\\epsilon \\Vert ^2\\nonumber \\\\&+\\Vert \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\eta ^\\epsilon _H\\Vert ^2)+\\Vert \\nabla P^\\epsilon _2\\Vert (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert +\\Vert \\eta ^\\epsilon _v\\Vert )+\\Vert \\nabla ^2P^\\epsilon _1\\Vert \\Vert \\eta ^\\epsilon _v\\Vert +\\Vert \\nabla P^\\epsilon \\Vert \\Vert \\eta ^\\epsilon _v\\Vert .$ Due to (REF ) and (REF ), we get that $\\epsilon \\Vert \\chi \\nabla ^2v^\\epsilon \\Vert _{m-1}&\\le C\\,\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert \\nabla v^\\epsilon \\Vert _m+\\Vert v^\\epsilon \\Vert _m).$ Similarly, we get $\\epsilon \\Vert \\chi \\nabla ^2H^\\epsilon \\Vert _{m-1}\\le C\\,\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert \\nabla H^\\epsilon \\Vert _m+\\Vert H^\\epsilon \\Vert _m).$ By using (REF ), (REF ) and Young's inequality, we have $\\frac{1}{2}\\frac{d}{dt}&(\\Vert \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\eta ^\\epsilon _H\\Vert ^2)+\\epsilon (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\nabla \\eta ^\\epsilon _H\\Vert ^2)\\nonumber \\\\\\le &\\, C\\big {\\lbrace }\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _1+\\Vert \\nabla H^\\epsilon \\Vert _1)(\\Vert \\eta ^\\epsilon _v\\Vert +\\Vert \\eta ^\\epsilon _H\\Vert )\\nonumber \\\\&+(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert ^2_1+\\Vert H^\\epsilon \\Vert ^2_1+\\Vert \\nabla v^\\epsilon \\Vert ^2+\\Vert \\nabla H^\\epsilon \\Vert ^2\\nonumber \\\\&+\\Vert \\eta ^\\epsilon _v\\Vert ^2+\\Vert \\eta ^\\epsilon _H\\Vert ^2)+\\epsilon ^{-1}\\Vert \\nabla P^\\epsilon _2\\Vert ^2+\\Vert \\eta ^\\epsilon _v\\Vert (\\Vert \\nabla P^\\epsilon \\Vert +\\Vert \\nabla ^2P^\\epsilon _1\\Vert )\\big {\\rbrace }.$ Since $\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _1+\\Vert \\nabla H^\\epsilon \\Vert _1)$ has been estimated in Lemma REF , this yields (REF ) for the case of $m=1$ .", "Now we assume that Lemma REF is true for $\|\\alpha \|\\le m-2$ and let us consider the situation of $\|\\alpha \|= m-1$ .", "By applying $Z^\\alpha $ to (REF )-(), we have $\\partial _tZ^\\alpha \\eta ^\\epsilon _v-&\\epsilon Z^\\alpha \\Delta \\eta ^\\epsilon _v+v^\\epsilon \\cdot \\nabla Z^\\alpha \\eta ^\\epsilon _v-H^\\epsilon \\cdot \\nabla Z^\\alpha \\eta ^\\epsilon _H\\nonumber \\\\&=Z^\\alpha F_v^b+Z^\\alpha F_v^\\chi +Z^\\alpha F_v^\\kappa -Z^\\alpha (\\chi \\Pi (\\nabla ^2P^\\epsilon n))+\\mathcal {C}_3,\\\\\\partial _tZ^\\alpha \\eta ^\\epsilon _H-&\\epsilon Z^\\alpha \\Delta \\eta ^\\epsilon _H+v^\\epsilon \\cdot \\nabla Z^\\alpha \\eta ^\\epsilon _H-H^\\epsilon \\cdot \\nabla Z^\\alpha \\eta ^\\epsilon _v\\nonumber \\\\&=Z^\\alpha F_H^b+Z^\\alpha F_H^\\chi +Z^\\alpha F_H^\\kappa +\\mathcal {C}_4,$ where $\\nonumber \\begin{split}&\\mathcal {C}_3:=-[Z^\\alpha ,v^\\epsilon \\cdot \\nabla ]\\eta ^\\epsilon _v+[Z^\\alpha ,H^\\epsilon \\cdot \\nabla ]\\eta ^\\epsilon _H,\\\\&\\mathcal {C}_4:=-[Z^\\alpha ,v^\\epsilon \\cdot \\nabla ]\\eta ^\\epsilon _H+[Z^\\alpha ,H^\\epsilon \\cdot \\nabla ]\\eta ^\\epsilon _v.\\end{split}$ From the standard energy estimate, we get $\\frac{1}{2}\\frac{d}{dt}&(\\Vert Z^\\alpha \\eta ^\\epsilon _v\\Vert ^2+\\Vert Z^\\alpha \\eta ^\\epsilon _H\\Vert ^2)\\nonumber \\\\\\le &\\,\\epsilon \\,(Z^\\alpha \\Delta \\eta ^\\epsilon _v,Z^\\alpha \\eta ^\\epsilon _v)+\\epsilon (Z^\\alpha \\Delta \\eta ^\\epsilon _H,Z^\\alpha \\eta ^\\epsilon _H)\\nonumber \\\\&+(C_1,Z^\\alpha \\eta ^\\epsilon _v)+(C_2,Z^\\alpha \\eta ^\\epsilon _H)-2(Z^\\alpha (\\chi \\Pi (\\nabla ^2P^\\epsilon n)),Z^\\alpha \\eta ^\\epsilon _v)\\nonumber \\\\&+(Z^\\alpha F_v^b+Z^\\alpha F_v^\\chi +Z^\\alpha F_v^\\kappa ,Z^\\alpha \\eta ^\\epsilon _v)+(Z^\\alpha F_H^b+Z^\\alpha F_H^\\chi +Z^\\alpha F_H^\\kappa ,Z^\\alpha \\eta ^\\epsilon _H).$ First, let us estimate $\\epsilon (Z^\\alpha \\Delta \\eta ^\\epsilon _v,Z^\\alpha \\eta ^\\epsilon _v)$ and $\\epsilon (Z^\\alpha \\Delta \\eta ^\\epsilon _H,Z^\\alpha \\eta ^\\epsilon _H)$ .", "We observe that $\\int _\\Omega Z^\\alpha \\partial _{ii}\\eta ^\\epsilon _v\\cdot Z^\\alpha \\eta ^\\epsilon _v=& -\\int _\\Omega \|\\partial _iZ^\\alpha \\eta ^\\epsilon _v\|^2-\\int _\\Omega [Z^\\alpha ,\\partial _i]\\eta ^\\epsilon _v\\cdot \\partial _iZ^\\alpha \\eta ^\\epsilon _v\\nonumber \\\\&+\\int _\\Omega [Z^\\alpha ,\\partial _i]\\partial _i\\eta ^\\epsilon _v\\cdot Z^\\alpha \\eta ^\\epsilon _v,$ where $i=1, 2, 3$ .", "To estimate the last two terms on the right hand side of (REF ), we use the structure of the commutator $[Z^\\alpha ,\\partial _i]$ and the expansion $\\partial _i=\\beta ^1\\partial _{y^1}+\\beta ^2\\partial _{y^2}+\\beta ^3\\partial _{y^3}$ in the local basis.", "We have the following expansion $[Z^\\alpha ,\\partial _i]\\eta ^\\epsilon _v=\\sum _{\\gamma ,\|\\gamma \|\\le \|\\alpha \|-1}c_\\gamma \\partial _zZ^\\gamma \\eta ^\\epsilon _v+\\sum _{\\beta ,\|\\beta \|\\le \|\\alpha \|}c_\\beta Z^\\beta \\eta ^\\epsilon _v.\\nonumber $ This yields the estimates $&\\Big {\|}\\int _\\Omega [Z^\\alpha ,\\partial _i]\\eta ^\\epsilon _v\\cdot \\partial _iZ^\\alpha \\eta ^\\epsilon _v\\Big {\|}\\le C\\,\\Vert \\nabla Z^{m-1}\\eta ^\\epsilon _v\\Vert (\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-2}+\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}),\\\\&\\Big {\|}\\int _\\Omega [Z^\\alpha ,\\partial _i]\\partial _i\\eta ^\\epsilon _v\\cdot Z^\\alpha \\eta ^\\epsilon _v\\Big {\|}\\le C\\,\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-1}(\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-2}+\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}).$ Taking the same argument as above, we have $&\\Big {\|}\\int _\\Omega [Z^\\alpha ,\\partial _i]\\eta ^\\epsilon _H\\cdot \\partial _iZ^\\alpha \\eta ^\\epsilon _H\\Big {\|}\\le C\\,\\Vert \\nabla Z^{m-1}\\eta ^\\epsilon _H\\Vert (\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-2}+\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}),\\\\&\\Big {\|}\\int _\\Omega [Z^\\alpha ,\\partial _i]\\partial _i\\eta ^\\epsilon _H\\cdot Z^\\alpha \\eta ^\\epsilon _H\\Big {\|}\\le C\\,\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-1}(\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-2}+\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}).$ Consequently, we get from (REF ), (REF )-() and Young's inequality that $\\frac{1}{2}\\frac{d}{dt}&(\\Vert Z^\\alpha \\eta ^\\epsilon _v\\Vert ^2+\\Vert Z^\\alpha \\eta ^\\epsilon _H\\Vert ^2)+\\frac{\\epsilon }{2}(\\Vert \\nabla Z^{m-1}\\eta ^\\epsilon _v\\Vert ^2+\\Vert \\nabla Z^{m-1}\\eta ^\\epsilon _H\\Vert ^2)\\nonumber \\\\\\le &\\,C\\epsilon \\,(\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-1}^2+\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}^2+\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-1}^2+\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}^2)\\nonumber \\\\&+(\\Vert F_v^b\\Vert _{m-1}+\\Vert F_v^\\chi \\Vert _{m-1}+\\Vert F_v^\\kappa \\Vert _{m-1})\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert \\mathcal {C}_3\\Vert \\Vert \\eta ^\\epsilon _v\\Vert _{m-1}\\nonumber \\\\&+(\\Vert F_H^b\\Vert _{m-1}+\\Vert F_H^\\chi \\Vert _{m-1}+\\Vert F_H^\\kappa \\Vert _{m-1})\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert \\mathcal {C}_4\\Vert \\Vert \\eta ^\\epsilon _H\\Vert _{m-1}\\nonumber \\\\&-2(Z^\\alpha (\\chi \\Pi (\\nabla ^2P^\\epsilon n)),Z^\\alpha \\eta ^\\epsilon _v).$ Second, we get from (REF )-(REF ), (REF ), and (REF ) that $\\Vert F_v^b&\\Vert _{m-1}+\\Vert F_v^\\chi \\Vert _{m-1}+\\Vert F_v^\\kappa \\Vert _{m-1}\\nonumber \\\\\\le \\,&C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1})\\nonumber \\\\&+\\epsilon \\, C\\,\\Vert \\nabla v^\\epsilon \\Vert _m+\\epsilon \\, C\\,\\Vert \\nabla \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert \\nabla P^\\epsilon \\Vert _{m-1},\\\\\\Vert F_H^b&\\Vert _{m-1}+\\Vert F_H^\\chi \\Vert _{m-1}+\\Vert F_H^\\kappa \\Vert _{m-1}\\nonumber \\\\\\le \\,&C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }}+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1})\\nonumber \\\\&+\\epsilon \\, C\\,\\Vert \\nabla H^\\epsilon \\Vert _m+\\epsilon \\, C\\,\\Vert \\nabla \\eta ^\\epsilon _H\\Vert _{m-1}.$ Next, we estimate $\\Vert \\mathcal {C}_3\\Vert $ and $\\Vert \\mathcal {C}_4\\Vert $ .", "In the local coordinates, we observe $f\\cdot \\nabla g=f_1\\partial _{y^1}g+f_2\\partial _{y^2}g+f\\cdot N\\partial _zg.\\nonumber $ Hence $&[Z^\\alpha ,v^\\epsilon \\cdot \\nabla ]\\eta ^\\epsilon _v\\nonumber \\\\=&\\sum _{i=1,2}\\sum _{\|\\beta \|\\ge 1,\|\\beta \|+\|\\gamma \|\\le \|\\alpha \|}Z^\\beta v_i^\\epsilon Z^\\gamma Z_i\\eta ^\\epsilon _v+\\sum _{\|\\beta \|\\ge 1,\|\\beta \|+\|\\gamma \|\\le \|\\alpha \|}Z^\\beta (v_3^\\epsilon \\cdot N) Z^\\gamma \\partial _z\\eta ^\\epsilon _v\\nonumber \\\\=&\\sum _{i=1,2}\\sum _{\|\\beta \|\\ge 1,\|\\beta \|+\|\\gamma \|\\le \|\\alpha \|}Z^\\beta v_i^\\epsilon Z^\\gamma Z_i\\eta ^\\epsilon _v+\\sum _{\|\\widetilde{\\beta }\|\\ge 1,\|\\widetilde{\\beta }\|+\|\\widetilde{\\gamma }\|\\le \|\\alpha \|}Z^{\\widetilde{\\beta }}( \\frac{v_3^\\epsilon \\cdot N}{\\varphi (z)}) Z^{\\widetilde{\\gamma }} Z_3\\eta ^\\epsilon _v.$ We can do similar caculations for other terms in $\\mathcal {C}_3$ and $\\mathcal {C}_4$ .", "Consequently, from (REF ), () and Lemma REF , we get $\\Vert \\mathcal {C}_3\\Vert \\le & \\,C\\,(\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert v^\\epsilon \\Vert _{w^{1,\\infty }}+\\Vert Z\\eta ^\\epsilon _v\\Vert _{L^\\infty })(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _m)\\nonumber \\\\&+C\\,(\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{w^{1,\\infty }}+\\Vert Z\\eta ^\\epsilon _H\\Vert _{L^\\infty })(\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _m),\\\\\\Vert \\mathcal {C}_4\\Vert \\le &\\, C\\,(\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert v^\\epsilon \\Vert _{w^{1,\\infty }}+\\Vert Z\\eta ^\\epsilon _H\\Vert _{L^\\infty })(\\Vert \\eta ^\\epsilon _H\\Vert _{m-1}+\\Vert v^\\epsilon \\Vert _m)\\nonumber \\\\&+C\\,(\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{w^{1,\\infty }}+\\Vert Z\\eta ^\\epsilon _v\\Vert _{L^\\infty })(\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}+\\Vert H^\\epsilon \\Vert _m).$ Final, it remains to deal with the terms involving the pressure $P^\\epsilon $ .", "As above, we use the split $P^\\epsilon =P^\\epsilon _1+P^\\epsilon _2$ and we integrate by parts the terms involving $P^\\epsilon _2$ .", "We have $\|\\big {(}Z^\\alpha (\\chi \\Pi (\\nabla ^2P^\\epsilon n)),Z^\\alpha \\eta ^\\epsilon _v\\big {)}\|\\le \\,&C\\,\\big {(}\\Vert \\nabla ^2P^\\epsilon _1\\Vert _{m-1}\\Vert \\eta ^\\epsilon _v\\Vert _{m-1}\\nonumber \\\\&+\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}(\\Vert \\nabla Z^{m-1}\\eta ^\\epsilon _v\\Vert +\\Vert \\eta ^\\epsilon _v\\Vert _{m-1})\\big {)}.$ By combining (REF ), (REF ), (), (REF ), (), (REF ) and using the induction assumption and Young's inequality, we complete the proof of Lemma REF ." ], [ "Pressure Estimates", "It remains to estimate the pressure terms and the $L^\\infty $ norms, the aim of this subsection is to give the pressure estimates.", "Lemma 3.4 For every $m\\ge 2$ , we have the following estimates: $\\Vert \\nabla P^\\epsilon _1\\Vert _{m-1}+\\Vert \\nabla ^2 P^\\epsilon _1\\Vert _{m-1}\\le \\, &C\\,(1+\\Vert v^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1})\\nonumber \\\\&+C\\,(1+\\Vert H^\\epsilon \\Vert _{W^{1,\\infty }})(\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}),\\\\\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}\\le &\\,C\\,\\epsilon \\,(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}).$ Recall that $P^\\epsilon =P^\\epsilon _1+P^\\epsilon _2$ and $P^\\epsilon _1$ , $P^\\epsilon _2$ are defined in (REF ) and (REF ), respectively.", "From the standard elliptic regularity results with Neumann boundary conditions, we obtain that $&\\Vert \\nabla P^\\epsilon _1\\Vert _{m-1}+\\Vert \\nabla ^2 P^\\epsilon _1\\Vert _{m-1}\\nonumber \\\\\\le \\, &C\\,\\big {(}\\Vert \\nabla v\\cdot \\nabla v-\\nabla H\\cdot \\nabla H\\Vert _{m-1}+\\Vert v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon \\Vert \\nonumber \\\\&+\|(v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon )\\cdot n\|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}\\big {)}.$ Due to $v^\\epsilon \\cdot n=0$ , $H^\\epsilon \\cdot n=0$ and Lemma REF , we get that $\|(v^\\epsilon \\cdot \\nabla v^\\epsilon -H^\\epsilon \\cdot \\nabla H^\\epsilon )\\cdot n\|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )})\\le \\,&C\\,(\\Vert \\nabla (v\\otimes v)\\Vert _{m-1}+\\Vert v\\otimes v\\Vert _{m}\\nonumber \\\\&+\\Vert \\nabla (H\\otimes H)\\Vert _{m-1}+\\Vert H\\otimes H\\Vert _{m}).$ Using Lemma REF , we get (REF ).", "It remains to estimate $P^\\epsilon _2$ .", "By using the standard elliptic regularity results with Neumann boundary conditions again, we obtain $\\Vert \\nabla P^\\epsilon _2\\Vert _{m-1}\\le C\\,\\epsilon \\, \|\\Delta v^\\epsilon \\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}.\\nonumber $ Since $\\Delta v^\\epsilon \\cdot n=2\\Big {(}\\nabla \\cdot (Sv^\\epsilon n)-\\sum _j(Sv^\\epsilon \\partial _jn)_j\\Big {)},\\nonumber $ we can get $\|\\Delta v^\\epsilon \\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\le C\\,\|\\nabla \\cdot (Sv^\\epsilon n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}+C\\,\|\\nabla v^\\epsilon \|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}.\\nonumber $ Due to (REF ) and (REF ), we can further arrive at $\|\\Delta v^\\epsilon \\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\le C\\,\|\\nabla \\cdot (Sv^\\epsilon n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}+C\\,\|v^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}.\\nonumber $ Let us estimate $\|\\nabla \\cdot (Sv^\\epsilon n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}$ .", "We can use (REF ) to obtain $\|\\nabla \\cdot (Sv^\\epsilon n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\le \\,&C\\,\|\\partial _n(Sv^\\epsilon n)\\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\nonumber \\\\&+C\\,(\|\\Pi (Sv^\\epsilon n)\|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}+\|\\nabla v^\\epsilon \|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}).$ Also, due to (REF ), (REF ) and the Navier boundary conditions, we get $\|\\nabla \\cdot (Sv^\\epsilon n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\le \\,C\\,\|\\partial _n(Sv^\\epsilon n)\\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}+\|v^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}.$ The first term of the right-hand side of (REF ) have the following estimates $\|\\partial _n(Sv^\\epsilon n)\\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}&\\le \\, C\\,\|\\partial _n(\\partial _nv^\\epsilon \\cdot n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}+C\\,\|\\nabla v^\\epsilon \|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\nonumber \\\\&\\le \\, C\\,\|\\partial _n(\\partial _nv^\\epsilon \\cdot n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}+C\\,\|v^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}.$ By taking the normal derivative of (REF ) and using (REF ), we obtain $\|\\partial _n(\\partial _nv^\\epsilon \\cdot n)\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}&\\le C\\,\|\\Pi \\partial _nv^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}+C \\,\|\\nabla v^\\epsilon \|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\nonumber \\\\&\\le C\\,\|v^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}.$ Consequently, we have $\|\\Delta v^\\epsilon \\cdot n\|_{H^{m-\\frac{3}{2}}(\\partial \\Omega )}\\le C\\,\|v^\\epsilon \|_{H^{m-\\frac{1}{2}}(\\partial \\Omega )}.\\nonumber $ By Lemma REF , we finally get () which complete the proof of Lemma REF .", "We can get from Lemmas REF -REF that $&\\Vert v^\\epsilon \\Vert _m^2+\\Vert H^\\epsilon \\Vert _m^2+\\Vert \\nabla v^\\epsilon \\Vert ^2_{m-1}+\\Vert \\nabla H^\\epsilon \\Vert ^2_{m-1}+\\epsilon \\int ^t_0(\\Vert \\nabla ^2v^\\epsilon \\Vert _{m-1}+\\Vert \\nabla ^2H^\\epsilon \\Vert _{m-1})\\nonumber \\\\\\le \\, &C\\,(\\Vert v^\\epsilon (0)\\Vert _m^2+\\Vert H^\\epsilon (0)\\Vert _m^2+\\Vert \\nabla v^\\epsilon (0)\\Vert ^2_{m-1}+\\Vert \\nabla H^\\epsilon (0)\\Vert ^2_{m-1})\\nonumber \\\\&+C\\,\\int ^t_0(1+\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty })\\nonumber \\\\&\\times (\\Vert v^\\epsilon \\Vert _m^2+\\Vert H^\\epsilon \\Vert _m^2+\\Vert \\nabla v^\\epsilon \\Vert ^2_{m-1}+\\Vert \\nabla H^\\epsilon \\Vert ^2_{m-1}).$" ], [ "$L^\\infty $ estimates", "In order to close the estimates in (REF ), we need to give the $L^\\infty $ estimates on $\\nabla v^\\epsilon $ and $\\nabla H^\\epsilon $ .", "We have Lemma 3.5 For $m_0>1$ , we have the following estimates: $\\Vert &v^\\epsilon \\Vert _{2,\\infty }\\le C(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1})\\le C N_m(t)^{\\frac{1}{2}}\\quad m\\ge m_0+3,\\\\\\Vert &H^\\epsilon \\Vert _{2,\\infty }\\le C(\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1})\\le C N_m(t)^{\\frac{1}{2}}\\quad m\\ge m_0+3,\\\\\\Vert &v^\\epsilon \\Vert _{W^{1,\\infty }}\\le C(\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}+\\Vert \\partial _zv^\\epsilon \\Vert _{L^\\infty })\\le C N_m(t)^{\\frac{1}{2}}\\quad m\\ge m_0+2,\\\\\\Vert &H^\\epsilon \\Vert _{W^{1,\\infty }}\\le C(\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}+\\Vert \\partial _zH^\\epsilon \\Vert _{L^\\infty })\\le C N_m(t)^{\\frac{1}{2}}\\quad m\\ge m_0+2,$ where $N_m(t)$ is defined in (REF ).", "By using lemma REF , we can obtain (REF )-(), and ()-() are obvious.", "Lemma 3.6 For $m>6$ , we have the following estimate: $\\Vert \\nabla v&^\\epsilon \\Vert _{1,\\infty }^2+\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty }^2\\le C\\Big {(}N_m(0)+(1+t+\\epsilon ^3t^2)\\int ^t_0(N_m(s)+N_m(s)^2)ds\\Big {)}.$ We observe that, away from the boundary, the following estimates hold: $\\Vert \\beta _i\\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert \\beta _i \\nabla H^\\epsilon \\Vert _{1,\\infty }\\le C\\,(\\Vert v^\\epsilon \\Vert _m+\\Vert H^\\epsilon \\Vert _m),\\quad m\\ge 4,\\nonumber $ where $\\lbrace \\beta _i\\rbrace $ is a partition of unity subordinated to the covering (REF ).", "In order to estimate the near boundary parts, we adopt the ideas in the Proposition 21 of [16].", "Here, we use a local parametrization in the vicinity of the boundary given by a normal geodesic system: $ \\Psi ^n(y,z)=\\left(\\begin{array}{c}y\\\\\\psi (y)\\\\\\end{array}\\right)-zn(y),$ where $n(y)=\\frac{1}{\\sqrt{1+\|\\nabla \\psi (y)\|^2}}\\left(\\begin{array}{c}\\partial _1\\psi (y)\\\\\\partial _2\\psi (y)\\\\-1\\end{array}\\right).$ Now, we can extend $n$ and $\\Pi $ in the interior by setting $n(\\Psi ^n(y,z))=n(y),\\quad \\Pi (\\Psi ^n(y,z))=\\Pi (y).$ We observe $\\partial _z=\\partial _n$ and $\\left(\\begin{array}{c}\\partial _{y^i}\\end{array}\\right)\\Big {\|}_{\\Psi ^n(y,z)}\\cdot \\left(\\begin{array}{c}\\partial _z\\end{array}\\right)\\Big {\|}_{\\Psi ^n(y,z)}=0.\\nonumber $ Hence, the Riemann metric $g$ has the following form $\\nonumber g(y,z)=\\left(\\begin{matrix}\\widetilde{g}(y,z)&0\\\\0&1\\\\\\end{matrix}\\right).$ Consequently, the Laplacian in this coordinate system reads: $\\nonumber \\Delta f=\\partial _{zz}f+\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _zf+\\Delta _{\\widetilde{g}}f,$ where $\|g\|$ is the determinant of the matrix $g$ and $\\Delta _{\\widetilde{g}}$ is defined by $\\Delta _{\\widetilde{g}}f=\\frac{1}{\|\\widetilde{g}\|^{\\frac{1}{2}}}\\sum _{1\\le i,j\\le 2}\\partial _{y^i}(\\widetilde{g}^{ij}\|\\widetilde{g}\|^{\\frac{1}{2}}\\partial _{y^j}f).$ Here, $\\lbrace \\widetilde{g}^{ij}\\rbrace $ is the inverse matrix to $g$ and (REF ) only involves tangential derivatives.", "With these preparation, we now turn to estimate the near boundary parts.", "Due to (REF ), (REF ) and () , we have $&\\Vert \\chi \\nabla v^\\epsilon \\Vert _{1,\\infty }\\le \\,C\\,(\\Vert \\chi \\Pi \\partial _nv^\\epsilon \\Vert _{1,\\infty }+\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}),\\\\&\\Vert \\chi \\nabla H^\\epsilon \\Vert _{1,\\infty }\\le \\,C\\,(\\Vert \\chi \\Pi \\partial _nH^\\epsilon \\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}).$ Hence, we need to estimate $\\Vert \\chi \\Pi \\partial _nv^\\epsilon \\Vert _{1,\\infty }$ and $\\Vert \\chi \\Pi \\partial _nH^\\epsilon \\Vert _{1,\\infty }$ .", "To this end, we first introduce the vorticity $\\omega _v^\\epsilon =\\nabla \\times v^\\epsilon ,\\quad \\omega _H^\\epsilon =\\nabla \\times H^\\epsilon .$ We find that $\\Pi (\\omega _v^\\epsilon \\times n)&=\\Pi (\\nabla v^\\epsilon -(\\nabla v^\\epsilon )^t)n\\nonumber \\\\&=\\Pi (\\partial _nv^\\epsilon -\\nabla (v^\\epsilon \\cdot n)+v^\\epsilon \\cdot \\nabla n+v^\\epsilon \\times (\\nabla \\times n)).$ Consequently, we have $\\Vert \\chi \\Pi \\partial _nv^\\epsilon \\Vert _{1,\\infty }\\le C\\,(\\Vert \\chi \\Pi (\\omega _v^\\epsilon \\times n)\\Vert _{1,\\infty }+\\Vert v^\\epsilon \\Vert _{2,\\infty }).$ By using (REF ) again, we get $\\Vert \\chi \\nabla v^\\epsilon \\Vert _{1,\\infty }\\le \\, C\\,(\\Vert \\chi \\Pi (\\omega _v^\\epsilon \\times n)\\Vert _{1,\\infty }+\\Vert v^\\epsilon \\Vert _m+\\Vert \\nabla v^\\epsilon \\Vert _{m-1}).$ Similar to $v^\\epsilon $ , we have the following estimates for $H^\\epsilon $ , $\\Vert \\chi \\nabla H^\\epsilon \\Vert _{1,\\infty }\\le \\,C\\,(\\Vert \\chi \\Pi (\\omega _H^\\epsilon \\times n)\\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _m+\\Vert \\nabla H^\\epsilon \\Vert _{m-1}).$ Below we estimate $\\Vert \\chi \\Pi (\\omega _v^\\epsilon \\times n)\\Vert _{1,\\infty }$ and $\\Vert \\chi \\Pi (\\omega _H^\\epsilon \\times n)\\Vert _{1,\\infty }$ .", "We know that $\\omega _v^\\epsilon $ and $\\omega _H^\\epsilon $ satisfy $&\\partial _t\\omega _v^\\epsilon -\\epsilon \\Delta \\omega _v^\\epsilon +v^\\epsilon \\cdot \\nabla \\omega _v^\\epsilon -H^\\epsilon \\cdot \\nabla \\omega _H^\\epsilon +\\omega _H^\\epsilon \\cdot \\nabla H^\\epsilon -\\omega _v^\\epsilon \\cdot \\nabla v^\\epsilon =0,\\\\&\\partial _t\\omega _H^\\epsilon -\\epsilon \\Delta \\omega _H^\\epsilon +v^\\epsilon \\cdot \\nabla \\omega _H^\\epsilon -H^\\epsilon \\cdot \\nabla \\omega _v^\\epsilon +[\\nabla \\times ,v^\\epsilon \\cdot \\nabla ] H^\\epsilon -[\\nabla \\times ,H^\\epsilon \\cdot \\nabla ] v^\\epsilon =0.$ By setting $&\\widetilde{\\omega }_v^\\epsilon (y,z):=\\omega _v^\\epsilon (\\Psi ^n(y,z)),\\quad \\,\\,\\,\\,\\widetilde{v}^\\epsilon (y,z):=v^\\epsilon (\\Psi ^n(y,z)),\\nonumber \\\\&\\widetilde{\\omega }_H^\\epsilon (y,z):=\\omega _H^\\epsilon (\\Psi ^n(y,z)),\\quad \\widetilde{H}^\\epsilon (y,z):=H^\\epsilon (\\Psi ^n(y,z)),\\nonumber $ we have $\\partial _t\\widetilde{\\omega }_v^\\epsilon +&(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\omega }_v^\\epsilon +(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\omega }_v^\\epsilon +\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{\\omega }_v^\\epsilon -(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\omega }_H^\\epsilon -(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\omega }_H^\\epsilon \\\\&-\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{\\omega }_H^\\epsilon =\\epsilon (\\partial _{zz}\\widetilde{\\omega }_v^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\omega }_v^\\epsilon +\\Delta _{\\widetilde{g}}\\widetilde{\\omega }_v^\\epsilon )+\\overline{F}^v,\\\\\\partial _t\\widetilde{\\omega }_H^\\epsilon +&(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\omega }_H^\\epsilon +(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\omega }_H^\\epsilon +\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{\\omega }_H^\\epsilon -(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\omega }_v^\\epsilon -(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\omega }_v^\\epsilon \\\\&-\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{\\omega }_v^\\epsilon =\\epsilon (\\partial _{zz}\\widetilde{\\omega }_H^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\omega }_H^\\epsilon +\\Delta _{\\widetilde{g}}\\widetilde{\\omega }_H^\\epsilon )+\\overline{F}^H,\\\\\\partial _t\\widetilde{v}^\\epsilon +&(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{v}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{v}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{v}-(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{H}-(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{H}-\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{H}\\\\&=\\epsilon (\\partial _{zz}\\widetilde{v}^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{v}^\\epsilon +\\Delta _{\\widetilde{g}}\\widetilde{v}^\\epsilon )-(\\nabla P^\\epsilon )\\circ \\Psi ^n,\\\\\\partial _t\\widetilde{H}^\\epsilon +&(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{H}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{H}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{H}-(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{v}-(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{v}-\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{v}\\\\&=\\epsilon (\\partial _{zz}\\widetilde{H}^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{H}^\\epsilon +\\Delta _{\\widetilde{g}}\\widetilde{H}^\\epsilon ),$ where $\\overline{F}^v:=\\omega _v^\\epsilon \\cdot \\nabla v^\\epsilon -\\omega _H^\\epsilon \\cdot \\nabla H^\\epsilon ,\\quad \\ \\overline{F}^H:=[\\nabla \\times ,H^\\epsilon \\cdot \\nabla ] v^\\epsilon -[\\nabla \\times ,v^\\epsilon \\cdot \\nabla ] H^\\epsilon .$ By using (REF ) and (REF ) on the boundary, we have $\\Pi (\\widetilde{\\omega }_v^\\epsilon \\times n)=2\\Pi (\\widetilde{v}^\\epsilon \\cdot \\nabla n-\\zeta \\widetilde{v}^\\epsilon ),\\quad \\Pi (\\widetilde{\\omega }_H^\\epsilon \\times n)=2\\Pi (\\widetilde{H}^\\epsilon \\cdot \\nabla n-\\zeta \\widetilde{H}^\\epsilon ),~~z=0.$ Consequently, we introduce the following quantities $&\\widetilde{\\eta }_v^\\epsilon (y,z):=\\chi \\Pi (\\widetilde{\\omega }_v^\\epsilon \\times n-2\\widetilde{v}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{v}^\\epsilon ),\\\\&\\widetilde{\\eta }_H^\\epsilon (y,z):=\\chi \\Pi (\\widetilde{\\omega }_H^\\epsilon \\times n-2\\widetilde{H}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{H}^\\epsilon ).$ Noting that $\\widetilde{\\eta }_v^\\epsilon (y,0)=0$ and $\\widetilde{\\eta }_H^\\epsilon (y,0)=0$ , we easily get $&\\!\\!\\!\\partial _t\\widetilde{\\eta }_v^\\epsilon +(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\eta }_v^\\epsilon +(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\eta }_v^\\epsilon +\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{\\eta }_v^\\epsilon -(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\eta }_H^\\epsilon -(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\eta }_H^\\epsilon \\nonumber \\\\&\\!\\!\\!\\ \\ -\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{\\eta }_H^\\epsilon =\\epsilon (\\partial _{zz}\\widetilde{\\eta }_v^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\eta }_v^\\epsilon )+\\chi \\Pi \\overline{F}^v\\times n+\\overline{F}_v^v+\\overline{F}_v^\\chi +\\overline{F}_v^\\kappa ,\\\\&\\!\\!\\!\\!\\partial _t\\widetilde{\\eta }_H^\\epsilon +(\\widetilde{v}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\eta }_H^\\epsilon +(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\eta }_H^\\epsilon +\\widetilde{v}^\\epsilon \\cdot n\\partial _z\\widetilde{\\eta }_H^\\epsilon -(\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\widetilde{\\eta }_v^\\epsilon -(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}\\widetilde{\\eta }_v^\\epsilon \\nonumber \\\\&\\!\\!\\!\\ \\ -\\widetilde{H}^\\epsilon \\cdot n\\partial _z\\widetilde{\\eta }_v^\\epsilon =\\epsilon (\\partial _{zz}\\widetilde{\\eta }_H^\\epsilon +\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\eta }_H^\\epsilon )+\\overline{F}_H^\\chi +\\overline{F}_H^\\kappa +\\chi \\Pi \\overline{F}^H\\times n,$ where $\\overline{F}_v^v=&\\,2\\chi \\Pi (\\nabla P^\\epsilon \\cdot \\nabla n-\\zeta \\nabla P^\\epsilon )\\circ \\Psi ^n,\\\\\\overline{F}_v^\\chi =&\\,(((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\chi )\\Pi (\\widetilde{\\omega }_v^\\epsilon \\times n-2\\widetilde{v}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{v}^\\epsilon )\\\\&-(((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\chi )\\Pi (\\widetilde{\\omega }_H^\\epsilon \\times n-2\\widetilde{H}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{H}^\\epsilon )\\\\&-\\epsilon (\\partial _{zz}\\chi +2\\partial _z\\chi \\partial _z+\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\chi )\\Pi (\\widetilde{\\omega }_v^\\epsilon \\times n-2\\widetilde{v}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{v}^\\epsilon ),\\\\\\overline{F}_v^\\kappa =&\\chi (((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})\\Pi )(\\widetilde{\\omega }_v^\\epsilon \\times n-2\\widetilde{v}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{v}^\\epsilon )+\\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{\\omega }_v^\\epsilon \\times n)\\\\&-\\chi (((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})\\Pi )(\\widetilde{\\omega }_H^\\epsilon \\times n-2\\widetilde{H}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{H}^\\epsilon )\\\\&+\\chi \\Pi ((((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})\\nabla n)\\widetilde{H}^\\epsilon )-\\chi \\Pi (\\widetilde{\\omega }_H^\\epsilon \\times ((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}\\\\&+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})n)-2\\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{v}^\\epsilon \\nabla n)-\\chi \\Pi ((((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})\\nabla n)\\widetilde{v}^\\epsilon )\\\\&+\\chi \\Pi (\\widetilde{\\omega }_v^\\epsilon \\times ((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})n)+2\\zeta \\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{v}^\\epsilon ),\\\\\\overline{F}_H^\\chi =&(((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\chi )\\Pi (\\widetilde{\\omega }_H^\\epsilon \\times n-2\\widetilde{H}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{H}^\\epsilon )\\\\&-(((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\chi )\\Pi (\\widetilde{\\omega }_v^\\epsilon \\times n-2\\widetilde{v}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{v}^\\epsilon )\\\\&-\\epsilon (\\partial _{zz}\\chi +2\\partial _z\\chi \\partial _z+\\frac{1}{2}\\partial _z(\\ln \|g\|)\\partial _z\\chi )\\Pi (\\widetilde{\\omega }_H^\\epsilon \\times n-2\\widetilde{H}^\\epsilon \\cdot \\nabla n+2\\zeta \\widetilde{H}^\\epsilon ),\\\\\\overline{F}_H^\\kappa =&\\chi (((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})\\Pi )(\\widetilde{\\omega }_H^\\epsilon \\times n-\\widetilde{H}^\\epsilon \\cdot \\nabla n+\\zeta \\widetilde{H}^\\epsilon )+2\\zeta \\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{H}^\\epsilon )\\\\&-\\chi (((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})\\Pi )(\\widetilde{\\omega }_v^\\epsilon \\times n-\\widetilde{v}^\\epsilon \\cdot \\nabla n+\\zeta \\widetilde{H}^\\epsilon )+\\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{\\omega }_H^\\epsilon \\times n)\\\\&+\\chi \\Pi ((((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})\\nabla n)\\widetilde{v}^\\epsilon )-\\chi \\Pi (\\widetilde{\\omega }_v^\\epsilon \\times ((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2})n)\\\\&-2\\epsilon \\chi \\Pi (\\Delta _{\\widetilde{g}}\\widetilde{H}^\\epsilon \\nabla n)-\\chi \\Pi ((((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})\\nabla n)\\widetilde{H}^\\epsilon )\\\\&+\\chi \\Pi (\\widetilde{\\omega }_v^\\epsilon \\times ((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2})n).$ We know that $\\Pi $ and $n$ do not dependent the normal variable.", "Due to $\\Delta _{\\widetilde{g}}$ only involving the tangential derivatives and the derivatives of $\\chi $ compactly supported away from the boundary, we easily obtain that $\\Vert \\overline{F}^v_v\\Vert _{1,\\infty }\\le \\,&C\\,(\\Vert \\Pi \\nabla P^\\epsilon \\Vert _{1,\\infty },\\\\\\Vert \\overline{F}^v_\\chi \\Vert _{1,\\infty }\\le \\,&C\\,(\\Vert v^\\epsilon \\Vert _{1,\\infty }\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{1,\\infty }\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\epsilon \\Vert v^\\epsilon \\Vert _{3,\\infty }),\\\\\\Vert \\overline{F}_v^\\kappa \\Vert _{1,\\infty }\\le \\, &C\\,(\\Vert v^\\epsilon \\Vert _{1,\\infty }\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _{1,\\infty }\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty }+\\Vert v^\\epsilon \\Vert _{1,\\infty }^2+\\Vert H^\\epsilon \\Vert _{1,\\infty }^2\\nonumber \\\\&+\\epsilon \\Vert v^\\epsilon \\Vert _{3,\\infty }+\\epsilon \\Vert \\nabla v^\\epsilon \\Vert _{3,\\infty }),\\\\\\Vert \\overline{F}^H_\\chi \\Vert _{1,\\infty }\\le \\, &C\\,(\\Vert v^\\epsilon \\Vert _{1,\\infty }\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{1,\\infty }\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\epsilon \\Vert H^\\epsilon \\Vert _{3,\\infty }),\\\\\\Vert \\overline{F}_H^\\kappa \\Vert _{1,\\infty }\\le \\, &C\\,(\\Vert v^\\epsilon \\Vert _{1,\\infty }\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty }+\\Vert H^\\epsilon \\Vert _{1,\\infty }\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert v^\\epsilon \\Vert _{1,\\infty }^2+\\Vert H^\\epsilon \\Vert _{1,\\infty }^2\\nonumber \\\\&+\\epsilon \\Vert H^\\epsilon \\Vert _{3,\\infty }+\\epsilon \\Vert \\nabla H^\\epsilon \\Vert _{3,\\infty }).$ A crucial estimate towards the proof of Lemma REF is the following: Lemma 3.7 ([16]) Let $\\rho $ is a smooth solution of $\\nonumber \\partial _t\\rho +u\\cdot \\nabla \\rho =\\epsilon \\partial _{zz}\\rho +f,\\quad z>0,\\quad \\rho (t,y,0)=0,$ where $u$ satisfies the divergence free condition and $u\\cdot n$ vanishes on the boundary.", "Assume that $\\rho $ and $f$ are compactly supported with respect to $z$ .", "Then, we have the estimate: $\\Vert \\rho \\Vert _{1,\\infty }\\le \\,C&\\Vert \\rho (0)\\Vert _{1,\\infty }+C\\int _0^t\\big {\\lbrace }(\\Vert u\\Vert _{2,\\infty }+\\Vert \\partial _zu\\Vert _{1,\\infty })\\nonumber \\\\&\\quad \\qquad \\qquad \\qquad \\times (\\Vert \\rho \\Vert _{1,\\infty }+\\Vert \\rho \\Vert _{m_0+3})+\\Vert f\\Vert _{1,\\infty }\\big {\\rbrace }\\quad \\text{for}\\quad m_0>2.$ In order to use Lemma REF , we shall eliminate $\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\eta }_v^\\epsilon $ in (REF ) and $\\partial _z(\\ln \|g\|)\\partial _z\\widetilde{\\eta }_H^\\epsilon $ in (), respectively.", "We set $\\widetilde{\\eta }_v^\\epsilon =\\frac{1}{\|g\|^{\\frac{1}{4}}}\\overline{\\eta }_v^\\epsilon =\\gamma \\overline{\\eta }_v^\\epsilon ,\\quad \\widetilde{\\eta }_H^\\epsilon =\\frac{1}{\|g\|^{\\frac{1}{4}}}\\overline{\\eta }_H^\\epsilon =\\gamma \\overline{\\eta }_H^\\epsilon .$ We note that $\\Vert \\widetilde{\\eta }_v^\\epsilon \\Vert _{1,\\infty }\\sim \\Vert \\overline{\\eta }_v^\\epsilon \\Vert _{1,\\infty },\\quad \\Vert \\widetilde{\\eta }_H^\\epsilon \\Vert _{1,\\infty }\\sim \\Vert \\overline{\\eta }_H^\\epsilon \\Vert _{1,\\infty }$ and $\\overline{\\eta }_v^\\epsilon $ and $\\overline{\\eta }_H^\\epsilon $ solve the equations $\\partial _t\\overline{\\eta }_v^\\epsilon +((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}&+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\overline{\\eta }_v^\\epsilon -((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\overline{\\eta }_H^\\epsilon \\nonumber \\\\-\\epsilon \\partial _{zz}\\overline{\\eta }_v^\\epsilon =&\\frac{1}{\\gamma }(\\chi \\Pi \\overline{F}^v\\times n+\\overline{F}_v^v+\\overline{F}_v^\\chi +\\overline{F}_v^\\kappa +\\epsilon \\partial _{zz}\\gamma \\overline{\\eta }_v^\\epsilon +\\frac{\\epsilon }{2}\\partial _z(\\ln \|g\|)\\partial _z\\gamma \\overline{\\eta }_v^\\epsilon \\nonumber \\\\&-(\\widetilde{v}^\\epsilon \\cdot \\nabla \\gamma )\\overline{\\eta }_v^\\epsilon +(\\widetilde{H}^\\epsilon \\cdot \\nabla \\gamma )\\overline{\\eta }_H^\\epsilon ):=S_1,\\\\\\partial _t\\overline{\\eta }_H^\\epsilon +((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}&+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\overline{\\eta }_H^\\epsilon -((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\overline{\\eta }_v^\\epsilon \\nonumber \\\\-\\epsilon \\partial _{zz}\\overline{\\eta }_H^\\epsilon =&\\frac{1}{\\gamma }(\\chi \\Pi \\overline{F}^H\\times n+\\overline{F}_H^\\chi +\\overline{F}_H^\\kappa +\\epsilon \\partial _{zz}\\gamma \\overline{\\eta }_H^\\epsilon +\\frac{\\epsilon }{2}\\partial _z(\\ln \|g\|)\\partial _z\\gamma \\overline{\\eta }_H^\\epsilon \\nonumber \\\\&-(\\widetilde{v}^\\epsilon \\cdot \\nabla \\gamma )\\overline{\\eta }_v^\\epsilon +(\\widetilde{H}^\\epsilon \\cdot \\nabla \\gamma )\\overline{\\eta }_H^\\epsilon ):=S_2.$ Finally, we set $\\eta _1:=\\overline{\\eta }_v^\\epsilon +\\overline{\\eta }_H^\\epsilon ,\\quad \\eta _2:=\\overline{\\eta }_v^\\epsilon -\\overline{\\eta }_H^\\epsilon $ and easily find $\\partial _t\\eta _1+((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}&+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\eta _1\\nonumber \\\\&-((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\eta _1-\\epsilon \\partial _{zz}\\eta _1=S_1+S_2,\\\\\\partial _t\\eta _2+((\\widetilde{v}^\\epsilon )^1\\partial _{y^1}&+(\\widetilde{v}^\\epsilon )^2\\partial _{y^2}+\\widetilde{v}^\\epsilon \\cdot n\\partial _z)\\eta _2\\nonumber \\\\&-((\\widetilde{H}^\\epsilon )^1\\partial _{y^1}+(\\widetilde{H}^\\epsilon )^2\\partial _{y^2}+\\widetilde{H}^\\epsilon \\cdot n\\partial _z)\\eta _2-\\epsilon \\partial _{zz}\\eta _1=S_1-S_2.$ By applying Lemma REF to (REF ), we directly obtain $\\Vert \\eta _1\\Vert _{1,\\infty }\\le & \\,C\\,\\Vert \\eta _1(0)\\Vert _{1,\\infty }+\\int _0^t\\big {\\lbrace }(\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty })\\nonumber \\\\&\\times (\\Vert \\eta _1\\Vert _{1,\\infty }+\\Vert \\eta _1\\Vert _{m_0+3})+\\Vert S_1\\Vert _{1,\\infty }+\\Vert S_2\\Vert _{1,\\infty }\\big {\\rbrace }\\quad \\text{for}\\quad m_0>2.$ From (REF )-() and (REF )-(), we get $\\Vert \\eta _1\\Vert _{1,\\infty }\\le & \\,C\\,\\Vert \\eta _1(0)\\Vert _{1,\\infty }+\\int _0^t\\big {\\lbrace }(\\Vert v^\\epsilon \\Vert _{2,\\infty }+\\Vert H^\\epsilon \\Vert _{2,\\infty }+\\Vert \\nabla v^\\epsilon \\Vert _{1,\\infty }+\\Vert \\nabla H^\\epsilon \\Vert _{1,\\infty })\\nonumber \\\\&\\times (\\Vert \\eta _1\\Vert _{1,\\infty }+\\Vert \\eta _1\\Vert _{m_0+3}+\\Vert \\eta _2\\Vert _{1,\\infty }+\\Vert \\eta _2\\Vert _{m_0+3}+N_m^{\\frac{1}{2}})+N_m\\nonumber \\\\&+N_m^{\\frac{1}{2}}+\\epsilon (\\Vert \\nabla v^\\epsilon \\Vert _{3,\\infty }+\\Vert \\nabla H^\\epsilon \\Vert _{3,\\infty })+\\Vert \\Pi \\nabla P^\\epsilon \\Vert _{1,\\infty }\\nonumber \\\\&+\\Vert \\Pi (\\nabla P^\\epsilon \\cdot \\nabla n)\\Vert _{1,\\infty }\\big {\\rbrace }\\quad \\text{for}\\quad m_0>2.$ Due to Lemmas REF and REF , we have $\\Vert \\Pi \\nabla P^\\epsilon \\Vert _{1,\\infty }\\le C( N_m^{\\frac{1}{2}}(t)+N_m(t))\\quad \\text{for}\\quad m\\ge 4.$ Now, we deal with the terms with the coefficient $\\epsilon $ .", "From Lemma REF , we get $&\\Big {(}\\epsilon \\int _0^t(\\Vert \\nabla v^\\epsilon \\Vert _{3,\\infty }+\\Vert \\nabla H^\\epsilon )\\Vert _{3,\\infty }\\Big {)}^2\\nonumber \\\\&\\le C\\epsilon ^2\\Big {(}\\int _0^t\\Vert \\nabla ^2v^\\epsilon \\Vert _{m-1}^{\\frac{1}{2}}+\\Vert \\nabla ^2H^\\epsilon )\\Vert _{m-1}^{\\frac{1}{2}})N_m^{\\frac{1}{4}}\\Big {)}^2+C\\epsilon ^2t\\int _0^tN_m\\nonumber \\\\&\\le C\\epsilon ^2t\\Big {(}\\int _0^t(\\Vert \\nabla ^2v^\\epsilon \\Vert _{m-1}^2+\\Vert \\nabla ^2H^\\epsilon \\Vert _{m-1}^2)\\Big {)}^{\\frac{1}{2}}\\Big {(}\\int _0^tN_m\\Big {)}^{\\frac{1}{2}}+C\\epsilon ^2t\\int _0^tN_m\\nonumber \\\\&\\le C\\epsilon \\int _0^t(\\Vert \\nabla ^2v^\\epsilon \\Vert _{m-1}^2+\\Vert \\nabla ^2H^\\epsilon \\Vert _{m-1}^2)+C(\\epsilon ^2t+\\epsilon ^3t^2)\\int _0^tN_m$ for $m\\ge m_0+4$ .", "Consequently, we get from (REF ), (REF )-() and (REF )-(REF ) that $\\Vert \\eta _1\\Vert _{1,\\infty }^2\\le C N(0)+C(1+t+\\epsilon ^3t^2)\\int _0^t(N_m^2+N_m).$ Similarly, we also get $\\Vert \\eta _2\\Vert _{1,\\infty }^2\\le C N(0)+C(1+t+\\epsilon ^4t^2)\\int _0^t(N_m^2+N_m).$ Therefore, we complete the proof of Lemma REF ." ], [ "Proof of Theorem ", "Based on Lemma REF , Lemma REF and (REF ), we can easily prove Theorem REF .", "We omit the details here." ], [ "Proof of Theorem ", "By smoothing the initial data and using the a priori estimates obtained in Theorem REF and the strong compactness argument, we can prove Theorem REF in the same spirit of [16].", "Hence we omit it here." ], [ "Proof of Theorem ", "In this section, we shall establish the convergence with a rate for the solution $(v^{\\epsilon },H^{\\epsilon })$ to $(v,H)$ .", "We start with the rate of convergence in $L^{2}$ .", "Lemma 4.1 Under the assumptions in the Theorem REF , we have $\\nonumber \\Vert v^{\\epsilon }-v \\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2+\\epsilon \\int _o^t(\\Vert v^{\\epsilon }-v\\Vert _{H^1}^2+\\Vert H^{\\epsilon }-H\\Vert _{H^1}^2)\\le C\\epsilon ^\\frac{3}{2}\\quad on\\quad [0,T_2],$ where $\\epsilon $ small enough and $T_2=\\min \\lbrace T_0,T_1\\rbrace $ .", "Consequently, we have $\\nonumber \\Vert v^{\\epsilon }-v\\Vert _{L^\\infty ([0,T_2]\\times \\Omega )}+\\Vert H^{\\epsilon }-H\\Vert _{L^\\infty ([0,T_2]\\times \\Omega )}\\le C\\epsilon ^{\\frac{3}{10}}.$ We note that $v^{\\epsilon }-v$ and $H^{\\epsilon }-H$ satisfy $& \\partial _t(v^{\\epsilon }-v)-\\epsilon \\Delta (v^{\\epsilon }-v)+\\Phi _1+\\nabla (p^{\\epsilon }-p)=\\epsilon \\Delta v \\quad \\text{in}\\quad \\Omega ,\\\\& \\partial _t(H^{\\epsilon }-H)-\\epsilon \\Delta (H^{\\epsilon }-H)+\\Phi _2=\\epsilon \\Delta H \\quad \\text{in}\\quad \\Omega ,\\\\& \\nabla \\cdot v^{\\epsilon }=0 ~~,~~ \\nabla \\cdot H^{\\epsilon } =0 \\quad \\text{in}\\quad \\Omega ,\\\\& (v^{\\epsilon }-v)\\cdot n=0,\\ \\ n\\times (\\omega ^{\\epsilon }_v-\\omega _v)=[B(v^{\\epsilon }-v)+Bv]_\\tau -n\\times \\omega _v\\ \\ \\text{on}\\ \\ \\partial \\Omega ,\\\\& (H^{\\epsilon }-H)\\cdot n=0,\\ \\ n\\times (\\omega ^{\\epsilon }_H-\\omega _H)=[B(H^{\\epsilon }-H)+BH]_\\tau -n\\times \\omega _H\\ \\ \\text{on}\\ \\ \\partial \\Omega ,$ where $\\omega ^{\\epsilon }_v=\\nabla \\times v^{\\epsilon },\\,\\,\\,\\omega ^{\\epsilon }_H=\\nabla \\times H^{\\epsilon },\\,\\,\\,\\omega _v=\\nabla \\times v,\\,\\,\\,\\omega _H=\\nabla \\times H$ , and $\\Phi _1:=\\,&v\\cdot \\nabla (v^{\\epsilon }-v)+(v^{\\epsilon }-v)\\cdot \\nabla v+(v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v)\\\\&-H\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla H-(H^{\\epsilon }-H)\\cdot \\nabla (H^{\\epsilon }-H)\\\\&+\\frac{1}{2}\\nabla (\|H^{\\epsilon }\|^2-\|H\|^2)-\\frac{1}{2}\\nabla (\|v^{\\epsilon }\|^2-\|v\|^2),\\\\\\Phi _2:=\\,&(v^{\\epsilon }-v)\\cdot \\nabla H+(v^{\\epsilon }-v)\\cdot \\nabla (H^{\\epsilon }-H)+v\\cdot \\nabla (H^{\\epsilon }-H)\\\\&-(H^{\\epsilon }-H)\\cdot \\nabla v-(H^{\\epsilon }-H)\\cdot \\nabla (v^{\\epsilon }-v)-H\\cdot \\nabla (v^{\\epsilon }-v).$ Doing basic $L^{2}$ -estimate, we obtain the following identity: $\\frac{1}{2}&\\frac{d}{dt}(\\Vert v^{\\epsilon }-v \\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2)+\\epsilon (\\Vert \\nabla \\times (v^{\\epsilon }-v) \\Vert ^2+\\Vert \\nabla \\times ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\&+(\\Phi _1,v^{\\epsilon }-v )+(\\Phi _2,H^{\\epsilon }-H)+B_1+B_2=(\\epsilon \\Delta v,v^{\\epsilon }-v)+(\\epsilon \\Delta H,H^{\\epsilon }-H),$ where $B_1&:=\\epsilon \\int _{\\partial \\Omega } n\\times (\\omega ^{\\epsilon }_v-\\omega _v)(v^{\\epsilon }-v )\\\\&=\\epsilon \\int _{\\partial \\Omega }(B(v^{\\epsilon }-v )+Bv-n\\times \\omega _v)(v^{\\epsilon }-v ),\\\\B_2&:=\\epsilon \\int _{\\partial \\Omega } n\\times (\\omega ^{\\epsilon }_H-\\omega _H)(H^{\\epsilon }-H )\\\\&=\\epsilon \\int _{\\partial \\Omega }(B(H^{\\epsilon }-H )+BH-n\\times \\omega _H)(H^{\\epsilon }-H ).$ First, we easily note that $\|(\\epsilon \\Delta v,v^{\\epsilon }-v)\|+\|(\\epsilon \\Delta H,H^{\\epsilon }-H)\|\\le C(\\Vert v^{\\epsilon }-v \\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2)+\\epsilon ^2.$ Next, we deal with the boundary terms $B_1$ and $B_2$ .", "For $B_1$ , we have $B_1=&\\,\\epsilon \\int _{\\partial \\Omega }(B(v^{\\epsilon }-v )+Bv-n\\times \\omega _v)(v^{\\epsilon }-v )\\\\\\le &\\, C\\,\\epsilon \\int _{\\partial \\Omega }(\|v^{\\epsilon }-v \|^2+\|v^{\\epsilon }-v \|).$ Due to the trace theorem: $\|u\|_{L^1(\\partial \\Omega )}\\le C\\,\|u\|_{L^2(\\partial \\Omega )}\\le C\\Vert u\\Vert _{H^\\frac{1}{2}}$ and the interpolation inequality: $\\Vert u\\Vert _{H^\\frac{1}{2}(\\Omega )}\\le C\\,\\Vert u\\Vert ^{\\frac{1}{2}}\\Vert u\\Vert _{H^1}^{\\frac{1}{2}},$ we further obtain that $B_1\\le \\,& C\\epsilon \\,(\\Vert v^{\\epsilon }-v \\Vert \\Vert \\omega ^{\\epsilon }_v-\\omega _v\\Vert +\|v^{\\epsilon }-v \|_{L^1(\\partial \\Omega )})\\nonumber \\\\\\le &\\,2\\delta \\epsilon \\Vert \\omega ^{\\epsilon }_v-\\omega _v\\Vert ^2+C_\\delta \\Vert v^{\\epsilon }-v \\Vert ^2+\\epsilon ^\\frac{3}{2}.$ Similarly, we also get that $B_2\\le 2\\delta \\epsilon \\Vert \\omega ^{\\epsilon }_H-\\omega _H\\Vert ^2+C_\\delta \\Vert H^{\\epsilon }-H \\Vert ^2+\\epsilon ^\\frac{3}{2}.$ Finally, we deal with $(\\Phi _1, v^{\\epsilon }-v)$ and $(\\Phi _2, H^{\\epsilon }-H)$ .", "We have $&\|(\\Phi _1, v^{\\epsilon }-v)+(\\Phi _2, H^{\\epsilon }-H)\|\\nonumber \\\\=\\,&\\big {\|}(v^{\\epsilon }-v,v\\cdot \\nabla (v^{\\epsilon }-v)+(v^{\\epsilon }-v)\\cdot \\nabla v+(v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v)-\\frac{1}{2}\\nabla (\|v^{\\epsilon }\|^2-\|v\|^2)\\nonumber \\\\&-H\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla H-(H^{\\epsilon }-H)\\cdot \\nabla (H^{\\epsilon }-H)\\nonumber \\\\&+\\frac{1}{2}\\nabla (\|H^{\\epsilon }\|^2-\|H\|^2))+( H^{\\epsilon }-H,(v^{\\epsilon }-v)\\cdot \\nabla H+(v^{\\epsilon }-v)\\cdot \\nabla (H^{\\epsilon }-H)\\nonumber \\\\&+v\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla v-(H^{\\epsilon }-H)\\cdot \\nabla (v^{\\epsilon }-v)-H\\cdot \\nabla (v^{\\epsilon }-v))\\big {\|}.$ We note that $&(\\frac{1}{2}\\nabla (\|v^{\\epsilon }\|^2-\|v\|^2)-\\frac{1}{2}\\nabla (\|H^{\\epsilon }\|^2-\|H\|^2),v^{\\epsilon }-v)=0,\\\\&(v^{\\epsilon }-v,v\\cdot \\nabla (v^{\\epsilon }-v))=0,\\quad (v^{\\epsilon }-v,(v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v))=0,\\\\&(H^{\\epsilon }-H,v\\cdot \\nabla (H^{\\epsilon }-H))=0,\\quad (H^{\\epsilon }-H,(v^{\\epsilon }-v)\\cdot \\nabla (H^{\\epsilon }-H) )=0,\\\\&((H^{\\epsilon }-H)\\cdot \\nabla (v^{\\epsilon }-v),H^{\\epsilon }-H)+((H^{\\epsilon }-H)\\cdot \\nabla H^{\\epsilon }-H,v^{\\epsilon }-v)=0,\\\\&(H^{\\epsilon }-H,H\\cdot \\nabla (v^{\\epsilon }-v))+(v^{\\epsilon }-v,H\\cdot \\nabla (H^{\\epsilon }-H))=0.$ Consequently, one has $\|(\\Phi _1, v^{\\epsilon }-v)+(\\Phi _2, H^{\\epsilon }-H)\|\\le C(\\Vert v^{\\epsilon }-v\\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2).$ From (REF ), (REF ), (REF ) and (REF ), we get $\\frac{1}{2}\\frac{d}{dt}(\\Vert v^{\\epsilon }-v\\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2)+\\epsilon (\\Vert \\nabla \\times (v^{\\epsilon }-v) \\Vert ^2+\\Vert \\nabla \\times ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\\\le C\\Vert v^{\\epsilon }-v\\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2+\\epsilon ^\\frac{3}{2}.$ Then, by using Gronwall's inequality, we arrive at $\\Vert v^{\\epsilon }-v \\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2+\\epsilon \\int _0^t(\\Vert v^{\\epsilon }-v\\Vert _{H^1}^2+\\Vert H^{\\epsilon }-H\\Vert _{H^1}^2)\\le C\\epsilon ^\\frac{3}{2}.$ Consequently, by using the Gagliardo-Nirenberg interpolation inequality, we have $\\Vert v^{\\epsilon }-v\\Vert _{L^\\infty }+\\Vert H^{\\epsilon }-H\\Vert _{L^\\infty }\\le &\\, C(\\Vert v^{\\epsilon }-v\\Vert ^{\\frac{2}{5}}\\Vert v^{\\epsilon }-v\\Vert ^{\\frac{3}{5}}_{W^{1,\\infty }}\\nonumber \\\\&+\\Vert H^{\\epsilon }-H\\Vert ^{\\frac{2}{5}}\\Vert H^{\\epsilon }-H\\Vert ^{\\frac{3}{5}}_{W^{1,\\infty }})\\le C\\epsilon ^{\\frac{3}{10}}.$ Before we go to prove the rate of the convergence in $H^1$ , we have the following observation.", "Lemma 4.2 We have $\\Vert u\\Vert _{H^2}\\le \\Vert P\\Delta u\\Vert +\\Vert u\\Vert ,\\quad \\forall \\,u\\in W_{B},\\nonumber $ where $W_{B}=\\big {\\lbrace }u\\in H^2(\\Omega )\\,\\big {\|}\\,\\nabla \\cdot u=0\\,\\,\\,\\text{in}\\,\\,\\,\\Omega ,\\,\\,u\\cdot n=0,\\,\\,\\,n\\times (\\nabla \\times u)=[Bu]_\\tau \\,\\,\\,\\text{on}\\,\\,\\,\\partial \\Omega \\big {\\rbrace }.\\nonumber $ We consider the following boundary value problem: $&\\gamma I-\\Delta u +\\nabla p=f\\quad \\text{in}\\quad \\Omega ,\\\\&\\nabla \\cdot u=0\\quad \\text{in}\\quad \\Omega ,\\\\&u\\cdot n=0,\\quad n\\times (\\nabla \\times u)=[Bu]_\\tau \\quad \\text{on}\\quad \\Omega ,$ where $\\gamma $ is a large enough positive constant.", "Define a bilinear form as $\\mathcal {B}(u,\\phi )=\\gamma (u,\\phi )+(\\nabla \\times u,\\nabla \\times \\phi )+\\int _{\\partial \\Omega }Au\\cdot \\phi $ with the domain $D(\\mathcal {B})=\\big {\\lbrace }u\\in H^1(\\Omega )\\,\\big {\|}\\,\\nabla \\cdot u=0\\,\\,\\,\\text{in}\\,\\,\\,\\Omega ,\\,\\,u\\cdot n=0\\,\\,\\,\\text{on}\\,\\,\\,\\partial \\Omega \\big {\\rbrace }.$ It is clear that $\\mathcal {B}(u,\\phi )$ with domain $D(\\mathcal {B})$ is a positive densely defined closed bilinear form.", "Let $\\mathcal {O}$ be the self-extension of $\\mathcal {B}(u,\\phi )$ .", "We find that $W_{B}\\subset D(\\mathcal {O})$ and $\\mathcal {O}u=\\gamma u+P(-\\Delta u)$ for any $u\\in D(\\mathcal {O})$ .", "Let $u\\in W_{B}$ and $\\mathcal {O}u=f$ .", "It follows from (REF ) and Lemma REF that $\\Vert u\\Vert _{H^1}\\le C \\,\\Vert f\\Vert .$ Now, let $n(x)$ and $B(x)$ be the internal smooth extensions of the normal vector $n$ and $B$ in ().", "Based on Lemma REF , we have $B(x)u\\times n(x)=\\nabla \\times k+\\nabla h+\\nabla g,\\nonumber $ where $k\\in FH\\cap H^2$ , $\\nabla h\\in HG$ and $\\nabla g\\in GG$ .", "We find $&\\Delta g=\\nabla \\cdot (B(x)u\\times n(x))\\quad \\text{in}\\quad \\Omega ,\\\\&g=0\\quad \\text{on}\\quad \\partial \\Omega .$ From the elliptic regularity theory, we obtain $\\Vert \\nabla g\\Vert _{H^1}\\le C\\Vert u\\Vert _{H^1}.\\nonumber $ Since $HG$ is finite dimensional, the following inequality holds $\\Vert \\nabla h\\Vert _{H^1}\\le C\\Vert u\\Vert .\\nonumber $ Further, it follows from Lemma REF and Poincar$\\acute{e}$ type inequality in Lemma 3.3 of [25] that $\\Vert k\\Vert _{H^2}\\le C\\Vert \\nabla \\times k\\Vert _{H^1}\\le C\\Vert u\\Vert _{H^1}\\le C\\Vert f\\Vert .$ Integrating by parts and noting that $n\\times \\nabla h=0$ , $n\\times \\nabla g=0$ on the boundary, we have $\\int _\\Omega (\\nabla \\times k)\\cdot (\\nabla \\times \\phi )+\\int _{\\partial \\Omega }n\\times (Bu\\times n)\\cdot \\phi =(-\\Delta k,\\phi )\\nonumber $ for any $\\phi \\in H^1$ .", "We observe that $n\\times (Bu\\times n)=Bu$ , so we have $\\int _\\Omega (\\nabla \\times (u-k))\\cdot (\\nabla \\times \\phi )=(P_{FH}(f-u+\\Delta k),\\phi ),\\quad \\forall \\,\\phi \\in H^1\\cap FH,\\nonumber $ where $P_{FH}$ denotes the projection on $FH$ .", "Further, due to $\\nabla \\times u=\\nabla \\times P_{FH}(u)$ , we get $\\int _\\Omega (\\nabla \\times (P_{FH}(u)-k))\\cdot (\\nabla \\times \\phi )=(P_{FH}(f-\\gamma u+\\Delta k),\\phi ),\\quad \\forall \\,\\phi \\in H^1\\cap FH\\nonumber $ From Theorem 3.1 in [25], we obtain $\\Vert P_{FH}(u)-k\\Vert _{H^2}\\le \\,C\\,(\\Vert f\\Vert +\\Vert \\Delta k\\Vert +\\Vert u\\Vert ).$ Since $HH$ is finite dimensional, the following inequality holds $\\Vert P_{HH}(u)\\Vert _{H^2}\\le \\,C\\,\\Vert u\\Vert ,$ where $&HH=\\big {\\lbrace }u\\in L^2(\\Omega )\\,\\big {\|}\\,\\nabla \\cdot u=0,\\,\\,\\nabla \\times u=0\\,\\,\\,\\text{in}\\,\\,\\,\\Omega ,\\,\\,u\\cdot n=0\\,\\,\\,\\text{on}\\,\\,\\,\\partial \\Omega \\big {\\rbrace },\\,\\,\\,\\,\\nonumber \\\\&\\mathbb {H}=HH\\oplus FH.\\nonumber $ We get from (REF ), (REF ), (REF ) and (REF ) that $\\Vert u\\Vert _{H^2}\\le \\,C\\,\\Vert f\\Vert .\\nonumber $ Consequently, we complete the proof of Lemma REF .", "Now we turn to prove the rate of convergence in $H^1(\\Omega )$ .", "Lemma 4.3 Under the assumptions in Theorem REF , we have $\\Vert v^{\\epsilon }-v \\Vert _{H^1}^2&+\\Vert H^{\\epsilon }-H\\Vert _{H^1}^2\\nonumber \\\\&+\\epsilon \\int _0^t(\\Vert v^{\\epsilon }-v\\Vert _{H^2}^2+\\Vert H^{\\epsilon }-H\\Vert _{H^2}^2)\\le C\\epsilon ^\\frac{1}{2}\\quad \\text{on}\\quad [0,T_2],$ where $\\epsilon $ small enough and $T_2=\\min \\lbrace T_0,T_1\\rbrace $ .", "Also, we have $\\nonumber \\Vert v^{\\epsilon }-v\\Vert ^p_{W^{1,p}}+\\Vert H^{\\epsilon }-H\\Vert ^p_{W^{1,p}}\\le C\\epsilon ^\\frac{1}{2}\\quad \\text{on}\\quad [0,T_2]$ for $2\\le p<\\infty .$ We note $\\nonumber \\partial _t(v^{\\epsilon }-v)\\cdot n=0,\\quad \\partial _t(H^{\\epsilon }-H)\\cdot n=0.$ It follows from (REF )-() that $\\frac{1}{2}&\\frac{d}{dt}(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2)+\\epsilon (\\Vert P\\Delta (v^{\\epsilon }-v) \\Vert ^2+\\Vert P\\Delta ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\=&(\\Phi _1,P\\Delta (v^{\\epsilon }-v ))+(\\Phi _2,P\\Delta (H^{\\epsilon }-H))+B_1+B_2-(\\epsilon \\Delta v,P\\Delta (v^{\\epsilon }-v))\\nonumber \\\\&-(\\epsilon \\Delta H,P\\Delta (H^{\\epsilon }-H)),$ where $\\Phi _1$ and $\\Phi _2$ are as same as these in Lemma REF , but $B_1$ and $B_2$ have different forms: $\\nonumber B_1:=\\int _{\\partial \\Omega }\\partial _t(v^{\\epsilon }-v )\\cdot (n\\times (\\omega ^{\\epsilon }_v-\\omega _v)),\\quad B_2:=\\int _{\\partial \\Omega }\\partial _t(H^{\\epsilon }-H )\\cdot (n\\times (\\omega ^{\\epsilon }_H-\\omega _H)).$ Now, let us deal with these two boundary terms as follows $&B_1+B_2\\nonumber \\\\=&\\int _{\\partial \\Omega }\\partial _t(v^{\\epsilon }-v )\\cdot (B(v^{\\epsilon }-v )+Bv-n\\times \\omega _v)\\nonumber \\\\&+\\int _{\\partial \\Omega }\\partial _t(H^{\\epsilon }-H )\\cdot (B(H^{\\epsilon }-H )+BH-n\\times \\omega _H)\\nonumber \\\\=&\\frac{1}{2}\\frac{d}{dt}\\Big {(}\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot (v^{\\epsilon }-v )+2\\int _{\\partial \\Omega }(v^{\\epsilon }-v)\\cdot (Bv-n\\times \\omega _v)\\Big {)}-\\widetilde{B}_1\\nonumber \\\\&+\\frac{1}{2}\\frac{d}{dt}\\Big {(}\\int _{\\partial \\Omega }B(H^{\\epsilon }-H )\\cdot (H^{\\epsilon }-H )+2\\int _{\\partial \\Omega }(H^{\\epsilon }-H)\\cdot (BH-n\\times \\omega _H)\\Big {)}-\\widetilde{B}_2,$ where $\\widetilde{B}_1:=\\int _{\\partial \\Omega }(v^{\\epsilon }-v )\\cdot \\partial _t(Bv-n\\times \\omega _v),\\quad \\widetilde{B}_2:=\\int _{\\partial \\Omega }(H^{\\epsilon }-H )\\cdot \\partial _t(BH-n\\times \\omega _H).$ It follows from Lemma REF , (REF ) and (REF ) that $\\begin{split}\|\\widetilde{B}_1+\\widetilde{B}_2\|\\le &\\, C\\Big {(}\\int _{\\partial \\Omega }\|v^{\\epsilon }-v\|^{2}\\Big {)}^\\frac{1}{2}+\\,C\\Big {(}\\int _{\\partial \\Omega }\|H^{\\epsilon }-H\|^2\\Big {)}^\\frac{1}{2}\\\\\\le &\\,\\delta \\,(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2)+C\\epsilon ^\\frac{1}{2}.\\end{split}$ We easily get $&\|(\\epsilon \\Delta v,P\\Delta (v^{\\epsilon }-v))+(\\epsilon \\Delta H,P\\Delta (H^{\\epsilon }-H))\|\\nonumber \\\\\\le &\\,\\frac{\\epsilon }{2}(\\Vert P\\Delta (v^{\\epsilon }-v)\\Vert ^2+\\Vert P\\Delta (H^{\\epsilon }-H)\\Vert ^2)+C\\,\\epsilon $ and $&-(\\Phi _1,P\\Delta (v^{\\epsilon }-v ))-(\\Phi _2,P\\Delta (H^{\\epsilon }-H))\\nonumber \\\\=&(P\\Phi _1,-\\Delta (v^{\\epsilon }-v ))+(P\\Phi _2,-\\Delta (H^{\\epsilon }-H))\\nonumber \\\\=&(\\nabla \\times \\Phi _1,\\omega _ v^{\\epsilon }-\\omega _v)+\\int _{\\partial \\Omega }n\\times (\\omega _ v^{\\epsilon }-\\omega _v)\\cdot P\\Phi _1\\nonumber \\\\&+(\\nabla \\times \\Phi _2,\\omega _ H^{\\epsilon }-\\omega _H)+\\int _{\\partial \\Omega }n\\times (\\omega _ H^{\\epsilon }-\\omega _H)\\cdot P\\Phi _2\\nonumber \\\\=&(\\nabla \\times \\Phi _1,\\omega _ v^{\\epsilon }-\\omega _v)+\\int _{\\partial \\Omega }(B(v^{\\epsilon }-v )+Bv-n\\times \\omega _v)\\cdot P\\Phi _1\\nonumber \\\\&+(\\nabla \\times \\Phi _2,\\omega _ H^{\\epsilon }-\\omega _H)+\\int _{\\partial \\Omega }(B(H^{\\epsilon }-H)+BH-n\\times \\omega _H)\\cdot P\\Phi _2.$ From (REF ), (REF ), (REF ), and (REF ), we arrive at $\\frac{1}{2}\\frac{d}{dt}E&+\\frac{\\epsilon }{2}(\\Vert P\\Delta (v^{\\epsilon }-v) \\Vert ^2+\\Vert P\\Delta ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\&\\le I_1 + I_2 + I_3 +C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2+\\epsilon ^\\frac{1}{2}),$ where $E:=&\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2\\\\&-\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot (v^{\\epsilon }-v )-2\\int _{\\partial \\Omega }(v^{\\epsilon }-v)\\cdot (Bv-n\\times \\omega _v)\\\\&-\\int _{\\partial \\Omega }B(H^{\\epsilon }-H )\\cdot (H^{\\epsilon }-H )-2\\int _{\\partial \\Omega }(H^{\\epsilon }-H)\\cdot (BH-n\\times \\omega _H),\\\\I_1:=&\|(\\nabla \\times \\Phi _1,\\omega _ v^{\\epsilon }-\\omega _v)+(\\nabla \\times \\Phi _2,\\omega _ H^{\\epsilon }-\\omega _H)\|,\\\\I_2:=&\\Big {\|}\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot P\\Phi _1+\\int _{\\partial \\Omega }B(H^{\\epsilon }-H)\\cdot P\\Phi _2\\Big {\|},\\\\I_3:=&\\Big {\|}\\int _{\\partial \\Omega }(Bv-n\\times \\omega _v)\\cdot P\\Phi _1+\\int _{\\partial \\Omega }(BH-n\\times \\omega _H)\\cdot P\\Phi _2\\Big {\|}.$ Now we estimate the terms $I_1$ , $I_2$ and $I_3$ in turn.", "The term $I_1$ can be estimated easily by using Sobolev inequalities and the obtained uniform bounds for $v^{\\epsilon }$ and $H^{\\epsilon }$ in Theorem REF .", "We have $\\nonumber I_1=\|(\\nabla \\times \\Phi _1,\\omega _ v^{\\epsilon }-\\omega _v)+(\\nabla \\times \\Phi _2,\\omega _ H^{\\epsilon }-\\omega _H)\|\\le I_{11}+I_{12},$ where $I_{11}=&\\big {\|}(v\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)+(v^{\\epsilon }-v)\\cdot \\nabla \\omega _v+(v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&-H\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)-(H^{\\epsilon }-H)\\cdot \\nabla \\omega _H-(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H),\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&+((v^{\\epsilon }-v)\\cdot \\nabla \\omega _H+(v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)+v\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)\\nonumber \\\\&-(H^{\\epsilon }-H)\\cdot \\nabla \\omega _v-(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)-H\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v),\\omega _ H^{\\epsilon }-\\omega _H)\\big {\|},\\\\I_{12}=&\\big {\|}([\\nabla \\times ,v\\cdot \\nabla ](v^{\\epsilon }-v)+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ] v+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](v^{\\epsilon }-v)\\nonumber \\\\ &-[\\nabla \\times ,H\\cdot \\nabla ](H^{\\epsilon }-H)-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ]H\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](H^{\\epsilon }-H),\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&+([\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ]H+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](H^{\\epsilon }-H)+[\\nabla \\times ,v\\cdot \\nabla ](H^{\\epsilon }-H)\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ] v-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](v^{\\epsilon }-v)\\nonumber \\\\&-[\\nabla \\times ,H\\cdot \\nabla ](v^{\\epsilon }-v),\\omega _ H^{\\epsilon }-\\omega _H)\\big {\|}.$ We observe that $&(v\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v),\\omega _ v^{\\epsilon }-\\omega _v)=0,\\quad \\quad ((v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v),\\omega _ v^{\\epsilon }-\\omega _v)=0,\\\\&(v\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H),\\omega _ H^{\\epsilon }-\\omega _H)=0,\\quad ((v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H),\\omega _ H^{\\epsilon }-\\omega _H)=0,\\\\&(H\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)+(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H),\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&\\quad \\quad \\qquad \\qquad +((H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)+H\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v),\\omega _ H^{\\epsilon }-\\omega _H)=0.$ Hence $I_1\\le C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2).$ Next, we estimate the term $I_2$ .", "We note that $\\nonumber P\\Phi =\\Phi +\\nabla \\phi $ holds for any function $\\Phi \\in L^2(\\Omega )$ , so we need to estimate the scalar function $\\phi $ which is difficult to estimate on the boundary.", "In order to overcome this difficulty, we need to transform it to an estimate on $\\Omega $ .", "First, we should extend $n$ and $B$ to the interior of $\\Omega $ as follows: $\\nonumber n(x)=\\varphi (r(x))\\nabla (r(x)),\\quad B(x)=\\varphi (r(x))B(\\Pi x),$ where $r(x)=\\min _{y\\in \\partial \\Omega }d(x,y),\\quad \\Pi x=y_x\\in \\partial \\Omega $ such that $\\nonumber r(x):=d(x,y_x)$ is well-defined in $\\Omega _\\sigma =\\lbrace x\\in \\Omega , r(x)\\le 2\\sigma \\rbrace $ for some $\\sigma >0$ and $\\varphi (s)\\in C_c^\\infty {[0,2\\sigma )}$ satisfying $\\nonumber \\varphi (s)=1\\quad \\text{in}\\quad [0,\\sigma ].$ Then, we can obtain that $I_2=\\,&\\Big {\|}\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot P\\Phi _1+\\int _{\\partial \\Omega }B(H^{\\epsilon }-H)\\cdot P\\Phi _2\\Big {\|}\\nonumber \\\\=\\,&\\Big {\|}\\int _{\\partial \\Omega }\\big {(}(n\\times B(v^{\\epsilon }-v )\\cdot (n\\times P\\Phi _1)+(n\\times B(H^{\\epsilon }-H))\\cdot (n\\times P\\Phi _2)\\big {)}\\Big {\|}\\nonumber \\\\=\\,&\\big {\|}(n\\times B(v^{\\epsilon }-v ),\\nabla \\times \\Phi _1)+(n\\times B(H^{\\epsilon }-H ),\\nabla \\times \\Phi _2)\\nonumber \\\\&-(\\nabla \\times (n\\times B(v^{\\epsilon }-v )),P\\Phi _1)-(\\nabla \\times (n\\times B(H^{\\epsilon }-H )),P\\Phi _2)\\big {\|}.$ We easily get that $&\|(\\nabla \\times (n\\times B(v^{\\epsilon }-v )),P\\Phi _1)+(\\nabla \\times (n\\times B(H^{\\epsilon }-H )),P\\Phi _2)\|\\nonumber \\\\\\le \\,&\\Vert n\\times B(v^{\\epsilon }-v )\\Vert \\Vert P\\Phi _1\\Vert +\\Vert n\\times B(H^{\\epsilon }-H )\\Vert \\Vert P\\Phi _2\\Vert \\nonumber \\\\\\le \\,& C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2).$ Now, we turn to estimate the remaining terms in (REF ): $\\nonumber \|(n\\times B(v^{\\epsilon }-v ),\\nabla \\times \\Phi _1)+(n\\times B(H^{\\epsilon }-H ),\\nabla \\times \\Phi _2)\|\\le I_{21}+I_{22},$ where $I_{21}=\\,&\\big {\|}(v\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)+(v^{\\epsilon }-v)\\cdot \\nabla \\omega _v+(v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&-H\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)-(H^{\\epsilon }-H)\\cdot \\nabla \\omega _H\\nonumber \\\\&-(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H),n\\times B(v^{\\epsilon }-v ))\\nonumber \\\\&+((v^{\\epsilon }-v)\\cdot \\nabla \\omega _H+(v^{\\epsilon }-v)\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)+v\\cdot \\nabla (\\omega _ H^{\\epsilon }-\\omega _H)\\nonumber \\\\&-(H^{\\epsilon }-H)\\cdot \\nabla \\omega _v-(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v)\\nonumber \\\\&-H\\cdot \\nabla (\\omega _ v^{\\epsilon }-\\omega _v),n\\times B(H^{\\epsilon }-H ))\\big {\|},\\\\I_{22}=\\,&\\big {\|}([\\nabla \\times ,v\\cdot \\nabla ](v^{\\epsilon }-v)+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ] v+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](v^{\\epsilon }-v)\\nonumber \\\\ &-[\\nabla \\times ,H\\cdot \\nabla ](H^{\\epsilon }-H)-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ]H\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](H^{\\epsilon }-H),n\\times B(v^{\\epsilon }-v ))\\nonumber \\\\&+([\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ]H+[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](H^{\\epsilon }-H)\\nonumber \\\\&+[\\nabla \\times ,v\\cdot \\nabla ](H^{\\epsilon }-H)-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ] v\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](v^{\\epsilon }-v)-[\\nabla \\times ,H\\cdot \\nabla ](v^{\\epsilon }-v),n\\times B(H^{\\epsilon }-H ))\\big {\|}.$ By using Hölder's inequality and Sobolev inequality, we obtain $\|(n\\times B(v^{\\epsilon }-v ),\\nabla \\times \\Phi _1)+(n\\times B(H^{\\epsilon }-H ),\\nabla \\times \\Phi _2)\|\\le C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2).$ Based on (REF ) and (REF ), we have $I_2\\le C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2).$ Finally, we need to estimate the term $I_3$ , i.e.", "$\\nonumber \\Big {\|}\\int _{\\partial \\Omega }(Bv-n\\times \\omega _v)\\cdot P\\Phi _1+\\int _{\\partial \\Omega }(BH-n\\times \\omega _H)\\cdot P\\Phi _2\\Big {\|}.$ We observe that the estimate is trivial if the ideal MHD satisfies the same boundary condition as that the MHD does.", "However, $[Bv]_\\tau -n\\times \\omega _v$ and $[BH]_\\tau -n\\times \\omega _H$ may be not equal to zero.", "As a result, the boundary layer may occur, so we will experience more complicate estimates.", "Similar to the above, we get $I_3=\\,&\\Big {\|}\\int _{\\partial \\Omega }(Bv-n\\times \\omega _v)\\cdot P\\Phi _1+\\int _{\\partial \\Omega }(BH-n\\times \\omega _H)\\cdot P\\Phi _2\\Big {\|}\\nonumber \\\\=\\,&\|\\int _{\\partial \\Omega }(n\\times (Bv-n\\times \\omega _v))\\cdot (n\\times P\\Phi _1)+\\int _{\\partial \\Omega }(n\\times (BH-n\\times \\omega _H))\\cdot (n\\times P\\Phi _2)\|\\nonumber \\\\=\\,&\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times \\Phi _1)+(n\\times (BH-n\\times \\omega _H),\\nabla \\times \\Phi _2)\\nonumber \\\\&-(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)-(\\nabla \\times (n\\times (BH-n\\times \\omega _H)),P\\Phi _2)\|\\nonumber \\\\\\le &\\,I_{31}+ I_{32},$ where $I_{31}=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times \\Phi _1)+(n\\times (BH-n\\times \\omega _H),\\nabla \\times \\Phi _2)\|,\\\\I_{32}=&\\,\|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)+(\\nabla \\times (n\\times (BH-n\\times \\omega _H)),P\\Phi _2)\|.$ We first deal with the term $I_{31}$ and note that $\\nonumber I_{31}\\le L_1+L_2+L_3+L_4+L_5+L_6,$ where $L_1=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times (v\\cdot \\nabla (v^{\\epsilon }-v))-\\nabla \\times (H\\cdot \\nabla (H^{\\epsilon }-H)))\|,\\\\L_2=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla v)-\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla H))\|,\\\\L_3=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v))\\nonumber \\\\&-\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla (H^{\\epsilon }-H)))\|,\\\\L_4=&\\,\|(n\\times (BH-n\\times \\omega _H),\\nabla \\times (v\\cdot \\nabla (H^{\\epsilon }-H))-\\nabla \\times (H\\cdot \\nabla (v^{\\epsilon }-v)))\|,\\\\L_5=&\\,\|(n\\times (BH-n\\times \\omega _H),\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla H)-\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla v))\|,\\\\L_6=&\\,\|(n\\times (BH-n\\times \\omega _H),\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla (v^{\\epsilon }-v))\\nonumber \\\\&-\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla (H^{\\epsilon }-H)))\|.$ We have $L_1=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times (v\\cdot \\nabla (v^{\\epsilon }-v))-\\nabla \\times (H\\cdot \\nabla (H^{\\epsilon }-H)))\|\\nonumber \\\\=&\\,\|(n\\times (Bv-n\\times \\omega _v),v\\cdot \\nabla (\\omega _v^{\\epsilon }-\\omega _v)-H\\cdot \\nabla (\\omega _H^{\\epsilon }-\\omega _H)\\nonumber \\\\&+[\\nabla \\times ,v\\cdot \\nabla ](v^{\\epsilon }-v)-[\\nabla \\times ,H\\cdot \\nabla ](H^{\\epsilon }-H))\|.$ Here, we first deal with the terms which contain higher derivatives and get that $&\\,\|(n\\times (Bv-n\\times \\omega _v),v\\cdot \\nabla (\\omega _v^{\\epsilon }-\\omega _v)-H\\cdot \\nabla (\\omega _H^{\\epsilon }-\\omega _H)\|\\\\=&\\,\|(v\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v)),\\omega _v^{\\epsilon }-\\omega _v)-(H\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v)),\\omega _H^{\\epsilon }-\\omega _H)\|\\nonumber \\\\\\le &\\,\\Big {\|}\\int _{\\partial \\Omega } n\\times (v^{\\epsilon }-v)\\cdot (v\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v)))\\nonumber \\\\&+(\\nabla \\times (v\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))),v^{\\epsilon }-v)\\Big {\|}\\nonumber \\\\&+\\Big {\|}\\int _{\\partial \\Omega } n\\times (H^{\\epsilon }-H)\\cdot (H\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v)))\\nonumber \\\\&+(\\nabla \\times (H\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))),H^{\\epsilon }-H))\\Big {\|}\\nonumber \\\\\\le &\\, C\\,(\|v^{\\epsilon }-v\|_{L^2(\\partial \\Omega )}+\|H^{\\epsilon }-H\|_{L^2(\\partial \\Omega )}+\\Vert v^{\\epsilon }-v\\Vert +\\Vert H^{\\epsilon }-H\\Vert )\\nonumber \\\\\\le &\\, C\\,(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}).$ We also note that each component of $[\\nabla \\times ,v\\cdot \\nabla ](v^{\\epsilon }-v)-[\\nabla \\times ,H\\cdot \\nabla ](H^{\\epsilon }-H)$ is a combination of such terms $\\partial _iv\\cdot \\nabla (v^{\\epsilon }-v)_j$ and $\\partial _kH\\cdot \\nabla (H^{\\epsilon }-H)_l$ .", "Without loss of generality, we consider the term $\\nonumber ((n\\times (Bv-n\\times \\omega _v))_m,\\partial _iv\\cdot \\nabla (v^{\\epsilon }-v)_j-\\partial _kH\\cdot \\nabla (H^{\\epsilon }-H)_l).$ Since $\\nabla \\cdot \\partial _iv=0$ and $\\nabla \\cdot \\partial _kH=0$ , we have $&\|((n\\times (Bv-n\\times \\omega _v))_m,\\partial _iv\\cdot \\nabla (v^{\\epsilon }-v)_j-\\partial _kH\\cdot \\nabla (H^{\\epsilon }-H)_l)\|\\nonumber \\\\=\\,&\\big {\|}(\\partial _iv,\\nabla ((v^{\\epsilon }-v)_j(n\\times (Bv-n\\times \\omega _v))_m))\\nonumber \\\\&-(\\partial _iv\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))_m,(v^{\\epsilon }-v)_j)\\nonumber \\\\&-(\\partial _kH,\\nabla ((H^{\\epsilon }-H)_l(n\\times (Bv-n\\times \\omega _v))_m))\\nonumber \\\\&+(\\partial _kH\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))_m,(H^{\\epsilon }-H)_l)\\big {\|}\\nonumber \\\\=\\,&\\Big {\|}\\int _{\\partial \\Omega }(v^{\\epsilon }-v)_j(n\\times (Bv-n\\times \\omega _v))_m\\partial _iv\\cdot n\\nonumber \\\\&-\\int _{\\partial \\Omega }(H^{\\epsilon }-H)_l(n\\times (Bv-n\\times \\omega _v))_m\\partial _kH\\cdot n\\nonumber \\\\&-(\\partial _iv\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))_m,(v^{\\epsilon }-v)_j)\\nonumber \\\\&+(\\partial _iH\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v))_m,(H^{\\epsilon }-H)_l)\\Big {\|}\\nonumber \\\\\\le & \\,C\\,(\|v^{\\epsilon }-v\|_{L^1(\\Omega )}+\|H^{\\epsilon }-H\|_{L^1(\\Omega )}+\\Vert v^{\\epsilon }-v\\Vert +\\Vert H^{\\epsilon }-H\\Vert )\\nonumber \\\\\\le &\\,C\\,(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}).$ Hence, we obtain $\\nonumber L_1\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}).$ Compared to $L_1$ , both $L_2$ and $L_3$ can be easily estimated.", "In fact, we have $L_2=&\\,\|(n\\times (Bv-n\\times \\omega _v),\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla v)-\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla H))\|\\nonumber \\\\=&\\,\\Big {\|}\\int _{\\partial \\Omega }(n\\times (Bv-n\\times \\omega _v))(n\\times ((v^{\\epsilon }-v)\\cdot \\nabla v)-n\\times ((H^{\\epsilon }-H)\\cdot \\nabla H))\\nonumber \\\\&+(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),(v^{\\epsilon }-v)\\cdot \\nabla v-(H^{\\epsilon }-H)\\cdot \\nabla H)\\Big {\|}\\nonumber \\\\\\le \\,& C(\|v^{\\epsilon }-v\|_{L^1(\\partial \\Omega )}+\|H^{\\epsilon }-H\|_{L^1(\\partial \\Omega )}+\\Vert v^{\\epsilon }-v\\Vert +\\Vert H^{\\epsilon }-H\\Vert )\\nonumber \\\\\\le \\,& C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}),\\\\L_3=\\,&\\,\\big {\|}(n\\times (Bv-n\\times \\omega _v),\\nabla \\times ((v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v))\\nonumber \\\\&-\\nabla \\times ((H^{\\epsilon }-H)\\cdot \\nabla (H^{\\epsilon }-H)))\\big {\|}\\nonumber \\\\=\\,&\\,\\big {\|}(n\\times (Bv-n\\times \\omega _v),[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](v^{\\epsilon }-v)\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](H^{\\epsilon }-H))\\nonumber \\\\&+(n\\times (Bv-n\\times \\omega _v),(v^{\\epsilon }-v)\\cdot \\nabla (\\omega _v^{\\epsilon }-\\omega _v)-(H^{\\epsilon }-H)\\cdot \\nabla (\\omega _H^{\\epsilon }-\\omega _H))\\big {\|}\\nonumber \\\\=\\,&\\,\\big {\|}\\big {(}n\\times (Bv-n\\times \\omega _v),[\\nabla \\times ,(v^{\\epsilon }-v)\\cdot \\nabla ](v^{\\epsilon }-v)\\nonumber \\\\&-[\\nabla \\times ,(H^{\\epsilon }-H)\\cdot \\nabla ](H^{\\epsilon }-H)\\big {)}\\nonumber \\\\&-\\big {(}(v^{\\epsilon }-v)\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v)),\\omega _v^{\\epsilon }-\\omega _v\\big {)}\\nonumber \\\\&+\\big {(}(H^{\\epsilon }-H)\\cdot \\nabla (n\\times (Bv-n\\times \\omega _v),\\omega _H^{\\epsilon }-\\omega _H\\big {)}\\big {\|}\\nonumber \\\\\\le \\,& C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2).$ We find that $L_4$ , $L_5$ and $L_6$ have similar structures to $L_1$ , $L_2$ and $L_3$ respectively, so we can get $&L_4\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}),\\\\&L_5\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}),\\\\&L_6\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2).$ From the estimates of $L_i \\,(i=1,\\cdots , 6)$ , we get $\\nonumber I_{31}\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}).$ Now, it remains to estimate the term $I_{32}$ , i.e.", "$\\nonumber \|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)+(\\nabla \\times (n\\times (BH-n\\times \\omega _H)),P\\Phi _2)\|.$ First, we consider $(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)$ .", "Because it involves Leray projection, some terms which contain higher derivatives of $v^{\\epsilon }-v$ or $H^{\\epsilon }-H$ can not be estimated easily.", "We have the observations $&v\\cdot \\nabla (v^{\\epsilon }-v)-(v^{\\epsilon }-v)\\cdot \\nabla v=\\nabla \\times ((v^{\\epsilon }-v)\\times v),\\\\&H\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla H=\\nabla \\times ((H^{\\epsilon }-H)\\times H).$ Since $(v^{\\epsilon }-v)\\cdot n=0$ , $v\\cdot n=0$ , $(H^{\\epsilon }-H)\\cdot n=0$ and $H\\cdot n=0$ , it means that $(v^{\\epsilon }-v)\\times v=\\lambda _1n,\\quad (H^{\\epsilon }-H)\\times H=\\lambda _2n.$ Due to (REF )-(REF ), we easily obtain $\\nonumber \\nabla \\times ((v^{\\epsilon }-v)\\times v)\\in \\mathbb {H},\\quad \\nabla \\times ((H^{\\epsilon }-H)\\times H)\\in \\mathbb {H},$ where $\\mathbb {H}$ is Leray projection space.", "Thus we have the following equality $P\\Phi _1=&v\\cdot \\nabla (v^{\\epsilon }-v)-(v^{\\epsilon }-v)\\cdot \\nabla v+P\\Phi ^1_v\\nonumber \\\\&-(H\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla H)-P\\Phi ^1_H,$ where $&P\\Phi ^1_v=P[2(v^{\\epsilon }-v)\\cdot \\nabla v+(v^{\\epsilon }-v)\\cdot \\nabla (v^{\\epsilon }-v)],\\\\&P\\Phi ^1_H=P[2(H^{\\epsilon }-H)\\cdot \\nabla H+(H^{\\epsilon }-H)\\cdot \\nabla (H^{\\epsilon }-H)].$ Hence, we have $&(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)\\nonumber \\\\=&(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi ^1_v)-(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi ^1_H)\\nonumber \\\\&+(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),v\\cdot \\nabla (v^{\\epsilon }-v)-(v^{\\epsilon }-v)\\cdot \\nabla v\\nonumber \\\\&-(H\\cdot \\nabla (H^{\\epsilon }-H)-(H^{\\epsilon }-H)\\cdot \\nabla H)).$ First, we have $\\Vert P\\Phi ^1_v\\Vert \\le C&\\,(\\Vert v\\Vert _{W^{1,\\infty }}+\\Vert v^{\\epsilon }-v\\Vert _{W^{1,\\infty }})\\Vert v^{\\epsilon }-v\\Vert ,\\\\\\Vert P\\Phi ^1_H\\Vert \\le C&\\,(\\Vert H\\Vert _{W^{1,\\infty }}+\\Vert H^{\\epsilon }-H\\Vert _{W^{1,\\infty }})\\Vert H^{\\epsilon }-H\\Vert ,\\\\\|(\\nabla \\times (n\\times (&Bv-n\\times \\omega _v)),P\\Phi ^1_v)-(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi ^1_H)\|\\nonumber \\\\&\\le C\\Vert v\\Vert _{H^2}(\\Vert P\\Phi ^1_v\\Vert +\\Vert P\\Phi ^1_H\\Vert ).$ From (REF )-() and Lemma REF , we get $\\nonumber \|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi ^1_v)-(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi ^1_H)\|\\le C\\epsilon ^\\frac{3}{4}.$ Next, note that $&\|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),v\\cdot \\nabla (v^{\\epsilon }-v)\|\\nonumber \\\\=&\|(v\\cdot \\nabla (\\nabla \\times (n\\times (Bv-n\\times \\omega _v))),v^{\\epsilon }-v)\|\\le C\\Vert v\\Vert _{H^3}\\Vert v^{\\epsilon }-v\\Vert \\le C\\epsilon ^\\frac{3}{4}.$ Similarly, we obtain $&\|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),H\\cdot \\nabla (H^{\\epsilon }-H)\|\\nonumber \\\\=&\|(H\\cdot \\nabla (\\nabla \\times (n\\times (Bv-n\\times \\omega _v))),H^{\\epsilon }-H)\|\\le C\\Vert v\\Vert _{H^3}\\Vert H^{\\epsilon }-H\\Vert \\le C\\epsilon ^\\frac{3}{4}.$ At the same time, we get directly that $\|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),(&v^{\\epsilon }-v)\\cdot \\nabla v+(H^{\\epsilon }-H)\\cdot \\nabla H))\|\\\\\\le &\\,C(\\Vert v^{\\epsilon }-v\\Vert +\\Vert H^{\\epsilon }-H\\Vert )\\le C\\epsilon ^\\frac{3}{4}.$ Therefore, $\\nonumber \|(\\nabla \\times (n\\times (Bv-n\\times \\omega _v)),P\\Phi _1)\|\\le C\\epsilon ^\\frac{3}{4}.$ By using the same methods as above, we observe $\\nonumber P\\Phi _2=\\Phi _2.$ Hence, we get $\|(\\nabla \\times (n\\times (BH-n\\times \\omega _H)),P\\Phi _2)\|=\|(\\nabla \\times (n\\times (BH-n\\times \\omega _H)),\\Phi _2)\|\\le C\\epsilon ^\\frac{3}{4}.$ Finally, we have $\\nonumber I_{32}\\le C\\epsilon ^\\frac{3}{4}.$ Thus, we conclude that $I_3\\le C(\\Vert \\omega _v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _H^{\\epsilon }-\\omega _H\\Vert ^2+\\epsilon ^\\frac{1}{2}).$ In conclusion, it follows from (REF ), (REF ) and (REF ) that $\\frac{1}{2}\\frac{d}{dt}E+\\frac{\\epsilon }{2}(\\Vert P\\Delta (v^{\\epsilon }&-v) \\Vert ^2+\\Vert P\\Delta ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\&\\le C(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2+\\epsilon ^\\frac{1}{2}).$ Now, we need to deal with the left terms in the above inequality.", "Let us recall that $E=\\,&\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2\\nonumber \\\\&-\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot (v^{\\epsilon }-v )-2\\int _{\\partial \\Omega }(v^{\\epsilon }-v)\\cdot (Bv-n\\times \\omega _v)\\nonumber \\\\&-\\int _{\\partial \\Omega }B(H^{\\epsilon }-H )\\cdot (H^{\\epsilon }-H )-2\\int _{\\partial \\Omega }(H^{\\epsilon }-H)\\cdot (BH-n\\times \\omega _H).$ We note that $&\\Big {\|}\\int _{\\partial \\Omega }B(v^{\\epsilon }-v )\\cdot (v^{\\epsilon }-v )+\\int _{\\partial \\Omega }B(H^{\\epsilon }-H )\\cdot (H^{\\epsilon }-H )\\Big {\|}\\nonumber \\\\\\le \\,&C\\,(\|v^{\\epsilon }-v\|^2_{L^2({\\partial \\Omega })}+\|H^{\\epsilon }-H\|^2_{L^2({\\partial \\Omega })})\\nonumber \\\\\\le \\,&\\delta \\,(\\Vert \\omega _ v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2)+C_\\delta (\\Vert v^{\\epsilon }-v\\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2),\\\\&\\Big {\|}2\\int _{\\partial \\Omega }(v^{\\epsilon }-v)\\cdot (Bv-n\\times \\omega _v)+2\\int _{\\partial \\Omega }(H^{\\epsilon }-H)\\cdot (BH-n\\times \\omega _H)\\Big {\|}\\nonumber \\\\\\le \\,& \\,C\\,(\|v^{\\epsilon }-v\|_{L^1({\\partial \\Omega })}+\|H^{\\epsilon }-H\|_{L^1({\\partial \\Omega })})\\nonumber \\\\\\le \\,&\\,\\delta \\,(\\Vert \\omega _ v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2)+C_\\delta (\\Vert v^{\\epsilon }-v\\Vert ^2+\\Vert H^{\\epsilon }-H\\Vert ^2)$ for some $\\delta $ small enough.", "Consequently, we get $&\\Vert \\omega _ v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2+\\frac{\\epsilon }{2}\\int _0^t(\\Vert P\\Delta (v^{\\epsilon }-v) \\Vert ^2+\\Vert P\\Delta ( H^{\\epsilon }-H)\\Vert ^2)\\nonumber \\\\\\le \\,& C\\int _0^t(\\Vert \\omega _ v^{\\epsilon }-\\omega _v \\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H \\Vert ^2)+C\\epsilon ^\\frac{1}{2}.$ By using Gronwall's inequality, we have $\\Vert \\omega _ v^{\\epsilon }-\\omega _v\\Vert ^2+\\Vert \\omega _ H^{\\epsilon }-\\omega _H\\Vert ^2\\le C\\epsilon ^\\frac{1}{2}\\quad \\text{on} \\quad [0,T_2].$ Thus $\\Vert v^{\\epsilon }-v\\Vert ^2_{H^1}&+\\Vert H^{\\epsilon }-H\\Vert ^2_{H^1}\\\\&+\\epsilon \\int _0^t(\\Vert P\\Delta (v^{\\epsilon }-v)\\Vert ^2+\\Vert P\\Delta ( H^{\\epsilon }-H)\\Vert ^2)\\le \\, C\\epsilon ^\\frac{1}{2}.$ From Lemmas REF and REF , we get $\\epsilon \\int _0^t(\\Vert v^{\\epsilon }-v \\Vert ^2_{H^2}+\\Vert H^{\\epsilon }-H\\Vert _{H^2}^2)\\le \\, C\\epsilon ^\\frac{1}{2}.$ Note that the following inequality holds $\\Vert \\nabla (u^{\\epsilon }-u)\\Vert ^p_{L^p}\\le C\\Vert \\nabla (u^{\\epsilon }-u)\\Vert ^{p-2}_{L^\\infty }\\Vert \\nabla (u^{\\epsilon }-u)\\Vert ^2.$ Hence, we obtain $\\Vert \\nabla (v^{\\epsilon }-v)\\Vert ^p_{L^p}+\\Vert \\nabla (H^{\\epsilon }-H)\\Vert ^p_{L^p}\\le C\\epsilon ^{\\frac{1}{2}}.$ This completes the proof of Lemma REF .", "From Lemmas REF and REF , we easily get Theorem $\\ref {Th3}$ .", "Acknowledgements: Li is supported partially by NSFC (Grant No.", "11271184) and PAPD." ] ]	1606.05038
[ [ "Adaptive Cross-Packet HARQ" ], [ "Abstract In this work, we investigate a coding strategy devised to increase the throughput in hybrid ARQ (HARQ) transmission over block fading channel.", "In our approach, the transmitter jointly encodes a variable number of bits for each round of HARQ.", "The parameters (rates) of this joint coding can vary and may be based on the negative acknowledgment (NACK) provided by the receiver or, on the past (outdated) information about the channel states.", "These new degrees of freedom allow us to improve the match between the codebook and the channel states experienced by the receiver.", "The results indicate that significant gains can be obtained using the proposed coding strategy, particularly notable when the conventional HARQ fails to offer throughput improvement even if the number of transmission rounds is increased.", "The new cross-packet HARQ is also implemented using turbo codes where we show that the theoretically predicted throughput gains materialize in practice, and we discuss the implementation challenges." ], [ "Introduction", "In this work, in order to improve the throughput of the harq transmission over block-fading channel, we propose to use joint coding of multiple information packets into the same channel block and we develop methods to optimize the coding rates.", "harq is used in modern communications systems to deal with unpredictable changes in the channel (due to fading), and with the distortion of the transmitted signals (due to noise).", "harq relies on the feedback/acknowledgement channel, which is used by the receiver to inform the transmitter about the decoding errors (via nack) and about the decoding success, via ack.", "After nack, the transmitter makes another transmission round which conveys additional information necessary to decode the packet.", "This continues till ack is receiver and then a new harq cycle starts again for another information packet.", "In so-called truncated harq, the cycle stops also if the maximum number of rounds is attained.", "As in many previous works, e.g., [1], [2], we will consider throughput as a performance measure assuming that residual errors are taken care of by the upper layers [3].", "We consider here the “canonical” problem defined in [1], where the csi is available at the receiver but not at the transmitter, which knows only its statistical description.", "The essential part of harq is channel coding, which is done over many channel blocks as long as nacks are obtained over the feedback channel.", "It was shown in [1] that harq's throughput may approach the ergodic capacity of the channel with sufficiently high “nominal” coding rate per round.", "However, such an approach is based on large number of harq rounds, and thus has a limited practical value: long buffers are required which becomes a limiting factor for implementation of harq [4].", "On the other hand, using finite nominal coding rate and truncated harq, the difference between the throughput achievable using harq and the theoretical limits may be large, especially, when we target throughput close to the nominal rate [2], [5].", "To address this problem, various adaptive versions of harq were proposed in the literature.", "For example, [6], [7], [8], [9], [10], [11], [12] suggested to vary the length of the codewords so as to strike the balance between the number of channel uses and the chances of successful decoding.", "Their obvious drawback is that the resources assigned to the various harq rounds are not constant which may leave an “empty” space within the block.", "To deal with this issue, it was proposed to share the block resources (power, time or bandwidth) between various packets in e.g., [13], [14], [15], [16], [3], to encode many packets into predefined size blocks as done in [17], [18], or to group variable-length codewords to fill the channel block [19], [11].", "A simplified approach was also proposed in [20] to transmit the redundancy using two-step encoding.", "These approaches implicitly implement a joint coding of many packets into a single channel block.", "Here, we want to address the issue of cross-packet coding explicitly.", "The idea of this xp is to get rid of the restricting assumptions proper to various heuristics developed before and to use a generic joint harq encoder accepting many information packets and encoding them into a common codeword which fills the channel block.", "The contributions of this work are the following: We propose a general framework to analyze joint encoding of multiple packets which allow us to derive the relationship between the coding rates and the throughput.", "Our approach to cross-packet coding is similar to the one shown in [21], [22], [23], [24], which, however, did not optimize the coding parameters.", "The optimization was proposed in [25], however, due to complex decoding rules, it was very tedious and thus limited to the case of a simple channel model.", "In our work we simplify the problem assuming asymptotically long codewords are used, which leads to a compact description of the decoding criteria and allows us to solve the rate-optimization problem.", "We consider the so-called multi-bit feedback to adapt the coding rates to the channel state experienced by the receiver in the past transmission rounds of harq.", "The same idea was exploited already e.g., in[26], [27], [11], [6], [28], [29], [9], [30], [31], [12], [3].", "The assumption of multi-bit feedback not only simplifies the optimization but also yields the results which may be treated as the ultimate performance limits of any adaptation schemes when the instantaneous csi is not available at the transmitter.", "We optimize the coding rates using the mdp formulation [32], and compare the proposed, xp to the conventional irharq from the perspective of attainable throughput.", "For the particular case of two transmission round, we obtain the optimal solution in closed-form.", "We also present an analytical formula for attainable throughput using heuristic rate-adaptation inspired by the numerical results and which presents a notable gain over the conventional irharq.", "To obtain an insight into the practical constraints on the system design, we also show the results obtained when a turbo coding is adopted.", "The remainder of the paper is organized as follows.", "We define the transmission model as well as the basic performance metrics in Sec. .", "The idea of cross-packet coding is explained in Sec. .", "The optimization of the rates in the proposed coding strategy is presented in Sec. .", "We discuss the effects of using a practical encoding/decoding schemes in Sec. .", "The numerical results are presented in form of short examples throughout the work to illustrate the main ideas.", "Conclusions are presented in Sec. .", "The optimization methods used to obtain the numerical results and the proof of decoding conditions are presented in appendices." ], [ "Channel model and HARQ", "We consider a point-to-point irharq transmission of a packet $\\mathsf {m}$ over a block fading channel.", "After each transmission, using a feedback/acknowledgement channel, the receiver tells the transmitter whether the decoding of $\\mathsf {m}$ succeeded (ack) or failed (nack).", "We thus assume that error detection is possible (e.g., via crc mechanisms) and that the feedback channel is error-free.", "For simplicity, we ignore any loss of resources due to the crc and the acknowledgement feedback.", "The transmission of a single packet may thus require many transmission rounds which continue till the $K$ th round is reached or till ack is received.", "When $K$ is finite, we say that harq is truncated, otherwise we say it is persistent.", "We define a harq cycle as the sequence of transmission rounds of the same packet $\\mathsf {m}$ .", "The received signal in the $k$ th round is given by $y_{k}=\\sqrt{\\mathsf {snr}_{k}}x_{k}+z_{k},\\quad k=1,\\ldots , K$ where $z_{k}$ and $x_k$ modelling, respectively, the noise and the transmitted codeword are ${\\mathop {N_{\\textnormal {s}}}}$ -dimensional vectors, each containing iid zero mean, unit-variance random variables; $\\mathsf {snr}_{k}$ is thus the snr at the receiver.", "The elements of $z_k$ are drawn from complex Gaussian distribution, and elements of $x_{k}$ – from the uniform distribution over the set (constellation) $\\mathcal {X}$ .", "During the $k$ th round, $\\mathsf {snr}_{k}$ is assumed to be perfectly known/estimated at the receiver and unknown at the transmitter; it varies from one round to another and we model $\\mathsf {snr}_{k}, k=1,\\ldots ,K$ as the iid random variables $\\mathsf {SNR}$ with distribution $p_{\\mathsf {SNR}}(\\mathsf {snr})$ ." ], [ "Conventional harq", "In the conventional irharq, a packet $\\mathsf {m}\\in \\lbrace 0,1\\rbrace ^{R{\\mathop {N_{\\textnormal {s}}}}}$ is firstly encoded into a codeword $x=\\Phi [\\mathsf {m}]\\in \\mathcal {X}^{K{\\mathop {N_{\\textnormal {s}}}}}$ composed of $K{\\mathop {N_{\\textnormal {s}}}}$ complex symbols taken from a constellation $\\mathcal {X}$ where $\\Phi [\\cdot ]$ is the coding function and $R$ denotes the nominal coding rate per block.We clearly define the nominal rate as the coding rate per channel block because harq is a variable-rate transmission: the number of used channel blocks is random, and the final transmission rate is random as well.", "Then, the codeword $x$ is divided into $K$ disjoint subcodewords $x_k$ composed of different symbols i.e., $x=[x_1,x_2,\\ldots ,x_K]$ .", "After each round $k$ , the receiver try to decode the packet $\\mathsf {m}$ concatenating all received channel outcomes till the $k$ th block $y_{[k]}=[y_{1},\\ldots ,y_{k-1},y_{k}].$ Following [1], [27], we assume ${\\mathop {N_{\\textnormal {s}}}}$ large enough to make the random coding limits valid.", "Then, knowing the mi $I_k=\\mathsf {I}(X_k;Y_k\|\\mathsf {snr}_k)$ between the random variables $X_k$ and $Y_k$ modeling respectively, the channel input and output in the $k$ th block, allows us to determine when the decoding is successful or not: the decoding failure occurs in the $k$ th round if the accumulated mi at the receiver is smaller than the coding rate $\\mathsf {NACK}_k&\\triangleq \\lbrace \\big (I_1<R\\big ) \\wedge \\big (I^\\Sigma _2<R\\big ) \\wedge \\ldots \\wedge \\big (I^\\Sigma _k<R\\big ) \\rbrace \\\\&= \\left\\lbrace I^\\Sigma _k<R\\right\\rbrace ,$ where $I^\\Sigma _k\\triangleq \\sum _{l=1}^k I_l$ is the mi accumulated in $k$ rounds.", "Of course, the mi depends on the snr, i.e., $I_k\\equiv I_k(\\mathsf {snr}_k)$ .", "irharq can be modelled as a Markov chain where the transmission rounds correspond to the states, and the harq cycle corresponds to a renewal cycle in the chain.", "Thus, the long-term average throughput, defined as the average number of correctly received bits per transmitted symbol, may be calculated from the renewal-reward theorem: it is a ratio between the average reward (number of bits successfully decoded per cycle) and the average renewal time (the expected number of transmissions needed to deliver the packet with up to $K$ transmission rounds) [1].", "Let $f_k\\triangleq \\Pr \\left\\lbrace \\mathsf {NACK}_k\\right\\rbrace , k\\ge 1$ be the probability of $k$ successive errors so the probability of successful decoding in the $k$ th round is given by $\\Pr \\left\\lbrace \\mathsf {NACK}_{k-1}\\wedge I^\\Sigma _k\\ge R\\right\\rbrace =f_{k-1}-f_k$ [1].", "The throughput is then calculated as follows [1] $\\eta ^{{\\textnormal {ir}}}_{K}&=\\frac{R(1-f_1)+ R(f_1-f_2)+ \\ldots +R(f_{K-1}-f_K)}{1\\cdot (1-f_1)+2\\cdot (f_1-f_2)+\\ldots +K\\cdot (f_{K-1})}\\\\&=\\frac{R(1-f_K)}{1+\\sum _{k=1}^{K-1}f_k}.$ Because the instantaneous csi is not available at the transmitter, the highest achievable throughput is given by the ergodic capacityWe use the term “capacity” to denote the achievable rate for a given distribution of $X$ .", "of the channel [1], [33] $\\overline{C}\\triangleq {E}_\\mathsf {SNR}[ I(\\mathsf {SNR}]).$ However, achieving $\\overline{C}$ is not obvious: as shown in [1], it can be done growing simultaneously $R$ and $K$ to infinity but this approach is impractical due to large memory requirements.", "Example 1 (Two-states channel) Consider a block-fading channel where the mi can only take two values, $I_{\\textnormal {a}}$ and $I_{\\textnormal {b}}$ , where $\\Pr \\left\\lbrace I=I_{\\textnormal {a}}\\right\\rbrace =1-p$ and $\\Pr \\left\\lbrace I=I_{\\textnormal {b}}\\right\\rbrace =p$ .", "The ergodic capacity is given by $\\overline{C}=I_{\\textnormal {a}}(1-p)+I_{\\textnormal {b}} p$ .", "We force the harq to deliver the packet at most in the last transmission, i.e., $f_K=0$ , which means that we impose the constraints on the coding rate $R\\le KI_{\\textnormal {a}}$ if we assume that $I_{\\textnormal {a}}<I_{\\textnormal {b}}$ .", "Assume $I_{\\textnormal {a}}=1, I_{\\textnormal {b}}=1.5$ , and $p=0.75$ so $\\overline{C}=1.375$ .", "For $K=2, 3$ we easily calculate the throughputFor $R\\le 1$ we obtain $f_1=0$ .", "For $1<R\\le 1.5$ – $f_1=1-p$ and $f_2=0$ .", "For $1.5<R\\le 2$ – $f_1=1$ , $f_2=0$ , etc.", "as $\\eta ^{{\\textnormal {ir}}}_2&={\\left\\lbrace \\begin{array}{ll}R, &\\text{if}\\quad R\\le 1\\\\0.8R, &\\text{if}\\quad 1 <R\\le 1.5\\\\0.5R, &\\text{if}\\quad 1.5 <R\\le 2\\end{array}\\right.", "},$ and $\\eta ^{{\\textnormal {ir}}}_3&={\\left\\lbrace \\begin{array}{ll}\\eta ^{{\\textnormal {ir}}}_2, &\\text{if}\\quad R\\le 2\\\\0.48R, &\\text{if}\\quad 2 <R\\le 2.5\\\\0.41R, &\\text{if}\\quad 2.5 <R\\le 3 \\end{array}\\right.", "}.$ The optimum throughput-rate pairs are then $(\\eta ^{{\\textnormal {ir}}}_2=1.2, R=1.5)$ and $(\\eta ^{{\\textnormal {ir}}}_3=1.23, R=3)$ .", "First, the benefit of using harq is clear: we are able to transmit without errors with a finite number of channel blocks and go beyond the obvious limit of $I_{\\textnormal {a}}$ .", "Second, we note that for $K=2$ , after two transmissions, the accumulated mi always satisfies $I^\\Sigma _2\\ge 2$ , while the condition $I^\\Sigma _2\\ge 1.5$ is sufficient to decode the packet.", "This may be seen as a “waste” which will be removed with the idea of cross-packet coding introduced in Sec. .", "Example 2 (16QAM over Rayleigh fading channel) Assume now that the transmission is done using symbols drawn uniformly from 16-points qam constellation $\\mathcal {X}$ [34] and that the channel gains follow Rayleigh distribution, i.e., $p_{\\mathsf {SNR}}(\\mathsf {snr})=1/\\overline{\\mathsf {snr}}\\exp (-\\mathsf {snr}/\\overline{\\mathsf {snr}}),$ where $\\overline{\\mathsf {snr}}$ is the average snr.", "We calculate $I(\\mathsf {snr})$ and the average $\\overline{C}$ using the numerical methods outlined in [34] and compare it in Fig.", "REF with the throughput $\\eta ^{{\\textnormal {ir}}}_{K}$ when $K\\in \\lbrace 2,\\infty \\rbrace $ .$\\eta ^{{\\textnormal {ir}}}_{\\infty }$ can be computed by taking $K$ large enough in (REF ) as suggested in [2] or by evaluating the throughput using the method outlined in the Appendix and considering the policy $\\pi (\\mathsf {s})=R$ if $\\mathsf {s}=(0,0)$ and $\\pi (\\mathsf {s})=0$ otherwise.", "We opt for the later method.", "The results indicate that i) there is a significant loss with respect to the ergodic capacity when using truncated harq, and ii) increasing the number of transmission rounds ($K=\\infty $ ) helps recovering the loss for a small-medium range of throughput (e.g., for $\\eta ^{{\\textnormal {ir}}}=1$ we gain $\\sim 3{\\textnormal {dB}}$ and the gap to $\\overline{C}$ is less than $1{\\textnormal {dB}}$ ), but it is less useful in the region of high $\\eta ^{{\\textnormal {ir}}}_K$ , i.e., in the vicinity of the maximum attainable throughput (e.g., for $\\eta ^{{\\textnormal {ir}}}=3$ , we gain $1{\\textnormal {dB}}$ but the gap to $\\overline{C}$ is still $\\sim 5{\\textnormal {dB}}$ ).", "We highlight this well-known effect [2] to emphasize later the gains of the new coding strategy.", "Figure: Throughput of the conventional irharq, compared to the ergodic capacity, C ¯\\overline{C}, in Rayleigh block-fading channel.", "The R opt R_\\textrm {opt} curve is an envelope of the throughputs η K ir \\eta ^{{\\textnormal {ir}}}_{K} obtained with different coding rates per block R∈{0.25,0.5,...,7.75}R\\in \\lbrace 0.25,0.5,\\ldots , 7.75\\rbrace ." ], [ "Cross-packet HARQ", "The examples shown previously indicate that the conventional coding cannot bring the throughput of harq close to the capacity unless the nominal coding rate $R$ and the number of rounds $K$ increase.", "We would like now to exploit a new coding possibility consisting in joint coding of packets during the harq cycle.", "Let us start with the case of two transmission rounds.", "In the first round, we use the nominal rate $R_1$ is used, i.e., the packet $\\mathsf {m}_1\\in \\lbrace 0,1\\rbrace ^{R_1{\\mathop {N_{\\textnormal {s}}}}}$ is encoded $x_1=\\Phi _1[\\mathsf {m}_1] \\in \\mathcal {X}^{{\\mathop {N_{\\textnormal {s}}}}},$ and transmitted over the channel (REF ) producing $y_1=\\sqrt{\\mathsf {snr}_1}x_1+z_1$ , where $\\Phi _k[\\cdot ]$ is the encoding at the $k$ th round.", "If the packet $\\mathsf {m}_1$ is decoded correctly (which occurs if $I_1\\ge R_1$ ), a new cycle harq starts by the transmission of a new packet.", "However, if the decoding fails, the packet $\\mathsf {m}_{[2]}=[\\mathsf {m}_1,\\mathsf {m}_2]\\in \\mathbb {B}^{(R_1+R_2) {\\mathop {N_{\\textnormal {s}}}}}$ is encoded using a conventional code designed independently of the codebook corresponding to the first transmission $x_2=\\Phi _2[\\mathsf {m}_1,\\mathsf {m}_2] \\in \\mathcal {X}^{{\\mathop {N_{\\textnormal {s}}}}},$ which yields the channel outcome $y_2=\\sqrt{\\mathsf {snr}_2}x_2+z_2$ as depicted in Fig.", "REF .This coding strategy is introduced without any claim of optimality.", "The undeniable advantage of using independently generated codebooks is the simplicity of implementation.", "We note that the idea of using $\\Phi _2$ independent of $\\Phi _1$ was also proposed in [24], [25].", "Figure: Model of the adaptive xp transmission: the harq controller uses the information ℱ\\mathcal {F} obtained over the feedback channel to choose the rate for the next round; ℱ\\mathcal {F} represent ack/nack acknowledgement in the case of one bit feedback, or, it carries the index of the coding rate in the case of rate-adaptive transmission (Sec.", ").Intuitively, by introducing $\\mathsf {m}_2$ we want to prevent the “waste” of mi, which happens if $I^\\Sigma _2$ is much larger than $R_1$ , cf.", "Example REF .", "After the second transmission, the receiver decodes the packets $[\\mathsf {m}_1, \\mathsf {m}_2]$ using the observations $y_{[2]}=[y_1, y_2]$ .", "The codebook obtained after two transmissions is illustrated in Fig.", "REF .", "The associated decoding conditions based on the channel outcomes $y_{[2]}$ are given by $I^\\Sigma _2=I_1+I_2&\\ge R_1+R_2,\\\\I_2&\\ge R_2,$ where (REF ) is a constraint over the sum-rate that guarantees the joint decoding of the packets pair $(\\mathsf {m}_1, \\mathsf {m}_2)$ while () ensures the correct decoding of the packet $\\mathsf {m}_2$ .", "This means, the mi must be accumulated to decode each of the packets even though the decoding is done jointly.", "The formal proof of (REF ) and () is presented in the Appendix .", "Similar decoding conditions were presented in the context of phy security in [35].", "While the event $\\mathsf {NACK}_1$ remains unchanged with respect to the conventional coding, the event $\\mathsf {NACK}_2$ means that $\\mathsf {NACK}_1$ occurred, as well as, that (REF ) and () are not satisfied $\\nonumber \\mathsf {NACK}_2&=\\Big \\lbrace \\bigl ( I_1<R_1 \\bigr ) \\wedge \\overline{\\bigl ((I^\\Sigma _2\\ge R^\\Sigma _2) \\wedge (I_2\\ge R_2) \\bigr )}\\Big \\rbrace \\\\&=\\Big \\lbrace ( I_1<R_1) \\wedge \\bigl ( (I^\\Sigma _2<R^\\Sigma _2) \\vee ( I_2<R_2) \\bigr ) \\Big \\rbrace \\\\&=\\Big \\lbrace \\bigl ( I_1<R_1 \\bigr ) \\wedge \\bigl (I^\\Sigma _2<R^\\Sigma _2\\bigr )\\Big \\rbrace ,$ where $R^\\Sigma _k\\triangleq \\sum _{l=1}^k R_l$ and the event $\\overline{E}$ is the complement of $E$ .", "To pass from (REF ) to () we used the decoding failure implication $\\nonumber \\lbrace I_1<R_1 \\wedge I_2<R_2\\rbrace \\Rightarrow \\lbrace I_1<R_1 \\wedge I^\\Sigma _2<R^\\Sigma _2\\rbrace ,$ which means that $\\mathsf {NACK}_1$ combined with (REF ) implies ().", "The above conditions generalize straightforwardly for any $k>1$ with $R_k$ being the rate of the packet $\\mathsf {m}_k$ added in the $k$ th round $\\mathsf {NACK}_k=\\lbrace \\mathsf {NACK}_{k-1} \\wedge \\bigl (I^\\Sigma _k<R^\\Sigma _k \\bigr )\\rbrace .$ To calculate the throughput of such an xp, we adopt a similar approach as in (REF ) but we must account for the reward in the $k$ transmission round given by $R^\\Sigma _k$ , which yields $\\nonumber \\eta ^{{\\textnormal {xp}}}_{K}&=\\frac{R^\\Sigma _1(1-f_1)+ R^\\Sigma _2(f_1-f_2)+ \\ldots +R^\\Sigma _K(f_{K-1}-f_K)}{(1-f_1)+2\\cdot (f_1-f_2)+\\ldots +K\\cdot (f_{K-1})}\\\\&=\\frac{\\sum _{k=1}^{K}R_k \\big (f_{k-1}-f_K\\big ) }{1+\\sum _{k=1}^{K-1} f_k}.$ Here, again $f_k=\\Pr \\left\\lbrace \\mathsf {NACK}_k\\right\\rbrace , k\\ge 1$ with $\\mathsf {NACK}_k$ defined by (REF ).", "As a sanity check we can set $R_k=0, k=2, \\ldots , K$ , and recover the conventional single-packet harq, i.e., (REF ) will be equivalent to ().", "The fundamental difference of the proposed xp with respect to the conventional harq appears now clearly in the numerator of (REF ) which expresses the idea of variable rate transmission due to encoding of multiple packets.", "Nevertheless, not only the numerator changed with respect to () but also the denominator is different due to the new definition of $\\mathsf {NACK}_k$ in (REF ).", "Figure: Illustration of the codebook defined through the coding function Φ 1 \\Phi _1 in () and the joint coding function Φ 2 \\Phi _2 in ().", "Each codeword composed of 2N s 2{\\mathop {N_{\\textnormal {s}}}} symbols is indexed by the packet 𝗆 [2] \\mathsf {m}_{[2]}.", "The first N s {\\mathop {N_{\\textnormal {s}}}} symbols are created without indexing by 𝗆 2 \\mathsf {m}_2 so we artificially repeat them 2 R 2 N s 2^{R_2{\\mathop {N_{\\textnormal {s}}}}} times to match the number of codewords in the codebook Φ 2 \\Phi _2.Example 3 (Two-state channel and xp) We consider now the proposed xp in the scenario of Example REF .", "Let us start, as before, with $K=2$ and $R_1=1.5$ .", "After a decoding failure (which means that we obtained $I_1=I_{\\textnormal {a}}=1$ ), we are free to define any rate $R_2$ .", "In the absence of any formal criterion (more on that in Sec.", "), we take the following auxiliary (and somewhat ad-hoc) condition: we want to guarantee a non-zero successful decoding probability, i.e., $f_2<1$ .", "Here, since $I^\\Sigma _2\\in (2,2.5)$ , any $R_2 \\le 1$ can ensure that $f_2<1$ .", "In particular, if the rate $R_2\\le 0.5$ we guarantee a much stronger condition $f_2=0$ .", "For the case when $K=2$ and using $R_2=0.5$ , we obtain $f_1=0.25$ and $f_2=0$ .", "The throughput is then given by $\\eta ^{{\\textnormal {xp}}}_2=\\frac{R_1+0.25R_2 }{1+0.25}=1.3.$ Thus, we used exactly the same channel resources as in the conventional harq, obtained the same guarantee of successful decoding ($f_2=0$ ) after two transmission rounds, but the throughput is larger.", "The difference is that, while we still have $I^\\Sigma _2\\in (2,2.5)$ , we now use $R^\\Sigma _2=2$ to eliminated the “waste” of mi in the conventional irharq, where $R^\\Sigma _2=1.5$ .", "The improvement may be seen as the increase in the throughput (from $\\eta ^{{\\textnormal {ir}}}_2=1.2$ to $\\eta ^{{\\textnormal {xp}}}_2=1.3$ ) or as the reduction in the memory requirements (i.e., we obtain a better throughput with smaller $K$ , see $\\eta ^{{\\textnormal {ir}}}_3=1.23$ in Example REF ).", "The price to pay for this advantages is the possible increase in complexity of cross-packet encoding/decoding.", "Similarly, for $K=3$ , we can use the larger value of $R_2$ (that guarantees our objective of decodability, $f_2<1$ ), i.e., $R_2=1$ .", "In this case, $f_1=0.25$ , and $f_2=\\Pr \\left\\lbrace I_1<1.5 \\wedge I^\\Sigma _2<2.5\\right\\rbrace =0.0625$ .", "In the third transmission we observe $I^\\Sigma _{3}\\in (3,3.5)$ so, using $R_3=0.5$ , we obtain $f_3=0$ and thus the throughput is calculated as $\\eta _3^{{\\textnormal {xp}}}=\\frac{R_1+0.25R_2 + 0.0625R_3}{1+0.25+0.0625}\\approx 1.36,$ which is already quite close to $\\overline{C}=1.375$ .", "The improvement of the throughput in xp is due to the way the codebook is constructed.", "While the conventional irharq, see Sec.", "REF , makes a rigid separation of the codewords into the fixed-content subcodewords – an approach which is blind to the channel realizations, in xp we match the information content of the codebook following the outcome of the transmissions." ], [ "Optimization of the coding rates", "Our goal now is to evaluate how well the xp can perform.", "To this end, we will have to find the optimal coding rates $R_1, R_2, \\ldots , R_K$ which maximize throughput (REF ).", "Since the objective function is highly non linear, we will use the exhaustive search: for a truncated harq this can be done with a manageable complexity.", "Example 4 (16QAM, Rayleigh fading – continued) In Fig.", "REF we show the results of the exhaustive-search optimization of ${\\eta }^{{\\textnormal {xp}}}_K$ with ${\\eta }^{{\\textnormal {ir}}}_K$ ; for implementability, we limited the search space: irharq uses $R_1\\in \\lbrace 0,0.25,\\ldots ,3.75\\rbrace $ and xp uses rates which satisfy $R^\\Sigma _{K} \\le R_{\\max }$ , with $R_{\\max }=8$; $R_{1}\\in \\lbrace 0.25,\\ldots ,3.75\\rbrace $ , $R_{k}\\in \\lbrace 0,0.25,\\ldots ,3.75\\rbrace ~ \\forall k\\in \\lbrace 2,\\ldots ,K\\rbrace $ .", "We used here an additional constraints requires each transmission to have non zero probability of being decodable, that is $R_{k} < \\log _{2} M, \\forall k=1,\\ldots ,K$ , where $M=16$ .", "In fact, these constraints were always satisfied in xp so they only affect irharq; we will relax them in the next example.", "In terms of snr required to attain $\\eta =3$ , the gain of xp over irharq varies from $1.5{\\textnormal {dB}}$ (for $K=2$ ) to $2.5{\\textnormal {dB}}$ (for $K =3$ ).", "Figure: Throughput of the conventional irharq (η K ir \\eta ^{{\\textnormal {ir}}}_K) compared to xp (η K xp \\eta ^{{\\textnormal {xp}}}_K) in Rayleigh block-fading channel.", "The ergodic capacity (C ¯\\overline{C}) is shown for reference." ], [ "Rate adaptation", "The possibility of varying the rates during the harq cycle opens new optimization space and we want to explore it fully following the idea of adapting the transmission parameters in harq on the basis of obsolete csi considered before, e.g., in [6], [8], [9], [36], [10], [11], [29].", "The idea is to adapt the coding rates using obsolete csis, $I_1, I_2, \\ldots , I_{k-1}$ ; this concept remains compatible with the assumption of transmitter operating without csi knowledge because the obsolete csis $I_1, I_2, \\ldots , I_{k-1}$ cannot be used in the $k$ th round to infer anything about $I_{k}$ (due to iid model of the snrs).", "Using this approach, the rate $R_k$ will not only depend on the mis $I_1,\\ldots , I_{k-1}$ but also – on the past rates $R_1, \\ldots , R_k$ .Through $R_1, R^\\Sigma _2,\\ldots , R^\\Sigma _{k-1}$ , which determine the probability of the decoding success, see (REF ).", "This recursive dependence may be dealt with using the mdp framework, where the states of the Markov chain not only indicate the transmission number but also gather all information necessary to decide on the rate, which in the language of the mdp is called an action.", "The state has to be defined so that i) knowing the action (chosen rate), the state-transition probability can be determined after each transmission, and ii) the reward may be calculated knowing the state and the action.", "The state defined as a pair $\\mathsf {s}_k=(R^\\Sigma _k,I^\\Sigma _k)$ satisfies these two requirements, where we only need to consider the pairs which satisfy $R^\\Sigma _k>I^\\Sigma _k$ , otherwise the decoding is successful and the harq cycle terminates.", "Thus, the rate adaptation consists in finding the functions (called policies), $R_{l}(\\mathsf {s}_{l-1})$ maximizing the throughput, which is found generalizing the expression (REF ) $\\hat{\\eta }^{{\\textnormal {xp}}}_{K}&=\\frac{{E}\\big [\\sum _{k=1}^{K}\\xi _{k} R^\\Sigma _k\\big ]}{1+\\sum _{k=1}^{K-1}f_{k}},$ where $\\xi _{k}={I}\\big [{I_1<R_1 \\wedge \\ldots \\wedge I^\\Sigma _{k-1}<R^\\Sigma _{k-1}\\wedge I^\\Sigma _{k}\\ge R^\\Sigma _k}\\big ],$ indicates the successful decoding in the $k$ th round, and $R^\\Sigma _k=R^\\Sigma _{k-1}+R_{k}(\\mathsf {s}_{k-1})$ is the accumulated rate depending in a recursive fashion on the states of the Markov chain.", "The probability of $k$ successive errors, $f_k$ , may be expressed as (REF ) considering the dependence of the rates on the states given by (REF ).", "All the expectations are taken with respect to the states – or equivalently – with respect to $I_1,\\ldots , I_{K}$ .", "The expression (REF ) will be useful in Sec.", "REF , however, its maximization with respect to the policies $R_{l}(\\mathsf {s}_{l-1}), l=1,\\ldots , K$ will be done using efficient specialized algorithms as explained in Appendix .", "In the particular case of two harq rounds ($K=2$ ), the optimal rate adaptation policy can be derived in closed form as shown in Appendix .", "To run the optimization algorithms outlined in Appendix , we need to discretize the variables involved (states and actions).", "As for the rates (actions), we use a relatively course discretization step equal to $0.25$ and define the action space as the set $\\mathcal {R}=\\lbrace 0.25,0.5,\\ldots , R_{\\max }\\rbrace $ .", "While the results are notably affected by $R_{\\max }$ , using a finer discretization step did not change the results significantly.", "Here, it is natural to ask a question about the signaling overhead due to proposed adaptation scheme.", "We thus note that while we assume the outdated mi, $I^\\Sigma _k$ is discretized with a high resolution when optimizing the throughput (cf.", "Appendix ), the feedback load is affected by the cardinality of the action space, $\\mathcal {R}$ : the receiver knows the accumulated mi but only transmits the index of the chosen rate.", "Example 5 (16QAM, Rayleigh fading channel – continued) The throughput of adaptive xp, $\\hat{\\eta }^{{\\textnormal {xp}}}$ , is compared to the throughput of the conventional irharq in Fig.", "REF for $K=\\infty $ , while Fig.", "REF shows the comparison for truncated harq.", "Here, for irharq, we removed the constraints on the initial coding rate, $R_1<\\log _2 M$ , which were applied in Example REF .", "It allows us to increase the throughput $\\eta ^{{\\textnormal {ir}}}_3$ at the cost of first transmission not being decodable.", "In our view this is a potentially serious drawback but we show such results to complement those already shown in Fig.", "REF , where the decodability condition was imposed.", "Again, xp was insensitive to the decodability constraints and always provided results with decodable transmissions.", "Figure: Optimal throughput of the conventional irharq (η ∞ ir \\eta ^{{\\textnormal {ir}}}_\\infty ) compared to the proposed xp (η ^ ∞ xp \\hat{\\eta }^{{\\textnormal {xp}}}_\\infty ) in Rayleigh block-fading channel.", "The ergodic capacity (C ¯\\overline{C}) is shown for reference.Figure: Throughput of the conventional irharq (η K ir \\eta ^{{\\textnormal {ir}}}_K) compared to the proposed xp (η ^ K xp \\hat{\\eta }^{{\\textnormal {xp}}}_K) for a truncated harq, K∈{2,3}K\\in \\lbrace 2,3\\rbrace in Rayleigh block-fading channel; R max =8R_{\\max }=8.", "The ergodic capacity (C ¯\\overline{C}) and the optimal throughput of the persistent conventional irharq (η ∞ ir \\eta ^{{\\textnormal {ir}}}_\\infty ) are shown for reference.The improvements due to adaptive xp are most notable for high values of the throughput.", "In particular we observe that The persistent xp halves the gap between the ergodic capacity and the conventional irharq.", "For example, the snr gap between $\\hat{\\eta }^{{\\textnormal {xp}}}_\\infty =3$ and the ergodic capacity, $\\overline{C}=3$ is reduced by more than $50\\%$ when comparing to the gap between $\\eta ^{{\\textnormal {ir}}}_\\infty =3$ and $\\overline{C}=3$ which is equal to $5{\\textnormal {dB}}$ when $R_{\\max }=8$ .", "We note that the throughput of xp increases when $R_{\\max }$ increases: the snr gap between $\\overline{C}$ and $\\hat{\\eta }^{{\\textnormal {xp}}}_\\infty $ is reduced by half when $R_{\\max }=16$ is used instead of $R_{\\max }=8$ .", "For any value of throughput $\\eta >3$ , two rounds of xp yield higher throughput than the conventional persistent irharq.", "Thus, in this operation range we may improve the performance and yet decrease the memory requirements at the receiver." ], [ "Heuristic adaptation policy", "Fig.", "REF shows the optimal rate adaptation as a function of $R^\\Sigma _{k-1}-I^\\Sigma _{k-1}$ for different values of $R^\\Sigma _{k-1}$ , where we note a quasi-linear behaviour of the adaptation function with the saturation which occurs to guarantee $R^\\Sigma _{k-1}+R_k\\le R_{\\max }$ .", "To exploit this very regular form, which was also observed solving the related problems in [11], [29], we propose to use the following heuristic function inspired by Fig.", "REF $R_k=R_1- (R^\\Sigma _{k-1}-I^\\Sigma _{k-1}),$ where only the rate $R_1$ needs to be optimized (from Fig.", "REF we find $R_1\\approx 3.5$ ).", "Furthermore, applying (REF ) recursively we obtain $R_2=I_1, R_3=I_2, \\ldots , R_k=I_{k-1}$ ; the identical rate-adaptation strategy may be derived from [20].", "The simplicity of the adaptation function allows us now to evaluate analytically the throughput of xp.", "To this end we need to calculate $f_l$ in the denominator of (REF ) and the expectation in its numerator.", "We first note that, from (REF ) we obtain $\\big (I^\\Sigma _k&<R^\\Sigma _k\\big ) \\iff (I_k<R_1),$ which means that the probability of decoding failure does not change with the index of the transmission round.", "Thus $f_{k}&=(f_{1})^k,$ and (REF ) may be formulated as $\\xi _{k}=\\Big (\\prod _{l=1}^{k-1}{I}\\big [{I_l<R_1}\\big ]\\Big ){I}\\big [{I_{k}\\ge R_1}\\big ].$ From (REF ) we also obtain $R^\\Sigma _k=R_1 + \\sum _{l=1}^{k-1}I_l$ , which allows us to calculate the expectation in the numerator of (REF ) as ${E}[\\xi _{k} R^\\Sigma _k]&={E}[\\xi _{k} (R_1 + I_1+\\ldots , I_{k-1})]\\\\&=\\big (R_1f_1 +(k-1)\\tilde{C}\\big )(f_1)^{k-2}(1-f_1),$ where $\\tilde{C}={E}_{I_1}\\big [I_{1}\\cdot {I}\\big [{I_{1}<R_{1}}\\big ]\\big ]$ is a “truncated” expected mi.", "Using () and (REF ) in (REF ), the throughput is calculated as $\\tilde{\\eta }^{\\textnormal {xp}}_K&=R_{1}(1-f_{1})+\\frac{\\tilde{C}(1-f_{1})}{1-f_{1}^K} \\nonumber \\\\&\\qquad \\times \\Big (-(K-1)f_{1}^{K-1}+\\frac{1-f_{1}^{K-1}}{1-f_{1}} \\Big ).$ In the limit, $K\\rightarrow \\infty $ , (REF ) becomes $\\tilde{\\eta }^{{\\textnormal {xp}}}_\\infty =R_{1}(1-f_{1})+\\tilde{C},$ which is the same as [20].", "Figure: Throughput of the optimal xp (η ^ K xp \\hat{\\eta }^{\\textnormal {xp}}_K) is compared to the throughput of xp with the heuristic policy (η ˜ K xp \\tilde{\\eta }^{\\textnormal {xp}}_K) in Rayleigh block-fading channel.", "The ergodic capacity (C ¯\\overline{C}) is shown for reference.Example 6 (16QAM, Rayleigh fading – continued) We compare in Fig.", "REF the throughput of optimal xp with the heuristic policy (REF ), which is optimized over $R_{1}$ .", "As expected, the optimal solution outperforms the heuristic policy but the gap is very small (less than $0.5{\\textnormal {dB}}$ ).", "Moreover, since $\\hat{\\eta }^{\\textnormal {xp}}_K$ was optimized over a finite set of rates $\\mathcal {R}=\\lbrace 0.25,0.5,\\ldots ,R_{\\max }\\rbrace $ , and the heuristic policy assumes that $\\mathcal {R}$ is continuous and unbounded, $\\tilde{\\eta }^{\\textnormal {xp}}_K$ slightly outperforms $\\hat{\\eta }^{\\textnormal {xp}}_K$ above $\\overline{\\mathsf {snr}}=20{\\textnormal {dB}}$ .", "This gap can be reduced increasing the value of $R_{\\max }$ ; decreasing the discretisation step below $0.25$ had much lesser influence on the results.", "The results are quite intriguing and suggesting that the strategy of [20] based on a double-layer encoding[20] proposes double-step encoding: to form $\\mathsf {m}_{[k]}$ the bits $\\mathsf {m}_{k}$ and the parity bits of $\\mathsf {m}_{[k-1]}$ are first “mixed”, and next, the channel encoder is used.", "and a transmission-by-transmission decoding (as opposed to the joint decoding required in xp), asymptotically yield the same throughput as the heuristic cross-packet harq, whose throughput is also very close to the optimal xp.", "We cannot follow that path here but this relationship should be studied in more details; in particular, the effect of removing the idealized assumption of using a continuous set of rates $\\mathcal {R}$ , necessary to implement (REF ), should be analyzed." ], [ "Example of a practical implementation", "Until now, we have adopted the perfect decoding assumption, i.e., the decoding error in the $k$ th round is equivalent to the event $\\lbrace I_1<R_1\\wedge \\ldots \\wedge I^\\Sigma _{k}<R^\\Sigma _{k}\\rbrace $ .", "We will remove now this idealization to highlight also the practical aspect of xp.", "We thus implement the cross-packet encoders in Fig.", "REF using turbo encoders.", "To this end, as shown in Fig.", "REF we separate each encoder $\\Phi _k$ into i) a bit-level multiplexer, $\\mathcal {M}$ , whose role is to interleave the input packets $\\mathsf {m}_1, \\ldots , \\mathsf {m}_k$ and produce the packet, $\\mathsf {m}_{[k]}$ , ii) a conventional turbo-encoder (TC), iii) the rate-matching puncturer, $\\mathcal {P}$ , which ensures that all binary codewords $c_k$ have the same length, $N_{\\textnormal {c}}$ , and iv) a modulator, which maps the codewords $c_k$ onto the codewords $x_k$ from the constellation $\\mathcal {X}$ ; since we use 16ary qam, $N_{\\textnormal {c}}={\\mathop {N_{\\textnormal {s}}}}\\log _2(M)$ .", "The multiplexers $\\mathcal {M}_k$ are implemented using pseudo-random interleaving.", "The encoders (TC) are constructed via parallel concatenation of two recursive convolutional encoders with polynomials $[13/15]_8$ .", "Each TC produces a $N_{\\textrm {b},[k]}={\\mathop {N_{\\textnormal {s}}}}R_1+\\ldots +{\\mathop {N_{\\textnormal {s}}}}R_k$ systematic (input) bits and $N_{\\textrm {p}}=2N_{\\textrm {b},[k]}$ parity bits $p_k$ .We neglects the effect of the trellis terminating bits.", "The bits $c_k$ are obtained concatenating “fresh” systematic bits $\\mathsf {m}_k$ (those which were not transmitted in the previous rounds) and the parity bits selected from $p_k$ via a periodic puncturing.", "Such a construction of the encoders is of course not optimal and better interleavers and puncturers may be sought; however, their optimal design represents a challenge of its own and must be considered out of scope of the example we present here.", "The encoding is rather straightforward and can be implemented using conventional elements.", "The decoding in the $k$ th round is slightly more involved because it is done using outcomes of all transmissions, $y_{[k]}$ .", "From this perspective, we may see the binary codewords $c_1, \\ldots , c_k$ as an outcome of $2k$ concatenated convolutional encoders (two encoders per harq round), each producing the sequence with increasing lengths.", "The decoding of multiple encoding units was already addressed before [37][38] and requires implementation of $2k$ bcjr decoders (one for each of the encoders) exchanging the extrinsic probabilities for the information bits.", "We implement the serial scheduling, that is, once a bcjr decoder is activated, it must wait till all other bcjr decoders are activated.", "One iteration is defined as $2k$ activations.", "The results we present are obtained using algorithm from the library [39]; we use ${\\mathop {N_{\\textnormal {s}}}}=1024$ and four decoding iterations.", "arrows shapes positioning calc shapes.misc Figure: Implementation of the encoders Φ k [·]\\Phi _k[\\cdot ] using turbo codes (TC), bit multiplexing (ℳ k \\mathcal {M}_k), puncturing (𝒫\\mathcal {P}), and modulation (𝒳\\mathcal {X}).Since we do not have the closed-form formula which describes the probability of error under particular channel conditions, especially when multiples transmissions are involved, the rate-adaptation approach seems to be out of reach and we focus on finding the fixed coding rates $R_k, k=1,\\ldots ,K$ .", "We use the brute search over the space of available coding rates which verifies the following conditions $\\sum _{k=1}^{K}R_{k}\\le 8$ , $R_{1}\\in \\lbrace 1.5,1.75,2,\\ldots ,3.75\\rbrace $ , $R_{k}\\in \\lbrace 0,0.25,\\ldots ,3.75\\rbrace , \\forall k>1$ .", "The results obtained are shown in Fig.", "REF where the snr gap (for the throughput $\\eta =3$ ) between xp and the conventional irharq is $\\sim 1.5{\\textnormal {dB}}$ for $K=2$ and $\\sim 2{\\textnormal {dB}}$ for $K=3{\\textnormal {dB}}$ .", "We attribute a small improvement of the throughput $\\eta ^{{\\textnormal {xp}}}_3$ over $\\eta ^{{\\textnormal {xp}}}_2$ to the suboptimal encoding scheme we consider in this example.", "We also note that the improvement of $\\eta ^{{\\textnormal {ir}}}_3$ with respect to $\\eta ^{{\\textnormal {ir}}}_2$ does not materialize.", "This is because irharq is optimized for $R_1$ but, due to limitation of the turbo encoder which generates only $3{\\mathop {N_{\\textnormal {b}}}}$ bits, a full redundancy cannot be always obtained and, in such a case, we are forced to repeat the systematic and parity bits.", "This explains why $\\eta ^{{\\textnormal {ir}}}_3$ and $\\eta ^{{\\textnormal {ir}}}_2$ are very similar for low throughput.", "On the other hand, they should be, indeed, similar for high throughput as we have seen in the numerical examples before.", "We show in Fig.", "REF the ergodic capacity where the gap to the throughput of the TC-based transmission is increased by additional $3{\\textnormal {dB}}$ which should be expected when using relatively-short codewords and practical decoders.", "Figure: Turbo-coded transmission: the conventional irharq (η K \\eta _K) is compared to xp (η K xp \\eta ^{{\\textnormal {xp}}}_K) in Rayleigh block-fading channel." ], [ "Conclusions", "In this work we proposed and analyzed a coding strategy tailored for harq protocol and aiming at the increase of the throughput for transmission over block fading channel.", "Unlike many heuristic coding schemes proposed previously, our goal was to address explicitly the issue of joint coding of many packets into the channel block of predefined length.", "With such a setup, the challenge is to optimize the coding rates for each packet which we do efficiently assuming existence of a multi-bits feedback channel which transmit the outdated csi experienced by the receiver.", "The throughput of the resulting xp is compared to the conventional irharq indicating that significant gains can be obtained using the proposed coding strategy.", "The gains are particularly notable in the range of high throughput, where the conventional harq fails to offer any improvement with increasing number of transmission rounds.", "The proposed encoding scheme may be seen as a method to increase the throughput, or as a mean to diminish the memory requirements at the receiver; the price for the improvements is paid by a more complex joint encoding/decoding.", "We also proposed an example of a practical implementation based on turbo codes.", "This example highlights the practical aspects of the proposed coding scheme, where the most important difficulties are i) the need of tailoring the encoder to provide the jointly coded symbols with the best decoding performance, and ii) the design of the simple decoder.", "Moreover, the real challenge is to leverage the possibility of adaptation to the outdated csi.", "To do so, simple techniques for performance evaluation (e.g., the per) based on the expected csi, must be used; such as, for example those studied in [40]." ], [ "Decoding conditions of xp", "We outline the proof of the decoding conditions (REF ) and (), stated in the following Lemma REF .", "The HARQ-code refers to the encoding functions stated in (REF ) and (REF ) and the joint decoding of the pair $[\\mathsf {m}_1, \\mathsf {m}_2]$ .", "Lemma 1 (Decoding conditions) For all $\\varepsilon >0$ , there exists $\\bar{n}\\in \\mathbb {N}$ such that for all $n\\ge \\bar{n}$ , there exists a HARQ-code $c^{\\star } $ such that for all snr realization $(\\mathsf {snr}_1,\\mathsf {snr}_2)$ that satisfy: $R_1 + R_2 &\\le & \\mathsf {I}(X_1 ; Y_1 \|\\mathsf {snr}_{1}) + \\mathsf {I}(X_2 ; Y_2 \|\\mathsf {snr}_{2}) - \\varepsilon , \\\\R_2 &\\le & \\mathsf {I}(X_2 ; Y_2 \|\\mathsf {snr}_{2}) - \\varepsilon , $ the error probability is bounded by $\\Pr \\left\\lbrace [ \\mathsf {m}_1, \\mathsf {m}_2] \\ne [ \\hat{\\mathsf {m}}_1, \\hat{\\mathsf {m}}_2] \\bigg \|c^{\\star }, \\mathsf {snr}_{1}, \\mathsf {snr}_{2} \\right\\rbrace \\le \\varepsilon .$ [Proof of Lemma REF ] We consider the random HARQ-code: $\\bullet $ Random codebook: we generate $ 2^{{\\mathop {N_{\\textnormal {s}}}}\\cdot R_1 } $ codewords $x_1$ and $2^{{\\mathop {N_{\\textnormal {s}}}}\\cdot ( R_1 + R_2 )} $ codewords $x_2$ , drawn from the uniform distribution over the constellation $\\mathcal {X}$ .", "$\\bullet $ Encoding function: as explained in Sec.", ", the encoder starts by sending $x_1$ which corresponds to the packet (or message in the language of information theory) $\\mathsf {m}_1$ .", "If the encoder receives a feedback $\\mathsf {NACK}_1$ , it sends $x_2$ corresponding to the pair of messages $[\\mathsf {m}_1 , \\mathsf {m}_2]$ .", "Otherwise a new transmission process starts.", "$\\bullet $ Decoding function: if the snr realizations $(\\mathsf {snr}_1,\\mathsf {snr}_2)$ satisfy equations () and (REF ), then the decoder finds a pair of messages $[\\mathsf {m}_1,\\mathsf {m}_2]$ such that the following sequences of symbols are jointly typical: $\\Big (\\Phi _1[\\mathsf {m}_1] , y_1 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}},\\;\\Big (\\Phi _2[\\mathsf {m}_1, \\mathsf {m}_2] , y_2 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}}.", "$ $\\bullet $ Error is declared when sequences are not jointly typical.", "Error events.", "We define the following error events: $\\bullet $ $E_0=\\bigg \\lbrace \\Big (\\Phi _1[\\mathsf {m}_1] , y_1 \\Big ) \\notin A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}}\\bigg \\rbrace \\cup \\bigg \\lbrace \\Big (\\Phi _2[\\mathsf {m}_1, \\mathsf {m}_2] , y_2 \\Big ) \\notin A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}}\\bigg \\rbrace $ , $\\bullet $ $E_{1} =\\bigg \\lbrace \\exists [\\mathsf {m}_1^{\\prime },\\mathsf {m}_2^{\\prime }] \\ne [\\mathsf {m}_1,\\mathsf {m}_2],\\text{ s.t.", "}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad \\Big \\lbrace \\Big (\\Phi _1[\\mathsf {m}_1^{\\prime }] , y_1 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}} \\Big \\rbrace \\cap \\Big \\lbrace \\Big (\\Phi _2[\\mathsf {m}_1^{\\prime }, \\mathsf {m}_2^{\\prime }] , y_2 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}} \\Big \\rbrace \\bigg \\rbrace $ , $\\bullet $ $E_{2} =\\bigg \\lbrace \\exists \\mathsf {m}_1^{\\prime }\\ne \\mathsf {m}_1,\\text{ s.t.", "}\\\\\\qquad \\qquad \\qquad \\qquad \\qquad \\Big \\lbrace \\Big (\\Phi _1[\\mathsf {m}_1^{\\prime }] , y_1 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}} \\Big \\rbrace \\cap \\Big \\lbrace \\Big (\\Phi _2[\\mathsf {m}_1^{\\prime }, \\mathsf {m}_2] , y_2 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}} \\Big \\rbrace \\bigg \\rbrace $ , $\\bullet $ $E_{3} =\\bigg \\lbrace \\exists \\mathsf {m}_2^{\\prime }\\ne \\mathsf {m}_2,\\text{ s.t. }", "\\Big (\\Phi _2[\\mathsf {m}_1, \\mathsf {m}_2^{\\prime }] , y_2 \\Big ) \\in A_{\\varepsilon }^{{\\star }{{\\mathop {N_{\\textnormal {s}}}}}} \\bigg \\rbrace $ .", "The properties of the typical sequences imply that, for ${\\mathop {N_{\\textnormal {s}}}}$ large enough, $\\Pr \\left\\lbrace E_0\\right\\rbrace \\le \\varepsilon $ , and the Packing Lemma [41] implies that the probabilities of the events $E_1$ , $E_2$ , $E_3$ are bounded by $\\varepsilon $ if the following conditions are satisfied $R_1 + R_2 &\\le & \\mathsf {I}(X_1 ; Y_1 \|\\mathsf {snr}_{1}) + \\mathsf {I}(X_2 ; Y_2 \|\\mathsf {snr}_{2}) - \\varepsilon , \\\\R_2 &\\le & \\mathsf {I}(X_2 ; Y_2 \|\\mathsf {snr}_{2}) - \\varepsilon ,\\\\R_1 &\\le & \\mathsf {I}(X_1 ; Y_1 \|\\mathsf {snr}_{1}) + \\mathsf {I}(X_2 ; Y_2 \|\\mathsf {snr}_{2}) - \\varepsilon , .$ Since (REF )-() are the hypothesis (REF )-() of Lemma REF , there exists HARQ-code $c^{\\star }$ with small error probability." ], [ "Optimization via mdp", "To obtain the mdp formulation it is convenient to replace packet-wise notation of (REF ) with a time-wise model $y[n]=\\sqrt{\\mathsf {snr}[n]}x[n]+z[n],$ where $n$ is the index of the channel block.", "At each time $n$ , the harq controller observes the state $\\mathsf {s}[n]$ , and takes an action $\\mathsf {a}[n]=\\pi (\\mathsf {a}[n])$ , according to the policy $\\pi $ .", "The transition probability matrix, $Q(\\mathsf {a})$ , has the elements $Q_{\\mathsf {s},\\mathsf {s}^{\\prime }}(\\mathsf {a})\\triangleq \\displaystyle {\\Pr \\lbrace \\mathsf {s}[n+1]=\\mathsf {s}^{\\prime }\|\\mathsf {s}[n]=\\mathsf {s}, \\mathsf {a}[n]=\\mathsf {a}\\rbrace },$ defining the probabilities of the system moving to the state $\\mathsf {s}^{\\prime } \\in \\mathcal {S}$ at time $n+1$ conditioned on the system being in the state $\\mathsf {s}\\in \\mathcal {S}$ at time $n$ and the controller taking the action $\\mathsf {a}\\in \\mathcal {A}(\\mathsf {s})$ , where $\\mathcal {A}(\\mathsf {s})$ is the set of actions allowed in a state $\\mathsf {s}$ and $\\underset{\\mathsf {s}\\in \\mathcal {S}}{\\bigcup }\\mathcal {A}(\\mathsf {s})=\\mathcal {A}$ .", "In our case, the actions are the coding rates, which we assume may take any positive value, and thus $\\mathcal {A}(\\mathsf {s})=\\mathbb {R}_+$ .", "A policy $\\pi $ is defined as a mapping $\\pi :\\mathcal {S}\\mapsto \\mathcal {A}$ between the state space, $\\mathcal {S}$ , and the action space, $\\mathcal {A}$ .", "We aim at finding a policy $\\pi $ which maximizes the long-term average throughput $\\eta (\\pi ) &= \\lim _{N \\rightarrow \\infty }\\frac{1}{N}\\sum _{n=1}^N {E}\\big [ \\mathsf {R}(\\mathsf {s}[n],\\pi (\\mathsf {s}[n])) \\big ],$ where $\\mathsf {R}(\\mathsf {s},\\mathsf {a})$ is the average reward obtained when taking action $\\mathsf {a}$ in the state $\\mathsf {s}$ and the expectations are taken with respect to the random states $\\mathsf {s}[n]$ .", "In our case the reward is the number of decoded bits normalized by the duration of the channel block, ${\\mathop {N_{\\textnormal {s}}}}$ .", "The optimal policy thus solves the following problem: $\\hat{\\eta }^{{\\textnormal {xp}}}_{K}=\\max _{\\pi (\\cdot )}\\eta (\\pi )$ and may be found solving the Bellman equations [32] $\\hat{\\eta }^{{\\textnormal {xp}}}_{K}+h(\\mathsf {s})&=\\max _{\\mathsf {a}\\in \\mathcal {A}(\\mathsf {s})}\\left[ \\mathsf {R}(\\mathsf {s},\\mathsf {a}) + \\sum _{\\mathsf {s}^{\\prime }\\in \\mathcal {S}}Q_{\\mathsf {s},\\mathsf {s}^{\\prime }}(\\mathsf {a}) h(\\mathsf {s}^{\\prime })\\right],\\quad \\forall \\mathsf {s}\\in \\mathcal {S},$ where $h(\\mathsf {s})$ is a difference reward associated with the state.", "To calculate the optimal $\\hat{\\eta }^{{\\textnormal {xp}}}_{K}$ , we use here the policy iteration algorithm whose details may be found in [32] and which guarantees to reach the solution after a finite number of iterations.", "The unique optimal throughput $\\hat{\\eta }^{{\\textnormal {xp}}}_{K}$ exists and is independent of the initial state, $\\mathsf {s}[0]$ if, for any state $\\mathsf {s}^{\\prime }[t]\\in \\mathcal {S}$ , we can find a policy, which starting with arbitrary state $\\mathsf {s}[0]$ reaches the state $\\mathsf {s}^{\\prime }[t]$ in a finite time $t<\\infty $ , with non-zero probability [32].", "For our problems, finding such a policy is indeed possible, proof of which we skip for sake of brevity.", "In order to define the state space and the average reward, we deal separately with the truncated and persistent xp but in both cases we must track the accumulated rate, $R^\\Sigma [n]$ (it defines the reward, $\\mathsf {R}(\\mathsf {s},\\mathsf {a})$ ), and the accumulated mi, $I^\\Sigma [n]$ (it defines the matrix $Q$ ).", "Thus these two variables must enter the definition of the state, $\\mathsf {s}[n]$ ." ], [ "Persistent HARQ", "For the persistent xp, the state can be defined as a pair $\\mathsf {s}[n]\\triangleq (I^\\Sigma [n],R^\\Sigma [n]),$ and the transition to the state at time $n+1$ is defined as $\\mathsf {s}[n+1]={\\left\\lbrace \\begin{array}{ll}\\big (I^\\Sigma [n]+I[n],R^\\Sigma [n]+R[n]\\big ), \\\\\\qquad ~\\quad \\text{if} \\quad R^\\Sigma [n]+R[n]\\ge I^\\Sigma [n]+I[n]\\\\\\big (0, 0\\big ), \\quad \\text{otherwise}.\\end{array}\\right.", "}.$ A non-zero reward is obtained only by terminating the harq cycle, i.e., moving to the state $\\mathsf {s}[n+1]=(0,0)$ , $\\mathsf {R}(\\mathsf {s}[n],\\mathsf {a})=&\\big (R^\\Sigma [n]+\\mathsf {a}\\big )F^{\\textnormal {c}}_I (R^\\Sigma [n]-I^\\Sigma [n]+\\mathsf {a}),$ where $F^{\\textnormal {c}}_I(x)\\triangleq 1-F_I(x)$ and $F_I(x)$ is the cdf of $I$ ." ], [ "Truncated HARQ", "In the truncated harq, a new harq cycle starts also if the maximum number of allowed rounds is attained (even if the message is not decoded correctly).", "Thus i) the index of the transmission round, $\\mathsf {k}$ , must enter the defining of the state, ii) we need to make a distinction between the decoding success/failure of the last round.", "We thus define the state as $\\mathsf {s}[n]\\triangleq (I^\\Sigma [n],R^\\Sigma [n],\\mathsf {k}[n],\\mathsf {M}[n]),$ where $\\mathsf {k}[n]$ and $\\mathsf {M}[n]\\in \\lbrace \\mathsf {ACK},\\mathsf {NACK}\\rbrace $ are respectively, the number of rounds and the decoding result after the transmission in block $n$ .", "The system dynamic is described as follows: $\\nonumber \\mathsf {s}[n+1]={\\left\\lbrace \\begin{array}{ll}\\big (0, 0,0,\\mathsf {ACK}\\big ), \\quad ~~\\text{if} \\quad \\mathcal {E}_{\\mathsf {ACK}}[n] \\\\\\big (0, 0,0,\\mathsf {NACK}\\big ), \\quad \\text{if} \\quad \\mathcal {E}_{\\mathsf {NACK}}[n] \\\\\\big (I^\\Sigma [n]+I[n],R^\\Sigma [n]+R[n],\\mathsf {k}[n]+1,\\mathsf {NACK}\\big ), \\\\\\qquad ~\\qquad \\text{otherwise}\\end{array}\\right.", "}$ where $\\nonumber \\mathcal {E}_{\\mathsf {ACK}}[n]&\\triangleq \\lbrace R^\\Sigma [n]+R[n]\\le I^\\Sigma [n]+I[n]\\rbrace \\\\\\nonumber \\mathcal {E}_{\\mathsf {NACK}}[n]&\\triangleq \\lbrace R^\\Sigma [n]+R[n]> I^\\Sigma [n]+I[n]~\\wedge ~\\mathsf {k}[n]+1=K\\rbrace $ are respectively, the conditions indicating a successful decoding and a decoding failure at the end of the harq cycle.", "Thus, the state space is defined as: $\\mathcal {S}=\\mathbb {R}_{+}\\times \\mathbb {R}_{+}\\times \\lbrace 0,1,\\ldots ,K-1\\rbrace \\times \\lbrace \\mathsf {ACK},\\mathsf {NACK}\\rbrace $ and the reward is defined by (REF )." ], [ "Optimal MDP for $K=2$", "Knowing the rate of the first transmission, $R_{1}$ , the optimization problem (REF ) may be solved analytically for $K=2$ using (REF ) $&\\displaystyle {\\hat{\\eta }^{\\textnormal {xp}}_2=\\max _{ R_{2}(I_{1})} \\frac{{E}\\Big [R_{1}{I}\\big [{I_{1}\\ge R_{1}}\\big ]\\Big ]}{1+f_{1}}+}\\nonumber \\\\&\\displaystyle {\\frac{{E}\\Big [(R_{1}+ R_{2}(I_{1})){I}\\big [{I_{1}\\le R_{1} \\wedge I^\\Sigma _2\\ge R_{1}+R_{2}(I_{1})}\\big ]\\Big ]}{1+f_{1}}}.$ Since $f_{1}$ is independent of $R_{2}(\\cdot )$ , solving (REF ) is equivalent to finding, for each value of $I_{1}<R_{1}$ , the optimal $R_{2}(\\cdot )$ as follows $R_{2}(I_{1})=\\mathop {\\mathrm {argmax}}_{R}~(R_{1}+ R) \\cdot F_{I_{2}}^{\\textrm {c}}(R_{1}+R-I_{1}).$ which is a one-dimension optimization problem, that can be solved analytically, provided $F_{I_{2}}^{\\textrm {c}}(\\cdot )$ is known.", "In the case of Gaussian codebook, i.e., when the mi is given by $I_{k}=\\log _{2}(1+\\mathsf {snr}_{k})$ , the optimal rate adaptation policy is given by the following closed-form $R_{2}(I_{1})=\\max \\big (0,\\frac{W(2^{I_{1}} \\overline{\\mathsf {snr}})}{\\log (2)}-R_{1}\\big ),$ where $W(.", ")$ is Lambert $W$ function defined as the solution of $x=W(x)\\mathrm {e}^{W(x)}$ ." ] ]	1606.05182
[ [ "Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex\n Optimization" ], [ "Abstract We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\\ast g$.", "This problem, known as {\\em blind deconvolution}, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications.", "A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima.", "We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on $f$ and $g$.", "The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise.", "To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions.", "Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework." ], [ "Introduction", "Suppose we are given the convolution of two signals, $y= f\\ast g$ .", "When, under which conditions, and how can we reconstruct $f$ and $g$ from the knowledge of $y$ if both $f$ and $g$ are unknown?", "This challenging problem, commonly referred to as blind deconvolution problem, arises in many areas of science and technology, including astronomy, medical imaging, optics, and communications engineering, see e.g.", "[14], [35], [9], [40], [5], [23], [10], [42].", "Indeed, the quest for finding a fast and reliable algorithm for blind deconvolution has confounded researchers for many decades.", "It is clear that without any additional assumptions, the blind deconvolution problem is ill-posed.", "One common and useful assumption is to stipulate that $f$ and $g$ belong to known subspaces [1], [25], [22], [27].", "This assumption is reasonable in various applications, provides flexibility and at the same time lends itself to mathematical rigor.", "We also adopt this subspace assumption in our algorithmic framework (see Section for details).", "But even with this assumption, blind deconvolution is a very difficult non-convex optimization problem that suffers from an overabundance of local minima, making its numerical solution rather challenging.", "In this paper, we present a numerically efficient blind deconvolution algorithm that converges geometrically to the optimal solution.", "Our regularized gradient descent algorithm comes with rigorous mathematical convergence guarantees.", "The number of measurements required for the algorithm to succeed is only slightly larger than the information theoretic minimum.", "Moreover, our algorithm is also robust against noise.", "To the best of our knowledge, the proposed algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions." ], [ "State of the art", "Since blind deconvolution problems are ubiquitous in science and engineering, it is not surprising that there is extensive literature on this topic.", "It is beyond the scope of this paper to review the existing literature; instead we briefly discuss those results that are closest to our approach.", "We gladly acknowledge that those papers that are closest to ours, namely [1], [7], [17], [34], are also the ones that greatly influenced our research in this project.", "In the inspiring article [1], Ahmed, Recht, and Romberg develop a convex optimization framework for blind deconvolution.", "The formal setup of our blind deconvolution problem follows essentially their setting.", "Using the meanwhile well-known lifting trick, [1] transforms the blind deconvolution problem into the problem of recovering a rank-one matrix from an underdetermined system of linear equations.", "By replacing the rank condition by a nuclear norm condition, the computationally infeasible rank minimization problem turns into a convenient convex problem.", "The authors provide explicit conditions under which the resulting semidefinite program is guaranteed to have the same solution as the original problem.", "In fact, the number of required measurements is not too far from the theoretical minimum.", "The only drawback of this otherwise very appealing convex optimization approach is that the computational complexity of solving the semidefinite program is rather high for large-scale data and/or for applications where computation time is of the essence.", "Overcoming this drawback was one of the main motivations for our paper.", "While [1] does suggest a fast matrix-factorization based algorithm to solve the semidefinite program, the convergence of this algorithm to the optimal solution is not established in that paper.", "The theoretical number of measurements required in [1] for the semidefinite program to succeed is essentially comparable to that for our non-convex algorithm to succeed.", "The advantage of the proposed non-convex algorithm is of course that it is dramatically faster.", "Furthermore, numerical simulations indicate that the empirically observed number of measurements for our non-convex approach is actually smaller than for the convex approach.", "The philosophy underlying the method presented in our paper is strongly motivated by the non-convex optimization algorithm for phase retrieval proposed in [7], see also [11], [3].", "In the pioneering paper [7] the authors use a two-step approach: (i) Construct in a numerically efficient manner a good initial guess; (ii) Based on this initial guess, show that simple gradient descent will converge to the true solution.", "Our paper follows a similar two-step scheme.", "At first glance one would assume that many of the proof techniques from [7] should carry over to the blind deconvolution problem.", "Alas, we quickly found out that despite some general similarities between the two problems, phase retrieval and blind deconvolution are indeed surprisingly different.", "At the end, we mainly adopted some of the general “proof principles” from [7] (for instance we also have a notion of local regularity condition - although it deviates significantly from the one in [7]), but the actual proofs are quite different.", "For instance, in [7] and [11] convergence of the gradient descent algorithm is shown by directly proving that the distance between the true solution and the iterates decreases.", "The key conditions (a local regularity condition and a local smoothness condition) are tailored to this aim.", "For the blind deconvolution problem we needed to go a different route.", "We first show that the objective function decreases during the iterations and then use a certain local restricted isometry property to transfer this decrease to the iterates to establish convergence to the solution.", "We also gladly acknowledge being influenced by the papers [17], [16] by Montanari and coauthors on matrix completion via non-convex methods.", "While the setup and analyzed problems are quite different from ours, their approach informed our strategy in various ways.", "In [17], [16], the authors propose an algorithm which comprises a two-step procedure.", "First, an initial guess is computed via a spectral method, and then a nonconvex problem is formulated and solved via an iterative method.", "The authors prove convergence to a low-rank solution, but do not establish a rate of convergence.", "As mentioned before, we also employ a two-step strategy.", "Moreover, our approach to prove stability of the proposed algorithm draws from ideas in [16].", "We also benefitted tremendously from [34].", "In that paper, Sun and Luo devise a non-convex algorithm for low-rank matrix completion and provide theoretical guarantees for convergence to the correct solution.", "We got the idea of adding a penalty term to the objective function from [34] (as well as from [17]).", "Indeed, the particular structure of our penalty term closely resembles that in [34].", "In addition, the introduction of the various neighborhood regions around the true solution, that are used to eventually characterize a “basin of attraction”, stems from [34].", "These correspondences may not be too surprising, given the connections between low-rank matrix completion and blind deconvolution.", "Yet, like also discussed in the previous paragraph, despite some obvious similarities between the two problems, it turned out that many steps and tools in our proofs differ significantly from those in [34].", "Also this should not come as a surprise, since the measurement matrices and setup differ significantlyAnyone who has experience in the proofs behind compressive sensing and matrix completion is well aware of the substantial challenges one can already face by “simply” changing the sensing matrix from, say, a Gaussian random matrix to one with less randomness..", "Moreover, unlike [34], our paper also provides robustness guarantees for the case of noisy data.", "Indeed, it seems plausible that some of our techniques to establish robustness against noise are applicable to the analysis of various recent matrix completion algorithms, such as e.g. [34].", "We briefly discuss other interesting papers that are to some extent related to our work.", "[22] proposes a projected gradient descent algorithm based on matrix factorizations and provide a convergence analysis to recover sparse signals from subsampled convolution.", "However, this projection step can be hard to implement, which does impact the efficiency and practical use of this method.", "As suggested in [22], one can avoid this expensive projection step by resorting to a heuristic approximate projection, but then the global convergence is not fully guaranteed.", "On the other hand, the papers [25], [24] consider identifiability issue of blind deconvolution problem with both $f$ and $g$ in random linear subspaces and achieve nearly optimal result of sampling complexity in terms of information theoretic limits.", "Very recently, [15] improved the result from [25], [24] by using techniques from algebraic geometry.", "The past few years have also witnessed an increasing number of excellent works other than blind deconvolution but related to nonconvex optimization [32], [41], [31], [3], [21].", "The paper [37] analyzes the problem of recovering a low-rank positive semidefinite matrix from linear measurements via a gradient descent algorithm.", "The authors assume that the measurement matrix fulfills the standard and convenient restricted isometry property, a condition that is not suitable for the blind deconvolution problem (besides the fact that the positive semidefinite assumption is not satisfied in our setting).", "In [12], Chen and Wainwright study various the solution of low-rank estimation problems by projected gradient descent.", "The very recent paper [43] investigates matrix completion for rectangular matrices.", "By “lifting”, they convert the unknown matrix into a positive semidefinite one and apply matrix factorization combined with gradient descent to reconstruct the unknown entries of the matrix.", "[4] considers an interesting blind calibration problem with a special type of measurement matrix via nonconvex optimization.", "Besides some general similarities, there is little overlap of the aforementioned papers with our framework.", "Finally, a convex optimization approach to blind deconvolution and self-calibration that extends the work of [1] can be found in [27], while [26] also covers the joint blind deconvolution-blind demixing problem." ], [ "Organization of our paper", "This paper is organized as follows.", "We introduce some notation used throughout the paper in the remainder of this section.", "The model setup and problem formulation are presented in Section .", "Section describes the proposed algorithm and our main theoretical result establishing the convergence of our algorithm.", "Numerical simulations can be found in Section .", "Section is devoted to the proof of the main theorem.", "Since the proof is quite involved, we have split this section into several subsections.", "Some auxiliary results are collected in the Appendix." ], [ "Notation", "We introduce notation which will be used throughout the paper.", "Matrices and vectors are denoted in boldface such as $Z$ and $z$ .", "The individual entries of a matrix or a vector are denoted in normal font such as $Z_{ij}$ or $z_i.$ For any matrix $Z$ , $\\Vert Z\\Vert _$ denotes its nuclear norm, i.e., the sum of its singular values; $\\Vert Z\\Vert $ denotes its operator norm, i.e., the largest singular value, and $\\Vert Z\\Vert _F$ denotes its the Frobenius norm, i.e., $\\Vert Z\\Vert _F =\\sqrt{\\sum _{ij} \|Z_{ij}\|^2 }$ .", "For any vector $z$ , $\\Vert z\\Vert $ denotes its Euclidean norm.", "For both matrices and vectors, $Z^\\top $ and $z^\\top $ stand for the transpose of $Z$ and $z$ respectively while $Z^$ and $z^$ denote their complex conjugate transpose.", "We equip the matrix space $\\hbox{{C}}^{K\\times N}$ with the inner product defined as $\\left\\langle U, V\\right\\rangle : =\\text{Tr}(U^V).$ A special case is the inner product of two vectors, i.e., $\\left\\langle u, v\\right\\rangle = \\text{Tr}(u^v) = u^v.$ For a given vector $v$ , $\\operatorname{diag}(v)$ represents the diagonal matrix whose diagonal entries are given by the vector $v$ .", "For any $z\\in \\hbox{{R}}$ , denote $z_+$ as $z_+ = \\frac{z +\|z\|}{2}.$ $C$ is an absolute constant and $C_{\\gamma }$ is a constant which depends linearly on $\\gamma $ , but on no other parameters." ], [ "Problem setup", "We consider the blind deconvolution model $y= \\mathbf {f} \\ast g+ n,$ where $y$ is given, but $f$ and $g$ are unknown.", "Here “$\\ast $ \" denotes circular convolutionAs usual, ordinary convolution can be well approximated by circulant convolution, as long as the function $f$ decays sufficiently fast [30].. We will usually consider $f$ as the “blurring function” and $g$ as the signal of interest.", "It is clear that without any further assumption it is impossible to recover $f$ and $g$ from $y$ .", "We want to impose conditions on $f$ and $g$ that are realistic, flexible, and not tied to one particular application (such as, say, image deblurring).", "At the same time, these conditions should be concrete enough to lend themselves to meaningful mathematical analysis.", "A natural setup that fits these demands is to assume that $f$ and $g$ belong to known linear subspaces.", "Concerning the blurring function, it is reasonable in many applications to assume that $f$ is either compactly supported or that $f$ decays sufficiently fast so that it can be well approximated by a compactly supported function.", "Therefore, we assume that $\\mathbf {f}\\in \\hbox{{C}}^L$ satisfies $\\mathbf {f}: =\\begin{bmatrix}h\\\\ {\\bf 0}_{L-K}\\end{bmatrix}$ where $h\\in \\hbox{{C}}^K$ , i.e., only the first $K$ entries of $f$ are nonzero and $f_l = 0$ for all $l = K+1, K+2, \\ldots , L$ .", "Concerning the signal of interest, we assume that $g$ belongs to a linear subspace spanned by the columns of a known matrix $C$ , i.e., $g= C\\bar{x}$ for some matrix $C$ of size $L \\times N$ .", "Here we use $\\bar{x}$ instead of $x$ for the simplicity of notation later.", "For theoretical purposes we assume that $C$ is a Gaussian random matrix.", "Numerical simulations suggest that this assumption is clearly not necessary.", "For example, we observed excellent performance of the proposed algorithm also in cases when $C$ represents a wavelet subspace (appropriate for images) or when $C$ is a Hadamard-type matrix (appropriate for communications).", "We hope to address these other, more realistic, choices for $C$ in our future research.", "Finally, we assume that $n\\sim \\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2}I_L) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2}I_L) $ as the additive white complex Gaussian noise where $d_0 = \\Vert h_0\\Vert \\Vert x_0\\Vert $ and $h_0$ and $x_0$ are the true blurring function and the true signal of interest, respectively.", "In that way $\\sigma ^{-2}$ actually serves as a measure of SNR (signal to noise ratio).", "For our theoretical analysis as well as for numerical purposes, it is much more convenient to express (REF ) in the Fourier domain, see also [1].", "To that end, let $F$ be the $L\\times L$ unitary Discrete Fourier Transform (DFT) matrix and let the $L \\times K$ matrix $B$ be given by the first $K$ columns of $F$ (then $B^{} B= I_{K}$ ).", "By applying the scaled DFT matrix $\\sqrt{L}F$ to both sides of (REF ) we get $\\sqrt{L}F{y}= \\operatorname{diag}(\\sqrt{L} F\\mathbf {f} ) (\\sqrt{L}Fg) + \\sqrt{L}Fn,$ which follows from the property of circular convolution and Discrete Fourier Transform.", "Here, $ \\operatorname{diag}(\\sqrt{L} F\\mathbf {f} ) (\\sqrt{L}Fg)= (\\sqrt{L} F\\mathbf {f} )\\odot (\\sqrt{L}Fg)$ where $\\odot $ denotes pointwise product.", "By definition of $f$ in (REF ), we have $F\\mathbf {f} = Bh.$ Therefore, (REF ) becomes, $\\frac{1}{\\sqrt{L}}\\hat{y} = \\operatorname{diag}(Bh) \\overline{Ax} + \\frac{1}{\\sqrt{L}}Fn$ where $\\overline{A} = FC$ (we use $\\overline{A}$ instead of $A$ simply because it gives rise to a more convenient notation later, see e.g.", "(REF )).", "Note that if $C$ is a Gaussian matrix, then so is $FC$ .", "Furthermore, $e= \\frac{1}{\\sqrt{L}} Fn\\sim \\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L) $ is again a complex Gaussian random vector.", "Hence by replacing $\\frac{1}{\\sqrt{L}}\\hat{y}$ in (REF ) by $y$ , we arrive with a slight abuse of notation at $y= \\operatorname{diag}(Bh) \\overline{Ax} + e.$ For the remainder of the paper, instead of considering the original blind deconvolution problem (REF ), we focus on its mathematically equivalent version (REF ), where $h\\in \\hbox{{C}}^{K\\times 1}$ , $x\\in \\hbox{{C}}^{N\\times 1}$ , $y\\in \\hbox{{C}}^{L\\times 1}$ , $B\\in \\hbox{{C}}^{L\\times K}$ and $A\\in \\hbox{{C}}^{L\\times N}$ .", "As mentioned before, $h_0$ and $x_0$ are the ground truth.", "Our goal is to recover $h_0$ and $x_0$ when $B$ , $A$ and $y$ are given.", "It is clear that if $(h_0, x_0)$ is a solution to the blind deconvolution problem.", "then so is $(\\alpha h_0, \\alpha ^{-1} x_0)$ for any $\\alpha \\ne 0$ .", "Thus, all we can hope for in the absence of any further information, is to recover a solution from the equivalence class $(\\alpha h_0, \\alpha ^{-1}x_0), \\alpha \\ne 0$ .", "Hence, we can as well assume that $\\Vert h_0\\Vert = \\Vert x_0\\Vert := \\sqrt{d}_0$ .", "As already mentioned, we choose $B$ to be the “low-frequency\" discrete Fourier matrix, i.e., the first $K$ columns of an $L\\times L$ unitary DFT (Discrete Fourier Transform) matrix.", "Moreover, we choose $A$ to be an $L\\times N$ complex Gaussian random matrix, i.e., $A_{ij} \\sim \\mathcal {N}\\left(0, \\frac{1}{2}\\right) + \\mathrm {i}\\mathcal {N}\\left(0, \\frac{1}{2}\\right),$ where “$\\mathrm {i}$ \" is the imaginary unit.", "We define the matrix-valued linear operator $\\mathcal {A}(\\cdot )$ via $\\mathcal {A}: \\mathcal {A}(Z) = \\lbrace b_l^Za_l\\rbrace _{l=1}^L,$ where $b_l$ denotes the $l$ -th column of $B^$ and $a_l$ is the $l$ -th column of $A^.$ Immediately, we have $\\sum _{l=1}^L b_lb_l^* = B^B= I_K$ , $\\Vert b_l\\Vert = \\frac{K}{L}$ and $\\operatorname{E}(a_la_l^) = I_N$ for all $1\\le l\\le L$ .", "This is essentially a smart and popular trick called “lifting\" [8], [1], [27], which is able to convert a class of nonlinear models into linear models at the costs of increasing the dimension of the solution space.", "It seems natural and tempting to recover $(h_0, x_0)$ obeying (REF ) by solving the following optimization problem $\\min _{(h, x)} \\,\\, F(h, x),$ where $F(h, x) := \\Vert \\operatorname{diag}(Bh)\\overline{Ax} - y\\Vert ^2 = \\Vert \\mathcal {A}(hx^* - h_0x_0^) - e\\Vert ^2.$ We also let $F_0(h, x) : = \\Vert \\mathcal {A}(hx^ - h_0x_0^)\\Vert _F^2$ which is a special case of $F(h, x)$ when $e= 0.$ Furthermore, we define $\\delta (h, x)$ , an important quantity throughout our discussion, via $\\delta (h, x) : = \\frac{\\Vert hx^ - h_0x_0^\\Vert _F}{d_0}.$ When there is no danger of ambiguity, we will often denote $\\delta (h, x)$ simply by $\\delta $ .", "But let us remember that $\\delta (h, x)$ is always a function of $(h, x)$ and measures the relative approximation error of $(h, x)$ .", "Obviously, minimizing (REF ) becomes a nonlinear least square problem, i.e., one wants to find a pair of vectors $(h,x)$ or a rank-1 matrix $hx^$ which fits the measurement equation in (REF ) best.", "Solving (REF ) is a challenging optimization problem since it is highly nonconvex and most of the available algorithms, such as alternating minimization and gradient descent, may suffer from getting easily trapped in some local minima.", "Another possibility is to consider a convex relaxation of (REF ) at the cost of having to solve an expensive semidefinite program.", "In the next section we will describe how to avoid this dilemma and design an efficient gradient descent algorithm that, under reasonable conditions, will always converge to the true solution." ], [ "Algorithm and main result", "In this section we introduce our algorithm as well as our main theorems which establish convergence of the proposed algorithm to the true solution.", "As mentioned above, in a nutshell our algorithm consists of two parts: First we use a carefully chosen initial guess, and second we use a variation of gradient descent, starting at the initial guess to converge to the true solution.", "One of the most critical aspects is of course that we must avoid getting stuck in local minimum or saddle point.", "Hence, we need to ensure that our iterates are inside some properly chosen basin of attraction of the true solution.", "The appropriate characterization of such a basin of attraction requires some diligence, a task that will occupy us in the next subsection.", "We will then proceed to introducing our algorithm and analyzing its convergence." ], [ "Building a basin of attraction", "The road toward designing a proper basin of attraction is basically paved by three observations, described below.", "These observations prompt us to introduce three neighborhoods (inspired by [17], [34]), whose intersection will form the desired basin of attraction of the solution.", "Observation 1 - Nonuniqueness of the solution: As pointed out earlier, if $(h,x)$ is a solution to (REF ), then so is $(\\alpha h, \\alpha ^{-1}x)$ for any $\\alpha \\ne 0$ .", "Thus, without any prior information about $\\Vert h\\Vert $ and/or $\\Vert x\\Vert $ , it is clear that we can only recover the true solution up to such an unknown constant $\\alpha $ .", "Fortunately, this suffices for most applications.", "From the viewpoint of numerical stability however, we do want to avoid, while $\\Vert h\\Vert \\Vert x\\Vert $ remains bounded, that $\\Vert h\\Vert \\rightarrow 0$ and $\\Vert x\\Vert \\rightarrow \\infty $ (or vice versa).", "To that end we introduce the following neighborhood: $ \\mathcal {N}_{d_0}:= \\lbrace (h, x) : \\Vert h\\Vert \\le 2\\sqrt{d_0}, \\Vert x\\Vert \\le 2\\sqrt{d_0} \\rbrace .$ (Recall that $d_0 = \\Vert h_0\\Vert \\Vert x_0\\Vert $ .)", "Observation 2 - Incoherence: Our numerical simulations indicate that the number of measurements required for solving the blind deconvolution problem with the proposed algorithm does depend (among others) on how much $h_0$ is correlated with the rows of the matrix $B$ — the smaller the correlation the better.", "A similar effect has been observed in blind deconvolution via convex programming [1], [26].", "We quantify this property by defining the incoherence between the rows of $B$ and $h_0$ via $\\mu ^2_h = \\frac{L \\Vert Bh_0\\Vert _{\\infty }^2}{\\Vert h_0\\Vert ^2}.$ It is easy to see that $1\\le \\mu _h^2 \\le K$ and both lower and upper bounds are tight; i.e., $\\mu ^2_h = K$ if $h_0$ is parallel to one of $\\lbrace b_l\\rbrace _{l=1}^L$ and $\\mu ^2_h = 1$ if $h_0$ is a 1-sparse vector of length $K$ .", "Note that in our setup, we assume that $A$ is a random matrix and $x_0$ is fixed, thus with high probability, $x_0$ is already sufficiently incoherent with the rows of $A$ and thus we only need to worry about the incoherence between $B$ and $h_0$ .", "It should not come as a complete surprise that the incoherence between $h_0$ and the rows of $B$ is important.", "The reader may recall that in matrix completion [6], [28] the left and right singular vectors of the solution cannot be “too aligned\" with those of the measurement matrices.", "A similar philosophy seems to apply here.", "Being able to control the incoherence of the solution is instrumental in deriving rigorous convergence guarantees of our algorithm.", "For that reason, we introduce the neighborhood $ \\mathcal {N}_{\\mu }:= \\lbrace h: \\sqrt{L} \\Vert Bh\\Vert _{\\infty } \\le 4\\sqrt{d_0}\\mu \\rbrace , $ where $\\mu _h \\le \\mu $ .", "Observation 3 - Initial guess: It is clear that due to the non-convexity of the objective function, we need a carefully chosen initial guess.", "We quantify the distance to the true solution via the following neighborhood $ \\mathcal {N}_{\\varepsilon }:= \\lbrace (h, x) : \\Vert hx^* - h_0x_0^\\Vert _F \\le \\varepsilon d_0 \\rbrace .", "$ where $\\varepsilon $ is a predetermined parameter in $(0, \\frac{1}{15}]$ .", "It is evident that the true solution $(h_0,x_0) \\in \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ .", "Note that $(h, x)\\in \\mathcal {N}_{d_0}\\bigcap \\mathcal {N}_{\\varepsilon }$ implies $\\Vert hx^\\Vert \\ge (1 - \\varepsilon )d_0$ and $\\frac{1}{\\Vert h\\Vert } \\le \\frac{\\Vert x\\Vert }{(1 - \\varepsilon )d_0} \\le \\frac{2}{(1 - \\varepsilon )\\sqrt{d_0}}$ .", "Therefore, for any element $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon },$ its incoherence can be well controlled by $\\frac{\\sqrt{L}\\Vert Bh\\Vert _{\\infty }}{\\Vert h\\Vert } \\le \\frac{4\\sqrt{d_0}\\mu }{ \\Vert h\\Vert } \\le \\frac{ 8\\mu }{1 - \\varepsilon }.$" ], [ "Objective function and key ideas of the algorithm", "Our approach consists of two parts: We first construct an initial guess that is inside the “basin of attraction” $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ .", "We then apply a carefully regularized gradient descent algorithm that will ensure that all the iterates remain inside $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ .", "Due to the difficulties of directly projecting onto $ \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ (the neigbourhood $ \\mathcal {N}_{\\varepsilon }$ is easier to manage) we add instead a regularizer $G(h,x)$ to the objective function $F(h, x)$ to enforce that the iterates remain inside $ \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ .", "While the idea of adding a penalty function to control incoherence is proposed in different forms to solve matrix completion problems, see e.g., [33], [20], our version is mainly inspired by [34], [17].", "Hence, we aim to minimize the following regularized objective function to solve the blind deconvolution problem: $\\widetilde{F}(h, x) = F(h, x) + G(h, x),$ where $F(h, x)$ is defined in (REF ) and $G(h, x)$ , the penalty function, is of the form $G(h, x) = \\rho \\left[ G_0\\left(\\frac{\\Vert h\\Vert ^2}{2d}\\right) + G_0\\left(\\frac{\\Vert x\\Vert ^2}{2d}\\right) + \\sum _{l=1}^L G_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) \\right],$ where $G_0(z) = \\max \\lbrace z - 1, 0 \\rbrace ^2$ and $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2$ .", "Here we assume $\\frac{9}{10}d_0 \\le d \\le \\frac{11}{10}d_0$ and $\\mu \\ge \\mu _h$ .", "The idea behind this, at first glance complicated, penalty function is quite simple.", "The first two terms in (REF ) enforce the projection of $(h, x)$ onto $ \\mathcal {N}_{d_0}$ while the last term is related to $ \\mathcal {N}_{\\mu }$ ; it will be shown later that any $(h, x)\\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}\\bigcap \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }$ gives $G(h, x) = 0$ if $\\frac{9}{10}d_0 \\le d\\le \\frac{11}{10}d_0$ .", "Since $G_0(z)$ is a truncated quadratic function, it is obvious that $G_0^{\\prime }(z) = 2\\sqrt{G_0(z)}$ and $G(h, x)$ is a continuously differentiable function.", "Those two properties play a crucial role in proving geometric convergence of our algorithm presented later.", "Another important issue concerns the selection of parameters.", "We have three unknown parameters in $G(h, x)$ , i.e., $\\rho $ , $d$ , and $\\mu $ .", "Here, $d$ can be obtained via Algorithm 1 and $\\frac{9}{10}d_0 \\le d\\le \\frac{11}{10}d_0$ is guaranteed by Theorem REF ; $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2 \\approx d^2 + 2\\sigma ^2 d_0^2$ because $\\Vert e\\Vert ^2 \\sim \\frac{\\sigma ^2 d_0^2}{2L}\\chi ^2_{2L}$ and $\\Vert e\\Vert ^2$ concentrates around $\\sigma ^2 d_0^2$ .", "In practice, $\\sigma ^2$ which is the inverse of the SNR, can often be well estimated.", "Regarding $\\mu $ , we require $\\mu _h \\le \\mu $ in order to make sure that $h_0\\in \\mathcal {N}_{\\mu }$ .", "It will depend on the specific application how well one can estimate $\\mu _h^2$ .", "For instance, in wireless communications, a very common channel model for $h_0$ is to assume a Rayleigh fading model [36], i.e., $h_0 \\sim \\mathcal {N}(0, \\frac{\\sigma ^2_h}{2}I_K) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2_h}{2}I_K)$ .", "In that case it is easy to see that $\\mu _h^2 = \\mathcal {O}(\\log L).$" ], [ "Wirtinger derivative of the objective function and algorithm", "Since we are dealing with complex variables, instead of using regular derivatives it is more convenient to utilize Wirtinger derivatives For any complex function $f(z)$ where $z= u+ \\mathrm {i}v\\in \\hbox{{C}}^n$ and $u, v\\in \\hbox{{R}}^n$ , the Wirtinger derivatives are defined as $\\frac{\\partial f}{\\partial z} = \\frac{1}{2}\\left( \\frac{\\partial f}{\\partial u} - \\mathrm {i}\\frac{\\partial f}{\\partial v}\\right)$ and $\\frac{\\partial f}{\\partial \\bar{z}} = \\frac{1}{2}\\left( \\frac{\\partial f}{\\partial u} + \\mathrm {i}\\frac{\\partial f}{\\partial v}\\right).$ Two important examples used here are $\\frac{\\partial \\Vert z\\Vert ^2}{\\partial \\bar{z}} = z$ and $\\frac{\\partial (z^w) }{\\partial \\bar{z}} = w$ ., which has become increasingly popular since [7], [3].", "Note that $\\widetilde{F}$ is a real-valued function and hence we only need to consider the derivative of $\\widetilde{F}$ with respect to $\\bar{h}$ and $\\bar{x}$ and the corresponding updates of $h$ and $x$ because a simple relation holds, i.e., $\\frac{\\partial \\widetilde{F}}{\\partial \\bar{h}} = \\overline{ \\frac{\\partial \\widetilde{F}}{\\partial h}} \\in \\hbox{{C}}^{K\\times 1}, \\quad \\frac{\\partial \\widetilde{F}}{\\partial \\bar{x}} = \\overline{ \\frac{\\partial \\widetilde{F}}{\\partial x}} \\in \\hbox{{C}}^{N\\times 1}.$ In particular, we denote $\\nabla \\widetilde{F}_{h} := \\frac{\\partial \\widetilde{F}}{\\partial \\bar{h}}$ and $\\nabla \\widetilde{F}_{x} := \\frac{\\partial \\widetilde{F}}{\\partial \\bar{x}}$ .", "We also introduce the adjoint operator of $\\mathcal {A}: \\hbox{{C}}^L \\rightarrow \\hbox{{C}}^{K\\times N}$ , given by $\\mathcal {A}^(z) = \\sum _{l=1}^L z_l b_la_l^.$ Both $\\nabla \\widetilde{F}_{h}$ and $\\nabla \\widetilde{F}_{x}$ can now be expressed as $\\nabla \\widetilde{F}_{h} = \\nabla F_{h} + \\nabla G_{h}, \\quad \\nabla \\widetilde{F}_{x} = \\nabla F_{x} + \\nabla G_{x},$ where each component yields $\\nabla F_{h} & = & \\mathcal {A}^(\\mathcal {A}(hx^) - y)x= \\mathcal {A}^(\\mathcal {A}(hx^ - h_0x_0^) -e)x, \\\\\\nabla F_{x} & = & [\\mathcal {A}^(\\mathcal {A}(hx^) - y)]^h= [\\mathcal {A}^(\\mathcal {A}(hx^ - h_0x_0^) - e)]^h, \\\\\\nabla G_{h}& = & \\frac{\\rho }{2d}\\left[G^{\\prime }_0\\left(\\frac{\\Vert h\\Vert ^2}{2d}\\right) h+ \\frac{L}{4\\mu ^2} \\sum _{l=1}^L G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) b_lb_l^h\\right], \\\\\\nabla G_{x} & = & \\frac{\\rho }{2d} G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) x.", "$ Our algorithm consists of two steps: initialization and gradient descent with constant stepsize.", "The initialization is achieved via a spectral method followed by projection.", "The idea behind spectral method is that $\\operatorname{E}\\mathcal {A}^(y) = \\operatorname{E}\\mathcal {A}^\\mathcal {A}(h_0x_0^) + \\operatorname{E}\\mathcal {A}^(e) = h_0x_0^$ and hence one can hope that the leading singular value and vectors of $\\mathcal {A}^(y)$ can be a good approximation of $d_0$ and $(h_0, x_0)$ respectively.", "The projection step ensures $u_0\\in \\mathcal {N}_{\\mu }$ , which the spectral method alone might not guarantee.", "We will address the implementation and computational complexity issue in Section .", "[h!]", "Initialization via spectral method and projection [1] Compute $\\mathcal {A}^(y).$ Find the leading singular value, left and right singular vectors of $\\mathcal {A}^(y)$ , denoted by $d$ , $\\hat{h}_0$ and $\\hat{x}_0$ respectively.", "Solve the following optimization problem: $u_0 := \\text{argmin}_{z} \\Vert z- \\sqrt{d}\\hat{h}_0\\Vert ^2, \\quad \\operatorname{\\text{subject to}}\\sqrt{L}\\Vert Bz\\Vert _{\\infty } \\le 2\\sqrt{d}\\mu $ and $v_0 = \\sqrt{d}\\hat{x}_0.$ Output: $(u_0, v_0).$ [h!]", "Wirtinger gradient descent with constant stepsize $\\eta $ [1] Initialization: obtain $(u_0, v_0)$ via Algorithm REF .", "$t = 1, 2, \\dots , $ $u_t = u_{t-1} - \\eta \\nabla \\widetilde{F}_{h}(u_{t-1}, v_{t-1})$ $v_t = v_{t-1} - \\eta \\nabla \\widetilde{F}_{x}(u_{t-1}, v_{t-1})$" ], [ "Main results", "Our main finding is that with a diligently chosen initial guess $(u_0, v_0)$ , simply running gradient descent to minimize the regularized non-convex objective function $\\widetilde{F}(h,x)$ will not only guarantee linear convergence of the sequence $(u_t, v_t)$ to the global minimum $(h_0, x_0)$ in the noiseless case, but also provide robust recovery in the presence of noise.", "The results are summarized in the following two theorems.", "Theorem 3.1 The initialization obtained via Algorithm REF satisfies $(u_0, v_0) \\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}\\bigcap \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu } \\bigcap \\mathcal {N}_{\\frac{2}{5}\\varepsilon },$ and $\\frac{9}{10}d_0 \\le d\\le \\frac{11}{10}d_0$ holds with probability at least $1 - L^{-\\gamma }$ if the number of measurements satisfies $L \\ge C_{\\gamma } (\\mu _h^2 + \\sigma ^2)\\max \\lbrace K, N\\rbrace \\log ^2 (L)/\\varepsilon ^2.$ Here $\\varepsilon $ is any predetermined constant in $(0, \\frac{1}{15}]$ , and $C_{\\gamma }$ is a constant only linearly depending on $\\gamma $ with $\\gamma \\ge 1$ .", "The proof of Theorem REF is given in Section REF .", "While the initial guess is carefully chosen, it is in general not of sufficient accuracy to already be used as good approximation to the true solution.", "The following theorem establishes that as long as the initial guess lies inside the basin of attraction of the true solution, regularized gradient descent will indeed converge to this solution (or to a solution nearby in case of noisy data).", "Theorem 3.2 Consider the model in (REF ) with the ground truth $(h_0, x_0)$ , $\\mu ^2_h = \\frac{L\\Vert Bh_0\\Vert _{\\infty }^2}{\\Vert h_0\\Vert ^2}\\le \\mu ^2$ and the noise $e\\sim \\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L)$ .", "Assume that the initialization $(u_0, v_0)$ belongs to $\\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}\\bigcap \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }\\bigcap \\mathcal {N}_{\\frac{2}{5}\\varepsilon }$ and that $L \\ge C_{\\gamma } (\\mu ^2 + \\sigma ^2)\\max \\lbrace K, N\\rbrace \\log ^2 (L)/\\varepsilon ^2,$ Algorithm REF will create a sequence $(u_t, v_t)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ which converges geometrically to $(h_0, x_0)$ in the sense that with probability at least $1 - 4L^{-\\gamma } - \\frac{1}{\\gamma }\\exp (-(K + N))$ , there holds $\\max \\lbrace \\sin \\angle (u_t, h_0), \\sin \\angle (v_t, x_0)\\rbrace \\le \\frac{1}{d_t}\\left( \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50\\Vert \\mathcal {A}^(e)\\Vert \\right)$ and $\|d_t - d_0\| \\le \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50 \\Vert \\mathcal {A}^(e)\\Vert ,$ where $d_t := \\Vert u_t\\Vert \\Vert v_t\\Vert $ , $\\omega > 0$ , $\\eta $ is the fixed stepsize and $\\angle (u_t, h_0)$ is the angle between $u_t$ and $h_0$ .", "Here $\\Vert \\mathcal {A}^(e)\\Vert \\le C_0\\sigma d_0 \\max \\Big \\lbrace \\sqrt{\\frac{(\\gamma + 1)\\max \\lbrace K,N\\rbrace \\log L}{L}}, \\frac{(\\gamma + 1)\\sqrt{KN}\\log ^2 L }{L} \\Big \\rbrace .$ holds with probability $1 - L^{-\\gamma }.$ Remarks: While the setup in (REF ) assumes that $B$ is a matrix consisting of the first $K$ columns of the DFT matrix, this is actually not necessary for Theorem REF .", "As the proof will show, the only conditions on $B$ are that $B^{\\ast } B= I_K$ and that the norm of the $l$ -th row of $B$ satisfies $\\Vert b_l \\Vert ^2 \\le C \\frac{K}{L}$ for some numerical constant $C$ .", "The minimum number of measurements required for our method to succeed is roughly comparable to that of the convex approach proposed in [1] (up to log-factors).", "Thus there is no price to be paid for trading a slow, convex-optimization based approach with a fast non-convex based approach.", "Indeed, numerical experiments indicate that the non-convex approach even requires a smaller number of measurements compared to the convex approach, see Section .", "The convergence rate of our algorithm is completely determined by $\\eta \\omega $ .", "Here, the regularity constant $\\omega = \\mathcal {O}(d_0)$ is specified in (REF ) and $\\eta \\le \\frac{1}{C_L}$ where $C_L = \\mathcal {O}(d_0 ( N\\log L + \\frac{\\rho L}{d_0^2 \\mu ^2}))$ .", "The attentive reader may have noted that $C_L$ depends essentially linearly on $\\frac{\\rho L}{\\mu ^2}$ , which actually reflects a tradeoff between sampling complexity (or statistical estimation quality) and computation time.", "Note that if $L$ gets larger, the number of constraints is also increasing and hence leads to a larger $C_L$ .", "However, this issue can be solved by choosing parameters smartly.", "Theorem REF tells us that $\\mu ^2$ should be roughly between $\\mu _h^2$ and $\\frac{L}{(K + N)\\log ^2L}$ .", "Therefore, by choosing $\\mu ^2 = \\mathcal {O}(\\frac{L}{(K + N)\\log ^2L})$ and $\\rho \\approx d^2 + 2\\Vert e\\Vert ^2$ , $C_L$ is optimized and $\\eta \\omega = \\mathcal {O}(((1 + \\sigma ^2)(K + N) \\log ^2L)^{-1}),$ which is shown in details in Section REF .", "Relations (REF ) and (REF ) are basically equivalent to the following: $\\Vert u_tv_t^ - h_0x_0^\\Vert _F \\le \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50 \\Vert \\mathcal {A}^(e)\\Vert ,$ which says that $(u_t, v_t)$ converges to an element of the equivalence class associated with the true solution $(h_0,x_0)$ (up to a deviation governed by the amount of additive noise).", "The matrix $\\mathcal {A}^(e) = \\sum _{l=1}^L e_l b_la_l^$ , as a sum of $L$ rank-1 random matrices, has nice concentration of measure properties under the assumption of Theorem REF .", "Asymptotically, $\\Vert \\mathcal {A}^(e)\\Vert $ converges to 0 with rate $\\mathcal {O}(L^{-1/2})$ , which will be justified in Lemma REF of Section REF (see also [3]).", "Note that $F(h, x) = \\Vert e\\Vert ^2 + \\Vert \\mathcal {A}(hx^ - h_0x_0^)\\Vert _F^2 - 2\\operatorname{Re}(\\left\\langle \\mathcal {A}^(e) , hx^* - h_0x_0^\\right\\rangle ).$ If one lets $L\\rightarrow \\infty $ , then $\\Vert e\\Vert ^2\\sim \\frac{\\sigma ^2 d_0^2}{2L} \\chi ^2_{2L}$ will converge almost surely to $\\sigma ^2d_0^2$ under the Law of Large Numbers and the cross term $\\operatorname{Re}(\\left\\langle hx^ - h_0x_0^, \\mathcal {A}^(e) \\right\\rangle )$ will converge to 0.", "In other words, asymptotically, $\\lim _{L\\rightarrow \\infty } F(h, x) = F_0(h, x) + \\sigma ^2 d_0^2$ for all fixed $(h, x)$ .", "This implies that if the number of measurements is large, then $F(h, x)$ behaves “almost like\" $F_0(h, x) = \\Vert \\mathcal {A}(hx^* - h_0x_0^)\\Vert ^2$ , the noiseless version of $F(h, x)$ .", "This provides the key insight into analyzing the robustness of our algorithm, which is reflected in the so-called “Robustness Condition\" in (REF ).", "Moreover, the asymptotic property of $\\mathcal {A}^(e)$ is also seen in our main result (REF ).", "Suppose $L$ is becoming larger and larger, the effect of noise diminishes and heuristically, we might just rewrite our result as $\\Vert u_tv_t^* - h_0x_0^\\Vert _F \\le \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + o_p(1)$ , which is consistent with the result without noise." ], [ "Numerical simulations", "We present empirical evaluation of our proposed gradient descent algorithm (Algorithm REF ) using simulated data as well as examples from blind deconvolution problems appearing in communications and in image processing." ], [ "Number of measurements vs size of signals", "We first investigate how many measurements are necessary in order for an algorithm to reliably recover two signals from their convolution.", "We compare Algorithm REF , a gradient descent algorithm for the sum of the loss function $F(h,x)$ and the regularization term $G(h,x)$ , with the gradient descent algorithm only applied to $F(h,x)$ and the nuclear norm minimization proposed in [1].", "These three tested algorithms are abbreviated as regGrad, Grad and NNM respectively.", "To make fair comparisons, both regGrad and Grad are initialized with the normalized leading singular vectors of $\\mathcal {A}^(y)$ , which are computed by running the power method for 50 iterations.", "Though we do not further compute the projection of $\\hat{h}_0$ for regGrad as stated in the third step of Algorithm REF , we emphasize that the projection can be computed efficiently as it is a linear programming on $K$ -dimensional vectors.", "A careful reader may notice that in addition to the computational cost for the loss function $F(h,x)$ , regGrad also requires to evaluate $G(h,x)$ and its gradient in each iteration.", "When $B$ consists of the first $K$ columns of a unitary DFT matrix, we can evaluate $\\left\\lbrace b_l^h\\right\\rbrace _{l=1}^L$ and the gradient of $\\sum _{l=1}^L G_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)$ using FFT.", "Thus the additional per iteration computational cost for the regularization term $G(h,x)$ is only $O(L\\log L)$ flops.", "The stepsizes in both regGrad and Grad are selected adaptively in each iteration via backtracking.", "As suggested by the theory, the choices for $\\rho $ and $\\mu $ in regGrad are $\\rho =d^2/100$ and $\\mu =6\\sqrt{L/(K+N)}/\\log L$ .", "We conduct tests on random Gaussian signals $h\\in \\hbox{{C}}^{K\\times 1}$ and $x\\in \\hbox{{C}}^{N\\times 1}$ with $K=N=50$ .", "The matrix $B\\in \\hbox{{C}}^{L\\times K}$ is the first $K$ columns of a unitary $L\\times L$ DFT matrix, while $A\\in \\hbox{{C}}^{L\\times N}$ is either a Gaussian random matrix or a partial Hadamard matrix with randomly selected $N$ columns and then multiplied by a random sign matrix from the left.", "When $A$ is a Gaussian random matrix, $L$ takes 16 equal spaced values from $K+N$ to $4(K+N)$ .", "When $A$ is a partial Hadamard matrix, we only test $L=2^s$ with $6\\le s\\le 10$ being integers.", "For each given triple $(K,N,L)$ , fifty random tests are conducted.", "We consider an algorithm to have successfully recovered $(h_0,x_0)$ if it returns a matrix $\\hat{X}$ which satisfies $\\frac{\\Vert \\hat{X}-h_0x_0^\\Vert _F}{\\Vert h_0x_0^\\Vert _F}<10^{-2}.$ We present the probability of successful recovery plotted against the number of measurements in Figure REF .", "Figure: Empirical phase transition curves when (a) AA is random Gaussian (b) AA is partial Hadamard.", "Horizontal axis L/(K+N)L/(K+N) andvertical axis probability of successful recovery out of 50 random tests.It can be observed that regGrad and Grad have similar performance, and both of them require a significantly smaller number of measurements than NNM to achieve successful recovery of high probability." ], [ "Number of measurements vs incoherence", "Theorem REF indicates that the number of measurements $L$ required for Algorithm REF to achieve successful recovery scales linearly with $\\mu _h^2$ .", "We conduct numerical experiments to investigate the dependence of $L$ on $\\mu _h^2$ empirically.", "The tests are conducted with $\\mu _h^2$ taking on 10 values $\\mu _h^2\\in \\lbrace 3,6,\\cdots ,30\\rbrace $ .", "For each fixed $\\mu _h^2$ , we choose $h_0$ to be a vector whose first $\\mu _h^2$ entries are 1 and the others are 0 so that its incoherence is equal to $\\mu _h^2$ when $B$ is low frequency Fourier matrix.", "Then the tests are repeated for random Gaussian matrices $A$ and random Gaussian vectors $x_0$ .", "The empirical probability of successful recovery on the $(\\mu _h^2,L)$ plane is presented in Figure REF , which suggests that $L$ does scale linearly with $\\mu _h^2$ .", "Figure: Empirical probability of successful recovery.", "Horizontal axis μ h 2 \\mu _h^2 and vertical axis LL.", "White: 100%100\\% success and black: 0%0\\% success." ], [ "A comparison when $\\mu _h^2$ is large", "While regGrad and Grad have similar performances in the simulation when $h_0$ is a random Gaussian signal (see Figure REF ), we investigate their performances on a fixed $h_0$ with a large incoherence.", "The tests are conducted for $K=N=200$ , $x_0\\in \\hbox{{C}}^{N\\times 1}$ being a random Gaussian signal, $A\\in \\hbox{{C}}^{L\\times N}$ being a random Gaussian matrix, and $B\\in \\hbox{{C}}^{L\\times K}$ being a low frequency Fourier matrix.", "The signal $h_0$ with $\\mu _h^2=100$ is formed in the same way as in Section REF ; that is, the first 100 entries of $h_0$ are one and the other entries are zero.", "The number of measurements $L$ varies from $3(K+N)$ to $8(K+N)$ .", "For each $L$ , 100 random tests are conducted.", "Figure REF shows the probability of successful recovery for regGrad and Grad.", "It can be observed that the successful recovery probability of regGrad is at least $10\\%$ larger than that of Grad when $L\\ge 6(K+N)$ .", "Figure: Empirical phase transition curves when μ h 2 \\mu _h^2 is large.", "Horizontal axis L/(K+N)L/(K+N) andvertical axis probability of successful recovery out of 50 random tests." ], [ "Robustness to additive noise", "We explore the robustness of Algorithm REF when the measurements are contaminated by additive noise.", "The tests are conducted with $K=N=100$ , $L\\in \\lbrace 500, 1000\\rbrace $ when $A$ is a random Gaussian matrix, and $L\\in \\lbrace 512,1024\\rbrace $ when $A$ is a partial Hadamard matrix.", "Tests with additive noise have the measurement vector $y$ corrupted by the vector $e=\\sigma \\cdot \\Vert y\\Vert \\cdot \\frac{\\mathbf {w}}{\\Vert \\mathbf {w}\\Vert },$ where $\\mathbf {w}\\in \\hbox{{C}}^{L\\times 1}$ is standard Gaussian random vector, and $\\sigma $ takes nine different values from $10^{-4}$ to 1.", "For each $\\sigma $ , fifty random tests are conducted.", "The average reconstruction error in dB plotted against the signal to noise ratio (SNR) is presented in Fig.", "REF .", "First the plots clearly show the desirable linear scaling between the noise levels and the relative reconstruction errors.", "Moreover, as desired, the relative reconstruction error decreases linearly on a $\\log $ -$\\log $ scale as the number of measurements $L$ increases.", "Figure: Performance of Algorithm under different SNR when (a) AA is random Gaussian (b) AA is partial Hadamard." ], [ "An example from communications", "In order to demonstrate the effectiveness of Algorithm REF for real world applications, we first test the algorithm on a blind deconvolution problem arising in communications.", "Indeed, blind deconvolution problems and their efficient numerical solution are expected to play an increasingly important role in connection with the emerging Internet-of-Things [42].", "Assume we want to transmit a signal from one place to another over a so-called time-invariant multi-path communication channel, but the receiver has no information about the channel, except its delay spread (i.e., the support of the impulse response).", "In many communication settings it is reasonable to assume that the receiver has information about the signal encoding matrix—in other words, we know the subspace $A$ to which $x_0$ belongs to.", "Translating this communications jargon into mathematical terminology, this simply means that we are dealing with a blind deconvolution problem of the form (REF ).", "These encoding matrices (or so-called spreading matrices) are often chosen to have a convenient structure that lends itself to fast computations and minimal memory requirements.", "One such choice is to let $A:=DH$ , where the $L \\times K$ matrix $H$ consists of $K$ (randomly or not randomly) chosen columns of an $L \\times L$ Hadamard matrix, premultiplied with an $L \\times L$ diagonal random sign matrix $D$ .", "Instead of a partial Hadamard matrix we could also use a partial Fourier matrix; the resulting setup would then closely resemble an OFDM transmission format, which is part of every modern wireless communication scheme.", "For the signal $x_0$ we choose a so-called QPSK scheme, i.e., each entry of $x_0\\in \\hbox{{C}}^{123\\times 1}$ takes a value from $\\lbrace 1,-1,\\mathrm {i},-\\mathrm {i}\\rbrace $ with equal probability.", "The actual transmitted signal is then $z= Ax_0$ .", "For the channel $h_0$ (the blurring function) we choose a real-world channel, courtesy of Intel Corporation.", "Here, $h_0$ represents the impulse response of a multipath time-invariant channel, it consists of 123 time samples, hence $h_0\\in \\hbox{{C}}^{123\\times 1}$ .", "Otherwise, the setup follows closely that in Sec.", "REF .", "For comparison we also include the case when $A$ is a random Gaussian matrix.", "The plots of successful recovery probability are presented in Figure REF , which shows that Algorithm REF can successfully recover the real channel $h_0$ with high probability if $L\\gtrsim 2.5(L+N)$ when $A$ is a random Gaussian matrix and if $L\\gtrsim 4.5(L+N)$ when $A$ is a partial Hadamard matrix.", "Thus our theory seems a bit pessimistic.", "It is gratifying to see that very little additional measurements are required compared to the number of unknowns in order to recover the transmitted signal.", "Figure: Empirical phase transition curves when (a) AA is random Gaussian (b) AA is partial Hadamard.", "Horizontal axis L/(K+N)L/(K+N) andvertical axis probability of successful recovery out of 50 random tests.Figure: MRI image deblurring: (a) Original 512×512512\\times 512 MRI image; (b) Blurring kernel; (c) Blurred image; (d) Initial guess; (e) Reconstructed image when the subspaces are known; (f) Reconstructed image without knowing the subspaces exactly." ], [ "An example from image processing", "Next, we test Algorithm REF on an image deblurring problem, inspired by [1].", "The observed image (Figure REF ) is a convolution of a $512\\times 512$ MRI image (Figure REF ) with a motion blurring kernel (Figure REF ).", "Since the MRI image is approximately sparse in the Haar wavelet basis, we can assume it belongs to a low dimensional subspace; that is, $g=Cx_0$ , where $g\\in \\hbox{{C}}^{L}$ with $L=262,144$ denotes the MRI image reshaped into a vector, $C\\in \\hbox{{C}}^{L\\times N}$ represents the wavelet subspace and $x_0\\in \\hbox{{C}}^N$ is the vector of wavelet coefficients.", "The blurring kernel $f\\in \\hbox{{C}}^{L\\times 1}$ is supported on a low frequency region.", "Therefore $\\widehat{f}=Bh_0$ , where $B\\in \\hbox{{C}}^{L\\times K}$ is a reshaped 2D low frequency Fourier matrix and $h_0\\in \\hbox{{C}}^{K}$ is a short vector.", "Figure REF shows the initial guess for Algorithm REF in the image domain, which is obtained by running the power method for fifty iterations.", "While this initial guess is clearly not a good approximation to the true solution, it suffices as a starting point for gradient descent.", "In the first experiment, we take $C$ to be the wavelet subspace corresponding to the $N=20000$ largest Haar wavelet coefficients of the original MRI image, and we also assume the locations of the $K=65$ nonzero entries of the kernel are known.", "Figure REF shows the reconstructed image in this ideal setting.", "It can be observed that the recovered image is visually indistinguishable from the ground truth MRI image.", "In the second experiment, we test a more realistic setting, where both the support of the MRI image is the wavelet domain and the support of the kernel are not known.", "We take the Haar wavelet transform of the blurred image (Figure REF ) and select $C$ to be the wavelet subspace corresponding to the $N=35000$ largest wavelet coefficients.", "We do not assume the exact support of the kernel is known, but assume that its support is contained in a small box region.", "The reconstructed image in this setting is shown in Figure REF .", "Despite not knowing the subspaces exactly, Algorithm REF is still able to return a reasonable reconstruction.", "Yet, this second experiment also demonstrates that there is clearly room for improvement in the case when the subspaces are unknown.", "One natural idea to improve upon the result depicted in Figure REF is to include an additional total-variation penalty in the reconstruction algorithm.", "We leave this line of work for future research." ], [ "Proof of the main theorem", "This section is devoted to the proof of Theorems REF and REF .", "Since proving Theorem REF is a bit more involved, we briefly describe the architecture of its proof.", "In Subsection REF we state four key conditions: The Local Regularity Condition will allow us to show that the objective function decreases; the Local Restricted Isometry Property enables us to transfer the decrease in the objective function to a decrease of the error between the iterates and the true solution; the Local Smoothness Condition yields the actual rate of convergence, and finally, the Robustness Condition establishes robustness of the proposed algorithm against additive noise.", "Armed with these conditions, we will show how the three regions defined in Section characterize the convergence neighborhood of the solution, i.e., if the initial guess is inside this neighborhood, the sequence generated via gradient descent will always stay inside this neighborhood as well.", "In Subsections REF –REF we justify the aforementioned four conditions, show that they are valid under the assumptions stated in Theorem REF , and conclude with a proof of Theorem REF ." ], [ "Four key conditions and the proof of Theorem ", "Condition 5.1 (Local RIP condition) The following local Restricted Isometry Property (RIP) for $\\mathcal {A}$ holds uniformly for all $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }:$ $\\frac{3}{4} \\Vert hx^* - h_0x_0^\\Vert _F^2 \\le \\Vert \\mathcal {A}(hx^ - h_0x_0^)\\Vert ^2 \\le \\frac{5}{4}\\Vert hx^ - h_0x_0^\\Vert _F^2$ Condition REF states that $\\mathcal {A}$ almost preserves the $\\ell _2$ -distance between $hx^ - h_0x_0^$ over a “local\" region around the ground truth $h_0x_0^$ .", "The proof of Condition REF is given in Lemma REF .", "Condition 5.2 (Robustness condition) For the noise $e\\sim \\mathcal {N}(0, \\frac{\\sigma ^2 d_0^2}{2L}I_L) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2 d_0^2}{2L}I_L)$ , with high probability there holds, $\\Vert \\mathcal {A}^(e) \\Vert \\le \\frac{\\varepsilon d_0}{10\\sqrt{2}}$ if $L \\ge C_{\\gamma }(\\frac{\\sigma ^2}{\\varepsilon ^2} + \\frac{\\sigma }{\\varepsilon })\\max \\lbrace K, N\\rbrace \\log L$ .", "This condition follows directly from (REF ).", "It is quite essential when we analyze the behavior of Algorithm REF under Gaussian noise.", "With those two conditions above in hand, the lower and upper bounds of $F(h, x)$ are well approximated over $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ by two quadratic functions of $\\delta $ , where $\\delta $ is defined in (REF ).", "A similar approach towards noisy matrix completion problem can be found in [16].", "For any $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ , applying Condition REF to (REF ) leads to $F(h, x) \\le \\Vert e\\Vert ^2 + F_0(h, x) + 2\\sqrt{2}\\Vert \\mathcal {A}^(e)\\Vert \\delta d_0 \\le \\Vert e\\Vert ^2 + \\frac{5}{4}\\delta ^2 d_0^2 + 2\\sqrt{2} \\Vert \\mathcal {A}^(e)\\Vert \\delta d_0$ and similarly $F(h, x) \\ge \\Vert e\\Vert ^2 + \\frac{3}{4}\\delta ^2 d_0^2 - 2\\sqrt{2} \\Vert \\mathcal {A}^(e)\\Vert \\delta d_0$ where $\|\\left\\langle \\mathcal {A}^(e), hx^ - h_0x_0^* \\right\\rangle \| \\le \\Vert \\mathcal {A}^(e)\\Vert \\Vert hx^ - h_0x_0^\\Vert _ \\le \\sqrt{2}\\Vert \\mathcal {A}^(e)\\Vert \\delta d_0,$ because $\\Vert \\cdot \\Vert $ and $\\Vert \\cdot \\Vert _$ is a pair of dual norm and $\\operatorname{\\text{rank}}(hx^* - h_0x_0^) \\le 2$ .", "Moreover, with the Condition REF , (REF ) and (REF ) yield the followings: $F(h, x) \\le \\Vert e\\Vert ^2 + \\frac{5}{4}\\delta ^2d_0^2 + \\frac{\\varepsilon \\delta d_0^2}{5}$ and $F(h, x) \\ge \\Vert e\\Vert ^2 + \\frac{3}{4}\\delta ^2 d_0^2 - \\frac{\\varepsilon \\delta d_0^2}{5}.$ The third condition is about the regularity condition of $\\widetilde{F}(h, x)$ , which is the key to establishing linear convergence later.", "The proof will be given in Lemma REF .", "Condition 5.3 (Local regularity condition) Let $\\widetilde{F}(h, x)$ be as defined in (REF ) and $\\nabla \\widetilde{F}(h, x) := (\\nabla \\widetilde{F}_{h}, \\nabla \\widetilde{F}_{x})\\in \\hbox{{C}}^{K + N}$ .", "Then there exists a regularity constant $\\omega = \\frac{d_0}{5000}>0$ such that $\\Vert \\nabla \\widetilde{F}(h, x)\\Vert ^2 \\ge \\omega \\left[ \\widetilde{F}(h, x) - c\\right]_+$ for all $(h, x) \\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ where $c = \\Vert e\\Vert ^2 + a \\Vert \\mathcal {A}^(e)\\Vert ^2 $ with $a = 1700$ .", "In particular, in the noiseless case, i.e., $e= 0$ , we have $\\Vert \\nabla \\widetilde{F}(h, x)\\Vert ^2 \\ge \\omega \\widetilde{F}(h, x).$ Besides the three regions defined in (REF ) to (REF ), we define another region $ \\mathcal {N}_{\\widetilde{F}}$ via $ \\mathcal {N}_{\\widetilde{F}}:= \\left\\lbrace (h, x): \\widetilde{F}(h, x) \\le \\frac{1}{3}\\varepsilon ^2 d_0^2 + \\Vert e\\Vert ^2\\right\\rbrace $ for proof technical purposes.", "$ \\mathcal {N}_{\\widetilde{F}}$ is actually the sublevel set of the nonconvex function $\\widetilde{F}$ .", "Finally we introduce the last condition called Local smoothness condition and its corresponding quantity $C_L$ which characterizes the choice of stepsize $\\eta $ and the rate of linear convergence.", "Condition 5.4 (Local smoothness condition) Denote $z:= (h, x)$ .", "There exists a constant $C_L $ such that $\\Vert \\nabla f(z+ t \\Delta z) - \\nabla f(z)\\Vert \\le C_L t\\Vert \\Delta z\\Vert , \\quad \\forall 0\\le t\\le 1,$ for all $\\lbrace (z, \\Delta z) \| z+t \\Delta z\\in \\mathcal {N}_{\\varepsilon }\\bigcap \\mathcal {N}_{\\widetilde{F}}, \\forall 0\\le t\\le 1\\rbrace $ , i.e., the whole segment connecting $z$ and $z+ \\Delta z$ , which is parametrized by $t$ , belongs to the nonconvex set $ \\mathcal {N}_{\\varepsilon }\\bigcap \\mathcal {N}_{\\widetilde{F}}.$ The upper bound of $C_L$ , which scales with $\\mathcal {O}(d_0(1 + \\sigma ^2)(K + N)\\log ^2L )$ , will be given in Section REF .", "We will show later in Lemma REF that the stepsize $\\eta $ is chosen to be smaller than $\\frac{1}{C_L}.$ Hence $\\eta = \\mathcal {O}((d_0(1 + \\sigma ^2)(K + N)\\log ^2L )^{-1}).$ Lemma 5.5 There holds $ \\mathcal {N}_{\\widetilde{F}}\\subset \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ ; under Condition REF and REF , we have $ \\mathcal {N}_{\\widetilde{F}}\\cap \\mathcal {N}_{\\varepsilon }\\subset \\mathcal {N}_{\\frac{9}{10}\\varepsilon }$ .", "If $(\\mathbf {h}, \\mathbf {x}) \\notin \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ , by the definition of $G$ in (REF ), at least one component in $G$ exceeds $\\rho G_0\\left(\\frac{2d_0}{d}\\right)$ .", "We have $\\widetilde{F}(h, x) & \\ge & \\rho G_0\\left(\\frac{2d_0}{d}\\right) \\ge (d^2 + 2\\Vert e\\Vert ^2) \\left( \\frac{2d_0}{d} - 1\\right)^2 \\\\& \\ge & (2d_0 - d)^2 + 2\\Vert e\\Vert ^2\\left( \\frac{2d_0}{d} - 1\\right)^2 \\\\& \\ge & 0.81 d_0^2 + \\Vert e\\Vert ^2 > \\frac{1}{3}\\varepsilon ^2 d_0^2 + \\Vert e\\Vert ^2,$ where $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2$ and $0.9d_0 \\le d \\le 1.1d_0.$ This implies $(h, x) \\notin \\mathcal {N}_{\\widetilde{F}}$ and hence $ \\mathcal {N}_{\\widetilde{F}}\\subset \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ .", "For any $(h, x) \\in \\mathcal {N}_{\\widetilde{F}}\\cap \\mathcal {N}_{\\varepsilon }$ , we have $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ now.", "By (REF ), $\\Vert e\\Vert ^2 + \\frac{3}{4} \\delta ^2 d_0^2 - \\frac{\\varepsilon \\delta d_0^2}{5} \\le F(h, x)\\le \\widetilde{F}(h, x)\\le \\Vert e\\Vert ^2 + \\frac{1}{3}\\varepsilon ^2 d_0^2.$ Therefore, $(\\mathbf {h}, \\mathbf {x}) \\in \\mathcal {N}_{\\frac{9}{10}\\varepsilon }$ and $ \\mathcal {N}_{\\widetilde{F}}\\cap \\mathcal {N}_{\\varepsilon }\\subset \\mathcal {N}_{\\frac{9}{10}\\varepsilon }$ .", "This lemma implies that the intersection of $ \\mathcal {N}_{\\widetilde{F}}$ and the boundary of $ \\mathcal {N}_{\\varepsilon }$ is empty.", "One might believe this suggests that $ \\mathcal {N}_{\\widetilde{F}}\\subset \\mathcal {N}_{\\varepsilon }$ .", "This may not be true.", "A more reasonable interpretation is that $ \\mathcal {N}_{\\widetilde{F}}$ consists of several disconnected regions due to the non-convexity of $\\widetilde{F}(h, x)$ , and one or several of them are contained in $ \\mathcal {N}_{\\varepsilon }$ .", "Lemma 5.6 Denote $\\mathbf {z}_1 = (\\mathbf {h}_1, \\mathbf {x}_1)$ and $\\mathbf {z}_2 = (\\mathbf {h}_2, \\mathbf {x}_2)$ .", "Let $\\mathbf {z}(\\lambda ):=(1-\\lambda )\\mathbf {z}_1 + \\lambda \\mathbf {z}_2$ .", "If $\\mathbf {z}_1 \\in \\mathcal {N}_{\\varepsilon }$ and $\\mathbf {z}(\\lambda ) \\in \\mathcal {N}_{\\widetilde{F}}$ for all $\\lambda \\in [0, 1]$ , we have $\\mathbf {z}_2 \\in \\mathcal {N}_{\\varepsilon }$ .", "Let us prove the claim by contradiction.", "If $\\mathbf {z}_2 \\notin \\mathcal {N}_{\\varepsilon }$ , since $\\mathbf {z}_1 \\in \\mathcal {N}_{\\varepsilon }$ , there exists $\\mathbf {z}(\\lambda _0):=(\\mathbf {h}(\\lambda _0), \\mathbf {x}(\\lambda _0)) \\in \\mathcal {N}_{\\varepsilon }$ for some $\\lambda _0 \\in [0, 1]$ , such that $\\Vert \\mathbf {h}(\\lambda _0)\\mathbf {x}(\\lambda _0)^* - h_0x_0^\\Vert _F = \\varepsilon d_0$ .", "However, since $\\mathbf {z}(\\lambda _0) \\in \\mathcal {N}_{\\widetilde{F}}$ , by lem:betamu, we have $\\Vert \\mathbf {h}(\\lambda _0)\\mathbf {x}(\\lambda _0)^ - h_0x_0^\\Vert _F \\le \\frac{9}{10}\\varepsilon d_0$ .", "This leads to a contradiction.", "Remark 5.7 Lemma REF tells us that if one line segment is completely inside $ \\mathcal {N}_{\\widetilde{F}}$ with one end point in $ \\mathcal {N}_{\\varepsilon }$ , then this whole line segment lies in $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }.$ Lemma 5.8 Let the stepsize $\\eta \\le \\frac{1}{C_L}$ , $z_t : = (u_t, v_t)\\in \\hbox{{C}}^{K + N}$ and $C_L$ be the constant defined in (REF ).", "Then, as long as $\\mathbf {z}_t \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ , we have $\\mathbf {z}_{t+1} \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ and $\\widetilde{F}(z_{t+1}) \\le \\widetilde{F}(z_t) - \\eta \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2.$ It suffices to prove eq:decreasing2.", "If $\\nabla \\widetilde{F}(z_t) = \\mathbf {0}$ , then $z_{t+1} = z_t$ , which implies eq:decreasing2 directly.", "So we only consider the case when $\\nabla \\widetilde{F}(z_t) \\ne \\mathbf {0}$ .", "Define the function $\\varphi (\\lambda ) := \\widetilde{F}(z_t - \\lambda \\nabla \\widetilde{F}(z_t)).$ Then $\\varphi ^{\\prime }(\\lambda )\|_{\\lambda = 0} = - 2\\Vert \\nabla \\widetilde{F}(z_t) \\Vert ^2 <0.$ since $\\varphi (\\lambda )$ is a real-valued function with complex variables (See (REF ) for details).", "By the definition of derivatives, we know there exists $\\eta _0 > 0$ , such that $\\varphi (\\lambda ) < \\varphi (0)$ for all $0<\\lambda \\le \\eta _0$ .", "Now we will first prove that $\\varphi (\\lambda ) \\le \\varphi (0)$ for all $0\\le \\lambda \\le \\eta $ by contradiction.", "Assume there exists some $\\eta _1 \\in (\\eta _0, \\eta ]$ such that $\\varphi (\\eta _1) > \\varphi (0)$ .", "Then there exists $\\eta _2 \\in (\\eta _0, \\eta _1)$ , such that $\\varphi (\\eta _2) = \\varphi (0)$ and $\\varphi (\\lambda ) < \\varphi (0)$ for all $0 < \\lambda < \\eta _2$ , since $\\varphi (\\lambda )$ is a continuous function.", "This implies $z_t - \\lambda \\nabla \\widetilde{F}(z_t) \\in \\mathcal {N}_{\\widetilde{F}}, \\quad \\forall 0 \\le \\lambda \\le \\eta _2$ since $\\widetilde{F}(z_t - \\lambda \\nabla \\widetilde{F}(z_t)) \\le \\widetilde{F}(z_t)$ for $0\\le \\lambda \\le \\eta _2.$ By lem:linesection and the assumption $z_t \\in \\mathcal {N}_{\\varepsilon }$ , we have $z_t - \\lambda \\nabla \\widetilde{F}(z_t) \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}, \\quad \\forall 0 \\le \\lambda \\le \\eta _2.$ Then, by using the modified descent lemma (Lemma REF ), $\\widetilde{F}(z_t - \\eta _2 \\nabla \\widetilde{F}(z_t)) &\\le \\widetilde{F}(z_t) - 2 \\eta _2 \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2 + C_L\\eta _2^2 \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2 \\nonumber \\\\& =\\widetilde{F}(z_t) + (C_L\\eta _2^2 - 2\\eta _2) \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2 < \\widetilde{F}(z_t),$ where the final inequality is due to $\\eta _2/\\eta <1$ , $\\eta _2 > \\eta _0 \\ge 0$ and $\\nabla \\widetilde{F}(z_t) \\ne \\mathbf {0}$ .", "This contradicts $ \\widetilde{F}(z_t - \\eta _2 \\nabla \\widetilde{F}(z_t)) = \\varphi (\\eta _2) = \\varphi (0) = \\widetilde{F}(z_t)$ .", "Therefore, there holds $\\varphi (\\lambda ) \\le \\varphi (0)$ for all $0 \\le \\lambda \\le \\eta $ .", "Similarly, we can prove $z_t - \\lambda \\nabla \\widetilde{F}(z_t) \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}, \\quad \\forall 0 \\le \\lambda \\le \\eta ,$ which implies $z_{t+1} = z_t - \\eta \\nabla \\widetilde{F}(z_t) \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ .", "Again, by using Lemma REF we can prove $\\widetilde{F}(z_{t+1}) = \\widetilde{F}(z_t - \\eta \\nabla \\widetilde{F}(z_t)) \\le \\widetilde{F}(z_t) - 2 \\eta \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2 + C_L\\eta ^2 \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2 \\le \\widetilde{F}(z_t) - \\eta \\Vert \\nabla \\widetilde{F}(z_t)\\Vert ^2,$ where the final inequality is due to $ \\eta \\le \\frac{1}{C_L}$ .", "We conclude this subsection by proving Theorem REF under the Local regularity condition, the Local RIP condition, the Robustness condition, and the Local smoothness condition.", "The next subsections are devoted to justifying these conditions and showing that they hold under the assumptions of Theorem REF .", "[of Theorem REF] Suppose that the initial guess $z_0 := (u_0, v_0)\\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}\\bigcap \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }\\bigcap \\mathcal {N}_{\\frac{2}{5}\\varepsilon }$ , we have $G(u_0, v_0) = 0$ .", "This holds, because $\\frac{\\Vert u_0\\Vert ^2}{2d} \\le \\frac{2d_0}{3d} < 1, \\quad \\frac{L\|b_l^ u_0\|^2}{8d\\mu ^2} \\le \\frac{L}{8d\\mu ^2} \\cdot \\frac{16d_0\\mu ^2}{3L} \\le \\frac{2d_0}{3d} < 1,$ where $\\Vert u_0\\Vert \\le \\frac{2\\sqrt{d_0}}{\\sqrt{3}}$ , $\\sqrt{L}\\Vert Bu_0\\Vert _{\\infty } \\le \\frac{4 \\sqrt{d_0}\\mu }{\\sqrt{3}}$ and $\\frac{9}{10}d_0 \\le d\\le \\frac{11}{10}d_0.$ Therefore $G_0\\left( \\frac{\\Vert u_0\\Vert ^2}{2d}\\right) = G_0\\left( \\frac{\\Vert v_0\\Vert ^2}{2d}\\right) = G_0\\left(\\frac{L\|b_l^u_0\|^2}{8d\\mu ^2}\\right) = 0$ for all $1\\le l\\le L$ and $G(u_0, v_0) = 0.$ Since $(u_0, v_0)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ , (REF ) combined with $\\delta (z_0) := \\frac{\\Vert u_0v_0^ - h_0x_0^\\Vert _F}{d_0} \\le \\frac{2\\varepsilon }{5}$ imply that $\\widetilde{F}(u_0, v_0) = F(u_0, v_0) \\le \\Vert e\\Vert ^2 + \\frac{5}{4}\\delta ^2(z_0) d_0^2 + \\frac{1}{5}\\varepsilon \\delta (z_0) d_0^2 < \\frac{1}{3}\\varepsilon ^2 d_0^2 + \\Vert e\\Vert ^2$ and hence $z_0 = (u_0, v_0)\\in \\mathcal {N}_{\\varepsilon }\\bigcap \\mathcal {N}_{\\widetilde{F}}.$ Denote $z_t : = (u_t, v_t).$ Combining Lemma REF by choosing $\\eta \\le \\frac{1}{C_L}$ with Condition REF , we have $\\widetilde{F}(z_{t + 1}) \\le \\widetilde{F}(z_t) - \\eta \\omega \\left[ \\widetilde{F}(z_t) - c \\right]_+$ with $c = \\Vert e\\Vert ^2 + a\\Vert \\mathcal {A}^(e)\\Vert ^2$ , $a = 1700$ and $z_t \\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ for all $t\\ge 0.$ Obviously, the inequality above implies $\\widetilde{F}(z_{t+1}) - c \\le (1 - \\eta \\omega ) \\left[ \\widetilde{F}(z_t) - c \\right]_+ ,$ and by monotonicity of $z_+ = \\frac{z + \|z\|}{2}$ , there holds $\\left[ \\widetilde{F}(z_{t+1}) - c\\right]_+ \\le (1 - \\eta \\omega ) \\left[ \\widetilde{F}(z_t) - c \\right]_+ .$ Therefore, by induction, we have $\\left[ \\widetilde{F}(z_t) - c\\right]_+ & \\le & \\left(1 - \\eta \\omega \\right)^t \\left[ \\widetilde{F}(z_0) - c\\right]_+ \\le \\frac{1}{3} (1 - \\eta \\omega )^{t} \\varepsilon ^2 d_0^2$ where $\\widetilde{F}(z_0) \\le \\frac{1}{3}\\varepsilon ^2d_0^2 + \\Vert e\\Vert ^2$ and hence $\\left[ \\widetilde{F}(z_0) - c \\right]_+ \\le \\left[ \\frac{1}{3}\\varepsilon ^2 d_0^2 - a\\Vert \\mathcal {A}^(e)\\Vert ^2 \\right]_+ \\le \\frac{1}{3}\\varepsilon ^2 d_0^2.$ Now we can conclude that $\\left[ \\widetilde{F}(z_t) - c\\right]_+$ converges to 0 geometrically.", "Note that over $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ , $\\widetilde{F}(z_t) - \\Vert e\\Vert ^2 \\ge F_0(z_t) - 2\\operatorname{Re}(\\left\\langle \\mathcal {A}^(e), u_tv_t^* - h_0x_0^* \\right\\rangle )\\ge \\frac{3}{4} \\delta ^2(z_t)d_0^2 - 2\\sqrt{2}\\Vert \\mathcal {A}^(e)\\Vert \\delta (z_t)d_0$ where $ \\delta (z_t) := \\frac{\\Vert u_tv_t^ - h_0x_0^\\Vert _F}{d_0}$ , $F_0$ is defined in (REF ) and $G(z_t) \\ge 0$ .", "There holds $\\frac{3}{4} \\delta ^2(z_t)d_0^2 - 2\\sqrt{2}\\Vert \\mathcal {A}^(e)\\Vert \\delta (z_t)d_0 - a\\Vert \\mathcal {A}^(e)\\Vert ^2 \\le \\left[ \\widetilde{F}(z_t) - c \\right]_+ \\le \\frac{1}{3}(1 - \\eta \\omega )^t \\varepsilon ^2 d_0^2$ and equivalently, $\\left\|\\delta (z_t)d_0 - \\frac{4\\sqrt{2}}{3} \\Vert \\mathcal {A}^(e)\\Vert \\right\|^2 \\le \\frac{4}{9} (1 - \\eta \\omega )^t \\varepsilon ^2 d_0^2 + \\left(\\frac{4}{3}a + \\frac{32}{9}\\right)\\Vert \\mathcal {A}^(e)\\Vert ^2.$ Solving the inequality above for $\\delta (z_t)$ , we have $\\delta (z_t) d_0 & \\le & \\frac{2}{3}(1 - \\eta \\omega )^{t/2} \\varepsilon d_0 +\\left(\\frac{4\\sqrt{2}}{3} + \\sqrt{\\frac{4}{3}a + \\frac{32}{9}} \\right)\\Vert \\mathcal {A}^(e)\\Vert \\nonumber \\\\& \\le & \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50 \\Vert \\mathcal {A}^(e)\\Vert .", "$ Let $d_t : = \\Vert u_t\\Vert \\Vert v_t\\Vert $ , $t\\ge 1.$ By (REF ) and triangle inequality, we immediately conclude that $\|d_t - d_0\| \\le \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50 \\Vert \\mathcal {A}^(e)\\Vert .$ Now we derive the upper bound for $\\sin \\angle (u_t, h_0)$ and $\\sin \\angle (v_t, x_0).$ Due to symmetry, it suffices to consider $\\sin \\angle (u_t, h_0)$ .", "The bound follows from standard linear algebra arguments: $\\sin \\angle (u_t, h_0) & = & \\frac{1}{\\Vert u_t\\Vert }\\left\\Vert \\left(I- \\frac{h_0h_0^}{d_0}\\right)u_t\\right\\Vert \\\\& = & \\frac{1}{\\Vert u_t\\Vert \\Vert v_t\\Vert }\\left\\Vert \\left(I- \\frac{h_0h_0^}{d_0}\\right)(u_t v_t^* - h_0x_0^* )\\right\\Vert \\\\& \\le & \\frac{1}{d_t} \\Vert u_tv_t^* - h_0x_0^\\Vert _F \\\\& \\le & \\frac{1}{d_t}\\left( \\frac{2}{3}(1 - \\eta \\omega )^{t/2}\\varepsilon d_0 + 50 \\Vert \\mathcal {A}^(e)\\Vert \\right),$ where the second equality uses $\\left(I- \\frac{h_0h_0^}{d_0}\\right) h_0 = 0.$" ], [ "Supporting lemmata", "This subsection introduces several lemmata, especially Lemma REF , REF and REF , which are central for justifying Conditions REF and REF .", "After that, we will prove the Local RIP Condition in Lemma REF based on those three lemmata.", "We start with defining a linear space $T$ , which contains $h_0x_0^$ , via $T := \\left\\lbrace \\frac{1}{\\sqrt{d_0}}h_0v^* + \\frac{1}{\\sqrt{d_0}}ux_0^,~ u\\in \\hbox{{C}}^K, v\\in \\hbox{{C}}^N \\right\\rbrace \\subset \\hbox{{C}}^{K\\times N}.$ Its orthogonal complement is given by $T^{\\bot }: = \\left\\lbrace \\left(I- \\frac{1}{d_0}h_0h_0^\\right)Z\\left(I- \\frac{1}{d_0}x_0x_0^\\right),~ Z\\in \\hbox{{C}}^{K\\times N} \\right\\rbrace .$ Denote $\\mathcal {P}_T$ to be the projection operator from $\\hbox{{C}}^{K\\times N}$ onto $T$ .", "For any $\\mathbf {h}$ and $\\mathbf {x}$ , there are unique orthogonal decompositions ${\\mathbf {h} = \\alpha _1 \\mathbf {h}_0 + \\tilde{\\mathbf {h}}, \\quad \\mathbf {x} = \\alpha _2 \\mathbf {x}_0 + \\tilde{\\mathbf {x}}},$ where $\\mathbf {h}_0 \\perp \\tilde{\\mathbf {h}}$ and $\\mathbf {x}_0 \\perp \\tilde{\\mathbf {x}}$ .", "More precisely, $\\alpha _1 = \\frac{h_0^h}{d_0} = \\frac{\\left\\langle h_0, h\\right\\rangle }{d_0}$ and $\\alpha _2 = \\frac{\\left\\langle x_0, x\\right\\rangle }{d_0}.$ We thereby have the following matrix orthogonal decomposition $\\mathbf {h}\\mathbf {x}^* - \\mathbf {h}_0 \\mathbf {x}_0^* = (\\alpha _1 \\overline{\\alpha _2} - 1)\\mathbf {h}_0\\mathbf {x}_0^* + \\overline{\\alpha _2} \\tilde{\\mathbf {h}} \\mathbf {x}_0^* + \\alpha _1 \\mathbf {h}_0 \\tilde{\\mathbf {x}}^* + \\tilde{\\mathbf {h}} \\tilde{\\mathbf {x}}^$ where the first three components are in $T$ while $\\tilde{h}\\tilde{x}^\\in T^{\\bot }$ .", "Lemma 5.9 Recall that $\\Vert \\mathbf {h}_0\\Vert = \\Vert \\mathbf {x}_0\\Vert = \\sqrt{d_0}$ .", "If $\\delta := \\frac{\\Vert \\mathbf {h}\\mathbf {x}^* - \\mathbf {h}_0 \\mathbf {x}_0^\\Vert _F}{d_0}<1$ , we have the following useful bounds $\|\\alpha _1\|\\le \\frac{\\Vert \\mathbf {h}\\Vert }{\\Vert \\mathbf {h}_0\\Vert }, \\quad \|\\alpha _1\\overline{\\alpha _2} - 1\|\\le \\delta ,$ and $\\Vert \\tilde{\\mathbf {h}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {h}\\Vert ,\\quad \\Vert \\tilde{\\mathbf {x}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {x}\\Vert ,\\quad \\Vert \\tilde{\\mathbf {h}}\\Vert \\Vert \\tilde{\\mathbf {x}}\\Vert \\le \\frac{\\delta ^2}{2(1 - \\delta )} d_0.$ Moreover, if $\\Vert \\mathbf {h}\\Vert \\le 2\\sqrt{d_0}$ and $\\sqrt{L}\\Vert \\mathbf {B} \\mathbf {h}\\Vert _\\infty \\le 4\\mu \\sqrt{d_0}$ , we have $\\sqrt{L}\\Vert \\mathbf {B} \\tilde{\\mathbf {h}}\\Vert _\\infty \\le 6 \\mu \\sqrt{d_0}$ .", "Remark 5.10 This lemma is actually a simple version of singular value/vector perturbation.", "It says that if $\\frac{\\Vert hx^ - h_0x_0^\\Vert _F}{d_0}$ is of $\\mathcal {O}(\\delta )$ , then the individual vectors $(h, x)$ are also close to $(h_0, x_0)$ , with the error of order $\\mathcal {O}(\\delta ).$ The equality eq:orth implies that $\\Vert \\alpha _1 \\mathbf {h}_0\\Vert \\le \\Vert \\mathbf {h}\\Vert $ , so there holds $\|\\alpha _1\|\\le \\frac{\\Vert \\mathbf {h}\\Vert }{\\Vert \\mathbf {h}_0\\Vert }$ .", "Since $\\Vert \\mathbf {h}\\mathbf {x}^ - \\mathbf {h}_0 \\mathbf {x}_0^\\Vert _F = \\delta d_0$ , by eq:decomposition, we have $\\delta ^2d_0^2 = (\\alpha _1 \\overline{\\alpha _2} -1)^2 d_0^2 + \|\\overline{\\alpha _2}\|^2 \\Vert \\tilde{\\mathbf {h}}\\Vert ^2d_0 + \|\\alpha _1\|^2 \\Vert \\tilde{\\mathbf {x}}\\Vert ^2d_0+ \\Vert \\tilde{\\mathbf {h}}\\Vert ^2\\Vert \\tilde{\\mathbf {x}}\\Vert ^2.$ This implies that $\\Vert \\mathbf {h}\\Vert ^2\\Vert \\tilde{\\mathbf {x}}\\Vert ^2 = (\\alpha _1^2d_0 + \\Vert \\tilde{\\mathbf {h}}\\Vert ^2)\\Vert \\tilde{\\mathbf {x}}\\Vert ^2 \\le \\delta ^2d_0^2.$ On the other hand, $\\Vert \\mathbf {h}\\Vert \\Vert \\mathbf {x}\\Vert \\ge \\Vert \\mathbf {h}_0\\Vert \\Vert \\mathbf {x}_0\\Vert - \\Vert \\mathbf {h}\\mathbf {x}^ - \\mathbf {h}_0 \\mathbf {x}_0^\\Vert _F = (1 - \\delta ) d_0.$ The above two inequalities imply that $\\Vert \\tilde{\\mathbf {x}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {x}\\Vert $ , and similarly we have $\\Vert \\tilde{\\mathbf {h}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {h}\\Vert $ .", "The equality eq:pythag implies that $\|\\alpha _1 \\overline{\\alpha _2} -1\| \\le \\delta $ and hence $\|\\alpha _1 \\overline{\\alpha _2}\| \\ge 1 - \\delta $ .", "Moreover, eq:pythag also implies $\\Vert \\tilde{\\mathbf {h}}\\Vert \\Vert \\tilde{\\mathbf {x}}\\Vert \|\\alpha _1\| \|\\overline{\\alpha _2}\| \\le \\frac{1}{2}( \|\\overline{\\alpha _2}\|^2 \\Vert \\tilde{\\mathbf {h}}\\Vert ^2 + \|\\alpha _1\|^2\\Vert \\tilde{\\mathbf {x}}\\Vert ^2) \\le \\frac{\\delta ^2 d_0}{2},$ which yields $\\Vert \\tilde{\\mathbf {h}}\\Vert _2 \\Vert \\tilde{\\mathbf {x}}\\Vert _2 \\le \\frac{\\delta ^2}{2(1 - \\delta )} d_0$ .", "If $\\Vert \\mathbf {h}\\Vert \\le 2\\sqrt{d_0}$ and $\\sqrt{L}\\Vert \\mathbf {B} \\mathbf {h}\\Vert _\\infty \\le 4\\mu \\sqrt{d_0}$ , there holds $\|\\alpha _1\| \\le \\frac{\\Vert \\mathbf {h}\\Vert }{\\Vert \\mathbf {h}_0\\Vert } \\le 2$ .", "Then $\\sqrt{L}\\Vert \\mathbf {B} \\tilde{\\mathbf {h}}\\Vert _\\infty & \\le & \\sqrt{L}\\Vert \\mathbf {B} \\mathbf {h}\\Vert _\\infty + \\sqrt{L}\\Vert \\mathbf {B}(\\alpha _1\\mathbf {h}_0)\\Vert _\\infty \\le \\sqrt{L}\\Vert \\mathbf {B} \\mathbf {h}\\Vert _\\infty + 2\\sqrt{L}\\Vert \\mathbf {B}\\mathbf {h}_0\\Vert _\\infty \\\\& \\le & 4 \\mu \\sqrt{d_0} + 2 \\mu _h \\sqrt{d_0} \\le 6\\mu \\sqrt{d_0}$ where $\\mu _h \\le \\mu $ .", "In the following, we introduce and prove a series of local and global properties of $\\mathcal {A}$ : Lemma 5.11 (Lemma 1 in [1]) For $\\mathcal {A}$ defined in (REF ), $\\Vert \\mathcal {A}\\Vert \\le \\sqrt{N\\log (NL/2) + \\gamma \\log L}$ with probability at least $1 - L^{-\\gamma }.$ Lemma 5.12 (Corollary 2 in [1]) Let $\\mathcal {A}$ be the operator defined in (REF ), then on an event $E_1$ with probability at least $1 - L^{-\\gamma }$ , $\\mathcal {A}$ restricted on $T$ is well-conditioned, i.e., $\\Vert \\mathcal {P}_T\\mathcal {A}^\\mathcal {A}\\mathcal {P}_T - \\mathcal {P}_T\\Vert \\le \\frac{1}{10}$ where $\\mathcal {P}_T$ is the projection operator from $\\hbox{{C}}^{K\\times N}$ onto $T$ , provided $L \\ge C_{\\gamma } \\max \\lbrace K, \\mu _h^2 N\\rbrace \\log ^2(L)$ .", "Now we introduce a property of $\\mathcal {A}$ when restricted on rank-one matrices.", "Lemma 5.13 On an event $E_2$ with probability at least $1 - L^{-\\gamma } - \\frac{1}{\\gamma }\\exp (-(K+N))$ , we have $\\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 \\le \\left(\\frac{4}{3}\\Vert \\mathbf {u}\\Vert ^2 + 2\\Vert \\mathbf {u}\\Vert \\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty \\sqrt{2(K+N)\\log L} + 8\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty ^2(K+N) \\log L \\right)\\Vert \\mathbf {v}\\Vert ^2,$ uniformly for any $\\mathbf {u}$ and $\\mathbf {v}$ , provided $L\\ge C_{\\gamma }(K+N)\\log L$ .", "Due to the homogeneity, without loss of generality we can assume $\\Vert \\mathbf {u}\\Vert =\\Vert \\mathbf {v}\\Vert =1$ .", "Define $f(\\mathbf {u}, \\mathbf {v}):=\\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 - 2\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty \\sqrt{2(K+N)\\log L} - 8\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty ^2(K+N)\\log L.$ It suffices to prove that $f(\\mathbf {u}, \\mathbf {v})\\le \\frac{4}{3}$ uniformly for all $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {S}^{K-1}\\times \\mathcal {S}^{N-1}$ with high probability, where $\\mathcal {S}^{K-1}$ is the unit sphere in $\\hbox{{C}}^K.$ For fixed $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {S}^{K-1}\\times \\mathcal {S}^{N-1}$ , notice that $\\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 = \\sum \\limits _{l=1}^L \|\\mathbf {b}_l^ \\mathbf {u}\|^2 \|\\mathbf {a}_l^* \\mathbf {v}\|^2$ is the sum of subexponential variables with expectation $\\hbox{{E}} \\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 = \\sum \\limits _{l=1}^L \|\\mathbf {b}_l^ \\mathbf {u}\|^2 = 1$ .", "For any generalized $\\chi _n^2$ variable $Y \\sim \\sum _{i=1}^n c_i \\xi _i^2$ satisfies $\\hbox{{P}}(Y - \\hbox{{E}}(Y) \\ge t) \\le \\exp \\left(- \\frac{t^2}{8\\Vert \\mathbf {c}\\Vert _2^2}\\right) \\vee \\exp \\left(- \\frac{t}{8\\Vert \\mathbf {c}\\Vert _\\infty }\\right),$ where $\\lbrace \\xi _i\\rbrace $ are i.i.d.", "$\\chi ^2_1$ random variables and $c= (c_1, \\cdots , c_n)^T\\in \\hbox{{R}}^n$ .", "Here we set $\|a_l^v\|^2 = \\frac{1}{2} \\xi _{2l-1}^2 + \\frac{1}{2}\\xi _{2l}^2$ , $c_{2l-1} = c_{2l} = \\frac{\|b_l^u\|^2}{2}$ and $n = 2L$ .", "Therefore, $\\Vert c\\Vert _{\\infty } = \\frac{\\Vert Bu\\Vert _{\\infty }^2}{2}, \\quad \\Vert c\\Vert _2^2 = \\frac{1}{2}\\sum _{l=1}^L \|b_l^u\|^4 \\le \\frac{\\Vert Bu\\Vert ^2_{\\infty }}{2}$ and we have $\\hbox{{P}}(\\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 \\ge 1 + t)\\le \\exp \\left(- \\frac{t^2}{4 \\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty ^2}\\right) \\vee \\exp \\left(- \\frac{t}{4\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty ^2}\\right).$ Applying (REF ) and setting $t = g(\\mathbf {u}):= 2\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty \\sqrt{2(K+N)\\log L} + 8\\Vert Bu\\Vert ^2_{\\infty }(K + N)\\log L,$ there holds $\\hbox{{P}}\\left(\\Vert \\mathcal {A}(\\mathbf {u}\\mathbf {v}^)\\Vert ^2 \\ge 1 + g(\\mathbf {u})\\right) \\le \\exp \\left( - 2 (K+N)(\\log L) \\right).$ That is, $f(\\mathbf {u}, \\mathbf {v}) \\le 1$ with probability at least $1 - \\exp \\left( - 2 (K+N)(\\log L) \\right)$ .", "We define $\\mathcal {K}$ and $\\mathcal {N}$ as $\\varepsilon _0$ -nets of $\\mathcal {S}^{K-1}$ and $\\mathcal {S}^{N-1}$ , respectively.", "Then, $\|\\mathcal {K}\|\\le (1+\\frac{2}{\\varepsilon _0})^{2K}$ and $\|\\mathcal {N}\|\\le (1+\\frac{2}{\\varepsilon _0})^{2N}$ follow from the covering numbers of the sphere (Lemma 5.2 in [39]).", "By taking the union bounds over $\\mathcal {K}\\times \\mathcal {N},$ we have $f(\\mathbf {u}, \\mathbf {v})\\le 1$ holds uniformly for all $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {K} \\times \\mathcal {N}$ with probability at least $1- \\left(1+\\frac{2}{\\varepsilon _0}\\right)^{2(K + N)} e^{ - 2 (K+N)\\log L } = 1- \\exp \\left( -2(K + N)\\left(\\log L - \\log \\left(1 + \\frac{2}{\\varepsilon _0}\\right)\\right) \\right).$ Our goal is to show that $f(\\mathbf {u}, \\mathbf {v})\\le \\frac{4}{3}$ uniformly for all $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {S}^{K-1}\\times \\mathcal {S}^{N-1}$ with the same probability.", "For any $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {S}^{K-1}\\times \\mathcal {S}^{N-1}$ , we can find its closest $(\\mathbf {u}_0, \\mathbf {v}_0) \\in \\mathcal {K} \\times \\mathcal {N}$ satisfying $\\Vert u- u_0 \\Vert \\le \\varepsilon _0$ and $\\Vert v- v_0\\Vert \\le \\varepsilon _0$ .", "By lem:A-UPBD, with probability at least $1 - L^{-\\gamma }$ , we have $\\Vert \\mathcal {A}\\Vert \\le \\sqrt{(N+\\gamma )\\log L}$ .", "Then straightforward calculation gives $\| f(\\mathbf {u}, \\mathbf {v}) - f(\\mathbf {u}_0, \\mathbf {v})\| & \\le &\\Vert \\mathcal {A}((\\mathbf {u} - \\mathbf {u}_0)\\mathbf {v}^)\\Vert \\Vert \\mathcal {A}((\\mathbf {u} + \\mathbf {u}_0)\\mathbf {v}^)\\Vert \\\\&& + 2\\Vert \\mathbf {B}(\\mathbf {u} - \\mathbf {u}_0)\\Vert _\\infty \\sqrt{2(K+N)\\log L}\\\\&& + 8(K+N) (\\log L) \\Vert \\mathbf {B}(\\mathbf {u} - \\mathbf {u}_0)\\Vert _\\infty (\\Vert \\mathbf {B}\\mathbf {u}\\Vert _\\infty + \\Vert \\mathbf {B}\\mathbf {u}_0\\Vert _\\infty ) \\\\& \\le & 2\\Vert \\mathcal {A}\\Vert ^2 \\varepsilon _0 + 2\\sqrt{2(K+N)\\log L}\\varepsilon _0 + 16(K + N)(\\log L) \\varepsilon _0 \\\\&\\le & (21N + 19K + 2\\gamma )(\\log L) \\varepsilon _0$ where the first inequality is due to $\|\|z_1\|^2 - \|z_2\|^2\| \\le \|z_1 - z_2\|\|z_1 + z_2\|$ for any $z_1, z_2 \\in \\hbox{{C}}$ , and the second inequality is due to $\\Vert B\\mathbf {z} \\Vert _{\\infty } \\le \\Vert B\\mathbf {z} \\Vert = \\Vert \\mathbf {z}\\Vert $ for any $\\mathbf {z} \\in \\hbox{{C}}^K$ .", "Similarly, $\|f(\\mathbf {u}_0, \\mathbf {v}) - f(\\mathbf {u}_0, \\mathbf {v}_0)\| & = & \\Vert \\mathcal {A}( \\mathbf {u}_0 (\\mathbf {v} + \\mathbf {v}_0)^)\\Vert \\Vert \\mathcal {A}( \\mathbf {u}_0 (\\mathbf {v} - \\mathbf {v}_0)^)\\Vert \\\\& \\le & 2\\Vert \\mathcal {A}\\Vert ^2\\varepsilon _0 \\le 2( N + \\gamma )(\\log L)\\varepsilon _0 .$ Therefore, if $\\varepsilon _0 = \\frac{1}{70(N + K + \\gamma )\\log L}$ , there holds $\|f(\\mathbf {u}_0, \\mathbf {v}) - f(\\mathbf {u}_0, \\mathbf {v}_0)\| \\le \\frac{1}{3}.$ Therefore, if $L \\ge C_{\\gamma }(K+N)\\log L$ with $C_{\\gamma }$ reasonably large and $\\gamma \\ge 1$ , we have $\\log L - \\log \\left(1 + \\frac{2}{\\varepsilon _0}\\right) \\ge \\frac{1}{2}(1 + \\log (\\gamma ))$ and $f(\\mathbf {u}, \\mathbf {v})\\le \\frac{4}{3}$ uniformly for all $(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {S}^{K-1}\\times \\mathcal {S}^{N-1}$ with probability at least $1- L^{-\\gamma } - \\frac{1}{\\gamma }\\exp (-(K+N))$ .", "Finally, we introduce a local RIP property of $\\mathcal {A}$ conditioned on the event $E_1\\cap E_2$ , where $E_1$ and $E_2$ are defined in Lemma REF and Lemma REF Lemma 5.14 Over $\\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ with $\\mu \\ge \\mu _h$ and $\\varepsilon \\le \\frac{1}{15}$ , the following RIP type of property holds for $\\mathcal {A}$ : $\\frac{3}{4} \\Vert hx^ - h_0x_0^\\Vert _F^2 \\le \\Vert \\mathcal {A}(hx^ - h_0x_0^)\\Vert ^2 \\le \\frac{5}{4}\\Vert hx^ - h_0x_0^\\Vert _F^2$ provided $L \\ge C\\mu ^2 (K+N)\\log ^2 L$ for some numerical constant $C$ and conditioned on $E_1\\bigcap E_2.$ Let $\\delta : = \\frac{\\Vert hx^ - h_0x_0^\\Vert _F}{d_0} \\le \\varepsilon \\le \\frac{1}{15}$ , and $hx^ - h_0x_0^* := U+ V.$ where $U= (\\alpha _1 \\overline{\\alpha _2} - 1)\\mathbf {h}_0\\mathbf {x}_0^* + \\overline{\\alpha _2} \\tilde{\\mathbf {h}} \\mathbf {x}_0^* + \\alpha _1 \\mathbf {h}_0 \\tilde{\\mathbf {x}}^* \\in T,\\quad V= \\tilde{\\mathbf {h}} \\tilde{\\mathbf {x}}^* \\in T^\\perp .$ By lem:orthdecomp, we have $\\Vert V\\Vert _F \\le \\frac{\\delta ^2}{2(1 - \\delta )} d_0$ and hence $\\left(\\delta - \\frac{\\delta ^2}{2(1 - \\delta )}\\right)d_0 \\le \\Vert U\\Vert _F \\le \\left(\\delta + \\frac{\\delta ^2}{2(1 - \\delta )}\\right)d_0.$ Since $U\\in T$ , by lem:ripu, we have $\\sqrt{\\frac{9}{10}}\\left(\\delta - \\frac{\\delta ^2}{2(1 - \\delta )}\\right)d_0 \\le \\Vert \\mathcal {A}(U) \\Vert \\le \\sqrt{\\frac{11}{10}}\\left(\\delta + \\frac{\\delta ^2}{2(1 - \\delta )}\\right)d_0.$ By lem:key, we have $\\Vert \\mathcal {A}(V)\\Vert ^2 \\le \\left(\\frac{4}{3}\\Vert \\tilde{\\mathbf {h}}\\Vert _2^2 + 2\\Vert \\tilde{\\mathbf {h}}\\Vert \\Vert \\mathbf {B}\\tilde{\\mathbf {h}}\\Vert _\\infty \\sqrt{2(K+N)\\log L} + 8\\Vert \\mathbf {B}\\tilde{\\mathbf {h}}\\Vert _\\infty ^2(K+N) (\\log L)\\right) \\Vert \\tilde{\\mathbf {x}}\\Vert ^2.$ By lem:orthdecomp, we have $\\Vert \\tilde{\\mathbf {h}}\\Vert \\Vert \\tilde{\\mathbf {x}}\\Vert \\le \\frac{\\delta ^2}{2(1 - \\delta )} d_0$ , $\\Vert \\tilde{\\mathbf {x}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {x}\\Vert \\le \\frac{2\\delta }{1 - \\delta } \\sqrt{d_0}$ , $\\Vert \\tilde{\\mathbf {h}}\\Vert \\le \\frac{\\delta }{1 - \\delta }\\Vert \\mathbf {h}\\Vert \\le \\frac{2\\delta }{1 - \\delta } \\sqrt{d_0}$ , and $\\sqrt{L}\\Vert \\mathbf {B} \\tilde{\\mathbf {h}}\\Vert _\\infty \\le 6 \\mu \\sqrt{d_0}$ .", "By substituting all those estimations into (REF ), it ends up with $\\Vert \\mathcal {A}(V)\\Vert ^2 \\le \\frac{\\delta ^4}{3(1-\\delta )^2} d_0^2 + C^{\\prime }\\left(\\frac{\\delta ^3}{\\sqrt{C\\log L}} + \\frac{\\delta ^2}{C\\log L} \\right)d_0^2,$ where $C^{\\prime }$ is a numerical constant and $L \\ge C\\mu ^2 (K + N)\\log ^2 L$ .", "Combining eq:AV and eq:AU together with $C$ sufficiently large, numerical computation gives $\\frac{3}{4}\\delta d_0 \\le \\Vert \\mathcal {A}(U)\\Vert - \\Vert \\mathcal {A}(V)\\Vert \\le \\Vert \\mathcal {A}(U+ V)\\Vert \\le \\Vert \\mathcal {A}(U)\\Vert + \\Vert \\mathcal {A}(V)\\Vert \\le \\frac{5}{4}\\delta d_0.$ for all $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ given $\\varepsilon \\le \\frac{1}{15}$ ." ], [ "Local regularity", "In this subsection, we will prove Condition REF .", "Throughout this section, we assume $E_1$ and $E_2$ holds where $E_1$ and $E_2$ are mentioned in Lemma REF and Lemma REF .", "For all $(\\mathbf {h}, \\mathbf {x}) \\in \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\varepsilon }$ , consider $\\alpha _1, \\alpha _2, \\tilde{\\mathbf {h}}$ and $\\tilde{\\mathbf {x}}$ defined in eq:orth and let $\\Delta h= h- \\alpha h_0, \\quad \\Delta x= x- \\overline{\\alpha }^{-1}x_0.$ where $\\alpha (\\mathbf {h}, \\mathbf {x})={\\left\\lbrace \\begin{array}{ll}(1 - \\delta _0)\\alpha _1, & \\text{~if~} \\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2 \\\\\\frac{1}{(1 - \\delta _0)\\overline{\\alpha _2}}, & \\text{~if~} \\Vert \\mathbf {h}\\Vert _2 < \\Vert \\mathbf {x}\\Vert _2\\end{array}\\right.", "}$ with $\\delta _0 := \\frac{\\delta }{10}$ .", "The particular form of $\\alpha (h, x)$ serves primarily for proving the local regularity condition of $G(h, x)$ , which will be evident in Lemma REF .", "The following lemma gives bounds of $\\Delta x$ and $\\Delta h$ .", "Lemma 5.15 For all $(\\mathbf {h}, \\mathbf {x}) \\in \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\varepsilon }$ with $\\varepsilon \\le \\frac{1}{15}$ , there holds $\\Vert \\Delta h\\Vert _2^2 \\le 6.1 \\delta ^2 d_0$ , $\\Vert \\Delta x\\Vert _2^2 \\le 6.1 \\delta ^2 d_0$ , and $\\Vert \\Delta h\\Vert _2^2 \\Vert \\Delta x\\Vert _2^2 \\le 8.4 \\delta ^4 d_0^2$ .", "Moreover, if we assume $(\\mathbf {h}, \\mathbf {x}) \\in \\mathcal {N}_{\\mu }$ additionally, we have $ \\sqrt{L}\\Vert \\mathbf {B}(\\Delta h)\\Vert _\\infty \\le 6\\mu \\sqrt{d_0}$ .", "We first prove that $\\Vert \\Delta h\\Vert _2^2 \\le 6.1 \\delta ^2 d_0$ , $\\Vert \\Delta x\\Vert _2^2 \\le 6.1 \\delta ^2 d_0$ , and $\\Vert \\Delta h\\Vert _2^2 \\Vert \\Delta x\\Vert _2^2 \\le 8.4 \\delta ^4 d_0^2$ : Case 1: $\\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = (1 - \\delta _0) \\alpha _1$ .", "In this case, we have $\\Delta h= \\tilde{\\mathbf {h}} + \\delta _0 \\alpha _1 \\mathbf {h}_0,\\quad \\Delta x= \\mathbf {x} - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha }_1} \\mathbf {x}_0 = \\left(\\alpha _2 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha }_1}\\right)\\mathbf {x}_0 + \\tilde{\\mathbf {x}}.$ First, notice that $\\Vert \\mathbf {h}\\Vert _2^2 \\le 4d_0$ and $\\Vert \\alpha _1 \\mathbf {h}_0\\Vert _2^2\\le \\Vert \\mathbf {h}\\Vert _2^2$ .", "By lem:orthdecomp, we have $\\Vert \\Delta h\\Vert _2^2 = \\Vert \\tilde{\\mathbf {h}}\\Vert _2^2 + \\delta _0^2\\Vert \\alpha _1 \\mathbf {h}_0\\Vert _2^2 \\le \\left(\\left(\\frac{\\delta }{1-\\delta }\\right)^2 + \\delta _0^2\\right)\\Vert \\mathbf {h}\\Vert _2^2 \\le 4.7 \\delta ^2 d_0.$ Secondly, we estimate $\\Vert \\Delta x\\Vert .$ Note that $\\Vert \\mathbf {h}\\Vert _2 \\Vert \\mathbf {x}\\Vert _2 \\le (1+\\delta )d_0$ .", "By $\\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2$ , we have $\\Vert \\mathbf {x}\\Vert _2 \\le \\sqrt{(1+\\delta )d_0}$ .", "By $\|\\alpha _2\| \\Vert \\mathbf {x}_0\\Vert _2 \\le \\Vert \\mathbf {x}\\Vert _2$ , we get $\|\\alpha _2\| \\le \\sqrt{1+\\delta }$ .", "By lem:orthdecomp, we have $\|\\overline{\\alpha _1} \\alpha _2 -1\|=\|\\alpha _1 \\overline{\\alpha _2} -1\|\\le \\delta $ , so $\\left\|\\alpha _2 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha _1}}\\right\| = \|\\alpha _2\| \\left\|\\frac{(1 - \\delta _0)(\\overline{\\alpha _1} \\alpha _2- 1) - \\delta _0}{(1 - \\delta _0)\\overline{\\alpha _1} \\alpha _2}\\right\| \\le \\frac{\\delta \\sqrt{1+ \\delta }}{1 - \\delta } + \\frac{\\sqrt{1+\\delta }\\delta _0}{(1 - \\delta _0)(1 - \\delta )}\\le 1.22 \\delta $ where $\|\\overline{\\alpha _1}\\alpha _2\| \\le \\frac{1}{1 - \\delta }.$ Moreover, by lem:orthdecomp, we have $\\Vert \\tilde{\\mathbf {x}}\\Vert _2 \\le \\frac{\\delta }{1-\\delta }\\Vert \\mathbf {x}\\Vert _2 \\le \\frac{2\\delta }{1-\\delta } \\sqrt{d_0}$ .", "Then we have $\\Vert \\Delta x\\Vert _2^2 = \\left\|\\alpha _2 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha }_1}\\right\|^2 d_0 + \\Vert \\tilde{\\mathbf {x}}\\Vert _2^2 \\le \\left(1.22^2+ \\frac{4}{(1 - \\delta )^2}\\right) \\delta ^2 d_0 \\le 6.1 \\delta ^2 d_0.$ Finally, lem:orthdecomp gives $\\Vert \\tilde{\\mathbf {h}}\\Vert _2 \\Vert \\tilde{\\mathbf {x}}\\Vert _2 \\le \\frac{\\delta ^2}{2(1 - \\delta )} d_0$ and $\|\\alpha _1\| \\le 2$ .", "Combining (REF ) and (REF ), we have $\\Vert \\Delta h\\Vert _2^2 \\Vert \\Delta x\\Vert _2^2 &\\le \\Vert \\tilde{h}\\Vert _2^2\\Vert \\tilde{x}\\Vert _2^2 + \\delta _0^2 \|\\alpha _1\|^2 \\Vert \\mathbf {h}_0\\Vert _2^2 \\Vert \\Delta x\\Vert _2^2 + \\left\|\\alpha _2 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha }_1}\\right\|^2 \\Vert \\mathbf {x}_0\\Vert _2^2 \\Vert \\Delta h\\Vert _2^2\\\\& \\le \\left(\\frac{\\delta ^4}{4(1 - \\delta )^2} d_0^2 + \\delta _0^2(4d_0) (6.1 \\delta ^2 d_0) + (1.22 \\delta )^2 d_0 (4.7 \\delta ^2 d_0 )\\right) \\le 8.4 \\delta ^4 d_0^2.$ where $\\Vert \\tilde{x}\\Vert \\le \\Vert \\Delta x\\Vert $ .", "Case 2: $\\Vert \\mathbf {h}\\Vert _2 < \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = \\frac{1}{(1-\\delta _0)\\overline{\\alpha _2}}$ .", "In this case, we have $\\Delta h= \\left(\\alpha _1 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha _2}}\\right)\\mathbf {h}_0 + \\tilde{\\mathbf {h}},\\quad \\Delta x= \\tilde{\\mathbf {x}} + \\delta _0 \\alpha _2 \\mathbf {x}_0.$ By the symmetry of $ \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\varepsilon }$ , we can prove $\|\\alpha _1\| \\le \\sqrt{1+\\delta }$ , $\\left\|\\alpha _1 - \\frac{1}{(1 - \\delta _0)\\overline{\\alpha _2}}\\right\| \\le \\frac{\\delta \\sqrt{1+ \\delta }}{1 - \\delta } + \\frac{\\sqrt{1+\\delta }\\delta _0}{(1 - \\delta _0)(1 - \\delta )} \\le 1.22 \\delta .$ Moreover, we can prove $\\Vert \\Delta h\\Vert _2^2 \\le 6.1 \\delta ^2 d_0$ , $\\Vert \\Delta x\\Vert _2^2 \\le 4.7 \\delta ^2 d_0$ and $\\Vert \\Delta h\\Vert _2^2 \\Vert \\Delta x\\Vert _2^2 \\le 8.4 \\delta ^4 d^2$ .", "Next, under the additional assumption $(\\mathbf {h}, \\mathbf {x}) \\in \\mathcal {N}_{\\mu }$ , we now prove $\\sqrt{L}\\Vert \\mathbf {B}(\\Delta h)\\Vert _\\infty \\le 6\\mu \\sqrt{d_0}$ : Case 1: $\\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = (1 - \\delta _0) \\alpha _1$ .", "By lem:orthdecomp gives $\|\\alpha _1\| \\le 2$ , which implies $\\sqrt{L}\\Vert \\mathbf {B}(\\Delta h)\\Vert _\\infty &\\le \\sqrt{L}\\Vert \\mathbf {B}\\mathbf {h}\\Vert _\\infty + (1 - \\delta _0) \|\\alpha _1\|\\sqrt{L}\\Vert \\mathbf {B}\\mathbf {h}_0\\Vert _\\infty \\\\&\\le 4\\mu \\sqrt{d_0} + 2(1 - \\delta _0)\\mu _h \\sqrt{d}_0 \\le 6\\mu \\sqrt{d_0}.$ Case 2: $\\Vert \\mathbf {h}\\Vert _2 < \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = \\frac{1}{(1-\\delta _0)\\overline{\\alpha _2}}$ .", "Notice that in this case we have $\|\\alpha _1\| \\le \\sqrt{1+\\delta }$ , so $\\frac{1}{\|(1 -\\delta _0)\\overline{\\alpha _2}\|}= \\frac{\|\\alpha _1\|}{\|(1 -\\delta _0)\\overline{\\alpha _2} \\alpha _1\|} \\le \\frac{\\sqrt{1+\\delta }}{\|1 - \\delta _0\|\|1 - \\delta \|}$ .", "Therefore $\\sqrt{L}\\Vert \\mathbf {B}(\\Delta h)\\Vert _\\infty &\\le \\sqrt{L}\\Vert \\mathbf {B}(\\mathbf {h})\\Vert _\\infty + \\frac{1}{(1 - \\delta _0) \|\\overline{\\alpha _2}\|} \\sqrt{L}\\Vert \\mathbf {B}(\\mathbf {h}_0)\\Vert _\\infty \\\\&\\le 4\\mu \\sqrt{d_0} + \\frac{\\sqrt{1+\\delta }}{\|1 - \\delta _0\|\|1 - \\delta \|}\\mu _h \\sqrt{d}_0 \\le 5.2 \\mu \\sqrt{d_0}.$ Lemma 5.16 For any $(h, x) \\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ with $\\varepsilon \\le \\frac{1}{15}$ , the following inequality holds uniformly: $\\operatorname{Re}\\left(\\left\\langle \\nabla F_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla F_{x}, \\Delta x\\right\\rangle \\right)\\ge \\frac{\\delta ^2 d_0^2}{8} - 2\\delta d_0 \\Vert \\mathcal {A}^(e)\\Vert ,$ provided $L \\ge C\\mu ^2 (K+N)\\log ^2 L$ for some numerical constant $C$ .", "In this section, define $U$ and $V$ as $U= \\alpha h_0\\Delta x^ + \\overline{\\alpha }^{-1}\\Delta hx_0^* \\in T, \\quad V= \\Delta h\\Delta x^,$ which gives $hx^ - h_0x_0^* = U+ V.$ Notice that generally $V\\in T^\\perp $ does not hold.", "Recall that $\\nabla F_{h} = \\mathcal {A}^(\\mathcal {A}(hx^ - h_0x_0^) - e) x, \\quad \\nabla F_{x} = [\\mathcal {A}^(\\mathcal {A}(hx^* - h_0x_0^) - e)]^ h.$ Define $I_0 := \\left\\langle \\nabla _{\\mathbf {h}} F, \\Delta h\\right\\rangle + \\overline{\\left\\langle \\nabla _{x} F, \\Delta x\\right\\rangle }$ and we have $\\operatorname{Re}(I_0)=\\operatorname{Re}\\left(\\left\\langle \\nabla _{\\mathbf {h}} F, \\Delta h\\right\\rangle + \\left\\langle \\nabla _{x} F, \\Delta x\\right\\rangle \\right)$ .", "Since $I_0& = & \\left\\langle \\mathcal {A}^(\\mathcal {A}(hx^ - h_0x_0^) - e), \\Delta hx^ + h\\Delta x^* \\right\\rangle \\\\& = & \\left\\langle \\mathcal {A}(hx^* - h_0x_0^) - e, \\mathcal {A}(hx^ - h_0x_0^* + \\Delta h\\Delta x^) \\right\\rangle \\\\& = & \\left\\langle \\mathcal {A}(U+ V), \\mathcal {A}(U+ 2V)\\right\\rangle - \\left\\langle \\mathcal {A}^(e), U+ 2V\\right\\rangle : = I_{01} + I_{02}$ where $\\Delta hx^* + h\\Delta x^* = hx^* - h_0x_0^* + \\Delta h\\Delta x^.$ By the Cauchy-Schwarz inequality, $\\operatorname{Re}(I_{01})$ has the lower bound $\\operatorname{Re}(I_{01})& \\ge & \\Vert \\mathcal {A}(U)\\Vert ^2 - 3\\Vert \\mathcal {A}(U)\\Vert \\Vert \\mathcal {A}(V)\\Vert + 2\\Vert \\mathcal {A}(V)\\Vert ^2 \\nonumber \\\\& \\ge & (\\Vert \\mathcal {A}(U)\\Vert - \\Vert \\mathcal {A}(V)\\Vert ) (\\Vert \\mathcal {A}(U)\\Vert - 2\\Vert \\mathcal {A}(V)\\Vert ).$ In the following, we will give an upper bound for $\\Vert \\mathcal {A}(V)\\Vert $ and a lower bound for $\\Vert \\mathcal {A}(U)\\Vert $ .", "Upper bound for $\\Vert \\mathcal {A}(V)\\Vert $ : By lem:DxDh and lem:key, we have $\\Vert \\mathcal {A}(V)\\Vert ^2 &\\le \\left(\\frac{4}{3}\\Vert \\Delta h\\Vert ^2 + 2\\Vert \\Delta h\\Vert \\Vert \\mathbf {B}\\Delta h\\Vert _\\infty \\sqrt{2(K+N)\\log L} + 8\\Vert \\mathbf {B}\\Delta h\\Vert _\\infty ^2 (K+N) (\\log L)\\right)\\Vert \\Delta x\\Vert ^2\\\\&\\le \\left(11.2 \\delta ^2 + C_0\\left(\\delta \\mu \\sqrt{\\frac{1}{L} (K+N)(\\log L)} +\\frac{1}{L}\\mu ^2 (K+N)(\\log L)\\right) \\right) \\delta ^2 d_0^2\\\\& \\le \\left(11.2\\delta ^2 + C_0\\left( \\frac{\\delta }{\\sqrt{ C\\log L}} + \\frac{1}{C\\log L}\\right)\\right) \\delta ^2 d_0^2$ for some numerical constant $C_0$ .", "Then by $\\delta \\le \\varepsilon \\le \\frac{1}{15}$ and letting $L \\ge C\\mu ^2 (K + N)\\log ^2 L$ for a sufficiently large numerical constant $C$ , there holds $\\Vert \\mathcal {A}(V)\\Vert ^2 <\\frac{\\delta ^2 d_0^2}{16} \\Rightarrow \\Vert \\mathcal {A}(V)\\Vert \\le \\frac{\\delta d_0}{4}.$ Lower bound for $\\Vert \\mathcal {A}(U)\\Vert $ : By lem:DxDh, we have $\\Vert V\\Vert _F = \\Vert \\Delta h\\Vert _2 \\Vert \\Delta x\\Vert _2 \\le 2.9 \\delta ^2 d_0,$ and therefore $\\Vert U\\Vert _F \\ge d_0 \\delta - 2.9\\delta ^2 d_0 \\ge \\frac{4}{5} d_0 \\delta .$ if $\\varepsilon \\le \\frac{1}{15}$ .", "Since $U\\in T$ , by lem:ripu, there holds $\\Vert \\mathcal {A}(U)\\Vert \\ge \\sqrt{\\frac{9}{10}}\\Vert U\\Vert _F \\ge \\frac{3}{4} d_0 \\delta .$ With the upper bound of $\\mathcal {A}(V)$ in (REF ), the lower bound of $\\mathcal {A}(U)$ in (REF ), and (REF ), we finally arrive at $\\operatorname{Re}(I_{01}) \\ge \\frac{\\delta ^2 d_0^2}{8}.$ Now let us give a lower bound for $\\operatorname{Re}(I_{02})$ , $\\operatorname{Re}(I_{02}) \\ge - \\Vert \\mathcal {A}^(e)\\Vert \\Vert U+ 2V\\Vert _* \\ge - \\sqrt{2}\\Vert \\mathcal {A}^(e)\\Vert \\Vert U+ 2V\\Vert _F \\ge -2\\delta d_0 \\Vert \\mathcal {A}^(e)\\Vert $ where $\\Vert \\cdot \\Vert $ and $\\Vert \\cdot \\Vert _$ are a pair of dual norms and $\\Vert U+ 2V\\Vert _F \\le \\Vert U+ V\\Vert _F + \\Vert V\\Vert _F \\le \\delta d_0+ 2.9\\delta ^2 d_0 \\le 1.2\\delta d_0$ if $\\delta \\le \\varepsilon \\le \\frac{1}{15}.$ Combining the estimation of $\\operatorname{Re}(I_{01})$ and $\\operatorname{Re}(I_{02})$ above leads to $\\operatorname{Re}( \\left\\langle \\nabla F_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla F_{x}, \\Delta x\\right\\rangle \\ge \\frac{\\delta ^2 d_0^2}{8} - 2\\delta d_0 \\Vert \\mathcal {A}^(e)\\Vert $ as we desired.", "Lemma 5.17 For any $(h, x) \\in \\mathcal {N}_{d_0}\\bigcap \\mathcal {N}_{\\varepsilon }$ with $\\varepsilon \\le \\frac{1}{15}$ and $\\frac{9}{10}d_0 \\le d\\le \\frac{11}{10}d_0$ , the following inequality holds uniformly $\\operatorname{Re}\\left(\\left\\langle \\nabla G_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla G_{x}, \\Delta x\\right\\rangle \\right)\\ge {\\frac{\\delta }{5}}\\sqrt{ \\rho G(h, x)},$ where $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2.$ Recall that $G_0^{\\prime }(z) = 2\\max \\lbrace z - 1, 0\\rbrace = 2\\sqrt{G_0(z)}$ .", "Using the Wirtinger derivative of $G$ in () and (), we have $\\left\\langle \\nabla G_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla G_{x}, \\Delta x\\right\\rangle := \\frac{\\rho }{2d} \\left(H_1 +H_2 +H_3\\right),$ where $H_1 = G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right) \\left\\langle h, \\Delta h\\right\\rangle , \\quad H_2 = G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) \\left\\langle x, \\Delta x\\right\\rangle ,$ and $H_3 = \\frac{L}{4\\mu ^2} \\sum _{l=1}^L G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) \\left\\langle b_lb_l^h, \\Delta h\\right\\rangle .$ We will give lower bounds for $H_1$ , $H_2$ and $H_3$ for two cases." ], [ "Case 1", "$\\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = (1 - \\delta _0) \\alpha _1$ ." ], [ "Lower bound of $H_1$ :", "Notice that $\\Delta h= h- \\alpha h_0 = h- (1 - \\delta _0)\\alpha _1 h_0 = h- (1-\\delta _0)(h- \\tilde{h}) = \\delta _0 h+ (1 - \\delta _0) \\tilde{h}.$ We have $\\left\\langle h, \\Delta h\\right\\rangle = \\delta _0 \\Vert h\\Vert _2^2 + (1 - \\delta _0) \\left\\langle h, \\tilde{h}\\right\\rangle = \\delta _0 \\Vert h\\Vert _2^2 + (1 - \\delta _0) \\Vert \\tilde{h}\\Vert _2^2 \\ge \\delta _0 \\Vert h\\Vert _2^2$ , which implies that $H_1 \\ge G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right) \\frac{\\delta }{10} \\Vert h\\Vert _2^2$ .", "We claim that $H_1 \\ge \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right).$ In fact, if $\\Vert h\\Vert _2^2 \\le 2d$ , we get $H_1 = 0 = \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right)$ ; If $\\Vert h\\Vert _2^2 > 2d$ , we get (REF ) straightforwardly." ], [ "Lower bound of $H_2$ :", "The assumption $\\Vert \\mathbf {h}\\Vert _2 \\ge \\Vert \\mathbf {x}\\Vert _2$ gives $\\Vert \\mathbf {x}\\Vert _2^2 \\le \\Vert \\mathbf {x}\\Vert _2 \\Vert \\mathbf {h}\\Vert _2 \\le (1+\\delta ) d_0 \\le 1.1(1 + \\delta )d_0 < 2d,$ which implies that $H_2 = G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) \\left\\langle x, \\Delta x\\right\\rangle = 0 = \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right).$" ], [ "Lower bound of $H_3$ :", "When $L\|b_l^h\|^2 \\le 8d\\mu ^2$ , $\\frac{L}{4\\mu ^2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) \\left\\langle b_lb_l^h, \\Delta h\\right\\rangle = 0 = \\frac{d}{2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$ When $L\|b_l^h\|^2 > 8d\\mu ^2$ , by lem:orthdecomp, there holds $\|\\alpha _1\| \\le 2$ .", "Then by $\\mu _h \\le \\mu $ , we have $\\operatorname{Re}(\\left\\langle b_lb_l^h, \\Delta h\\right\\rangle ) & = & \\operatorname{Re}(\|b_l^h\|^2- \\alpha \\left\\langle b_l^h,b_l^h_0\\right\\rangle )\\\\& \\ge & \|b_l^h\| (\|b_l^h\| - (1-\\delta _0)\|\\alpha _1\| \|b_l^h_0\|)\\\\&\\ge & \|b_l^h\| (\|b_l^h\| - 2\\mu \\sqrt{d_0/L}) \\\\& \\ge & \\sqrt{\\frac{8d\\mu ^2}{L}} \\left(\\sqrt{\\frac{8d\\mu ^2}{L}} - 2\\mu \\sqrt{\\frac{10d}{9L}}\\right) \\ge \\frac{2d\\mu ^2}{L},$ where $(1 - \\delta _0)\|\\alpha _1\| \|b_l^h_0\| \\le \\frac{2\\mu _h\\sqrt{d_0}}{\\sqrt{L}} \\le \\frac{2\\mu \\sqrt{10d}}{\\sqrt{9L}}.$ This implies that $\\frac{L}{4\\mu ^2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)\\operatorname{Re}(\\left\\langle b_lb_l^h, \\Delta h\\right\\rangle ) \\ge \\frac{d}{2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$ So we always have $\\operatorname{Re}(H_3) \\ge \\sum _{l=1}^L \\frac{d}{2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$" ], [ "Case 2:", "$\\Vert \\mathbf {h}\\Vert _2 < \\Vert \\mathbf {x}\\Vert _2$ and $\\alpha = \\frac{1}{(1-\\delta _0)\\overline{\\alpha _2}}$ ." ], [ "Lower bound of $H_1$ :", "The assumption $\\Vert \\mathbf {h}\\Vert _2 < \\Vert \\mathbf {x}\\Vert _2$ gives $\\Vert \\mathbf {h}\\Vert _2^2 \\le \\Vert \\mathbf {x}\\Vert _2 \\Vert \\mathbf {h}\\Vert _2 \\le (1+\\delta ) d_0 < 2d,$ which implies that $H_1 = G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right) \\left\\langle h, \\Delta h\\right\\rangle = 0 = \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right).$" ], [ "Lower bound of $H_2$ :", "Notice that $\\Delta x= x- \\overline{\\alpha }^{-1} x_0 = x- (1 - \\delta _0)\\alpha _2 x_0 = x- (1-\\delta _0)(x- \\tilde{x}) = \\delta _0 x+ (1 - \\delta _0) \\tilde{x}.$ We have $\\left\\langle x, \\Delta x\\right\\rangle = \\delta _0 \\Vert x\\Vert _2^2 + (1 - \\delta _0) \\left\\langle x, \\tilde{x}\\right\\rangle = \\delta _0 \\Vert x\\Vert _2^2 + (1 - \\delta _0) \\Vert \\tilde{x}\\Vert _2^2 \\ge \\delta _0 \\Vert x\\Vert _2^2$ , which implies that $H_2 \\ge G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) \\frac{\\delta }{10} \\Vert x\\Vert _2^2$ .", "We claim that $H_2 \\ge \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right).$ In fact, if $\\Vert x\\Vert _2^2 \\le 2d$ , we get $H_2 = 0 = \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right)$ ; If $\\Vert x\\Vert _2^2 > 2d$ , we get (REF ) straightforwardly." ], [ "Lower bound of $H_3$ :", "When $L\|b_l^h\|^2 \\le 8d \\mu ^2$ , $\\frac{L}{4\\mu ^2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) \\left\\langle b_lb_l^h, \\Delta h\\right\\rangle = 0 = \\frac{d}{4} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$ When $L\|b_l^h\|^2 > 8d \\mu ^2$ , by lem:orthdecomp, there hold $\|\\alpha _1\\overline{\\alpha _2} -1\|\\le \\delta $ and $\|\\alpha _1\| \\le 2$ , which implies that $\\frac{1}{(1-\\delta _0)\|\\overline{\\alpha _2}\|} = \\frac{\|\\alpha _1\|}{(1 - \\delta _0)\|\\alpha _1 \\overline{\\alpha _2}\|} \\le \\frac{2}{(1-\\delta _0)(1-\\delta )}.$ By $\\mu _h \\le \\mu $ and $\\delta \\le \\varepsilon \\le \\frac{1}{15}$ , similarly we have $\\operatorname{Re}(\\left\\langle b_lb_l^h, \\Delta h\\right\\rangle ) &\\ge \|b_l^h\| \\left(\|b_l^h\| - \\frac{2}{(1-\\delta _0)(1-\\delta )} \|b_l^h_0\|\\right)\\\\&\\ge \\left(8 - 4\\sqrt{\\frac{20}{9}}\\frac{1}{(1-\\delta _0)(1-\\delta )}\\right) \\frac{d\\mu ^2 }{L} > \\frac{d\\mu ^2}{L}.$ This implies that for $1\\le l\\le L$ , $\\frac{L}{4\\mu ^2} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) \\operatorname{Re}\\left\\langle b_lb_l^h, \\Delta h\\right\\rangle \\ge \\frac{d}{4} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$ So we always have $\\operatorname{Re}(H_3) \\ge \\sum _{l=1}^L \\frac{d}{4} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right).$ To sum up the two cases, we have $\\operatorname{Re}(H_1 + H_2 + H_3) &\\ge \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right) + \\frac{\\delta d}{5}G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) + \\sum _{l=1}^L \\frac{d}{4} G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)\\\\&\\ge \\frac{2\\delta d}{5}\\left( \\sqrt{G_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right)} + \\sqrt{G_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right)} + \\sum _{l=1}^L \\sqrt{G_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)} \\right)\\\\&\\ge \\frac{2\\delta d}{5}\\left( \\sqrt{G_0\\left( \\frac{\\Vert h\\Vert ^2}{2d}\\right) + G_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right)+ \\sum _{l=1}^L G_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)} \\right)$ where $G_0^{\\prime }(z) = 2\\sqrt{G_0(z)}$ and it implies eq:Gregularity.", "Lemma 5.18 Let $\\widetilde{F}$ be as defined in (REF ), then there exists a positive constant $\\omega $ such that $\\Vert \\nabla \\widetilde{F}(h, x)\\Vert ^2 \\ge \\omega \\left[ \\widetilde{F}(h, x) - c \\right]_+$ with $c = \\Vert e\\Vert ^2 + 1700 \\Vert \\mathcal {A}^(e)\\Vert ^2$ and $\\omega = \\frac{d_0}{5000}$ for all $(h, x) \\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ .", "Here we set $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2.$ Following from Lemma REF and Lemma REF , we have $\\operatorname{Re}( \\left\\langle \\nabla F_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla F_{x}, \\Delta x\\right\\rangle )& \\ge & \\frac{\\delta ^2 d_0^2}{8} - 2\\delta d_0 \\Vert \\mathcal {A}^(e)\\Vert \\\\\\operatorname{Re}( \\left\\langle \\nabla G_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla G_{x}, \\Delta x\\right\\rangle )& \\ge & \\frac{\\delta d}{5} \\sqrt{ G(h, x)} \\ge \\frac{9\\delta d_0}{50}\\sqrt{G(h, x)}$ for $\\alpha = (1 - \\delta _0)\\alpha _1$ or $\\frac{1}{(1 - \\delta )\\overline{\\alpha _2}}$ and $\\forall (h, x) \\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }$ where $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2 \\ge d^2$ and $\\frac{9}{10}d_0 \\le d \\le \\frac{11}{10}d_0$ .", "Adding them together gives $\\frac{\\delta ^2 d_0^2}{8} + \\frac{ 9\\delta d_0}{50}\\sqrt{ G(h, x)} - 2\\delta d_0 \\Vert \\mathcal {A}^(e) \\Vert & \\le & \\operatorname{Re}\\left(\\left\\langle \\nabla F_{h}+ \\nabla G_{h}, \\Delta h\\right\\rangle + \\left\\langle \\nabla F_{x} + \\nabla G_{x} \\right\\rangle \\right) \\nonumber \\\\& \\le & \\Vert \\nabla \\widetilde{F}_{h}\\Vert \\Vert \\Delta h\\Vert + \\Vert \\nabla \\widetilde{F}_{x} \\Vert \\Vert \\Delta x\\Vert \\nonumber \\\\& \\le & \\sqrt{2} \\Vert \\nabla \\widetilde{F}(h, x)\\Vert \\max \\lbrace \\Vert \\Delta h\\Vert , \\Vert \\Delta x\\Vert \\rbrace \\nonumber \\\\& \\le & 3.6 \\delta \\sqrt{d_0} \\Vert \\nabla \\widetilde{F}(h, x)\\Vert $ where both $\\Vert \\Delta h\\Vert $ and $\\Vert \\Delta x\\Vert $ are bounded by $2.5\\delta \\sqrt{d_0}$ in Lemma REF .", "Note that $\\sqrt{2\\left[ \\operatorname{Re}(\\left\\langle \\mathcal {A}^(e), hx^ - h_0x_0^\\right\\rangle ) \\right]_+} \\le \\sqrt{ 2\\sqrt{2} \\Vert \\mathcal {A}^(e)\\Vert \\delta d_0} \\le \\frac{\\sqrt{5}\\delta d_0}{4} + \\frac{4}{\\sqrt{5}}\\Vert \\mathcal {A}^(e)\\Vert .$ Dividing both sides of (REF ) by $\\delta d_0$ , we obtain $\\frac{3.6}{\\sqrt{d_0}} \\Vert \\nabla \\widetilde{F}(h, x)\\Vert \\ge \\frac{\\delta d_0}{12} + \\frac{9}{50}\\sqrt{G(h, x)} + \\frac{\\delta d_0 }{24} - 2\\Vert \\mathcal {A}^(e)\\Vert .$ The Local RIP condition implies $F_0(h, x) \\le \\frac{5}{4}\\delta ^2 d_0^2$ and hence $\\frac{\\delta d_0}{12} \\ge \\frac{1}{6\\sqrt{5}}\\sqrt{F_0(h, x)}$ , where $F_0$ is defined in (REF ).", "Combining the equation above and (REF ), $\\frac{3.6}{\\sqrt{d_0}} \\Vert \\nabla \\widetilde{F}(h, x)\\Vert & \\ge & \\frac{1}{6\\sqrt{5}} \\Big [ \\left(\\sqrt{F_0(h, x)} +\\sqrt{2\\left[ \\operatorname{Re}(\\left\\langle \\mathcal {A}^(e), hx^ - h_0x_0^\\right\\rangle ) \\right]_+} + \\sqrt{G(h, x)}\\right) \\\\&& + \\frac{\\sqrt{5}\\delta d_0}{4} - \\left( \\frac{\\sqrt{5}\\delta d_0}{4} + \\frac{4}{\\sqrt{5}}\\Vert \\mathcal {A}^(e)\\Vert \\right)\\Big ] - 2\\Vert \\mathcal {A}^(e)\\Vert \\\\& \\ge & \\frac{1}{6\\sqrt{5}} \\left[ \\sqrt{ \\left[\\widetilde{F}(h, x) - \\Vert e\\Vert ^2\\right]_+} - 29\\Vert \\mathcal {A}^(e)\\Vert \\right]$ where $\\widetilde{F}(h, x) - \\Vert e\\Vert ^2 \\le F_0(h, x) + 2 [\\operatorname{Re}(\\left\\langle \\mathcal {A}^(e), hx^ - h_0x_0^\\right\\rangle )]_+ + G(h, x)$ follows from definition and (REF ).", "Finally, we have $\\Vert \\nabla \\widetilde{F}(h, x)\\Vert ^2 \\ge \\frac{d_0}{2500} \\left[ \\sqrt{\\left[\\widetilde{F}(h, x) - \\Vert e\\Vert ^2\\right]_+} - 29 \\Vert \\mathcal {A}^(e)\\Vert \\right]_+^2$ for all $(h, x)\\in \\mathcal {N}_{d_0} \\cap \\mathcal {N}_{\\mu } \\cap \\mathcal {N}_{\\varepsilon }.$ For any nonnegative fixed real numbers $a$ and $b$ , we have $[\\sqrt{(x - a)_+} - b ]_+ + b \\ge \\sqrt{(x - a)_+}$ and it implies $( x - a)_+ \\le 2 ( [\\sqrt{(x - a)_+} - b ]_+^2 + b^2) \\Longrightarrow [\\sqrt{(x - a)_+} - b ]_+^2 \\ge \\frac{(x - a)_+}{2} - b^2.$ Therefore, by setting $ a = \\Vert e\\Vert $ and $b = 30\\Vert \\mathcal {A}^(e)\\Vert $ , there holds $\\Vert \\nabla \\widetilde{F}(h, x)\\Vert ^2& \\ge & \\frac{d_0}{2500} \\left[ \\frac{\\widetilde{F}(h, x) - \\Vert e\\Vert ^2 }{2} - 850 \\Vert \\mathcal {A}^(e)\\Vert ^2 \\right]_+ \\\\& \\ge & \\frac{d_0}{5000} \\left[ \\widetilde{F}(h, x) - (\\Vert e\\Vert ^2 + 1700 \\Vert \\mathcal {A}^(e)\\Vert ^2) \\right]_+.$" ], [ "Local smoothness", "Lemma 5.19 For any $z: = (h, x)$ and $w: = (u, v)$ such that $\\mathbf {z}, \\mathbf {z}+\\mathbf {w} \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ , there holds $\\Vert \\nabla \\widetilde{F}(z+ w) - \\nabla \\widetilde{F}(z) \\Vert \\le C_L \\Vert w\\Vert ,$ with $C_L \\le \\sqrt{2}d_0\\left[ 10 \\Vert \\mathcal {A}\\Vert ^2 + \\frac{\\rho }{d^2} \\left(5 + \\frac{3L}{2\\mu ^2}\\right)\\right]$ where $\\rho \\ge d^2 + 2\\Vert e\\Vert ^2$ and $\\Vert \\mathcal {A}\\Vert ^2\\le \\sqrt{N\\log (NL/2) + \\gamma \\log L}$ holds with probability at least $1 - L^{-\\gamma }$ from Lemma REF .", "In particular, $L = \\mathcal {O}((\\mu ^2 + \\sigma ^2) (K + N)\\log ^2 L)$ and $\\Vert e\\Vert ^2 = \\mathcal {O}(\\sigma ^2d_0^2)$ follows from $\\Vert e\\Vert ^2 \\sim \\frac{\\sigma ^2d_0^2}{2L} \\chi ^2_{2L}$ and (REF ).", "Therefore, $C_L$ can be simplified into $C_L = \\mathcal {O}(d_0(1 + \\sigma ^2)(K + N)\\log ^2 L )$ by choosing $\\rho \\approx d^2 + 2\\Vert e\\Vert ^2.$ By lem:betamu, we have $\\mathbf {z}=(\\mathbf {h}, \\mathbf {x}), \\mathbf {z}+\\mathbf {w}=(\\mathbf {h}+\\mathbf {u}, \\mathbf {x}+\\mathbf {v}) \\in \\mathcal {N}_{d_0}\\cap \\mathcal {N}_{\\mu }$ .", "Note that $\\nabla \\widetilde{F}= (\\nabla \\widetilde{F}_{h}, \\nabla \\widetilde{F}_{x}) = (\\nabla F_{h} + \\nabla G_{h}, \\nabla F_{x} + \\nabla G_{x}),$ where $\\nabla F_{h} = \\mathcal {A}^(\\mathcal {A}(hx^* - h_0x_0^) - e) x, \\quad \\nabla F_{x} = (\\mathcal {A}^(\\mathcal {A}(hx^* - h_0x_0^) - e))^ h,$ and $\\nabla G_{h}= \\frac{\\rho }{2d}\\left[G^{\\prime }_0\\left(\\frac{\\Vert h\\Vert ^2}{2d}\\right) h+ \\frac{L}{4\\mu ^2} \\sum _{l=1}^L G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) b_lb_l^h\\right],\\quad \\nabla G_{x} = \\frac{\\rho }{2d}G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) x.$" ], [ "Step 1:", "we estimate the upper bound of $\\Vert \\nabla F_{h}(z+ w) - \\nabla F_{h}(z) \\Vert $ .", "A straightforward calculation gives $\\nabla F_{h}(z+ w) - \\nabla F_{h}(z) = \\mathcal {A}^\\mathcal {A}( ux^ + hv^* + uv^) x+ \\mathcal {A}^\\mathcal {A}((h+u)(x+v)^* - h_0x_0^) v- \\mathcal {A}^(e)v.$ Note that $z, z+w\\in \\mathcal {N}_{d_0}$ directly implies $\\Vert ux^* + hv^* + uv^\\Vert _F \\le \\Vert \\mathbf {u}\\Vert \\Vert \\mathbf {x}\\Vert + \\Vert \\mathbf {h}+\\mathbf {u}\\Vert \\Vert \\mathbf {v}\\Vert \\le 2\\sqrt{d_0} (\\Vert u\\Vert + \\Vert v\\Vert )$ where $\\Vert h+ u\\Vert \\le 2\\sqrt{d_0}.$ Moreover, $\\mathbf {z} + \\mathbf {w} \\in \\mathcal {N}_{\\varepsilon }$ implies $\\Vert (h+u)(x+v)^ - h_0x_0^\\Vert _F \\le \\varepsilon d_0.$ Combined with $\\Vert \\mathcal {A}^(e)\\Vert \\le \\varepsilon d_0$ and $\\Vert x\\Vert \\le 2\\sqrt{d_0}$ , we have $\\Vert \\nabla F_{h}(z+ w) - \\nabla F_{h}(z) \\Vert & \\le & 4d_0 \\Vert \\mathcal {A}\\Vert ^2(\\Vert u\\Vert + \\Vert v\\Vert ) + \\varepsilon d_0 \\Vert \\mathcal {A}\\Vert ^2 \\Vert v\\Vert + \\varepsilon d_0 \\Vert v\\Vert \\nonumber \\\\& \\le & 5d_0 \\Vert \\mathcal {A}\\Vert ^2 ( \\Vert u\\Vert + \\Vert v\\Vert ).$" ], [ "Step 2:", "we estimate the upper bound of $\\Vert \\nabla F_{x}(z+ w) - \\nabla F_{x}(z)\\Vert $ .", "Due to the symmetry between $\\nabla F_{h}$ and $\\nabla F_{x}$ , we have, $\\Vert \\nabla F_{x}(z+ w) - \\nabla F_{x}(z) \\Vert \\le 5d_0\\Vert \\mathcal {A}\\Vert ^2 ( \\Vert u\\Vert + \\Vert v\\Vert ).$" ], [ "Step 3:", "we estimate the upper bound of $\\Vert \\nabla G_{x}(z+ w) - \\nabla G_{x}(z)\\Vert $ .", "Notice that $G_0^{\\prime }(z) = 2\\max \\lbrace z - 1, 0\\rbrace $ , which implies that for any $z_1, z_2, z\\in \\hbox{{R}}$ , there holds $\|G^{\\prime }_0(z_1) - G^{\\prime }_0(z_2)\| \\le 2\|z_1 - z_2\|, \\quad G^{\\prime }(z) \\le 2\|z\|,$ although $G^{\\prime }(z)$ is not differentiable at $z = 1$ .", "Therefore, by (REF ), it is easy to show that $\\left\| G^{\\prime }_0\\left( \\frac{\\Vert x+ v\\Vert ^2}{2d}\\right) - G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) \\right\| \\le \\frac{\\Vert x+ v\\Vert + \\Vert x\\Vert }{d} \\Vert v\\Vert \\le \\frac{4\\sqrt{d_0}}{d}\\Vert v\\Vert $ where $\\Vert x+ v\\Vert \\le 2\\sqrt{d_0}.$ Therefore, by $z, z+w\\in \\mathcal {N}_{d_0}$ , we have $\\Vert \\nabla G_{x}(z+ w) - \\nabla G_{x}(z) \\Vert &\\le \\frac{\\rho }{2d} \\left\| G^{\\prime }_0\\left( \\frac{\\Vert x+ v\\Vert ^2}{2d}\\right) - G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right)\\right\| \\Vert x+ v\\Vert + \\frac{\\rho }{2d} G^{\\prime }_0\\left( \\frac{\\Vert x\\Vert ^2}{2d}\\right) \\Vert v\\Vert \\nonumber \\\\& \\le \\frac{\\rho }{2d} \\left( \\frac{8d_0 \\Vert v\\Vert }{d} + \\frac{2d_0\\Vert v\\Vert }{d} \\right) \\le \\frac{5d_0 \\rho }{d^2} \\Vert v\\Vert .", "$" ], [ "Step 4:", "we estimate the upper bound of $\\Vert \\nabla G_{h}(z+ w) - \\nabla G_{h}(z)\\Vert $ .", "Denote $\\nabla G_{h}(z+ w) - \\nabla G_{h}(z) &= \\frac{\\rho }{2d}\\left[G^{\\prime }_0\\left(\\frac{\\Vert h+ u\\Vert ^2}{2d}\\right) (h+ u) - G^{\\prime }_0\\left(\\frac{\\Vert h\\Vert ^2}{2d}\\right) h\\right]\\\\&+ \\frac{\\rho L}{8d\\mu ^2 }\\sum _{l=1}^L \\left[G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right) b_l^(h+ u) - G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) b_l^h\\right]b_l\\\\&:= \\mathbf {j}_1 + \\mathbf {j}_2.$ Similar to eq:LipGx, we have $\\Vert \\mathbf {j}_1\\Vert \\le \\frac{5d_0 \\rho }{d^2} \\Vert u\\Vert .$ Now we control $\\Vert \\mathbf {j}_2\\Vert $ .", "Since $\\mathbf {z}, \\mathbf {z}+\\mathbf {w} \\in \\mathcal {N}_{\\mu }$ , we have $\\left\| G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right) - G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)\\right\|& \\le & \\frac{L}{4d\\mu ^2}\\left(\|\\mathbf {b}_l^(\\mathbf {h} + \\mathbf {u})\| + \|\\mathbf {b}_l^\\mathbf {h}\|\\right) \|b_l^u\| \\nonumber \\\\& \\le & \\frac{2\\sqrt{d_0 L}}{d\\mu } \|\\mathbf {b}_l^\\mathbf {u}\|.", "$ and $\\left\|G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right)\\right\| \\le 2\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2} \\le \\frac{4d_0}{d}$ where both $\\max _l \|b_l^(h+ u)\|$ and $\\max _l \|b_l^h\|$ are bounded by $\\frac{4 \\sqrt{d_0}\\mu }{\\sqrt{L}}$ .", "Let $\\alpha _l$ be $\\alpha _l &:=G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right) b_l^(h+ u) - G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right) b_l^h\\\\& = \\left(G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right) - G^{\\prime }_0\\left(\\frac{L\|b_l^h\|^2}{8d\\mu ^2}\\right)\\right)\\mathbf {b}_l^\\mathbf {h} + G^{\\prime }_0\\left(\\frac{L\|b_l^(h+ u)\|^2}{8d\\mu ^2}\\right)\\mathbf {b}_l^\\mathbf {u}.$ Applying (REF ) and (REF ) leads to $\|\\alpha _l\| \\le \\frac{2\\sqrt{d_0 L}}{d \\mu }\|\\mathbf {b}_l^\\mathbf {u}\|\\left(4\\mu \\sqrt{\\frac{d_0}{L}}\\right) + \\frac{4d_0}{d}\|\\mathbf {b}_l^\\mathbf {u}\| = \\frac{12 d_0}{d}\|\\mathbf {b}_l^\\mathbf {u}\|.$ Since $\\sum _{l=1}^L \\alpha _l \\mathbf {b}_l = \\mathbf {B}^* \\begin{bmatrix} \\alpha _1 \\\\ \\vdots \\\\ \\alpha _L \\end{bmatrix}$ and $\\Vert \\mathbf {B}\\Vert =1$ , there holds $\\left\\Vert \\sum _{l=1}^L \\alpha _l \\mathbf {b}_l \\right\\Vert ^2 \\le \\sum _{l=1}^L \|\\alpha _l\|^2 \\le \\left(\\frac{12 d_0}{d}\\right)^2 \\sum _{l=1}^L \|\\mathbf {b}_l^\\mathbf {u}\|^2 = \\left(\\frac{12 d_0}{d}\\right)^2 \\Vert \\mathbf {B}\\mathbf {u}\\Vert ^2 \\le \\left(\\frac{12 d_0}{d}\\right)^2 \\Vert \\mathbf {u}\\Vert ^2.$ This implies that $\\Vert \\mathbf {j}_2\\Vert = \\frac{\\rho L}{8d\\mu ^2} \\left\\Vert \\sum _{l=1}^L \\alpha _l \\mathbf {b}_l \\right\\Vert \\le \\frac{\\rho L}{8d\\mu ^2}\\frac{12 d_0}{d}\\Vert \\mathbf {u}\\Vert = \\frac{3\\rho Ld_0}{2d^2\\mu ^2}\\Vert \\mathbf {u}\\Vert .$ In summary, by combining eq:LipFh, eq:LipFx, eq:LipGx, eq:LipG1, and eq:LipG2, we conclude that $\\Vert \\nabla \\widetilde{F}(z+ w) - \\nabla \\widetilde{F}(z)\\Vert \\le 10d_0 \\Vert \\mathcal {A}\\Vert ^2 (\\Vert \\mathbf {u}\\Vert + \\Vert \\mathbf {v}\\Vert ) + \\frac{5d_0\\rho }{d^2}(\\Vert \\mathbf {u}\\Vert + \\Vert \\mathbf {v}\\Vert ) + \\frac{3\\rho Ld_0}{2\\mu ^2d^2}\\Vert \\mathbf {u}\\Vert .$ With $\\Vert u\\Vert + \\Vert v\\Vert \\le \\sqrt{2}\\Vert w\\Vert $ , there holds $\\Vert \\nabla \\widetilde{F}(z+ w) - \\nabla \\widetilde{F}(z)\\Vert \\le \\sqrt{2}d_0\\left[ 10 \\Vert \\mathcal {A}\\Vert ^2 + \\frac{\\rho }{d^2} \\left(5 + \\frac{3L}{2\\mu ^2}\\right)\\right] \\Vert w\\Vert .$" ], [ "Initialization", "This section is devoted to justifying the validity of the Robustness condition and to proving Theorem REF , i.e., establishing the fact that Algorithm REF constructs an initial guess $(u_0, v_0)\\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}\\bigcap \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }\\bigcap \\mathcal {N}_{\\frac{2}{5}\\varepsilon }.$ Lemma 5.20 For $e\\sim \\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L) + \\mathrm {i}\\mathcal {N}(0, \\frac{\\sigma ^2d_0^2}{2L}I_L)$ , there holds $\\Vert \\mathcal {A}^(y) - h_0x_0^* \\Vert \\le \\xi d_0,$ with probability at least $1 - L^{-\\gamma }$ if $L \\ge C_{\\gamma } (\\mu ^2_h + \\sigma ^2) \\max \\lbrace K, N\\rbrace \\log L /\\xi ^2.$ Moreover, $\\Vert \\mathcal {A}^(e)\\Vert \\le \\xi d_0$ with probability at least $1 - L^{-\\gamma }$ if $L \\ge C_{\\gamma }(\\frac{\\sigma ^2}{\\xi ^2} + \\frac{\\sigma }{\\xi })\\max \\lbrace K, N\\rbrace \\log L.$ In particular, we fix $\\xi = \\frac{\\varepsilon }{10\\sqrt{2}}$ and then Robustness condition REF holds, i.e., $\\Vert \\mathcal {A}^(e)\\Vert \\le \\frac{\\varepsilon d_0}{10\\sqrt{2}}$ .", "In this proof, we can assume $d_0 = 1$ and $\\Vert h_0\\Vert = \\Vert x_0\\Vert = 1$ , without loss of generality.", "First note that $\\operatorname{E}(\\mathcal {A}^y) = \\operatorname{E}(\\mathcal {A}^\\mathcal {A}(h_0x_0^* ) + \\mathcal {A}^(e))= h_0x_0^.$ We will use the matrix Bernstein inequality to show that $\\Vert \\mathcal {A}^(y) - h_0x_0^ \\Vert \\le \\xi .$ By definition of $\\mathcal {A}$ and $\\mathcal {A}^$ in (REF ) and (REF ), $\\mathcal {A}^(y) - h_0x_0^* = \\sum _{l=1}^L \\left[ b_lb^_lh_0x_0^ (a_la_l^* - I_N) + e_l b_la_l^\\right] = \\sum _{l=1}^L \\mathcal {Z}_l,$ where $\\mathcal {Z}_l : = b_lb_l^h_0x_0^(a_l a_l^ - I_N) + e_l b_la_l^$ and $\\sum _{l=1}^L b_lb_l^ = I_K$ .", "In order to apply Bernstein inequality (REF ), we need to estimate both the exponential norm $\\Vert \\mathcal {Z}_l\\Vert _{\\psi _1}$ and the variance.", "$\\Vert \\mathcal {Z}_l\\Vert _{\\psi _1} & \\le & C \\Vert b_l\\Vert \|b_l^h_0\| \\Vert x_0^(a_la_l^* - I_N)\\Vert _{\\psi _1} + C \\Vert e_l b_la_l^\\Vert _{\\psi _1}\\\\& \\le & C \\frac{\\mu _h\\sqrt{KN}}{L} + C\\frac{\\sigma \\sqrt{KN}}{L} \\le C\\frac{(\\mu _h + \\sigma ) \\sqrt{KN}}{L},$ for some constant $C$ .", "Here, (REF ) of Lemma REF gives $\\Vert x_0^ (a_la_l^* - I_N)\\Vert _{\\psi _1} \\le C\\sqrt{N}$ and $\\Vert e_l b_la_l^\\Vert _{\\psi _1}\\le C\\sqrt{\\frac{K}{L}} (\|e_l\| \\Vert a_l\\Vert )_{\\psi _1} \\le C\\sqrt{ \\frac{K}{L}} \\frac{\\sigma }{\\sqrt{L}}\\sqrt{N} \\le \\frac{C\\sigma \\sqrt{KN}}{L}$ follows from (REF ) where both $\|e_l\|$ and $\\Vert a_l\\Vert $ are sub-gaussian random variables.", "Now we give an upper bound of $\\sigma _0^2 : = \\max \\lbrace \\Vert E\\sum _{l=1}^L \\mathcal {Z}_l^\\mathcal {Z}_l\\Vert , \\Vert \\operatorname{E}\\sum _{l=1}^L \\mathcal {Z}_l\\mathcal {Z}_l^\\Vert \\rbrace $ .", "$\\left\\Vert \\operatorname{E}\\sum _{l=1}^l \\mathcal {Z}_l\\mathcal {Z}_l^ \\right\\Vert & = & \\left\\Vert \\sum _{l=1}^L \|b_l^h_0\|^2 b_lb_l^ \\operatorname{E}\\Vert (a_la_l^* - I_N)x_0 \\Vert ^2 + \\sum _{l=1}^L \\operatorname{E}(\|e_l\|^2 \\Vert a_l\\Vert ^2)b_lb_l^* \\right\\Vert \\\\& \\le & C N \\left\\Vert \\sum _{l=1}^L \|b_l^h_0\|^2 b_lb_l^ \\right\\Vert ^2 + C\\frac{\\sigma ^2 N}{L}\\\\& \\le & C\\frac{\\mu ^2_hN}{L} + C\\frac{\\sigma ^2 N}{L}\\le C\\frac{(\\mu ^2_h + \\sigma ^2)N}{L}.$ where $\\operatorname{E}\\Vert (a_la_l^* - I_N)x_0 \\Vert ^2 = x_0^* \\operatorname{E}(a_l a_l^* - I_N)^2 x_0 = N$ follows from (REF ) and $\\operatorname{E}( \|e_l\|^2) = \\frac{\\sigma ^2}{L}.$ $\\left\\Vert \\operatorname{E}\\sum _{l=1}^l \\mathcal {Z}^_l\\mathcal {Z}_l \\right\\Vert & \\le & \\left\\Vert \\sum _{l=1}^L \|b_l^h_0\|^2 \\Vert b_l\\Vert ^2 \\operatorname{E}\\left[ (aa_l^* - I_N)x_0x_0^* (a_la_l^* - I_N)\\right] \\right\\Vert \\\\& & + \\left\\Vert \\operatorname{E}\\sum _{l=1}^L e_l^2 \\Vert b_l\\Vert ^2 a_la_l^* \\right\\Vert \\\\& \\le & C\\frac{K}{L} \\left\\Vert \\sum _{l=1}^L \|b_l^h_0\|^2 I_N\\right\\Vert + C\\frac{\\sigma ^2K}{L^2} \\left\\Vert \\sum _{l=1}^L \\operatorname{E}(a_la_l^) \\right\\Vert = C\\frac{(\\sigma ^2 + 1) K}{L}$ where we have used the fact that $\\sum _{l=1}^L \|b_l^h_0\|^2 = \\Vert h_0\\Vert ^2 = 1.$ Therefore, we now have the variance $\\sigma ^2_0$ bounded by $\\frac{(\\mu ^2_h + \\sigma ^2)\\max \\lbrace K, N\\rbrace }{L}.$ We apply Bernstein inequality (REF ), and obtain $\\left\\Vert \\sum _{l=1}^L \\mathcal {Z}_l \\right\\Vert & \\le & C_0 \\max \\Big \\lbrace \\sqrt{\\frac{( \\mu ^2_h + \\sigma ^2)\\max \\lbrace K,N\\rbrace (\\gamma + 1) \\log L}{L}}, \\\\&& \\quad \\frac{\\sqrt{KN}(\\mu _h + \\sigma )(\\gamma + 1)\\log ^2 L }{L} \\Big \\rbrace \\le \\xi $ with probability at least $1 - L^{-\\gamma }$ if $L \\ge C_{\\gamma } (\\mu ^2_h + \\sigma ^2 )\\max \\lbrace K, N\\rbrace \\log ^2L /\\xi ^2.$ Regarding the estimation of $\\Vert \\mathcal {A}^(e)\\Vert $ , the same calculations immediately give $R : = \\max _{1\\le l\\le L} \\Vert e_l b_la_l^\\Vert _{\\psi _1} \\le \\sqrt{\\frac{K}{L}}\\max _{1\\le l\\le L} ( \|e_l\| \\Vert a_l\\Vert )_{\\psi _1}\\le \\frac{C\\sigma \\sqrt{KN}}{L}$ and $\\max \\left\\lbrace \\left\\Vert \\operatorname{E}\\left[ \\mathcal {A}^(e) (\\mathcal {A}^(e))^\\right] \\right\\Vert , \\left\\Vert \\operatorname{E}\\left[ (\\mathcal {A}^(e))^\\mathcal {A}^(e)\\right] \\right\\Vert \\right\\rbrace \\le \\frac{\\sigma ^2\\max \\lbrace K, N\\rbrace }{L}.$ Applying Bernstein inequality (REF ) again, we get $\\Vert \\mathcal {A}^(e)\\Vert \\le C_0\\sigma \\max \\Big \\lbrace \\sqrt{\\frac{(\\gamma + 1)\\max \\lbrace K,N\\rbrace \\log L}{L}}, \\frac{(\\gamma + 1)\\sqrt{KN}\\log ^2 L }{L} \\Big \\rbrace \\le \\xi $ with probability at least $1 - L^{-\\gamma }$ if $L \\ge C_{\\gamma } (\\frac{\\sigma ^2}{\\xi ^2} + \\frac{\\sigma }{\\xi })\\max \\lbrace K, N\\rbrace \\log ^2L.$ Lemma REF lays the foundation for the initialization procedure, which says that with enough measurements, the initialization guess via spectral method can be quite close to the ground truth.", "Before moving to the proof of Theorem REF , we introduce a property about the projection onto a closed convex set.", "Lemma 5.21 (Theorem 2.8 in [13]) Let $Q := \\lbrace w\\in \\hbox{{C}}^K \| \\sqrt{L}\\Vert Bw\\Vert _{\\infty } \\le 2\\sqrt{d}\\mu \\rbrace $ be a closed nonempty convex set.", "There holds $\\operatorname{Re}( \\left\\langle z- \\mathcal {P}_Q(z) , w- \\mathcal {P}_Q(z) \\right\\rangle ) \\le 0, \\quad \\forall \\, w\\in Q, z\\in \\hbox{{C}}^K $ where $\\mathcal {P}_{Q}(z)$ is the projection of $z$ onto $Q$ .", "This is a direct result from Theorem 2.8 in [13], which is also called Kolmogorov criterion.", "Now we present the proof of Theorem REF .", "[of Theorem REF] Without loss of generality, we again set $d_0 = 1$ and by definition, all $h_0,$ $x_0$ , $\\hat{h}_0$ and $\\hat{x}_0$ are of unit norm.", "Also we set $\\xi = \\frac{\\varepsilon }{10\\sqrt{2}}.$ By applying the triangle inequality to (REF ), it is easy to see that $1 - \\xi \\le d \\le 1 + \\xi , \\quad \|d - 1\| \\le \\xi \\le \\frac{\\varepsilon }{10\\sqrt{2}} < \\frac{1}{10},$ which gives $\\frac{9}{10}d_0 \\le d \\le \\frac{11}{10}d_0.$ It is easier to get an upper bound for $\\Vert v_0\\Vert $ here, i.e., $\\Vert v_0 \\Vert = \\sqrt{d} \\Vert \\hat{x}_0\\Vert = \\sqrt{d} \\le \\sqrt{1 + \\xi } \\le \\frac{2}{\\sqrt{3}},$ which implies $v_0 \\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}.$ The estimation of $u_0$ involves Lemma REF .", "In our case, $u_0$ is the minimizer to the function $f(z) = \\frac{1}{2} \\Vert z- \\sqrt{d} \\hat{h}_0 \\Vert ^2$ over $Q = \\lbrace z\| \\sqrt{L}\\Vert Bz\\Vert _{\\infty } \\le 2\\sqrt{d}\\mu \\rbrace .$ Therefore, $u_0$ is actually the projection of $\\sqrt{d} \\hat{h}_0$ onto $Q$ .", "Note that $u_0\\in Q$ implies $\\sqrt{L}\\Vert Bu_0\\Vert _{\\infty } \\le 2\\sqrt{d}\\mu \\le \\frac{4\\mu }{\\sqrt{3}}$ and hence $u_0\\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }.$ Moreover, $u_0$ yields $\\Vert \\sqrt{d}\\hat{h}_0 - w\\Vert ^2& = & \\Vert \\sqrt{d}\\hat{h}_0 - u_0\\Vert ^2 + 2\\operatorname{Re}(\\left\\langle \\sqrt{d}\\hat{h}- u_0, u_0 - w\\right\\rangle ) + \\Vert u_0 - w\\Vert ^2 \\nonumber \\\\& \\ge & \\Vert \\sqrt{d}\\hat{h}_0 - u_0\\Vert ^2 + \\Vert u_0 - w\\Vert ^2 $ for all $w\\in Q$ because the cross term is nonnegative due to Lemma REF .", "Let $w= 0\\in Q$ and we get $\\Vert u_0\\Vert ^2 \\le d \\le \\frac{4}{3}.$ So far, we have already shown that $(u_0, v_0) \\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{d_0}$ and $u_0 \\in \\frac{1}{\\sqrt{3}} \\mathcal {N}_{\\mu }$ .", "Now we will show that $\\Vert u_0v_0^* - h_0x_0^\\Vert _F \\le 4\\xi .$ First note that $\\sigma _i(\\mathcal {A}^(y)) \\le \\xi $ for all $i\\ge 2$ , which follows from Weyl's inequality [29] for singular values where $\\sigma _i(\\mathcal {A}^(y))$ denotes the $i$ -th largest singular value of $\\mathcal {A}^(y)$ .", "Hence there holds $\\Vert d \\hat{h}_0\\hat{x}_0^* - h_0x_0^* \\Vert \\le \\Vert \\mathcal {A}^(y) - d \\hat{h}_0\\hat{x}_0^ \\Vert + \\Vert \\mathcal {A}^(y) - h_0x_0^ \\Vert \\le 2\\xi .$ On the other hand, $\\Vert (I- h_0h_0^)\\hat{h}_0 \\Vert & = & \\Vert (I- h_0h_0^)\\hat{h}_0\\hat{x}_0^\\hat{x}_0\\hat{h}_0^ \\Vert \\\\& = & \\Vert (I- h_0h_0^)( \\mathcal {A}^(y) - d \\hat{h}_0\\hat{x}_0^* + \\hat{h}_0\\hat{x}_0^* - h_0x_0^* ) \\hat{x}_0\\hat{h}_0^* \\Vert \\\\& = & \\Vert (I- h_0h_0^)( \\mathcal {A}^(y) - h_0x_0^* ) \\hat{x}_0\\hat{h}_0^* \\Vert \\le \\xi $ where the second equation follows from $ (I- h_0h_0^) h_0x_0^ = 0$ and $(\\mathcal {A}^(y) - d \\hat{h}_0\\hat{x}_0^)\\hat{x}_0\\hat{h}_0^* = 0$ .", "Therefore, we have $\\Vert \\hat{h}_0 - h_0^\\hat{h}_0 h_0 \\Vert \\le \\xi ,\\quad \\Vert \\sqrt{d} \\hat{h}_0 - \\alpha _0 h_0 \\Vert \\le \\sqrt{d}\\xi ,$ where $\\alpha _0 = \\sqrt{d}h_0^\\hat{h}_0$ .", "If we substitute $w$ by $\\alpha _0 h_0\\in Q$ into (REF ), $\\Vert \\sqrt{d}\\hat{h}_0 - \\alpha _0 h_0\\Vert \\ge \\Vert u_0 - \\alpha _0 h_0\\Vert .$ where $\\alpha _0 h_0\\in Q$ follows from $\\sqrt{L} \|\\alpha _0\|\\Vert Bh_0\\Vert _{\\infty } \\le \|\\alpha _0\| \\mu _h \\le \\sqrt{d} \\mu _h \\le \\sqrt{d} \\mu < \\sqrt{2d}\\mu $ .", "Combining (REF ) and (REF ) leads to $\\Vert u_0 - \\alpha _0h_0\\Vert \\le \\sqrt{d}\\xi .$ Now we are ready to estimate $\\Vert u_0v_0^* - h_0x_0^* \\Vert _F$ as follows, $\\Vert u_0v_0^* - h_0x_0^* \\Vert _F & \\le &\\Vert u_0v_0^* - \\alpha _0h_0v_0^* \\Vert _F + \\Vert \\alpha _0h_0v_0^* - h_0x_0^* \\Vert _F \\\\& \\le & \\Vert u_0 - \\alpha _0h_0\\Vert \\Vert v_0\\Vert + \\Vert d h_0 h^_0 \\hat{h}_0 \\hat{x}_0^ - h_0x_0^* \\Vert _F \\\\& \\le & \\xi \\sqrt{d} \\Vert v_0\\Vert + \\Vert d \\hat{h}_0\\hat{x}_0^* - h_0x_0^\\Vert _F \\\\& \\le & \\xi d + 2 \\sqrt{2}\\xi \\le \\xi (1 + \\xi ) + 2\\sqrt{2}\\xi \\\\& \\le & 4\\xi \\le \\frac{2}{5}\\varepsilon ,$ where $\\Vert v_0\\Vert = \\sqrt{d}$ , $v_0 = \\sqrt{d}\\hat{x}_0$ and $\\Vert d \\hat{h}_0\\hat{x}_0^ - h_0x_0^\\Vert _F \\le \\sqrt{2}\\Vert d \\hat{h}_0\\hat{x}_0^ - h_0x_0^\\Vert \\le 2\\sqrt{2} \\xi $ follows from (REF )." ], [ "Descent Lemma", "Lemma 6.1 If $f(z, \\bar{z})$ is a continuously differentiable real-valued function with two complex variables $z$ and $\\bar{z}$ , (for simplicity, we just denote $f(z, \\bar{z})$ by $f(z)$ and keep in the mind that $f(z)$ only assumes real values) for $z:= (h, x) \\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ .", "Suppose that there exists a constant $C_L$ such that $\\Vert \\nabla f(z+ t \\Delta z) - \\nabla f(z)\\Vert \\le C_L t\\Vert \\Delta z\\Vert , \\quad \\forall 0\\le t\\le 1,$ for all $z\\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ and $\\Delta z$ such that $z+ t\\Delta z\\in \\mathcal {N}_{\\varepsilon }\\cap \\mathcal {N}_{\\widetilde{F}}$ and $0\\le t\\le 1$ .", "Then $f(z+ \\Delta z) \\le f(z) + 2\\operatorname{Re}( (\\Delta z)^T \\overline{\\nabla } f(z)) + C_L\\Vert \\Delta z\\Vert ^2$ where $\\overline{\\nabla } f(z) := \\frac{\\partial f(z, \\bar{z})}{\\partial z}$ is the complex conjugate of $\\nabla f(z) = \\frac{\\partial f(z, \\bar{z})}{\\partial \\bar{z}}$ .", "The proof simply follows from proof of descent lemma (Proposition A.24 in [2]).", "However it is slightly different since we are dealing with complex variables.", "Denote $g(t) := f(z+ t\\Delta z).$ Since $f(z, \\bar{z})$ is a continuously differentiable function, we apply the chain rule $\\frac{d g(t)}{dt} = (\\Delta z)^T\\frac{\\partial f}{\\partial z} (z+ t\\Delta z) + (\\Delta \\bar{z})^T \\frac{\\partial f}{\\partial \\bar{z}} (z+ t\\Delta z) = 2\\operatorname{Re}( (\\Delta z)^T \\overline{\\nabla } f (z+ t\\Delta z) ).$ Then by the Fundamental Theorem of Calculus, $f(z+ t\\Delta z) - f(z) & = & \\int _0^1 \\frac{dg(t)}{dt} dt = 2\\int _0^1 \\operatorname{Re}( (\\Delta z)^T \\overline{\\nabla } f (z+ t\\Delta z) ) dt \\\\& \\le & 2\\operatorname{Re}((\\Delta z)^T \\overline{\\nabla } f(z_0)) + 2\\int _0^1 \\operatorname{Re}((\\Delta z)^T(\\overline{\\nabla } f(z+ t\\Delta z) - \\overline{\\nabla } f(z))) dt \\\\& \\le & 2\\operatorname{Re}((\\Delta z)^T \\overline{\\nabla } f(z)) + 2 \\Vert \\Delta z\\Vert \\int _0^1 \\Vert \\nabla f(z+ t\\Delta z) - \\nabla f(z)) \\Vert dt \\\\& \\le & 2\\operatorname{Re}((\\Delta z)^T \\overline{\\nabla } f(z)) + C_L\\Vert \\Delta z\\Vert ^2.$" ], [ "Some useful facts", "The key concentration inequality we use throughout our paper comes from Proposition 2 in [18], [19].", "Theorem 6.2 Consider a finite sequence of $\\mathcal {Z}_l$ of independent centered random matrices with dimension $M_1\\times M_2$ .", "Assume that $\\Vert \\mathcal {Z}_l\\Vert _{\\psi _1} \\le R$ where the norm $\\Vert \\cdot \\Vert _{\\psi _1}$ of a matrix is defined as $\\Vert Z\\Vert _{\\psi _1} := \\inf _{u \\ge 0} \\lbrace \\operatorname{E}[ \\exp (\\Vert Z\\Vert /u)] \\le 2 \\rbrace .$ and introduce the random matrix $S= \\sum _{l=1}^L \\mathcal {Z}_l.$ Compute the variance parameter $\\sigma _0^2 := \\max \\lbrace \\Vert \\operatorname{E}(SS^)\\Vert , \\Vert \\operatorname{E}(S^S)\\Vert \\rbrace = \\max \\Big \\lbrace \\Vert \\sum _{l=1}^L \\operatorname{E}(\\mathcal {Z}_l\\mathcal {Z}_l^)\\Vert , \\Vert \\sum _{l=1}^L \\operatorname{E}(\\mathcal {Z}_l^* \\mathcal {Z}_l)\\Vert \\Big \\rbrace ,$ then for all $t \\ge 0$ , we have the tail bound on the operator norm of $S$ , $\\Vert S\\Vert \\le C_0 \\max \\lbrace \\sigma _0 \\sqrt{t + \\log (M_1 + M_2)}, R\\log \\left( \\frac{\\sqrt{L}R}{\\sigma _0}\\right)(t + \\log (M_1 + M_2)) \\rbrace $ with probability at least $1 - e^{-t}$ where $C_0$ is an absolute constant.", "For convenience we also collect some results used throughout the proofs.", "Lemma 6.3 Let $z$ be a random variable which obeys $\\Pr \\lbrace \|z\| > u \\rbrace \\le a e^{-b u }$ , then $\\Vert z\\Vert _{\\psi _1} \\le (1 + a)/b.$ which is proven in Lemma 2.2.1 in [38].", "Moreover, it is easy to verify that for a scalar $\\lambda \\in \\hbox{{C}}$ $\\Vert \\lambda z\\Vert _{\\psi _1} = \|\\lambda \| \\Vert z\\Vert _{\\psi _1}.$ Lemma 6.4 ( Lemma 10-13 in [1], Lemma 12.4 in [26]) Let $u\\in \\hbox{{C}}^n \\sim \\mathcal {N}(0, \\frac{1}{2}I_n) + \\mathrm {i}\\mathcal {N}(0, \\frac{1}{2}I_n) $ , then $\\Vert u\\Vert ^2 \\sim \\frac{1}{2}\\chi ^2_{2n}$ and $\\Vert \\Vert u\\Vert ^2 \\Vert _{\\psi _1} = \\Vert \\left\\langle u, u\\right\\rangle \\Vert _{\\psi _1} \\le C n$ and $\\operatorname{E}\\left[(uu^* - I_n)^2 \\right]= nI_n.$ Let $q\\in \\hbox{{C}}^n$ be any deterministic vector, then the following properties hold $\\Vert (uu^* - I)q\\Vert _{\\psi _1} \\le C\\sqrt{n}\\Vert q\\Vert ,$ $\\operatorname{E}\\left[(uu^* - I)qq^* (uu^* - I)\\right]= \\Vert q\\Vert ^2 I_n.$ Let $v\\sim \\mathcal {N}(0, \\frac{1}{2}I_m) + \\mathrm {i}\\mathcal {N}(0, \\frac{1}{2}I_m) $ be a complex Gaussian random vector in $\\hbox{{C}}^m$ , independent of $u$ , then $\\left\\Vert \\Vert u\\Vert \\cdot \\Vert v\\Vert \\right\\Vert _{\\psi _1} \\le C\\sqrt{mn}.$" ], [ "Acknowledgement", "S. Ling, T. Strohmer, and K. Wei acknowledge support from the NSF via grant DTRA-DMS 1322393." ] ]	1606.04933
[ [ "A Novel Approach to Fine-Tuned Supersymmetric Standard Models -- Case of\n Non-Universal Higgs Masses model" ], [ "Abstract Discarding the prejudice about fine tuning, we propose a novel and efficient approach to identify relevant regions of fundamental parameter space in supersymmetric models with some amount of fine tuning.", "The essential idea is the mapping of experimental constraints at a low energy scale, rather than the parameter sets, to those of the fundamental parameter space.", "Applying this method to the non-universal Higgs masses model, we identify a new interesting superparticle mass pattern where some of the first two generation squarks are light whilst the stops are kept heavy as 6TeV.", "Furthermore, as another application of this method, we show that the discrepancy of the muon anomalous magnetic dipole moment can be filled by a supersymmetric contribution within the 1 {\\sigma} level of the experimental and theoretical errors, which was overlooked by the previous studies due to the required terrible fine tuning." ], [ "Introduction", "Although the discovery of the Higgs boson in July 2012 verifies our thought that the physics up to the electroweak scale should be well described by the standard model (SM) of particle physics [1], the SM itself suffers from the uncomfortably large disparity between the electroweak scale and the fundamental physics scale which is supposedly close to the Planck scale.", "Supersymmetry (SUSY) has been recognized as a promising candidate to solve this unease.", "The fact that superparticles have not yet been discovered, however, constrains their mass spectra, if exists: e.g.", "colored superparticles should weigh at least around 1 TeV [2].", "In the minimal supersymmetric standard model (MSSM), the measured Higgs boson mass of about 125 GeV requires large radiative corrections due to supersymmetry breaking (SUSY-breaking) to raise its tree-level mass below the Z boson mass [3].", "As the Higgs boson strongly couples to the top-stop sector, this typically requires that the stop mass has to be around 6 TeV or so, unless a SUSY-breaking trilinear coupling is parametrically large [4].", "In this case, there will be a little hierarchy between the electroweak scale and SUSY-breaking mass parameters, and thus some amount of fine tuning among these parameters may be requisite in order that the electroweak symmetry breakdown takes place at the correct energy scale.", "This situation does not mean that nature rejects SUSY, but implies that we should not have prejudice against the amount of fine tuning.", "Since SUSY is still a promising candidate for physics beyond the SM, we should study the supersymmetric SM with some amount of fine tuning (FT-SUSY: fine-tuned supersymmetry).", "To identify an experimentally viable region or an interesting region of a model, the scatter plot method has been widely used.", "This method represents a relevant region by a collection of discretized points in the fundamental parameter space, just like a “pointillism\".", "The collection of points is selected from a large number of initially chosen points in the fundamental parameter space to satisfy the experimental (and other) constraints at the experimental scale.", "However, in a FT-SUSY the relevant region might be too tiny to be represented in this way.", "In this paper, we propose a novel approach to a FT-SUSY regardless of the amount of fine tuning.", "In Sec., we propose a method to identify the relevant region of a FT-SUSY.", "In contrast to the ordinary top-down renormalization group (RG) picture, in which a point chosen in the fundamental parameter space at the fundamental scale flows to that at the experimental scale, we map a constraint for the parameter space at the experimental scale to that at the fundamental scale.", "Then, we can directly identify the restricted space by the mapped constraints as the relevant region written in the fundamental parameters.", "This procedure is like a “coloring\".", "This procedure allows us to identify the whole relevant region as well as its outlines in the fundamental parameter space.", "Furthermore, the area near an outline can be easily identified as a phenomenologically interesting region, if this outline corresponds to the boundary of a constraint given by an on-going experiment.", "Since the constraints we map can also include the requirement of a characteristic property, if we choose a suitable requirement, a fine-tuned region is identified.", "In Sec., to illustrate our idea and to show its efficiency, we apply this procedure to the non-universal Higgs masses model (NUHM) [5] which has the MSSM particle contents with universal SUSY breaking masses except for the Higgs masses at the GUT scale.", "We identify the experimental viable region of the NUHM and argue its features.", "We find an interesting region with a new superparticle mass pattern, where some of the first two generation squarks are light (Sec.REF ).", "This mass pattern is a consequence of the RG running where a negative Higgs mass squared dominantly raises the third generation squark masses due to their rather large Yukawa couplings.", "Since this effect never happens in the CMSSM, this region should be one of the characters of the NUHM (Sec.REF ).", "In Sec., using another application of our method, we find there is a terribly fine-tuned region that explains the anomaly of the muon anomalous magnetic dipole moment (muon $g-2$ ) [6], [7], [8] within the $1~\\sigma $ experimental and theoretical errors in the NUHM.", "Furthermore, with sufficiently large $\\tan \\beta $ , we also show that there is a parameter region explaining the muon $g-2$ anomaly with most of the 1st and 2nd generation sfermions light.", "These regions were overlooked by the previous studies using the scatter plot method which is not practical to find such a tiny and terribly fine-tuned region.", "This fact shows the power of our approach to a FT-SUSY." ], [ "A Novel Approach to FT-SUSY", "We propose a novel approach to tackle a FT-SUSY.In fact, the method developed here can apply to many models even without SUSY.", "However, for ease of explanation, we only apply our method to a FT-SUSY in this paper.", "In this approach, we can directly identify the relevant region in the FT-SUSY without being bothered with some amount of fine tuning.", "For the sake of simplicity, suppose that a fundamental supersymmetric model, such as a grand unified theory (GUT), can be described as a generic MSSM.", "The generic MSSM is defined as an effective theory with most general SUSY-breaking soft mass parameters of the particle contents, below the fundamental scale $t_f$ of the fundamental model.", "$t_f$ could be $\\log {\\left( 10^{{\\mathcal {O}(10)}}{\\rm GeV} \\over m_z \\right)}$ depending on the model we consider, where $m_z$ is the Z boson mass.", "In the parameter space of the generic MSSM, $\\mathcal {M}_f^{\\rm gen}$ , a point is specified by a set of $\\mathcal {O}(100)$ dimensionful parameters, $g^f_i$ at the scale $t_f$ , where we have assumed $\\tan \\beta $ is a given constant and no parameters are dimensionless.", "This assumption is only for simplicity and the generalization is straightforward.", "In contrast the fundamental model has a restricted parameter space, the fundamental parameter space $\\mathcal {M}_f^{\\rm fund}$ , with coordinates of much fewer fundamental parameters $G_a$ .", "Since below the scale $t_f$ , the fundamental model is described by the generic MSSM, $\\mathcal {M}_f^{\\rm fund}$ is embedded into a subspace of $\\mathcal {M}_f^{\\rm gen}$ by a set of relations, $g_i^f=f_i^f(G_a).$ This defines a map, $f^f: \\mathcal {M}_f^{\\rm fund} \\rightarrow \\mathcal {M}_f^{\\rm gen}, ~{\\rm and}~ f^f(\\mathcal {M}_f^{\\rm fund})=\\left\\lbrace g^f_i \\in \\mathcal {M}_f^{\\rm gen} \| g^f_i=f^f_i(G_a),~ G_a \\in \\mathcal {M}_f^{\\rm fund} \\right\\rbrace .$ On the other hand, the solution of the RG equation [9] which gives correspondence among the parameters of the same theory at different scales can also be considered as a map $f_{RG}$ .For simplicity, we suppose that $f_{RG}$ is a bijection, so that the image satisfies the equality, ${\\rm im}f_{RG}\\equiv f_{RG}(\\mathcal {M}_f^{\\rm gen})=\\mathcal {M}_e^{\\rm gen}$ , and the inverse map, $f^{-1}_{RG}$ , can be defined.Since the fundamental model can be described by the generic MSSM, we consider $f_{RG}$ in the context of the generic MSSM: $f_{RG}: \\mathcal {M}_f^{\\rm gen} \\rightarrow \\mathcal {M}_e^{\\rm gen},~{\\rm and}~ f_{RG}(\\mathcal {M}_f^{\\rm gen})=\\left\\lbrace g^e_i \\in \\mathcal {M}_e^{\\rm gen} \| g^e_i=g^{sol}_i(t_e; t_f, g^f_j),~ g_i^f \\in \\mathcal {M}^{\\rm gen}_f \\right\\rbrace .$ $\\mathcal {M}_e^{\\rm gen}$ is the parameter space at the experimental scale $t_e=\\log {({\\mathcal {O}(100) {\\rm GeV} \\over m_z})}$ , and a set of $g^{sol}_i(t_e; t_f, g^f_j)$ is the solution of the RG equation [9] in the generic MSSM at $t_e$ with an initial condition of a set of parameters, $g_j^f$ at $t_f$ .", "Suppose that $\\tilde{\\mathcal {M}}_{e}^{\\rm gen}$ denotes the region of interest of the generic MSSM at $t_e$ , which may be either a viable region, i.e.", "the part of the parameter space that survives the experimental constraints, or a phenomenologically interesting region with some characteristic properties.", "$\\tilde{\\mathcal {M}}_e^{\\rm gen}$ is characterized by a set of conditions expressed as $\\phi _l(g^e_i)>0$ or $\\phi _l(g^e_i)=0$ : $\\mathcal {M}_e^{\\rm gen} \\supset \\tilde{\\mathcal {M}}_e^{\\rm gen}=\\left\\lbrace g^e_i \\in \\mathcal {M}_e^{\\rm gen} \| \\phi _1(g^e_i)>0, \\phi _2(g^e_i)>0,.... , \\phi _n(g^e_i)>0 \\right\\rbrace .$ Here $\\phi _l(g^e_i)$ is a condition function for the generic MSSM parameters at $t_e$ , which could either correspond to a fitted function of an experimental constraint or a requirement to have a characteristic property.", "The conditions in equalities, such as the ones for correct electroweak symmetry breaking and the Higgs boson mass, reduce the dimension of $\\mathcal {M}_e^{\\rm gen}$ .", "On the other hand, the conditions in inequalities, such as the mass bounds for superparticles, restrict the parameter space $\\mathcal {M}_e^{\\rm gen}$ and hence constitute the outlines of $\\tilde{\\mathcal {M}}_e^{\\rm gen}$ .", "In Eq.", "(REF ), we have written down only the conditions in inequalities for illustrative purpose.", "What we would like to do is to identify the region of interest at $t_f$ , $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ in the parameter space of the fundamental model, $\\mathcal {M}_f^{\\rm fund}$ .", "A conventional definition of $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ is given as $\\mathcal {M}_f^{\\rm fund}\\supset \\tilde{\\mathcal {M}}_f^{\\rm fund}=\\left\\lbrace G_a \\in \\mathcal {M}_f^{\\rm fund} \| {f_{RG}} \\circ f^{f}(G_a) \\in \\tilde{\\mathcal {M}}_e^{\\rm gen} \\right\\rbrace .$ Namely, given a set of the fundamental parameters, $G_a \\in \\mathcal {M}_f^{\\rm fund}$ , we apply the RG procedure to obtain the corresponding parameters at $t_e$ , and check whether they satisfy the conditions characterizing the region of interest of the generic MSSM $\\tilde{\\mathcal {M}}^{\\rm gen}_e$ .", "The ordinary scatter plot method follows this procedure recursively by using sample points, $S={\\left\\lbrace G^{(1)}_a, G^{(2)}_a, ... G^{(N)}_a \\right\\rbrace }$ , which are chosen in some way from the fundamental parameter space.", "Here $N$ is the total number of the sample points.", "The region of interest of the fundamental model, $\\tilde{\\mathcal {M}}^{\\rm fund}_f$ is approximated as a collection of discretized points, like a “pointillism\".", "Therefore, when applying to a FT-SUSY, in which the region of interest is so tiny, the ordinary method requires a huge number of sample points, $N$ , as well as luck, and hence is time-consuming in numerical computation and sometimes misleading.", "We now propose a novel approach to identify the region of interest, $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ , in a fundamental model regardless of the amount of fine tuning $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ has.", "Our definition of $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ can be written, $\\tilde{\\mathcal {M}}_f^{\\rm fund}=\\left\\lbrace G_a \\in \\mathcal {M}_f^{\\rm fund} \| \\psi _1(G_a)>0, \\psi _2(G_a)>0,.... , \\psi _n(G_a)>0 \\right\\rbrace ,$ where $\\psi _l(G_a)$ is a condition function for the fundamental parameter space, expressed as $\\psi _l(G_a)=\\phi _l \\left(g_i^{sol}\\left(t_e; t_f, f^f_j(G_a)\\right)\\right)= \\phi _l \\circ f_{RG}\\circ f^f(G_a),$ and hence should be equivalent to Eq.", "(REF ).", "However, what we would like to obtain is not the correspondence among points in $\\mathcal {M}^{\\rm fund}_f$ and $\\tilde{\\mathcal {M}}^{\\rm gen}_e$ , but the correspondence between the two condition functions, $\\phi _l(g_i^f)$ and $\\psi _l(G_a)$ .", "Namely, we map the given set of conditions, $\\phi _l(g_i^e)>0$ that characterizes $\\tilde{\\mathcal {M}}_e^{\\rm gen}$ , to the corresponding one, $\\phi _l \\circ f_{RG}(g_i^f)>0$ , for the parameter space $\\mathcal {M}_f^{\\rm gen}$ at $t_f$ within the generic MSSM, and transform the latter into the corresponding conditions in $\\mathcal {M}^{\\rm fund}_f$ .", "Since the boundaries of these mapped conditions constitute the outlines of $\\mathcal {M}^{\\rm fund}_f$ and what we identify as $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ is the interior of the outlines, our procedure is like a “coloring\".", "Since the map, $f^f$ , is given, what we would like to know is the RG map of the condition function, $\\phi _l(g_i^e)$ , within the generic MSSM, $\\phi _l \\circ f_{RG}(g_j^f)=\\phi _l \\left(g_i^{sol}(t_e; t_f, g_j^f)\\right),$ and we will show how to derive the explicit form of this.", "One way is to solve the RG equation of the generic MSSM so that we can express $g_i^{sol}(t_e;t_f,g_j^f)$ in terms of the set of the parameters ${g}_j^f$ at $t_f$ .", "Alternatively, we can solve the differential equation which follows the RG map of the condition function, $\\Phi _l(g_j,t) \\equiv \\phi _l \\left(g_i^{sol}(t_e; t, g_j)\\right)$ , by varying $t$ : $\\biggl (\\frac{\\partial }{\\partial t}+\\sum _i \\beta _i \\frac{\\partial }{\\partial g_i}\\biggr )\\Phi _l(g_j,t)=0,$ where $\\beta _i$ is the RG beta function for $g_i$ [9].", "If the perturbative expansion $\\Phi _l (m_j,t)=\\sum _{n=0}{1 \\over n!}", "\\phi ^{i_1,i_2...i_n}_l(t)g_{i_1}g_{i_2}...g_{i_n}$ is allowed, a set of linear differential equations $\\frac{\\partial }{\\partial t}\\phi ^{i_1,i_2...i_n}_l(t)&= \\sum _{m=1}^{n} \\tilde{\\beta }^{i_1,i_2...i_n}_{j_1,j_2...j_m}\\phi ^{j_1,j_2...j_m}_l(t),$ for the coefficients, $\\phi ^{i_1,i_2...i_n}(t),$ are derived by requiring the vanishing of each Taylor coefficient in Eq.", "(REF ).", "The upper limit of the summation in Eq.", "(REF ) comes from the fact that a perturbative RG beta function always contains parameters of total exponents $\\geqslant 1$ .", "Eq.", "(REF ) is the running equation for coefficients of constraint (RECC) for $\\phi _l(g^e_i)$ .", "If all the parameters are dimensionful as in our case, $ \\tilde{\\beta }^{i_1,i_2...i_n}_{j_1,j_2...j_m}$ in Eq.", "(REF ) can be non-zero only if the dimension of $g_i$ , $d(i)$ , satisfies $\\sum _{l}^n{d(i_l)}=\\sum _{l}^m{d(j_l)}$ .", "Hence the coefficients can be evaluated by numerically solving the derived linear differential equation.If dimensionless parameters are included, we can also solve Eq.", "(REF ) but only perturbatively, namely the solution can approximately represent a mapped constraint function up to a precision depending on the order of the couplings we take into account.", "We show the explicit derivation of RECC in the generic MSSM in Appendix .", "Notice that $g^{sol}_i(t_e; t_f, g^f_i)$ in terms of a set of parameters, $g^f_i$ , is obtained, once we choose a condition function $\\phi _i(g^e_j)=g^e_i$ .", "We also note that in the derivation of RECC, $\\beta _i$ and $\\tilde{\\beta }^{i_1,i_2...i_n}_{j_1,j_2...j_m}$ can even depend on the scale $t$ .", "This is a convenient fact because we may take a shortcut to derive RECC with some parameters approximately treated as constants.", "Namely, if possible, we can numerically solve the RG equations for these parameters in advance, and substitute the numerical solutions as constants in the remaining RG equations.", "Then we can derive RECC from these remaining RG equations which explicitly depend on $t$ .", "This is what we do in Appendix .", "Solving the corresponding RECCs, we can obtain the set of conditions in terms of $g^f_i$ , $\\phi _l\\circ f_{RG}(g^f_i)>0$ , and applying the given map, $f^f$ , a set of $\\psi _l(G_a)>0$ is derived from Eq.", "(REF ).", "Therefore the whole region of interest, $\\tilde{\\mathcal {M}}^{\\rm fund}_f$ , is identified from Eq.", "(REF ).", "There are two additional advantages in our approach.", "Since a boundary of a constraint could correspond to an outline of $\\tilde{\\mathcal {M}}^{\\rm fund}_f$ , the viable region near such an outline may be testable if this constraint is given by an ongoing experiment.", "This implies that a viable region near an outline can be a phenomenologically interesting region.", "Therefore, checking the boundary profiles, namely the constraints the boundaries correspond to, we can guess some of the phenomenological interesting regions even without any additional requirement of characteristic properties.", "On the other hand, the boundary profiles of a phenomenologically interesting region in turn suggest the predictions that can be accompanied with the characteristic property.", "The second advantage is due to the fact that in our procedure the RG map of a condition function is followed within the MSSM.", "In fact, we can define the region of interest of the generic MSSM at $t_f$ , $\\tilde{\\mathcal {M}}_f^{\\rm gen}=\\left\\lbrace g_i^f \\in \\mathcal {M}_f^{\\rm gen} \| \\phi _1\\circ f_{RG}(g^f_i)>0, \\phi _2\\circ f_{RG}(g^f_i)>0,.... , \\phi _n\\circ f_{RG}(g^f_i)>0 \\right\\rbrace .$ Since the viable region of the generic MSSM at $t_f$ directly responds to the fundamental model, with stringent enough experimental constraints in future, $\\tilde{\\mathcal {M}}_f^{\\rm gen}$ can be a probe of the fundamental model.", "This approach may clarify the fundamental model directly." ], [ "Region of Interest in the Non-Universal Higgs Masses Model", "Using the method advocated in the previous section, we would like to analyze the non-universal-Higgs masses model (NUHM)[5] as an example of a fundamental model.", "The NUHM is an extension of the CMSSM motivated by GUT and has universal masses for sfermions at the GUT scale $t_f\\sim \\log { \\left( 2 \\times 10^{16} {\\rm GeV} \\over m_z\\right)}$ .", "The only difference from the CMSSM is that in the NUHM the SUSY breaking Higgs mass squared parameters are allowed to vary from that of sfermions at the GUT scale.", "This may be a natural assumption as the origin of the Higgs particles may be different from those of sfermions.", "The NUHM has a fundamental parameter space, $\\mathcal {M}_f^{\\rm fund}$ , where a point is specified by the fundamental parameters, ${ G_a= \\left\\lbrace m_0^2,m_{\\rm Hu0}^2,m_{\\rm Hd0}^2, M_0,A_0, \\mu _0,B_0\\right\\rbrace .", "}$ As we have noted, $\\tan \\beta $ , as well as the other dimensionless couplings, is taken to be a given constant that is not included in the parameter set, Eq.", "(REF ).", "Since the NUHM can be described by the generic MSSM below $t_f$ , a set of fundamental parameters in $\\mathcal {M}_f^{\\rm fund}$ is related to the parameters in $\\mathcal {M}_f^{\\rm gen}$ : ${\\bf m}_{\\rm \\tilde{Q}}^2={\\bf m}_{\\rm \\tilde{u}}^2={\\bf m}_{\\rm \\tilde{d}}^2={\\bf m}_{\\rm \\tilde{L}}^2={\\bf m}_{\\rm \\tilde{e}}^2=m_0^2 {\\bf 1}$ $M_{\\rm 1}=M_{\\rm 2}=M_{\\rm 3}=M_0$ ${\\bf A}_u={\\bf A}_d={\\bf A}_e=A_0 {\\bf 1}$ $m_{\\rm Hu}^2=m_{\\rm Hu0}^2,~ m_{\\rm Hd}^2=m_{\\rm Hd0}^2$ $B=B_0, \\mu =\\mu _0.$ Eqs.", "(REF ), (REF ) and (REF ) are the conditions of universal sfermion mass, gaugino mass and $A-$ term, respectively, where the bold characters are understood as three by three matrices of generation.", "Eq.", "(REF ) expresses the condition of the non-universal Higgs masses at $t_f$ which is the only difference from the CMSSM, and also is the character of this fundamental model.", "The Higgs mixing parameter, $\\mu $ , and the $B-$ term are taken to be free at $t_f$ in Eq.", "(REF ).", "The experimental constraints of the generic MSSM at the experimental scale, $t_e \\sim \\log ({100 {\\rm GeV} \\over m_z})$ , that restrict $\\mathcal {M}_e^{\\rm gen}$ to its viable region $\\tilde{\\mathcal {M}}_e^{\\rm gen}$ are given: $\\begin{split}&2B\\mu -(m^2_{{\\rm {\\rm Hu}}}+m^2_{{\\rm Hd}}+2\\mu ^2)\\sin 2\\beta =0\\\\&\\mu ^2-\\frac{(m^2_{{\\rm Hd}}-m^2_{{\\rm Hu}}\\tan ^2{\\beta } )}{\\tan ^2{\\beta } -1}-\\frac{m^2_{z}}{2}=0\\end{split}\\\\&m^2_{{\\rm \\tilde{Q}3}}\\cdot m^2_{{\\rm \\tilde{u}3}} \\equiv m_{{\\rm soft}}^4 \\sim (6{\\rm TeV})^4\\\\\\begin{split}&m^2_{{\\rm \\tilde{Q}}i,{\\rm \\tilde{u}}i,{\\rm \\tilde{d}}i} > (1 {\\rm TeV})^2,m^2_{{\\rm \\tilde{L}}i,{\\rm \\tilde{e}}i}>(300{\\rm GeV})^2 \\\\&M_3^2>(2 {\\rm TeV})^2, \\mu ^2> (300{\\rm GeV})^2,m_{\\rm A}^2 >(300 {\\rm GeV})^2.", "\\end{split}$ Here Eqs.", "(REF ) are the constraints to obtain a correct electroweak vacuum at the tree level.", "Eq.", "() is a rough requirement for the SM Higgs boson mass around $\\sim 125$ GeV suggested by FeynHiggs 2.11.2 [4].", "Eqs.", "() are the LHC and LEP bounds of the superparticles in the MSSM [2].", "Neglecting the 1st and 2nd generation Yukawa couplings, the SU(2) flavor symmetry suppresses flavor violation in the sfermion sector, and we do not consider the constraints from flavor physics.", "Also we assume that the parameters Eq.", "(REF ) are real and do not consider constraints of CP violation.", "Since we will solve the RECCs at the 1loop level, which can be derived from the given 1loop RG equations of the MSSM as in Appendix , we ignore the threshold corrections to the parameters.", "To apply a higher loop analysis, we can include the threshold corrections in these constraints and solve the RECCs derived from the higher loop RG equations." ], [ "Whole Viable Region and Phenomenologically Interesting Regions", "Solving the RECCs, which is derived from the given 1loop RG equations in [9] as in Appendix , we evaluate the Taylor coefficients of the conditions, Eqs.", "(REF ), () and (), in terms of the fundamental parameters Eq.", "(REF ).", "Directly solving these constraints, we obtain $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ , namely the whole viable region of the NUHM.", "$\\tilde{\\mathcal {M}}_f^{\\rm fund}$ is characterized by four independent parameters, as the seven fundamental parameters $G_a$ are constrained by these three equations in Eqs.", "(REF ) and ().", "A two dimensional slice in the four dimensional viable region, $\\mathcal {M}^{\\rm fund}_f$ , is shown in Fig.REF .", "Also shown are the mass bounds of superparticles and the pseudoscalar Higgs boson (A Higgs), where we have used indices “12\" and “3\" to denote the 1st/2nd and 3rd generations, respectively.", "Figure: The viable region and the boundaries of the experimental constraint on the slice of M 0 M_0=750GeV, A 0 A_0=500GeV with tanβ\\tan \\beta =10 (left).", "The region with green is experimentally allowed with the measured Higgs boson mass and a correct electroweak vacuum.", "Magnified view near an outline is presented in the right-hand side, where the boundary profiles are shown in detail.The black dot, (--15356iiGeV, 8836GeV), represents the low energy parameters given in Table.There is a new interesting region near the red solid line in Fig.REF , the boundary of which is given by the mass bound of the first two generation up-type squarks.", "We note that the stops are kept heavy to reproduce the measured Higgs boson mass.", "We call this region an inverted light squark (ILSQ) region.", "Here “inverted\" stands for the new superparticle mass pattern characterizing this region where the first two generation squarks are light, contrary to the ordinary light stop.", "Table.REF illustrates the low energy parameters corresponding to the point represented by the black dot shown in Fig.REF , where the up-type squarks in the first two generations may be within the reach of forth coming experiments.", "We have confirmed that the Higgs boson mass 124(1)GeV is evaluated from FeynHiggs 2.10.2 [4] with the parameters in Table.REF as an input.", "We have also confirmed that the ILSQ region still exists including 2loop RG running by solving RECCs at the two-loop level.", "Furthermore, we can observe a special point in the left-hand side of Fig.REF , where most of the contours including the gray dotted lines concentrate.", "Since they correspond to the mass bounds for the sfermions, the concentrating point, if near enough to the viable region, implies a surprising possibility.", "That is most of the scalar masses are just above the experimental bounds in spite of the heavy stops which weigh around 6TeV.", "However, as in the right figure the approximately concentrating point is too far away from the viable region, e.g.", "$\\mathcal {O}((10 {\\rm TeV} )^2)$ in mass squareds.", "In fact, the point is not excluded by the experimental mass bounds but excluded due to the unstable electroweak vacuum namely the A Higgs is tachyonic, $m_A^2<0$ .", "We will discuss the instability in detail in Sec.REF , and show the instability can be alleviated with large $\\tan \\beta $ in Sec.REF .", "Table: The low energy parameters corresponding to the black dot,(m Hu 0 m_{\\rm Hu0}, m Hd 0 m_{\\rm Hd0}, M 0 M_0,A 0 A_0)=(--15400iiGeV, 8840GeV, 750GeV, 500GeV) with tanβ=10\\tan \\beta =10, shown in Fig..", "The other fundamental parameters, (m 0 m_0, μ 0 \\mu _0, B 0 B_0)=(3200GeV, 12600GeV, 1800GeV), are evaluated from the solution of Eqs.", "() and ().", "“EW\" stands for electroweak." ], [ "Mechanism for the inverted light squark", "In this subsection we will explain how the characteristic mass pattern in the ILSQ region is generated in spite of the universal sfermion mass condition, Eq.", "(REF ), and the requirement of heavy stops, Eq.().", "Since these two conditions are imposed at two different scales, the RG running should be essential.", "The RG equation for a right-handed up-type squark mass is, ${d \\over dt}{m_{\\rm \\tilde{u}i}^2}& \\sim \\frac{2}{16 \\pi ^2} \\biggl \\lbrace 2 y_{\\rm t}^2X_{\\rm t}\\delta _{ i3}+Yg^{\\prime 2} S-{16 \\over 3}g_3^2M_3^2-{4}g^{\\prime 2}Y^2M_1^2\\biggr \\rbrace ,\\\\S \\equiv &\\left( m_{{\\rm {\\rm H_u}}}^2-m_{{\\rm H_{{\\rm d}}}}^2+ {\\rm Tr}[m_{{\\rm \\tilde{Q}}}^2 -m_{{\\rm \\tilde{L}}}^2-2m_{{\\rm \\tilde{u}}}^2+ m_{{\\rm \\tilde{d}}}^2 +m_{{\\rm \\tilde{e}}}^2] \\right),\\\\X_{\\rm t} \\equiv & m^2_{\\rm H_u}+m_{\\rm \\tilde{Q}_3}^2+m_{\\rm \\tilde{u}_3}^2+\\left\|{A_{\\rm t}}\\right\|^2,$ where $i$ represents 12 or 3, and $Y$ is the hypercharge, $-$ 2/3 [9].", "$y_t$ is the top- Yukawa coupling while $g^{\\prime }$ and $g_3$ are gauge couplings of $\\rm U(1)_Y$ and $\\rm SU(3)_c$ , respectively.", "The split between the 3rd and 1st/2nd generation up-type squarks should originate from the Yukawa term, the first term in Eq.", "(REF ).", "To raise the stop mass $m_{\\tilde{u}3}$ large enough for the inverted hierarchy, we need a negative and large $X_t$ that dominates over the other terms in Eq.", "(REF ).", "This is realized when $m_{\\rm H_u}^2$ is large and negative.", "For sufficiently large $\\tan \\beta $ , the bottom and tau Yukawa couplings can be effectively large in spite of the observed small mass ratio of bottom/tau to top.", "If $m_{\\rm Hd}^2$ is large and negative, the same argument applies to down-type squarks and sleptons.", "In summary, the inverted hierarchy can be generated by the Yukawa contribution through the RG running due to a negative and large Higgs mass squared parameter.", "Furthermore, we would like to explain the sfermion mass splitting within a generation.", "This originates in the gauge interactions, especially for $\\rm U(1)_Y$ gauge symmetry.", "In particular, the value of $S$ in Eq.", "() can largely deviate from zero in the NUHM contrary to the CMSSM case where $S$ is always zero.", "We find that when our mechanism of the ILSQ works, $S$ is non-zero in order to avoid the unstable electroweak vacuum, unless $\\tan \\beta $ is substantially large.", "We show this by reductio ad absurdum.", "Suppose that $S=0$ is satisfied with a large and negative $m_{\\rm Hu0}^2$ , namely $m_{\\rm Hu0}^2=m^2_{\\rm Hd0}\\ll 0$ at the GUT scale.", "Since $X_t$ dominates over the other terms in Eq.", "(REF ), the RG equations of $m^2_{\\rm H_u}$ and $m^2_{\\rm H_d}$ are also controlled by the large and negative Higgs masses [9], $\\beta _{m_{\\rm Hu}^2} \\sim {1 \\over 16\\pi ^2 } 6y_t^2 X_t^2 \\sim {1 \\over 16\\pi ^2 } 6y_t^2 m_{\\rm Hu}^2,$ $\\beta _{m_{\\rm Hd}^2} \\sim {1 \\over 16\\pi ^2 } \\left( 6y_b^2 m_{\\rm Hd}^2+2y_\\tau ^2 m_{\\rm Hd}^2 \\right).$ Hence, the RG running effect decreases the absolute value of $m_{\\rm Hu}^2$ at the experimental scale, while $m_{\\rm Hd}^2$ does not change so much due to the smaller Yukawa couplings.", "This implies that at low energy ${ 0\\gg m_{\\rm Hu}^2 > m_{\\rm Hd}^2}$ is satisfied and the problem of tachyonic instability of the electroweak vacuum occurs, as $m_A^2<0$ by solving Eqs.", "(REF ).", "Therefore, we need a greater $m_{\\rm H{d0}}^2$ to stabilize the electroweak vacuum and hence $S<0$ .", "Notice that this argument depends on the sign of the Higgs mass squareds and the size of $\\tan \\beta $ .", "Since Eq.", "(REF ) decreases the absolute value of $m_{\\rm Hu}^2$ , if $S$ is cancelled by positive and large $m_{\\rm Hu0}^2$ and $m_{\\rm Hd0}^2$ , Eq.", "(REF ) is replaced by $0 \\ll m_{\\rm Hu}^2 < m_{\\rm Hd}^2$ and the instability problem does not occur.", "This is the reason why the CMSSM is allowed to have a stable electroweak vacuum.", "Also if $\\tan \\beta $ is large enough, the RG running of $m_{\\rm Hd}^2$ , Eq.", "(REF ), can be effective and the vacuum instability problem should be alleviated with $S\\sim 0$ in the ILSQ region.", "For not so large $\\tan \\beta $ , the property of $S<0$ in the ILSQ region makes $\\tilde{u}_{12}$ the lightest squark as in Table.REF , because it has the least hypercharge.", "The requirement of $S<0$ is also the reason why the approximately concentrating point in the right-hand side of Fig.REF is far away from the viable region.", "In fact, the approximately concentrating point lies on the line of $S=0$ where the first two generation sfermion masses can be small without splitting.", "On the other hand, since $S\\sim 0$ may be allowed when $\\tan \\beta $ is sufficiently large, in this case there is a possibility to have an ILSQ region including most of the first two sfermion masses just above the experimental bounds.", "This is the case in Sec.REF .", "As we have discussed, the ILSQ region has a new superparticle mass pattern essentially related to the non-universal Higgs masses.", "Therefore, this mass pattern should be one of the most characteristic phenomenon of the NUHM, as the CMSSM never realize this." ], [ "Terribly Fine-Tuned but Important Region", "In Sec., we have presented the whole viable region of the NUHM by solving the experimental constraints in terms of the fundamental parameters which can be obtained from the method advocated in Sec.. We have found a new phenomenologically interesting region near the boundary corresponding to a squark mass bound.", "However, an interesting region with a characteristic property is not necessarily around the outlines, rather inside the viable region.", "This is particularly the case when the characteristic property requires a fairly large amount of fine tuning among the parameters.", "Notice that it would be difficult for the scatter plot method to identify such a fine-tuned region.", "In this section we will show how to analyze a large and complicated viable region to find out the characteristic properties localized inside of it, and argue this analysis has an advantage when applied to a FT-SUSY.", "To illustrate the idea, consider here the muon anomalous magnetic dipole moment (muon $g-2$ ), $\\alpha _\\mu $ .", "The muon $g-2$ anomaly is a hint of new physics, as the discrepancy between the theoretical and experimental values exceeds the $3~\\sigma $ level of these errors [6], [7], [8] $\\alpha _{\\rm \\mu }^{\\rm exp} -\\alpha _{\\rm \\mu }^{\\rm SM} = (26.1\\pm 8.0)\\times 10^{-10}.$ In the generic MSSM at least three light superparticles, weigh around 300GeV, are needed to generate a large enough contribution [10], [11].", "Therefore, the region of the generic MSSM, $\\tilde{\\mathcal {M}}_e^{\\rm gen}$ , that fills the discrepancy of the muon $g-2$ , should be characterized by three parameters with very tiny values compared to the stop mass parameters around $6 {\\rm TeV} $ .", "Therefore, if exists, the region of interest of the NUHM now, $\\tilde{\\mathcal {M}}_f^{\\rm fund}$ , should be fine-tuned from the whole viable region in the previous section.", "The particular diagram we consider for the SUSY contribution to the muon $g-2$ is made of a loop including a bino and smuons of both chiralities [10], [11].", "Following [10], we obtain an approximated formula of the SUSY contribution to the muon $g-2$ , $\\delta \\alpha _{\\rm \\mu } &\\equiv \\alpha _{\\rm \\mu }^{\\rm MSSM}-\\alpha _{\\rm \\mu }^{\\rm SM} \\nonumber \\\\&\\sim \\frac{1}{16\\pi ^2}\\frac{g^{\\prime 2} m_{\\mu }^2 M_1 \\mu \\tan \\beta \\min {\\lbrace m_{\\rm \\tilde{L}12}^2,m_{\\rm \\tilde{e}12}^2,M_1^2\\rbrace }}{2m_{\\rm \\tilde{L}12}^2m_{\\rm \\tilde{e}12}^2M_1^2} \\nonumber \\\\&\\sim 5 \\times 10^{-10} \\frac{(100{\\rm GeV})^2 M_1 \\mu \\tan \\beta \\min {\\lbrace m_{\\rm \\tilde{L}12}^2,m_{\\rm \\tilde{e}12}^2,M_1^2\\rbrace }}{m_{\\rm \\tilde{L}12}^2m_{\\rm \\tilde{e}12}^2M_1^2}.$ This rough approximation will be corrected by fitting the results evaluated in FeynHiggs 2.11.2 [4] by varying the overall coefficient of this function." ], [ "How large can the muon $g-2$ be in the NUHM?", "In order to clarify whether the NUHM can explain the muon $g-2$ anomaly, an efficient way is to evaluate the maximal value of the muon $g-2$ in the NUHM.", "This immediately draws a conclusion of whether the muon $g-2$ anomaly can be explained.", "Therefore we impose a condition, ${\\rm maximize}{[\\delta \\alpha _{\\rm \\mu }]} \\ &{\\rm by \\ varying \\ } m_0,$ in addition to Eqs.", "(REF )–().", "In fact, we can vary all the free parameters to maximize the muon $g-2$ , however, for the illustrative purpose we only vary one parameter.", "This reduces a free parameter, $m_0$ , in $\\mathcal {M}_f^{\\rm fund}$ .", "By solving the conditions, Eqs.", "(REF )–() and (REF ), in terms of $A_0, M_{0}$ , and $\\mu _0$ , we obtain a three dimensional region of interest, $\\mathcal {\\tilde{M}}_f^{\\rm fund}$ .", "An $A_0=0$ slice of the solution is shown in Fig.REF with $\\tan \\beta =35$ .", "The contours represent the maximized total muon $g-2$ by varying $m_0$ .", "We find the maximized muon $g-2$ can exceed $18.1 \\times 10^{-10}$ with $\\tan \\beta \\gtrsim 25$ , namely the NUHM is able to fill the discrepancy of the muon $g-2$ within the $1~\\sigma $ level error.", "In fact, there are two kinds of regions, namely type 1 and type 2 regions, depending on the lightest particle in the loop diagram corresponding to Eq.", "(REF ).", "The type 1 region explains the muon $g-2$ anomaly with either $\\tilde{e}_{12}$ or $\\tilde{L}_{12}$ as the lightest particle in this diagram, while the type 2 region has bino as the lightest particle.", "Fig.REF corresponds to the former one, while the region of type 2 appears for $\\tan \\beta \\gtrsim 55$ as we will show in Sec.REF .", "Figure: The contour plot of the maximized muon g-2g-2 by varying m 0 m_0 on a slice at A 0 =0A_0=0 with tanβ\\tan \\beta =35 (left).", "Magnified view of the yellow region is presented in the right-hand side.", "The red (green, blue) region represents that the total muon g-2g-2, including the maximal NUHM correction, is within the 1σ1~\\sigma (2σ2~\\sigma , 3σ3~\\sigma ) level error of the observed value.", "The black dot, (8700GeV, 2010GeV), represents the low energy parameters given in Table..Table: The low energy parameters corresponding to the black dot, (M 0 ,μ 0 ,A 0 )(M_0,\\mu _0,A_0)=(2010GeV, 8700GeV, 0GeV), in Fig.", "with tanβ\\tan \\beta =35.", "The other fundamental parameters, (m 0 ,m Hu 0 ,m Hd 0 ,B 0 )(m_0, m_{\\rm Hu0}, m_{\\rm Hd0}, B_0) = (1186iiGeV, 9375iiGeV, 8326iiGeV, --768GeV), are the solution of Eqs.", "(), () and ()." ], [ "The region explaining the muon $g-2$ anomaly and the mechanism", "Since the NUHM has a region where the muon $g-2$ anomaly is explained, now we would like to explore the features of this region.", "Since we know that an outline may have some phenomenological information about the nearby region, it is meaningful to show the outlines of the region where the muon $g-2$ is just at the experimental central value, rather than the maximal value.", "We impose a condition instead of Eq.", "(REF ), $\\frac{(100{\\rm GeV})^2 M_1 \\mu \\tan \\beta \\min {\\lbrace m_{\\rm \\tilde{L}12}^2,m_{\\rm \\tilde{e}12}^2,M_1^2 \\rbrace }}{m_{\\rm \\tilde{L}12 }^2m_{\\rm \\tilde{e}12}^2M_1^2} =\\left\\lbrace \\begin{array}{ll}&10 \\ (\\tan \\beta \\sim 35)\\\\&7.5 \\ (\\tan \\beta \\sim 60).\\end{array} \\right.$ The factor 10 and 7.5 in the right-hand side are fitted experimentally using FeynHiggs 2.11.2 [4].", "The solutions by solving Eqs.", "(REF )–(), and (REF ) in terms of $A_0, M_{0}$ , and $\\mu _0$ , are presented in Fig.REF and Fig.REF corresponding to type 1 and type 2 regions, respectively.", "Also shown are the boundary profiles.", "If the muon $g-2$ anomaly is explained by the region in Fig.REF , from the boundary profiles some of light smuons, selectrons, right-handed stau and A-Higgs might be within the reach of the forth coming experiments.", "On the other hand, Fig.REF implies that many superparticles might be within the reach of the forth coming experiments, as many boundaries of the experimental constraints are within the distances of $\\lesssim \\mathcal {O}( {\\rm TeV} )$ from this interesting region.", "A set of low energy parameters corresponding to the type 2 region is given in Table.REF .", "We have confirmed that the Higgs mass and the muon $g-2$ anomaly are explained using FeynHiggs 2.10.2 [4] with an input of these parameters.", "We have also confirmed that the parameter region does not vanish but shifts to a different area in the parameter space by solving the RECCs at the 2loop level with some of the 1loop threshold corrections included in the conditions Eqs.().", "Chosen in this parameter region, some sets of fundamental parameters are inputted into SOFTSUSY 3.7.1 [12] and a similar pattern of the low energy parameters to those in Table.REF is produced.", "For example, with an input $\\lbrace \\tan \\beta =61, M_{\\rm 0}=919 {\\rm GeV} , m_{\\rm Hd0}^2=-1.835\\times 10^{8} {\\rm GeV} ^2, m_{\\rm Hu0}^2=-2.200\\times 10^{8} {\\rm GeV} ^2, m_0=623.0 {\\rm GeV} , \\mu _0>0 \\rbrace $ , SOFTSUSY 3.7.1We have set the number of loops in the Higgs boson mass computation to be one, as otherwise the computation would not converge due to the terrible fine tuning.", "shows a set of low energy parameters from which the measured Higgs boson mass and the muon $g-2$ within the 1$\\sigma $ error are evaluated by FeynHiggs [12], [4].", "Figure: A 0 A_0=0 slice of the region of type 1 with tanβ\\tan \\beta =35.", "The region with greenexplains the muon g-2g-2 anomaly within its 1σ1\\sigma level error where one of the smuonsis lighter than the bino.The magnified view of the corresponding colored region is also shown with the boundary profiles in detail.The profiles of the boundaries that do not constitute the outlines are turned off.Figure: A 0 A_0=0 slice of the region of type 2 with tanβ=60\\tan \\beta =60.", "The region with green explains the muon g-2g-2 anomaly within its 1σ1\\sigma level error where the bino is lighter than the smuons.The boundary profiles are also shown.", "The blue solid and dashed lines denote the boundary of the condition to have type 2 region which constrains the bino to be the lighter than the smuons.The black dot, (15120GeV, 700GeV), has low energy parameters given in Table.Now we would like to consider how the muon $g-2$ anomaly is explained.", "From Eq.", "(REF ), we find that large $\\mu -$ term and $\\tan \\beta $ are favored for a large value of the muon $g-2$ .", "This requires large and negative Higgs masses to have a correct electroweak vacuum due to Eqs.", "(REF ).", "Therefore the mechanism of the ILSQ in Sec.REF can be applied.", "As noted, a negative $S-$ term is required to stabilize the electroweak vacuum when $\\tan \\beta $ is not substantially large.", "Neglecting RG running due to gauginos, $\\tilde{u}_{12}$ becomes the lightest sfermion in most of the cases as in Table.REF .", "This excludes a smuon around 300GeV, as it should be heavier than $\\tilde{u}_{12}$ which is bounded below by 1TeV.", "If we would like to decrease the smuon masses in the ILSQ region, what we should do is to increase the gluino mass.", "The RG effect from a heavy gluino, as the third term in Eq.", "(REF ), raises the masses of squarks universally from those of sleptons.", "However the universal gaugino mass condition, Eq.", "(REF ), implies a heavy bino.", "Hence, the lightest particle in the loop diagram corresponding to Eq.", "(REF ) becomes a smuon which can be seen from the boundary profiles in Fig.", "REF .", "This is the case for type 1 region.", "The type 2 region where bino is the lightest particle in the loop diagram corresponding to Eq.", "(REF ) can be realized with sufficiently large $\\tan \\beta $ .", "This is consistent with the argument noted in Sec.REF : $S\\sim 0$ can be realized with large $\\tan \\beta $ .", "In this case, we do not need a heavy gluino to raise the squark masses and the bino can be as light as the smuons.", "Furthermore, all the 1st and 2nd generation sfermions will be light because small gaugino masses and $S-$ term imply the degeneration of the sfermions in the first two generations, and the mechanism of ILSQ works for all of them.", "This is actually the case in Table.REF .", "As we can see in Fig.REF , Fig.REF , and Fig.REF , the relevant regions have small sizes due to fine tuning.", "In fact, fine tuning is already alleviated in these figures as we have reduced a free parameter that is fine-tuned contrary to the stop mass scale by solving Eq.", "(REF ) or Eq.", "(REF ).", "On the other hand, after the discovery of the Higgs boson the muon $g-2$ correction in the NUHM is discussed in several studies [13] by using the scatter plot method.", "In these studies, the tiny regions we found in this paper had been overlooked, as the “pointillism\" is not practical to find such a tiny and terribly fine-tuned region.", "Therefore, we have shown that the method advocated in this paper has a strong advantage in identifying a fine-tuned region and this should be an efficient approach to a FT-SUSY.", "Table: The low energy parameters corresponding to the black dot,(M 0 ,μ 0 ,A 0 M_0, \\mu _0, A_0) =(700GeV, 15120GeV, 0GeV), represented in Fig.", "with tanβ\\tan \\beta =60.", "The other fundamental parameters, (m 0 ,m Hu 0 ,m Hd 0 ,B 0 m_0,m_{\\rm Hu0},m_{\\rm Hd0},B_0) =(898GeV, 14910iiGeV, 13650iiGeV,21GeV), are obtained by solving the conditions corresponding to Eqs.", "(), () and ()." ], [ "Conclusions", "In this paper, we proposed a novel and efficient approach to the supersymmetric models with some amount of fine tuning, in which the commonly used approach of scatter plot is inefficient and sometimes even fails to find relevant regions in the parameter space of superparticle masses with the limited number of plotted points.", "The essential idea of our approach is to directly map the (experimental or other) constraints at low energy to those in the fundamental parameter space.", "We can identify the relevant region in the fundamental parameter space by filling the interior of the mapped constraints as the boundaries of the constraints will form the outlines of the relevant region.", "Furthermore the areas near the boundaries of the experimental constraints which are rather easily identified in our method can be phenomenologically interesting as they will be within forth coming experiments.", "We applied this method to the non-universal Higgs masses (NUHM) model.", "The features of the NUHM model are the same as the CMSSM except that the SUSY-breaking Higgs masses differ from the universal sfermion mass at the GUT scale.", "By using our method, we identified the phenomenologically viable regions of the parameter space and argued some interesting features of the model.", "Among other things, we found, in some cases, that the inverted squark masses are realized, where the renormalization group effects raise the third generation squark masses compared to those of the first two generations.", "This mass pattern is a characteristic phenomenon of the NUHM model and is never realized in the CMSSM.", "Another application of our method is to identify, within the NUHM model, the existing but tiny region in the parameter space, where the SUSY contribution explains the discrepancy of the muon $g-2$ within the $1~\\sigma $ level of experimental and theoretical errors.", "The price to pay is the terrible fine tuning among the parameters, and therefore the previous studies with the conventional scatter plot method failed to find this region, drawing misleading conclusions.", "This example illustrates the power of our method in particular when the required fine tuning is severe.", "The relevance of our approach will even increase when the forth coming experiments will give null results in superparticle searches and more fine tuning will be required to correctly produce the electroweak scale.", "The method advocated in this paper has a variety of applications, some of which was given in [14] and also will be discussed elsewhere." ], [ "Acknowledgement", "We would like to thank Yutaro Shoji for collaboration at an early stage of this work.", "This work is supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No.23104008." ], [ "RECCs for the generic MSSM", "We would like to show the derivation of RECCs in the generic MSSM with given dimensionless couplings at $t_e$ .", "In fact, the following argument can apply to the derivation of RECCs at any loop order with given RG equation at the same order.", "The parameters $g_i^f$ and constants of the generic MSSM are classified by their dimensions: D=2 Sfermion masses, Higgs masses: $m^2_{i}$ D=1 Gaugino masses, $\\mu -$ and $A-$ terms: $M_{i}$ D=0 Gauge couplings and Yukawa couplings: $y_{i}$ where the set of $m^2_i$ is understood to include the off-diagonal elements of usual three by three sfermion mass matrices etc.", "A perturbative RG equation for the parameters of dimension $d$ is written in terms of the parameters with dimension$\\le d$ : $\\begin{split}{d \\over dt}{m^2_{i}} &=a_{2,i}^{k}(y_{j} )m^2_{k}+a_{1,i}^{kl}(y_{j})M_{k}M_{m},\\\\{d \\over dt}{M}_{i} &=b_{i}^{k}(y_{j})M_{k}, \\\\{d \\over dt}{y}_{i} &=c_{i}(y_{j}).\\end{split}$ $a^k_{2,i}(y_j),a^{kl}_{1,i}(y_j), b_i^k(y_j)$ and $c_i(y_j)$ are given functions that can be derived from loop calculations in the generic MSSM [9].", "The summation is understood.", "Firstly, we can solve the RG equation for dimensionless constants numerically with given $y_{i}$ at $t_e$ .", "Substituting this numerical solution for $y_i$ , the (effective) RG equation for dimensionful parameters become, $\\begin{split}{d \\over dt}{m^2_i} &=a_{2,i}^{k}(t)m^2_{k}+a_{1,i}^{kl}(t)M_{k}M_{l}\\\\{d \\over dt}{M}_{i} &=b_{i}^{k}(t)M_{k}.\\end{split}$ Secondly, using Eq.", "(REF ), we can derive the RECCs for condition functions, ${ \\phi _2(m_i^2)=m_{1}^2(t_e)-(10^3 {\\rm GeV} )^2~ {\\rm and} ~ \\phi _1(M_i)=M_{1}(t_e)-(10^3 {\\rm GeV} ),}$ corresponding to simplified experimental mass bounds of a scalar and a fermion, respectively.", "Dimensional analysis allows us to guess the forms of the mapped condition functions at an arbitrary scale, $t$ , to be $\\Phi _2(t) &\\equiv \\phi _2^{i}(t)m^2_{i}+\\phi _2^{ij}(t)M_{i}M_{j}+c_2(t)\\\\\\Phi _1(t) &\\equiv \\phi _1^{i}(t)M_{i}+c_1(t).$ Therefore, employing Eq.", "(REF ) the RECCs for Eqs.", "(REF ) and () are derived as $\\nonumber {d \\over dt}{\\phi }_{2}^{i}(t) &=-\\phi _{2}^{k}(t) a_{2,k}^{i}(t)\\\\\\nonumber {d \\over dt}{\\phi }_{2}^{ij}(t)&=-\\phi _{2}^{k}(t)a_{1,k}^{ij}(t)-\\phi _{2}^{kj}(t)b_{k}^{i}(t)-\\phi _{2}^{ik}(t)b^{j}_{k}(t)\\\\{d \\over dt}{\\phi _1^{i}}(t)&=-\\phi _1^{k}(t) b^{i}_{k}(t)\\\\\\nonumber {d \\over dt}{c_1}(t)&={d \\over dt}{c_2}(t)=0.$ With the initial condition, $\\phi _2^i(t_e)= \\delta _1^i, \\phi _2^{ij}(t_e)=0, c_2(t_e)=-(10^3 {\\rm GeV} )^2$ and $\\phi _1^i(t_e)=\\delta _1^i, c_1(t_e)=-10^3 {\\rm GeV} ,$ taken from Eq.", "(REF ), we obtain the mapped condition functions as the solutions of Eq.", "(REF ).", "Notice that a mapped condition function which equals to a dimensionful parameter at $t_e$ , as ${\\phi (t_e)=m_i^2 ~ {\\rm or}~ \\phi (t_e)=M_i, }$ is a solution of the RG equation with an explicit form in terms of the dimensionful parameters at $t$ .", "These solutions actually can be obtained by solving Eq.", "(REF ) with proper initial conditions imposed where $c_1(t)=c_2(t)=0$ .", "Finally, substituting all the solutions corresponding to the parameters at $t_e$ , we can map any condition functions to the parameter space at an arbitrary scale $t$ ." ] ]	1606.04953
[ [ "The Leptonic Higgs Portal" ], [ "Abstract An extended Higgs sector may allow for new scalar particles well below the weak scale.", "In this work, we present a detailed study of a light scalar $S$ with enhanced coupling to leptons, which could be responsible for the existing discrepancy between experimental and theoretical determinations of the muon anomalous magnetic moment.", "We present an ultraviolet completion of this model in terms of the lepton-specific two-Higgs doublet model and an additional scalar singlet.", "We then analyze a plethora of experimental constraints on the universal low energy model, and this UV completion, along with the sensitivity reach at future experiments.", "The most relevant constraints originate from muon and kaon decays, electron beam dump experiments, electroweak precision observables, rare $B_d$ and $B_s$ decays and Higgs branching fractions.", "The properties of the leptonic Higgs portal imply an enhanced couplings to heavy leptons, and we identify the most promising search mode for the high-luminosity electron-positron colliders as $e^+{+}e^-\\to\\tau^+{+}\\tau^-{+}S \\to \\tau^+{+}\\tau^-{+}\\ell{+}\\bar \\ell$, where $\\ell =e,\\mu$.", "Future analyses of existing data from BaBar and Belle, and from the upcoming Belle II experiment, will enable tests of this model as a putative solution to the muon $g-2$ problem for $m_S < 3.5$ GeV." ], [ "Introduction", "The LHC discovery of a new particle of mass $\\sim $ 125 GeV, with properties consistent with those of the Standard Model Higgs boson [1], , provides compelling evidence for the picture of the electroweak symmetry, and its spontaneous breakdown, encapsulated in the Standard Model (SM).", "It remains an important question to understand whether the entire Higgs sector is minimal, as in the SM, or contains additional states as would be required by supersymmetry, or may be motivated by other scenarios including, for example, models of dark matter.", "While the existence of new physics at the TeV scale is still a distinct possibility (see e.g.", "[3]), in recent years, independent empirical motivations related to dark matter and neutrino masses have pointed to the possibility of a hidden sector, weakly coupled to the SM [4].", "The mass scales in the hidden sector can be considered free parameters, and therefore particles much lighter than the electroweak or TeV scales are plausible.", "On general effective field theory grounds, the leading interactions with a neutral light hidden sector would be through the relevant and marginal interactions involving SM gauge singlets, which have been dubbed “portals\" [5] and are the subject of considerable theoretical and experimental study.", "In several cases, hypothetical light particles may help to explain certain experimental anomalies and deviations from the SM.", "It has been appreciated that a rather minimal extension of the SM via an additional vector particle $V$ (often termed the “dark photon\") that kinetically mixes with the photon through the interaction $(\\epsilon /2)V^{\\mu \\nu }F_{\\mu \\nu }$ , where $V^{\\mu \\nu }$ and $F^{\\mu \\nu }$ are the $V$ and photon field strengths respectively, can generate an appreciable shift of the muon anomalous magnetic moment [6], $\\Delta a_\\mu \\simeq \\frac{ \\alpha \\epsilon ^2}{2 \\pi }~~{\\rm ~ when }~ m_V\\ll m_\\mu .$ For $\\epsilon \\sim 10^{-3}$ , such a model offers a correction on the order of the existing discrepancy in $a_\\mu $ , with the right sign to alleviate the tension between theory and experiment [7].", "A subsequent painstaking search for light dark photons in both old data and in dedicated new experiments has resulted in upper limits on $\\epsilon $ that now render the minimal dark photon model unable to explain the existing discrepancy.", "(The last remaining portion of the parameter space able to account for the discrepancy was excluded by the NA48/2 experiment [8].)", "However, modifications of the minimal vector portal model, for example dark photons decaying to other dark sector states, and gauge groups based on $L_\\mu - L_\\tau $ , are still able to shift $a_\\mu $ by $3\\times 10^{-9}$ (the scale of the experimental discrepancy), and be consistent with all other constraints (see, e.g., [9], [10], [11], [12]).", "Figure: Left panel: Constraints on the coupling to leptons (in terms of both ξ ℓ S =g ℓ (v/m ℓ )\\xi ^S_\\ell =g_\\ell (v/m_\\ell ) and ϵ eff =g e /e\\epsilon _{\\rm eff}=g_e/e) as a function of the scalar mass, based purely on the effective theory in Eq. ().", "The region where (g-2) μ (g-2)_\\mu is discrepant at 5σ5\\sigma is shaded in red, while the green shaded band shows where the current discrepancy is brought below 2σ2\\sigma .", "We show constraints from the beam dumps E137, Orsay, and E141.", "The projected sensitivities from μ→3e\\mu \\rightarrow 3e, NA48/2, NA62, HPS, analyses of existing data from COMPASS and BB-factories, as well as a projected sensitivity at BELLE II are also shown.", "(See Section 3 for details.)", "Right panel: Constraints on the L2HDM+ϕ\\varphi UV completion of the effective theory in Eq.", "(), as described in Sec. .", "Model independent results are as in the left panel.", "In addition, for this particular UV completion, there are constraints on the model from searches for h→SS→2μ2τh\\rightarrow SS\\rightarrow 2\\mu 2\\tau , B→K () ℓ + ℓ - B\\rightarrow K^{(\\ast )}\\ell ^+\\ell ^-, and B s →μ + μ - B_s\\rightarrow \\mu ^+\\mu ^-.", "We have set tanβ=200\\tan \\beta =200, m H =m H ± =500 GeV m_{H}=m_{H^\\pm }=500~\\rm GeV, and m 12 =1 TeV m_{12}=1~\\rm TeV.", "(See Section 4 for details.", ")In this paper, we concentrate on light scalars coupled to leptons as a prospective solution to the muon $g-2$ anomaly.", "The relevant observation was originally made by Kinoshita and Marciano [13]: a SM-like Higgs boson with a very light mass, $m_h \\ll m_\\mu $ (excluded by now via numerous experiments culminating in the discovery of the Higgs at the LHC), gives the following positive shift of the muon anomalous magnetic moment, $\\Delta a_\\mu = \\frac{3}{16 \\pi ^2} \\times \\left( \\frac{m_\\mu }{v} \\right)^2 \\simeq 3.5 \\times 10^{-9},$ which is very close to the existing discrepancy.", "In this expression, $v=246~\\rm GeV$ is related to the vacuum expectation value of the Higgs doublet, $H$ , via $\\langle H\\rangle =v/\\sqrt{2}$ .", "The lesson of this observation is that if a new light scalar particle couples to leptons with a coupling strength on the order of the SM lepton Yukawa couplings, which in the case of the muon is $m_\\mu /v \\simeq 4\\times 10^{-4}$ , the muon $g-2$ problem can be solved.", "Thus we are motivated to study the effective Lagrangian of an elementary scalar $S$ , ${\\cal L}_{\\rm eff} = \\frac{1}{2}(\\partial _\\mu S)^2 - \\frac{1}{2}m_S^2 S^2 +\\sum _{l = e,\\mu ,\\tau } g_\\ell S \\overline{\\ell }\\ell ,$ with $g_l \\sim m_l/v$ as a promising phenomenological model.", "Given that $S$ is not the SM Higgs boson, the interaction terms in (REF ) may appear to contradict SM gauge invariance.", "Thus, at minimum, Eq.", "(REF ) requires an appropriate UV completion, generically in the form of new particles at the electroweak (EW) scale charged under the SM gauge group.", "On the other hand, if a UV-complete model is found that represents a consistent generalization of (REF ), the light scalar solution to the muon $g-2$ problem deserves additional attention.", "Another impetus for studying very light beyond-the-SM (BSM) scalars comes from the existing discrepancy of the muon- and electron-extracted charge radius of the proton [14].", "This paper presents a detailed study of light scalars with enhanced coupling to leptons, and provides a viable UV-completion of Eq.", "(REF ) through what we dub the `leptonic Higgs portal'.", "We also analyze a variety of phenomenological consequences of the model.", "The phenomenology of a light scalar coupled to leptons resembles in many ways the phenomenology of the dark photon, but with the distinct feature that the couplings to individual flavors are non-universal and proportional to the mass.", "As a result, at any given energy the production of such a scalar is most efficient using the heaviest kinematically accessible lepton.", "We identify the most important search modes for the scalar that could decisively explore its low mass regime.", "Our main conclusion is that an elementary scalar with coupling to leptons $\\ell $ scaling as $m_\\ell $ can be very efficiently probed, and in particular the whole mass range consistent with a solution of the muon $g-2$ discrepancy can be accessed through an analysis of existing data and in upcoming experiments.", "Our full UV-complete model is based on the lepton-specific two Higgs doublet model with an additional light scalar singlet.", "The mixing of the singlet with components of the electroweak doublets results in the effective Lagrangian of Eq.", "(REF ).", "The model also induces additional observables, and thus constraints, due to the fact that $S$ receives small but nonvanishing couplings to the SM quarks and gauge bosons.", "We note that the UV completion presented in this work is not unique.", "For an alternative UV completion of the same model utilizing vector-like fermions at the weak scale, see Ref. [15].", "While many aspects of the low-energy phenomenology based on the effective Lagrangian (REF ) are similar in both approaches, the UV-dependent effects are markedly different (especially for flavor-changing observables).", "This paper is organized as follows.", "In the next section we discuss light scalars coupled to leptons and a possible UV completion of such models via the leptonic Higgs portal.", "In Sec.", "we analyze the constraints and sensitivity levels to light scalars coupled to leptons that are universal, and independent of the UV completion (resulting from muon decays, leptonic kaon decays, electron beam dumps and high-intensity $e^+e^-$ colliders); the results are shown in the left panel of Fig.", "REF .", "In Sec.", "we analyze the constraints and sensitivities that are tied to the specific UV-completion involving the leptonic Higgs portal.", "These include rare $B$ and Higgs decays; the results are shown in the right panel of Fig.", "REF .", "We present some additional discussion and reach our conclusions in Sec.", "." ], [ "Leptonic Higgs portal", "In this section, we discuss a concrete UV-completion of the low-energy Lagrangian in Eq.", "(REF ).", "A simple starting point to couple a singlet field $\\varphi $ to the SM is through the Higgs portal, ${\\cal L}_{\\rm int} =(A \\varphi +\\lambda \\varphi ^2)H^\\dagger H,$ where $H$ is the SM Higgs doublet and $A,\\lambda $ are coupling constants.", "The trilinear term induces mixing between the singlet and the ordinary Higgs boson $h$ after electroweak symmetry breaking, where $H^0 = (v+h)/\\sqrt{2}$ .", "The mixing angle is given by $\\theta = \\frac{Av}{m_h^2 - m_\\varphi ^2},$ and after field diagonalization the coupling of the light scalar $S$ (mostly comprised of the singlet $\\varphi $ ) to SM fermions is simply given by their SM Yukawa coupling times this mixing angle.", "Low mass singlets are constrained by $B$ and $K$ meson decays (see, e.g.", "a collection of theoretical and experimental studies in Refs.", "[16], [17], [18], [19], [20], [21], [22], [23]), and for $m_S < 4~\\rm GeV$ the mixing angle is limited to $\|\\theta \| < 10^{-3}$ .", "Significant further advances in sensitivity to $\\theta $ are possible with the planned SHiP experiment [24].", "Therefore, there is no room to accommodate $\\theta \\sim O(1)$ , and consequently no large correction to the muon $g-2$ is allowed within this simple model.", "To circumvent this obstacle, we modify the SM by not only adding a singlet but also by introducing a second Higgs doublet that mixes with the singlet.", "In particular, we are interested in the so-called `lepton-specific' representation of a generic two Higgs doublet model (L2HDM) [25], [26], [27], [28], [29].", "Calling the two doublets with SM Higgs charge assignments $\\Phi _1$ and $\\Phi _2$ , we assume that $\\Phi _1$ couples exclusively to leptons, while $\\Phi _2$ couples to quarks.", "Moreover, we assume that all physical compenents of $\\Phi _{1,2}$ are at the weak scale or above.", "Taking $\\langle \\Phi _2\\rangle /\\langle \\Phi _1 \\rangle \\equiv \\tan \\beta $ very large, as well as arranging for the physical bosons of $\\Phi _1$ to be heavier than those of $\\Phi _2 $ , we arrive at an “almost SM-like\" limit, but with the set of heavier Higgses that couple to leptons possessing couplings enhanced by $\\tan \\beta $ .", "The mixing term $A_{12}(\\Phi _1^\\dagger \\Phi _2 +\\Phi _2^\\dagger \\Phi _1)\\varphi $ will then efficiently mix $\\varphi $ with $\\Phi _1$ , resulting in the light scalar $S$ coupling to leptons with strength $g_\\ell = \\frac{m_\\ell }{v} \\times \\tan \\beta \\times \\theta _\\ell ,$ where $\\theta _\\ell $ is the mixing between $S$ and $\\Phi _1$ .", "It is then clear that the desirable outcome of $g_\\ell \\sim m_\\ell /v$ can be achieved in the regime $\\tan \\beta \\gg 1$ , $\\theta _\\ell \\ll 1$ , and $\\tan \\beta \\times \\theta _\\ell \\sim O(1)$ .", "We now elaborate on this simple idea and present details of the model.", "The scalar potential we consider is given by $V(\\Phi _1, \\Phi _2,\\varphi ) = V_{2\\rm HDM} + V_\\varphi + V_{\\rm portal}.$ $V_{2\\rm HDM}$ is the main part of the potential that determines the pattern of electroweak symmetry breaking.", "Its CP-conserving version is given by the familiar expression, $V_{2\\rm HDM}&=m_{11}^2\\Phi _1^\\dagger \\Phi _1+m_{22}^2\\Phi _2^\\dagger \\Phi _2-m_{12}^2\\left(\\Phi _1^\\dagger \\Phi _2+\\Phi _2^\\dagger \\Phi _1\\right) \\nonumber \\\\&\\hspace{-19.91684pt}+\\frac{\\lambda _1}{2}\\left(\\Phi _1^\\dagger \\Phi _1\\right)^2+\\frac{\\lambda _2}{2}\\left(\\Phi _2^\\dagger \\Phi _2\\right)^2 +\\lambda _3\\left(\\Phi _1^\\dagger \\Phi _1\\right)\\left(\\Phi _2^\\dagger \\Phi _2\\right) \\nonumber \\\\&\\hspace{-19.91684pt}+\\lambda _4\\left(\\Phi _1^\\dagger \\Phi _2\\right)\\left(\\Phi _2^\\dagger \\Phi _1\\right)+\\frac{\\lambda _5}{2}\\left[\\left(\\Phi _1^\\dagger \\Phi _2\\right)^2+\\left(\\Phi _2^\\dagger \\Phi _1\\right)^2\\right].$ The singlet potential in (REF ) is a generic polynomial with positive $\\varphi ^4$ term, $V_S=B\\varphi +\\frac{1}{2}m_0^2\\varphi ^2+\\frac{A_\\varphi }{2}\\varphi ^3+\\frac{\\lambda _\\varphi }{4}\\varphi ^4.$ In the portal part of the potential we are most interested in the trilinear terms, $\\nonumber V_{\\rm portal}=\\left[A_{11}\\Phi _1^\\dagger \\Phi _1+A_{22}\\Phi _2^\\dagger \\Phi _2 \\right.~~~~~~~~~\\\\\\left.+A_{12}\\left(\\Phi _1^\\dagger \\Phi _2+\\Phi _2^\\dagger \\Phi _1\\right)\\right]\\varphi .$ Generically, the $A_{11}$ portal term leads to a $1/\\tan \\beta $ suppressed mixing between $\\varphi $ and the electroweak scalars, while, for $\\tan \\beta \\gg 1$ , the $A_{22}$ portal coupling is strongly constrained by existing limits on the $A\\varphi H^\\dagger H$ operator.", "On the other hand, the $A_{12}$ portal is less constrained and leads to efficient mixing with $\\varphi $ .", "In what follows we will ignore $A_{11,22}$ .", "The spectrum of the theory at the electroweak scale is dominated by $V_{2\\rm HDM}$ , while $V_\\varphi $ and $V_{\\rm portal}$ can be regarded as small perturbations.", "In determining the spectrum at the weak scale, we decompose the doublets assuming each obtains a vacuum expectation value, $\\langle \\Phi _a\\rangle \\equiv v_a$ , $\\Phi _a&\\supset v_a+\\rho _a$ for $a=$ 1, 2 with $\\rho _a$ a real scalar field (we work in unitary gauge and ignore charged components of the doublets for now).", "The ratio of the VEVs is $v_2/v_1=\\tan \\beta $ with $v_1^2+v_2^2\\equiv v^2=(246~{\\rm GeV})^2$ .", "Furthermore, through a $\\varphi $ field redefinition, the coefficient $B$ in Eq.", "(REF ) can be chosen so that $\\varphi $ does not obtain a VEV.", "The elements of the mass matrix of the neutral CP-even scalars in the basis $\\left(\\rho _1,\\,\\rho _2,\\,\\varphi \\right)$ are $M^2_{11}&=m_{12}^2\\tan \\beta +\\lambda _1v^2\\cos ^2\\beta ,\\\\M^2_{22}&=m_{12}^2\\cot \\beta +\\lambda _2v^2\\sin ^2\\beta ,\\\\M^2_{12}&=-m_{12}^2+\\lambda _{345}v^2\\cos \\beta \\sin \\beta ,\\\\M^2_{13}&=v A_{12}\\sin \\beta ,~M^2_{23}=vA_{12}\\cos \\beta ,~M^2_{33}=m_{0}^2.$ with $\\lambda _{345}\\equiv \\lambda _3+\\lambda _4+\\lambda _5$ .", "In the limit that $A_{12}\\ll v,\\, m_{12}$ , we can rotate to the mass basis perturbatively, $\\left(\\begin{array}{c}\\rho _1 \\\\\\rho _2 \\\\\\varphi \\end{array}\\right)\\simeq \\left(\\begin{array}{ccc}-\\sin \\alpha & \\cos \\alpha & \\delta _{13} \\\\\\cos \\alpha & \\sin \\alpha & \\delta _{23} \\\\\\delta _{31} & \\delta _{32} & 1\\end{array}\\right)\\left(\\begin{array}{c}h \\\\H \\\\S\\end{array}\\right),$ with small mixing angles $\\delta _{ij}$ , and $\\alpha $ satisfying $\\tan 2\\alpha =\\frac{2M_{12}^2}{M_{11}^2-M_{22}^2}.$ The masses of the physical states $h$ and $H$ are $m_{h,H}^2&=\\frac{1}{2}\\bigg [M_{11}^2+M_{22}^2\\\\&\\quad \\quad \\mp \\sqrt{\\left(M_{11}^2-M_{22}^2\\right)^2+4\\left(M_{12}^2\\right)^2}\\bigg ],\\nonumber $ while the mass of $S$ is $m_S^2&\\simeq m_0^2+\\delta _{13}M^2_{13}+\\delta _{23}M^2_{23}.$ We will see that $S$ can be rendered light while coupling dominantly to leptons (when $\\tan \\beta $ is large) below, putting off questions of fine-tuning for the time being.", "In the L2HDM, the Yukawa interactions of $\\Phi _1$ and $\\Phi _2$ with fermions are given by $-{\\cal L}_Y=\\overline{L} Y_e\\Phi _1 e_R+\\overline{Q} Y_d\\Phi _2 d_R+\\overline{Q} Y_u\\tilde{\\Phi }_2 u_R+{\\rm h.c.},$ suppressing generational indices and using first generation notation.", "The Yukawa content of this model is exactly the same as in the SM, ensuring a pattern of minimal flavor violation (MFV).", "In particular, there are no flavor-changing neutral currents (FCNCs) mediated by either of the Higgs fields at tree level.", "The only difference with the SM is through the appearance of the vacuum angle $\\beta $ in the mass-Yukawa coupling relation, $m_e = \\cos \\beta \\times \\frac{Y_e v}{\\sqrt{2}},~~ m_{u(d)} = \\sin \\beta \\times \\frac{Y_{u(d)} v}{\\sqrt{2}}.$ In the large $\\tan \\beta $ regime, the size of the Yukawa couplings in the quark sector is almost the same as in the SM, but in the lepton sector all Yukawa couplings are enhanced by $\\tan \\beta $ .", "Upon diagonalization of the Higgs mass matrix, the Yukawa interactions of the physical states are $-{\\cal L}_Y&\\supset \\sum _{\\begin{array}{c}\\phi =S,\\,h,\\,H\\\\ \\psi =\\ell ,\\,q\\end{array}}\\xi ^\\phi _{\\psi }\\frac{m_\\psi }{v}\\phi \\overline{\\psi }\\psi $ where $\\ell $ labels each generation of lepton fields and $q$ those of the quarks.", "The couplings to the weak gauge bosons can be found by expanding the kinetic terms of the doublets in the Lagrangian or by expanding $v^2$ about the vacuum: ${\\cal L}&\\supset \\sum _{\\phi =S,\\,h,\\,H}\\xi ^\\phi _{V}\\frac{\\phi }{v}\\left(2m_W^2W_\\mu ^+W^{\\mu -}+m_Z^2Z_\\mu Z^\\mu \\right).$ Defined this way, $\\xi ^\\phi _{\\psi ,V}=1$ is a coupling of SM Higgs strength.", "In Table REF , we show these couplings in terms of the angles $\\alpha $ and $\\beta $ , and in Table REF provide approximate values in the regime of interest.", "Table: Values of ξ ψ φ \\xi ^\\phi _{\\psi } for φ=S\\phi =S, hh, HH, ψ=ℓ\\psi =\\ell , qq, WW, ZZ in the L2HDM+ϕ\\varphi .Table: Approximate values of ξ ψ φ \\xi ^\\phi _{\\psi } when tanβ≫1\\tan \\beta \\gg 1 for φ=S\\phi =S, hh, HH, ψ=ℓ\\psi =\\ell , qq, WW, ZZ, with α\\alpha chosen so that ξ ℓ h ≃1\\xi ^h_{\\ell }\\simeq 1, r≡m h 2 /m H 2 r\\equiv m_h^2/m_{H}^2 and x≡1+ξ ℓ h 1-rx\\equiv 1+\\xi ^h_{\\ell }\\left\|1-r\\right\| in the L2HDM+ϕ\\varphi for m h <m H m_h<m_H (m h >m H m_h>m_H).We assume that $h$ has SM-like couplings to the gauge bosons and quarks, which means that $\\cos \\left(\\beta -\\alpha \\right)\\simeq 0$ and $\\cos \\alpha \\simeq \\sin \\beta $ .", "Furthermore, if $\\tan \\beta \\gg 1$ , then $H$ and $S$ will couple much more strongly to leptons than to quarks.", "This can be accomplished by choosing $\\alpha \\simeq 0$ (and negative) and $\\beta \\simeq \\pi /2$ .", "In this case, we can make $h$ arbitrarily SM-like, consistent with the observations of the ATLAS and CMS experiments, while allowing $m_H$ and $\\tan \\beta $ to vary (again ignoring questions of fine tuning for now).", "Given this pattern of masses and couplings, we can find the singlet mixing angles, $\\delta _{13}&\\simeq -\\frac{vA_{12}}{m_{H}^2},~\\delta _{23}\\simeq -\\frac{vA_{12}}{m_h^2}\\left[1+\\xi ^h_{\\ell }\\left(1-\\frac{m_h^2}{m_H^2}\\right)\\right]\\cot \\beta ,$ or $\\xi ^S_\\ell &\\simeq -\\frac{vA_{12}}{m_{H}^2}\\tan \\beta ,\\\\\\xi ^S_q&\\simeq -\\frac{vA_{12}}{m_h^2}\\left[1+\\xi ^h_{\\ell }\\left(1-\\frac{m_h^2}{m_H^2}\\right)\\right]\\cot \\beta .$ Recall that the Yukawa couplings of $S$ are $g_{\\ell ,q}=\\xi ^S_{\\ell ,q} m_{\\ell ,q}/v$ .", "We can re-express the mass shift of the lightest scalar from Eq.", "(REF ) due to electroweak symmetry breaking in terms of more physical parameters, $m_S^2&\\simeq m_0^2-\\left(\\frac{m_H \\xi ^S_\\ell }{\\tan \\beta }\\right)^2.$ The cancellation between $\\delta m_S^2$ and $m_0^2$ to obtain a GeV-scale value of $m_S$ represents a (mild) fine-tuning in this theory.", "We have checked that the hierarchy of the mass scales, $m_S \\ll m_{h,H}$ is indeed possible without inducing an instability of the corresponding minimum in the scalar potential." ], [ "Universal constraints on the (leptonic) light scalar", "We subdivide all the possible constraints on the light scalar $S$ into two groups.", "The first, model independent, group relies exclusively on the coupling to leptons in Eq.", "(REF ), and comes mostly from low and medium energy processes, and does not use any of the additional particles brought in by the UV completion.", "We present the second, model dependent, group of constraints in the next Section.", "Although we introduced the notation $g_\\ell =\\xi ^S_\\ell m_\\ell /v$ in describing a particular UV completion in Sec.", ", we will make use of this parameterization and present results in this Section in terms of $\\xi ^S_\\ell $ , i.e.", "normalizing $g_\\ell $ on the SM Higgs Yukawa coupling." ], [ "Lifetimes and decay modes of $S$", "We will concentrate on the masses in the range $1~\\rm MeV$ to a few GeV for $m_S$ .", "(A region from $\\sim $ 200 keV to $2m_e\\simeq 1$ MeV may represent an interesting blind spot [30], [31], but is not treated in this paper.)", "In this mass range, the dominant decay modes of $S$ are to leptons, with partial width given by $\\Gamma _{S\\rightarrow \\ell \\overline{\\ell }} = g_\\ell ^2 \\times \\frac{m_S}{8 \\pi } \\left(1-\\frac{4 m_\\ell ^2}{m_S^2}\\right)^{3/2}.$ Depending on the coupling strength and the boost of the $S$ particle produced, the decay length of $S$ can be macroscopic, or rather prompt.", "For example, for $m_S = 1$ GeV, the proper decay length is $c\\tau (m_S= 1\\,{\\rm GeV}) \\simeq 3 \\times 10^{-6} {\\rm cm} \\times \\left( \\frac{1}{\\xi ^S_\\ell } \\right)^2,$ and the decay is prompt.", "The $\\gamma \\gamma $ decay fraction may become noticeable (up to $\\sim 20\\%$ just below $m_S = 2m_\\mu $ ) due to the loop-induced coupling to photons.", "In our model, the scaling $g_\\ell \\propto m_\\ell $ allows for unambiguous determinations of the corresponding branching ratios.", "We plot the branching ratios of $S$ as a function of its mass in Fig.", "REF noting that the decay is always dominated by the heaviest kinematically allowed lepton pair.", "Figure: Branching ratios for S→γγS\\rightarrow \\gamma \\gamma , e + e - e^+e^-, μ + μ - \\mu ^+\\mu ^-, τ + τ - \\tau ^+\\tau ^- as a function of m S m_S." ], [ "Muon anomalous magnetic moment", "A loop of light scalars contributes to the anomalous magnetic moments of fermions.", "A straightforward calculation gives $a_\\ell = \\frac{g_\\ell ^2}{8\\pi ^2}\\int _0^1\\frac{(1-z)^2(1+z)}{(1-z)^2 + z(m_S/m_\\ell )^2},$ which, in the limits of a very light and a very heavy scalar, reduces to $3g_\\ell ^2/(16\\pi ^2)$ and $g_\\ell ^2/(4\\pi ^2)\\times (m_\\ell ^2/m_S^2) \\log (m_S/m_\\ell )$ respectively.", "Equation (REF ) and the $g_\\ell \\propto m_\\ell $ dependence lead to $a_\\ell $ scaling as the second (fourth) power of lepton mass in the limit of a light (heavy) scalar.", "The tau lepton $g-2$ receives the largest contribution from scalar exchange, but is not measured to the required precision (and, in fact, the $a_\\tau $ sign is not experimentally determined).", "The strongest constraints come from $g-2$ of the muon, and if the the current discrepancy, which we take to be $\\left(26.1\\pm 8.0\\right)\\times 10^{-10}$ [32], is interpreted as new physics, it suggests a non-zero range for $\\xi ^S_\\mu $ shown in Fig.", "REF .", "Notice that, in contrast to the dark photon case, the highly precise measurements of electron $g-2$ do not provide competitive sensitivity.", "For the rest of the paper, we will treat the suggested muon $g-2$ band as a target of opportunity, and investigate other observables that could provide complementary sensitivity to $g_\\mu $ in this range.", "To facilitate comparison with the dark photon case, we show results in Fig.", "REF (left panel) in terms of both $\\xi ^S_\\ell =g_e(v/m_e)$ and $\\epsilon _{\\rm eff}\\equiv g_e/e$ , where $-e$ is the charge of the electron, which is the coupling strength to the electron of a dark photon with kinetic mixing angle $\\epsilon _{\\rm eff}$ .", "Expressed in terms of $\\epsilon _{\\rm eff}$ , regions determined by the coupling to the electron are in roughly the same place as those in the dark photon case (modulo small differences due to scalar vs. vector properties), while those determined by couplings to $\\mu $ and $\\tau $ move to smaller values of $\\epsilon _{\\rm eff}$ by factors of $\\sim m_{\\mu ,\\tau }/m_e$ .", "Note that in our UV-completion via the leptonic Higgs portal, there are additional contributions to $a_\\mu $ from the heavy neutral and charged Higgs states.", "These contributions are subdominant to that of $S$ , unless some of the neutral scalars are light, below the mass of the weak bosons.", "In this work, we will assume the heavy Higgs bosons are much heavier than this, so that the dominant contribution to $a_\\mu $ comes from $S$ , but see e.g.", "Refs.", "[33], [28], [29] for a recent study exploring this region of parameter space in the lepton-specific 2HDM." ], [ " Beam-dump and fixed target constraints", "The coupling of the scalar $S$ to electrons is considerably smaller than to muons, $g_e/g_\\mu = (m_e/m_\\mu ) \\simeq 0.005$ .", "Consequently, low mass scalars with $m_S < 2m_\\mu $ can have displaced decays, or even travel a macroscopic distance before decaying.", "Fig.", "REF shows constraints from older beam dump experiments, such as E137 and E141.", "In both cases, the scalars $S$ are produced in an underlying bremsstrahlung-like process, $e + {\\rm Nucleus} \\rightarrow e + S + {\\rm Nucleus}$ .", "Notice that these experiments firmly rule out scalars with masses below 30 MeV as candidates for the solution of muon $g-2$ discrepancy.", "Consequently, for the rest of the constraints, we will concentrate on $m_S > 10$ MeV.", "It is also important to note the modification of the shape of the excluded region compared to the case of dark photons, universally coupled to all leptons.", "In the scalar model above $m_S=210$ MeV, there is no sensitivity in the beam dump experiments due to abrupt shortening of the lifetime of $S$ by the muon pair decay channel.", "The JLab experiment HPS [34] utilizes a fixed target, scattering electrons on tungsten, producing scalars through their couplings to electrons.", "It has the capability to detect displaced decays within a few cm from the target, and will be sensitive to the scalar $S$ in the relevant mass range.", "Translating the projected sensitivity to the dark photon parameter space to the case of the leptonic scalar, we arrive at the sensitivity reach of HPS shown in Fig.", "REF .", "Above the muon threshold, the scalar decays are too prompt to be detected in this fashion.", "At the same time, muon fixed target experiments have a chance of probing this parameter space for the model.", "This possibility was discussed in Ref.", "[35], in connection with a possible search for an axion-like particle in $\\mu + {\\rm Nucleus} -> \\mu + {\\rm Nucleus} +a(\\rightarrow \\mu ^+\\mu ^-)$ at the COMPASS facility at CERN [36].", "Recasting the projected sensitivity in the case of the scalar particles, we obtain an $O(1)$ sensitivity to $\\xi _l^S$ , shown in Fig.", "REF .", "It is also possible that proton beam dump and fixed target experiments could be sensitive to $S$ .", "Indeed, primary mesons produced subsequently lead to muons, which in turn can radiate the scalar using a larger coupling, $g_\\mu $ .", "The challenge in such a set-up would be to identify a clean way of detecting electron-positron pairs (or for the case of the fixed target experiments, possibly muon pairs) that result from scalar decays.", "A planned high-energy proton beam dump experiment, SHiP [24], as well as the existing Fermilab experiment SeaQuest [37], may present advantageous venues, as the high-energy and relatively short distance to the detector will increase chances for detecting displaced decays.", "As a separate note, it is worth mentioning that recent studies of the LHCb sensitivity to dark photons [38] may open a new pathway to probe dark scalars as well.", "The search suggested in [38] will not directly apply to a leptophilic scalar $S$ .", "Nonetheless, LHCb provides an attractive opportunity to search for $S$ via its production in association with muons.", "The large boosts available at LHCb may facilitate such searches via displaced decays of $S$ ." ], [ " Future sensitivity from muon decay", "Flavor-violating muon decays will be scrutinized in a series of upcoming experiments.", "Of particular interest for the model discussed in this paper is the $\\mu ^+ \\rightarrow e^+ e^+e^-$ search, planned at the Paul Scherrer Institute [39], which will have exquisite energy resolution for the final state leptons.", "In the present model, the flavor-violating decays of muons are absent, but the exotic scalars $S$ can be radiated on-shell in the process $\\mu ^+ \\rightarrow \\nu \\overline{\\nu }e^+ S \\rightarrow \\nu \\overline{\\nu }e^+ e^+ e^-$ .", "The momenta for the electron and one of the two positrons in the final state must reconstruct the mass of the scalar, $(p_{e^+} + p_{e^-})^2 = m_S^2$ .", "Therefore, a scalar signal would be a bump in the invariant mass of the electron-positron pairs, superimposed on the SM background $\\mu ^+ \\rightarrow \\nu \\overline{\\nu }e^+ e^+ e^-$ .", "Making use of the recent study of a future dark photon search in this set-up [40], we recast the projected sensitivity for the case of the leptonic scalar $S$ .", "The signals for $S$ and $V$ were simulated using MadGraph.", "For the scalar, emission from the initial muon line dominates, since $g_\\mu \\gg g_e$ .", "The resulting sensitivity reach is shown in Fig.", "REF .", "Note that the projections of Ref.", "[40] assume a prompt decay of the intermediate $e^+ e^-$ resonance.", "However, for a small portion of the low mass, small $\\xi ^{S}_\\ell $ parameter space where the experiment has sensitivity, the decay length of the $S$ particle can be longer than ${\\cal O}$ (cm), which is approximately the radius of the innermost silicon detector.", "Thus, a more careful study must be carried out to assess the sensitivity in this region.", "The displaced decays may in fact help to reduce the level of background if, of course, the vertex can be cleanly reconstructed.", "See also Ref.", "[40] for further discussion of a potential search involving displaced decays." ], [ " Kaon decays", "Another well-studied source of muons is via kaon decays.", "A new particle coupled to muons can be emitted in the decay $K ^+ \\rightarrow \\mu ^+ \\nu S$ .", "Note that charge conjugated processes are understood to be implicitly included throughout this section.", "(For recent discussions of scalar and vector emission in similar processes, see Refs.", "[41], [42].)", "For this study, we will concentrate on the past experiment NA48/2 [43] and the on-going experiment NA62 [44].", "Depending on the mass of the scalar, it will decay to either $\\mu ^+\\mu ^-$ or $e^+e^-$ .", "The first case is relatively straightforward.", "The SM rate for a similar process, $K ^+ \\rightarrow \\mu ^+ \\nu \\mu ^+\\mu ^-$ , was beyond the reach of previous experiments, and only upper limits on the corresponding branching fraction exist.", "On the other hand, for the electron-positron decays of $S$ there are significant sources of known background.", "The first source is due to a rare SM decay $K ^+ \\rightarrow \\mu ^+ \\nu e^+e^-$ .", "This process has been measured for the invariant mass of a pair in excess of 150 MeV [45] with a branching ratio of $7\\times 10^{-8}$ .", "Below 150 MeV, there is a significant background due to the SM process $K ^+ \\rightarrow \\mu ^+ \\nu \\pi ^0$ , with subsequent Dalitz decay of the neutral pion $\\pi ^0 \\rightarrow e^+e^-\\gamma $ that would mimic the signal if the photon is not detected.", "Finally, there is also some background from pion/muon mis-identification in the underlying $K^+ \\rightarrow \\pi ^+ \\pi ^0$ decay and the Dalitz decay of $\\pi ^0$ .", "Even though NA48/2 data has been collected, the corresponding analysis has not yet been done, and therefore both experiments need to be viewed in terms of potential future sensitivity levels.", "We derive them using the calculated signal rate in our model, and the published detector resolution for electron-positron pairs.", "To estimate the backgrounds, we use known kaon branching ratios and assume that the probability of missing a photon is $\\sim 10^{-3}$ .", "We also extend $K ^+ \\rightarrow \\mu ^+ \\nu e^+e^-$ to the entire range of $m_{ee}$ using simulations.", "Above muon threshold we set the rate of the signal to 5 events to derive the corresponding sensitivity limits.", "The projected sensitivity is shown in Fig.", "REF ." ], [ " Associated production of scalars with $\\tau \\overline{\\tau }$ at lepton colliders", "High-luminosity $B$ -factories, such as BaBar and Belle, have collected an integrated luminosity of $\\sim 1 ~{\\rm ab}^{-1}$ , and among other things have produced a significant sample of $\\tau ^+\\tau ^-$ pairs.", "The upcoming experiment Belle II is aiming to expand this dataset by a factor of ${\\cal O}(100)$ .", "Given lepton couplings proportional to mass, the associated production of scalars $S$ from the taus, $e^+ e^- \\rightarrow \\tau ^+\\tau ^- + \\left(S \\rightarrow e^+e^- ~{\\rm or }~ \\mu ^+\\mu ^-\\right),$ may represent the best chance for discovering or limiting the parameter space for such particles.", "The search for exotic particles in association with taus is a relatively unexplored subject, with only one specific case analyzed to date [46], [47].", "Figure: Production rate for SS in association with taus at BB factories, as a function of m S m_S.", "The cross section is proportional to (ξ ℓ S ) 2 (\\xi ^S_\\ell )^2, and we have set ξ ℓ S =1\\xi ^S_\\ell =1.The production cross section for (REF ) can be calculated analytically.", "We present the corresponding result as a function of the scalar mass in Fig.", "REF .", "To set the scale of the expected event rate for a 1 GeV mass scalar, we take parameters within the muon $g-2$ band, and translate to the scale of the coupling to $\\tau $ -leptons, $g_\\tau ^2 \\sim 1.3\\times 10^{-3}$ .", "This leads to a very large number of produced scalars in the combined BaBar and Belle dataset, on the order of $5\\times 10^4$ .", "Simulating the QED backgrounds using MadGraph, and requiring that at least one of the taus decay leptonically, we arrive at the sensitivity curves shown in Fig.", "REF .", "These sensitivity projections rely on a “bump hunt\" in $\\mu ^+\\mu ^-$ (or $e^+e^-$ ) over the smoothly distributed QED background.", "Notice that for $m_S > 2m_\\tau $ the dominant decay mode of the scalar is the tau pair, and the sensitivity is reduced due to the lack of stable leptons reconstructing to the invariant mass $m_S$ .", "The decay to muons in this mass range is suppressed by $(m_\\mu /m_\\tau )^2$ .", "Also, for scalar masses below $2m_\\mu $ the decay length of scalars become comparable to the size of the detector, leading to reduced sensitivity.", "We account for this by introducing a requirement that the $S$ decays occur within $25~{\\rm cm}$ of the beam pipe.", "It is worth emphasizing that an analysis of process (REF ) represents perhaps the most effective way of probing the parameter space of the leptonic scalar model in a wide mass range, from a few MeV to $\\sim 3.5$ GeV." ], [ "Constraints on light scalars due to their electroweak properties", "In this Section we analyze constraints that depend on the embedding of the simple framework of Eq.", "(REF ) into the SM.", "We focus on those that are a consequence of our choice of the L2HDM+$\\varphi $ scenario outlined in Sec.", "; in other models, constraints could differ." ], [ "Higgs decays", "The SM-like Higgs $h$ can decay to pairs of light scalars through both $V_{\\rm 2HDM}$ and $V_{\\rm portal}$ after electroweak symmetry breaking via the operator $C_{hSS}hSS$ .", "In the SM-like limit, $C_{hSS}\\simeq \\left(\\frac{m_h^2}{2\\tan \\beta }+2m_{12}^2\\right)\\frac{\\left(\\xi ^S_\\ell \\right)^2}{v \\tan \\beta }.$ The decays $h\\rightarrow SS\\rightarrow 4\\tau $ [48], and $h\\rightarrow SS\\rightarrow 2\\mu 2\\tau $ [50] have been probed at the LHC, but not observed.", "These null results can be interpreted as an upper limit on $\\xi ^S_\\ell $ .", "As suggested in [51], the $2\\mu 2\\tau $ final state offers better reach.", "In Fig.", "REF (right panel), we show the limit from this search for $\\tan \\beta =200$ , $m_{12}=1~\\rm TeV$ .", "The constraints become important for the muon $g-2$ -motivated parameter space once $m_S$ is in the multi-GeV regime." ], [ "$B$ -meson decays", "Although its coupling to quarks and $W$ bosons is suppressed, the scalar mediates quark flavor-changing transitions at one-loop, leading to, for instance, rare $B$ decays like $B\\rightarrow K\\mu ^+\\mu ^-$ (or more generically, $B\\rightarrow X_s\\mu ^+\\mu ^-$ ) and $B_s\\rightarrow \\mu ^+\\mu ^-$ .", "At large $\\tan \\beta $ and $\\xi ^h_\\ell =1$ , the leading term in the effective Lagrangian mediating $b\\rightarrow s$ transitions relevant for these decays is ${\\cal L}_{b\\rightarrow s}\\simeq -\\frac{3V_{ts}^\\ast V_{tb}}{16\\pi ^2}\\frac{m_b m_t^2}{v^3}\\frac{m_H^2\\xi ^S_\\ell }{m_h^2\\tan ^2\\beta } S\\overline{s}_R b_L+{\\rm h.c.}$ This operator can mediate the decay $B_s\\rightarrow S^\\ast \\rightarrow \\mu ^+\\mu ^-$ through an off-shell $S$ and can lead to the decay $B\\rightarrow KS^{(\\ast )}\\rightarrow K\\mu ^+\\mu ^-,Ke^+e^-$ .", "If $2m_e<m_S<2m_\\tau $ , the decays $B\\rightarrow KS$ and $B\\rightarrow K^S$ can proceed with $S$ decaying to $\\mu ^+\\mu ^-$ subsequently; this is subject to strong constraints from the lack of a bump in the $\\mu ^+\\mu ^-$ invariant mass in $B\\rightarrow K^\\ast \\mu ^+\\mu ^-$ at LHCb [22].", "We show limits on $\\xi ^S_\\ell $ that result from these decay modes in Fig.", "REF , taking $\\tan \\beta =200$ , $m_{H}=m_{H^\\pm }=500~\\rm GeV$ .", "(The degeneracy of the heavy Higgs masses weakens electroweak precision constraints.)", "Notice that for the mass range $2 m_\\tau < m_S < m_B - m_K^{(*)}$ the sensitivity is degraded as $S$ would primarily decay to a tau pair.", "We note in passing that the constraint on $\\xi ^S_\\ell $ could be weakened by a factor $\\sim m_H^2/m_h^2$ if $\\xi ^h_\\ell \\sim -1$ [cf.", "Eq.", "(REF )] which is consistent with the data on Higgs properties.", "For $m_S< 2 m_\\mu $ the important search channels are $B\\rightarrow X_se^+e^-$ .", "These modes are better suited for searches at Belle II, and sensitivities below branchings of $10^{-8}$ will also cover the remaining `triangular' parameter space in Fig.", "REF (right panel)." ], [ "Electroweak precision constraints", "Enhanced couplings of the lepton-specific Higgses will also induce one-loop corrections to leptonic branching ratios of the $Z$ -boson.", "Here we analyze $R_\\tau $ , defined as $R_\\tau \\equiv {\\Gamma (Z\\rightarrow {\\rm hadrons})}/{\\Gamma (Z\\rightarrow \\tau \\overline{\\tau })}$ , where $\\Gamma (Z\\rightarrow {\\rm hadrons}) \\propto \\sum _{q=u,d,s,c,b} (\|g_{q_L}\|^2+\|g_{q_R}\|^2)$ and $\\Gamma (Z\\rightarrow \\tau \\overline{\\tau }) \\propto (\|g_{\\tau _L}\|^2 + \|g_{\\tau _R}\|^2)$ , with $g_L = I_3 -Q s^2_W$ and $g_R=-Q s^2_W$ .", "$s_W$ stands for the sine of the weak mixing angle.", "Perturbations to $R_\\tau $ can be expressed in terms of corrections to $s^2_W $ and modifications of the $Z\\tau \\tau $ vertices by the scalar loops, $\\frac{\\Delta R_\\tau }{R_\\tau } &=& 4.3 \\delta g_{\\tau _L} - 3.7 \\delta g_{\\tau _R} - 0.8 \\delta s^2_W \\\\&\\simeq & 4\\delta g_A^\\tau + 1.9\\times 10^{-3} T $ with $g_A = g_L - g_R$ .", "Interpreting the PDG fit, $ R_\\tau = 20.764 \\pm 0.045 $ as the constraint, $-2\\times 10^{-3} \\le \\Delta R_\\tau /R_\\tau \\le 2 \\times 10^{-3}$ , we compare it to the result of the one-loop calculation in our model.", "The corrections to $\\delta g_{\\tau _L}$ and $\\delta g_{\\tau _R}$ can be obtained in the L2HDM model following [52], [53], and we present the ensuing constraint in the right panel of Fig.", "REF .", "The contributions due to loops of scalars that are (mostly) components of electroweak doublets are negligible for $m_{H,H^\\pm ,A}\\gtrsim 300~\\rm GeV$ , even for $\\tan \\beta $ as large as 200, as taken in Fig.", "REF .", "Additionally, we mention that as long as there is some degeneracy in the masses of at least two heavy scalars (at the order of $\\sim 50~\\rm GeV$ ), corrections to the oblique electroweak parameters $S$ , $T$ , and $U$ are not constraining." ], [ "Discussion and conclusions", "We have analyzed a simplified model of a light `dark scalar' that couples predominantly to leptons.", "This hidden sector model has a very distinct phenomenology, differing in several ways from the phenomenology of the canonical dark photon model.", "It is interesting that the coupling of a light scalar $S$ to leptons can still be of order $m_\\mu /v$ , and thus capable of inducing a large shift in the anomalous magnetic moment of the muon, without being excluded by direct searches.", "This is because the coupling to electrons relative to muons is suppressed by $m_e/m_\\mu $ , and many constraints that have ruled out the minimal version of the dark photon model as an explanation of the muon $g-2$ discrepancy do not have any constraining power.", "The simplified model (REF ) does not, however, respect the SU(2)$\\times $ U(1) gauge symmetry of the SM and needs a UV completion.", "This implies that either the field $S$ or the fermion fields in (REF ) cannot have well defined charge assignments.", "One possible UV completion, investigated in this paper, defines $S$ predominantly as a singlet scalar with a small admixture of an SU(2) doublet.", "On the other hand, one can consider the possibility of lepton fields in (REF ) arising from a mixing between the `normal' SM fields and heavy vector-like leptons [15], so that mixing with a pure singlet $S$ becomes possible.", "The UV completion of the model proposed here is based on the lepton-specific two Higgs doublet model, augmented by an additional light singlet.", "In the large $\\tan \\beta $ regime, the Yukawa couplings of the lepton-specific Higgs bosons $h_l \\supset (H,A, H^\\pm )$ to leptons are enhanced relative to their SM values.", "If an additional singlet field $\\varphi $ mixes with $h_l$ , the end result can be a new light boson $S$ with couplings to leptons that scale as $m_l$ and are of order the SM Yukawa couplings, proportional to the product of a small mixing angle $\\theta $ and large $\\tan \\beta $ .", "At the same time, the couplings of $S$ to quarks and weak gauge bosons are suppressed, which softens all constraints from the FCNC processes derived from $K$ , $B$ physics.", "Moreover, there are no charged lepton flavor violating processes, since flavour conservation is built into the Yukawa structure of the model.", "(For the alternate UV completion with vector-like fermions [15], flavor symmetry in the charged lepton sector is likely to be broken.", "At the same time, the pure singlet nature of $S$ in this type of UV completion may allow flavor changing processes to be kept separate for the quark and lepton sectors, thus avoiding strong constraints from hadronic FCNC.)", "We have analyzed a wide selection of constraints and sensitivity limits from the existing experiments, and from upcoming searches.", "The production of scalars is enhanced in processes that involve muons and tau leptons.", "We have studied muon and kaon decays, and shown that future experiments and analyses of the existing data (e.g.", "by NA48/2, BaBar and Belle experiments) are capable of reaching the levels of sensitivity to the parameter space suggested by the muon $g-2$ discrepancy.", "The mass range $m_S <2 m_\\mu $ naturally leads to longer lived bosons, and may be probed through experiments that have sensitivity to displaced decays, such as the HPS experiment at JLab.", "Perhaps the most sensitive current search for a leptonic dark scalar can be performed by the BaBar and Belle collaborations, using existing data.", "The process of interest involves tau pair production with an associated emission of the scalar.", "The large datasets generated by the two experiments will allow a sensitive analysis of $\\tau ^+\\tau ^-\\mu ^+\\mu ^-$ and $\\tau ^+\\tau ^-e^+e^-$ production, looking for a peak in the invariant mass of electrons and muons.", "Even without extra data that should be collected at Belle II, the two $B$ -factories should comprehensively test the dark scalar model in the wide mass range spanning almost three orders of magnitude.", "The constraints and projected sensitivity reach for many experiments are summarized in the two panels of Fig.", "REF .", "The results in the left panel are based only on the simplified model (REF ) and use only the $g_l \\propto m_l$ scaling and absence of invisible decay channels for $S$ .", "Much stronger constraints are derived for $m_S>2 m_\\mu $ using quark flavor physics, within the lepton-specific 2HDM UV completion.", "One should still keep in mind that the strong constraints shown in the right panel of Fig.", "REF are indeed very sensitive to the type of UV completion, and can in principle be avoided with a different microscopic model of (REF ).", "Note added: Following the completion of this work, the BaBar Collaboration released a preprint [54] with an analysis that constrains any light vector particle ($V$ ) in the $e^+e^-\\rightarrow \\mu ^+\\mu ^-V \\rightarrow \\mu ^+\\mu ^-\\mu ^+\\mu ^-$ channel.", "This limit can be appropriately recast for the scalar model, and we show the resulting constraint as the solid black line in Fig.", "1.", "This is now the strongest model-independent constraint over a large region of the $2 m_\\mu < m_S < 2 m_\\tau $ mass range.", "However, unlike the case of a vector coupled to $L_\\mu -L_\\tau $ , the limit from [54] does not rule out the $g-2$ band in that region.", "The reason is that the scalar contribution to $g-2$ is somewhat larger than that of the vector and its production cross section is smaller at the same mass and coupling to muons.", "The constraint can be improved even further if BaBar performs the corresponding $e^+e^-\\rightarrow \\tau ^+\\tau ^-S$ analysis.", "We would like to thank B. Echenard, E. Goudzovski, I. Nugent, M. Roney and B. Shuve for helpful discussions.", "The work of M. P. and A. R. is supported in part by NSERC, Canada, and research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT.", "The work of B. B.", "is supported in part by the U.S. Department of Energy under grant No.", "DE-SC0015634.", "The work of D. M. is supported by the U.S. Department of Energy under grant No.", "DE-FG02-96ER40956." ] ]	1606.04943
[ [ "A practical local tomography reconstruction algorithm based on known\n subregion" ], [ "Abstract We propose a new method to reconstruct data acquired in a local tomography setup.", "This method uses an initial reconstruction and refines it by correcting the low frequency artifacts known as the cupping effect.", "A basis of Gaussian functions is used to correct the initial reconstruction.", "The coefficients of this basis are iteratively optimized under the constraint of a known subregion.", "Using a coarse basis reduces the degrees of freedom of the problem while actually correcting the cupping effect.", "Simulations show that the known region constraint yields an unbiased reconstruction, in accordance to uniqueness theorems stated in local tomography." ], [ "Introduction", "In this section, we briefly recall the Region of Interest (ROI) tomography problem and review the related work.", "Region of Interest (ROI) tomography, also called local tomography, naturally arises when imaging objects that are too large for the detector field of view (FOV), as depicted on Figure REF .", "It notably occurs in medical imaging, where only a small part of a body is imaged.", "Local tomography can also originate from a radiation dose concern in medical imaging.", "Figure: Local tomography setup when the detector covers only a ROI of the object.", "Image: Since the projection data does not cover the entire object, it is said to be truncated with respect to a scan that would cover the entire object.", "The aim is then to reconstruct the ROI from this “truncated\" data.", "However, due to the nature of the tomography acquisition, the acquired data is not sufficient to reconstruct the ROI in general: for each angle, rays go through the entire object, not only the ROI.", "Thus, the data does not only contain information on the ROI, but also contribution from the parts of the object external to the ROI.", "For example, on Figure REF , the detector gets data from parts of the object located at the left of the ROI.", "These contributions from the external parts actually preclude from reconstructing exactly the ROI from the acquired data in general.", "The problem of reconstructing the interior of an object from truncated data is referred as the interior problem.", "It is well known that the interior problem does not have a unique solution in general.", "If $P$ denotes the projection operator, $d$ the acquired data and $x$ a solution of the problem $P(x) = d$ , then $x$ is defined up to a set of ambiguity functions $u$ such that $P(x + u) = d$ .", "An example is given in where $u$ is non-zero in the ROI, but $P(u) = 0$ in the detector zone corresponding to the ROI : two solutions differing by $u$ would produce the identical interior data.", "In , it is emphasized that the ambiguity is an infinitely differentiable function whose variation increases when going outside the ROI.", "The non-uniqueness of the solution of the interior problem prevents quantitative analysis of the reconstructed slices.", "Methods tackling the ROI tomography problem can mainly be classified in two categories.", "The first category methods aim at completing the data by extrapolating the sinogram.", "There are often oriented toward easy and practical use, although having no theoretical guarantees.", "The second category of methods rely on prior knowledge on the object.", "Many theoretical efforts were made on these methods, providing for example uniqueness and stability results.", "Other works use wavelets to localize the Radon transform or focus on the detection of discontinuities, the best known being probably Lambda-tomography ." ], [ "Sinogram extrapolation methods", "In a classical tomography acquisition, the whole object is imaged.", "If nothing is surrounding the object, the rays are not attenuated by the exterior of the object ; thus the sinogram values for each angle go to zero on the left and right parts (after taking the negative logarithm of the normalized intensity).", "In a local tomography acquisition, however, the data is “truncated\" with respect to what would have been a whole scan.", "The incompleteness of the data induces artifacts on the reconstructed image.", "The first obvious artifact is visible as a bright rim on the exterior of the image.", "This bright rim is the result of the abrupt transition in the truncated sinogram: the filtration process suffers from a Gibbs phenomenon.", "Another artifact is referred as the cupping effect: an unwanted background appears in the reconstructed image, which makes further analysis like segmentation challenging.", "These two artifacts occur simultaneously, but they have different causes.", "The bright rim comes from the truncation, while the cupping comes from the contribution of the external part.", "Figure REF and REF illustrates these artifacts.", "A synthetic slice is projected, and the resulting sinogram is truncated to simulate a ROI tomography setup.", "The filtering step enhances the transition between the ROI and the truncated part which is set to zero.", "The difference between the filtered whole sinogram and the filtered truncated sinogram also shows the cupping effect, which appears as a low-frequency bias.", "Figure: Illustration of the truncation artifacts on a line of the sinogram of the Shepp-Logan phantom.", "(a): Whole sinogram corresponding to a scan where all the object is imaged (green), and truncated sinogram (blue).", "(b): After the ramp-filtering.Figure: (a): Reconstruction of the truncated sinogram with filtered back-projection.The contrast has been modified to visualize the interior of the slice.", "(b): Line profile of the reconstructionSinogram extrapolation methods primarily aim at eliminating the bright rim resulting from the truncation by ensuring a smooth transition between the ROI and the external part.", "Besides, efforts have been put into the estimation of the missing data in order to reduce the cupping effect.", "These techniques are referred as sinogram extrapolation methods: the external part is estimated from the truncated data with some extrapolating function.", "Extrapolating function can be for example constant (the outermost left/right values are replicated), polynomial, $\\cos ^2$ .", "In , a mixture of exponential and quadratic functions are used to estimate the external part, possibly iteratively.", "Projection of a circle, for which a closed-form formula is known, can also be used .", "A common approach is using the values of the left/right part of the sinogram to estimate the external part, that is, replicating the borders values.", "In general, sinogram extrapolation methods do not take into account the sinogram theoretical properties.", "For example, given an object being nonzero only inside a circle of a given radius, the sinogram decreases to zero at the left and right boundaries.", "Generally speaking, a sinogram of complete measurements satisfies the Helgason-Ludwig consistency conditions (REF ) : $H_n (\\theta ) = \\int _{-\\infty }^\\infty s^n p(\\theta , s) \\operatorname{d}\\!s$ is a homogeneous polynomial of degree $n$ in $\\sin \\theta $ and $\\cos \\theta $ , for all $n \\ge 0$ .", "An alternative formulation is given by equation (REF ) : $H_{n, k} (\\theta ) = \\int _0^\\pi \\int _{-\\infty }^\\infty s^n e^{j k \\theta } p(\\theta , s) \\operatorname{d}\\!s \\operatorname{d}\\!\\theta \\; = \\, 0$ for $k > n \\ge 0$ and $k - n$ even.", "In , (REF ) is used as a quantitative measure of the sinogram consistency, and is optimized as an objective function.", "For many applications, constant extrapolation provides acceptable results , although cupping artifact makes the segmentation challenging." ], [ "Prior knowledge based interior tomography", "It was long believed that ROI tomography cannot be solved exactly, because of the nature of Radon inversion through FBP: the reconstruction of each voxel requires the knowledge of all the (complete) lines passing through this voxel.", "However, in the last decade, it has been shown that multiple nonequivalent reconstruction formulas allow partial reconstruction from partial data in the 2D case .", "Alternatively to Filtered Back Projection reconstruction, which requires complete data, Virtual Fan Beam (VFB) and Differentiated Back-Projection (DBP) were developed based on the Hilbert projection equality .", "Moreover, uniqueness theorems based on analytical continuation of the Hilbert Transform were stated and progressively refined in , , , , , , , .", "They ensure an exact and stable reconstruction of the ROI given some assumptions.", "These assumptions can be of geometric nature, or in the form of a prior knowledge.", "Figure: (a): Setup where the DBP can reconstruct the ROI.As the scanner field of view extends the ROI on both sides, the finite inverse Hilbert Transform can be computed.", "(b): Setup of interior tomography when the FOV does not extend the object.", "Only the knowledge of a sub-region can provide an exact reconstruction.Images: Geometry-based prior knowledge is related to the acquisition geometry.", "For example, in DBP based reconstruction, a point can be reconstructed if it lies on a line segment extending outside the object on both sides, and all lines crossing the segment are measured , as shown in Figure REF (a).", "Similar results were obtained under less restrictive assumptions, for example the field of view extending the ROI on only one side .", "These geometry-based methods do not work, however, when the FOV does not extend the object (Figure REF (b)).", "In this case, it has been shown that a prior knowledge on the function inside the ROI enables exact and stable reconstruction of the ROI.", "This knowledge can be in the form of the function values inside a sub-region of the ROI or can be about the properties of the function to reconstruct, for example sparsity in some domain.", "This latest kind of knowledge has led to compressive sensing based ROI tomography.", "In , , , Total Variation method is used to reconstruct the ROI.", "In , the function is assumed to be sparse in the wavelet domain, and a multi-resolution scheme reduces the number of unknown by keeping only fine-scale wavelet coefficients inside the ROI.", "In , it is shown that piecewise constant functions are determined everywhere by their ROI data, the underlying hypothesis being formulated as sparsity in the Haar domain." ], [ "Low frequencies artifacts correction with Gaussian blobs", "Sinogram extrapolation usually copes well with the correction of discontinuities in the truncated sinogram, but does not correct the cupping effect in general.", "This cupping effect appears as a low frequency bias in the reconstructed image.", "Sinogram extrapolation and other background correction techniques do not give guarantees that the low frequency bias will actually be removed without distorting the reconstruction.", "In this section, we describe a new method using prior knowledge on a subregion of the reconstructed volume to eliminate the low frequency cupping bias.", "The starting point of this method is an initial reconstruction, hereby denoted $x_0$ , which can be obtained for example with the padded FBP method.", "This initial reconstruction is then refined with an additive correction term.", "This correction term uses the known sub-region as a constraint which should be sufficient, according to uniqueness theorems stated in the references given in REF , to accurately reconstruct the region of interest.", "As $x_0$ bears the high frequencies features, the correction term is expressed as a linear combination of Gaussian functions to counterbalance the low frequency artifacts.", "The coefficients are optimized subject to the knowledge of the subregion, hereby denoted $\\Omega $ .", "To constrain all the Gaussian coefficients by the knowledge of the image values in $\\Omega $ , a reduced set of coefficients is firstly computed inside $\\Omega $ .", "Then, the Gaussian coefficients are iteratively optimized to fit the reconstruction error of the whole image, using the coefficients computed inside $\\Omega $ as a constraint.", "Let $u_0$ denote the “true\" object values in the known region $\\Omega $ .", "The proposed method can be summarized as follows: The reconstruction error in $\\Omega $ , denoted $e_{\|\\Omega } = \\left( (x_0)_{\|\\Omega } - u_0 \\right)$ , is expressed as a linear combination of two dimensional Gaussians.", "The resulting Gaussian coefficients are denoted $g_0$ .", "The error in the whole image is iteratively fitted with Gaussians coefficients $g$ , subject to $g_{\|\\Omega } = g_0$ , to build a consistent reconstruction error in the whole image.", "Details of each step are described in the following parts." ], [ "Capturing the low frequencies of the error in the known zone", "The key assumption of this method is that $\\Omega $ is large enough to bear sufficient information on the low frequencies artifacts (cupping effect) of a classical reconstruction.", "The reconstruction error in $\\Omega $ is approximated as a linear combination of Gaussian functions.", "This function is chosen for computational convenience, more details are given in REF .", "Equation (REF ) gives the expression of the approximation.The error estimation $\\widehat{e_{\|\\Omega }}$ is a linear combination of translated Gaussians of weights $c_{i,j}$ , with a spacing $s$ .", "$\\begin{aligned}\\widehat{e_{\|\\Omega }} &= \\sum _{i, j} c_{i,j} \\psi _\\sigma (x - i \\cdot s, y - j \\cdot s)\\\\\\psi _\\sigma (x, y) &= \\frac{1}{\\sigma \\sqrt{2\\pi }} \\exp \\left( -\\frac{x^2 + y^2}{2\\sigma ^2} \\right)\\end{aligned}$ For simplicity, all the Gaussians have the same standard deviation $\\sigma $ and all have the same spacing $s$ between them.", "Their location on the image grid is also fixed, so that the fit turns into a linear inverse problem : $g_0 = \\underset{g}{\\operatorname{argmin}} \\; \\left\\lbrace \\frac{1}{2}\\left\\Vert G g - \\left( (x_0)_{\|\\Omega } - u_0 \\right) \\right\\Vert _2^2 \\right\\rbrace $ where $G$ is the operator taking as an input the coefficients $c_i$ , stacked in a vector $g$ ; and producing an image tiled with Gaussians (here in region $\\Omega $ ).", "The norm $\\left\\Vert \\cdot \\right\\Vert _2^2$ is the squared Frobenius norm, that is, the sum of the squares of all components.", "Choosing the same standard deviation $\\sigma $ and the same spacing $s$ for all Gaussians enables to implement $G$ as a convolution.", "More precisely, given an image being zero everywhere, coefficients $c_i$ are placed every $s$ pixels in $\\Omega $ and this image is convolved with two dimensional $\\psi _\\sigma $ .", "The reconstruction error in $\\Omega $ is thus estimated in the least squares sense: $g_0$ is the vector of Gaussian coefficients giving the best estimation $\\widehat{e_{\|\\Omega }}$ of the reconstruction error in the L2 sense.", "This vector $g_0$ will be used in the second part of the algorithm.", "Outside the known region $\\Omega $ , the reconstruction error is not known.", "Like some other methods described in REF , this algorithm aims at using the known region information to accurately reconstruct the whole ROI.", "However, this approach focuses on correcting an initial reconstruction: the reconstruction error in $\\Omega $ is fitted by as a linear combination of Gaussians, then the whole image is corrected in a coarse Gaussian basis whose coefficients are constrained in the known subregion.", "The Filtered Backprojection (FBP) with sinogram extrapolation is widely used in local tomography because it is both simple and gives satisfactory results in general .", "Theoretical investigations found that FBP provides a reconstructed function bearing the same discontinuities as the reference function , although the cupping effect can make the segmentation challenging.", "In this method, FBP with padding is used to obtain an initial estimate of the reconstruction ; the aim is to correct the local tomography artifacts on this image using the prior knowledge.", "Equation (REF ) gives the expression of the estimate $x$ where $x_0$ is the initial reconstruction, $G$ is the operator described in REF and $\\hat{g}$ is a linear combination of Gaussian functions aiming at counterbalancing the low frequencies artifacts.", "$x = \\tilde{x_0} + G \\hat{g}$ The vector $\\hat{g}$ is found by minimizing an objective function which is built as follows.", "A new image $x = \\tilde{x_0} + G g$ , containing the initial reconstruction, is created.", "The image $\\tilde{x_0}$ is an extension of the initial reconstruction $x_0$ .", "This new image is projected with a projector $P$ adapted to the bigger size.", "To compare with the acquired sinogram $d$ , the computed sinogram $P (\\tilde{x_0} + G g)$ is truncated by a cropping operator $C$ .", "The data fidelity is then given by Equation (REF ).", "$\\frac{1}{2} \\left\\Vert C P ( \\tilde{x_0} + G g ) - d \\right\\Vert _2^2$ We emphasize that this approach differs from the full estimation of the ROI based on a subregion.", "The variables $g$ are in a coarse basis while $\\tilde{x_0}$ is fixed, which is notably reducing the degrees of freedom of the problem.", "The operation $G g$ has two goals: a coarse estimation of the exterior (outside the $x_0$ support) and a correction of the low frequencies error inside the $x_0$ support.", "As the minimization is on $g$ , the initial estimate of the ROI $x_0$ is constant, and the data fidelity term (REF ) can be rewritten as in Equation (REF ) $\\frac{1}{2}\\left\\Vert C P G g - d_e \\right\\Vert _2^2$ where $d_e = d - \\tilde{P} x_0$ is the difference between the acquired sinogram $d$ and the projection of the initial reconstruction $x_0$ , and $\\tilde{P}$ is the projector adapted to the size of $x_0$ .", "The optimization problem is given by Equation (REF ).", "$\\hat{g} = \\underset{g}{\\operatorname{argmin}} \\; \\left\\lbrace \\frac{1}{2}\\left\\Vert C P G g - d_e \\right\\Vert _2^2\\quad \\text{subject to } \\quad g_{\|\\Omega _g} = g_0 \\right\\rbrace $ $g_0$ is the vector of Gaussian coefficients found in (REF ), such that $G g_0$ approximates the error in the known zone.", "The set $\\Omega _g$ denotes the subset of the Gaussian basis corresponding to $\\Omega $ in the pixel basis: if a coefficient $c_i$ of $g$ lies in $\\Omega _g$ in the Gaussian basis, then $(G g)_i$ lies in $\\Omega $ in the pixel basis.", "Equation (REF ) boils down to finding coefficients $g$ minimizing the reconstruction error in the whole image, under the constraint that $G g$ should give the (known) reconstruction error in $\\Omega $ .", "This local constraint is propagated in all the variables by the projection operator involved in the optimization process.", "Uniqueness theorems mentioned in REF state that the knowledge of a subregion of the ROI is sufficient to yield an exact reconstruction.", "However, when using a pixel basis without space constraints, the number of degrees of freedom might be too high ; leading to a slow convergence.", "Using a coarse basis for correcting the low frequencies reduces this number of degrees of freedom." ], [ "Details on the involved operations", "In this part, more details are given on the different steps of the algorithm.", "We start by computing a padded FBP reconstruction $x_0$ which gives an initial estimate of the ROI of size $(N, N)$ .", "This image is extended to a bigger image $\\tilde{x_0}$ of size $(N_2, N_2)$ where $N_2 > N$ , and $x_0$ is placed in the center of the image.", "At each iteration $k$ , the image $G g_k$ , where $g_k$ is the Gaussian coefficients vector at iteration $k$ , is computed.", "The operator $G$ consists in placing the coefficients $g$ on a regular grid and convolving with the Gaussian kernel $(x, y) \\mapsto \\frac{1}{\\sigma \\sqrt{2\\pi }}\\exp \\left( -\\frac{x^2+y^2}{2\\sigma ^2}\\right)$ .", "Thus, the operator $G$ can be written $G = C_\\sigma U_s$ where $C_\\sigma $ is the convolution by the aforementioned Gaussian kernel of standard deviation $\\sigma $ , and $U_s$ is an operator upsampling an image by a factor of $s$ .", "As both are linear operators, $G$ is a linear operator and $G^T = U_s^T C_\\sigma ^T$ where $U_s^T$ is the $s$ -downsampling operator and $C_\\sigma ^T$ is a convolution by the matched Gaussian kernel, which is the same kernel due to symmetry.", "In our implementation, the Gaussian kernel has a size of $\\lfloor 8\\sigma + 1\\rfloor $ , i.e the Gaussian is truncated at $4 \\sigma $ .", "The resulting image is given by Equation (REF ) $\\begin{aligned}x(i_0, j_0) &= (G g)(i_0, j_0) = (C_\\sigma U_s g)(i_0, j_0) = (C_\\sigma z)(i_0, j_0) \\\\&= \\sum _{i, j} z(i, j) \\psi _\\sigma (i_0 - i, j_0 - j)\\end{aligned}$ where $\\psi _\\sigma $ is the discrete Gaussian kernel, $z$ is the image containing the Gaussian coefficients $g$ placed on the grid of size $(N_2, N_2)$ after upsampling, that is, zeros almost everywhere except coefficients every $s$ pixel.", "The summation in Equation (REF ) is done on the convolution kernel support.", "If $s < 4 \\sigma $ , the Gaussian functions supports can overlap once placed on the grid.", "In practice, these Gaussians should overlap to appropriately fit constant regions : for $s$ close to $\\sigma $ , the Gaussians almost yield a partition of unity .", "Once the coefficients are placed on a grid and convolved by the 2D Gaussian function, the image $x$ is projected.", "The projection operator adapted to the new geometry (the bigger image $G g_k$ ) is denoted $P$ .", "This is a standard Radon transform.", "This process is illustrated in Figure REF .", "As the new image $x$ is bigger than $x_0$ , the sinogram $P x$ and the acquired data $d$ cannot be directly compared.", "The computed sinogram $P x$ is thus cropped to the region corresponding to the ROI.", "The cropping operator is denoted by $C$ ; it is also a linear operator whose transpose consists in extending the sinogram by inserting zeros on both sides.", "The resulting sinogram aims at fitting the error between the acquired sinogram $d$ and the (cropped) projection of the object.", "As the object is unknown except inside $\\Omega $ , the reconstruction error is only known in $\\Omega $ .", "The Gaussian coefficients $g$ are constrained by those found by fitting the error inside $\\Omega $ in REF .", "The projection operator involved in the process propagates the constraint to all the other coefficients.", "Figure: Description of the operator PGPG (first line).A grid of points is created ; one coefficient is assigned to each point on the grid.The result is convolved by a two dimensional Gaussian function (depicted as green circles), and projected to obtain a sinogram.An equivalent approach (second line) consists in first projecting the “point coefficients\" with an appropriate projector,and convolving each line of the sinogram by a one dimensional Gaussian function.Coefficients $\\hat{g}$ from Equation (REF ) are computed with an iterative solver.", "As this objective function is quadratic, efficient minimization algorithms like conjugate gradient can be used.", "The final image is obtained with $\\hat{x} = x_0 + G \\hat{g}$ and is cropped to the region of interest." ], [ "Computational aspects", "Using Gaussians as functions to iteratively express the error has several computational advantages.", "Gradient-based algorithms for solving (REF ) involve the computation of the forward operator $P G$ and its adjoint $G^T P^T$ .", "They are usually the computationally expensive steps of iterative solvers.", "In this case, these operators can be computed in an efficient way.", "The Gaussian kernel has an interesting property: it is the only (nonzero) one to be both rotationally invariant and separable .", "In our case, the convolution by a Gaussian followed by a projection (forward Radon transform) is equivalent to projecting first and convolving each line of the sinogram by the corresponding one dimensional Gaussian, as illustrated on Figure REF .", "The first advantage is of theoretical nature.", "In many implementations, the projector and backprojector pair are usually not adjoint of each other for performances reasons.", "Although giving satisfying results in most practical applications, this raises theoretical issues on convergence of algorithms using iteratively forward and backward operators .", "By using a point-projector and a point-backprojector implementation, the pair can be exactly adjoint, besides giving more accurate results.", "The second advantage is on the computational side.", "As the operator $P G$ consists in projecting 2D Gaussians disposed on the image, it is equivalent to placing one-pixel coefficients (Dirac functions in the continuous case) in the image on a grid denoted $\\mathcal {I}$ , projecting the image (with a point-projector) and convolving the sinogram by a one-dimensional Gaussian kernel.", "The same goes for the adjoint operator $G^T P^T$ consisting in retrieving the Gaussians coefficients from a sinogram.", "The standard way to compute this operator would be backprojecting the sinogram, convolving by the two dimensional Gaussian (which is its own matched filter due to symmetry), and sampling the image on the grid $\\mathcal {I}$ to get the coefficients.", "Here, the convolution can be first performed in one dimension along the sinogram lines.", "The sinogram is sampled at locations corresponding to points $\\mathcal {I}$ in the image domain.", "The resulting sampled sinogram is then backprojected with a point-backprojector.", "In this section, the different steps of the proposed method are summarized in two algorithms.", "The first performs the fitting of the error in the known zone as described in section REF , the second builds the resulting image as described in section REF .", "A complete implementation of the proposed method is available at .", "It contains comments on the different steps and can be tuned for various setups.", "This implementation relies on the ASTRA toolbox , the point-projector scheme described in REF is not implemented for readability ; but this approach would be more suited to a production reconstruction algorithm where performances are an issue.", "[H]1.35 Algorithm 1.", "Known zone fitting.", "Known zone fitting $d$ : acquired sinogram $\\Omega $ : location and size of the known zone $u_0$ : known values in the zone $\\Omega $ [1] fitknown$d$ , $\\Omega $ , $u_0$ $x_0 = \\operatorname{padded}\\_\\operatorname{FBP}(d)$ Padded FBP of $d$ $e_{\|\\Omega } = (x_0)_{\|\\Omega } - u_0$ Error in the known zone $g_0 = \\underset{g}{\\operatorname{argmin}} \\; \\left\\lbrace \\left\\Vert G g - e_{\|\\Omega } \\right\\Vert _2^2 \\right\\rbrace $ Fit the error with Gaussians $\\tilde{x_0} = \\operatorname{extend}(x_0)$ Extend to a bigger image $\\tilde{x_0},\\, g_0$ The location of the known zone $\\Omega $ can be simply implemented as a tuple of pixels $(i_0, j_0)$ and a radius $r$ for a circular zone.", "[H]1.35 Algorithm 2.", "Error correction Error correction $d$ : acquired sinogram $\\Omega $ : location and size of the known zone $u_0$ : known values in the zone $\\Omega $ $N_2$ : size of the extended image $\\sigma $ : standard deviation of the Gaussian functions $s$ : grid spacing [1] localtomo$d$ , $\\Omega $ , $u_0$ , $N_2$ , $\\sigma $ , $s$ $\\tilde{x_0},\\, g_0 = \\operatorname{FITKNOWN}(d, \\Omega , u_0)$ Compute $\\tilde{x_0}$ and $g_0$ with Algorithm 1$d_e = C P \\tilde{x_0} - d$ Difference between the cropped projection of $x_0$ and $d$ $\\hat{g} = \\underset{g}{\\operatorname{argmin}} \\; \\left\\lbrace \\frac{1}{2}\\left\\Vert C P G g - d_e \\right\\Vert _2^2\\quad \\text{s.t.}", "\\quad g_{\|\\Omega _g} = g_0 \\right\\rbrace $ Operators are described in REF $\\hat{x} = \\tilde{x_0} + G \\hat{g}$ $\\hat{x}$ In practice, the final image $\\hat{x}$ is cropped to the region of interest.", "In algorithm 2, the optimization (line 5) can be done with a gradient algorithm, as differentiating the quadratic error term requires only the operators and their adjoints.", "In this section, results and discussions on three test cases are presented.", "Synthetic sinograms are generated by projecting an object and truncating the sinogram to the radius of a given region of interest in the image.", "The following notations are used: $\\sigma $ is the standard deviation of the Gaussians of the basis, $s$ is the grid spacing, $N_2$ is the size (width or height in pixels) of the extended image and $R$ is the radius (in pixels) of the known region.", "In practice, the size of the “original image\" (which corresponds to the size of an image that would contain the whole object in practice) is unknown, hence $N_2$ is always chosen different from the width of the original test image.", "In all cases, the input image is projected with a projector covering the entire object.", "The resulting sinogram is then truncated to the radius of the region of interest.", "The truncated sinogram is the input of the methods.", "The proposed method is compared to the padded FBP.", "As the padded FBP is used as an initial reconstruction by the proposed method, the benchmark is mainly about checking that the cupping effect is actually removed, and that the correction does not induce distortion to the final image.", "The first test involves the standard Shepp-Logan phantom (Figure REF ), $256 \\times 256$ pixels.", "The region of interest is embedded inside the “absorbing outer material\" (ellipse with the highest gray values) to simulate a local tomography acquisition.", "For an easier interpretation of the line profiles in the final reconstructed images, the values of the standard phantom are multiplied by 250 so that all the values are between 0 and 250.", "The width of the extended image is $N_2 = 260$ .", "Figure: (a): Shepp-Logan phantom of size 256×256256 \\times 256.The outer circle is the region of interest, the inner circle is the known subregion.The dashed lines indicate the profiles which are to be plotted in the reconstructed slice.", "(b): View of the region of interest.Figure REF shows the result of the reconstruction with padded FBP and with the proposed method.", "The Gaussian coefficients were computed with $\\sigma = 4$ on a grid of spacing $s = 6$ .", "The known region radius is $R = 20$ pixels, and the extended image width is $N_2 = 260$ pixels.", "By visual inspection, this method do not induce new artifacts in the reconstruction.", "Figure REF shows a line profile of this reconstruction.", "The cupping effect is visible for the padded FBP, and it has been removed with the proposed method.", "More importantly, the average reconstructed values are distributed around the true interior values.", "This provides an illustration of the uniqueness theorem: knowing the values of a subregion of the ROI enables to exactly reconstruct (up to numerical errors) the ROI.", "The reconstruction with the proposed method bears the same high frequencies as the FBP with full data, which is a good indication that this method do not induce new artifacts.", "The fact that the reconstruction has the same mean values than the true interior could enable quantitative analysis of the reconstructed volume, which is not easily achievable in local tomography.", "Figure: Results of reconstructions.", "(a) Proposed (b) padded FBPFigure: Line profiles of reconstructions.", "(a): Middle line.", "(b): Middle columnFigure REF shows the difference between the reconstructions and the interior values (denoted $x^\\sharp $ ).", "As expected, the cupping effect is visible for padded FBP, while being almost entirely suppressed in the reconstruction with proposed method.", "Figure: Difference between the reconstruction and the true volume x ♯ x^\\sharp (a): along the middle line,(b): along the middle columnThe second test involves the test image “Lena\", $512 \\times 512$ pixels, bearing both smooth regions and high frequencies textures.", "Figure REF shows the test setup.", "The known region has be chosen as slowly varying as possible, as in real acquisitions the known region is likely to be air or coarse features.", "The width of the extended image is $N_2 = 520$ .", "Figure: (a) Phantom “Lena\".", "An ellipse with high gray values has been added to accentuate the local tomography setup.The outer circle is the ROI, and the inner circle is the known region.The dashed lines indicate the profiles which are to be plotted in the reconstructed slice.", "(b) View of the region of interest.Figure REF shows the difference between the true interior and the reconstruction with the proposed method, with varying values of the radius $R$ of the known region.", "The parameters $\\sigma = s = 3$ has been used for these reconstructions.", "As it can be expected, the cupping effect removal is better when the known region is wide.", "Figure: Profiles of difference between the reconstruction and the true interior for the Lena image.x 0 x_0, x ^\\hat{x} and x ♯ x^\\sharp are the padded FBP, the proposed reconstruction and the true interior, respectively.In blue: difference between the padded FBP and the true interior.", "In green: difference between the reconstruction with the proposed method with σ=s=3\\sigma = s = 3 and the true interior.First row: profiles of the middle line of the image for (a) R=35R = 35, (b) R=15R = 15, (c) R=50R = 50.Second row: profiles of the middle column for (d) R=35R = 35, (e) R=15R = 15, (f) R=50R = 50Figure REF highlights the quality improvement for higher values of $R$ : the reconstruction profile get closer to the true interior or the full FBP as $R$ increases.", "Figure: Line profiles of reconstructions with parameters σ=s=3\\sigma = s = 3.", "(a) For radius = 15 pixels (b) For radius = 50 pixels.It is also interesting to visualize the reconstruction of the whole extended image.", "As it can be seen on Figure REF , the Gaussian basis even yields an approximation of the exterior.", "This approximation is actually important for modeling the contribution of the external part in the acquired sinogram.", "The bias correction is thus closely related to the modeling of the external part.", "Figure: Extended image after solving (), without cropping to the region of interest.The parameters used were σ=s=3\\sigma = s = 3 and R=35R = 35.The third test involves the image of a pencil resulting from a scan at the ESRF ID19 beamline, $512 \\times 512$ pixels, shown on Figure REF .", "The width of the extended image is $N_2 = 520$ .", "Figure: (a) Pencil test image.", "In red: region of interest.", "In green: known sub-region.", "(b) View of the region of interest.Figure REF shows profiles of the difference between the reconstruction and the true interior.", "On this image, a greater radius also improves the cupping removal.", "The profile of a line through the reconstructed image is depicted on Figure REF .", "Figure: Profiles of difference between the reconstruction and the true interior for the pencil image.x 0 x_0, x ^\\hat{x} and x ♯ x^\\sharp are the padded FBP, the proposed reconstruction and the true interior, respectively.In blue: difference between the padded FBP and the true interior.", "In green: difference between the reconstruction with the proposed method with σ=s=3\\sigma = s = 3 and the true interior.First row: profiles of the middle line of the image for (a) R=20R = 20, (b) R=10R = 10, (c) R=40R = 40.Second row: profiles of the middle column for (d) R=20R = 20, (e) R=10R = 10, (f) R=40R = 40Figure: Line profiles for the pencil image.", "The proposed method were applied with parameters σ=s=3\\sigma = s = 3 and R=40R = 40.As a last remark, Figure REF shows the result of this method without using the known zone constraint, that is, without applying the constraint $g_{\|\\Omega _g} = g_0$ in (REF ).", "As expected, there is a not-null mean bias, even if it has been reduced with respect to padded FBP.", "Figure: Profiles of difference between the reconstruction without known zone constraint and the true interior for the pencil image.", "(a) Line profile.", "(b) Column profile.Beside visual inspection, reconstructions can be quantitatively compared to the true interior of the test image.", "Table REF shows the comparison with two image metrics: peak signal to noise ratio (PSNR) and the structural similarity index (SSIM).", "As these metrics are indicators of an average distance between two images, we believe it is well suited for this purpose of evaluating how the low frequencies are corrected by the proposed method.", "Table: Metrics of reconstruction quality for the three test images, computed inside the reconstructed ROI.These results suggest that the proposed method yield better overall reconstruction quality than padded FBP.", "In particular, it does not induce critical distortion in the reconstruction.", "The following advantages of this method can be highlighted.", "Using a Gaussian basis allows for efficient computation: the correction can be implemented as a convolution with a simple kernel.", "This basis can also be extended to a multi-resolution grid, where big Gaussian functions are placed outside the ROI and small Gaussian functions are placed inside the ROI, to reduce the degrees of freedom even further.", "This approach would be similar to the method proposed in , although here only the correction is expressed in a multi resolution grid, not the image variables.", "What can also be noted is that no assumption is done on the shape or location of the ROI and the known region: the known region can for example be several regions of various shapes, corresponding to pores in a sample.", "Finally, by using the correction in the forward model (REF ), the Helgason-Ludwig conditions are naturally fulfilled.", "The proposed method depends on some parameters.", "The first is the size of the extended image, which should be big enough to model the contribution of the external part.", "The other parameters are the Gaussian standard deviation $\\sigma $ and the spacing $s$ of the grid.", "Both are related in a way that the Gaussian functions should slightly overlap to approximate constant functions: if $s$ value is high, then $\\sigma $ should also be high and conversely.", "These parameters essentially tune how coarse is the Gaussian basis: high values would yield fast convergence but coarse result, while small values would yield slow convergence and fine result.", "Using a Gaussian basis does not yield an exact correction of the error, as Gaussian functions defined in Equation (REF ) do not form a basis.", "For example, Gaussian functions do not yield a partition of unity, although a very close approximation of this property can be achieved .", "Thus, the final reconstruction cannot be exact due to the basis coarseness, but can provide results quite close to FBP with full data as seeing Figures REF , REF and REF .", "The fact that only a known zone of the ROI is enough to guarantee an almost-exact reconstruction might be counterintuitive, especially in our case where this constraint is expressed in a coarse basis.", "In the Gaussian basis, local constraints are propagated to the global image by the projection and backprojection operators involved in the process.", "Using a coarse basis greatly reduces the degrees of freedom of problem (REF ).", "The classical tomographic reconstruction problem $P x = d$ turned into a least squares optimization $\\underset{x}{\\operatorname{argmin}} \\; {\\left\\Vert P x - d \\right\\Vert _2^2}$ is ill-posed, even for complete data.", "In a local tomography setup, the ill-posedness is even worse .", "Iterative solvers dealing with problem (REF ) $\\underset{x}{\\operatorname{argmin}} \\; {\\left\\Vert C P x - d \\right\\Vert _2^2 \\quad \\text{s.t.}", "\\quad x_{\|\\Omega } = u_0}$ for $x$ in the pixel space, have very slow convergence in general due to the high number of degrees of freedom, even with spatial constraints.", "The importance of reducing the degrees of freedom of (REF ) is highlighted for example in and ." ], [ "Conclusion", "We presented a new technique of local tomography reconstruction based on the knowledge of a zone of the region of interest.", "This technique corrects the cupping effect in an initial reconstruction by expressing the error in a coarse basis of Gaussian functions.", "In accordance to local tomography uniqueness theorems, this method yield almost exact reconstructions, in spite of being only a correction with a coarse basis.", "Besides, practical considerations are given for an efficient implementation suitable for reconstruction of real data.", "A commented implementation of this method can be found at ." ] ]	1606.04940
[ [ "The Schur-Horn theorem for unbounded operators with discrete spectrum" ], [ "Abstract We characterize diagonals of unbounded self-adjoint operators on a Hilbert space H that have only discrete spectrum, i.e., with empty essential spectrum.", "Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space, analogous to Kadison's theorem for orthogonal projections, Kaftal and Weiss' results for positive compact operators, and Bownik and Jasper's characterization for operators with finite spectrum.", "Furthermore, we show that if a symmetric unbounded operator E on H has a nondecreasing unbounded diagonal, then any sequence that weakly majorizes this diagonal is also a diagonal of E." ], [ "Introduction", "The classical Schur-Horn theorem characterizes diagonals of hermitian matrices in terms of their eigenvalues.", "An infinite dimensional extensions of this result has been a subject of intensive study in recent years.", "This line of research was jumpstarted by the influential work of Kadison [17], [18], who discovered a characterization of diagonals of orthogonal projections on separable Hilbert space, and by Arveson and Kadison [6] who extended the Schur-Horn theorem to positive trace class operators.", "This has been preceded by earlier work of Gohberg and Markus [13] and by Neumann [29].", "The Schur-Horn theorem has been extended to compact positive operators by Kaftal and Weiss [21] and Loreaux and Weiss [27] in terms of majorization inequalities [20].", "Lebesgue type majorization was used by Bownik and Jasper [10], [11], [16] to characterize diagonals of self-adjoint operators with finite spectrum operators.", "Other notable progress includes the work of Arveson [5] on diagonals of normal operators with finite spectrum and Antezana, Massey, Ruiz, and Stojanoff's results [1].", "Finally, there is a rapidly growing body of literature on the corresponding problems for von Neumann algebras [2], [3], [4], [12], [22], [31], [32].", "The goal of this paper is to prove an infinite dimensional variant of the Schur-Horn theorem for unbounded self-adjoint operators with discrete spectrum.", "This represents a new direction in extending the Schur-Horn theorem to infinite dimensional setting since previous results dealt only with bounded operators.", "Assume that an unbounded self-adjoint operator $E$ on a separable Hilbert space $\\mathcal {H}$ is bounded from below and has discrete spectrum.", "That is, the essential spectrum $\\sigma _{ess}(E) = \\emptyset $ , and hence, every point $\\lambda \\in \\sigma (E)$ is an isolated eigenvalue of finite multiplicity.", "Since $E$ is bounded from below, its eigenvalues can be listed by a nondecreasing sequence $\\lambda = \\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ according to their multiplicities.", "Since $\\sigma _{ess}(E) = \\emptyset $ , we must necessarily have $\\lim _{i\\rightarrow \\infty } \\lambda _i =\\infty $ , and thus $E$ is unbounded from above.", "Consequently, $E$ is diagonalizable, i.e., there exists an orthonormal basis $\\lbrace v_i\\rbrace _{i\\in \\mathbb {N}}$ of eigenvectors $Ev_{i}=\\lambda _{i}v_{i}$ for all $i\\in \\mathbb {N}$ , and the domain of $E$ is given by $\\mathcal {D} = \\bigg \\lbrace f\\in \\mathcal {H}: \\sum _{i\\in \\mathbb {N}} \|\\lambda _i\|^2 \|\\langle f, v_i \\rangle \|^2 < \\infty \\bigg \\rbrace .$ In order to emphasize this point we will use the notation $E=\\operatorname{diag}\\lambda $ to denote the operator which has eigenvalues $\\lambda $ and domain (REF ) as above.", "If $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}} \\subset \\mathcal {D}$ is any other orthonormal basis of $\\mathcal {H}$ , then the diagonal $d_i= \\langle Ee_i, e_i\\rangle $ of $E$ with respect to $\\lbrace e_{i}\\rbrace $ satisfies $\\sum _{i=1}^{n}\\lambda _{i} \\le \\sum _{i=1}^{n}d_{i} \\qquad \\text{for all } n\\in \\mathbb {N}.$ In particular, the same inequality holds true when $\\lbrace d_i\\rbrace _{i\\in \\mathbb {N}}$ is replaced by its increasing rearrangement $\\lbrace d_i^\\uparrow \\rbrace _{i\\in \\mathbb {N}}$ .", "The necessity of condition (REF ) is often attributed to Schur [33].", "Our main result says that (REF ) is also sufficient, thus generalizing Horn's theorem [15].", "Theorem 1.1 Suppose that $\\lambda =\\lbrace \\lambda _i\\rbrace _{i\\in \\mathbb {N}}$ and $\\lbrace d_i\\rbrace _{i\\in \\mathbb {N}}$ are two nondecreasing and unbounded sequences.", "Let $E=\\operatorname{diag}\\lambda $ be a self-adjoint operator with eigenvalues $\\lambda $ and eigenvectors $\\lbrace v_i\\rbrace _{i\\in \\mathbb {N}}$ .", "If the majorization inequality (REF ) holds, then there exists an orthonormal basis $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ , which lies in the linear span of $\\lbrace v_i\\rbrace _{i\\in \\mathbb {N}}$ , such that $d_{i} = \\langle E e_{i},e_{i}\\rangle $ for all $i\\in \\mathbb {N}$ .", "The remarkable consequence of our main theorem is that majorization inequality (REF ) is the only condition that a sequence $\\lbrace d_i\\rbrace _{i\\in \\mathbb {N}}$ must satisfy in order to be diagonal of $\\operatorname{diag}\\lambda $ .", "Moreover, the required diagonal is achieved with respect to an o.n.", "basis $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}}$ , whose elements are finite linear combinations of eigenvectors $\\lbrace v_i\\rbrace _{i\\in \\mathbb {N}}$ .", "In particular, it is possible that $\\lambda _i=d_i$ for all but finitely many $i\\in \\mathbb {N}$ , the trace condition is violated, i.e., $\\sum _{i=1}^\\infty (d_i-\\lambda _i) \\ne 0$ , but yet the conclusion of Theorem REF still holds.", "Despite the simplicity of the statement of Theorem REF , its proof is far from trivial as it needs to deal with two major cases.", "The majorization inequality (REF ) can be equivalently stated as $\\delta _{k} = \\sum _{i=1}^{k}(d_{i}-\\lambda _{i})\\ge 0\\quad \\text{for all }k\\in \\mathbb {N}.$ After dealing with elementary reductions in Section , the first case deals with the conservation of mass scenario $\\liminf _{k\\rightarrow \\infty } \\delta _k =0.$ The second case deals with vanishing mass at infinity scenario $\\alpha =\\liminf _{k\\rightarrow \\infty } \\delta _k >0.$ This further splits in two subcases: $\\delta _k \\ge \\alpha $ for sufficiently large $k$ , and $\\delta _k<\\alpha $ for infinitely many $k$ , shown by Theorems REF and REF , respectively.", "The proofs of these cases require careful application of an infinite sequence of convex moves, also known as $T$ -transforms [21], to guarantee that the limiting o.n.", "sequence is a basis.", "In addition, we need to ensure that the constructed basis is contained in the dense domain $\\mathcal {D}$ .", "This constraint was not present in earlier work on bounded operators and requires new techniques of moving from a prescribed diagonal into a desired diagonal configuration.", "Our methods work not only for self-adjoint operators with discrete spectrum as in Theorem REF , but also for unbounded symmetric operators (possibly with continuous spectrum) as in Theorem REF .", "Indeed, “eigenvalue to diagonal” Theorem REF is an immediate consequence of a more general “diagonal to diagonal” Theorem REF .", "We end the paper by giving several examples illustrating Theorem REF in Section .", "Laplacians, or more generally elliptic differential operators, provide a broad and interesting class of operators falling into the scope of this paper." ], [ "Diagonal to diagonal elementary reductions", "In this section we show several reductions that are employed in the proof of Theorem REF .", "To achieve this we formulate a generalization of Theorem REF for unbounded symmetric operators which are not necessarily diagonalizable.", "Recall that a linear operator $E$ defined on a dense domain $\\mathcal {D} \\subset \\mathcal {H}$ is symmetric if $\\langle Ef,g \\rangle =\\langle f, Eg \\rangle \\qquad \\text{for all }f,g \\in \\mathcal {D}.$ Theorem REF is an immediate consequence of the following diagonal to diagonal theorem.", "Theorem 2.1 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D} \\subset \\mathcal {H}$ .", "Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be two nondecreasing unbounded sequences satisfying (REF ).", "If there exists an orthonormal sequence $\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}\\subset \\mathcal {D}$ such that $\\langle Ef_{i},f_{i}\\rangle = \\lambda _i\\qquad \\text{for all } i\\in \\mathbb {N},$ then there exists an orthonormal sequence $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}\\subset \\operatorname{span}\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}$ such that $\\overline{\\operatorname{span}}\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}=\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\langle Ee_{i},e_{i}\\rangle = d_{i}\\qquad \\text{for all }i\\in \\mathbb {N}.$ In the special case when $\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}}$ is an orthonormal basis of eigenvectors with eigenvalues $\\lbrace \\lambda _i\\rbrace _{i\\in \\mathbb {N}}$ of a self-adjoint operator $E=\\operatorname{diag}\\lambda $ , Theorem REF immediately yields Theorem REF .", "To facilitate statements of reduction results, we shall make some formal definitions.", "Definition 2.2 Let $\\lambda = \\lbrace \\lambda _{i}\\rbrace _{i\\in I}$ and $\\mathbf {d}=\\lbrace d_i\\rbrace _{i\\in I}$ be two real sequences, where $I$ is countable.", "Let $E$ be unbounded (here it means not necessarily bounded) linear operator defined on a dense domain $\\mathcal {D}$ of a Hilbert space $\\mathcal {H}$ .", "We say that an operator $E$ has diagonal $\\lambda $ if there exists an orthonormal sequence $\\lbrace f_{i}\\rbrace _{i\\in I}$ contained in $\\mathcal {D}$ such that $\\langle Ef_{i},f_{i}\\rangle = \\lambda _{i}\\qquad \\text{for all }i\\in I.$ We say that $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from diagonal $\\lambda $ , if there exists an orthonormal sequence $\\lbrace e_{i}\\rbrace _{i\\in I}$ in $\\mathcal {D}$ satisfying $\\langle Ee_{i},e_{i}\\rangle = d_{i}$ for all $i\\in I$ , $\\overline{\\operatorname{span}}\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}=\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}\\qquad \\text{and}\\qquad \\forall k\\in I \\quad e_k \\in \\operatorname{span}\\lbrace f_i\\rbrace _{i\\in I}.$ Suppose $\\lbrace \\lambda _i\\rbrace _{i=1}^N$ and $\\lbrace d_i\\rbrace _{i=1}^N$ are two real sequences.", "Let $\\lbrace \\lambda _i^\\downarrow \\rbrace _{i=1}^N$ and $\\lbrace d_i^\\downarrow \\rbrace _{i=1}^N$ be their decreasing rearrangements.", "Following [28] we define a majorization order $\\lbrace d_i\\rbrace \\preccurlyeq \\lbrace \\lambda _i\\rbrace $ if and only if $\\sum _{i=1}^{N}d^\\downarrow _i =\\sum _{i=1}^{N}\\lambda ^\\downarrow _{i} \\quad \\text{ and }\\quad \\sum _{i=1}^{n}d^\\downarrow _{i} \\le \\sum _{i=1}^{n}\\lambda ^\\downarrow _{i} \\quad \\text{for all } 1\\le n \\le N.$ The classical Schur-Horn theorem [15], [33] characterizes diagonals of self-adjoint (Hermitian) matrices with given eigenvalues.", "It can be stated as follows, where $\\mathcal {H}_N$ is an $N$ dimensional Hilbert space over $\\mathbb {R}$ or $\\mathbb {C}$ , i.e., $\\mathcal {H}_N=\\mathbb {R}^N$ or $\\mathbb {C}^N$ .", "Theorem 2.3 (Schur-Horn theorem) There exists a self-adjoint operator $E:\\mathcal {H}_N \\rightarrow \\mathcal {H}_N$ with eigenvalues $\\lbrace \\lambda _{i}\\rbrace _{i=1}^N$ and diagonal $\\lbrace d_{i}\\rbrace _{i=1}^N$ if and only if $\\lbrace d_i\\rbrace \\preccurlyeq \\lbrace \\lambda _i\\rbrace $ .", "As a consequence of Theorem REF we have the following block diagonal lemma.", "Lemma 2.4 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D} \\subset \\mathcal {H}$ .", "Suppose that $\\lbrace d_i\\rbrace _{i\\in I}$ and $\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in I}$ are two sequence of real numbers such that: there exists a collection of disjoint finite subsets $\\lbrace I_{j}\\rbrace _{j\\in J}$ of the index set $I$ , $\\lbrace d_i\\rbrace _{i\\in I_j}\\preccurlyeq \\lbrace \\widetilde{d}_i\\rbrace _{i\\in I_j}$ for each $j\\in J$ , $\\widetilde{d}_{i}=d_{i}$ for all $i\\in I\\setminus \\left(\\bigcup _{j\\in J}I_{j}\\right)$ .", "Suppose that $E$ has diagonal $\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in I}$ with respect to an orthonormal sequence $\\lbrace f_{i}\\rbrace _{i\\in I}$ .", "Then, $\\lbrace d_{i}\\rbrace _{i\\in I}$ is a finitely derived diagonal of $E$ .", "That is, there exists an orthonormal sequence $\\lbrace e_{i}\\rbrace _{i\\in I}$ satisfying (REF ) with respect to which $E$ has diagonal $\\lbrace d_{i}\\rbrace _{i\\in I}$ .", "Let $P_j$ be the orthogonal projection of $\\mathcal {H}$ onto finite dimensional block subspace $\\mathcal {H}_j = \\operatorname{span}\\lbrace f_i: i \\in I_j\\rbrace $ .", "Observe that a finite dimensional self-adjoint operator $E_j:=(P_j E)\|_{\\mathcal {H}_j}$ has diagonal $\\lbrace \\widetilde{d}_i\\rbrace _{i\\in I_j}$ with respect to $\\lbrace f_i\\rbrace _{i\\in I_j}$ .", "By (ii) and the Schur-Horn Theorem, there exists a unitary operator $U_j$ on $\\mathcal {H}_j$ such that $U_j E_j (U_j)^$ has diagonal $\\lbrace d_i\\rbrace _{i\\in I_j}$ with respect to $\\lbrace f_i\\rbrace _{i\\in I_j}$ .", "Define an o.n.", "basis $\\lbrace e_i\\rbrace _{i\\in I}$ by $e_{i}= {\\left\\lbrace \\begin{array}{ll} U_j f_j & i \\in I_j,\\\\f_i & i\\in I\\setminus \\left(\\bigcup _{j\\in J}I_{j}\\right).\\end{array}\\right.", "}$ For $i\\in I_j$ we have $\\langle E e_i, e_i \\rangle = \\langle E (U_j)^ f_i, (U_j)^* f_i \\rangle = \\langle P_j E (U_j)^* f_i, (U_j)^* f_i \\rangle = \\langle U_j E_j (U_j)^* f_i, f_i \\rangle =d_i.$ The same identity holds trivially for $i\\notin \\bigcup _{j\\in J}I_{j}$ , which shows that $E$ has diagonal $\\lbrace d_i\\rbrace $ with respect to $\\lbrace e_i\\rbrace $ .", "This completes the proof of the lemma.", "As an application of Lemma REF we can show the special case of Theorem REF .", "Lemma 2.5 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D} \\subset \\mathcal {H}$ .", "Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be two nondecreasing sequences such that $\\delta _{k} := \\sum _{i=1}^{k}(d_{i}-\\lambda _{i})\\ge 0\\quad \\text{for all }k\\in \\mathbb {N}.$ Suppose that there are infinitely many $k\\in \\mathbb {N}$ such that $\\delta _{k} = 0$ .", "If $\\lambda $ is diagonal of $E$ , then $\\mathbf {d}$ is a finitely derived diagonal of $E$ .", "Set $k_{1}=0$ and let $\\lbrace k_{j}\\rbrace _{j=2}^{\\infty }$ be a strictly increasing sequence in $\\mathbb {N}$ such that $\\delta _{k_{j}}=0$ for all $j\\ge 2$ .", "For each $j\\in \\mathbb {N}$ set $I_{j} = \\lbrace k_{j}+1,\\ldots ,k_{j+1}\\rbrace $ .", "For each $j\\in \\mathbb {N}$ and $k\\in I_{j}$ $\\sum _{i=k_{j}+1}^{k}(d_{i}-\\lambda _{i}) = \\delta _{k}-\\delta _{k_{j}} = \\delta _{k}\\ge 0.$ Since $\\delta _{k_{j+1}}=0$ we have $\\lbrace d_{i}\\rbrace _{i\\in I_{j}}\\preccurlyeq \\lbrace \\lambda _{i}\\rbrace _{i\\in I_{j}}$ .", "By our assumption, $E$ has diagonal $\\lambda $ with respect to some o.n.", "sequence $\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}}$ .", "By Lemma REF there is an o.n.", "sequence $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) with respect to which $E$ has diagonal $\\mathbf {d}$ .", "In the proof of Theorem REF it is convenient to make the reducing assumption (REF ) about nondecreasing sequences $\\lambda $ and $\\mathbf {d}$ .", "Theorem 2.6 If Theorem REF holds under an additional assumption $\\delta _k = \\sum _{i=1}^{k}(d_{i}-\\lambda _{i}) >0 \\qquad \\text{for all }k\\in \\mathbb {N},$ then it holds in a full generality.", "Suppose that $E$ has diagonal $\\lambda $ with respect to o.n.", "sequence $\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}}$ .", "The case when $\\delta _k=0$ for infinitely many $k\\in \\mathbb {N}$ is covered by Lemma REF .", "Hence, we can assume that there are finitely many $k\\in \\mathbb {N}$ such that $\\delta _{k} = 0$ .", "Let $N\\in \\mathbb {N}$ be the largest such integer.", "Define the spaces $\\mathcal {H}_{0}= \\overline{\\operatorname{span}} \\lbrace f_{i}\\rbrace _{i=1}^{N}\\qquad \\text{and}\\qquad \\mathcal {H}_{1}=\\overline{\\operatorname{span}} \\lbrace f_i\\rbrace _{i=N+1}^\\infty .$ Applying Theorem REF to the sequences $\\lbrace d_{i}\\rbrace _{i=N+1}^{\\infty }$ and $\\lbrace \\lambda _{i}\\rbrace _{i=N+1}^{\\infty }$ , and noting that for $k\\ge N+1$ $\\sum _{i=N+1}^{k}(d_{i}-\\lambda _{i}) = \\delta _{k} - \\delta _{N} = \\delta _{k}>0$ we obtain an orthonormal basis $\\lbrace e_{i}\\rbrace _{i=N+1}^{\\infty }$ of $\\mathcal {H}_1$ such that $\\langle E e_{i},e_{i}\\rangle = d_{i}$ for all $i\\ge N+1$ .", "Then the operator $E$ has diagonal $\\lambda _{1},\\ldots ,\\lambda _{N},d_{N+1},d_{N+2},\\ldots $ with respect to o.n.", "basis $\\lbrace f_{1},\\ldots ,f_{N},e_{N+1},e_{N+2},\\ldots \\rbrace $ of $\\mathcal {H}_0 \\oplus \\mathcal {H}_1$ .", "Since $\\delta _{N}=0$ we have $\\lbrace d_{i}\\rbrace _{i=1}^{N}\\preccurlyeq \\lbrace \\lambda _{i}\\rbrace _{i=1}^{N}$ .", "Applying Lemma REF we obtain an o.n.", "sequence $\\lbrace e_i\\rbrace _{i=1}^\\infty $ with respect to which $E$ has diagonal $\\mathbf {d}$ and (REF ) holds.", "We end this section with a basic linear algebra lemma about convex moves of $2\\times 2$ hermitian matrices.", "Lemma REF generalizes the corresponding well-known result for matrices with zero off-diagonal entries.", "Lemma 2.7 Let $E$ be a symmetric operator on $\\mathcal {D} \\subset \\mathcal {H}$ .", "Assume that real numbers $d_1$ , $d_2$ , $\\widetilde{d}_1$ , $\\widetilde{d}_2$ satisfy $\\widetilde{d}_{1}\\le d_{1},d_{2} \\le \\widetilde{d}_{2},\\qquad \\widetilde{d}_{1}\\ne \\widetilde{d}_{2}, \\qquad \\text{and}\\qquad \\widetilde{d}_{1}+\\widetilde{d}_{2}=d_{1}+d_{2}.$ If there exists an orthonormal set $\\lbrace f_{1},f_{2}\\rbrace \\subset \\mathcal {D}$ such that $\\langle Ef_{i},f_{i}\\rangle = \\widetilde{d}_{i}$ for $i=1,2$ , then there exists $\\frac{\\widetilde{d}_{2}-d_{1}}{\\widetilde{d}_{2}-\\widetilde{d}_{1}}\\le \\alpha \\le 1$ and $\\theta \\in [0,2\\pi )$ such that $\\langle Ee_{i},e_{i}\\rangle = d_{i}$ for $i=1,2$ , where $e_{1}=\\sqrt{\\alpha }f_{1} + \\sqrt{1-\\alpha }e^{i\\theta }f_{2}\\qquad \\text{and}\\qquad e_{2}=\\sqrt{1-\\alpha }f_{1}-\\sqrt{\\alpha }e^{i\\theta }f_{2}.$ Moreover, if $\\mathcal {H}$ is a real Hilbert space, then $e^{i\\theta } = \\pm 1$ .", "If the inequalities in (REF ) are strict, then $\\alpha <1$ .", "Set $\\beta :=\\langle Ef_{1},f_{2}\\rangle .$ Choose $\\theta \\in [0,2\\pi )$ such that $e^{-i\\theta }\\beta \\le 0$ .", "For $x\\in [0,1]$ define $e_{1}^{x} = \\sqrt{x}f_{1} + \\sqrt{1-x}e^{i\\theta }f_{2}\\qquad \\text{and} \\qquad e_{2}^{x} = \\sqrt{1-x}f_{1}-\\sqrt{x}e^{i\\theta }f_{2}.$ We calculate $\\langle Ee_{1}^{x},e_{1}^{x}\\rangle = x\\widetilde{d}_{1} + (1-x)\\widetilde{d}_{2} + 2e^{-i\\theta }\\beta \\sqrt{x(1-x)}$ so that $\\langle Ee_{1}^{1},e_{1}^{1}\\rangle = \\widetilde{d}_{1}\\ge d_{1}$ and for $\\alpha _{0} = (d_{1}-\\widetilde{d}_{2})/(\\widetilde{d}_{1}-\\widetilde{d}_{2})$ , since $e^{-i\\theta }\\beta \\le 0$ we have $\\langle Ee_{1}^{\\alpha _{0}},e_{1}^{\\alpha _{0}}\\rangle & = \\alpha _{0}(\\widetilde{d}_{1}-\\widetilde{d}_{2}) + \\widetilde{d}_{2} + 2e^{-i\\theta }\\beta \\sqrt{\\alpha _{0}(1-\\alpha _{0})}\\\\& = d_{1}-\\widetilde{d}_{2} + \\widetilde{d}_{2} + 2e^{-i\\theta }\\beta \\sqrt{\\alpha _{0}(1-\\alpha _{0})}\\le d_{1}.$ Since $x\\mapsto \\langle Ee_{1}^{x},e_{1}^{x}\\rangle $ is continuous on $[\\alpha _{0},1]$ there is some $\\alpha \\ge \\alpha _{0}$ such that $\\langle Ee_{1}^{\\alpha },e_{1}^{\\alpha }\\rangle = d_{1}$ .", "Finally, using the assumption that $\\widetilde{d}_{1}+\\widetilde{d}_{2}=d_{1}+d_{2}$ , we have $\\langle Ee_{2}^{\\alpha },e_{2}^{\\alpha }\\rangle & = (1-\\alpha )\\widetilde{d}_{1} + \\alpha \\widetilde{d}_{2} - 2e^{-i\\theta }\\beta \\sqrt{\\alpha (1-\\alpha )}\\\\& = \\widetilde{d}_{1}+\\widetilde{d}_{2} - \\Big (\\alpha \\widetilde{d}_{1}+(1-\\alpha )\\widetilde{d}_{2} + 2e^{-i\\theta }\\beta \\sqrt{\\alpha (1-\\alpha )}\\Big )\\\\& = \\widetilde{d}_{1}+\\widetilde{d}_{2} - \\langle Ee_{1}^{\\alpha },e_{1}^{\\alpha }\\rangle = \\widetilde{d}_{1}+\\widetilde{d}_{2} - d_{1} = d_{2}.$ This completes the proof of the lemma." ], [ "Conservation of mass scenario", "In this section we will establish Theorem REF under additional conservation of mass assumption $\\liminf _{k\\rightarrow \\infty } \\delta _k =0, \\qquad \\text{where }\\delta _k= \\sum _{i=1}^{k}(d_{i}-\\lambda _{i}).$ It is remarkable that we achieve this goal without assuming that the sequence $\\lbrace \\lambda _i\\rbrace $ is unbounded.", "This requires a careful application of an infinite sequence of convex moves, also known as $T$ -transforms [21], to the original o.n.", "basis of eigenvectors $\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}}$ .", "The key Lemma REF guarantees that the limiting o.n.", "sequence is complete.", "Lemma 3.1 Let $\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}$ be an orthonormal set, and let $\\lbrace \\alpha _{i}\\rbrace _{i\\in \\mathbb {N}}$ be a sequence in $[0,1]$ .", "Set $\\widetilde{e}_{1}=f_{1}$ and inductively define for $i\\in \\mathbb {N}$ , $e_{i} =\\sqrt{\\alpha _{i}}\\,\\widetilde{e}_{i} + \\sqrt{1-\\alpha _{i}}f_{i+1}\\qquad \\text{and}\\qquad \\widetilde{e}_{i+1} =\\sqrt{1-\\alpha _{i}}\\widetilde{e}_{i} - \\sqrt{\\alpha _{i}}f_{i+1}.$ If for each $n\\in \\mathbb {N}$ $\\prod _{i=n}^{\\infty }(1-\\alpha _{i})=0,$ then $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ is an orthonormal basis for $\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}$ and (REF ) holds.", "In particular, if $\\alpha _{i}<1$ for all $i$ and $\\sum _{i=1}^{\\infty }\\frac{\\alpha _{i}}{1-\\alpha _{i}}=\\infty $ , then $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ is an orthonormal basis for $\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}$ .", "By induction, we see that for each $i\\in \\mathbb {N}$ , $\\lbrace e_1,e_2, \\ldots , e_{i-1}, \\widetilde{e}_i, f_{i+1}, f_{i+2}, \\ldots \\rbrace $ is an orthonormal sequence.", "Hence, $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ is orthonormal and it is enough to show that $f_{j}\\in \\overline{\\operatorname{span}}\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ for all $j\\in \\mathbb {N}$ .", "Note that $e_{i}\\in \\operatorname{span}\\lbrace f_{j}\\rbrace _{j=1}^{i+1}$ and $\\widetilde{e}_{i}\\in \\operatorname{span}\\lbrace f_{j}\\rbrace _{j=1}^{i}$ .", "Thus, $\\langle f_{j},e_{i}\\rangle = 0$ for $i\\le j-2$ and $\\langle f_{j},\\widetilde{e}_{i}\\rangle = 0$ for $i\\le j-1$ .", "Also note that for each $n\\in \\mathbb {N}$ the sequence $\\lbrace e_{1},e_{2},\\ldots ,e_{n},\\widetilde{e}_{n+1}\\rbrace $ is an orthonormal basis for $\\operatorname{span}\\lbrace f_{i}\\rbrace _{i=1}^{n+1}$ .", "Thus, for $n\\ge j-1$ we have $1-\|\\langle f_{j},\\widetilde{e}_{n+1}\\rangle \|^{2} = \\sum _{i=1}^{n}\|\\langle f_{j},e_{i}\\rangle \|^{2}.$ If we set $\\alpha _{0}=1$ , then $\\langle f_{j},\\widetilde{e}_{j}\\rangle = -\\sqrt{\\alpha _{j-1}}$ for all $j\\in \\mathbb {N}$ .", "For $n\\ge 0$ we have $\\langle f_{j},\\widetilde{e}_{j+n}\\rangle = \\sqrt{1-\\alpha _{j+n-1}}\\langle f_{j},\\widetilde{e}_{j+n-1}\\rangle ,$ so that, by induction for $n\\ge 0$ we have $\\langle f_{j},\\widetilde{e}_{j+n}\\rangle = -\\left(\\alpha _{j-1}\\prod _{k=j}^{j+n-1}(1-\\alpha _{j+k})\\right)^{\\frac{1}{2}}.$ Letting $n\\rightarrow \\infty $ in (REF ), we see from (REF ) that $\\lim _{n\\rightarrow \\infty } \\langle f_{j},\\widetilde{e}_{n}\\rangle =0$ .", "Hence, (REF ) implies that for each $j\\in \\mathbb {N}$ $\\sum _{i=1}^{\\infty }\|\\langle f_{j},e_{i}\\rangle \|^{2} = 1,$ That is, $f_{j}\\in \\overline{\\operatorname{span}}\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ , which completes the proof.", "Finally, consider the case that $\\alpha _{i}<1$ for all $i\\in \\mathbb {N}$ , and $\\sum _{i=1}^{\\infty }\\frac{\\alpha _{i}}{1-\\alpha _{i}}=\\infty $ .", "In this case we have $\\sum _{i=n}^{k}\\frac{\\alpha _{i}}{1-\\alpha _{i}}\\le \\prod _{i=n}^{k}\\Big (1+\\frac{\\alpha _{i}}{1-\\alpha _{i}}\\Big ) = \\frac{1}{\\prod _{i=n}^{k}(1-\\alpha _{i})}.$ Letting $k\\rightarrow \\infty $ we obtain (REF ).", "Lemma 3.2 If $\\lbrace t_{n}\\rbrace $ is a positive nonincreasing sequence with limit zero, then $\\sum _{n=1}^{\\infty }\\frac{t_{n}-t_{n+1}}{t_{n+1}}=\\infty .$ Since $(t_{n}-t_{n+1})/t_{n+1} = t_{n}/t_{n+1}-1$ , we may assume $t_{n+1}/t_{n}\\rightarrow 1$ as $n\\rightarrow \\infty $ .", "Since $t_{n}/t_{n+1}\\ge 1$ we have $\\sum _{n=1}^{k}\\frac{t_{n}-t_{n+1}}{t_{n+1}} = \\sum _{n=1}^{k}\\Big (\\frac{t_{n}}{t_{n+1}}-1\\Big ) \\ge \\sum _{n=1}^{k}\\log \\Big (\\frac{t_{n}}{t_{n+1}}\\Big ) = \\log (t_{1})-\\log (t_{k+1})\\rightarrow \\infty \\quad \\text{as }k\\rightarrow \\infty .$ Next, we prove the first preliminary version of Theorem REF under the additional assumption that $\\lbrace \\delta _{k}\\rbrace $ is strictly decreasing to 0.", "However, we do not assume in Lemma REF that $\\lbrace d_{i}\\rbrace $ is arranged in nondecreasing order.", "Also in all subsequent results in Section we do not assume that $\\lbrace \\lambda _i\\rbrace $ is an unbounded sequence.", "Lemma 3.3 Let $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be a nondecreasing sequence.", "Let $E$ be a symmetric operator with diagonal $\\lambda $ as in Definition REF .", "If $\\mathbf {d}= \\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ is a sequence such that the following two properties hold: $\\lambda _{1}\\le d_{n}<\\lambda _{n}\\quad \\text{for all}\\ n\\ge 2,$ $ d_{1}= \\lambda _{1} + \\sum _{i=2}^{\\infty }(\\lambda _{i}-d_{i})< \\lambda _{2},$ then $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\lambda $ .", "For each $n\\in \\mathbb {N}$ set $\\widetilde{\\lambda }_{n} := d_{n} - \\sum _{i=n+1}^{\\infty }(\\lambda _{i}-d_{i}) = \\lambda _{n} -\\sum _{i=n}^{\\infty }(\\lambda _{i}-d_{i}).$ From (REF ) for each $n\\ge 2$ , we have $\\widetilde{\\lambda }_{n} < d_{n}<\\lambda _{n}\\le \\lambda _{n+1}.$ From (REF ) we have $\\widetilde{\\lambda }_{1} = \\lambda _{1}< d_{1}< \\lambda _{2}.$ Thus, for all $n\\in \\mathbb {N}$ we have $\\widetilde{\\lambda }_n<d_n,\\widetilde{\\lambda }_{n+1}<\\lambda _{n+1} \\qquad \\text{and}\\qquad \\widetilde{\\lambda }_n+\\lambda _{n+1}=d_n+\\widetilde{\\lambda }_{n+1}.$ We conclude that for all $n\\in \\mathbb {N}$ $\\widetilde{\\alpha }_{n} := \\frac{\\lambda _{n+1}-d_{n}}{\\lambda _{n+1}-\\widetilde{\\lambda }_{n}} = \\frac{\\lambda _{n+1}-d_{n}}{\\lambda _{n+1}-d_{n} + \\sum _{i=n+1}^{\\infty }(\\lambda _{i}-d_{i})}\\in (0,1).$ For each $n\\in \\mathbb {N}$ set $t_{n} = \\sum _{i=n}^{\\infty }(\\lambda _{i}-d_{i}).$ Note that $t_{1}=0$ , and $\\lbrace t_{i}\\rbrace _{i=2}^{\\infty }$ is a positive, nonincreasing sequence with limit zero.", "By Lemma REF we have $\\sum _{n=1}^{\\infty }\\frac{\\widetilde{\\alpha }_{n}}{1- \\widetilde{\\alpha }_{n}} = \\sum _{n=1}^{\\infty }\\frac{\\lambda _{n+1}-d_{n}}{\\sum _{i=n+1}^{\\infty }(\\lambda _{i}-d_{i})}\\ge \\sum _{n=1}^{\\infty }\\frac{\\lambda _{n}-d_{n}}{\\sum _{i=n+1}^{\\infty }(\\lambda _{i}-d_{i})} = \\sum _{n=1}^{\\infty }\\frac{t_{n}-t_{n+1}}{t_{n+1}}=\\infty .$ Let $\\lbrace f_{n}\\rbrace _{n\\in \\mathbb {N}}$ be an orthonormal sequence with respect to which $E$ has diagonal $\\lambda $ .", "We shall now define an orthonormal sequence $\\lbrace e_{n}\\rbrace _{n\\in \\mathbb {N}}$ as in Lemma REF for an appropriate choice of the sequence $\\lbrace \\alpha _n\\rbrace _{n\\in \\mathbb {N}}$ .", "We have $\\langle Ef_{1},f_{1}\\rangle = \\lambda _1=\\widetilde{\\lambda }_1$ , and $\\langle Ef_{2},f_{2}\\rangle = \\lambda _2$ .", "By Lemma REF there exist $\\alpha _1\\in [\\widetilde{\\alpha }_1,1)$ and $\\theta _{2}\\in [0,2\\pi )$ such that vectors $e_{1} = \\sqrt{\\alpha _{1}}f_{1} + \\sqrt{1-\\alpha _{1}}e^{i\\theta _{2}}f_2 \\quad \\text{and}\\quad \\widetilde{e}_2 = \\sqrt{1-\\alpha _{1}}f_{1} - \\sqrt{\\alpha _{1}}e^{i\\theta _{2}}f_{2}$ form an orthonormal basis for $\\operatorname{span}\\lbrace f_{1},f_{2}\\rbrace $ and $\\langle Ee_{1},e_{1}\\rangle = d_{1}\\quad \\text{and}\\quad \\langle E\\widetilde{e}_{2},\\widetilde{e}_{2}\\rangle = \\widetilde{\\lambda }_2.$ Now, we may inductively assume that for some $n\\ge 2$ we have an orthonormal basis $\\lbrace e_1,\\ldots ,e_{n-1},\\widetilde{e}_n\\rbrace $ for $\\operatorname{span}\\lbrace f_{j}\\rbrace _{j=1}^n$ such that $\\langle Ee_j,e_j \\rangle = d_j \\quad \\text{for }j\\le n-1 \\qquad \\text{and}\\qquad \\langle E\\widetilde{e}_n,\\widetilde{e}_n \\rangle = \\widetilde{\\lambda }_n.$ Using (REF ), by Lemma REF there exist $\\alpha _n\\in [\\widetilde{\\alpha }_n,1)$ and $\\theta _{n+1}\\in [0,2\\pi )$ such that the vectors $e_n = \\sqrt{\\alpha _n}\\widetilde{e}_n + \\sqrt{1-\\alpha _n}e^{i\\theta _n}f_{n+1} \\quad \\text{and}\\quad \\widetilde{e}_{n+1} = \\sqrt{1-\\alpha _n}\\widetilde{e}_n - \\sqrt{\\alpha _n}e^{i\\theta _{n+1}}f_{n+1}$ form an orthonormal basis for $\\operatorname{span}\\lbrace \\widetilde{e}_n,f_{n+1}\\rbrace $ and $\\langle Ee_n,e_n\\rangle = d_n\\quad \\text{and}\\quad \\langle E\\widetilde{e}_{n+1},\\widetilde{e}_{n+1}\\rangle = \\widetilde{\\lambda }_{n+1}.$ The fact that $\\alpha _n<1$ for all $n\\in \\mathbb {N}$ is a consequence of strict inequalities in (REF ).", "Observe that the above procedure yields an orthonormal sequence $\\lbrace e_n\\rbrace _{n=1}^{\\infty }$ that is obtained by applying Lemma REF to $\\lbrace e^{i\\theta _n}f_n \\rbrace _{n\\in \\mathbb {N}}$ with $\\lbrace \\alpha _n\\rbrace _{n\\in \\mathbb {N}}$ and $\\lbrace \\theta _n\\rbrace _{n\\in \\mathbb {N}}$ as already defined and $\\theta _1=0$ .", "Since for all $n\\in \\mathbb {N}$ , $\\alpha _n\\in [\\widetilde{\\alpha }_n,1)$ , by (REF ) we have $\\sum _{n=1}^\\infty \\frac{\\alpha _n}{1-\\alpha _n}\\ge \\sum _{n=1}^\\infty \\frac{\\widetilde{\\alpha }_n}{1-\\widetilde{\\alpha }_n}=\\infty .$ Hence, by Lemma REF $\\lbrace e_n\\rbrace _{n\\in \\mathbb {N}}$ is an orthonormal basis for $\\overline{\\operatorname{span}}\\lbrace f_n\\rbrace _{n\\in \\mathbb {N}}$ .", "By (REF ) each vector $e_n$ is a linear combination $f_1,\\ldots ,f_{n+1}$ .", "Therefore, $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\lambda $ .", "The following is the second preliminary version of the main result of this section.", "The final result of this section, which is Theorem REF , will be identical with the exception of the extra assumption that $d_{1}<\\lambda _{2}$ .", "Lemma 3.4 Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be nondecreasing sequences such that (REF ) and (REF ) hold.", "Let $E$ be a symmetric operator with diagonal $\\lambda $ .", "If $\\lambda _{2}>d_{1}$ , then $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\lambda $ .", "By Theorem REF we may assume that $\\delta _{k}>0$ for all $k\\in \\mathbb {N}$ .", "Inductively define the sequence $\\lbrace m_{j}\\rbrace $ as follows.", "Set $m_{1}=1$ and for $j\\ge 2$ set $m_{j}=\\min \\lbrace n>m_{j-1}\\colon \\delta _{n}< \\delta _{m_{j-1}}\\rbrace $ .", "For each $j\\in \\mathbb {N}$ and $i=m_{j}+1,m_{j+1},\\ldots ,m_{j+1}$ set $\\widetilde{d}_{i} = \\frac{\\delta _{m_{j+1}}-\\delta _{m_{j}}}{m_{j+1}-m_{j}} + \\lambda _{i}.$ Also set $\\widetilde{d}_{1}=d_{1}$ and define $\\widetilde{\\delta }_{k}:=\\sum _{i=1}^{k}(\\widetilde{d}_{i}-\\lambda _{i}).$ By induction, for each $j\\in \\mathbb {N}$ and $k=m_{j}+1,\\ldots ,m_{j+1}$ we have $\\widetilde{\\delta }_{k} = \\delta _{m_{j}} + \\frac{\\delta _{m_{j+1}}-\\delta _{m_{j}}}{m_{j+1}-m_{j}}(k-m_{j}).$ In particular, we have $\\widetilde{\\delta }_{m_{j}}=\\delta _{m_{j}} \\qquad \\text{for all }j\\in \\mathbb {N}.$ Since $\\delta _{k}\\ge \\delta _{m_{j}}>\\delta _{m_{j+1}}$ for all $m_j<k< m_{j+1}$ , we have $\\delta _{m_{j}}+\\sum _{i=m_{j}+1}^{k}(\\widetilde{d}_{i}-\\lambda _{i}) = \\widetilde{\\delta }_{k} = \\delta _{m_{j}} + \\frac{\\delta _{m_{j+1}}-\\delta _{m_{j}}}{m_{j+1}-m_{j}}(k-m_{j}) \\le \\delta _{m_{j}}\\le \\delta _{k} = \\delta _{m_{j}}+\\sum _{i=m_{j}+1}^{k}(d_{i}-\\lambda _{i}).$ Combining this with (REF ) shows that $\\lbrace d_{i}\\rbrace _{i=m_{j}+1}^{m_{j+1}}\\preccurlyeq \\lbrace \\widetilde{d}_{i}\\rbrace _{i=m_{j}+1}^{m_{j+1}}$ for each $j\\in \\mathbb {N}$ .", "Using $\\delta _{m_{j+1}}-\\delta _{m_{j}}< 0$ , (REF ), and (REF ) we deduce that the sequence $\\lbrace \\widetilde{\\delta }_{k}\\rbrace $ is decreasing and $\\lim _{k\\rightarrow \\infty }\\widetilde{\\delta }_k=0$ .", "Moreover, we have $\\widetilde{d}_{1} = d_{1} <\\lambda _{2}$ and $\\lambda _1 \\le \\widetilde{d}_n$ for all $n\\ge 2$ .", "Applying Lemma REF to the sequences $\\lambda $ and $\\widetilde{\\mathbf {d}} := \\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ shows that $E$ has diagonal $\\widetilde{\\mathbf {d}}$ , which is finitely derived from $\\lambda $ .", "Finally, since the sets $I_{j}=\\lbrace m_{j},\\ldots ,m_{j+1}-1\\rbrace $ are disjoint, and $\\lbrace d_{i}\\rbrace _{i\\in I_{j}}\\preccurlyeq \\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in I_{j}}$ , Lemma REF shows that $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\widetilde{\\mathbf {d}}$ , and hence finitely derived from $\\lambda $ .", "Lemma 3.5 Let $\\mathbf {d} = \\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda = \\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be nondecreasing sequences such that such that (REF ) and (REF ) hold.", "Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D}$ .", "If the following two conditions hold: there exists $N\\in \\mathbb {N}$ such that $\\delta _{N}\\le \\delta _{k}$ for all $k\\le N$ , $E$ has diagonal $\\widetilde{\\mathbf {d}}:=\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ , where $\\widetilde{d}_{i}: = {\\left\\lbrace \\begin{array}{ll} \\lambda _{1}+\\delta _{N} & i=1,\\\\\\lambda _{i} & i=2,\\ldots ,N,\\\\d_{i} & i>N,\\end{array}\\right.", "}$ then $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\widetilde{\\mathbf {d}}$ .", "Let $I_{1}=\\lbrace 1,\\ldots ,N\\rbrace $ .", "In light of Lemma REF it is enough to show that $\\lbrace d_{i}\\rbrace _{i\\in I_{1}}\\preccurlyeq \\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in I_{1}}$ .", "Let $\\lbrace \\widetilde{d}_{i}^{\\,\\,\\uparrow }\\rbrace _{i=1}^{N}$ denote the nondecreasing rearrangement of $\\lbrace \\widetilde{d}_{i}\\rbrace _{i=1}^{N}$ , then for $k=1,\\ldots ,N$ $\\sum _{i=1}^{k}\\widetilde{d}_{i}^{\\,\\,\\uparrow }\\le \\sum _{i=1}^{k}\\widetilde{d}_{i} = \\delta _{N} + \\sum _{i=1}^{k}\\lambda _{i} = \\sum _{i=1}^{k}d_{i}+\\delta _{N}-\\delta _{k}\\le \\sum _{i=1}^{k}d_{i}.$ Together with the observation that both of the inequalities above become equality when $k=N$ demonstrates the desired majorization.", "We are now ready to show Theorem REF under the additional hypothesis (REF ), but without the assumption that $\\lbrace \\lambda _i\\rbrace $ is unbounded.", "Theorem 3.6 Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i\\in \\mathbb {N}}$ be nondecreasing sequences such that (REF ) and (REF ) hold.", "Let $E$ be a symmetric operator with diagonal $\\lambda $ as in Definition REF .", "Then, $E$ has diagonal $\\mathbf {d}$ , which is finitely derived from $\\lambda $ .", "By Theorem REF we may assume that $\\delta _{k}>0$ for all $k\\in \\mathbb {N}$ .", "We also claim that $\\lambda $ is not a constant sequence.", "On the contrary, suppose $\\lambda _{i}=L$ for all $i\\in \\mathbb {N}$ .", "Since $\\mathbf {d}$ is nondecreasing and $\\liminf _{k\\rightarrow \\infty }\\delta _{k}=0$ we conclude that $d_{i}\\nearrow L$ as $i\\rightarrow \\infty $ .", "The assumption that $\\delta _{1}>0$ implies $d_{1}>L$ , which is a contradiction.", "Since $\\lambda $ is not constant, there is some $M\\in \\mathbb {N}$ such that $\\lambda _{1}<\\lambda _{M}$ .", "Choose $N>M$ such that $\\delta _{N}\\le \\delta _{k}\\quad \\text{for all}\\ k\\le N.$ and $\\delta _{N}<\\lambda _{M}-\\lambda _{1}$ .", "Since $\\lambda _{M}\\le \\lambda _{N+1}$ we also have $ \\lambda _{N+1}>\\delta _{N}+\\lambda _{1}.$ Define the sequence $\\widetilde{\\mathbf {d}}=\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ as in (REF ).", "Define the sequences $\\lbrace c_{i}\\rbrace $ and $\\lbrace \\mu _{i}\\rbrace $ by $c_{i}={\\left\\lbrace \\begin{array}{ll}\\widetilde{d}_{1} & i=1,\\\\ \\widetilde{d}_{i+N-1} & i\\ge 2,\\end{array}\\right.", "}\\qquad \\text{and}\\qquad \\mu _{i}={\\left\\lbrace \\begin{array}{ll}\\lambda _{1} & i=1,\\\\ \\lambda _{i+N-1} & i\\ge 2.\\end{array}\\right.", "}$ Note that $\\widetilde{\\delta }_{k}:=\\sum _{i=1}^{k}(c_{i}-\\mu _{i}) = \\delta _{N+k-1} \\qquad \\text{for all }k\\in \\mathbb {N}.$ From (REF ) we see that $c_{1}=\\delta _{N}+\\lambda _{1}< \\lambda _{N+1}=\\mu _{2}$ .", "By our hypothesis, $E$ has diagonal $\\lbrace \\mu _i\\rbrace $ with respect to o.n.", "sequence $\\lbrace f_i\\rbrace _{i=1,i>N}$ .", "Applying Lemma REF yields an o.n.", "basis $\\lbrace \\widetilde{e}_i\\rbrace _{i=1,i>N}$ of $\\overline{\\operatorname{span}}\\lbrace f_i\\rbrace _{i=1,i>N}$ with respect to which $E$ has diagonal $\\lbrace c_{i}\\rbrace $ , which is finitely derived from $\\lbrace \\mu _i\\rbrace $ .", "Letting $\\widetilde{e}_i=f_i$ for $2\\le i \\le N$ , yields an o.n.", "sequence $\\lbrace \\widetilde{e}_i\\rbrace _{i\\in \\mathbb {N}}$ with respect to which $E$ has diagonal $\\widetilde{\\mathbf {d}}$ .", "By (REF ) we can apply Lemma REF to obtain a desired o.n.", "sequence $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}}$ , with respect to which $E$ has diagonal $\\mathbf {d}$ .", "Moreover, $\\mathbf {d}$ is finitely derived from $\\widetilde{\\mathbf {d}}$ , and hence from $\\lambda $ ." ], [ "Mass vanishing at infinity scenario", "In this section we will show Theorem REF under complementary assumption to (REF ).", "This involves a construction of an infinite sequence of convex moves continually transforming a diagonal sequence, where some of the mass must necessarily vanish at infinity.", "First we handle the strong domination case $\\lambda _k\\le d_k$ for every $k\\in \\mathbb {N}$ .", "Equivalently, the sequence $\\lbrace \\delta _k\\rbrace _{k\\in \\mathbb {N}}$ is assumed to be nondecreasing in Lemma REF .", "Lemma 4.1 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D}$ .", "Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i=1}^{\\infty }$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i=1}^{\\infty }$ be nondecreasing unbounded sequences with $d_i\\ge \\lambda _i$ for every $i$ .", "If there exists an orthonormal sequence $\\lbrace f_{i}\\rbrace _{i\\in \\mathbb {N}}\\subset \\mathcal {D}$ such that $\\langle Ef_{i},f_{i}\\rangle = \\lambda _i\\qquad \\text{for all } i\\in \\mathbb {N},$ then there exists an orthonormal sequence $\\lbrace e_{i}\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) and $\\langle Ee_{i},e_{i}\\rangle = d_{i}\\qquad \\text{for all } i\\in \\mathbb {N}.$ Suppose that $I$ is an infinite subset of $\\mathbb {N}$ .", "For any such subset we define inductively an increasing sequence $\\lbrace i_{k}\\rbrace _{k=1}^{\\infty }$ in $I$ by letting $i_1=\\min I$ and choosing $i_k\\in I$ large enough to have $\\lambda _{i_{k}}>2d_{i_{k-1}} \\qquad k\\ge 2.$ In addition, we require that $I \\setminus \\lbrace i_k: k\\in \\mathbb {N}\\rbrace $ is infinite.", "This is possible since the sequence $\\lbrace \\lambda _i\\rbrace $ is not bounded.", "Now recursively define another sequence by $x_{i_1}=\\lambda _{i_1}$ and $x_{i_{k+1}}=\\lambda _{i_{k+1}}+x_{i_k}-d_{i_k} \\qquad k\\ge 1.$ Note that $x_{i_2}\\le \\lambda _{i_2}$ and $x_{i_2}-d_{i_1}>0$ (using condition (REF )).", "By induction we get that for any $k\\ge 1$ $x_{i_k}\\le d_{i_k}<x_{i_{k+1}}\\le \\lambda _{i_{k+1}}.$ Furthermore, $\\widetilde{\\alpha }_k:=\\frac{\\lambda _{i_{k+1}}-d_{i_k}}{\\lambda _{i_{k+1}}-x_{i_k}}>\\frac{\\lambda _{i_{k+1}}/2}{\\lambda _{i_{k+1}}}=\\frac{1}{2}$ Now we are ready to start constructing an o.n.", "sequence $\\lbrace e_{i_k}\\rbrace _{k=1}^\\infty $ .", "We have $\\langle Ef_{i_1},f_{i_1}\\rangle = x_{i_1}$ , and $\\langle Ef_{i_{2}},f_{i_{2}}\\rangle = \\lambda _{i_{2}}$ .", "By Lemma REF there exist $\\alpha _1\\in [\\widetilde{\\alpha }_1,1]$ and $\\theta _{2}\\in [0,2\\pi )$ such that vectors $e_{i_1} = \\sqrt{\\alpha _{1}}f_{i_1} + \\sqrt{1-\\alpha _{1}}e^{i\\theta _{2}}f_{i_{2}} \\quad \\text{and}\\quad \\widetilde{e}_{i_{2}} = \\sqrt{1-\\alpha _{1}}f_{i_1} - \\sqrt{\\alpha _{1}}e^{i\\theta _{2}}f_{i_{2}}$ form an orthonormal basis for $\\operatorname{span}\\lbrace f_{i_1},f_{i_{2}}\\rbrace $ and $\\langle Ee_{i_1},e_{i_1}\\rangle = d_{i_1}\\quad \\text{and}\\quad \\langle E\\widetilde{e}_{i_{2}},\\widetilde{e}_{i_{2}}\\rangle = x_{i_2}.$ Now, we may inductively assume that for some $k\\ge 2$ we have an orthonormal basis $\\lbrace e_{i_{1}},\\ldots ,e_{i_{k-1}},\\widetilde{e}_{i_{k}}\\rbrace $ for $\\operatorname{span}\\lbrace f_{i_{j}}\\rbrace _{j=1}^{k}$ such that $\\langle Ee_{i_{j}},e_{i_{j}}\\rangle = d_{i_{j}}\\quad \\text{for }j\\le k-1 \\qquad \\text{and}\\qquad \\langle E\\widetilde{e}_{i_{k}},\\widetilde{e}_{i_{k}}\\rangle = x_{i_{k}}.$ Using (REF ), by Lemma REF there exist $\\alpha _k\\in [\\widetilde{\\alpha }_k,1]$ and $\\theta _{k+1}\\in [0,2\\pi )$ such that the vectors $e_{i_{k}} = \\sqrt{\\alpha _{k}}\\widetilde{e}_{i_{k}} + \\sqrt{1-\\alpha _{k}}e^{i\\theta _{k}}f_{i_{k+1}} \\quad \\text{and}\\quad \\widetilde{e}_{i_{k+1}} = \\sqrt{1-\\alpha _{k}}\\widetilde{e}_{i_{k}} - \\sqrt{\\alpha _{k}}e^{i\\theta _{k+1}}f_{i_{k+1}}$ form an orthonormal basis for $\\operatorname{span}\\lbrace \\widetilde{e}_{i_{k}},f_{i_{k+1}}\\rbrace $ and $\\langle Ee_{i_{k}},e_{i_{k}}\\rangle = d_{i_{k}}\\quad \\text{and}\\quad \\langle E\\widetilde{e}_{i_{k+1}},\\widetilde{e}_{i_{k+1}}\\rangle = x_{i_{k+1}}.$ This completes the inductive step, and thus we have an orthonormal sequence $\\lbrace e_{i_{k}}\\rbrace _{k=1}^{\\infty }$ .", "Observe that this is exactly the orthonormal sequence obtained by applying Lemma REF to $\\lbrace e^{i\\theta _{k}}f_{i_{k}}\\rbrace _{k\\in \\mathbb {N}}$ with $\\lbrace \\alpha _{k}\\rbrace _{k\\in \\mathbb {N}}$ and $\\lbrace \\theta _{k}\\rbrace _{k\\in \\mathbb {N}}$ as already defined with $\\theta _1=0$ .", "By (REF ) we have $\\alpha _{k}>1/2$ for all $k\\in \\mathbb {N}$ .", "Hence, by Lemma REF $\\lbrace e_i\\rbrace _{i\\in I_1}$ is an orthonormal basis for $\\mathcal {H}_1=\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in I_1}$ , with respect to which $E$ has diagonal $\\lbrace d_{i}\\rbrace _{i\\in I_1}$ , where $I_1=\\lbrace i_k:k\\in \\mathbb {N}\\rbrace $ .", "Moreover, diagonal $\\lbrace d_{i}\\rbrace _{i\\in I_1}$ is finitely derived from $\\lbrace \\lambda _{i}\\rbrace _{i\\in I_1}$ .", "In the initial step we run the above construction starting with the full index set $I=\\mathbb {N}$ to obtain the required diagonal subsequence indexed by $I_1$ .", "Then, we repeat the above construction inductively with respect to the unused index set $I=\\mathbb {N}\\setminus (I_1\\cup \\ldots \\cup I_{k-1})$ , $k\\ge 2$ , to obtain the required diagonal subsequence indexed by $I_k$ .", "Since we always include the smallest unused element in $I$ and we leave out infinitely many unused indices, the family $\\lbrace I_k\\rbrace _{k\\in \\mathbb {N}}$ is a partition of $\\mathbb {N}$ .", "Thus, we obtain an orthogonal decomposition $\\overline{\\operatorname{span}}\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}} = \\bigoplus _{k=1}^\\infty \\mathcal {H}_k, \\qquad \\text{where }\\mathcal {H}_k=\\overline{\\operatorname{span}}\\lbrace f_{i}\\rbrace _{i\\in I_k}.$ For each subspace $\\mathcal {H}_k$ we have constructed an orthonormal basis $\\lbrace e_{i}\\rbrace _{i\\in I_k}$ , with respect to which $E$ has diagonal $\\lbrace d_{i}\\rbrace _{i\\in I_k}$ , that is finitely derived from $\\lbrace \\lambda _{i}\\rbrace _{i\\in I_k}$ .", "This defines the required orthonormal basis $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}}$ of $\\overline{\\operatorname{span}}\\lbrace f_i\\rbrace _{i\\in \\mathbb {N}}$ with respect to which $E$ has diagonal $\\mathbf {d}$ .", "We are now ready to show Theorem REF in the case when sequence $\\lbrace \\delta _k\\rbrace _{k\\in \\mathbb {N}} $ as in (REF ), eventually stays above its $\\liminf _{k\\rightarrow \\infty }\\delta _k$ .", "Theorem 4.2 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D}$ .", "Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i=1}^{\\infty }$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i=1}^{\\infty }$ be nondecreasing unbounded sequences such that (REF ) holds.", "Assume that there exists $M\\ge 0$ such that $\\delta _{k}\\ge \\alpha :=\\liminf _{i\\rightarrow \\infty }\\delta _i \\qquad \\text{for all }k\\ge M.$ If $\\lambda $ is a diagonal of $E$ , then $\\mathbf {d}$ is a finitely derived diagonal of $E$ .", "By Lemma REF we may assume $\\delta _{k}>0$ for all $k\\in \\mathbb {N}$ .", "Fix $N\\in \\mathbb {N}$ such that $N> \\max _{k\\le M-1}\\lbrace \\frac{k\\alpha }{\\delta _{k}},M\\rbrace $ .", "Hence, $\\delta _{k}\\ge \\frac{k\\alpha }{N} \\qquad \\text{for }k\\le M-1.$ Define $\\widetilde{d}_{i} = {\\left\\lbrace \\begin{array}{ll} d_{i} - \\frac{\\alpha }{N} & i=1,\\ldots ,N,\\\\ d_{i} & i\\ge N+1.\\end{array}\\right.", "}$ Observe that $\\sum _{i=1}^{k}(\\widetilde{d}_{i} - \\lambda _{i}) = {\\left\\lbrace \\begin{array}{ll}\\delta _{k} - \\frac{k\\alpha }{N}\\ge 0 & k\\le M-1,\\\\\\delta _{k} - \\frac{k\\alpha }{N}\\ge \\alpha -\\frac{k\\alpha }{N}\\ge 0 & M\\le k\\le N,\\\\\\delta _{k}-\\alpha \\ge 0 & k\\ge N+1.\\end{array}\\right.", "}$ The last equation implies that $\\liminf _{k\\rightarrow \\infty }\\sum _{i=1}^{k}(\\widetilde{d}_{i}-\\lambda _{i})=0$ .", "We may apply Theorem REF to deduce that $E$ has diagonal $\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ , which is finitely derived from $\\lambda $ .", "Since $d_i \\ge \\widetilde{d}_i$ for all $i\\in \\mathbb {N}$ , Lemma REF yields the desired diagonal $\\lbrace {d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ .", "Finally, we are left we the case when the sequence $\\lbrace \\delta _k\\rbrace _{k\\in \\mathbb {N}}$ dips infinitely many times below its $\\liminf _{k\\rightarrow \\infty }\\delta _k$ .", "Theorem 4.3 Let $E$ be a symmetric operator defined on a dense domain $\\mathcal {D}$ .", "Let $\\mathbf {d}=\\lbrace d_{i}\\rbrace _{i\\in \\mathbb {N}}$ and $\\lambda =\\lbrace \\lambda _{i}\\rbrace _{i=1\\in \\mathbb {N}}$ be nondecreasing unbounded sequences such that (REF ) holds.", "Assume that $\\delta _{k}< \\alpha := \\liminf _{i\\rightarrow \\infty } \\delta _i\\qquad \\text{for infinitely many }k.$ If $\\lambda $ is a diagonal of $E$ , then $\\mathbf {d}$ is a finitely derived diagonal of $E$ .", "We define inductively the index sequence $\\lbrace m_j\\rbrace _{j=0}^\\infty $ as follows.", "Let $m_0=0$ .", "For $j\\ge 1$ set $m_j = \\min \\lbrace n>m_{j-1}: \\forall k \\ge n\\quad \\delta _n \\le \\delta _k \\rbrace .$ That is, the sequence $\\lbrace m_j\\rbrace $ records consecutive global minima of the tail $\\lbrace \\delta _n\\rbrace _{n>m_{j-1}}$ .", "In particular, using the convention that $\\delta _0=0$ , we have $\\delta _{m_{j-1}} \\le \\delta _{m_j} \\le \\delta _k\\qquad \\text{for all } m_{j-1} < k \\le m_j,\\quad j\\ge 1.$ Define the sequence $\\lbrace \\widetilde{d}_i \\rbrace _{i\\in \\mathbb {N}}$ by $\\widetilde{d}_i = {\\left\\lbrace \\begin{array}{ll}\\lambda _i + (\\delta _{m_j} - \\delta _{m_{j-1}}) & \\text{for }i=m_j, \\ j \\ge 1,\\\\\\lambda _i & \\text{otherwise.}\\end{array}\\right.", "}$ Set $\\widetilde{\\delta }_{k}=\\sum _{i=1}^{k}(\\widetilde{d}_{i}-\\lambda _{i}).$ For $j\\ge 1$ , set $I_j=\\lbrace m_{j-1}+1,\\ldots ,m_j\\rbrace $ .", "By (REF ) and (REF ), for any $k\\in I_j$ we have $\\delta _{m_{j-1}}+ \\sum _{i=m_{j-1}+1}^k (d_i-\\lambda _i) = \\delta _k \\ge \\delta _{m_{j+1}} \\ge \\delta _{m_{j-1}}+ \\sum _{i=m_{j-1}+1}^k (\\widetilde{d}_i-\\lambda _i)$ with equalities when $k=m_j$ .", "This shows that $\\lbrace d_i\\rbrace _{i\\in I_j} \\preccurlyeq \\lbrace \\widetilde{d}_i\\rbrace _{i\\in I_j}$ .", "Since the sets $\\lbrace I_j\\rbrace _{j\\in \\mathbb {N}}$ form a partition of $\\mathbb {N}$ , we can apply Lemma REF to reduce the problem to showing that $E$ has diagonal $\\lbrace \\widetilde{d}_{i}\\rbrace _{i\\in \\mathbb {N}}$ .", "This case is already covered by Lemma REF since the sequence $\\lbrace \\widetilde{\\delta }_i\\rbrace _{i\\in \\mathbb {N}}$ is nondecreasing.", "Theorem REF now follows immediately by combining Theorems REF and REF ." ], [ "Diagonals and eigenvalues of inverse operators", "It is worth observing how our main result, Theorem REF , is related to the result of Kaftal and Weiss [21] who characterized the diagonals of positive compact operators.", "The earlier result of Arveson and Kadison [6] characterized diagonals of positive trace class operators.", "In the case of positive compact operators that are not trace class, the trace condition is not present both in [21] and in Theorem REF .", "Hence, one might attempt to deduce Theorem REF from [21].", "For simplicity assume that the first eigenvalue of $E$ is $\\lambda _1>0$ .", "Then, the inverse $E^{-1}$ is a compact positive operator with eigenvalues $1/\\lambda _1 \\ge 1/\\lambda _2 \\ge \\ldots \\searrow 0$ .", "Conversely, the inverse of positive self-adjoint operator with trivial kernel is unbounded with discrete spectrum.", "However, the diagonal does not behave in such controlled way when taking inverses.", "Thus, Theorem REF does not follow from [21] in any obvious way.", "For the converse direction, Theorem REF implies a special case of [21] when $\\liminf _{k\\rightarrow \\infty } \\delta _k=0$ .", "Nevertheless, it is possible to deduce majorization for sums of inverses from the majorization of sums of eigenvalues as follows.", "We say that a sequence $\\lbrace a_i\\rbrace _{i\\in \\mathbb {N}}$ is (weakly) majorized by a sequence $\\lbrace b_i\\rbrace _{i\\in \\mathbb {N}}$ , and write $\\lbrace a_i\\rbrace \\prec \\lbrace b_i\\rbrace $ , if $\\sum _{i=1}^n a_i \\le \\sum _{i=1}^n b_i \\qquad \\text{for all }n\\in \\mathbb {N}.$ Note that unlike (strong) majorization order $\\preccurlyeq $ , we do not alter the order of elements of the sequences.", "Recall the classical Hardy-Littlewood-Pólya majorization theorem [14].", "Theorem 5.1 (Hardy-Littlewood-Pólya majorization) Assume that $\\lbrace a_i\\rbrace $ and $\\lbrace b_i\\rbrace $ are nondecreasing sequences of positive real numbers such that $\\lbrace a_i\\rbrace \\prec \\lbrace b_i\\rbrace $ .", "Then for any concave increasing function $\\Phi : \\mathbb {R}_+ \\rightarrow \\mathbb {R}$ we have $\\lbrace \\Phi (a_i)\\rbrace \\prec \\lbrace \\Phi (b_i)\\rbrace $ .", "Similarly, when $\\lbrace a_i\\rbrace $ and $\\lbrace b_i\\rbrace $ are nonincreasing, then the result holds for convex increasing functions $\\Phi $ .", "Let $a_i=\\lambda _i$ and $b_i=d_i$ with the sequences coming from the unbounded operator $E$ as in Theorem REF .", "Now choose $\\Phi (x)=-1/x$ to get that $\\lbrace 1/d_i\\rbrace \\prec \\lbrace 1/\\lambda _i\\rbrace $ .", "Therefore, whenever $\\lbrace d_i\\rbrace $ is a possible diagonal for $E$ , the sequence of inverses is a valid diagonal for the compact operator $E^{-1}$ .", "Interestingly, the inverse procedure does not work.", "Even if $\\lbrace \\widetilde{d}_i\\rbrace $ is majorized by $\\lbrace 1/\\lambda _i\\rbrace $ , the sequence of inverses $\\lbrace 1/\\widetilde{d}_i\\rbrace $ does not need to majorize $\\lbrace \\lambda _i\\rbrace $ , since $\\Phi (x)=-1/x$ is not convex.", "As another consequence of Hardy-Littlewood-Pólya majorization we get that whenever $\\lbrace d_i\\rbrace $ is a valid diagonal for $E$ , the sequence of eigenvalues $\\lbrace e^{-\\lambda _i t}\\rbrace $ of the heat operator $e^{-tE}$ majorizes $\\lbrace e^{-d_i t}\\rbrace $ .", "Therefore the heat operator associated with $E$ admits diagonal $\\lbrace e^{-d_i t}\\rbrace $ ." ], [ "Examples using Laplacians", "Elliptic differential operators provide a broad and interesting class of operators falling into the scope of this paper.", "In particular, Laplace operators on domains $\\Omega \\subset \\mathbb {R}^d$ imposed with various boundary conditions can be closed in $L^2(\\Omega )$ leading to essentially self-adjoint operators with discrete spectrum.", "This follows from classical considerations involving compactness of their inverses and compactness of the Sobolev embeddings.", "For more details see Bandle [7] or Blanchard-Brüning [8].", "To be more specific, consider two Laplace operators defined (weakly) on Sobolev spaces, via the corresponding quadratic forms: Neumann Laplacian $\\Delta _N$ : domain $H^1(\\Omega )$ , quadratic form $\\langle \\Delta _N u,v\\rangle = \\int _\\Omega \\nabla u\\cdot \\nabla v \\,dA$ ; Dirichlet Laplacian $\\Delta _D$ : domain $H^1_0(\\Omega )$ , quadratic form $\\int _\\Omega \\nabla u\\cdot \\nabla v \\,dA$ .", "It turns out that the eigenfunctions for these operators satisfy appropriate classical boundary conditions: Neumann $\\partial _n u=0$ on $\\partial \\Omega $ , and Dirichlet $u=0$ on $\\partial \\Omega $ , respectively.", "See Chapters 5 and 6 of Laugesen [23] for a nice overview.", "Let $\\mu _j$ and $\\lambda _j$ denote the eigenvalues (in nondecreasing order) for the Neumann and Dirichlet Laplacians, resp.", "It is easy to see (via operator domain inclusion) that for any $j$ we have $\\mu _j\\le \\lambda _j$ , see [7] or [23].", "Therefore we have two sequences exhibiting strong domination as in Lemma REF .", "Interestingly, these operator are not self-adjoint, or even symmetric, according to the theory of unbounded operators.", "They are defined on a dense subspace $H^1(\\Omega )$ of $L^2(\\Omega )$ , however their adjoints have much smaller domain.", "One can however consider the same operators restricted to $H^2(\\Omega )$ .", "Assuming that $\\Omega $ is somewhat smooth (locally Lipschitz boundary is enough), elliptic regularity theory implies that domain of the adjoint is now the same as for the operator.", "Hence we get self-adjoint operators on $H^2(\\Omega )$ which agree with the weak formulation on their domains.", "See [23] for a detailed exposition." ], [ "Dirichlet eigenvalues and Neumann Laplacian", "We can ask for an orthonormal basis of $L^2(\\Omega )$ such that the diagonal entries of the Neumann Laplacian equal the Dirichlet eigenvalues $\\lambda _j\\ge \\mu _j$ .", "Theorem REF asserts that such a basis must exist.", "In the simplest possible case of an interval, $\\Omega =[0,\\pi ]$ , the Dirichlet eigenfunctions equal $\\lbrace u_j=\\sin (j x)\\rbrace _{j\\ge 1}$ and they form an o.n.", "basis of $L^2$ .", "These functions certainly belong to $H^1(\\Omega )$ (or even $H^2(\\Omega )$ ), so we already have the required o.n.", "basis for $L^2(\\Omega )$ (Fourier sine series).", "However, we are acting on these functions using Neumann Laplacian.", "This is irrelevant for the quadratic form definition, but the pointwise action is not simply the second derivative.", "In order to compute the Neumann Laplacian of $\\sin (j x)$ we must first find the Fourier cosine series expansion of that function, since $\\lbrace \\cos (j x)\\rbrace _{j \\ge 0}$ is the o.n.", "basis formed by the eigenfunction of the Neumann Laplacian.", "Therefore our transformations amount to constructing a cosine series for sine functions." ], [ "Domain monotonicity for Dirichlet Laplacian", "It is also easy to see that if $\\Omega _1\\subset \\Omega _2$ , then $\\lambda _j(\\Omega _1)\\ge \\lambda _j(\\Omega _2)$ , simply because $H^1_0(\\Omega _1)\\subset H^1_0(\\Omega _2)$ (by setting functions equal 0 outside).", "If $\\Omega _1$ is a relatively compact subset of $\\Omega _2$ then the eigenfunctions of the Dirichlet Laplacian on $\\Omega _1$ are concentrated on a compact subset of $\\Omega _2$ , hence they cannot form an o.n.", "basis for $L^2(\\Omega _2)$ .", "Theorem REF still asserts that there is an o.n.", "basis of $L^2(\\Omega _2)$ such that the diagonal of the Dirichlet Laplacian on $\\Omega _2$ equals $\\lbrace \\lambda _j(\\Omega _1)\\rbrace $ .", "However, it is not at all clear how to find such a basis." ] ]	1606.05236
[ [ "Partial $W^{2,p}$ regularity for optimal transport maps" ], [ "Abstract We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \\geq 1$ outside of a closed singular set of measure zero.", "We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost.", "The latter result is new even when the cost is exactly the quadratic cost." ], [ "Introduction", "Regularity of optimal transport maps is a very important problem that has been studied extensively in the recent years.", "For the special case when the cost function is given by $c(x,y)=\\frac{1}{2}\|x-y\|^2$ (or equivalently $c(x,y)=-x\\cdot y$ , see the discussion in [13]), Caffarelli [1], [2], [3], [4], [5] developed a deep regularity theory.", "However, for general costs functions the situation was much more complicated.", "A major breakthrough happened in 2005 when Ma, Trudinger, and Wang [34] introduced a fourth order condition on the cost function (now known as MTW condition) that guarantees the smoothness of optimal transport map under suitable global assumptions on the data.", "Later, it was shown by Loeper [31] that the MTW condition is actually a necessary condition.", "Motivated by these results, a lot of efforts have been devoted to understanding the regularity properties of optimal map under the MTW condition, see for instance [20], [28], [37], [38], [21], [32], [33], [29], [30], [22], [27], [24], [23], [18], [19].", "Unfortunately, as observed by Loeper in [31] and further noticed in many subsequent works, the MTW condition is extremely restrictive and many interesting costs do not satisfy this condition.", "Hence, a natural and important question became the following: What can we say about the regularity of optimal transport maps when the MTW condition fails?", "A first major answer was given by De Philippis and Figalli [12]: there, the authors proved that, without assuming neither the MTW condition nor any convexity on the domains, for the optimal transport problem with positive continuous (resp.", "positive smooth) densities, the potential function is always $C_{loc}^{1,\\alpha }$ (resp.", "smooth) outside a closed singular set of measure zero.", "In a related direction, Caffarelli, Gonzáles, and Nguyen [7] obtained an interior $C_{loc}^{2,\\alpha }$ regularity result of optimal transport problem when the densities are $C^\\alpha $ and the cost function is of the form $c(x,y)=\\frac{1}{p}\|x-y\|^p$ with $2<p<2+\\epsilon $ for some $\\epsilon \\ll 1$ (or, $p>1$ and the distance between source and target is sufficiently large).", "This interior regularity result was later extended by us to a global one [10].", "The aim of this work is to further develop the techniques introduced in [10], [11], [12] and prove a partial $W^{2,p}$ regularity result.", "More precisely we show that, for the optimal transport problem with positive continuous densities, there exists a closed singular set of measure zero outside which the potential function is of class $W^{2,p}_{loc}$ for any $p>1$ (in particular, the singular set is independent of the exponent $p$ ).", "As a corollary of our techniques together with an argument due to Savin [36], we are able to obtain global $W^{2,p}$ estimates when the domains are convex and the cost function is $C^2$ -close to $-x\\cdot y$ .", "The paper is organized as follows.", "In section 2 we introduce some notation and state our main results.", "Then, in section 3 we prove our key Proposition REF , and finally in the last section we prove our main results." ], [ "Preliminaries and main results", "First, we introduce some conditions which should be satisfied by the cost.", "Let $X$ and $Y$ be two bounded open subsets of $\\mathbb {R}^n$ .", "(C0) The cost function $c:X\\times Y\\rightarrow \\mathbb {R}$ is of class $C^3$ , with $\\Vert c\\Vert _{C^3(X\\times Y)}<\\infty .$ (C1) For any $x\\in X$ , the map $Y \\ni y\\mapsto D_xc(x,y)\\in \\mathbb {R}^n$ is injective.", "(C2) For any $y\\in Y$ , the map $X \\ni x\\mapsto D_yc(x,y)\\in \\mathbb {R}^n$ is injective.", "(C3) $\\det (D_{xy}c)(x,y)\\ne 0$ for all $(x,y)\\in X\\times Y.$ A function $u: X\\rightarrow \\mathbb {R}$ is said $c$ -convex if it can be written as $ u(x)=\\underset{y\\in Y}{\\sup }\\lbrace -c(x,y)+\\lambda _y\\rbrace $ for some family of constants $\\lbrace \\lambda _y\\rbrace _{y \\in Y}\\subset \\mathbb {R}$ .", "Note that $(\\textbf {C0})$ and (REF ) imply that a $c$ -convex function is semiconvex, namely, there exists a constant $K$ depending only on $\\Vert c\\Vert _{C^2(X\\times Y)}$ such that $u+K\|x\|^2$ is convex.", "One immediate consequence of the semiconvexity is that $u$ is twice differentiable almost everywhere.", "Thanks to $(\\textbf {C0})$ and $(\\textbf {C1})$ it is well known (see for instance [40]) that there exists a unique optimal transport map.", "Also, there exists a $c$ -convex function $u$ such that the optimal map is a.e.", "uniquely characterized in terms of $u$ (and for this reason we denote it by $T_u$ ) via the relation $-D_xc(x, T_u (x))= \\nabla u(x) \\qquad \\text{for a.e.", "}x.$ As explained for instance in [12] (see also [13]), the transport condition $(T_u)_\\#f=g$ implies that $u$ solves at almost every point the Monge-Ampère type equation $\\det \\Bigl (D^2u(x)+D_{xx}c\\bigl (x,\\operatorname{c-exp}_x(\\nabla u(x))\\bigr ) \\Bigr )=\\left\|\\det \\left(D_{xy}c\\bigl (x,\\operatorname{c-exp}_x(\\nabla u(x))\\bigr ) \\right) \\right\| \\frac{f(x)}{g(\\operatorname{c-exp}_x(\\nabla u(x)))},$ where $\\operatorname{c-exp}$ denotes the $c$ -exponential map defined as $\\text{for any $x\\in X$, $y \\in Y$, $p \\in \\mathbb {R}^n$},\\qquad \\operatorname{c-exp}_x(p)=y \\quad \\Leftrightarrow \\quad p=-D_xc(x,y).$ Notice that, with this notation, $T_u(x)= \\operatorname{c-exp}_x(\\nabla u(x))$ .", "For a $c$ -convex function, in analogy with the subdifferential for convex functions, we can talk about its $c$ -subdifferential: If $u: X\\rightarrow \\mathbb {R}$ is a $c$ -convex function, the $c$ -subdifferential of $u$ at $x$ is the (nonempty) set $\\partial _cu(x):=\\bigl \\lbrace y\\in \\overline{Y}: u(z)\\ge -c(z,y)+c(x,y)+u(x)\\qquad \\forall \\,z\\in X \\bigr \\rbrace .$ We also define Frechet subdifferential of $u$ at $x$ as $\\partial ^-u(x):=\\bigl \\lbrace p\\in \\mathbb {R}^n: u(z)\\ge u(x)+p\\cdot (z-x)+o(\|z-x\|)\\bigr \\rbrace .$ It is easy to check that $y\\in \\partial _cu(x)\\quad \\Longrightarrow \\quad -D_xc(x,y)\\in \\partial ^-u(x).$ Also, it is a well-known fact (see for instance [40]) that the transport map $T_u$ and the $c$ -subdifferential $\\partial _cu$ are related by the inclusion $T_u(x) \\in \\partial _cu(x).$ In particular, since $\\partial _cu(x)$ is a singleton at every differentiability point of $u$ (this follows by (REF )), we deduce that $\\partial _cu(x)=\\lbrace T_u(x)\\rbrace \\qquad \\text{whenever $u$ is differentiable at $x$.", "}$ The analogue of sublevels of a convex functions is played by the sections: given $y_0 \\in \\partial _cu(x_0)$ , we define $S(x_0, y_0, u, h):=\\lbrace x: u(x)\\le -c(x, y_0)+c(x_0, y_0)+c(x_0, y_0)+u(x_0)+h\\rbrace .$ Note that, whenever $u$ is differentiable at $x_0$ then $y_0=T_u(x_0)$ .", "To simplicity the notation, we will use $S_h(x_0)$ to denote $S(x_0, y_0, u, h)$ when no confusion arises.", "Finally, we recall that given $u$ $c$ -convex, its $c$ -transform $u^c$ is defined as $u^c(y):=\\sup _{x \\in X}\\lbrace -c(x,y)-u(x)\\rbrace .$ With this definition, $u^c$ plays the role of $u$ for the transportation problem from $g$ to $f$ .", "Our first main result states that, if $f$ and $g$ are positive continuous densities, then $u$ is of class $W^{2,p}_{loc}$ for any $p\\ge 1$ outside a closed set of measure zero.", "A crucial fact in our proof is to show that the singular set $\\Sigma $ is independent of $p$ .", "Theorem 2.1 Let $u$ be the potential function for the optimal transport problem from $(X, f)$ to $(Y, g)$ with cost $c$ satisfying $(\\textbf {C0})$ -$(\\textbf {C3})$ .", "Suppose $f:X\\rightarrow \\mathbb {R}^+$ and $g:Y\\rightarrow \\mathbb {R}^+$ are positive continuous densities.", "Then there exists a closed set $\\Sigma \\subset X$ of measure zero such that $u\\in W^{2,p}_{loc}(X\\setminus \\Sigma )$ for any $p\\ge 1.$ By a localization argument, the above theorem yields the following: Corollary 2.2 Let $(M, \\mathcal {G})$ be a smooth closed Riemannian manifold, and denote by $d$ the Riemannian distance induced by $\\mathcal {G}$ .", "Let $f$ and $g$ be two positive continuous densities, and let $T$ be the optimal transport map for the cost $c=\\frac{d^2}{2}$ sending $f$ onto $g$ .", "Then there exist two closed sets $\\Sigma _1,\\Sigma _2\\subset M$ of measure zero, such that $T:M\\setminus \\Sigma _1 \\rightarrow M\\setminus \\Sigma _2$ is a diffeomorphism of class $W^{1,p}_{loc}$ for any $p\\ge 1.$ In the next result we show that if the cost function is sufficiently close to the “quadratic” cost $-x\\cdot y$ (recall that this cost is equivalent to $\\frac{1}{2}\|x-y\|^2$ ), then the potential is $W^{2,p}$ up to the boundary.", "Observe that the smallness parameter $\\hat{\\delta }$ is independent of $p$ , and that this result is new even in the case $c(x,y)=-x\\cdot y$ .", "Theorem 2.3 Suppose $X$ and $Y$ are two $C^2$ uniformly convex bounded domains in $\\mathbb {R}^n$ .", "Assume $f:X\\rightarrow \\mathbb {R}^+$ and $g:Y\\rightarrow \\mathbb {R}^+$ are two continuous positive densities, and let $u$ be the $c$ -convex function associated to the optimal transport problem between $f$ and $g$ with cost $c(x,y)$ .", "Suppose $c$ satisfies $(\\textbf {C0})$ -$(\\textbf {C3})$ and $\\Vert c+x\\cdot y\\Vert _{C^2(X\\times Y)} \\le \\delta .$ Then there exists $\\hat{\\delta }>0$ , depending only on $n$ , the modulus of continuity of $f$ and $g$ , and the uniform convexity and $C^2$ -smoothness of $X$ , and $Y$ , such that $u\\in W^{2, p}(\\overline{X})$ for any $p\\ge 1$ provided $\\delta \\le \\hat{\\delta }$ .", "The proof of above results is based on the following proposition.", "Proposition 2.4 Let $f$ and $g$ be two densities supported in $B_{1/K}\\subset \\mathcal {C}_1\\subset B_K$ and $B_{1/K}\\subset \\mathcal {C}_2\\subset B_K$ , respectively.", "Suppose that $\\mathcal {C}_2$ is convex, $\\Vert f-\\textbf {1}\\Vert _{L^\\infty (\\mathcal {C}_1)}+\\Vert g-\\textbf {1}\\Vert _{L^\\infty (\\mathcal {C}_2)}\\le \\delta ,$ and $\\Vert c(x,y)+x\\cdot y\\Vert _{C^2(B_K \\times B_K)}\\le \\delta .$ Then, for any $p\\ge 1$ there exists $\\bar{\\delta }>0$ , depending only on $n$ , $K$ , and $p$ , such that $u\\in W^{2,p}(B_{\\frac{1}{4K}})$ provided $\\delta \\le \\bar{\\delta }$ .", "Note that, in the result above, the smallness of the parameter $\\delta $ depends on $p$ .", "So, for the proof of Theorems REF and REF and Corollary REF , it will be crucial to prove that actually $\\delta $ can be chosen independently of $p$ (see Lemma REF ).", "Also, as explained in Section REF below, to prove Proposition REF we shall first approximate $u$ with smooth solutions and then obtain $W^{2,p}$ a priori estimates that are independent of the regularization.", "We note that, in this context, such a regularization procedure is nontrivial and require some attention.", "Remark 2.5 As we shall also observe later, the condition “$\\mathcal {C}_2$ is convex\" in Proposition REF can be replaced by the assumption $\\biggl \\Vert u-\\frac{1}{2}\|x\|^2\\biggr \\Vert _{L^\\infty (B_{\\eta _0})}\\le \\delta $ for some fixed $\\eta _0\\le 1/K.$ Under this assumption, for any $p\\ge 1$ there exists $\\bar{\\delta }>0$ , depending only on $n$ , $K$ , $\\eta _0$ , and $p$ , such that $u\\in W^{2,p}(B_{\\frac{1}{2}\\eta _0})$ provided $\\delta \\le \\bar{\\delta }$ .", "Moreover, in the above condition, the function $\\frac{1}{2}\|x\|^2$ can be replaced by a $C^2$ convex function $v$ such that $\\frac{1}{M}{\\rm Id}\\le D^2v \\le M{\\rm Id}$ , in which case $\\bar{\\delta }$ depends also on $M$ and the modulus of continuity of $D^2 v$ ." ], [ "Proof of Proposition ", "We begin by observing that, under our assumptions, it follows by [12] and the argument in the proof of [10] that $u \\in C^{1,\\alpha }(B_{\\frac{1}{2K}})$ for some $\\alpha >0$ .", "Hence, up to replace $K$ by $2K$ , we can assume that $u \\in C^{1,\\alpha }(B_{1/K})$ .", "In particular it follows by (REF ) that $S(x_0,y_0,u,h)=S(x_0,T_u(x_0),u,h)$ , and we can use the notation $S_h(x_0)=S(x_0,T_u(x_0),u,h)$ ." ], [ "Engulfing property of sections", "The first step consists in establish the engulfing property for sections of $u$ , which is stated as the following lemma.", "Lemma 3.1 (Engulfing property) There exist universal constants $r_0>0$ and $C>1$ such that, for $h\\le r_0$ and $x_0\\in B_{\\frac{2}{3K}},$ $x_1\\in S_h(x_0)\\qquad \\Longrightarrow \\qquad S_h(x_0)\\subset S_{Ch}(x_1).$ Without loss of generality we may assume $x_0=0, y_0=T_u(x_0)=0$ , and $u(0)=0$ .", "Up to performing the transformations $c(x,y)\\mapsto \\tilde{c}(x,y):=c(x,y)-c(x,0)-c(0,y)+c(0,0),\\qquad u(x)\\mapsto \\tilde{u}(x):=u(x)+c(x,0),$ we may assume $c(x,y)=-x\\cdot y+O(\|x\|^2\|y\|+\|x\|\|y\|^2).$ Set $\\rho :=\\left(\\frac{\|\\mathcal {C}_1\|}{\|\\mathcal {C}_2\|}\\right)^{1/n}$ so that $\|\\rho \\mathcal {C}_2\|=\|\\mathcal {C}_1\|$ , and let $v$ be a convex function satisfying $(\\nabla v)_\\sharp \\textbf {1}_{\\mathcal {C}_1}=\\textbf {1}_{\\rho \\mathcal {C}_2}$ with $v(0)=0$ (we note that $\\nabla v$ is the optimal transport map from $\\frac{1}{\|\\mathcal {C}_1\|}\\textbf {1}_{\\mathcal {C}_1}$ to $\\frac{1}{\|\\mathcal {C}_2\|}\\textbf {1}_{\\mathcal {C}_2}$ for the quadratic cost).", "By a compactness argument similar to the proof of [12] we have $\\Vert u-v\\Vert _{L^\\infty (B_{\\frac{1}{K}})}\\le \\omega (\\delta ),$ where $\\omega :\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ satisfies $\\omega (r)\\rightarrow 0$ as $r\\rightarrow 0.$ Also, since $\\rho \\mathcal {C}_2$ is convex, it follows by [4] that $v$ is smooth and uniformly convex in $B_{\\frac{3}{4K}}.$ Thanks to (REF ), (REF ), and (REF ), we can follow the proof of [12] to show that, for small $h$ , there exists an affine transform $A$ such that $A(B_{\\frac{1}{3}\\sqrt{h}})\\subset S_h\\subset A(B_{3\\sqrt{h}}),$ $A^{\\prime -1}(B_{\\frac{1}{3}\\sqrt{h}})\\subset T_u(S_h)\\subset A^{\\prime -1}(B_{3\\sqrt{h}})$ and $\\biggl \|u(Ax)-\\frac{1}{2}\|x\|^2\\biggr \|\\le \\eta h \\quad \\text{in $B_{3\\sqrt{h}}$}$ with $\\Vert A\\Vert , \\Vert A^{-1}\\Vert \\le h^{-\\theta },$ where $\\eta , \\theta >0$ can be as small as we want, provided $\\delta $ is sufficiently small.", "Note that (REF ) plays the same role as the fact that $u$ is close to a quadratic function, which is used in the proof of [12].", "Furthermore, (REF ) and (REF ) imply that $\\text{diam}(S_h(x))\\le Ch^{\\frac{1}{2}-\\theta }.$ Hence, if we choose $r_0$ small enough so that $Cr_0^{\\frac{1}{2}-\\theta }\\le \\frac{1}{4K}$ , we see that for $h\\le r_0$ and $x\\in B_{\\frac{2}{3K}}$ we have $S_h(x)\\subset B_{\\frac{3}{4K}}.$ Now we perform the transformations $c_1(x,y):=c(Ax, A^{\\prime -1}y)$ and $u_1(x):=u(Ax)$ , and we use the notation $S_h^1=S(0,0, u_1, h)$ .", "By (REF ) and (REF ) we have $B_{\\frac{1}{3}\\sqrt{h}}\\subset S_h^1\\subset B_{3\\sqrt{h}}$ and $B_{\\frac{1}{3}\\sqrt{h}}\\subset T_{u_1}(S_h^1)\\subset B_{3\\sqrt{h}}.$ Note that $0\\le u_1(x)+c_1(x,0)-c_1(0,0)-u_1(0)\\le h\\qquad \\text{for any $x\\in S_h^1$.", "}$ Also, by (REF ) and (REF ) we see that $\\Vert c_1\\Vert _{C^2(B_{3\\sqrt{h}}\\times B_{3\\sqrt{h}})} \\le C$ for some universal constant $C$ .", "Therefore, thanks to (REF ) and (REF ), for any $x,x_1 \\in S_h^1$ and $y_1 =T_{u_1}(x_1) \\in T_{u_1}(S_h^1)$ , $\|c_1(x, y_1)-c_1(x_1, y_1) + c_1(x_1, 0) -c_1(x, 0)\| \\le \\Vert D_{xy}c_1\\Vert _{C^0(B_{3\\sqrt{h}}\\times B_{3\\sqrt{h}})}\|x_1-x\|\\,\|y_1\| \\le C_1h$ for some universal constant $C_1$ .", "Hence, by (REF ) applied to both $x$ and $x_1$ we get $u_1(x)+c_1(x,y_1)-c_1(x_1,y_1)-u_1(x_1)&=&u_1(x)+c_1(x,0)-c_1(0,0)-u_1(0)\\\\&&-(u_1(x_1)+c_1(x_1,0)-c_1(0,0)-u_1(0))\\\\&&+ c_1(x, y_1) - c_1(x_1, y_1) + c_1(x_1, 0) -c_1(x, 0)\\\\&\\le & h+C_1h.$ Since $x \\in S_h^1=S(0,0,u_1,h)$ was arbitrary, this proves that $S(0,0,u_1,h) \\subset S(x_1,y_1,u_1, (1+C_1)h).$ Recalling the relation between $u_1$ and $u$ , this proves the desired result.", "As a consequence of this result, one gets the following: Corollary 3.2 There exists a small constant $r_1$ such that for $h\\le r_1$ and $x,y\\in B_{\\frac{1}{2K}}$ the following holds: suppose $S_h(x)\\cap S_t(y)\\ne \\emptyset ,$ $t\\le h.$ Then there exists an universal constant $C^{\\prime }$ such that $S_t(y)\\subset S_{C^{\\prime }h}(x).$ Fix $z\\in S_h(x)\\cap S_t(y)$ .", "By Lemma REF we have that $S_t(y)\\subset S_{Ch}(z)\\qquad \\text{and}\\qquad x\\in S_h(x)\\subset S_{Ch}(z)$ for some universal constant $C.$ Also, by the argument in the proof of Lemma REF , $z \\in B_{\\frac{2}{3K}}$ for $h$ small enough.", "Hence, using Lemma REF again we have $S_{Ch}(z)\\subset S_{C^2h}(x),$ thus $S_t(y)\\subset S_{C^{\\prime }h}(x)$ with $C^{\\prime }:=C^2.$ It is well known that the property of sections stated in Corollary REF implies the following Vitali covering lemma (see for instance [16] for a proof): Lemma 3.3 (Vitali covering) Under the assumptions of Proposition REF , let $D$ be a compact subset of $B_{\\frac{1}{2K}},$ and let $\\lbrace S_{h_x}(x)\\rbrace _{x\\in D}$ be a family of sections with $h_x\\le r_1.$ Then, there exists a finite number of sections $\\lbrace S_{h_{x_i}}(x_i)\\rbrace _{i=1,\\ldots , m}$ such that $D\\subset \\bigcup _{i=1}^mS_{h_{x_i}}(x_i)$ with $\\lbrace S_{\\sigma h_{x_i}}(x_i)\\rbrace _{i=1,\\ldots , m}$ disjoint, where $\\sigma >0$ is a universal constant." ], [ "Approximation argument", "In the next sections we will prove our $W^{2,p}$ estimates by controlling the measure of the super-level sets of the Hessian of $u$ .", "Because we shall need to use the pointwise value of $D^2u$ , we need an approximation argument in order to work with $C^2$ convex functions.", "Since in this setting this is not a standard procedure, we now provide the details.", "Given $u$ as in Proposition REF , we set $\\rho :=\\left(\\frac{\|\\mathcal {C}_1\|}{\|\\mathcal {C}_2\|}\\right)^{1/n}$ so that $\|\\rho \\mathcal {C}_2\|=\|\\mathcal {C}_1\|$ , and let $v$ be a convex function satisfying $(\\nabla v)_\\sharp \\textbf {1}_{\\mathcal {C}_1}=\\textbf {1}_{\\rho \\mathcal {C}_2}$ with $v(0)=0$ .", "Since $\\Vert u-v\\Vert _{L^\\infty (B_{\\frac{1}{K}})} \\rightarrow 0\\qquad \\text{as $\\delta \\rightarrow 0$}$ (see (REF )), as in the proof of Lemma REF we can choose $\\delta $ small enough so that, for any $x \\in B_{\\frac{1}{2K}}$ and $h>0$ small but universal, the section $S_h(x)$ satisfies (REF ), (REF ), (REF ), and (REF ).", "We now consider $f_\\epsilon :\\mathcal {C}_1\\rightarrow \\mathbb {R}$ and $g_\\epsilon :\\mathcal {C}_2\\rightarrow \\mathbb {R}$ as sequence of $C^\\infty $ densities that approximate $f$ and $g$ respectively, and denote by $u_\\epsilon $ the potential function for the optimal transport problem from $f_\\epsilon $ to $g_\\epsilon $ with cost $c$ .", "Without loss of generality, we can assume that $u_\\epsilon (0)=u(0)$ .", "Then, by a compactness argument it follows that $\\Vert u_\\epsilon -u\\Vert _{L^\\infty (B_{\\frac{1}{K}})} \\rightarrow 0\\qquad \\text{as $\\epsilon \\rightarrow 0$}$ Since $u$ is strictly convex, choosing $\\epsilon $ sufficiently small we see that the sections $S_h^\\epsilon (x)=S(x,T_{u_\\epsilon }(x),u_\\epsilon ,h)$ satisfy (REF ), (REF ), (REF ), and (REF ) with bounds independent of $\\epsilon $ .", "In particular, assuming $\\delta $ is small enough, by [12] applied to $\\frac{1}{h}u_\\epsilon (A\\sqrt{h}x)$ we deduce that ${u}_\\epsilon $ is of class $C^{1, 6/7}$ in $A(B_{\\frac{1}{4}\\sqrt{h}})$ .", "By duality, similarly we also have that its $c$ -transform ${u}_\\epsilon ^{c}$ is of class $C^{1, 6/7}$ inside $A^{\\prime -1}(B_{\\frac{1}{4}\\sqrt{h}}).$ Hence, by [11] we deduce that ${u}_\\epsilon $ is of class $C^2$ in a neighborhood of $x$ .", "Since $x \\in B_{\\frac{1}{2K}}$ was arbitrary, this proves that $u_\\epsilon \\in C^2(B_{\\frac{1}{2K}})$ for any $\\epsilon >0$ small enough.", "Hence, up to proving our $W^{2,p}$ estimates with $u_\\epsilon $ in place of $u$ and then letting $\\epsilon \\rightarrow 0$ , in the next sections we shall directly assume that $u \\in C^2$ ." ], [ "Density estimates", "The goal here is to show that, given a section $S_h(x)\\subset B_{\\frac{1}{2K}},$ the density of “bad points” where the Hessian of $u$ is large has measure that goes to zero as $\\delta \\rightarrow 0.$ Fix $x_0\\in B_{\\frac{1}{2K}},$ and let $y_0=T_u(x_0).$ Without loss of generality, we may assume $x_0=y_0=0.$ Also, as in the proof of Lemma REF we can assume that (REF ) holds.", "In this way it follows that, for $h$ small, (REF ), (REF ), (REF ), and (REF ) hold.", "Perform the transformations $c(x,y)\\mapsto \\bar{c}(x,y):=\\frac{1}{h}c(\\sqrt{h}Ax, \\sqrt{h}A^{\\prime -1}y);$ $u(x)\\mapsto \\bar{u}(x):=\\frac{1}{h}u(\\sqrt{h}Ax);$ $f(x)\\mapsto \\bar{f}(x):=f(\\sqrt{h}Ax),\\qquad g(y)\\mapsto \\bar{g}(y)=g(\\sqrt{h}A^{\\prime -1}y).$ Note that, by (REF ) and (REF ), we have $\\Vert \\bar{c}+x\\cdot y\\Vert _{C^2(B_8\\times B_8)}\\le \\delta $ provided $h$ is sufficiently small.", "Also, it follows by (REF ), (REF ), and (REF ) that $B_{\\frac{1}{3}}\\subset S(0,0, \\bar{u}, 1)\\subset B_{3},$ $B_{\\frac{1}{3}}\\subset T_{\\bar{u}}(S(0,0, \\bar{u}, 1))\\subset B_{3},$ and $\\biggl \\Vert \\bar{u}(x)-\\frac{1}{2}\|x\|^2\\biggr \\Vert _{L^\\infty (B_3)}\\le \\eta .$ We now construct a smooth function $w$ that well approximates $\\bar{u}$ .", "Denote $X_1:=S(0,0, \\bar{u}, h)$ and $Y_1:=T_{\\bar{u}}(S(0,0, \\bar{u}, h)).$ Lemma 3.4 Set $\\rho :=\\left(\\frac{\|X_1\|}{\|Y_1\|}\\right)^{1/n},$ and let $w$ be a convex function such that $(\\nabla w)_\\sharp \\textbf {1}_{X_1}=\\textbf {1}_{\\rho Y_1}$ and $w(0)=u(0).$ Then, for any $\\gamma >0,$ there exist $\\delta _\\gamma , \\eta _\\gamma >0$ such that $\\Vert \\bar{u}-w\\Vert _{L^\\infty (B_{1/4})}\\le \\gamma $ and $\\Vert w\\Vert _{C^3(B_{1/6})}\\le C$ provided $\\delta \\le \\delta _\\gamma $ and $\\eta \\le \\eta _\\gamma ,$ where $C$ is a universal constant.", "The bound (REF ) follows from a compactness argument similar to the proof of [12].", "Also, taking $\\gamma \\le \\eta $ , (REF ) and (REF ) imply that $\\biggl \\Vert w(x)-\\frac{1}{2}\|x\|^2\\biggr \\Vert _{L^\\infty (B_{1/4})}\\le 2\\eta .$ Thanks to (REF ), as in the proof of Lemma REF (see also Step 1 in the proof of [12]) we can apply [4] to deduce that $ \\Vert w\\Vert _{C^3(B_{1/6})}\\le C$ for some universal constant $C.$ Let $L$ be the operator defined by $L\\bar{u}(x):=D^2\\bar{u}(x)+D_{xx}\\bar{c}\\bigl (x, T_{\\bar{u}}(x)\\bigr ).$ By (REF ) and (REF ), we have $\\det (L\\bar{u}(x))=\\left\|\\det \\left(D_{xy}\\bar{c}\\bigl (x,T_{\\bar{u}}(x)\\bigr ) \\right) \\right\| \\frac{\\bar{f}(x)}{\\bar{g}(T_{\\bar{u}}(x))}=1+O(\\delta ).$ We now follow the argument in [2] to establish the density estimate.", "Since the argument is rather standard, we shall just emphasize the main points, referring to [2] or [16] for more details.", "Lemma 3.5 Let $\\bar{u}, w$ be as above, and denote by $\\Gamma \\left(\\bar{u}-\\frac{w}{2}\\right)$ the convex envelope of $\\bar{u}-\\frac{w}{2}.$ Then, for any Borel set $E\\subset B_{1/6},$ we have $\\biggl \|\\nabla \\Gamma \\Bigl (\\bar{u}-\\frac{w}{2}\\Bigr )(E)\\biggr \|\\le \\biggl (\\frac{1}{2^n}+O(\\delta )\\biggr ) \\biggl \|E\\cap \\biggl \\lbrace \\Gamma \\Bigl (\\bar{u}-\\frac{w}{2}\\Bigr )=\\bar{u}-\\frac{w}{2}\\biggl \\rbrace \\biggl \|$ Noticing that $\\det D^2w= 1$ , $\\det D^2\\bar{u}=1+O(\\delta )$ , and $D_{xx}\\bar{c}=O(\\delta )$ , since $w$ is uniformly convex and $\\det D^2\\Gamma (\\bar{u}-\\frac{w}{2})$ is a measure supported on $\\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace $ , it follows by the Area Formula (see for instance [16]) that $\\biggl \|\\nabla \\Gamma \\Bigl (\\bar{u}-\\frac{w}{2}\\Bigr )(E)\\biggr \|&=&\\int _{E\\cap \\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace }\\det D^2\\Bigl (\\bar{u}-\\frac{w}{2}\\Bigr )\\\\&=&\\int _{E\\cap \\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace } \\det \\biggl [ L\\bar{u}-\\Bigl (D^2\\frac{w}{2}+D_{xx}\\bar{c}\\bigl (x, T_{\\bar{u}}x\\bigr )\\Bigr )\\biggr ]\\\\&\\le & \\int _{E\\cap \\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace }\\biggl (\\det (L\\bar{u})^{1/n}- \\det \\Bigl [\\Bigl (D^2\\frac{w}{2}+O(\\delta )\\Bigr )\\Bigr ]^{1/n}\\biggr )^n\\\\&\\le & \\int _{E\\cap \\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace } \\biggl (1+O(\\delta )-\\Bigl (\\frac{1}{2}(\\det D^2w)^{1/n}-O(\\delta )\\Bigr )\\biggr )^n\\\\&\\le & \\biggl (\\frac{1}{2}+O(\\delta )\\biggr )^n\\biggl \|E\\cap \\biggl \\lbrace \\Gamma \\Bigl (\\bar{u}-\\frac{w}{2}\\Bigr )=\\bar{u}-\\frac{w}{2}\\biggr \\rbrace \\biggr \|,$ where we used the inequality $[\\det (A+B)]^{1/n}\\ge (\\det A)^{1/n}+(\\det B)^{1/n}\\qquad \\forall \\,A,B \\text{ symmetric, nonnegative definite}$ (see for instance [16] for a proof).", "By using Lemma REF , we can follow the lines of proof of [2] (see also the proof of [16]) to establish the estimate $\\frac{\|\\lbrace \\Gamma (\\bar{u}-\\frac{w}{2})=\\bar{u}-\\frac{w}{2}\\rbrace \\cap B_{1/8}\|}{\|B_{1/8}\|}\\ge 1-C\\delta ^{1/2},$ from which one immediately obtain the following bound (see [2] or Step 6 in the proof of [16]): Lemma 3.6 (Density estimate) Let $\\bar{u}$ be as above.", "Then there exist universal constants $N>1, \\eta >0$ such that $\\left\|\\left\\lbrace x\\in S^{\\bar{u}}_{\\eta }(0): \\Vert D^2\\bar{u}(x)\\Vert \\ge N\\right\\rbrace \\right\|\\le N\\delta ^{1/2}\\left\|S^{\\bar{u}}_{\\eta }(0)\\right\|.$" ], [ "$W^{2,p}$ estimate", "We now prove our $W^{2,p}$ interior estimates.", "Recall that we are assuming that $u \\in C^2$ .", "As in the proof of Lemma REF , for any $x\\in B_{\\frac{1}{2K}}$ and $h>0$ small enough, there exists an affine transformation $A$ with $\\det A=1$ such that $A(B_{\\frac{1}{3}\\sqrt{h}})\\subset S_h(x)\\subset A(B_{3\\sqrt{h}}).$ We define the normalized size of the section $S_h(x)$ as $\\mathbf {a}(S_h(x)):=\\Vert A^{-1}\\Vert ^{2}.$ Although $A$ is not unique, if $A_1$ and $A_2$ are two affine transformations that satisfy (REF ) then both $\\Vert A_1^{-1}A_2\\Vert $ and $\\Vert A_2^{-1}A_1\\Vert $ are universally bounded, thus the normalized size is well defined up to universal constants.", "With the notation from the previous section, we see that the estimate (REF ) can be rewritten in terms of $u$ and becomes $\\bigl \|S_h(x)\\cap \\bigl \\lbrace \\Vert D^2u(x)\\Vert \\ge N\\mathbf {a}\\bigl (S_h(x)\\bigr )\\bigr \\rbrace \\bigr \|\\le C\\delta ^{1/2}\|S_{h}(x)\|$ for any $h$ small enough.", "Also, since $\\det D^2 \\bar{u}=1+O(\\delta )$ , it follows that $\\Vert D^2u\\Vert \\le N\\qquad \\Longrightarrow \\qquad D^2 u \\ge \\frac{1}{2N^{n-1}}{\\rm Id}.$ Thus, up to enlarging $N$ and using Lemma REF again, we deduce that $ \|S_{h}(u)\|\\le C\\biggl \|S_{\\sigma h}\\cap \\biggl \\lbrace \\frac{\\mathbf {a}\\bigl (S_h(x)\\bigr )}{N}\\le \\Vert D^2u\\Vert \\le N\\mathbf {a}\\bigl (S_h(x)\\bigr )\\biggr \\rbrace \\biggr \|,$ that combined with (REF ) yields $\\bigl \|S_h(x)\\cap \\bigl \\lbrace \\Vert D^2u\\Vert \\ge N\\mathbf {a}\\bigl (S_h(x)\\bigr )\\bigr \\rbrace \\bigr \|\\le C\\delta ^{1/2}\\biggl \|S_{\\sigma h}\\cap \\biggl \\lbrace \\frac{\\mathbf {a}\\bigl (S_h(x)\\bigr )}{N}\\le \\Vert D^2u\\Vert \\le N\\mathbf {a}\\bigl (S_h(x)\\bigr )\\biggl \\rbrace \\biggr \|.$ Also, by (REF ) we have $\\text{diam}(S_h(x))\\le Ch^{1/2}\\Vert A\\Vert \\le Ch^{1/2-\\theta }\\le \\hat{C}\\mathbf {a}\\bigl (S_h(x)\\bigr )^{-\\beta },$ where $\\beta :=\\frac{1}{4\\theta }-\\frac{1}{2}.$ Let $M\\gg 1$ to be fixed later, set $\\rho _0:=\\frac{1}{2K},$ and for $m\\ge 1$ we define $\\rho _m$ inductively by $\\rho _m:=\\rho _{m-1}-\\hat{C}M^{-m\\beta },$ where the constants $\\hat{C}, \\beta $ are as those in (REF ).", "Note that, by taking $M$ large enough so that $\\sum _{m=1}^\\infty \\hat{C}M^{-m\\beta }<\\frac{1}{4K},$ we can ensure that $\\rho _m\\ge \\frac{1}{4K}$ for all $m\\ge 1.$ Now, for $k \\ge 0$ we set $D_k:=\\lbrace x\\in B_{\\rho _k}: \\Vert D^2u\\Vert \\ge M^k\\rbrace .$ We shall prove the following lemma.", "Lemma 3.7 $\|D_{k+1}\|\\le N\\delta ^{1/2}\|D_k\|.$ Let $M\\gg N$ to be chosen later, and for any $x\\in D_{k+1}$ choose a section $S_{h_x}(x)$ such that $\\textbf {a}(S_{h_x}(x))=NM^k.$ Such a section always exists because $\\textbf {a}(S_{h})\\approx 1<NM^k$ when $h=h_0$ is a small but fixed universal constant, while $\\textbf {a}(S_{h})\\approx \\Vert D^2u(x)\\Vert \\ge M^{k+1}>NM^k \\qquad \\text{as $h\\rightarrow 0$}$ (the estimate $\\textbf {a}(S_{h})\\approx \\Vert D^2u(x)\\Vert $ follows by a simple Taylor expansion, see for instance [16]).", "Hence, by continuity there exists $h_x\\in (0, h_0)$ such that (REF ) holds.", "Now, by Lemma REF , we can find a finite number of sections $\\lbrace S_{h_{x_i}}(x_i)\\rbrace _{i=1,\\ldots ,m}$ covering $D_{k+1}$ such that $\\lbrace S_{\\sigma h_{x_i}}(x_i)\\rbrace _{i=1,\\ldots , m}$ are disjoint.", "Then, it follows by (REF ) that $\\bigl \|S_{h_i}(x_i)\\cap \\bigl \\lbrace \\Vert D^2u\\Vert \\ge N^2M^k\\bigr \\rbrace \\bigr \|\\le N\\delta ^{1/2}\\bigl \|S_{\\sigma h_i}(x_i)\\cap \\bigl \\lbrace M^k\\le \\Vert D^2u\\Vert \\le N^2M^k\\bigr \\rbrace \\bigr \|.$ Hence, recalling (REF ) and (REF ), we obtain $\|D_{k+1}\|&\\le & \\sum _{i=1}^m \\bigl \|S_{h_i}(x_i)\\cap \\bigl \\lbrace \\Vert D^2u\\Vert \\ge N^2M^k\\bigr \\rbrace \\bigr \|\\\\&\\le &N\\delta ^{1/2}\\sum _{i=1}^m\\bigl \|S_{\\sigma h_i}(x_i)\\cap \\bigl \\lbrace M^k\\le \\Vert D^2u\\Vert \\le N^2M^k\\bigr \\rbrace \\bigr \|\\\\&\\le &N\\delta ^{1/2}\|D_k\|$ provided $M\\ge N^2.$ Thanks to Lemma REF , we have $\|D_k\|\\le (N\\delta ^{1/2})^k\|D_0\|\\le \\frac{1}{M^{k(p+1)}}\|B_{\\frac{1}{2K}}\|$ provided $\\delta \\le \\frac{1}{N^2M^{2(p+1)}}.$ Therefore $\\int _{B_{\\frac{1}{4K}}}\\Vert D^2u\\Vert ^p&=&p\\int _{B_{\\frac{1}{4K}}}t^{p-1}\|B_{\\frac{1}{4K}}\\cap \\lbrace \\Vert D^2u\\Vert \\ge t\\rbrace \|\\\\&\\le &C\\sum _{k=1}^\\infty M^{kp}\|D_k\|\\le C,$ as desired." ], [ "Proof of Theorem ", "By the argument in [12], we only need to establish the following result, which is a strengthened version of Proposition REF for continuous densities.", "Indeed, the lemma shows that the exponent $p$ in the $W^{2,p}$ estimate is independent of the parameter $\\delta .$ This is crucial in showing that the singular set $\\Sigma $ can be chosen independently of $p.$ Lemma 4.1 Let $f, g$ be two continuous densities supported in $B_{1/K}\\subset X_1\\subset B_K$ and $B_{1/K}\\subset Y_1\\subset B_K$ respectively.", "Suppose that $\\Vert f-\\textbf {1}\\Vert _{L^\\infty (X_1)}+\\Vert g-\\textbf {1}\\Vert _{L^\\infty (Y_1)}\\le \\delta ,$ $\\biggl \\Vert u-\\frac{1}{2}\|x\|^2\\biggr \\Vert _{L^\\infty (B_K)}\\le \\delta $ and $\\Vert c(x,y)+x\\cdot y\\Vert _{C^2(B_K\\times B_K})\\le \\delta .$ Then there exists $\\bar{\\delta }>0$ , depending only on $n$ and $K$ , such that $u\\in W^{2,p}(B_{\\frac{1}{2K}})$ for any $p\\ge 1$ provided $\\delta \\le \\bar{\\delta }$ .", "Fix $x_0\\in B_{\\frac{1}{2K}},$ and without loss of generality assume $x_0=0,$ $T_u(x_0)=0$ , and $u(x_0)=0$ .", "For small $h,$ similarly to the proof of Lemma REF , there exists an affine transformation $A$ with $\\det A=1, \\Vert A\\Vert , \\Vert A^{-1}\\Vert \\le h^{-\\theta },$ such that (REF ), (REF ), and (REF ) hold, where $\\theta $ can be as small as we want provided $\\delta $ is sufficiently small.", "Also, we may assume (REF ) holds.", "Given a set $E$ , let $[E]$ denote its convex hull.", "By [10] we have that ${\\rm dist}(S_h, [S_h])\\le Ch^{1-6\\theta }.$ Also, by $C^{1,\\alpha }$ regularity of $u$ (hence, $C^{0,\\alpha }$ regularity of $T_u$ ), we have ${\\rm dist}\\bigl (T_u(S_h), T_u([S_h])\\bigr )\\le Ch^{(1-6\\theta )\\alpha }.$ Perform the transformations $u(x)\\mapsto \\frac{1}{h}u(\\sqrt{h}A^{-1}x):=u_1(x);$ $c(x, y) \\mapsto \\frac{1}{h}c(\\sqrt{h}A^{-1}x, \\sqrt{t}A^{\\prime }y):=c_1(x,y);$ $f(x) \\mapsto f_1(x):=f(\\sqrt{h}A^{-1}x),\\ \\qquad g(y)\\mapsto g_1(y):=g(\\sqrt{h}A^{\\prime }y);$ $S_h\\mapsto \\tilde{S}:=\\frac{1}{\\sqrt{h}}A (S_h).$ Also, set $\\mathcal {C}_1:=[\\tilde{S}],$ $\\mathcal {C}_2:=T_{u_1}([\\tilde{S}])$ , $\\bar{f}:=f_1\\textbf {1}_{\\mathcal {C}_1},$ and $\\bar{g}:=g_1\\textbf {1}_{\\mathcal {C}_2}.$ By (REF ), (REF ), (REF ), (REF ) we have $B_{\\frac{1}{4}}\\subset \\mathcal {C}_1\\subset B_{4};$ $B_{\\frac{1}{4}}\\subset \\mathcal {C}_2\\subset B_{4};$ $\\Vert \\bar{f}-\\textbf {1}_{\\mathcal {C}_1}\\Vert _{L^\\infty (B_{4})}=o(1),\\quad \\Vert \\bar{g}-\\textbf {1}_{\\mathcal {C}_2}\\Vert _{L^\\infty (B_{4})}=o(1) \\rightarrow 0\\ \\ \\ \\text{as}\\ h\\rightarrow 0.$ It is also easy to check that $\\Vert c_1+x\\cdot y\\Vert _{C^2(B_{4}\\times B_{4})}=o(1) \\rightarrow 0\\ \\ \\ \\text{as}\\ h \\rightarrow 0,$ Since $\\mathcal {C}_1$ is convex, we can apply Proposition REF (switch the role of $x$ and $y$ ) to deduce that, given any $p \\ge 1$ , we can choose $h$ small enough so that $u^c_1$ , the $c$ -transform of $u_1$ , belongs to $W^{2,p}(B_{\\frac{1}{8}})$ provided $h$ is sufficiently small.", "By a symmetric argument (or using that $D^2 u$ and $D^2 u^c$ are related), one gets that, given any $p\\ge 1,$ $u_1\\in W^{2,p}(B_{\\frac{1}{8} })$ provided $h$ is sufficiently small.", "Rescaling back to $u$ this proves that, given $p\\ge 1,$ $u\\in W^{2,p}(B_r)$ provided $r$ is small enough (the smallness depending on $h$ ).", "Thanks to this fact, Lemma REF follows from a standard covering argument.", "Theorem REF is an easy consequence of Lemma REF , following the argument in [12].", "Corollary REF follows by the same reasoning as the proof of [12]." ], [ "Proof of Theorem ", "Since interior $W^{2,p}$ estimates follows from Lemma REF and [11], we focus on the estimate near the boundary.", "Under the assumptions of Theorem REF , it is proved in [9] that, for any $\\alpha <1$ , there exists $\\bar{\\delta }>0$ such that $u\\in C^{1,\\alpha }(\\bar{X})$ provided $\\delta \\le \\bar{\\delta }$ .", "Let $\\bar{h}(x):=\\max \\lbrace h>0: S_h(x)\\subset X\\rbrace ,$ and set $S_x:=S_{\\bar{h}(x)}(x)$ .", "As in the proof of Lemma REF , there exists an affine transformation $A$ with $\\det A=1, \\Vert A\\Vert , \\Vert A^{-1}\\Vert \\le \\bar{h}(x)^{-\\theta },$ such that $B_{\\frac{1}{3}\\sqrt{\\bar{h}(x)}} \\subset A(S_x)\\subset B_{3\\sqrt{\\bar{h}(x)}}.$ Hence, since $\\Vert A\\Vert , \\Vert A^{-1}\\Vert \\le \\bar{h}(x)^{-\\theta }$ , it follows by (REF ) and the definition of $\\bar{h}(x)$ that $\\text{dist}(x, \\partial X)\\le C\\bar{h}(x)^{\\frac{1}{2}-\\theta },$ which proves that $S_{x}\\subset X_{C\\bar{h}(x)^{\\frac{1}{2}-\\theta }}:=\\left\\lbrace z \\in X: \\text{dist}(z, \\partial X)\\le C\\bar{h}(x)^{\\frac{1}{2}-\\theta }\\right\\rbrace .$ Fix $h_0>0$ small but universal.", "Similarly to the proof of Lemma 3.1, we can find a Vitali covering of $X_{h_0},$ denoted by $\\lbrace S_{\\bar{h}(x_i)}(x_i)\\rbrace ,$ such that the sections $\\lbrace S_{\\sigma \\bar{h}(x_i)}(x_i)\\rbrace $ are disjoint.", "Now, fix $x_0\\in X_{h_0}$ a point close to $\\partial X.$ Without loss of generality we may assume $x_0=0,$ $T_u(x_0)=0,$ and $u(x_0)=0$ .", "Consider the section $S_{\\bar{h}}:=S(0, 0, u, \\bar{h}(0)).$ As in the proof Lemma REF , we perform the transformations (REF ), (REF ), (REF ), and (REF ), and we set $\\mathcal {C}_1:=[\\tilde{S}]$ , $\\mathcal {C}_2:=T_{u_1}([\\tilde{S}]),$ $\\bar{f}:=f_1\\textbf {1}_{\\mathcal {C}_1},$ and $\\bar{g}:=g_1\\textbf {1}_{\\mathcal {C}_2},$ so that $B_{\\frac{1}{4}}\\subset \\mathcal {C}_1\\subset B_{4};$ $B_{\\frac{1}{4}}\\subset \\mathcal {C}_2\\subset B_{4};$ $\\Vert \\bar{f}-\\textbf {1}_{\\mathcal {C}_1}\\Vert _{L^\\infty (B_{4})}=o(1),\\quad \\Vert \\bar{g}-\\textbf {1}_{\\mathcal {C}_2}\\Vert _{L^\\infty (B_{4})}=o(1) \\rightarrow 0\\ \\ \\ \\text{as}\\ \\bar{h}, \\delta \\rightarrow 0;$ $\\Vert c_1+x\\cdot y\\Vert _{C^2(B_{4}\\times B_{4})}=o(1) \\rightarrow 0\\ \\ \\ \\text{as}\\ \\bar{h}, \\delta \\rightarrow 0.$ Note that, by (REF ), $u_1$ is arbitrarily close to the function $\\frac{1}{2}\|x\|^2$ .", "Let $v$ be the convex function solving $(\\nabla v)_\\sharp \\textbf {1}_{\\mathcal {C}_1}=\\textbf {1}_{\\rho \\mathcal {C}_2}$ with $v(0)=u(0)$ and $\\rho :=\\left(\\frac{\|\\mathcal {C}_1\|}{\|\\mathcal {C}_2\|}\\right)^{1/n}.$ By a compactness argument we have that $\\Vert u_1-v\\Vert _{L^\\infty (B_{\\frac{1}{4}})}\\le \\omega (\\delta ),$ where $\\omega :\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ satisfies $\\omega (r)\\rightarrow 0$ as $r\\rightarrow 0.$ This implies that also $v$ is uniformly close to the function $\\frac{1}{2}\|x\|^2$ inside $B_{\\frac{1}{4}}$ , hence [4] yields that $\\Vert v\\Vert _{C^3(B_{\\frac{1}{5}})}\\le C$ for some universal constant $C$ , and that $v$ is uniformly convex in $B_{\\frac{1}{5}}$ .", "Thus, if we set $S_t^1:=S(0,0,u_1,t),$ we can apply Proposition REF to deduce that $\\Vert u_1\\Vert _{W^{2,p}(S^{u_1}_t)}\\le C$ for some universal constants $t, C,$ .", "Rescaling back to $u,$ this proves that $\\int _{S_{t\\bar{h}(x_0)}} \\Vert D^2u\\Vert ^p\\le C\\bar{h}(x_0)^{-2p\\theta }\|S_{\\sigma \\bar{h}(x_0)}\|.$ Now, consider the family of sections $\\mathcal {F}_k:=\\lbrace S_{\\bar{h}(x_i)}(x_i):h_02^{-k-1}\\le \\bar{h}(x_i)\\le h_02^{-k}\\rbrace .$ Then, since $\|X_{r}\|\\approx r$ for $r$ small and the sections $\\lbrace S_{\\sigma \\bar{h}(x_i)}(x_i)\\rbrace $ are disjoint, it follows by (REF ) and (REF ) that $\\sum _{S_{\\bar{h}(x_i)}(x_i)\\in \\mathcal {F}_k}\\int _{S_{\\bar{h}(x_i)}(x_i)}\\Vert D^2u\\Vert ^p&\\le & C\\sum _{S_{\\bar{h}(x_i)}(x_i)\\in \\mathcal {F}_k}\\bar{h}(x_i)^{-2p\\theta }\|S_{\\sigma \\bar{h}(x_i)}\|\\\\&\\le & C2^{2kp\\theta } \|X_{C(h_02^{-k})^{\\frac{1}{2}-\\theta }}\|\\\\&\\le & C2^{2kp\\theta }(h_02^{-k})^{\\frac{1}{2}-\\theta }\\\\&\\le & C2^{-k(\\frac{1}{2}-3p\\theta )}$ Choosing $\\theta $ small enough so that $3p\\theta \\le \\frac{1}{4}$ , we can sum the above estimate with respect to $k$ to get $\\int _{X_{h_0}}\\Vert D^2u\\Vert ^p \\le C.$ Since $\\int _{X\\setminus X_{h_0}}\\Vert D^2u\\Vert ^p \\le C$ by interior regularity, this concludes the proof.", "$\\Box $" ] ]	1606.05173
[ [ "Origin and evolution of two-component debris discs and an application to\n the q$^1$ Eridani system" ], [ "Abstract Many debris discs reveal a two-component structure, with an outer Kuiper-belt analogue and a warm inner component whose origin is still a matter of debate.", "One possibility is that warm emission stems from an \"asteroid belt\" closer in to the star.", "We consider a scenario in which a set of giant planets is formed in an initially extended planetesimal disc.", "These planets carve a broad gap around their orbits, splitting up the disc into the outer and the inner belts.", "After the gas dispersal, both belts undergo collisional evolution in a steady-state regime.", "This scenario is explored with detailed collisional simulations involving realistic physics to describe a long-term collisional depletion of the two-component disc.", "We find that the inner disc may be able to retain larger amounts of material at older ages than thought before on the basis of simplified analytic models.", "We show that the proposed scenario is consistent with a suite of thermal emission and scattered light observational data for a bright two-temperature debris disc around a nearby solar-type star q$^1$ Eridani.", "This implies a Solar System-like architecture of the system, with an outer massive \"Kuiper belt\", an inner \"asteroid belt\", and a few Neptune- to Jupiter-mass planets in between." ], [ "Introduction", "Debris discs, dusty belts of planetesimals around stars, are natural by-products of planet formation.", "While far-infrared observations of the past decades typically revealed cold, distant dust stemming from Kuiper belt analogues, recent observations and analyses demonstrate that many discs possess a rich radial structure.", "There are systems such as Fomalhaut and Vega where dust appears to be present all the way through from more than a hundred au to the sublimation zone at a few stellar radii , .", "The presence of material in the cavities of the main, cold discs seems to be a rule rather than an exception.", "Indeed, about two-thirds of debris disc systems around late-type stars may exhibit an additional warm component , [1], , , .", "The nature of that warm component remains a matter of debate.", "One possibility is to attribute the warm dust emission to an inner planetesimal belt, i.e.", "to an asteroid belt analogue.", "By analogy with the Solar System with its Kuiper belt and asteroid belt, such a two-component structure could be created by a set of giant planets.", "If these succeeded to form in the disc, they would carve a hole in the protoplanetary disc by removing neighbouring dust and gas .", "After the gas dispersal, nascent planets would swiftly remove planetesimals from their chaotic zones.", "At the same time, these planets would dynamically excite planetesimals in the zones bracketing the planetary region, preventing their further growth to full-size planets.", "All this would generate a broad gap in the planetary region, splitting up the disc into two distinct debris belts.", "So far, the presumed inner belt may have been marginally resolved for two systems, $\\varepsilon $ Eri and HD 107146 .", "Instead of being locally produced in an “asteroid belt”, warm dust can be transported inward from outer production zones by Poynting-Robertson and stellar wind drag.", "For low-density systems where collisional timescales are longer than the one for transport, the inner disc region is filled by dust, leading to a nearly uniform density profile , .", "This scenario is able to explain the observed emission of discs around late-type stars such as $\\varepsilon $ Eri (K2 V), HIP 17439 (K2 V), and AU Mic (M1 V) as shown previously by means of collisional modelling , , .", "However, this may not be a viable alternative for discs around earlier-type stars where no strong stellar winds are expected.", "In this paper, we consider the debris disc around the nearby F-star q$^\\mathrm {1}$ Eridani.", "Among the discs observed with Spitzer and Herschel, the q$^\\mathrm {1}$ Eri system is outstanding for having a strong infrared excess with a fractional luminosity (infrared luminosity of the disc divided by the stellar luminosity) of about $(2-3)\\times 10^{-4}$ , .", "The disc has been spatially resolved in the far infrared , and in the scattered light (K. Stapelfeldt, priv. comm.).", "The images reveal a bright disc with a radius of $\\sim 100$ au.", "The mid-infrared Spitzer/IRS spectrum between 20 and 30 clearly hints at the presence of an additional warm debris component, which is unresolved in the images.", "The fractional luminosity of the warm component is $\\sim 10^{-4}$, which is only a factor of several lower than that of the main disc.", "Attempts to reproduce the data with a two-component model require the warm dust to be located at several au from the star , .", "Apart from the two-component disc, a Jupiter-mass planet with a semi-major axis of 2 au and an eccentricity of 0.15 has been detected by radial velocity measurements [4], .", "The region inside the planetary orbit seems to be largely free of dust as there is no detection of near-infrared excess, associated with the presence of hot exozodiacal dust .", "Explaining the architecture of the q$^\\mathrm {1}$ Eri disc is a challenge.", "Since q$^\\mathrm {1}$ Eri is an F star, we do not expect strong stellar winds, and the Poynting-Robertson drag alone would be inefficient in delivering an amount of material sufficient to explain the warm dust emission.", "This might reinforce the planetary scenario described above, in which the system would be reminiscent of the Solar System's two-belt architecture with a set of invisible planets between the belts.", "However, this explanation is questionable, too.", "It is well known that the fractional luminosity of a debris belt collisionally evolving in a steady-state regime cannot exceed a certain value for a given system's age , .", "That limit is more stringent for older systems and for belts closer to the star where the collisional decay is faster.", "Assuming plausible parameters, the model by suggests that the warm dust in the q$^\\mathrm {1}$ Eri disc may be indeed too bright, and the system too old, to be compatible with a steady-state collisional cascade in an “asteroid belt”.", "If true, this would probably leave us with a possibility that the observed warm dust is an aftermath of a recent major collision between two big planetesimals.", "However, this comes with the caveat that such events in gigayear-old systems may not be likely either.", "In this paper, we show that the Solar System-like planetary scenario may work, contrary to what simplified models of the long-term collisional decay may suggest.", "We present a model to explain the formation of both components of the q$^\\mathrm {1}$ Eri disc self-consistently.", "To this end, we simulate the collisional evolution of an initially extended planetesimal distribution, assumed to be typical for the end stage of a protoplanetary disc, and let it evolve over the age of the q$^\\mathrm {1}$ Eri system.", "Our simulations predict the radial and size distribution of the produced dust as a function of time, from which thermal emission properties are calculated to compare with observations.", "The goal is to see whether a sufficient amount of material can survive in the inner region of the system against collisional depletion to reproduce the available observational data.", "Section presents the long-term collisional evolution of fiducial debris discs.", "In Sect.", ", we give an overview of the thermal emission data, obtained for the q$^\\mathrm {1}$ Eri system, and apply the collisional modelling to the q$^\\mathrm {1}$ Eri disc.", "Section contains a discussion and Sect.", "lists our conclusions.", "We perform the modelling with the ACE code , , , which simulates the collisional evolution of debris discs.", "Starting from an assumed initial distribution of planetesimals, the code calculates the production and loss of material in a collisional cascade by solving the Smoluchowski-Boltzmann kinetic equation.", "Radiative and corpuscular forces acting on the disc particles are also included.", "ACE provides coupled spatial and size distributions of the objects within the disc, ranging from $\\sim 100$ km-sized planetesimals down to submicron-sized dust grains.", "The code uses object masses, pericentric distances, and eccentricities as phase-space variables and assumes the disc to be rotationally symmetric.", "Particle densities are averaged over the inclination, from zero to the disc's semi-opening angle, which is fixed to half the maximum eccentricity of the planetesimals.", "The outcome of a two-particle collision is controlled by the specific energy threshold for disruption and dispersal.", "We use the specific disruption energy, $Q_\\mathrm {D}^\\ast $ , as expressed in equation (7) of , which is a sum of two power laws for the particle size ranges where $Q_\\mathrm {D}^\\ast $ is dominated by tensile strength and self-gravity.", "Following the values of basalt presented in [2], we assume $Q_\\mathrm {D,s}=Q_\\mathrm {D,g}=5\\times 10^6~\\mathrm {erg}\\,\\mathrm {g}^{-1}$ , $b_\\mathrm {s}=-0.37$ , and $b_\\mathrm {g}=1.38$ .", "If the specific impact energy of a collision exceeds the threshold $Q_\\mathrm {D}^\\ast $ , the target breaks apart and the collision is treated as disruptive.", "Beside disruptive collisions, the code also considers cratering, where the target is cratered only while the projectile is disrupted, and bouncing collisions, where the target stays intact and the projectile is cratered.", "The mass distribution of fragments for each collision is assumed to follow $m^{-1.83}$ , which is the equilibrium value averaged over many collisions .", "Dust transport within the q$^\\mathrm {1}$ Eri disc is assumed to play a minor role.", "Hence, we do not take Poynting-Robertson and stellar wind drag into account in the simulations presented in this paper.", "Figure: Radially extended debris disc at different time instants after thebeginning of the collisional evolution.Thick black lines are ACE results,thin blue lines represent the model of .Top: radial profiles of the surface density (left)and the normal optical depth (right).The blue dots on the straight lines indicate the distances where the collisionallifetime of the largest planetesimals equals the current system age.Beyond these distances the lines turn into the initial profiles.Bottom: size distributions in terms of optical depthsat 4 au (left) and 94 au (right).At 94 au, the lines of the model are the same atall time instants, since the collisional depletion of the largest planetesimalsat that distance has not yet started by 1 Gyr." ], [ "Extended planetesimal disc", "We first consider the long-term collisional evolution of an initially extended planetesimal disc, which is a likely remnant of the planetesimal formation process.", "In this section, we seek to understand the general features of collisional evolution predicted by our simulations.", "Therefore, we generated a representative setup by using a set of standard assumptions that are not yet adjusted to the q$^\\mathrm {1}$ Eri system.", "We assumed a planetesimal distribution between 1 and 100 au around a solar-type star, having a solid surface density of the standard Minimum Mass Solar Nebula (MMSN) model: $\\Sigma =1\\,\\mathrm {M}_\\oplus \\, \\mathrm {au}^{-2} \\, (r/\\mathrm {au})^{-1.5}$ , .", "The disc was filled with planetesimals up to 100 km in radius.", "We set the orbital eccentricities of the planetesimals to a mean value of 0.05, in agreement with values expected for discs around solar-type stars .", "Also, smaller bodies were already present within the disc at the beginning of the simulation.", "The initial size distribution of the bodies followed $s^{-3.7}$ , which is the equilibrium distribution of dust in an infinite collisional cascade for the material parameters in the strength regime given in Sect.", "REF .", "This particular choice does not really matter, since the simulation promptly drives the system to an equilibrium size distribution that does not depend much on the one assumed initially.", "This comes with the caveat that the kilometre-sized and larger planetesimals in the outer disc do not reach a collisional equilibrium by the end of the simulation and may retain their primordial distribution set by their formation process (which is poorly known anyway).", "The smallest particles considered in the simulation had radii of $\\approx 0.5\\,$ that corresponds to the blowout radius, $s_\\mathrm {blow}$ , of silicate grains around a solar-type star.", "Grains smaller than $s_\\mathrm {blow}$ were neglected because they are pushed on unbound orbits through the direct radiation pressure and are expelled from the system on dynamical timescales.", "Top panels in Fig.", "REF depict the evolution of the mass surface density, $\\Sigma $ , and the normal optical depth, $\\tau _\\perp $ , as functions of distance from the star.", "While $\\Sigma $ reflects the distribution of kilometre-sized, massive bodies, $\\tau _\\perp $ traces the distribution of micron-sized dust grains that dominate the cross section.", "This difference is best illustrated by considering the region beyond 100 au.", "In this zone, the disc mass surface density is negligible, while the optical depth shows a halo that contains small grains with radii slightly above $s_\\mathrm {blow}$ .", "These grains are in eccentric orbits with pericentres within the planetesimal zone and apocentres outside.", "Bottom panels in Fig.", "REF depict the size distributions close to the inner (4 au) and outer (94 au) edge of the planetesimal disc.", "The size distributions exhibit a wavy pattern which is more pronounced towards the dust size range (radii $<$ 1 mm), since there are no grains below $s_\\mathrm {blow}$ that could work as destructive projectiles [5], , , .", "Both the radial profiles and the size distributions change with time.", "This evolution is described and discussed in detail below, and we start with the initial stage.", "The top right panel of Figure REF reveals an extremely high initial optical depth ($>10$ at 1 au), in the innermost regions of the disc.", "It arises because ACE takes the initial slope of the $\\Sigma $ profile and populates the disc initially by small, micron-sized grains in accord with the $s^{-3.7}$ power law.", "However, as explained above, that power law tacitly assumes that the system has an equilibrium distribution of an idealised, infinite collisional cascade from the very beginning.", "Obviously, this assumption is not valid for a debris disc right after its birth.", "Instead, the initial distribution of dust in a young debris disc must be the one set by physical processes that operated in the preceding protoplanetary disc by the time of the gas dispersal.", "On any account, ACE gets rid of the “unphysically” overabundant dust particles swiftly and the optical depth drops to the equilibrium values after a few thousand years.", "Already at 1 Myr, the $\\tau _\\perp $ curve in Fig.", "REF is flat, regardless of the profile assumed initially.", "The subsequent evolution of the disc depends on the collisional lifetimes of the disc particles and can be understood as follows.", "Since collisional lifetimes get longer with increasing particle size and distance from the star, modifications of the size distributions go from small to large objects and from inner to outer disc regions.", "The small dust that dominates the optical depth depletes and reaches collisional equilibrium first, larger objects follow.", "As a result, at times shorter than the collisional lifetime of the largest objects at a given distance, $\\tau _\\perp $ decreases faster than $\\Sigma $ does.", "It is this gradual inside-out, bottom-up transition of the size distribution from the initial to the quasi-equilibrium state combined with the initial radial distribution that explains why the radial profiles of $\\tau _\\perp $ are almost flat, even though the $\\Sigma $ profiles are not.", "Once the lifetimes of the biggest objects are reached at a given distance, the local size distribution keeps its shape, while $\\Sigma $ and $\\tau _\\perp $ both deplete with their mutual ratio staying constant.", "After tens of megayears, material is sufficiently eroded to generate positive slopes of both $\\Sigma $ and $\\tau _\\perp $ in the inner regions.", "After 1 Gyr, almost all material is reprocessed and only the largest planetesimals in the outer disc have not yet started to deplete collisionally and follow their primordial size distribution.", "This is seen for objects larger than 10 km at 94 au.", "Figure: Evolution of a two-component disc in comparison with that of an extended one.Thin black lines correspond to the radiallyextended planetesimal model presented in Fig.", ".Thick red lines show two narrow planetesimal belts,between 3–10 au and 70–100 au.For the sake of comparison, the analytic steady-state evolution model of is overplotted in Fig.", "REF .", "Their model was designed to describe the evolution of a narrow debris ring, but can readily be applied to an extended debris disc, assuming it to consist of non-interacting concentric annuli .", "To enable direct comparison, we converted the maximum disc mass $M_\\mathrm {max}$ and the maximum fractional luminosity of the dust $f_\\mathrm {max}$ of to the surface density and the normal optical depth: $\\Sigma (r,t) &= \\frac{M_\\mathrm {max}}{2\\pi \\, r\\, dr}, \\\\\\tau _\\perp (r,t) &= 2\\frac{r}{dr} f_\\mathrm {max}(r,t),$ where $r$ is the radius and $dr$ the width of a planetesimal ring.", "We also calculated the evolution of the size distributions by means of the equations (1), (13), and (14) of .", "The parameters of the model were chosen in such a way that it matches the ACE run as closely as possible.", "Accordingly, all parameters that are shared by the analytic model and the ACE simulation were set to the same values.", "These include the stellar mass and luminosity, the initial disc mass, the eccentricity of the planetesimal orbits, and the diameter $D_\\mathrm {c}$ of the largest planetesimals in the collisional cascade.", "There are two important parameters, however, for which the best choice is less obvious.", "One is the slope $(2-3q)$ of the size distribution, which is not an input parameter of the ACE simulation.", "We assumed $q=11/6$ , or $2-3q = -7/2$ .", "Another parameter is the specific disruption energy $Q_\\mathrm {D}^\\ast $ , which must be independent of size in the model.", "We chose $Q_\\mathrm {D}^\\ast =10,000$ J/kg, which is representative for two different particle populations.", "Indeed, since $Q_\\mathrm {D}^\\ast $ used in ACE has a V-like shape in a log-log scale with a minimum at around 100 m, any constant value above this minimum intersects the two branches in the small and large particle size range.", "The assumed $10,000$ J/kg corresponds to the material strength of planetesimals with a few tens of kilometres in radius as well as of millimetre-sized grains.", "The latter are important sources for the production of yet smaller dust that mainly determines the optical depth.", "Thus, our choice of $Q_\\mathrm {D}^\\ast $ ensures that particles from the lower and upper end of the wide size range are taken into account.", "As can be seen in Fig.", "REF , the ACE simulation predicts a significantly higher mass surface density at later times than the analytic model of .", "This is a consequence of a slower depletion of the planetesimals best visible in the size distributions at 4 au.", "In contrast to the analytic reference, the ACE simulation involves a size-dependent $Q_\\mathrm {D}^\\ast $ .", "The lifetimes of the largest planetesimals are longer, because their $Q_\\mathrm {D}^\\ast $ 's exceed the value of 10,000 J/kg adopted in the analytic model by a factor of several.", "Furthermore, the $Q_\\mathrm {D}^\\ast $ 's of the planetesimals increase with object size as a result of self-gravity.", "The latter effect enhances the abundance of larger planetesimals relative to their disruptive projectiles.", "The slope of the resulting steady-state size distribution of the largest planetesimals flattens with respect to that in the strength regime or the canonical $s^{-3.5}$ .", "Thus, at later times, the ACE simulation retains a reservoir of large planetesimals that is more massive and is able to sustain a given optical depth or dust mass for a longer period of time than the analytic model." ], [ "Two narrow planetesimal discs", "We now consider the evolution of a disc whose radial structure was additionally sculpted by the formation of planets.", "The primary effect of giant planet formation, which we only consider here, is the opening of a planetary gap that splits up the disc.", "Since such a gap is already produced before the debris disc phase, we can take it into account through a suitable choice of the initial density profile.", "Accordingly, we cut out two regions of the extended planetesimal disc presented in the previous section.", "The inner disc was assumed to lie between 3 and 10 au, the outer disc between 70 and 100 au.", "Other initial parameters were the same as explained for the extended disc.", "In particular, the initial surface density of both belts followed the MMSN description.", "The two belts were then evolved with two independent ACE runs.", "Figure REF compares the two-component system with the extended disc.", "At any given age, the surface densities in the two models are very close (Fig.", "REF , top left).", "There are, however, some deviations between the shape of the size distributions in the two models, especially in the outer zone (Fig.", "REF , bottom right).", "These deviations are related to the collisional interaction of different disc regions.", "In the extended disc model, the dust grains with sizes just above the blowout size, produced at a certain distance, are sent by radiation pressure to eccentric orbits, and therefore, enter the outer disc zones where they collide with the local material.", "This causes an overabundance of grains just above the blowout limit, which enhances the depletion of larger grains.", "This effect favours the formation of distinct dips in the size distributions, as seen around 5 at 94 au.", "Since the two-component model simulates the two belts independently, such a collisional interaction is neglected, and so, the dips are absent.", "As a consequence, the planetesimal rings of the two-component model tend to have somewhat higher optical depths than in the extended disc simulation (Fig.", "REF , top right).", "We emphasize that our model treats the evolution of the inner and outer planetesimal belts self-consistently since both originate from a common initial surface density profile.", "Possible interactions with one or more planets within the gap that separates the two components are not directly included.", "However, such interactions can be taken into account through the choice of initial disc parameters.", "For instance, the gap width reflects the number and mass of planets." ], [ "Star", "We took stellar parameters from the Herschel/DUNES final archive .", "The star has the spectral type F8 V, a luminosity of 1.5 L$_\\odot $ , an effective temperature of $6155\\,$ K, a mass of 1.12 M$_\\odot $ , a metallicity of $[\\mathrm {Fe/H}]=-0.04$ , and a surface gravity of $\\log (g)=4.48$ (CGS).", "The star is at a distance of 17.4 pc .", "We adopted the stellar photosphere model from , which has been obtained by an interpolation in the PHOENIX/GAIA model grid [3] with a normalisation to optical and WISE fluxes.", "Age estimations of q$^\\mathrm {1}$ Eri span a wide range, from 0.3 to 4.8 Gyr .", "However, the strength of chromospheric activity and the luminosity in X-rays suggest that the star is $\\gtrsim 1\\,$ Gyr old.", "In this paper, we assume an age of around 1 Gyr." ], [ "Observational data", "Photometric data have been collected from the Herschel/DUNES archivehttp://sdc.cab.inta-csic.es/dunes/jsp/homepage.jsp and the literature.", "From the set of available data, we use the measurements that are at least $1\\sigma $ above the stellar photosphere model, thus indicating the infrared excess (Table REF ).", "A Spitzer/IRS spectrum, reduced by the c2d Legacy Team Pipeline, was also taken from the DUNES archive.", "In addition, we extracted radial surface brightness profiles from the Herschel/PACS images provided in the DUNES archive.", "At 70, 100, and 160 , cuts were performed along the minor and major axes of the observed disc, with position angles of 144 and 54 east of north, respectively.", "The stellar position was assumed to coincide with the intensity-weighted centroid of the disc.", "Profiles of the minor and major axes were obtained by averaging the two sides of the disc.", "Uncertainties in intensity are dominated by differences between opposing branches of both axes, added in quadrature to a minor contribution from background noise that was estimated from an annular aperture.", "The resulting profiles and error bars are depicted in Fig.", "REF .", "Horizontal error bars reflect the bin widths in the underlying images: 1 at 70 and 100 , 2 at 160 .", "Table: Photometry of the q 1 ^\\mathrm {1} Eri disc." ], [ "Two-component model", "As explained in Sect.", ", the q$^1$ Eri system shows strong indications for harbouring a two-component debris disc.", "We performed additional tests to confirm that the presence of a warm component is certain.", "One test was to vary the effective temperature of the star by $\\pm 100$ K, which is the scatter in the literature values , .", "For each stellar temperature, we generated a photospheric model and then fitted two-component disc models to the SED, assuming the dust distributions to follow power laws.", "The warm components we inferred were moderately different.", "However, in all the cases a warm component was clearly needed to explain the data.", "Another test was to overplot the q$^\\mathrm {1}$ Eri system in figure 5 of that presents a $\\chi ^2$ map calculated from fitting a single-temperature blackbody (i.e., a one-component disc) to fiducial two-temperature (i.e., two-component) discs.", "We find q$^\\mathrm {1}$ Eri to lie close to the contour line $\\chi ^2_\\text{red} \\approx 20$ , meaning that this system is highly inconsistent with a one-component structure.", "Finally, we estimated an uncertainty in the normalisation of the photospheric model to the optical and WISE fluxes to be about 2%.", "A two-component fitting similar to that described above, but with the photospheric model made by 2% brighter, also showed that the warm component was needed to reproduce the SED.", "Given these results, we applied the ACE-based two-component model presented in Sect.", "REF to explain the SED and the Herschel surface brightness profiles of the q$^\\mathrm {1}$ Eri system.", "We assumed the stellar properties as described in Sect.", "REF and simulated a set of several two-component models where we varied the following parameters: the location, width, and initial mass (in terms of the MMSN multiples) of the inner and outer components, the eccentricities of the planetesimal orbits, the size of the largest planetesimals, and the chemical composition of the disc particles.", "For the purpose of surface profile modelling, we generated synthetic images from the ACE models at 70, 100, and 160 with a position angle of 54.2 (east of north), which is the mean value seen in the three Herschel/PACS images.", "The synthetic images were then convolved with $\\alpha $ Boo PSFs.", "The PSFs were rotated by 83 counter clockwise to align the maps with the telescope pupil orientation on the sky during the q$^\\mathrm {1}$ Eri observations.", "Fitting the model to the observational data is challenging, because we can only vary the parameters of the parent planetesimals and not the dust distributions.", "The impact of parameter variations on the resulting dust distributions, and thus, on the thermal emission properties is difficult to predict quantitatively.", "Several time-consuming trial-and-error ACE simulations are necessary to change the dust distribution in a desired way.", "Furthermore, different combinations of disc parameters can produce similar SEDs and brightness profiles, and therefore, a unique solution does not exist.", "We aimed at finding one plausible model that agrees with the observations reasonably well.", "We probed various maximum orbital eccentricities of planetesimals (0.03, 0.05, and 0.1) and their largest radii (50 and 100 km).", "Inner and outer edges of the inner planetesimal belt were varied between 3 and 20 au, and those of the outer belt between 65 and 135 au.", "We assumed dust particles to be a mixture of astrosilicate and water ice and chose the volume fraction of ice to be a free parameter.", "Ice fractions of 0, 50, 70, and 90 per cent were explored.", "Figure: Thermal emission of a two-component modelthat we found in satisfying agreement with the observational data (dots).Top: the full SED (solid) along with contributionsfrom the inner (dotted) and outer (dashed-dotted) component.The inset in the upper right corner shows how the model matches the ATCA measurement at 6.8 mm.Bottom: surface brightness profiles across the minor andmajor axes for 70 (blue), 100 (green), and 160 (red).For 70 , contributions of the star (dashed line), inner component (dotted),outer component (dashed-dotted), and the sum of all three (solid) are shown.At 100 and 160 , only the total profilesare depicted since they come almost entirely from the outer component.For better visibility, the 70 and 100 profiles were shifted verticallyby constant values of +4 and +2 mJy/arcsec 2 ^2, respectively.Table: Parameters of the two-component model shown in Fig.", ".We finally achieved the model presented in Fig.", "REF .", "The corresponding parameters are summarised in Table REF .", "The model profiles overpredict the 160 emission in the central region by $\\approx 20$ per cent.", "This can reflect some inaccuracy of the collisional model.", "Also, the modelled SED predicts submm and mm flux densities to be by a factor of 2–3 lower than those measured by LABOCA, SCUBA-2, and ATCA.", "Since the images from SCUBA-2 (Holland et al, in prep.)", "and LABOCA are extended eastwards, the most likely reason for the discrepancy is a background galaxy.", "The flux densities plotted are aperture fluxes.", "The peak fluxes – for instance, $20.1 \\pm 2.7$ mJy/beam of the SCUBA-2 measurement – is more in line with the model prediction.", "Overall, we deem the resulting model satisfactory for the purposes of this paper.", "The initial surface density of both components corresponds to 5.3 times the MMSN model.", "The position of the outer component is consistent with the ring-like disc found through a deconvolution of the Herschel 100 image by .", "They quote a surface brightness maximum at 85 au and a belt width of about 35 to 45 au.", "The location of the inner disc is beyond the orbit of the reported RV planet HD 10647b.", "Although there is a large gap between both components, the overall synthetic surface brightness profiles show no signs of a dip and are close to the observed ones.", "The best disc inclination is 70 (from face on), which is close to 76, the value inferred from the HST scattered light image (K. Stapelfeldt, priv.", "comm.).", "The planetesimal orbits have eccentricities $\\le 0.05$, which implies relatively low impact velocities in the disc and a rather depressed production of small dust.", "However, planets that reside within the disc can push planetesimals to more eccentric, and also inclined, orbits through secular perturbations .", "In particular, this planetary stirring is likely to operate in the inner component, which is close to HD 10647b.", "There is also the possibility that further, yet undiscovered, planets in the gap between both planetesimal belts enhance the eccentricities in the outer component.", "Higher values of the eccentricity are not excluded by our modelling since there are degeneracies between the eccentricity and other model parameters (see Sect. ).", "The dust composition determines the grain temperatures and emission properties.", "During the modelling process, we noticed that the rise of the SED in the mid infrared shows a strong dependence on the ice fraction.", "This allows us to improve the agreement with the characteristic bump (“shoulder”) that is exhibited by the IRS spectrum in the 10 to 30 range.", "Larger fractions of ice reduce the emission of the inner component, whereas the one of the outer component remains nearly unchanged.", "This is due to the fact that the mid-infrared emission mostly comes from small grains (1-10 radius), while the far-infrared emission is produced by larger grains ($>10$ ) with an emissivity that is not strongly dependent on the chemical composition.", "The higher the ice fraction, the steeper the rise of the total SED.", "However, the distinct silicate feature around 18 is still well pronounced and is able to reproduce the bump seen in the IRS spectrum.", "A mixture of 30 per cent silicate and 70 per cent water ice for inner and outer component provides a good overall match to the SED and the surface brightness profiles." ], [ "Discussion", "To verify the scenario considered here, it will be crucial to resolve the proposed inner belt.", "At 0.17-0.57, the inner belt would be difficult to see with Hubble due to the large inner working angle.", "The latest extreme adaptive optics instruments allow us to image much closer to the star.", "For instance, SPHERE/IRDIS on the VLT has an inner working angle of 0.15 in its standard mode, opening up these regions to scattered light observations for the first time.", "We performed a test by generating scattered light radial profiles of the inner belt at 1.6 .", "We compared the profiles with the IRDIS detection limit provided by the SPHERE Exposure Time Calculatorhttps://www.eso.org/observing/etc/.", "The surface brightness along the major axis, $\\sim 0.2$ mJy/arcsec$^2$, is about three orders of magnitude below the detection limit.", "For SPHERE/ZIMPOL with a smaller working angle at 0.55 , the inner belt could be traced along the minor axis where it appears brighter in polarized light, $\\sim 4$ mJy/arcsec$^2$.", "However, even in that case we are still one order of magnitude below the detection limit.", "This indicates a big challenge in resolving the inner belt with present-day observational facilities.", "However, the coming JWST might offer better chances for such a detection.", "The proposed two-component scenario also implies a set of alleged planets between the inner and outer belts.", "The number of planets is unknown, as are their orbits and masses.", "Nevertheless, rough estimates can be made by assuming that the presumed planets are in nearly-circular, nearly-coplanar orbits with a uniform logarithmic spacing of semi-major axes .", "The set of planets should be tightly packed dynamically to ensure that no dust-producing planetesimal belts have survived between the orbits, while the planets themselves should be at least marginally stable against mutual perturbations over the age of the system of $\\sim 1$ Gyr.", "Using equations (3) and (6) of , we estimate that the cavity between $\\sim 10$ au and $\\sim 80$ au could be populated by 4–5 planets of at least Neptune mass, with a semi-major axis ratio between neighbouring planets of $\\sim 1.6$ (i.e., similar to the orbital radius ratio of Earth and Mars in the Solar System).", "The minimum total mass of these planets would amount to $\\sim 0.2$ Jupiter masses.", "There are currently two planetary systems known to have an architecture similar to the one proposed for q$^1$ Eri.", "One is our own Solar System.", "Another is a prominent system around HR 8799, with its four directly imaged massive planets , , an outer massive “Kuiper belt” exterior to and an inner debris belt interior to the planetary orbits , .", "However, there are also fine differences between these cases.", "The q$^1$ Eri system is known to contain a close-in Jupiter-mass radial velocity planet whereas our Solar System harbours four terrestrial planets.", "Around HR 8799, no planets have been discovered so far inside the inner dusty belt.", "It is interesting to note that the known planet of q$^1$ Eri system is orbiting at $\\sim 2$ au, whereas the proposed inner belt must be located farther out, at 3 to 10 au.", "Thus, the belt was likely located outside the snow line when it formed which, assuming an ice sublimation temperature at the protoplanetary phase of 170 K , must have been at $\\sim 3.3$ au from the star.", "This means that the planet could have comfortably formed in the zone interior to the “asteroid belt”.", "No migration through the belt is required here – unlike in the systems such as HD 69830 where the planets and the belt are all located inside the snow line .", "Due to numerous degeneracies, the model found in our study is by far not unique.", "For instance, we cannot constrain tightly a set of gas or icy giants that separates the two components, and consequently, it is unknown to what degree these would stir the components.", "However, we expect that at least the inner component is rather strongly affected by the known nearby giant planet HD 10647b which is on a close-in ($a_\\mathrm {pl}\\approx 2$ au), eccentric orbit ($e_\\mathrm {pl}\\approx 0.2$ ).", "Equation (6) of yields a forced eccentricity of $\\sim 0.1$ that could be imposed by HD 10647b on planetesimals at 5 au.", "Note that this is merely a rough estimate since only one planetary perturber is assumed, but it shows that the inner planetesimal disc may have a distinctly higher level of dynamical excitation than assumed in our model.", "If so, there would be a faster collisional depletion, resulting in a lower amount of material in the inner disc.", "However, the size of the largest planetesimals and their critical fragmentation energy are not well constrained, too.", "If there are larger and harder planetesimals within the disc, the collisional depletion would be slower, which might ensure that the inner system contains sufficient dust after $\\approx 1$ Gyr to explain the observations.", "The collisional timescales can also be prolonged if the inner disc were located farther out and/or had a larger width.", "Furthermore, the slope of the assumed initial density profile of a protoplanetary disc is completely uncertain , , .", "Steeper profiles are preferred for in-situ formation scenarios of close-in giant planets, where the disc mass is more centrally concentrated.", "The chemical composition of the dust is another crucial point.", "In our model, the dust contains 30 per cent silicates and 70 per cent water ice in inner and outer component.", "However, the inner component lies between 3 and 10 au.", "Since, in contrast to protoplanetary discs, the gas pressure in debris discs is negligible, ice sublimates at lower temperatures of $\\sim 100$ K , suggesting the ice line for blackbody grains to lie at $\\sim 10$ au.", "Thus the presence of icy grains in the inner component is questionable.", "Other materials, e.g.", "porous silicate, for the inner component may be a better choice.", "Such a scenario agrees, at least qualitatively, with the compositional model of ." ], [ "Conclusions", "We consider a scenario that might naturally explain the two-component structure observed in many debris discs.", "This includes the formation of giant planets in an initially extended planetesimal disc that is still immersed in gas (protoplanetary phase).", "The nascent planets forming within the disc open the gap around their orbits by scattering the planetesimals away.", "Several planets together can generate a single broad gap and, as a consequence, the initially extended disc splits up into an inner and an outer region.", "After the gas dispersal, both planetesimal discs start to evolve collisionally.", "Even though the dust produced in collisions between planetesimals is spread over large distances by radiative and corpuscular forces, the structure of the underlying planetesimal distribution is that of a two-component disc.", "In this paper, a detailed collisional model involving realistic physics is used to simulate the long-term collisional evolution of a two-component disc around a solar-type star.", "Our conclusions are as follows: From a comparison of the collisional simulations with predictions of a simplified analytic model of , we show that the results are quite different.", "The surface density provided by the collisional simulations reveals much more material close to the star after an evolution period of 1 Gyr.", "The optical depth does not rise as steeply with distance from the star as predicted by the analytic model.", "This means that the discs evolving in a steady-state regime may be able to retain larger amounts of material in the inner region at older ages than thought before.", "That scenario implies the presence of as yet undiscovered planets.", "Since it is the only scenario proposed so far for systems without significant transport (which is the case for stars of earlier spectral types), and since there are many two-component discs around such stars, this suggests the Solar System-like architecture with sets of planets between the Kuiper belts and asteroid belts may be quite common.", "If no planets populate the gap between both components, the only remaining explanation for the formation of a two-component disc will be that planetesimals, for whatever reason, did not form in a certain region of the disc.", "The two-component model is able to reproduce the thermal emission of the q$^1$ Eri disc over a wide wavelength range.", "This is achieved with an inner planetesimal belt between 3 to 10 au and an outer one between 75 to 125 au.", "The initial surface density of both components is approximately five times the Minimum Mass Solar Nebula model.", "The belts contain objects up to 50 km in radius and the planetesimal orbits have a maximum eccentricity of 0.05.", "Silicate grains with 70 per cent ice content are assumed to reproduce the thermal emission properties.", "Due to a number of degeneracies, this is not a unique solution.", "To check the scenario itself and break some of the degeneracies, it would be vital to resolve the inner belt.", "There must also be a chance in the future to detect the expected planets." ], [ "Acknowledgements", "We thank Karl Stapelfeldt for sharing with us the radial brightness profile derived from the scattered light observations of q$^1$ Eri, Nicole Pawellek for helpful discussions on SED fitting, and Mark Wyatt for commenting on the manuscript draft.", "We also thank the referee for insightful comments that helped to improve the paper significantly.", "A.V.K., T.L., F.K., and S.W.", "acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) through projects Kr 2164/13-1, Kr 2164/15-1, Lo 1715/2-1, and Wo 857/15-1.", "M.B.", "acknowledges support from a FONDECYT Postdoctral Fellowship, project no.", "3140479." ] ]	1606.05187
[ [ "Reconstructing a Bounded-Degree Directed Tree Using Path Queries" ], [ "Abstract We present a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$, by asking at most $O(dn\\log^2 n)$ path queries.", "Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target.", "Regarding lower bounds, we show that any randomized algorithm requires at least $\\Omega(n\\log n)$ queries, while any deterministic algorithm requires at least $\\Omega(dn)$ queries.", "Additionally, we present a $O(dn\\log^3 n)$ randomized algorithm for noisy queries, and a $O(dn\\log^2 n)$ randomized algorithm for additive queries on weighted trees." ], [ "Introduction", "Scientists in diverse areas, such as statistics, epidemiology and economics, aim to unveil relationships within variables from collected data.", "This process can be seen as the task of reconstructing a graph (i.e., finding the hidden edges of a graph) by asking queries to an oracle.", "In this graph, each vertex is a variable, and each edge denotes the relationship between two variables.", "For instance, in cancer research, biologists try to discover the causal relationships between genes.", "By providing a specific treatment to a particular gene (origin), biologists can observe whether there is an effect in another gene (target).", "This effect can be either direct (if the two genes are connected with a directed edge) or indirect (if there is a directed path from the origin to the target gene.)", "In the above example, we can think of nature as the oracle that is answering whether there is a directed path from an origin to a target node.", "Arguably, the cost of asking queries is very high in several application domains.", "Thus, we are interested on the reconstruction of graphs that do not require the trivial $n^2$ queries for $n$ nodes (i.e., one query for every possible pair of nodes.)", "Prior work on reconstructing graphs has exclusively focused on undirected graphs.", "In [11], a $\\mathcal {O}(dn\\log n)$ algorithm was provided for recovering undirected trees of $n$ nodes and maximum node degree $d$ , by using queries that return the path length between two given nodes.", "Authors of [5] provide a $\\mathcal {O}(dn)$ algorithm for reconstructing undirected trees of $n$ nodes and maximum node degree $d$ , and for a query that returns the distance from a given node to every other node.", "The results in [3], [2], [4] pertain to queries that answer whether there exists at least one edge between a given set of nodes.", "While [3], [2] focused on matchings and stars, the work of [4] provides a $\\mathcal {O}(m\\log n)$ algorithm for undirected graphs of $n$ nodes and $m$ edges.", "The results in [10], [14] pertain to queries that return the number of edges between a given set of nodes.", "A $\\mathcal {O}(dn)$ algorithm was provided in [10] for undirected graphs of $n$ nodes and maximum node degree $d$ , while a $\\mathcal {O}(m\\log n)$ algorithm was given in [14] for undirected graphs of $n$ nodes and $m$ edges.", "Other works have focused on the recovery of weighted undirected graphs.", "The work of [9] provides a $O(dn\\log n)$ algorithm for recovering weighted undirected trees of $n$ nodes and maximum node degree $d$ , by using queries that return the sum of edge weights on the path between two given nodes.", "The work of [6], [13] pertains to the reconstruction of weighted undirected graphs of $n$ nodes and $m$ edges.", "A $\\mathcal {O}(m\\log n)$ algorithm was provided for a query that gives the sum of edge weights between a given set of nodes.", "The closest work to ours is [12], which provides a $\\mathcal {O}(dn\\log ^2 n)$ randomized algorithm for undirected trees of $n$ nodes and maximum node degree $d$ , and for separator queries.", "A separator query takes three nodes $i$ , $k$ , $j$ as input, and answers whether $k$ is on the undirected path between $i$ and $j$ .", "In contrast, our work pertains to directed rooted trees.", "Furthermore, we use a different type of query which we call path query.", "A path query takes an ordered pair of nodes $i$ , $j$ as input, and answers whether there exists a directed path from $i$ to $j$ .", "We provide a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$ , in $\\mathcal {O}(dn\\log ^2 n)$ time.", "To the best of our knowledge, there is no simple reduction to transform our problem to the problem of [12] or any of the above mentioned literature.", "Our algorithm relies on the divide and conquer approach, the use of even separators [7] and sorting.", "Regarding lower bounds, we show that any randomized algorithm requires at least $\\Omega (n\\log n)$ queries, while any deterministic algorithm requires at least $\\Omega (dn)$ queries.", "We also present a $\\mathcal {O}(dn\\log ^3 n)$ randomized algorithm for a noisy regime, in which the bit that represents the oracle's answer gets flipped with some probability, by an adversary, before it is revealed to the algorithm.", "Furthermore, we present a $\\mathcal {O}(dn\\log ^2 n)$ randomized algorithm for reconstructing weighted trees by using additive queries that return the sum of the edge weights on the directed path between two given nodes.", "We finish the paper by showing some negative results that provide some motivation for our assumptions.", "We show that any deterministic or randomized algorithm requires at least $\\Omega (n^2)$ queries in order to recover more general directed acyclic graphs.", "We also show that any deterministic algorithm requires at least $\\Omega (n^2)$ queries for recovering a family of sparse disconnected graphs, as well as a family of sparse connected graphs." ], [ "Preliminaries", "In this section, we provide several formal definitions which will be useful later for the detailed description of our algorithm.", "For clarity, we also provide some preliminary introduction to the main aspects of our algorithm.", "Let $G=(V,E)$ be a directed acyclic graph with vertex set $V$ and edge set $E$ .", "For clarity, when $G$ is a directed rooted tree, we will use $T$ instead of $G$ .", "In this paper, we assume that $T$ has $n$ nodes, i.e., $\|V\|=n$ .", "Furthermore, we also assume that the node degree is at most $d$ .", "(In a directed acyclic graph, the node degree is the sum of the indegree and the outdegree of the node.)", "Recall that a path in $G$ from node $i$ to node $j$ (both in $V$ ) is a sequence of nodes $i, x_1, x_2, \\dots x_k, j$ such that $\\lbrace (i, x_1), (x_1, x_2), \\dots (x_{k-1}, x_k), (x_k, j)\\rbrace $ is a subset of the edge set $E$ .", "Our algorithm reconstructs a directed rooted tree, by using path queries.", "Next, we formally define path queries.", "Definition 1 Let $G=(V,E)$ be a directed acyclic graph.", "A path query is a function $Q_G : V \\times V \\rightarrow \\lbrace 0,1\\rbrace $ such that $Q_G(i, j) = 1$ if there exists a path in $G$ from $i$ to $j$ , and $Q_G(i, j) = 0$ otherwise.", "Note that the above query only reveals a single bit of information, and it does not provide any information regarding the length of the path, thus making graph reconstruction a nontrivial task.", "In this paper, we assume that the node set $V$ is known, while edge set $E$ is unknown.", "Our main problem is indeed to reconstruct $E$ by using path queries.", "We will use $Q(i,j)$ to denote $Q_T(i,j)$ since for our problem, the directed rooted tree $T$ is fixed (but unknown).", "Figure: (a) A directed rooted tree with multidirectional path of nodes 1,2,3,4,51, 2, 3, 4, 5.Nodes and edges in the multidirectional path are shown in red.", "(b) Skeleton graph of the directed rooted tree on the left, with path of nodes 1,2,3,4,51, 2, 3, 4, 5.Nodes and edges in the path are shown in red.A key step in our algorithm is the recovery of what we call multidirectional paths.", "A multidirectional path consists of the directed edges associated to an undirected path in the skeleton graph (i.e., the graph obtained by replacing each directed edge with an undirected edge in the original graph.)", "Figure REF provides a visual illustration for intuitive understanding.", "Next, we formally define multidirectional paths.", "Definition 2 Let $G=(V,E)$ be a directed acyclic graph.", "A multidirectional path of $G$ between $i$ and $j$ is a sequence of nodes $i, x_1, x_2, \\dots , x_{k-1}, x_k, j$ such that each node in $V$ appears at most once in the sequence, and that there is an edge on either direction between each pair of adjacent nodes in the sequence.", "That is, either $(i, x_1) \\in E$ or $(x_1, i) \\in E$ , either $(x_1, x_2) \\in E$ or $(x_2, x_1) \\in E$ , $\\dots $ either $(x_{k-1}, x_k) \\in E$ or $(x_k, x_{k-1}) \\in E$ , and either $(x_k, j) \\in E$ or $(j, x_k) \\in E$ .", "Next, we show that for directed rooted trees, a multidirectional path between any two arbitrary nodes always exists and is unique.", "More importantly, we show that a multidirectional path is either a directed path, or two directed paths that share the same origin (i.e., the lowest common ancestor.)", "Later, we leverage this property for recovering multidirectional paths.", "Lemma 1 Let $T=(V,E)$ be a directed rooted tree.", "Given any two arbitrary nodes $i$ and $j$ , a multidirectional path of $T$ between $i$ and $j$ always exists and is unique.", "Furthermore, a multidirectional path of $T$ between $i$ and $j$ , is either a path from $i$ to $j$ , or a path from $j$ to $i$ , or two paths (one from $k$ to $i$ , and one from $k$ to $j$ , for some $k \\in V - \\lbrace i,j\\rbrace $ .)", "In this case, node $k$ is the lowest common ancestor of $i$ and $j$ .", "(See Appendix for detailed proofs.)", "Figure: A directed rooted tree with multidirectional path of nodes 1,2,3,4,51, 2, 3, 4, 5.Nodes and edges in the multidirectional path are shown in red.The edge (2,3)(2,3) in the multidirectional path is an even separator.Our algorithm will recursively recover each of the two generated subtrees (shown as blue boxes.", ")Our algorithm relies on the divide and conquer approach.", "In order to apply the above approach in our problem, it is important to introduce the concepts of even separators and bags.", "Next, we introduce even separators.", "Definition 3 Let $T=(V,E)$ be a directed rooted tree of bounded degree $d$ and let $n=\|V\|$ .", "An even separator of $T$ is an edge $e \\in E$ that when removed from $T$ , divides $T$ into two subtrees $T_1$ and $T_2$ , where each of the subtrees have a number of nodes between $n/d$ and $(d-1)n/d$ .", "The existence of even separators is pivotal for using divide and conquer in our problem.", "Corollary 2.3 in [7] shows that if a graph is a bounded-degree directed tree, then an even separator exists.", "For our graph reconstruction problem, once the even separator is identified, we cut the tree through the even separator.", "This operation splits the tree into two subtrees.", "We then recursively call the algorithm for both subtrees.", "We illustrate this in Figure REF .", "While even separators exist [7], it remains to know whether they can be efficiently found.", "We show later that (on average) there is an even separator in the multidirectional path between two nodes chosen independently and uniformly at random (See Theorem REF .)", "Figure: Four different directed rooted trees, and their bags with respect to the multidirectional path of nodes 1,2,3,4,51, 2, 3, 4, 5.Nodes and edges in the multidirectional path are shown in red.The lowest common ancestor of nodes 1 and 5 in the multidirectional path is shown in darker red.A tree root is shown as a thick circle.Bags are shown as blue boxes.Note that a multidirectional path can be a path (a,c) or two paths (b,d).The root can be in the multidirectional path (a,b).Otherwise, the root can be an ancestor of a node in the multidirectional path (c,d).In what follows, we formally define bags, which are also important for the divide and conquer approach taken here.", "Definition 4 Let $T=(V,E)$ be a directed rooted tree, and $S$ be the set of edges in a multidirectional path of $T$ .", "Define $T_S=(V, E - S)$ as the subgraph of $T$ after we remove all edges in $S$ .", "A bag with respect to a node $i$ in a multidirectional path with edges $S$ , is a subset of nodes in $V$ that contains $i$ and all the nodes that are reachable from $i$ in the (undirected) skeleton graph of $T_S$ .", "Intuitively speaking, we can think of edges in a tree as “ropes”.", "If we “nail” all nodes of a multidirectional path into the “wall”, then all other nodes will “hang” from one of the nodes in the path.", "Nodes that hang from the same particular node belong to the same bag.", "We include a visual example in Figure REF .", "In our algorithm, we recover exactly all the directed edges in a multidirectional path.", "Bags are used to count the number of nodes associated to each node in the multidirectional path (without the need to recover all the directed edges.)", "For each edge in the multidirectional path, one can then count the number of nodes on the two subtrees that would be generated if we were to cut the tree through the given edge.", "This process is used for identifying even separators.", "Finally, our algorithm also performs sorting of nodes with a properly defined order relation, which is used for instance in the recovery of the directed edges in multidirectional paths.", "Definition 5 Define the order relation of two nodes $i$ and $j$ as follows.", "If $Q(i, j) = 1$ we say that $i$ is “less than” $j$ , and “greater than” otherwise." ], [ "Algorithm", "In this section, we present our randomized algorithm and analyze its time complexity.", "Our algorithm is similar in spirit to [12] which applies to undirected trees and separator queries.", "In this paper, we focus on directed rooted trees and path queries.", "We remind the reader that path queries only reveal a single bit of information, and they do not provide any information regarding the length of the path.", "(We discuss noisy and additive extensions in Section .)", "Figure: Main algorithm (in red), subroutines, and their time complexity.In what follows, we explain our divide and conquer approach in our main Algorithm .", "A high-level overview is shown in Figure REF .", "(See Appendix for detailed algorithms.)", "First, we randomly pick two nodes and recover the sequence of nodes in the multidirectional path between those two randomly chosen nodes (Algorithm .)", "Then, we divide the rest of the nodes (not in the multidirectional path) into bags defined by the multidirectional path (Algorithm .)", "Later, we count the number of nodes on each bag, and determine if there exists an even separator in the multidirectional path.", "We repeat the whole procedure (i.e., from choosing a new pair of random nodes) until we find an even separator.", "Finally, we cut the tree through the even separator, thus effectively splitting the tree into two subtrees (Algorithm .)", "We then recursively call our Algorithm for both subtrees (until the input tree contains a single node.)", "On a more technical side, reconstructing a multidirectional path that consists of two paths that share the same origin, is more involved than reconstructing single paths.", "The former requires finding the lowest common ancestor of the two randomly chosen nodes (Algorithm ) as well as finding the root of the tree (Algorithm .)", "In our main Algorithm , finding out the bag of a particular node with respect to a directed path is relatively easier than with respect to a multidirectional path.", "Note that by Lemma REF , a multidirectional path is either a directed path, or two directed paths that share the same origin (i.e., the lowest common ancestor.)", "In Algorithm , we simplify the bag assignment task, by splitting a multidirected path (that is not a single path) into its two constituent directed paths.", "The algorithm first recovers the lowest common ancestor, and then breaks the multidirectional path into two paths.", "For instance, the multidirectional path $1,2,3,4,5$ in Figure REF (d) with lowest common ancestor 3 would become two paths $3,2,1$ and $3,4,5$ .", "Next, we explain each subroutine used in our main Algorithm .", "The first key step is to recover the multidirectional path between two randomly chosen nodes.", "Algorithm recovers the sequence of nodes in the multidirectional path between any two nodes.", "The idea behind Algorithm is as follows.", "Let $i,j \\in V$ be two arbitrary nodes in the directed rooted tree $T=(V,E)$ .", "Recall that some multidirectional paths are a single directed path.", "We can easily detect this case by asking whether there is a path from $i$ to $j$ (i.e., whether $Q(i,j)=1$ ), or whether there is a path from $j$ to $i$ (i.e., whether $Q(j,i)=1$ .)", "Without loss of generalization, assume there is a path in the directed rooted tree $T$ from $i$ to $j$ .", "(We could similarly assume that there is a path from $j$ to $i$ .)", "We can find out the set of all nodes $\\lbrace x_1, x_2, \\dots x_k\\rbrace $ on the path from $i$ to $j$ , by using path queries.", "For all nodes $k \\in V - \\lbrace i,j\\rbrace $ , we ask the oracle about $Q(i, k)$ and $Q(k, j)$ .", "Note that $k$ is on the path from $i$ to $j$ , if and only if $Q(i, k)=Q(k, j)=1$ .", "After finding out the set of all nodes $\\lbrace x_1, x_2, \\dots x_k\\rbrace $ on the path from $i$ to $j$ , it remains to sort the nodes in order to obtain the correct sequence, thus recovering the path.", "We sort the list of nodes $\\lbrace i, x_1, x_2, \\dots x_k, j\\rbrace $ by using the order relation given in Definition REF .", "Some multidirectional paths consist of two directed paths that share the same origin $m$ .", "In this case, Algorithm first recovers the lowest common ancestor $m$ and then reconstructs two directed paths: one from $m$ to $i$ , and one from $m$ to $j$ , by following the approach explained before.", "In Algorithm , reconstructing a multidirectional path that consists of two paths requires finding the lowest common ancestor of the two randomly chosen nodes.", "Our Algorithm finds the lowest common ancestor of a multidirectional path between any two arbitrary nodes.", "Algorithm works as follows.", "Let $i,j \\in V$ be two arbitrary nodes of the directed rooted tree $T=(V,E)$ .", "First, we recover the directed path from the root to $i$ .", "We assume that the order of the nodes in the above path follow the order relation given in Definition REF .", "Thus, the tree root is the first element on such path.", "Then, we iterate through all nodes in the path, in order to find the last ancestor of $j$ in the path from the root.", "This last ancestor is indeed the lowest common ancestor of $i$ and $j$ .", "In Algorithm , in order to find the lowest common ancestor, one has to find the root of the directed tree.", "Our Algorithm identifies the path from the root to a given arbitrary node.", "The inner workings of Algorithm are as follows.", "Let $i \\in V$ be an arbitrary node in the directed rooted tree $T = (V,E)$ .", "For each node $j \\in V - \\lbrace i\\rbrace $ , we ask the oracle about $Q(j,i)$ .", "If $Q(j,i)=1$ then there is a path from $j$ to $i$ , and therefore we add node $j$ to the list of nodes that reach $i$ .", "In order to recover the directed path from the root to $i$ , we sort the aforementioned list of nodes, by using the order relation given in Definition REF .", "That is, the first element on the sorted list is the tree root.", "The second key step in our main Algorithm is to divide the nodes (which are not in the multidirectional path) into bags defined by the multidirectional path.", "As argued before, if the multidirectional path consists of two directed paths, then Algorithm breaks the multidirectional path into its two constituent directed paths, by using the least common ancestor.", "Thus, the bag assigment task needs only to consider directed paths, as we do in Algorithm .", "Here we give an intuitive explanation of the bag assignment task in Algorithm .", "For instance, consider finding out the bag of node 10 in the directed path $1,2,3,4,5$ in Figure REF (a).", "We see that nodes 1, 2 and 3 are all ancestors of node 10 (i.e., $Q(1,10)=Q(2,10)=Q(3,10)=1$ .)", "We also see that nodes 4 and 5 are not ancestors of node 10 (i.e., $Q(4,10)=Q(5,10)=0$ .)", "Note that node 10 belongs to the bag of node 3.", "The above suggests that the task of finding out the bag of node 10 can be done by searching for a node $i$ for which $Q(i,10)=1$ and $Q(j,10)=0$ where $(i,j)$ is an edge in the path.", "This can be efficiently done by performing binary search.", "There are two exceptions to the above rule: when queries for all nodes in the path return 1, and when queries for all nodes in the path return 0.", "As an example for the first case (when all queries return 1), consider finding out the bag of node 11 in the directed path $1,2,3,4,5$ in Figure REF (a).", "We see that nodes 1, 2, 3, 4 and 5 are all ancestors of node 11 (i.e., $Q(1,11)=Q(2,11)=Q(3,11)=Q(4,11)=Q(5,11)=1$ .)", "In this case, we assign node 11 to the bag of the last node in the path, i.e., 5.", "As an example for the second case (when all queries return 0), consider finding out the bag of node 10 in the directed path $3,4,5$ in Figure REF (d).", "We see that neither nodes 3, 4 or 5 are ancestors of node 10 (i.e., $Q(3,10)=Q(4,10)=Q(5,10)=0$ .)", "In this case, we assign node 10 to the bag of the lowest common ancestor, i.e., 3.", "Fortunately, all the cases analyzed above are naturally handled by binary search.", "The final step in our main Algorithm is to cut the tree through the even separator, which splits the tree into two subtrees.", "Our Algorithm splits a directed rooted tree into two subtrees, by cutting the original tree through any arbitrary edge.", "The idea behind Algorithm is as follows.", "Let $e = (i,j) \\in E$ be an arbitrary edge in the directed rooted tree $T=(V,E)$ .", "Let $T_1$ and $T_2$ be two subtrees that result from removing $e$ from $T$ .", "Note that every node must belong to either $T_1$ or $T_2$ .", "Without loss of generality, let $i \\in T_1$ and $j \\in T_2$ .", "Since $T$ is a directed rooted tree, $j$ has one parent in $T$ , which is indeed $i$ .", "Removing $(i, j)$ from $T$ makes $j$ have no parents in $T_2$ .", "Therefore, $j$ is the root of $T_2$ .", "For all nodes $k \\in V - \\lbrace j\\rbrace $ , we ask the oracle about $Q(j, k)$ .", "If $Q(j, k) = 1$ , then $k$ belongs to $T_2$ , otherwise $k$ belongs to $T_1$ .", "We finish the section by analyzing the time complexity of our randomized algorithm.", "Theorem 1 Algorithm takes $\\mathcal {O}(dn \\log ^2 n)$ expected time, in order to reconstruct a directed rooted tree of $n$ nodes and node degree at most $d$ .", "Furthermore, for a fixed probability of error $\\delta \\in (0,1)$ , Algorithm takes at most $\\mathcal {O}(\\frac{1}{\\delta } \\, dn \\log ^2 n)$ time, with probability at least $1-\\delta $ .", "(See Appendix for detailed proofs.)" ], [ "Lower Bound and Extensions", "In this section, we study lower bounds for reconstructing directed rooted trees from path queries.", "We also extend our original algorithm for the case of noisy queries, as well as the reconstruction of weighted trees from additive queries.", "Finally, we provide negative results for directed acyclic graphs that provide some motivation for our assumptions." ], [ "Lower Bounds", "Here, we present lower bounds for reconstructing directed rooted trees from path queries.", "Theorem 2 In order to reconstruct a directed rooted tree of $n$ nodes and node degree at most $d$ , any randomized algorithm requires at least $\\Omega ((1-\\delta ) \\, n\\log n)$ queries, otherwise it would fail with probability at least $\\delta $ .", "Theorem 3 In order to reconstruct a directed rooted tree of $n$ nodes and node degree at most $d$ , any deterministic algorithm requires at least $\\Omega (dn)$ queries.", "Given the above results and Theorem REF , our algorithm is only a factor of $\\mathcal {O}(d\\log n)$ from the lower bound for any randomized algorithm, and a factor of $\\mathcal {O}(\\log ^2 n)$ from the lower bound for any deterministic algorithm.", "Both of these factors are small when compared to the time complexity of our algorithm, which is $\\mathcal {O}(dn\\log ^2 n)$ ." ], [ "Noisy Queries", "Here, we analyze a noisy regime, in which the bit that represents the oracle's answer gets flipped with some probability, by an adversary, before it is revealed to the algorithm.", "Next, we formally define noisy queries.", "Definition 6 Let $G=(V,E)$ be a directed acyclic graph, and let $Q_G$ be a path query.", "A noisy path query with noise parameter $\\varepsilon \\in (0,1/2)$ is a function $\\widetilde{Q}_G : V \\times V \\rightarrow \\lbrace 0,1\\rbrace $ such that $\\widetilde{Q}_G(i, j) = Q_G(i,j)$ with probability $1-\\varepsilon $ , and $\\widetilde{Q}_G(i, j) = 1 - Q_G(i,j)$ with probability $\\varepsilon $ .", "In order to make use of our original algorithm, we proceed with the following strategy.", "Algorithm works as follows.", "For each node pair, we will perform majority voting on $m$ noisy path queries.", "If $m$ is large enough, noise will be removed with high probability.", "(See Appendix for details.)", "The above opens up a question on the number of queries $m$ per node pair, that are sufficient to guarantee graph recovery success.", "A second question is whether $m$ depends on the number of nodes $n$ and maximum node degree $d$ , thus affecting the time complexity of our randomized algorithm.", "The following theorem answers both questions.", "Theorem 4 For a fixed probability of error $\\delta \\in (0,1)$ and noise parameter $\\varepsilon \\in (0,1/2)$ , Algorithm takes at most $\\mathcal {O}(\\frac{1}{\\delta } \\, \\frac{1}{(1/2-\\varepsilon )^2} \\, dn \\log ^2 n(\\log d + \\log n + \\log \\frac{1}{\\delta }))$ time, with probability at least $1-\\delta $ , in order to reconstruct a directed rooted tree of $n$ nodes and node degree at most $d$ , provided that the number of queries per node pair fulfills $m \\in \\Theta (\\frac{1}{(1/2-\\varepsilon )^2} \\, (\\log d + \\log n + \\log \\frac{1}{\\delta }))$ .", "In practice we do not need to know the exact value of the noise parameter $\\varepsilon $ .", "A lower bound of $\\varepsilon $ suffices to define the number of queries $m$ per node pair." ], [ "Additive Queries on Weighted Trees", "Here, we focus on the reconstruction of weighted directed rooted trees by using additive queries.", "An additive path query returns the sum of the edge weights on the directed path between two given nodes, if such a path exists, or zero otherwise.", "Next, we formally define additive path queries.", "Definition 7 Let $T=(V,E,W)$ be a weighted directed rooted tree, with positive weights for each edge, i.e., $w_{i,j} > 0$ for all $(i,j) \\in E$ , and $w_{i,j} = 0$ for all $(i,j) \\notin E$ .", "An additive path query is a function $\\widehat{Q}_T : V \\times V \\rightarrow [0,+\\infty )$ such that if there exists a path in $T$ from $i$ to $j$ then $\\widehat{Q}_T(i, j)$ is the sum of the weights of the edges in the path, and $\\widehat{Q}_T(i, j) = 0$ otherwise.", "Note that the above query reveals much more information compared to the path query in Definition REF which only reveals a single bit of information.", "In Algorithm , our strategy is to convert the additive query problem into our original problem for recovering the edge set.", "Afterwards, we recover the edge weights by calling the additive queries for each edge.", "(See Appendix for details.)", "The time complexity of the above algorithm is as follows.", "Theorem 5 Algorithm takes $\\mathcal {O}(dn \\log ^2 n)$ expected time, in order to reconstruct a weighted directed rooted tree of $n$ nodes and node degree at most $d$ .", "Furthermore, for a fixed probability of error $\\delta \\in (0,1)$ , Algorithm takes at most $\\mathcal {O}(\\frac{1}{\\delta } \\, dn \\log ^2 n)$ time, with probability at least $1-\\delta $ ." ], [ "Negative Results for Directed Acyclic Graphs", "As argued before, the cost of asking queries is very high in several application domains.", "Thus, we are interested on the reconstruction of graphs that do not require the trivial $n^2$ queries for $n$ nodes (i.e., one query for every possible pair of nodes.)", "A natural question is whether more general directed acyclic graphs could be recovered efficiently by asking less than $\\Omega (n^2)$ queries.", "Here, we provide a negative answer for the above.", "First note that some directed acyclic graphs are non-identifiable by using path queries.", "For instance, consider the two graphs shown in Figure REF .", "In both graphs, we have that $Q(1,2)=Q(2,3)=Q(1,3)=1$ .", "Thus, by using path queries, it is impossible to discern whether the edge $(1,3)$ exists or not.", "Next, we formalize the above intuition.", "Figure: Two directed acyclic graphs that produce the same answers when using path queries.Definition 8 Let $G=(V,E)$ be a directed acyclic graph.", "We say that an edge $(i,j) \\in E$ is transitive if there exists a directed path from $i$ to $j$ of length greater than 1.", "In Figure REF , edge $(1,3)$ is transitive, since there is a directed path $1,2,3$ (i.e., a directed path of length 2.)", "Transitive edges are not possible to be recovered by using path queries for the following reason.", "Let $i$ and $j$ be two fixed nodes.", "Assume there is a directed path from $i$ to $j$ of length greater than 1.", "Due to this path, we have that $Q(i,j)=1$ , regardless of whether $(i,j) \\in E$ or $(i,j) \\notin E$ .", "A trivial algorithm for reconstructing directed acyclic graphs of $n$ nodes, without transitive edges, is to ask $n^2$ queries (i.e., one query for every possible pair of nodes.)", "Next, we show that this trivial algorithm is indeed optimal.", "Theorem 6 In order to reconstruct a directed acyclic graph of $n$ nodes, without transitive edges, any deterministic algorithm requires at least $\\Omega (n^2)$ queries.", "Furthermore, any randomized algorithm requires at least $\\Omega ((1-\\delta ) \\, n^2)$ , otherwise it would fail with probability at least $\\delta $ .", "Recall that our algorithm pertains to directed rooted trees with a maximum node degree.", "These graphs are not only sparse but also weakly connected (i.e., their undirected skeleton graphs are connected.)", "One could ask whether connectedness is a necessary condition, and whether sparsity makes graph reconstruction easier.", "Next, we show that an algorithm requires $\\Omega (n^2)$ queries for recovering a family of sparse disconnected graphs, as well as a family of sparse connected graphs.", "Theorem 7 In order to reconstruct a sparse disconnected directed acyclic graph of $n$ nodes, any deterministic algorithm requires at least $\\Omega (n^2)$ queries.", "Theorem 8 In order to reconstruct a sparse connected directed acyclic graph of $n$ nodes, any deterministic algorithm requires at least $\\Omega (n^2)$ queries." ], [ "Concluding Remarks", "There are several ways of extending this research.", "The analysis of the reconstruction of other families of graphs in $\\mathcal {O}(n\\log n)$ time would be of great interest.", "Given our results for directed rooted trees, it would be interesting to analyze other families of sparse connected graphs, such as graphs with bounded tree-width as well as graphs with bounded arboricity." ], [ "Detailed Algorithms", "In this section, we show the pseudocode for all the algorithms in our manuscript.", "First, we present our main Algorithm for reconstructing a directed rooted tree by using path queries.", "[H] Reconstruct tree [1] Input: vertex set $V$ of the directed rooted tree $T$ $\|V\|=1$ $E \\leftarrow \\emptyset $ true Pick $i, j \\in V$ independently and uniformly at random (multidirpath, lowestancestoridx) $\\leftarrow $ Reconstruct multidirectional path($V, i, j$ ) Assume multidirpath = $[x_1,\\dots ,x_P]$ pathleft $\\leftarrow $ reverse(multidirpath[1 to lowestancestoridx]) pathright $\\leftarrow $ multidirpath[lowestancestoridx to $P$ ] bagsize $\\leftarrow [1,\\dots ,1]$ , an array of size $P$ each node $k$ that is not on multidirpath bagidxleft $\\leftarrow $ Find bag(pathleft, $k$ ) bagidxright $\\leftarrow $ Find bag(pathright, $k$ ) bagidxleft = lowestancestoridx bagidx $\\leftarrow $ bagidxright + length(pathleft) $-$ 1 bagidx $\\leftarrow $ bagidxleft bagsize[bagidx] $\\leftarrow $ bagsize[bagidx] + 1 leftsize $\\leftarrow $ 0 evenseparator $\\leftarrow $ None $r = 1,\\dots ,P-1$ leftsize $\\leftarrow $ leftsize + bagsize[r] leftsize $ \\in [n/d, (d-1)n/d]$ evenseparator $\\leftarrow $ $(x_r, x_{r+1})$ breakevenseparator $\\ne $ None break($V_1$ , $V_2$ ) $\\leftarrow $ Split tree($V$ , evenseparator) $E_1$ $\\leftarrow $ Reconstruct tree($V_1$ ) $E_2$ $\\leftarrow $ Reconstruct tree($V_2$ ) $E \\leftarrow E_1 \\cup E_2 \\cup \\lbrace $ evenseparator $\\rbrace $ Output: edge set $E$ Next, we present Algorithm for recovering the sequence of nodes in the multidirectional path between any two nodes.", "[H] Reconstruct multidirectional path [1] Input: vertex set $V$ of the directed rooted tree $T$ , two nodes $i, j \\in V$ multidirpath = [], lowestancestoridx = None $Q(i, j) = 1$ each node $k \\in V - \\lbrace i,j\\rbrace $ where $Q(i, k) = 1$ and $Q(k, j) = 1$ multidirpath $\\leftarrow $ append(multidirpath, $k$ ) Sort multidirpath with the order relation given in Definition REF multidirpath $\\leftarrow $ append($i$ , multidirpath, $j$ ) lowestancestoridx $\\leftarrow $ 1 $Q(j, i) = 1$ each node $k \\in V - \\lbrace i,j\\rbrace $ where $Q(j, k) = 1$ and $Q(k, i) = 1$ multidirpath $\\leftarrow $ append(multidirpath, $k$ ) Sort multidirpath with the order relation given in Definition REF multidirpath $\\leftarrow $ append($j$ , multidirpath, $i$ ) lowestancestoridx $\\leftarrow $ 1 $m$ $\\leftarrow $ Find lowest common ancestor($V, i, j$ ) pathleft $\\leftarrow $ [], pathright $\\leftarrow $ [] each node $k \\in V - \\lbrace m,i\\rbrace $ where $Q(m, k) = 1$ and $Q(k, i) = 1$ pathleft $\\leftarrow $ append(pathleft, $k$ ) Sort pathleft with the order relation given in Definition REF each node $k \\in V - \\lbrace m,j\\rbrace $ where $Q(m, k) = 1$ and $Q(k, j) = 1$ pathright $\\leftarrow $ append(pathright, $k$ ) Sort pathright with the order relation given in Definition REF multidirpath $\\leftarrow $ append($i$ , reverse(pathleft), $m$ , pathright, $j$ ) lowestancestoridx $\\leftarrow $ 2+length(pathleft) Output: multidirpath, lowestancestoridx In what follows, we present Algorithm for finding the lowest common ancestor of a multidirectional path between any two arbitrary nodes.", "[H] Find lowest common ancestor [1] Input: vertex set $V$ of the directed rooted tree $T$ , two nodes $i, j \\in V$ pathfromroot $\\leftarrow $ Find path from root($V$ , $i$ ) each node $k$ in pathfromroot $Q(k, j) = 1$ lowestcommonancestor $\\leftarrow k$ breakOutput: lowestcommonancestor Next, we present Algorithm for identifying the path from the root to a given arbitrary node.", "[H] Find path from root [1] Input: vertex set $V$ of the directed rooted tree $T$ , node $i \\in V$ pathfromroot $\\leftarrow $ [] each node $j \\in V - \\lbrace i\\rbrace $ $Q(j, i) = 1$ pathfromroot $\\leftarrow $ append(pathfromroot, $j$ ) Sort pathfromroot with the order relation given in Definition REF Output: pathfromroot Next, we present Algorithm for finding out the bag of a node with respect to an arbitrary directed path.", "[H] Find bag [1] Input: path $x_1, x_2,\\dots , x_k$ , node $i \\in V$ where $V$ is the vertex set of the directed rooted tree $T$ $l \\leftarrow 1$ , $r \\leftarrow k$ $l < r$ $m \\leftarrow (l+r)/2$ $Q(x_m, i) = 1$ $l \\leftarrow m$ $r \\leftarrow m$ Output: bag index $l \\in \\lbrace 1, \\dots , k\\rbrace $ In what follows, we present Algorithm for splitting a directed rooted tree into two subtrees, by cutting the original tree through any arbitrary edge.", "[H] Split tree [1] Input: vertex set $V$ of the directed rooted tree $T=(V,E)$ , edge $(i,j) \\in E$ $V_1 \\leftarrow \\emptyset , V_2 \\leftarrow \\lbrace j\\rbrace $ each node $k \\in V - \\lbrace j\\rbrace $ $Q(j, k) = 1$ $V_2 \\leftarrow V_2 \\cup \\lbrace k\\rbrace $ $V_1 \\leftarrow V_1 \\cup \\lbrace k\\rbrace $ Output: partitions $V_1$ , $V_2$ of $V$ Next, we present Algorithm for reconstructing a directed rooted tree by using noisy path queries.", "We will use $\\widetilde{Q}^{(k)}(i,j)$ to denote the $k$ -th call to the query $\\widetilde{Q}_T(i,j)$ , since for our problem, the directed rooted tree $T$ is fixed (but unknown).", "[H] Reconstruct tree from noisy queries [1] Input: vertex set $V$ of the directed rooted tree $T$ , number of queries $m$ per node pair Define the path query $Q^{\\prime }$ as follows.", "Let $Q^{\\prime }(i,j) = 1$ if $\\sum _{k=1}^m\\widetilde{Q}^{(k)}(i,j) > m/2$ , and $Q^{\\prime }(i,j) = 0$ otherwise.", "$E$ $\\leftarrow $ Reconstruct tree($V$ ) Output: edge set $E$ Finally, we present Algorithm for reconstructing a weighted directed rooted tree by using additive path queries.", "We will use $\\widehat{Q}(i,j)$ to denote $\\widehat{Q}_T(i,j)$ since for our problem, the weighted directed rooted tree $T$ is fixed (but unknown).", "[H] Reconstruct weighted tree from additive queries [1] Input: vertex set $V$ of the weighted directed rooted tree $T$ Define the path query $Q$ as follows.", "Let $Q(i,j) = 1$ if $\\widehat{Q}(i,j) > 0$ , and $Q(i,j) = 0$ otherwise.", "$E$ $\\leftarrow $ Reconstruct tree($V$ ) $W \\leftarrow 0$ each edge $(i,j) \\in E$ $w_{i,j}$ = $\\widehat{Q}(i,j)$ Output: edge set $E$ , edge weights $W$" ], [ "Detailed Proofs", "In this section, we state the proofs of all the theorems and lemmas in our manuscript." ], [ "Proof of Lemma ", "Existence and uniqueness follows straightforwardly from the fact that the directed rooted tree $T$ is weakly connected (i.e., the undirected skeleton graph of $T$ is connected.)", "For the second claim, note that each node in $T=(V,E)$ has at most one parent.", "Given any three different nodes $p,q,r \\in V$ , the following condition holds $\\lnot ( (p,r) \\in E \\, \\wedge \\, (q,r) \\in E )$ .", "Thus, we can only construct the three cases provided in the statement, otherwise we would violate the latter condition." ], [ "Proof of Theorem ", "Let a “round” be a repetition of the “while” loop at Line of Algorithm .", "We show that there are $\\mathcal {O}(d)$ rounds in expectation, and that each round takes $\\mathcal {O}(n \\log n)$ time.", "We then finish the proof by the application of the master theorem.", "First, we analyze the expected number of rounds for finding an even separator in the multidirectional path between two nodes chosen independently and uniformly at random.", "Recall that by Lemma REF for directed rooted trees, a multidirectional path between any two arbitrary nodes always exists and is unique.", "Thus, it remains to analyze the probability that two randomly chosen nodes lie on a different subtree defined by the even separator.", "Recall that Corollary 2.3 in [7] shows if the tree $T$ has node degree at most $d$ , then an even separator exists.", "More specifically, from Definition REF , we know the even separator splits the tree into two subtrees, where each of the subtrees have a number of nodes between $n/d$ and $(d-1)n/d$ .", "Next, we reason about the two randomly selected nodes on each round.", "Let $q$ be the proportion of nodes on the first subtree, and $1-q$ be the proportion of nodes on the second subtree.", "We know that $q \\in [1/d, (d-1)/d]$ and similarly $1-q \\in [1/d, (d-1)/d]$ .", "Since both nodes are selected independently and uniformly at random from the set of $n$ nodes, then $q^2$ is the probability that both nodes fall in the first subtree.", "Similarly, $(1-q)^2$ is the probability that both nodes fall in the second subtree.", "The probability $p$ that the two nodes lie on a different subtree is $p & \\ge \\min _{q \\in [1/d, (d-1)/d]}{\\left( 1-q^2-(1-q)^2 \\right)} \\\\& = 1 - (1/d)^2 - ((d-1)/d)^2 \\\\& = 2 (d-1) / d^2 \\\\& \\in \\Omega (1/d) \\; .$ Therefore, the expected number of rounds $r$ until we successfully find two nodes lying on a different subtree is $E[r] = \\sum _{r=1}^\\infty {r(1-p)^{r-1}p} = 1/p$ which is $\\mathcal {O}(d)$ .", "We now show that each round takes $\\mathcal {O}(n \\log n)$ time.", "First, we derive a sequence of conclusions regarding the subroutines: Algorithm takes $\\mathcal {O}(n\\log n)$ time, since the most time-consuming step is sorting in Line .", "Algorithm takes $\\mathcal {O}(n\\log n)$ time, since the most time-consuming step is the call to Algorithm in Line , which takes $\\mathcal {O}(n\\log n)$ time.", "Algorithm takes $\\mathcal {O}(n\\log n)$ time, since the most time-consuming steps are sorting in Lines , , and , and the call to Algorithm in Line , which takes $\\mathcal {O}(n\\log n)$ time.", "Algorithm takes $\\mathcal {O}(\\log n)$ time, since it performs binary search.", "Algorithm takes $\\mathcal {O}(n)$ time, since the “for” loop in Line iterates for at most $n$ times.", "Recall that each round calls Algorithms , and .", "It can be observed then, that the most time-consuming step on each round is the call to Algorithm , which takes $\\mathcal {O}(n \\log n)$ time.", "Thus, so far we know that the time complexity of the “while” loop at Line is $\\mathcal {O}(dn \\log n)$ .", "To finalize the proof, note that Algorithm exits the “while” loop at Line when it finds an even separator.", "Recall from Definition REF that the even separator splits the tree into two subtrees, where each of the subtrees have a number of nodes between $n/d$ and $(d-1)n/d$ .", "The total running time for Algorithm is given by the recursive formula $C(n) = C(n/d) + C((d-1)n/d) + \\mathcal {O}(dn \\log n) \\; .$ For clarity, we rewrite $C(n)$ in terms of the master theorem in [1].", "That is, $C(n) = \\alpha _1 C(\\beta _1 n) + \\alpha _2 C(\\beta _2 n) + \\gamma (n)$ , for $\\alpha _1 = \\alpha _2 = 1$ , $\\beta _1 = 1/d$ , $\\beta _2 = (d-1)/d$ and $\\gamma (n) \\in \\mathcal {O}(dn \\log n)$ .", "By invoking the master theorem in [1], we have that $C(n) \\in \\mathcal {O}(n^s \\int _1^n{ \\gamma (z) / z^{s+1} dz })$ , where $s$ is the value for which $\\alpha _1 \\beta _1^s + \\alpha _2 \\beta _2^s = 1$ .", "In our case $s=1$ and thus $C(n) \\in \\mathcal {O}(dn \\log ^2 n)$ .", "For the remainder of the proof, we will use $C$ to denote $C(n)$ since $n$ is a constant.", "Note that $C$ is a non-negative random variable with expectation $\\mathbb {E}[C] \\in \\mathcal {O}(dn \\log ^2 n)$ .", "By Markov's inequality, we have that $\\mathbb {P}[C > a] \\le \\mathbb {E}[C]/a$ .", "By letting $a = \\mathbb {E}[C]/\\delta $ we have $\\mathbb {P}[C > \\mathbb {E}[C]/\\delta ] \\le \\delta $ .", "Therefore $\\mathbb {P}[C \\le \\mathbb {E}[C]/\\delta ] \\ge 1-\\delta $ , and we prove our claim that $\\mathbb {P}[C \\in \\mathcal {O}(\\frac{1}{\\delta } \\, dn \\log ^2 n)] \\ge 1-\\delta $ ." ], [ "Proof of Theorem ", "Here we provide an information-theoretic lower bound for any randomized algorithm, based on Fano's inequality [8].", "Let ${n,d}$ be the set of directed rooted trees of $n$ nodes and node degree at most $d$ .", "Next, we show that $\\log \|{n,d}\| \\in \\Theta (n\\log n)$ .", "Interestingly, the latter (tight) bound does not depend on $d$ .", "For a lower bound, note that the number of directed rooted trees with at most one children is equal to the number of permutations of $n$ nodes, which is $n!$ .", "We have $\\log \|{n,d}\| \\ge \\log n!", "\\ge n(\\log n - 1) \\ge \\frac{1}{4} n\\log n$ for $n \\ge 4$ and therefore $\\log \|{n,d}\| \\in \\Omega (n\\log n)$ .", "For an upper bound, note that the number of directed rooted trees with no constraint in the number of children is equal to the number of directed spanning trees, which is $n^{n-1}$ by Cayley's formula.", "We have $\\log \|{n,d}\| \\le \\log n^{n-1} \\le n\\log n$ and therefore $\\log \|{n,d}\| \\in \\mathcal {O}(n\\log n)$ .", "From the above, we conclude that $\\log \|{n,d}\| \\in \\Theta (n\\log n)$ .", "Assume that nature picks a directed rooted tree $T^* \\in {n,d}$ uniformly at random.", "Any mechanism for finding out the correct tree is allowed to make $C$ possibly-dependent queries (for $C$ node pairs) for which the binary responses are $Q_1,\\dots ,Q_C$ .", "Let $T \\in {n,d}$ be the directed rooted tree that is guessed on the basis of the $C$ query responses.", "The above defines a Markov chain $T^* \\rightarrow (Q_1,\\dots ,Q_C) \\rightarrow T$ .", "By properties of the mutual information, and since the joint responses $(Q_1,\\dots ,Q_C)$ can take up to $2^C$ possible values, we have: $I(T^;Q_1,\\dots ,Q_C) & \\le \\mathbb {H}(Q_1,\\dots ,Q_C) \\\\& \\le C\\log 2 \\; .$ By the Fano's inequality [8] on the Markov chain $T^ \\rightarrow (Q_1,\\dots ,Q_C) \\rightarrow T$ we have $\\mathbb {P}[T \\ne T^] & \\ge 1 - \\frac{ I(T^;Q_1,\\dots ,Q_C) + \\log {2} }{ \\log {\|{n,d}\|} } \\\\& \\ge 1 - \\frac{ C\\log 2 + \\log {2} }{ n(\\log n - 1) } \\\\& \\equiv \\delta \\; .$ By solving for $C$ , we have that if $C \\le (1-\\delta ) \\, n(\\log n - 1)/\\log 2 - 1$ then $\\mathbb {P}[T \\ne T^*] \\ge \\delta $ and we prove our claim." ], [ "Proof of Theorem ", "The proof relies on constructing a family of “parallel-chain” trees.", "We assume that the algorithm knows that the directed rooted tree to be reconstructed is “parallel-chain”.", "We also assume that the algorithm has additional side information (described later for clarity.)", "Assume that $n-1$ is divisible by $d$ , and let $k=\\frac{n-1}{d}$ .", "Assume the node set $V$ is partitioned into $k+1$ fixed sets $V_0,V_1,\\dots ,V_k$ , such that $\|V_0\|=1$ and $\|V_l\|=d$ for $l=1,\\dots ,k$ .", "We then create a tree $T=(V,E)$ with $kd$ edges such that: the single node in $V_0$ has $d$ children (each node in $V_1$ ), $n-d-1$ nodes have one child (each node in $V_l$ has one children in $V_{l+1}$ for $l=1,\\dots ,k-1$ ), and the remaining $d$ nodes (in $V_k$ ) are leaves.", "The above creates a “parallel-chain” directed rooted tree as shown in Figure REF .", "Figure: A “parallel-chain” directed rooted tree.Moreover, we assume that the algorithm also knows the node sets $V_0,V_1,\\dots ,V_k$ .", "What remains to be recovered is the “branch” (from 1 to $d$ ) to which the nodes in $V_1,\\dots ,V_k$ belong to.", "In order to do this, the algorithm has to ask a query $Q(i,j)$ for each $i \\in V_l$ , $j \\in V_{l+1}$ and $l=1,\\dots ,k-1$ .", "The total number of queries is $(k-1)d^2 = (\\frac{n-1}{d}-1)d^2 \\ge \\frac{dn}{2}$ for $n \\ge 2(d+1)$ .", "Thus, a deterministic algorithm requires at least $\\frac{dn}{2}$ queries." ], [ "Proof of Theorem ", "Our first goal is to find the condition for which the path query $Q$ is equal to its noisy approximation $Q^{\\prime }$ .", "We start by making some observations for a single fixed node pair $(i,j)$ and later extends our observations for several node pairs.", "Define the random variable $R(i,j) \\equiv \\frac{1}{m} \\sum _{k=1}^m\\widetilde{Q}^{(k)}(i,j)$ .", "Recall that we defined the path query $Q^{\\prime }$ as follows.", "Let $Q^{\\prime }(i,j) = 1$ if $R(i,j) > 1/2$ , and $Q^{\\prime }(i,j) = 0$ otherwise.", "Assume that $Q(i,j) = 1$ .", "By Definition REF , we have that $\\widetilde{Q}^{(k)}(i, j) = 1$ with probability $1-\\varepsilon $ , and $\\widetilde{Q}^{(k)}(i, j) = 0$ with probability $\\varepsilon $ .", "Clearly, $\\mathbb {E}[\\widetilde{Q}^{(k)}(i, j)] = 1-\\varepsilon $ and therefore $\\mathbb {E}[R(i,j)] = 1-\\varepsilon $ .", "By using Hoeffding's inequality, we have $\\mathbb {P}[Q(i,j) \\ne Q^{\\prime }(i,j)] & = \\mathbb {P}[R(i,j) < 1/2] \\\\& = \\mathbb {P}[R(i,j) - \\mathbb {E}[R(i,j)] < 1/2 - \\mathbb {E}[R(i,j)]] \\\\& = \\mathbb {P}[R(i,j) - \\mathbb {E}[R(i,j)] < \\varepsilon - 1/2] \\\\& \\le e^{-2m(1/2-\\varepsilon )^2} \\; .$ Now assume that $Q(i,j) = 0$ .", "By Definition REF , we have that $\\widetilde{Q}^{(k)}(i, j) = 0$ with probability $1-\\varepsilon $ , and $\\widetilde{Q}^{(k)}(i, j) = 1$ with probability $\\varepsilon $ .", "Clearly, $\\mathbb {E}[\\widetilde{Q}^{(k)}(i, j)] = \\varepsilon $ and therefore $\\mathbb {E}[R(i,j)] = \\varepsilon $ .", "By using Hoeffding's inequality, we have $\\mathbb {P}[Q(i,j) \\ne Q^{\\prime }(i,j)] & = \\mathbb {P}[R(i,j) > 1/2] \\\\& = \\mathbb {P}[R(i,j) - \\mathbb {E}[R(i,j)] > 1/2 - \\mathbb {E}[R(i,j)]] \\\\& = \\mathbb {P}[R(i,j) - \\mathbb {E}[R(i,j)] > 1/2 - \\varepsilon ] \\\\& \\le e^{-2m(1/2-\\varepsilon )^2} \\; .$ Assume that Algorithm in Line of Algorithm makes $C$ path queries.", "That is, assume that queries were made for $C$ node pairs $(i_1,j_1),(i_2,j_2),\\dots ,(i_C,j_C)$ .", "By the union bound and the previous observations, we have: $\\mathbb {P}[(\\exists k = 1,\\dots ,C) \\, Q(i_k,j_k) \\ne Q^{\\prime }(i_k,j_k)] & \\le C e^{-2m(1/2-\\varepsilon )^2} \\\\& \\equiv \\delta /2 \\; .$ By solving for $m$ , we have that if $m \\ge \\frac{1}{2(1/2-\\varepsilon )^2} \\, (\\log C + \\log \\frac{2}{\\delta })$ then $\\mathbb {P}[(\\forall k = 1,\\dots ,C) \\, Q(i_k,j_k) = Q^{\\prime }(i_k,j_k)] \\ge 1-\\delta /2 \\; .$ That is, with probability at least $1-\\delta /2$ , the path query $Q$ is equal to its noisy approximation $Q^{\\prime }$ and thus Algorithm reconstructs the tree $T$ correctly, provided that $m$ is large enough.", "By Theorem REF , with probability at least $1-\\delta /2$ , we have that $C \\in \\mathcal {O}(\\frac{2}{\\delta } \\, dn \\log ^2 n)$ .", "From the above, we have that $m \\in \\Theta (\\frac{1}{(1/2-\\varepsilon )^2} \\, (\\log d + \\log n + \\log \\frac{1}{\\delta }))$ fulfills $m \\ge \\frac{1}{2(1/2-\\varepsilon )^2} \\, (\\log C + \\log \\frac{2}{\\delta })$ .", "Finally, the total number of queries is given by $C m \\in \\mathcal {O}(\\frac{1}{\\delta } \\, \\frac{1}{(1/2-\\varepsilon )^2} \\, dn \\log ^2 n (\\log d + \\log n + \\log \\frac{1}{\\delta }))$ and we prove our claim." ], [ "Proof of Theorem ", "Straightforwardly, by the fact that the most time-consuming step in Algorithm is the call to Algorithm in Line , and by the result in Theorem REF ." ], [ "Proof of Theorem ", "Let $\\mathcal {G}_n$ be the set of directed acyclic graphs of $n$ nodes, without transitive edges.", "Next, we show that $\\log \|\\mathcal {G}_n\| \\in \\Theta (n^2)$ .", "For a lower bound, we focus on a particular class of “two-layered” graphs.", "First, the node set $V=\\lbrace 1,\\dots ,n\\rbrace $ is partitioned into two sets $V_1 = \\lbrace 1,\\dots ,\\lfloor n/2 \\rfloor \\rbrace $ and $V_2 = \\lbrace \\lfloor n/2 \\rfloor +1,\\dots ,n\\rbrace $ .", "We then allow only for edges from nodes in $V_1$ to nodes in $V_2$ .", "That is, each node in $V_2$ can have as parents any subset of the nodes in $V_1$ , thus, there are $2^{\|V_1\|}$ choices of edge sets for each of the $\|V_2\|$ nodes in $V_2$ .", "We have $\\log \|\\mathcal {G}_n\| \\ge \\log 2^{\|V_1\|\|V_2\|} = \\log 2^{\\lfloor n/2 \\rfloor (n-\\lfloor n/2 \\rfloor )} \\ge \\log 2^{n(n-1)/4} \\ge \\frac{\\log 2}{5} n^2$ for $n \\ge 5$ and therefore $\\log \|\\mathcal {G}_n\| \\in \\Omega (n^2)$ .", "For an upper bound, note that the number of directed acyclic graphs (without transitive edges) is less than the number of directed graphs of $n$ nodes, which is $3^{\\binom{n}{2}}$ .", "This follows from the fact that for any two nodes $i$ and $j$ , we have three cases in a directed graph $G=(V,E)$ .", "Either $(i,j) \\in E$ , or $(j,i) \\in E$ or $\\lbrace (i,j),(j,i)\\rbrace \\lnot \\subset E$ .", "Now, we have $\\log \|\\mathcal {G}_n\| \\le \\log 3^{\\binom{n}{2}} \\le \\frac{\\log 3}{2} n^2$ and therefore $\\log \|\\mathcal {G}_n\| \\in \\mathcal {O}(n^2)$ .", "From the above, we conclude that $\\log \|\\mathcal {G}_n\| \\in \\Theta (n^2)$ .", "Since each query only reveals a single bit of information, the lower bound of $\\Theta (n^2)$ for any deterministic algorithm follows.", "For the lower bound for any randomized algorithm, we proceed as in Theorem REF by using Fano's inequality [8]." ], [ "Proof of Theorem ", "The proof relies on constructing a family of graphs with a single edge.", "We assume that the algorithm knows that the directed acyclic graph to be reconstructed has a single edge.", "Assume two fixed nodes $i,j \\in V$ .", "The directed graph to be reconstructed is $G=(V,E)$ where $E=\\lbrace (i,j)\\rbrace $ .", "Note that $Q(i,j)=1$ .", "Furthermore $Q(k,l)=0$ for every node pair $(k,l) \\ne (i,j)$ .", "That is only one query returns 1, while $n^2-1$ queries return 0.", "Thus, a deterministic algorithm does not obtain any information from the $n^2-1$ queries in order to guess the edge $(i,j)$ , and therefore it requires at least $n^2$ queries in the worst case." ], [ "Proof of Theorem ", "The proof relies on constructing a family of “v-structured two-layered” graphs.", "We assume that the algorithm only knows that the directed acyclic graph to be reconstructed has $n-1$ edges (i.e., the algorithm does not know that the graph is “v-structured two-layered”.)", "Assume the node set $V$ is partitioned into two fixed sets $V_1$ and $V_2$ .", "For simplicity assume that there is an odd number of nodes, and that $\|V_2\|=\|V_1\|+1$ .", "Thus, $\|V_1\| = \\lfloor n/2 \\rfloor $ and $\|V_2\| = \\lfloor n/2 \\rfloor + 1$ .", "We then create the graph $G=(V,E)$ with $n-1$ edges such that each node in $V_1$ is the source of 2 edges, and each node in $V_2$ is the target of at most 2 edges.", "The above creates a “v-structured two-layered” graph as shown in Figure REF .", "Figure: A “v-structured two-layered” directed acyclic graph.Note that $Q(i,j)=1$ for $(i,j) \\in E$ , while $Q(i,j)=0$ for $(i,j) \\notin E$ .", "That is only $n-1$ queries return 1, while $n^2-n+1$ queries return 0.", "Thus, a deterministic algorithm does not obtain any information from the $n^2-n+1$ queries in order to guess the edge set $E$ , and therefore it requires at least $n^2-n+2$ queries in the worst case.", "(Since the algorithm knows that there are $n-1$ edges, it can stop asking queries as soon as the first bit 1 is returned.)" ], [ "Experiments", "In this section, we present our experimental validation.", "We performed 10 repetitions for different number of nodes, and node degrees.", "For each repetition, we generate a random bounded-degree directed rooted tree, in order to test whether our Algorithm can successfully recover the tree by using path queries, as well as to count the number of queries needed in practice.", "We experimentally found that all trees were successfully recovered in practice.", "Thus, we focused on the question regarding the number of queries needed by Algorithm in practice.", "Figure REF (a) shows the number of queries for different number of nodes, while Figure REF (b) shows the number of queries for different node degrees.", "From our results, it can be observed that the number of queries used by Algorithm is less than what Theorem REF predicted.", "Furthermore, for a constant degree $d$ , the experimental results are much better for a large number of nodes $n$ , as shown in Figure REF (a).", "Figure: (a) Number of queries for different number of nodes nn, for a maximum node degree d=5d=5.", "(b) Number of queries for different node degrees dd, for a number of nodes n=1000n=1000.The upper bound for the number of queries in Theorem is shown in blue.The actual number of queries used by Algorithm is shown in red.", "(Error bars were computed for 10 repetitions, at 95%95\\% significance level.)" ] ]	1606.05183
[ [ "Modulated magnetism and anomalous electronic transport in $\\rm\n Ce_3Cu_4As_4O_2$" ], [ "Abstract The complex magnetism and transport properties of tetragonal Ce$_3$Cu$_4$As$_4$O$_2$ were examined through neutron scattering and physical properties measurements on polycrystalline samples.", "The lamellar structure consists of alternating layers of $\\rm CeCu_4As_4$ with a single square Ce lattice and oxygen-linked Ce bi-layer $\\rm Ce_2O_2$.", "Extending along $\\bf c$, a tube-like Fermi surface from DFT calculations points to a quasi-two-dimensional electronic system.", "Peaks in the specific heat at the Ne\\'{e}l temperature $T_{N}=24$ $\\rm K$, $T_{2}~=~16 $ $\\rm K$ and $T_{3}~=~1.9$ $ \\rm K$ indicate three magnetic phase transitions or distinct cross-over phenomena.", "For $T<T_{N}$ neutron diffraction indicates the development of ferromagnetic ab sheets for both Ce sites, with alternating polarization along $\\bf{c}$, a wave vector ${\\bf k}_{1}={\\bf c}^$.", "For $T<T_{2}$, quasi-two-dimensional low-energy spin fluctuations with ${\\bf k}_{2}=\\frac{1}{2}{\\bf a}^$ and polarized perpendicular to ${\\bf k}_{2}$ are suppressed.", "The data are consistent with quasi-two-dimensional antiferromagnetic order in the $\\rm CeCu_4As_4$ planes polarized along the ${\\bf k}_{2}$ wave vector.", "$T_{3}$ marks a spin-flop transition where the ${\\bf k}_{1}$ staggered magnetization switches to in-plane polarization.", "While the narrow 4f bands lie deep below the Fermi surface, there are significant transport anomalies associated with the transitions; in particular a substantial reduction in resistivity for $T<T_{N}$.", "At $T=100$ $ \\rm mK$ the ${\\bf k}_1$ modulated staggered moment is $0.85~\\mu_B$, which matches the $0.8~\\mu_B$ saturation magnetization achieved for H $~=~7$ $ \\rm T$ at $T~=~2$ $ \\rm K$.", "From low T Lorentzian fits the correlation length is in excess of 75 \\AA.", "We argue the unusual sequence of magnetic transitions results from competing interactions and anisotropies for the two Ce sites." ], [ "Introduction", "As spin degeneracy is lifted through crystal field and exchange interactions, intermetallic compounds containing rare earth ions display intricate thermodynamic, magnetic, and transport anomalies.", "In the Kondo effect dilute rare earth impurities in a metal give rise to a minimum in the resistivity due to resonant electron scattering from the impurity spin[1]$^,$ [8].", "When the rare-earth ions form a full crystalline lattice and when hybridization with more dispersive bands is sufficiently strong, a full band gap can open in so-called Kondo insulators which may support topologically protected surface states[2]$^,$[3]$^,$[4].", "The Kondo lattice regime between these limits is defined by an intricate balance between inter-site super-exchange and intra-site Kondo screening[5].", "When the former interactions prevail, magnetic order occurs concomitant with transport anomalies, while the ground state in the latter case is a heavy Fermi Liquid (FL).", "At the quantum critical point (QCP) where these phases meet, there are non-Fermi-Liquid (NFL) characteristics and in some cases superconductivity[7]$^,$[6].", "The search for materials to expose this strongly correlated regime recently led to the discovery of a family of rare earth bearing quasi-two-dimensional metals of the form $\\rm R_3T_4As_4O_{2-\\delta }$ [13], [12], [9] that is structurally related to the iron superconductors.", "Here R indicates a rare earth ion and T a transition metal ion.", "While T appears to be non-magnetic in this structure,[9] it strongly influences the sequence of transitions so there are three transitions for $\\rm Ce_3Cu_4As_4O_2$ but just a single low temperature transition for $\\rm Ce_3Ni_4As_4O_2$ .", "The presence of two distinct rare earth sites appears to underlie the complicated magnetic and transport properties of these materials.", "Focusing on $\\rm Ce_3Cu_4As_4O_2$ , we seek in this paper to understand the physics underlying the previously reported sequence of phase transitions in this class of materials.", "While the absence of single crystalline samples limits the specificity of our conclusions, the combination of thermodynamic, transport and neutron scattering data that we shall report, as well as ab-initio band structure calculations, provides a first atomic scale view of the intricate electronic properties of these materials.", "It is also apparent however, that the complexity of the material is such that a full understanding will require much more detailed experiment that can only be carried out on single crystalline samples.", "So far our experiments indicate that incompatible in-plane magnetic interactions and magnetic anisotropies produce separate as well as coordinated magnetic phase transitions reminiscent of phase transitions in magnetic multilayer thin-film structures.", "The strongly anisotropic nature of the cerium spins and/or their interactions also plays an important role by allowing for a thermodynamic phase transition for an isolated 2D layer.", "The highest temperature transition at $T_{N}=24$ K is to an antiferromagnetic (AFM) stacking of ferromagnetically (FM) aligned tetragonal layers of spins oriented along the c axis, with characteristic magnetic wave vector ${\\bf k}_1=1.0(2){\\bf c^}$ .", "This order is predominantly associated with the $\\rm Ce_2O_2$ layers.", "The $T_2$ = 16 K transition is associated with the loss of low energy spin fluctuations in the square sublattice $\\rm CeCu_4As_4$ .", "While we do not have direct diffraction evidence for this, the paramagnetic spin fluctuations indicate that the spin structure in $\\rm CeCu_4As_4$ layers for $T<T_{2}$ has a characteristic wave vector ${\\bf k}_2=0.50(2){\\bf a}^$ with spins along ${\\bf k}_2$ .", "Such longitudinally striped structures are also found among iron spins in the parent compounds of iron superconductors[10]$^,$[11], and is supported by our DFT calculations.", "The lowest temperature transition at $T_{3}=1.9$ K is associated with the development of a component of the ${\\bf k}_1$ -type magnetic order polarized within the basal plane.", "Indeed the data is consistent with the rotation of the entire polarization of $\\rm Ce_2O_2$ layers into the basal plane and perpendicular to an anisotropic striped AFM of the $\\rm CeCu_4As_4$ layers.", "The paper thus exposes an intricate interplay between distinct forms of rare earth magnetism in the two different layers that make up $\\rm Ce_3Cu_4As_4O_2$ .", "After the methods section, our experimental results and initial observations are presented in section followed by analysis in section .", "Section draws together a physical picture of $\\rm Ce_3Cu_4As_4O_2$ and in the concluding section () we put the results into the broader context of the $\\rm R_3T_4As_4O_2$ family of compounds and rare earth based strongly correlated electron systems in general.", "Polycrystalline Ce$_3$ Cu$_4$ As$_4$ O$_2$ was synthesized using a previously described solid state method[9].", "The sample employed for neutron scattering was a loose powder with a total mass of 5.8 g. Magnetization measurements for temperatures between 2 K and 300 K were performed in a Quantum Design (QD) Magnetic Property Measurement System (MPMS) [20] in magnetic fields up to 7 T. Specific heat measurements were performed in a QD Physical Property Measurement System (PPMS) [20], using an adiabatic relaxation method for temperature down to 0.4 K and fields up to 9 T. DC electrical resistivity was measured on dense samples in the QD PPMS [20], using a standard four point contact method." ], [ "Neutron scattering", "All neutron scattering experiments were conducted on instrumentation at the NIST Center for Neutron Research.", "For determination of the chemical structure powder diffraction data was acquired on BT1 using the Ge(311) monochromator $\\lambda = 1.5398 ~ Å$ , with 60$^\\prime $ in-pile collimation and the standard 20$^\\prime $ and 7$^\\prime $ collimation before and after the sample respectively.", "In this measurement the powder sample was held in a thin walled vanadium can with $^4$ He as exchange gas and the sample was cooled by a closed cycle compressor system.", "Rietveld refinement of the BT1 data was carried out using the General Structure Analysis System (GSAS) [14].", "To detect magnetic diffraction from the relatively small moment magnetism of Ce$_3$ Cu$_4$ As$_4$ O$_2$ , high intensity diffraction data were acquired for temperatures between 50 mK and 45 K on the Multi-Axis Crystal Spectrometer (MACS) at NIST[15].", "For these long wave length measurements the powder was held in a thin-walled aluminum can with $^4$ He as exchange gas.", "A total of three different MACS experiments employing as many cryogenic systems contributed to the paper: In May 2013 we used an \"Orange\" $^4$ He flow cryostat with access to temperatures from 1.5 K to 50 K. In November 2013 we used an Oxford Instruments dilution fridge with an 11.5 T magnet for the temperature range from 0.05 K to 14 K. Finally in May 2014 a dilution insert with access to temperatures from 0.1 K to 40 K was employed.", "The incident and final neutron energies on MACS were defined by PG(002) monochromatization to be 5 meV.", "The full double focusing configuration of the MACS monochromator was used for temperature scans while higher resolution data were collected with the monochromator set for vertical focusing only and the beam width limited to 10 cm at the pre-monochromator beam aperture.", "The horizontal angular divergence of the incident neutron beam was $3.8^\\circ $ and $0.6^\\circ $ for the doubly and singly focused configurations respectively.", "A second set of energy integrating detectors on MACS were utilized in this experiment.", "These “two-axis” detectors are behind the PG(002) double bounce analyzers and detect scattered neutrons irrespectively of the final energy of scattering.", "With an incident energy $E_i= 5$ meV and a beryllium filter between the powder sample and the analyzer the measured quantity can be written as follows[16]: $\\frac{d\\sigma }{d\\Omega }=r_0^2\\int _0^{E_i}\\sqrt{1-\\frac{\\hbar \\omega }{E_i}}\|\\frac{g}{2}F(Q_\\omega )\|^2 2\\tilde{\\cal S}(Q_\\omega ,\\omega ) \\hbar d\\omega $ Here $Q_\\omega =\|{\\bf k}_i-{\\bf k}_f\|$ , ${\\bf k}_i$ and ${\\bf k}_f$ being the wave vectors of incident and scattered neutrons respectively, and $\\tilde{\\cal S}(Q_\\omega ,\\omega )$ is the spherically averaged dynamic correlation function defined so as to include elastic scattering.", "For the ultimate sensitivity to weak temperature-dependent scattering we plot and analyze temperature difference intensity data from the spectroscopic detectors on MACS.", "While this removes $T-$ independent scattering on average, thermal expansion produces peak derivative line shapes in place of nuclear Bragg peaks.", "These features can be accounted for quantitatively based on thermal expansion coefficients and a Rietveld refinement [17] of the nuclear diffraction data as follows.", "Using angular variables as is customary for monochromatic beam diffraction data, the Rietveld refined nuclear diffraction pattern is written as follows: ${\\cal I}(\\theta )={\\cal C}\\sum _{\\tau }\\frac{I(\\tau )}{2\\sigma (\\tau )}g\\left(\\frac{\\theta -\\Theta (\\tau )}{\\sigma (\\tau )}\\right).$ Here ${\\cal C}$ is the ratio between the experimental count rate and the macroscopic scattering cross section of the sample, $2\\theta $ is the scattering angle, $\\tau $ indicates distinct reciprocal lattice vectors, $2\\sigma (\\tau )$ is the standard deviation for $2\\theta $ near the Bragg peak at scattering angle $2\\Theta (\\tau )$ , and $g(x)=\\exp (-x^2/2)/\\sqrt{2\\pi }$ is a unity normalized gaussian distribution.", "$I(\\tau )$ is the $2\\theta -$ integrated powder cross section for $\|\\vec{\\tau }\|=\\tau $ , which can be expressed as: $I(\\tau )=\\frac{4\\pi ^2\\tan \\Theta }{v_0\\tau ^3} N\\sum _{\|\\vec{\\tau }\|=\\tau }\|F(\\vec{\\tau })\|^2.$ Here $v_0$ is the unit cell volume, $N$ is the number of unit cells in the sample, and $F(\\vec{\\tau })$ is the unit cell structure factor including Debye Waller factors.", "Thermal expansion gives rise to a small shift in the scattering angle that can be written as follows: $\\Delta \\Theta =\\frac{\\Delta \\tau }{\\tau }{\\tan \\Theta },$ Neglecting changes in the intensity of nuclear Bragg peaks, the resulting difference intensity is given by $\\Delta {\\cal I}(\\theta ) &=& \\frac{\\partial {\\cal I}(\\theta )}{\\partial \\Theta }\\Delta \\Theta \\\\&=&{\\cal C}\\sum _{\\tau }\\frac{I(\\tau )}{2\\sigma (\\tau )} g^{\\prime }\\left(\\frac{\\theta -\\Theta (\\tau )}{\\sigma (\\tau )}\\right)\\nonumber \\\\&&\\times \\left(-\\frac{1}{\\sigma (\\tau )}\\right) \\frac{\\Delta \\tau }{\\tau } \\tan \\Theta (\\tau ).$ Here $g^{\\prime }(x)=-xg(x)$ .", "In general we have $\\frac{\\Delta \\tau }{\\tau }=\\sum _i\\frac{\\partial \\ln \\tau }{\\partial a_i}\\Delta a_i,$ where the summation is over all temperature dependent lattice parameters including angular variables.", "In the present case of persistent tetragonal symmetry we have $\\frac{\\Delta \\tau }{\\tau }=-\\left[ (1-(\\hat{\\tau }_{\\parallel })^2)\\frac{\\Delta a}{a} +(\\hat{\\tau }_\\parallel )^2\\frac{\\Delta c}{c}\\right]$ where $\\hat{\\tau }_\\parallel $ is the projection of the unit vector $\\hat{\\tau }$ on the tetragonal c-axis.", "Having determined $\\sigma (\\tau )$ and $I(\\tau )$ from Rietveld refinement in the paramagnetic phase, we then Rietveld refined the difference between the low and high $T$ data with a functional consisting of the magnetic diffraction profile plus Eq.", "REF allowing only for adjustable thermal expansion coefficients $\\Delta a/a$ and $\\Delta c/c$ (Eq.", "REF ).", "The magnetic diffraction profile was developed using representation analysis as implemented in SARAh [18] and FULLPROF [19]." ], [ "Electronic Structure", "Band structure calculations were performed using the full-potential linearized augmented plane wave (FP-LAPW) method implemented in the WIEN2K package [21].", "The Perdew, Burke, Ernzerhof version of the generalized gradient approximation [22] (PBE-GGA) was used for the exchange correlation potential and an onsite Coulomb repulsion of 5 eV was added to Ce sites to approximately account for electronic correlations in the narrow 4f bands[24], [23].", "Because the Ce states do not contribute to the electronic states at the Fermi level, spin-orbit coupling was not included in the calculation.", "The lattice parameters and atomic positions used for the calculation were taken from the neutron diffraction refinement at T = 300 K and are listed in Table REF ." ], [ "Chemical Structure", "The low temperature (T = 5 K) crystal structure was determined from the Rietveld refinement of the powder neutron diffraction data shown in Fig.", "REF .", "A single phase fit can account for the data with space group I4/mmm and $\\chi ^2$ = 2.7.", "Powder X-ray diffraction data had indicated small oxygen defficiency in most reported $\\rm R_3T_4As_4O_{2-\\delta }$ compounds (T = Ni or Cu)[9].", "However, the inset to Fig.", "REF shows the refinement of the oxygen stoichiometry in $\\rm Ce_3Cu_4As_4O_{2-\\delta }$ based on the present neutron data, which yields the best refinement (lowest $\\chi ^2$ ) for $\\delta =0.00(5)$ and therefore indicates the compound is stoichiometric.", "No magnetic diffraction is visible in the BT1 data.", "This is however not inconsistent with the MACS data where the strongest magnetic peak at T = 5 K has an integrated intensity of just 0.4% of the strongest nuclear peak, which is too weak to be detected by BT1.", "A summary of the structural information is provided in table REF and REF .", "Table: T = 300 K atomic positions for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 with space group I 4/mmm determined by the Rietveld analysis of neutron powder diffraction data.", "The corresponding reduced chi-square measure of goodness of fit is χ 2 \\chi ^2 = 2.7.", "The room temperature lattice constants are a = 4.07733(10) Å and c = 27.4146(9) Å.Table: T = 4.5 K Atomic positions for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 in space group I 4/mmm determined by the Rietveld analysis of powder diffraction data.", "The corresponding reduced chi-square measure of goodness of fit is χ 2 \\chi ^2 = 3.4.", "The corresponding lattice constants are a = 4.06344(9) Å and c = 27.3365(7) Å.Illustrated in Fig.", "REF (a), the structure of $\\rm Ce_3Cu_4As_4O_2$ consists of alternating layers of $\\rm CeCu_4As_4$ (blue box) and $\\rm Ce_2O_2$ (red box).", "In $\\rm CeCu_4As_4$ , Ce atoms lie at the vertices of a simple square lattice sandwiched between $\\rm Cu_2As_2$ layers.", "This structural unit is found in the $\\rm ThCr_2Si_2$ structure, which is familiar from 122 iron superconductors such as $\\rm BaFe_2As_2$ where Ba occupies the site equivalent to Ce in the $\\rm CeCu_4As_4$ layer [10].", "There are also a number of heavy fermion systems with a magnetic ion on this site including $\\rm CeT_2Si_2$ [26]$^,$[27]$^,$[28]$^,$[29] (T=Pd, Ru, Rh, Cu) and $\\rm URu_2Si_2$ [25].", "Figure: Color image of (a) elastic and (b) energy integrated neutron scattering versus temperature and wave vector.", "The data were acquired on the MACS instrument at NIST using the spectroscopic and energy integrating detectors respectively.", "Corresponding data sets acquired at T=32T=32 K were subtracted from each to expose small temperature dependent effects.", "The incident neutron energy was 5 meV.", "Gray vertical regions indicate temperature intervals within 5% of the peak maxima in the magnetic specific heat.", "Horizontal dashes lines indicate the locations of allowed nuclear Bragg peaks.", "(c) Temperature dependence of the average scattering intensity in select ranges of wave vector transfer.", "Black symbols are from the data presented in frame (b).", "Red and blue symbols are from (a).", "In all the figures where data are from neutron scattering experiments, the error bars indicate one standard deviation.In the $\\rm Ce_2O_2$ layer, a 45$^0$ rotated square lattice of O with lattice parameter $a/\\sqrt{2}$ is sandwiched by square lattices of Ce with lattice parameter $a$ , such that half of the squares of the O lattice have Ce above and the other half have Ce below.", "This structural element is also found in the 1111 type Fe superconductors and specifically in CeFeAsO[30] and CeNiAsO[31].", "Three peaks in the specific heat of polycrystalline $\\rm Ce_3Cu_4As_4O_2$ were previously reported and associated with magnetic transitions[9].", "In order of decreasing temperature, we shall denote the corresponding phases by I ($T_{2}<T<T_{N}$ ), II ($T_{3}<T<T_{2}$ ), and III ($T<T_{3}$ ) respectively.", "In search for the associated atomic scale magnetic correlations, we acquired neutron diffraction data for temperatures between 1.5 K ($<$ T$_3$ ) and 40 K ($>$ T$_N$ ) using a high intensity configuration of the MACS spectrometer.", "For sensitivity to weak temperature dependent scattering we subtract a high statistics data set acquired at $T=32$ K and plot the difference data as a color image in Fig.", "REF .", "The MACS instrument offers simultaneous energy resolved data and final energy integrated data shown respectively in frames Fig.", "REF (a) and Fig.", "REF (b).", "Coincident with each of the transition regimes inferred from specific heat data (vertical grey regions) are anomalies in the temperature dependent neutron scattering data.", "Fig.", "REF (c) provides a detailed view of the temperature dependent intensity averaged over relevant ranges of wave vector transfer.", "Below $T_{N}$ we observe the development of Bragg diffraction for 1.5 $Å^{-1}<Q<1.9~Å^{-1}$ in Fig.", "REF (a).", "In being associated with a net increase in intensity, these features in the “thermo-diffractogram\" are distinguished from the effects of thermal expansion, which produce matched minima and maxima surrounding each nuclear Bragg peak (thin dashed lines).", "Fig.", "REF (b) and Fig.", "REF (c) (black symbols and line) show a distinct reduction in inelastic scattering for temperatures below the characteristic temperature $T_{2}$ extracted from bulk properties.", "This observation was reproduced in experiments using both a flow cryostat and a dilution fridge, and the temperature and momentum regimes are inconsistent with contributions from cryogenic fluids such as nitrogen or helium.", "We therefore associate this feature with a reduction in inelastic magnetic scattering for $T<T_{2}$ .", "The total scattering sum-rule requires that reduced inelastic scattering appears elsewhere in ${\\cal S}(Q\\omega )$ .", "One option is a shift of all the lost spectral weight beyond the 5 meV cut-off of the experiment (see Eq.", "REF ).Alternatively a reduction in inelastic scattering can be associated with the development of elastic scattering and static correlations.", "In that case a possible explanation for the absence of elastic magnetic scattering associated with $T_{2}$ (see Fig.", "REF (a)) is that the associated component in ${\\cal S}(Q\\omega )$ is extinguished or severely weakened by the so-called polarization factor, which removes scattering associated with spin components parallel to wave vector transfer.", "This would be the case for a modulated in-plane spin structure in phase II where the staggered moment is oriented along the characteristic wave vector of the magnetic structure.", "Figure: (a) Image of elastic neutron scattering versus temperature and wave vector transfer in the low temperature regime.", "Data collected at T = 15 K has been subtracted to remove scattering intensities of nuclear Bragg peaks.", "(b) Intensity and (c) inverse correlation length versus temperature extracted from fits of a Lorentzian convoluted with a Gaussian of the resolution width (horizontal bar within inset) to the most intense magnetic peak.", "An example of such a fit is shown in the inset to (c).The clearest anomaly in scattering is associated with the lowest temperature transition.", "The temperature regime for $T<T_{3}$ (phase III) is detailed in Fig.", "REF .", "Phase III is associated with the development of strong Bragg peaks and we shall argue that only in this phase is the spin direction perpendicular to the direction of modulation so the polarization factor does not extinguish the low $Q$ peaks.", "Fig.", "REF (b) shows the integrated intensity, that develops in an order parameter like fashion for $T<T_{3}$ and Fig.", "REF (c) shows the half width at half maximum of the Lorentzian fit (inset to Fig.", "REF (c)).", "After correcting for resolution effects the low T limit corresponds to a correlation length above 75 $Å$ .", "While we have identified distinct anomalies in the $T$ dependence of neutron scattering at each transition, all except perhaps those associated with $T_{3}$ are unusually broad in temperature compared to a standard second order phase transition observed through diffraction with a cold neutron beam.", "This mirrors the specific heat anomalies[9] and suggests we are dealing with cross-over phenomena rather than critical phase transitions, which could result from disorder.", "Figure: Neutron diffraction patterns for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 obtained at MACS with T = (a) 17 K, (b) 6 K and (c) 30 mK.", "The fitted lines in (a)-(c) include a conventional Rietveld model for the magnetic scattering plus the peak derivative profile (dashed line in (c)) described in Section II(B).", "The pattern in panel (b) includes the calculated diffraction pattern corresponding to the proposed spin structure in Phase II (Fig. )", "with moment size listed in table .", "Such contribution would go undetected experimentally due to the polarization factor when spins are almost parallel to the wave vector.", "The line in (c) is a Rietveld fit to the structural peaks." ], [ "Diffraction in each phase", "To elucidate the nature of each magnetic phase, we isolate the $Q$ dependence of the associated scattering by subtracting data collected for $T>T_{N}$ from high statistics data acquired in each phase.", "For phase I the difference pattern is shown in Fig REF (a).", "Apart from the characteristic zig-zag anomalies associated with thermal expansion, two distinct Bragg peaks are associated with Phase I.", "A detailed view of these peaks is provided in Fig.", "REF .", "Discounting peak derivative line shapes indicated with the dashed lines described by Eq.", "REF , we identify two peaks (shadowed in bright green lines) whose intensities are less than $1\\%$ of nuclear ones at wave vector transfer $Q\\sim 1.57 ~ Å^{-1}$ and $Q\\sim 1.63 ~ Å^{-1}$ .", "Nearby there are no strong nuclear peaks and is therefore possible to obtain reliable magnetic diffraction.The additional intensity at $Q\\sim 1.8 Å^{-1}$ could be misleading due to nuclear subtraction.", "The figure shows the peaks are resolution limited and this sets a low T limit of 75 Å on the correlation length associated with translational symmetry breaking.", "The position of the peaks is indicated by the two vertical hatched regions in Fig.", "REF for $Q>1.5 Å^{-1}$ .", "Assuming that a single propagation vector is associated with Phase I, Fig.", "REF shows that this must be $\\textbf {k}_1\\equiv (0, 0, 1.0(2))$ .", "The absence of magnetic Bragg peaks of the form $(0, 0,2n+1)$ , where $n$ is an integer, indicates these peaks are extinguished by the polarization factor and so calls for spins polarization along $\\bf c$ .", "Figure: Detailed Q-dependence of neutron scattering data characterizing the magnetic order in (a) phase I (16K<T<24K16K < T < 24K) and (b) phase II (2K<T<16K2K < T < 16K).", "A background data set acquired at T=40T=40 K was subtracted to expose temperature dependent features associated with the phase transitions.", "The lines through the data include the peak derivative features associated with thermal expansion (Eq.", "and the calculated magnetic diffraction associated with the structures proposed in Fig.", ").Figure: Relationship between high symmetry magnetic modulation wave vectors distinguished by colors and the location of magnetic satellite peaks in the powder diffraction pattern for Ce 3 Cu 4 As 4 O 2 \\rm Ce_3Cu_4As_4O_2.", "The vertical grey regions show the locations of the experimentally observed magnetic Bragg peaks.", "The low TT phase III has all indicated Bragg peaks whereas only the peaks near 1.5 Å -1 Å^{-1} are visible in Fig.", "for phases I-II.", "The calculated wave vector transfer associated with the horizontal yellow line in (a) 𝐤 1 =𝐜 * {\\bf k}_1={\\bf c}^* or equivalently 𝐤 1 =𝐚 * {\\bf k}_1={\\bf a}^* are consistent with the observed magnetic Bragg peaks.", "Frames (b) and (c) show the remaining singly indexed high symmetry magnetic wave vectors that fail to account for the magnetic Bragg peaks observed in phase III.The diffraction pattern for Phase II is shown in Fig REF (b) with the detail in Fig.", "REF (b).", "Surprisingly there is virtually no change in the position or widths of the Bragg peaks observed for Phase I.", "The black symbols in Fig.", "3(c) however, show that Phase II is associated with the loss of inelastic scattering.", "To better understand the associated dynamic correlations, Fig.", "REF shows the $Q-$ dependence of the energy integrated scattering that is lost upon cooling into phase II.", "It takes the form of asymmetric peaks, which, in the direction of increasing $Q$ , rise more abruptly than they decay.", "This Warren-like line shape[33] is a clear indication of low dimensional correlations.", "We shall later show the correlations to be quasi-two-dimensional.", "In that case rods of scattering extend perpendicular to the plane of long range correlations.", "The sharp leading edge is associated with these rods becoming tangents to the Ewald sphere and the long trailing tail results from the fraction of the Ewald sphere pierced by that rod decreasing in inverse proportion to the area of the sphere.", "From Fig.", "REF we see that the characteristic wave vector of the rod is 0.8 $Å^{-1}$ .", "The proximity of this number to $a^/2=0.77 ~ Å^{-1}$ indicates that the spin fluctuations of phase II are associated with doubling the unit cell in the basal plane.", "Possible explanations for the absence of a corresponding magnetic Bragg peak in Fig.", "REF (c) are (1) that a competing instability prevails so that static long range AFM correlations of the ${\\bf k}_2={\\bf a}^/2$ variety never materialize and (2) that a polarization factor extinguishes the magnetic Bragg peaks associated with this order in the low $Q$ regime where the experiment has sufficient sensitivity to detect them.", "Figure: Q ˜\\tilde{Q}-dependence difference intensity 20 K and 10 K from the energy integrating detectors on the MACS spectrometer.", "The data is plotted as a function of wave vector transfer Q ˜(θ,ω)\\tilde{Q}(\\theta ,\\omega ) for energy transfer 〈ℏω〉=1.2(3)\\langle \\hbar \\omega \\rangle =1.2(3) meV inferred from the fitting analysis (see section ).", "The upper horizontal axis shows wave vector transfer for elastic scattering.", "The solid blue line is a Warren-like line shape for quasi-two-dimensional AFM correlations with characteristic wave vector 𝐤 2 =(1/2,0,0){\\bf k}_2=(1/2,0,0) and in-plane dynamic correlation length ξ=8.2(6)\\xi =8.2(6) Å.", "The dashed line is a Q 2 Q^2 background ascribed to the difference in the Debye Waller factor for incoherent scattering at the two temperatures involved.", "The inset shows the pattern of quasi-two-dimensional fluctuation associated with 𝐤 2 {\\bf k}_2, with black and grey spots representing alternative spin .", "The proposed ordered state for phase II has spins parallel to 𝐤 2 {\\bf k}_2 (Fig.", "(b)).Shown in Fig.", "REF (c), the strongest magnetic diffraction peaks are associated with Phase III.", "As indicated in Fig.", "REF all observed peaks associated with phase III can be accounted for by a wave vector ${\\bf k}_1={\\bf c}^$ or equivalently ${\\bf k}_1={\\bf a}^$ .", "The appearance of Bragg peaks of the form $(0 0,2n+1)$ in Phase III indicate spin components within the basal plane and a spin-flop type transition relative to phases I and II.", "To be quantified in section , the magnetic peaks are slightly broader than the nuclear peaks even at 30 mK indicating an element of disorder in the magnetic structure or strong thermal diffuse magnetic scattering with a length scale shorter than the coherence length of the chemical structure.", "Upon heating, the intensity of the magnetic peaks decreases precipitously above 2 K, though a broad feature remains at all magnetic Bragg positions up to T = 2.6 K. This indicates dynamic diffuse scattering near the critical temperature of the spin-flop transition." ], [ "Magnetization", "The magnetization versus temperature was measured in different fields and the corresponding data are shown in Fig.", "REF (a).", "All magnetization measurements were conducted on polycrystalline samples, so the data represent the spherical average of the longitudinal response to the magnetic field $H$ .", "For temperatures below $\\sim $ 30 K, the bifurcation between zero field cooled (ZFC) (solid lines) and field cooled (FC) (dashed lines) magnetization data occurs for fields up to H = 0.4 T. To identify potential magnetic transitions, Fig.", "REF (b) shows the derivatives $H^{-1}d(MT)/dT$ [34], for which peaks are expected at magnetic phase transitions.", "Indeed, anomalies in $H^{-1}d(MT)/dT$ , marked by vertical arrows, occur near each of the transitions previously identified in specific heat data [9].", "As the field increases, $H^{-1}d(MT)/dT$ data (Fig.", "REF (b)) indicate that the lowest transition temperature T$_{3}$ increases and reaches $\\approx $ 7 K for H = 7 T, while the upper two transitions at $T_{N}$ and $T_{2}$ are largely unaffected by $H~<~7$ T. Figure: The resistivity of Ce 3 _3Cu 4 _4As 4 _4O 2 _2 for H = 0 (squares) and H = 9 T (circles).", "Inset: enlarged view of the low temperature resistivity in this field range.", "Solid orange line is an example of polynomial fits for H = 0 T to determine the transition temperature." ], [ "Resistivity", "The field dependence of the resistivity measured on a dense polycrystalline sample is shown in Fig.", "REF .", "The resistivity increases upon cooling below room temperature, reaching a maximum at T $\\approx $ 150 K. The increase is typically associated with the single ion Kondo effect, with the maximum signaling the onset of inter-site coherence.", "Below 150 K, poor metallic behavior is observed, as the resistivity decreases with temperature, albeit with large resistivity values $\\rho > 10~{\\rm m}\\Omega $ cm.", "As the temperature decreases below 50 K, a difference between H = 0 (full symbols) and H = 9 T (open symbols) data develops, well above the upper transition temperature $T_{N}$ inferred from specific heat data[9].", "Notably however, the magnetoresistance reaches a maximum near $T_{3}$ , as will be shown explicitly below.", "The inset of Fig.", "REF shows the low temperature resistivity measured in different magnetic fields up to H = 9 T. The resistivity decreases monotonically with increasing magnetic field.", "For H = 0, a broad peak around 4 K is observed, which shifts to higher temperature with increasing field.", "In phase III, Fermi liquid (FL) behavior is evidenced by the quadratic temperature dependence of the resistivity (Fig.", "REF a) up to H $\\approx $ 3 T. This behavior is reminiscent of a Kondo lattice system, in which the competition between magnetic coupling and Kondo screening gives rise to a peak around the coherence temperature T$_{coh}$ and FL behavior below T$_N$[35].", "For fields beyond H $\\approx 2.5 - 3$ T however, a finite-T minimum in the resistivity develops, followed by a logarithmic increase of the resistivity upon cooling below $\\approx $ 1 K as apparent in the semi-log plot in Fig.", "REF (b).", "The $\\rho \\propto \\log T$ behavior persists up to a temperature $T^*$ that increases with magnetic field.", "Figure: (a): The quadratic dependence of resistivity for small magnetic fields between H = 0 T and H = 3 T. (b): The logarithmic dependence of resistivity for large fields between H = 2.5 T and H = 9 T.The origin of the logarithmic temperature dependence of resistivity in Ce$_3$ Cu$_4$ As$_4$ O$_2$ is not clear: either the formation of Kondo singlet [36] or weak Anderson localization in a 2D systems [37] could lead to $\\rho (T) \\propto \\log T$ behavior, however neither phenomenon is relevant for Ce$_3$ Cu$_4$ As$_4$ O$_2$ .", "For the former scenario, a magnetic field breaks up the Kondo singlet, which is inconsistent with the observation that the logarithmic temperature dependence extends over a larger T-regime for higher magnetic fields.", "In the latter scenario, the magnetic field may tune the localization-delocalization transition.", "The upturn in resistivity could be observed in a system with weak localization[38].", "However, this localization usually increases the resistivity compared to the delocalized state.", "The magnetic field induced localization is not consistent with the observation of negative magnetoresistance in Ce$_3$ Cu$_4$ As$_4$ O$_2$ .", "A complication in the interpretation of these data is that the polycrystalline nature of the sample implies the transport measurement is an average over all possible field and current directions with $ \\bf H \\parallel J$ in the tetragonal structure.", "For an anisotropic magnetic material such as Ce$_3$ Cu$_4$ As$_4$ O$_2$ the high field low $T$ behavior could therefore be uncharacteristic and dominated by specific field directions.", "Figure: Magnetoresistance data for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 (symbols) for various temperatures, with the magnetoresistance of non-magnetic La 3 _3Cu 4 _4As 4 _4O 2 _2 (line) at T = 2 K shown for comparison.The magnetoresistance is shown in Fig.", "REF for Ce$_3$ Cu$_4$ As$_4$ O$_2$ (symbols) and the non-magnetic analogue La$_3$ Cu$_4$ As$_4$ O$_2$ (line).", "For the La compound, the magnetoresistance is positive and small (less than 5%), as expected for a normal metal in which the Lorentz force introduces electron scattering.", "For the Ce compound, the T = 2 K magnetoresistance is negative and reaches $\\approx $ –30% at H = 9 T. This suggests that the magnetism associated with the Ce$^{3+}$ ions is responsible for the large magnetoresistance in Ce$_3$ Cu$_4$ As$_4$ O$_2$ .", "The magnetoresistance for Ce$_3$ Cu$_4$ As$_4$ O$_2$ is largest at T = 2 K, close to T$_3$ .", "Similar negative large magnetoresistance was also observed in the Kondo lattice systems Ce$_2$ Ni$_3$ Ge$_5$ , CeB$_6$ and CeRhSn$_2$ .", "[39], [40], [41] Above the magnetic ordering temperature, an important contribution to the resistivity is the Kondo scattering of conduction electrons off local moments.", "The magnetic field could effectively suppress the Kondo scattering and induce a negative magnetoresistance.", "This negative magnetoresistance will reach its maximum at the magnetic ordering temperature where the magnetic fluctuations dominate all other contributions to the resistivity.", "Figure: (a) The specific heat of Ce 3 _3Cu 4 _4As 4 _4O 2 _2 measured from T = 0.4 K to T = 14 K, in applied magnetic fields between H = 0 and H = 9 T. (b) Quadratic temperature dependence of C p C_p in magnetic fields between H = 2 T and H = 9 T." ], [ "Specific Heat", "Additional information relevant to understanding the unusual low $T$ behaviors indicated by magnetization and transport measurements is provided by field-dependent specific heat data in Fig.", "REF .", "From Fig.", "REF (a), the broad peak associated with low $T$ magnetic ordering at $T_{3}~=~2.0$ K shifts to higher temperatures and broadens for lager fields applied to the polycrystalline sample.", "$C_p$ is quadratic in temperature for H $~\\ge ~2.5$ T (Fig.", "REF (b)).", "At H = 9 T, the quadratic dependence extends up to T = 4 K. This behavior can be associated with gapless excitations with a quasi-two-dimensional linear dispersion relation as in a two dimensional antiferromagnet.", "This indication of reduced dimensionality is consistent with the tube-like Fermi surface inferred from DFT (see section IV (C)).", "In zero field the low $T$ state indicated in Fig.", "REF would be expected to have an excitation gap similar in magnitude to $T_{3}$ .", "The high field $C_p(T)~\\propto ~T^2$ regime might be associated with specific field directions in the polycrystalline experiment that are transverse to the easy axes and induce gapless behavior." ], [ "Phase Diagram", "From the thermal anomalies observed in the wide range of experimental data presented, a phase diagram can be sketched.", "As previously emphasized, all anomalies are broad, indicative of cross-overs between different regimes as opposed to sharp symmetry breaking phase transitions.", "It also means that different measurements will display slightly different characteristic temperatures.", "In combination though such measurements lead us to the definition of well defined cross over lines that we have labeled $T_{N}$ , $T_{2}$ , and $T_{3}$ bordering the corresponding phases I-III.", "The upper cross-over lines are remarkably field independent.", "The predominant effect of field appears to be broadening that can be associated with a spherically averaged result on an anisotropic material.", "At low temperatures a distinct non-Fermi liquid (NFL) regime is present for H $\\ge $ 2.5 T where $\\rho (T)~\\propto ~\\log T$ and $C_p~\\propto ~T^2$ .", "Figure: Phase diagram of temperature vs. magnetic field for Ce 3 _3Cu 4 _4As 4 _4O 2 _2, with phase boundaries determined by features in the physical property measurements and neutron diffractions." ], [ "Magnetic Structures", "From the neutron scattering data presented in section REF we have inferred the following principal features of magnetism in $\\rm Ce_3Cu_4As_4O_2$ : For $T< T_{N}=24$ K Ce spins are polarized along the tetragonal $\\bf c$ axis, with a characteristic propagation vector $\\textbf {k}_{1} = 001$ .", "This structure persists in the second phase for $T~<~T_{2}~=~16$ K, which is marked by the loss of low energy spin fluctuations that have quasi-two-dimensional spatial correlations, a characteristic wave vector ${\\bf k}_2=(0.5,0,0)$ , and a component of polarization perpendicular to ${\\bf k}_2$ .", "Finally, for $T~<~T_{3}~=~2~K$ there is a spin-flop transition where the orientation of spins modulated in accordance with ${\\bf k}_1$ rotates from the $\\bf c$ axis to the tetragonal basal plane.", "While the the present powder diffraction data are insufficient to definitely determine the complex spin structures of $\\rm Ce_3Cu_4As_4O_2$ , we shall develop a specific picture of the ordered spin structures that is consistent with the diffraction profiles and provides a logical sequence of phase transitions.", "One of the main outcomes that is supported by Rietveld analysis of the data is an association of the two wave vectors with the two different Ce sites in $\\rm Ce_3Cu_4As_4O_2$ .", "The sequence of spin structures inferred from this analysis is shown in Fig.", "REF ." ], [ "Phase I", "We use representation theory to classify the magnetic structures with wave vector ${\\bf k}_1=(001)$ that can develop from the paramagnetic phase of space group I4/mmm in a second order phase transitions.", "Using Kovalev notation [42], the representations associated with ${\\bf k}_1$ and each of the two cerium sites decompose into irreducible representations (IR) as follows.", "For the Ce1 cite associated with $\\rm CeRu_4As_4$ layers: $\\Gamma _{mag}(\\rm Ce1) = \\Gamma _3 + 2\\Gamma _9$ .", "For the Ce2 sites associated with the $\\rm Ce_2O_2$ layers: $\\Gamma _{mag}(\\rm Ce2) = \\Gamma _3 + \\Gamma _2 + 2\\Gamma _9 + 2\\Gamma _{10}$ .", "There are a total of nine basis vectors (BVs) and these are listed in Table REF .", "These were obtained using the SARAh program [18].", "For both cerium sites the two-dimensional $\\Gamma _9$ and $\\Gamma _{10}$ IRs describe magnetic structures with in-plane moments while the one-dimensional $\\Gamma _3$ and $\\Gamma _2$ IRs describe structures with moments along the tetragonal c-axis.", "Table: The nine basis vectors associated with magnetic structures that transform according to irreducible representations of the I /4mmm space group with propagation vector k=(0,0,1).", "The irreducible representations and basis vectors were obtained using SARAh and the notation is based on Kovalev.Because no magnetic peaks of the form $(0,0,2n+1)$ are observed for $T_{2}<T<T_{N}$ (Fig.", "REF ), the moment is oriented along the c-axis and we must focus on the one-dimensional IRs $\\Gamma _2$ and $\\Gamma _3$ .", "For the Ce1 sites this leaves just one option namely $\\Gamma _3$ , which corresponds to the centered site antiparallel to the nearest neighbors (Fig.", "REF ).", "For Ce2 sites there are two options corresponding to FM ($\\Gamma _2$ ) or AFM ($\\Gamma _3$ ) alignment of spins within a $\\rm Ce_2O_2$ bi-layer.", "Assuming the $\\Gamma _2$ representation for Ce2 sites only (and no order on Ce1 sites) leads to $\\chi ^2=23.7$ , which is larger than $\\chi ^2=17.5$ corresponding to $\\Gamma _3$ with order on both cerium sites.", "Here the reduced $\\chi ^2$ goodness is reported for difference data in the range $1.5Å^{-1}<Q<1.8Å^{-1}$ .", "Fig.", "REF shows three fit lines corresponding to $\\Gamma _3$ order on Ce2, on Ce1 sites only and on both sites.The constrained differential line shape associated with thermal expansion is included with $\\Delta a/a=1.3\\%$ and $\\Delta c/c=0.6\\%$ .", "The fit with order on both sites provides the best account for the relative intensity of the two peaks and the corresponding ordered moments inferred are $\\mu _{Ce_1} =0.14(6)~\\mu _B $ and $\\mu _{Ce_2} = 0.18(2)~\\mu _B$ .", "A sketch of the corresponding spin configuration is shown as phase I in Fig.", "REF (b).", "The weak nuclear Bragg peak (101) is close to the magnetic peaks.", "Subtle changes in the chemical structure thus could also play a role in the temperature dependence of diffraction in this Q-range." ], [ "Phase II", "While the present experiment detects no new elastic peaks upon entering phase II, a distinct loss of low energy inelastic scattering is observed in the energy integrating detectors of the MACS instrument for $T<T_{2}$ (Fig.", "REF (b-c)).", "In Fig.", "REF the broad anisotropic peaks associated with the scattering that vanishes for $T<T_{\\rm 2}$ is compared to the spherical average of the following dynamic correlation function associated with quasi-two-dimensional magnetic correlations: ${\\cal S}(\\textbf {Q})= \\sum _m \\frac{\\langle S^2\\rangle \\cdot \\xi ^2/\\pi }{[1+(\\xi \|{\\bf Q_{\\perp }-Q_m}\|)^2]^2}.$ Here ${\\bf Q}_m = \\tau \\pm \\textbf {k}_{2}$ indicates the location of critical points in the 2D reciprocal lattice.", "Because the incident neutron energy is similar to the energy scale of the spin system, actual wave vector transfer differs from that calculated neglecting energy transfer as follows $\\tilde{Q}(\\theta ,\\omega ) = k_{i}\\sqrt{2-\\frac{\\hbar \\omega }{E_i}-2\\sqrt{1-\\frac{\\hbar \\omega }{E_i}}\\cos 2\\theta }.$ With $\\hbar \\omega =1.2(3)$ meV, in-plane correlations length $\\xi = 8.2(6) Å$ and an effective fluctuating moment of $(g_J\\mu _B)^2\\langle S^2\\rangle \\le 0.3~\\mu _B^2$ this model provides a satisfactory account of the difference data in Fig.", "REF .", "Here we have plotted the data versus the inferred wave vector transfer for inelastic scattering $\\tilde{Q}$ showing also wave vector transfer for elastic scattering on the upper horizontal axis.", "Ce2 sites play the dominant role both in Phase I and Phase III.", "We therefore pursue the hypothesis that the ${\\bf k}_2=(1/2,0,0)$ type scattering in Fig.", "REF is associated with Ce1 sites.", "The total scattering can decrease if the kinematically accessible Bragg peaks of the corresponding order are extinguished by the polarization factor.", "Fig.", "REF (b) includes the calculated elastic scattering that would result from the longitudinally polarized in-plane antiferromagnetic spin structure indicated in Fig.", "REF (b) for Phase II.", "Because spins are almost parallel to ${\\bf k}_2$ at low wave vector transfer, this structure produces very little magnetic powder diffraction and would go undetected in the present experiment.", "This structure is therefore a viable candidate to account for the loss of critical scattering with wave vector ${\\bf k}_2$ without the appearance of appreciable magnetic Bragg scattering." ], [ "Phase III", "The appearance of low-$Q$ peaks in Phase III of the ${\\bf k}_1=(001)$ variety is indicative of in-plane magnetic moments that are described by IR $\\Gamma _9$ or $\\Gamma _{10}$ (Table REF ).", "Of these $\\Gamma _{10}$ is inconsistent with the observed relative intensities while $\\Gamma _9$ provides for the excellent fit shown in Fig.", "REF (c).", "The contribution to magnetic scattering from out of plane moments (Fig.", "REF (a-b)) is so weak that we cannot place meaningful limits on how much might remain in Phase III.", "The experimental data in Fig.", "REF (c) is consistent with an AFM spin structure of the $\\Gamma _9$ variety (Fig.", "REF (b)) with moments within the tetragonal basal plane.", "The total moment obtained is ($\\mu _{Ce_1}+\\mu _{Ce_2}) =0.85(2) \\mu _B$ .", "Consistent with the assignment of Ce1 with moment $\\le 0.45 \\mu _B$ to the ${\\bf k}_2$ structure, the Rietveld refinement displayed in Fig.", "REF (c) indicates the Ce1 site contributes less than 0.1 $\\mu _B$ to the ${\\bf k}_1$ structure of Phase III.", "In a field of 7 Tesla the magnetization per cerium site reaches a value of 0.6 $\\mu _B$ corresponding to 2/3 of the low $T$ ordered moment on the Ce2 sites, which make up 2/3 of the cerium in $\\rm Ce_3Cu_4As_4O_2$ .", "This is consistent with the magnetization in 7 T of only the Ce2 sites that consist of FM layers leaving the Ce1 layers in the proposed in-plane ${\\bf k}_2=(0.5,0,0)$ type AFM order.", "This hypothesis could be examined by higher field magnetization measurements.", "Density functional calculations were done on Ce$_3$ Cu$_4$ As$_4$ O$_2$ for three types of magnetic configurations.", "The first is the ferromagnetic (FM) state.", "The second is the AFM (0, 0, 1) periodicity, which corresponds to phase I in Fig.", "REF and is referred to as AFM1.", "The third magnetic configuration is to model phase III in Fig.", "REF , with the same magnetic moment arrangements for Ce2 sites as in the AFM1 state and (1/2, 0, 0) periodicity for Ce1 sites.", "This structure is referred to as AFM2.", "The magnetic unit cell for the AFM2 state is $2\\times 1\\times 1$ times the crystallographic unit cell.", "Among all three magnetic configuration considered, AFM2 has the lowest energy.", "The relative energy for AFM2 state is E(AFM2) - E(FM) = - 127 meV/F.U., while the relative energy for AFM1 is E(AFM1) - E(FM) = 2 meV/F.U.", "The energy comparison directly supports AFM2 as the ground state, which is consistent with the neutron scattering result.", "For the AFM2 state, the magnetic moments on both Ce sites are between 0.96 $\\sim $ 0.97 $\\mu _B$ , significantly larger than the experimentally observed values on Ce1 sites (0.45 $\\mu _B$ ), but close to Ce2 sites (0.85 $\\mu _B$ ).", "One possible reason for the reduced moment on Ce1 compared to DFT is that the Ce1 atoms and As2 atoms are both spatially and energetically close to each other, resulting in Kondo hybridization between them.", "Such Kondo hybridization can strongly reduce the sizes of magnetic moments [43], [44].", "Figure: Energy versus magnetic moment from the fixed spin moment calculation for La 3 _3Cu 4 _4As 4 _4O 2 _2, the non-magnetic analogue of Ce 3 _3Cu 4 _4As 4 _4O 2 _2.", "Inset: M(H) isotherm for both Ce 3 _3Cu 4 _4As 4 _4O 2 _2 and La 3 _3Cu 4 _4As 4 _4O 2 _2.Figure: The total and atom-projected DOS calculated for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 in the AFM2 state.", "The spin down DOS is multiplied by -1.", "Inset is an enlarged view for DOS close to the Fermi level.Figure: The band structure calculated along an in-plane k-path for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 in the AFM2 state, with only the spin up component shown.Figure: The Fermi surface for Ce 3 _3Cu 4 _4As 4 _4O 2 _2 in the AFM2 state(spin up component only).To examine the magnetism on the Cu site, a fixed spin moment calculation was performed for La$_3$ Cu$_4$ As$_4$ O$_2$ (Fig.", "REF ), the non-magnetic analogue of Ce$_3$ Cu$_4$ As$_4$ O$_2$ , with methods similar to those for the Ce compound.", "The energy of La$_3$ Cu$_4$ As$_4$ O$_2$ increases monotonically with increasing spin moment, indicative of a paramagnetic ground state.", "This monotonic behavior implies that there is no tendency for the 3d electrons to become magnetic, which is consistent with the experimental observations[9].", "The density of states (DOS) for Ce$_3$ Cu$_4$ As$_4$ O$_2$ in the AFM2 state is shown in Fig.", "REF .", "Unlike for Fe-pnictides case, the majority of the Cu bands lie in the energy interval close to the O bands, despite the fact that O has lower electronegativity than Cu.", "The increased energy of O bands might be due to their proximity to Ce atoms.", "At the Fermi level, Cu and As contribute most to the DOS.", "The non-zero DOS at the DFT Fermi level is indicative of a metal.", "The Ce1 atoms have a lower energy, and their bands are broader than for the Ce2 atoms.", "The Cu 3d band is very broad, with a bandwidth of about 10 eV.", "Although centered around -3 eV, the tail of the 3d band is visible up to 5 eV, and the Fermi level just crosses this tail.", "This indicates a weakly correlated 3d band, consistent with a non-magnetic state for Cu.", "The band structure for Ce$_3$ Cu$_4$ As$_4$ O$_2$ in the AFM2 state is shown in Fig.", "REF .", "A quasi-two-dimensional nature is apparent from the similarity of the bands along the in-plane $\\Gamma $ -X-S-$\\Gamma $ path and Z-U-R-Z paths.", "As for the iron pnictides, most bands crossing the Fermi levels are anti-bonding [45] and associated with Cu or As.", "The flat band near -2.1 eV is from Ce1, while the less visible Ce2 band is centered around -1.0 eV.", "The Fermi surface is shown in Fig.", "REF plotted in the folded Brillouin zone corresponding to the orthorhombic magnetic unit cell which is $2\\times 1\\times 1$ times the tetragonal crystallographic unit cell.", "Extended along the c-axis with almost no corrugation, the Fermi surface indicates electron hopping is mainly confined to the tetragonal basal plane so that that the electronic state for Ce$_3$ Cu$_4$ As$_4$ O$_2$ is very two dimensional.", "The pockets near the $\\Gamma $ points are however more dispersive along c-axis than the pockets far away the $\\Gamma $ points.", "Table: Our working hypothesis for the magnetic structure in the three distinct phases of Ce 3 Cu 4 As 4 O 2 \\rm Ce_3Cu_4As_4O_2: phase I ( 16K<T<2416~{\\rm K} < T <24 K), and phase II ( 2K<T<16 2~{\\rm K} < T < 16 K ) and phase III (T<2 T < 2 K ).", "In the table, 𝐤 1 \\bf k_1 = (0, 0, 1) and 𝐤 2 \\bf k_2 = (1/2, 0, 0).", "One dimensional IR Γ 2 \\Gamma _2 associated with 𝐤 2 \\bf k_2 represents Basis Vectors as (100).", "The hypothesis is consistent with the experimental data but more specific than can be proven on their basis." ], [ "Discussion", "Having presented and analyzed comprehensive data for the magnetic and electronic properties of $\\rm Cu_3Cu_4As_4O_2$ we now discuss the overall physical picture of this material.", "Table REF provides a summary of the spin structures proposed for the three magnetic phases.", "While the upper two transitions mark the development of distinct magnetic modulations with wave vectors ${\\bf k}_1$ = (001) and ${\\bf k}_2$ = (1/2, 0, 0) in the $\\rm Ce_2O_2$ and the $\\rm CeCu_4As_4$ layers respectively, the transition to phase III is a spin flop transition where the staggered magnetization already established for $\\rm Ce_2O_2$ in phase I rotates into the basal plane.", "This last transition is by far the most prominent as observed by magnetic neutron scattering because it allows for diffraction at low $Q$ which is extinguished by the polarization factor in the higher temperature phases.", "The lower transition to phase III might be understood as follows.", "If we allow for isotropic bilinear spin exchange interactions between different cerium sites then the effect of the $\\rm Ce_2O_2$ layers on the $\\rm CeCu_4As_4$ layers is analogous to a uniform magnetic field, which in phase II is oriented perpendicular to the staggered magnetization and perpendicular to the basal plane.", "Rotation of the $\\rm Ce_2O_2$ magnetization into the basal plane can then be understood as the result of a competition between incompatible single ion and/or exchange anisotropy for the Ce1 and Ce2 sites.", "Having ordered antiferromagnetically with spins within the basal plane in phase II, $\\rm CeCu_4As_4$ layers apparently have an easy plane character.", "Phase I on the other hand demonstrates an easy axis character to $\\rm Ce_2O_2$ layers.", "The final spin flop transition might then be understood as the easy-plane character prevailing if $\\rm Ce_2O_2$ spins rotate into the basal plane and perpendicular to ${\\bf k}_2$ .", "This is consistent with the diffraction data though the diffraction data do not establish the orientation of $\\rm Ce_2O_2$ spins within the basal plane.", "Comparison to $\\rm Pr_3Cu_4As_4O_{2-\\delta }$ is instructive.", "The strength of RKKY exchange interactions can be expected to vary between rare earth ions in accordance with the so-called de Gennes factor $F=(g-1)^2J(J+1))$ , which is 0.18 for cerium and 0.8 for praseodymium for a ratio of $F({\\rm Pr})/F({\\rm Ce})=4.4$ .", "While the ordered spin structures for $\\rm Pr_3Cu_4As_4O_{2-\\delta }$ are as yet unknown, there are two phase transitions at $T_{N}=35(3)$ K and $T_{2}=22$ K [9].", "The enhancement of $T_N$ is not nearly as large as might be expected based on de Gennes scaling but the similar ratio for $T_{2}/T_{N}$ of approximately 1.5 for Ce and 1.6 for Pr is consistent with an analogy between phases I and II of the two compounds.", "The absence of a third transition for the praseodymium compound is consistent with $T_{3}$ for $\\rm Ce_3Cu_4As_4O_{2}$ being contingent on competing anisotropies that might be absent for $\\rm Pr_3Cu_4As_4O_{2-\\delta }$ .", "Replacing Ni for Cu in $\\rm CeNi_4As_4O_{2-\\delta }$ results in just a single peak in the specific heat at $T_{N}=1.7$ K[9].", "The change in entropy up to 40 K, the effective moment, and the Weiss temperature are however, similar to $\\rm CeCu_4As_4O_2$ , which indicates the cerium moment persists with similar interaction strengths.", "Increased magnetic frustration and quenched chemical or structural disorder are potentially relevant factors to account for the suppression relative to $\\rm Ce_3Cu_4As_4O_{2}$ of the higher temperature transitions.", "Here we note that the rare earth layers are separated by transition metal ions so that all inter-layer interactions must proceed through the copper or nickel layer.", "The dramatic effect of changing these layers from copper to nickel thus might indicate the upper transitions are driven by interlayer interactions.", "A curious feature of $\\rm Ce_3Cu_4As_4O_{2}$ are the broad nature of the specific anomalies.", "The diffraction experiments presented here reveal magnetic order above the 75 $Å$ length scale but in harmony with the specific heat data, the corresponding staggered magnetization emerges gradually upon cooling and without a critical onset (Fig.", "REF and Fig.", "REF ).", "These specific heat and scattering data consistently point to cross-over phenomena rather than actual phase transitions and thus an absence of true symmetry breaking.", "Potentially relevant features that could lead to this unusual situation are (1) disorder, particularly for the weak interactions between layers (2) frustration (3) the interplay of alternating layers of spins with distinct quasi-2D Ising transitions and (4) Kondo screening.", "To distinguish between these scenarios and achieve a deeper understanding of the unusual sequence of transitions in $\\rm Ce_3Cu_4As_4O_{2}$ will require single crystalline samples from which more specific information can be obtained via magnetic neutron diffraction." ], [ "Conclusions", "In conclusion, field dependent physical properties and neutron scattering measurements as well as band structure calculations were carried out on the layered transition metal pnictide compound $\\rm Ce_3Cu_4As_4O_{2}$ .", "All investigations point to a quasi-two-dimensional electronic state and modulated complex magnetic structure.", "The crystal structure resembles an interleaving 122 and 1111 Fe pnictide structures, resulting in two rare earth sites with very different local environments, which in turn lead to the complex magnetic structure.", "The neutron scattering measurements reveal three successive transitions, consistent with the magnetization and specific heat measurements.", "For the first transition at T$_{N}$ = 24 K, neutron scattering indicates alternative FM layers with spins oriented along c. Below the second transition T$_{2}$ = 16 K neutron scattering reveals the loss of spin fluctuations which we can account for by in-plane AFM ordering of Ce1 site.", "The third transition $T_{3}$ = 1.9 K appears to be a spin-flop transition where all the magnetic moment directions switch to in-plane polarization.", "Band structure calculation confirm the AFM2 state, which models the magnetic phase below $T_{3}$ .", "A notably large negative magneto-resistance ($\\approx $ -30 $\\%$ ) around $T_{3}$ is observed up to H = 9 T, implying significant impact for magnetic fluctuations on the transport behavior.", "Under high magnetic field H $>$ 3 T, the temperature dependence for both resistivity and heat capacity violate Fermi liquid behavior.", "This is possibly due to the anisotropic quasi-two-dimensional nature of the electronic state as indicated by neutron data and the band structure calculations." ], [ "Acknowledgements", "The work at Rice University was supported by AFOSR MURI.", "Work at IQM was supported by the US Department of Energy, office of Basic Energy Sciences, Division of Material Sciences and Engineering under grant DE-FG02-08ER46544.", "This work utilized facilities supported in part by the National Science Foundation under Agreement No.", "DMR-0944772.", "The authors thank Andriy Nevidomskyy, Meigan Aronson and Liang Zhao for useful discussions." ] ]	1606.04937
[ [ "Monte Carlo Set-Membership Filtering for Nonlinear Dynamic Systems" ], [ "Abstract When underlying probability density functions of nonlinear dynamic systems are unknown, the filtering problem is known to be a challenging problem.", "This paper attempts to make progress on this problem by proposing a new class of filtering methods in bounded noise setting via set-membership theory and Monte Carlo (boundary) sampling technique, called Monte Carlo set-membership filter.", "The set-membership prediction and measurement update are derived by recent convex optimization methods based on S-procedure and Schur complement.", "To guarantee the on-line usage, the nonlinear dynamics are linearized about the current estimate and the remainder terms are then bounded by an optimization ellipsoid, which can be described as a semi-infinite optimization problem.", "In general, it is an analytically intractable problem when dynamic systems are nonlinear.", "However, for a typical nonlinear dynamic system in target tracking, we can analytically derive some regular properties for the remainder.", "Moreover, based on the remainder properties and the inverse function theorem, the semi-infinite optimization problem can be efficiently solved by Monte Carlo boundary sampling technique.", "Compared with the particle filter, numerical examples show that when the probability density functions of noises are unknown, the performance of the Monte Carlo set-membership filter is better than that of the particle filter." ], [ "Introduction", "Filtering techniques for dynamic systems are widely used in applied fields such as target tracking, signal processing, automatic control, computer vision and economics, just to name a few.", "The Kalman filter [1] is well known as the recursive best linear unbiased state estimator, which is clearly established as a fundamental tool for analyzing and solving a broad class of filtering problems with linear dynamic systems.", "When dynamic systems are nonlinear, a few well-known generalizations are the extended Kalman filter (EKF), Gaussian sum filters and unscented Kalman filtering (UKF) (see, e.g., [2], [3]).", "These methods are based on local linear approximations of the nonlinear system where the higher order terms are ignored.", "Most recently, researchers have been attracted to a new class of filtering methods based on the sequential Monte Carlo approach for nonlinear and non-Gaussian dynamic systems.", "Sequential Monte Carlo methods achieve the filtering task by recursively generating weighted Monte Carlo samples of the state variables by importance sampling.", "The samples and their weights are then used to estimate expectation, covariance and other system characteristics.", "The earliest two methods is the particle filter (also called the bootstrap filter) [4] and sequential imputation for general missing data problems [5].", "Subsequently, a lot of methods have been developed in different situations.", "A sequential importance sampling framework [6] has been proposed to unify and generalize these methods.", "Monte Carlo filtering techniques have caught the attention of researchers in many different fields.", "Many excellent results in different situations can be found in, e.g., [7], [8], [9], [10], [11], and references therein.", "Most of these methods are based on the assumptions that probability density functions of the state noise and measurement noise are known.", "When underlying probability density functions (pdf) are unknown, the filtering problem for nonlinear dynamic systems is known to be a difficult problem.", "Actually, when the underlying probabilistic assumptions are not realistic (e.g., the main perturbation may be deterministic), it seems more natural to assume that the state noise and measurement noise are unknown but bounded and to characterize the set of all values of the parameter or state vector that are consistent with this hypothesis [12].", "The set-membership estimation was considered first at end of 1960s and early 1970s (see [13], [14]).", "The idea of propagating bounding ellipsoids (or boxes, polytopes, simplexes, parallelotopes, and polytopes) for systems with bounded noises has also been extensively investigated, for example, see recent papers [15], [16], [12], [17], the book [18], and references therein.", "Most of these methods concentrate on the linear dynamic systems.", "The set-membership filtering for nonlinear dynamic systems is known to be a challenging problem.", "Based on ellipsoid-bounded, fuzzy-approximated or Lipschitz-like nonlinearities, several results have been made [19], [20], [21], [22].", "These results assume that the ellipsoid bounds, the coefficients of fuzzy-approximation or Lipschitz constants are known before filtering, which limit them in real-time implementation.", "For example, for a typical nonlinear dynamic system in a radar, the bounds of the remainder depends on the past estimates so that they cannot be obtained before filtering.", "As far as we know, [23] develops a nonlinear set-membership filtering which can estimate ellipsoid bounds of nonlinearities in real-time and is capable of being on-line usage, and the filter is called the extended set-membership filter (ESMF).", "Specifically, the nonlinear dynamics are linearized about the current estimate and the state bounding ellipsoid is relaxed to an outer bounding box by the ellipsoid projection method, the remainder terms are then bounded using interval mathematics [24], and finally the output interval box is bounded using an outer bounding ellipsoid by minimizing the volume of the bounding ellipsoid.", "Moreover, the set-membership filtering algorithm is derived based on the linear set-membership filtering in the earliest work [13].", "It is not difficult to see that the outer bounding ellipsoids of both the remainder and the state is conservative.", "The cumulative effect of the conservative bounding ellipsoid at each time step may yield disconvergence of a filtering.", "In fact, if the state bounding ellipsoid were not relaxed to an outer bounding box by the ellipsoid projection method and using some recent linear set-membership filtering techniques [25], it should be possible to derive the tighter outer bounding ellipsoids for both the remainder and the state of the nonlinear dynamic system.", "More details will be clarified in Remark REF and Figure REF .", "In this paper, when underlying pdfs of nonlinear dynamic systems are unknown, we attempt to make progress on the corresponding filtering problem in the bounded noise setting.", "We propose a new class of filtering methods via set-membership estimation theory and Monte Carlo (boundary) sampling technique, denoted by MCSMF.", "The set-membership prediction and measurement update of MCSMF are derived by recent convex optimization methods based on S-procedure and Schur complement.", "To guarantee the on-line usage, the nonlinear dynamics are linearized about the current estimate and the remainder terms are then bounded by an ellipsoid, which can be described as a semi-infinite optimization problem.", "In general, it is an analytically intractable problem when dynamic systems are nonlinear.", "However, for a typical nonlinear dynamic system in target tracking, we can analytically derive some regular properties for the remainder.", "Moreover, based on the remainder properties and the inverse function theorem, we prove that the boundary of the remainder set must be from the the boundary of a set $\\lbrace \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ when we linearize the nonlinear equations by Taylor's Theorem.", "Thus, when we take samples from the set $\\lbrace \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ , the samples on the boundary $\\lbrace \|\|{\\bf u}_k\|\|=1\\rbrace $ are sufficient to derive the outer bounding ellipsoids of the remainder set.", "The samples in $\\lbrace \|\|{\\bf u}_k\|\|<1\\rbrace $ is not necessary.", "Therefore, the computation complexity can be reduced much more.", "Compared with the particle filter and ESMF in [23], numerical examples show that when the probability density functions of noises are known, the performance of the particle filter is better than that of ESMF and MCSMF.", "Nevertheless, when the probability density functions of noises are unknown, the performance of MCSMF is better than that of the other two filters.", "The rest of the paper is organized as follows.", "Preliminaries are given in Section .", "In Section , the prediction step and the measurement update step of the set-membership filtering for nonlinear dynamic systems are derived by solving an SDP problem based on S-procedure and Schur complement, respectively.", "In Section REF , the bounding ellipsoid of the remainder set is described as a semi-infinite optimization problem and the steps of MCSMF is summarized.", "In Section REF , for a typical nonlinear dynamic system in target tracking, some regular properties for the remainder is derived.", "Based on the remainder properties and the inverse function theorem, the semi-infinite optimization problem can be efficiently solved by Monte Carlo boundary sampling technique.", "In Section , numerical examples are given and discussed.", "In Section , concluding remarks are provided." ], [ "Problem formulation", "We consider a nonlinear dynamic system ${\\bf x}_{k+1}&=&f_k({\\bf x}_k)+{\\bf w}_k,\\\\[3mm]{\\bf y}_k&=&h_k({\\bf x}_k)+{\\bf v}_k,$ where ${\\bf x}_k\\in \\mathcal {R}^n$ is the state of system at time $k$ ; ${\\bf y}_k\\in \\mathcal {R}^{n_1}$ is the measurement.", "$f_k({\\bf x}_k)$ and $h_k({\\bf x}_k)$ are nonlinear functions of ${\\bf x}_k$ , ${\\bf w}_k\\in \\mathcal {R}^n$ is the uncertain process noise and ${\\bf v}_k\\in \\mathcal {R}^{n_1}$ is the uncertain measurement noise.", "They are assumed to be confined to specified ellipsoidal sets $\\nonumber {\\bf W}_k&=&\\lbrace {\\bf w}_k: {\\bf w}_k^T{\\bf Q}_k^{-1}{\\bf w}_k\\le 1\\rbrace \\\\\\nonumber {\\bf V}_k&=&\\lbrace {\\bf v}_k: {\\bf v}_k^T{\\bf R}_k^{-1}{\\bf v}_k\\le 1\\rbrace ,$ where ${\\bf Q}_k$ and ${\\bf R}_k$ are the shape matrix of the ellipsoids ${\\bf W}_k$ and ${\\bf V}_k$ , respectively, which are known symmetric positive-definite matrices.", "Moreover, we assume that when the nonlinear functions are linearized, the remainder terms can be bounded by an ellipsoid.", "Specifically, by Taylor's Theorem, $f_k$ and $h_k$ can be linearized to $ f_k(\\hat{{\\bf x}}_k+{\\bf E}_{f_k}{\\bf u}_k)=f_k(\\hat{{\\bf x}}_k)+{\\bf J}_{f_k}{\\bf E}_{f_k}{\\bf u}_k+\\Delta f_k({\\bf u}_k),\\\\[3mm] h_k(\\hat{{\\bf x}}_k+{\\bf E}_{h_k}{\\bf u}_k)=h_k(\\hat{{\\bf x}}_k)+{\\bf J}_{h_k}{\\bf E}_{h_k}{\\bf u}_k+\\Delta h_k({\\bf u}_k),$ where ${\\bf J}_{f_k}=\\frac{\\partial f_k({\\bf x}_k)}{\\partial {\\bf x}}\|_{\\hat{{\\bf x}}_k}$ and ${\\bf J}_{h_k}=\\frac{\\partial h_k({\\bf x}_k)}{\\partial {\\bf x}}\|_{\\hat{{\\bf x}}_k}$ are Jacobian matrices, $\\Delta f_k({\\bf u}_k)$ and $\\Delta h_k({\\bf u}_k)$ are high-order remainders, which can be bounded in an ellipsoid for all $\|\|{\\bf u}_k\|\|\\le 1$ , respectively, i.e., $ \\Delta f_k({\\bf u}_k)\\in \\mathcal {E}_{f_k} &=&\\lbrace {\\bf x}\\in R^n:({\\bf x}-{\\bf e}_{f_k})^T{({\\bf P}_{f_k})}^{-1}({\\bf x}-{\\bf e}_{f_k})\\le 1\\rbrace ,\\\\[3mm]&=&\\lbrace {\\bf x}\\in R^n: {\\bf x}={\\bf e}_{f_k}+{\\bf B}_{f_k}\\Delta _{f_k}, {\\bf P}_{f_k}={\\bf B}_{f_k}{\\bf B}_{f_k}^T, \\parallel \\Delta _{f_k}\\parallel \\le 1\\rbrace ,\\\\ \\Delta h_k({\\bf u}_k)\\in \\mathcal {E}_{h_k} &=&\\lbrace {\\bf x}\\in R^{n_1}:({\\bf x}-{\\bf e}_{h_k})^T{({\\bf P}_{h_k})}^{-1}({\\bf x}-{\\bf e}_{h_k})\\le 1\\rbrace ,\\\\[3mm]&=&\\lbrace {\\bf x}\\in R^{n_1}: {\\bf x}={\\bf e}_{h_k}+{\\bf B}_{h_k}\\Delta _{h_k}, {\\bf P}_{h_k}={\\bf B}_{h_k}{\\bf B}_{h_k}^T, \\parallel \\Delta _{h_k}\\parallel \\le 1\\rbrace ,$ where ${\\bf e}_{f_k}$ and ${\\bf e}_{h_k}$ are the center of the ellipsoids $\\mathcal {E}_{f_k}$ and $\\mathcal {E}_{h_k}$ , respectively; ${\\bf P}_{f_k}$ and ${\\bf P}_{h_k}$ are the shape matrices of the ellipsoids $\\mathcal {E}_{f_k}$ and $\\mathcal {E}_{h_k}$ respectively.", "Note that we do not assume that the ellipsoids $\\mathcal {E}_{f_k}$ and $\\mathcal {E}_{h_k}$ are given before filtering.", "Both of them are predicated in real time.", "The corresponding set-membership filtering problem can be formulated as follows.", "Suppose that the initial state ${\\bf x}_0$ belongs to a given bounding ellipsoid: $ \\mathcal {E}_0&=&\\lbrace {\\bf x}\\in R^n:({\\bf x}-\\hat{{\\bf x}}_0)^T({\\bf P}_0)^{-1}({\\bf x}-\\hat{{\\bf x}}_0)\\le 1\\rbrace ,$ where $\\hat{{\\bf x}}_0$ is the center of ellipsoid $\\mathcal {E}_0$ ; ${\\bf P}_0$ is the shape matrix of the ellipsoid $\\mathcal {E}_0$ which is a known symmetric positive-definite matrix.", "At time $k$ , given that ${\\bf x}_k$ belongs to a current bounding ellipsoid: $ \\mathcal {E}_k&=&\\lbrace {\\bf x}\\in R^n:({\\bf x}-\\hat{{\\bf x}}_k)^T({\\bf P}_k)^{-1}({\\bf x}-\\hat{{\\bf x}}_k)\\le 1\\rbrace \\\\&=&\\lbrace {\\bf x}\\in R^n: {\\bf x}=\\hat{{\\bf x}}_k+{\\bf E}_k{\\bf u}, {\\bf P}_k={\\bf E}_k{\\bf E}_k^T, \\parallel {\\bf u}\\parallel \\le 1\\rbrace ,$ where $\\hat{{\\bf x}}_k$ is the center of ellipsoid $\\mathcal {E}_k$ ; ${\\bf P}_k$ is a known symmetric positive-definite matrix.", "The goal of the set-membership filtering is to determine a bounding ellipsoid $\\mathcal {E}_{k+1}$ based on the measurement ${\\bf y}_{k+1}$ at time $k+1$ , i.e, look for $\\hat{{\\bf x}}_{k+1}, {\\bf P}_{k+1}$ such that the state ${\\bf x}_{k+1}$ belongs to $\\mathcal {E}_{k+1} &=&\\lbrace {\\bf x}\\in R^n:({\\bf x}-\\hat{{\\bf x}}_{k+1})^T{({\\bf P}_{k+1})}^{-1}({\\bf x}-\\hat{{\\bf x}}_{k+1})\\le 1\\rbrace ,$ whenever I) ${\\bf x}_k$ is in $\\mathcal {E}_k$ , II) the process and measurement noises ${\\bf w}_k, {\\bf v}_{k+1}$ are bounded in ellipsoids, i.e.", "${\\bf w}_k\\in {\\bf W}_k$ , ${\\bf v}_{k+1}\\in {\\bf V}_{k+1}$ , and III) the remainders $\\Delta f_k({\\bf u}_k)\\in \\mathcal {E}_{f_k} $ and $\\Delta h_k({\\bf u}_k)\\in \\mathcal {E}_{h_k}$ .", "The key problem is how to determine the bounding ellipsoids $\\mathcal {E}_{f_k}$ and $\\mathcal {E}_{h_k}$ in real-time so that the filtering algorithm can be on-line usage.", "Moreover, we provide a state estimation ellipsoid by minimizing its “size\" which is a function of the shape matrix $P$ and is denoted by $f(P)$ .", "It is well known that $tr(P)$ corresponds to the sum of squares of semiaxes lengths of the ellipsoid, and $logdet(P)$ is related to the volume of the ellipsoid.", "More discussion on size of the ellipsoid can be seen in [17]." ], [ "Set-membership prediction and measurement update", "In this section, we derive the prediction step and the measurement step of the set-membership filtering.", "Both of them can be converted to solve an SDP problem based on S-procedure and Schur complement.", "The main results are summarized to Theorems 1-2.", "The proofs are given in Appendix." ], [ "Prediction step", "Theorem 3.1 At time $k+1$ , based on measurements ${\\bf y}_{k}$ , the bounding ellipsoids $\\mathcal {E}_{f_k}$ and $\\mathcal {E}_{h_k}$ , a predicted bounding ellipsoid $ \\mathcal {E}_{k+1\|k}=\\lbrace {\\bf x}:({\\bf x}-\\hat{{\\bf x}}_{k+1\|k})^T({\\bf P}_{k+1\|k})^{-1}({\\bf x}-\\hat{{\\bf x}}_{k+1\|k})\\le 1\\rbrace $ can be obtained by solving the optimization problem in the variables ${\\bf P}_{k+1\|k}$ , $\\hat{{\\bf x}}_{k+1\|k}$ , nonnegative auxiliary variables $\\tau ^u\\ge 0, \\tau ^w\\ge 0,\\tau ^v\\ge 0,\\tau ^f\\ge 0, \\tau ^h\\ge 0$ , $ &&\\min ~~ f({\\bf P}_{k+1\|k}) \\\\[5mm]&&~~\\mbox{subject to}~~ -\\tau ^u\\le 0,~ -\\tau ^w\\le 0, ~-\\tau ^v\\le 0,-\\tau ^f\\le 0,~ -\\tau ^h\\le 0,\\\\[5mm] && -{\\bf P}_{k+1\|k}\\prec 0,\\\\[5mm]&&\\left[\\begin{array}{cc}-{\\bf P}_{k+1\|k}&\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }\\\\[3mm](\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot })^T& ~~-(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }^T\\Xi (\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }\\\\\\end{array}\\right]\\preceq 0,$ where $ \\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})&=&[f_k(\\hat{{\\bf x}}_{k})+{\\bf e}_{f_k}-\\hat{{\\bf x}}_{k+1\|k},~{\\bf J}_{f_k}{\\bf E}_{k},{\\bf I}, ~0, ~{\\bf B}_{f_k}, ~0], ~~0\\in \\mathcal {R}^{n,n_1},\\\\[3mm]\\Psi _{k+1\|k}({\\bf y}_{k})&=& [h_{k}(\\hat{{\\bf x}}_{k})+{\\bf e}_{h_{k}}-{\\bf y}_{k}, ~{\\bf J}_{h_{k}}{\\bf E}_k, ~0,~{\\bf I}, ~0, ~{\\bf B}_{h_{k}}].$ $(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }$ is the orthogonal complement of $\\Psi _{k+1\|k}({\\bf y}_{k})$ .", "${\\bf E}_{k}$ is the Cholesky factorization of ${\\bf P}_{k}$ , i.e, ${\\bf P}_{k}={\\bf E}_{k}({\\bf E}_{k})^T$ .", "${\\bf e}_{f_k}$ , ${\\bf e}_{h_k}$ , ${\\bf B}_{f_k}$ , ${\\bf B}_{h_k}$ are denoted by () and (), respectively.", "${\\bf J}_{f_k}=\\frac{\\partial f_k({\\bf x}_k)}{\\partial {\\bf x}}\|_{\\hat{{\\bf x}}_k}$ and ${\\bf J}_{h_k}=\\frac{\\partial h_k({\\bf x}_k)}{\\partial {\\bf x}}\|_{\\hat{{\\bf x}}_k}$ .", "$\\Xi &=&\\qopname{}{o}{diag}(1-\\tau ^u-\\tau ^w-\\tau ^v-\\tau ^f-\\tau ^h,\\tau ^uI,\\tau ^w{\\bf Q}_k^{-1}, \\tau ^v{\\bf R}_k^{-1},\\tau ^fI,\\tau ^hI).$ Proof: See Appendix." ], [ "Measurement update step", "Theorem 3.2 At time $k+1$ , based on measurements ${\\bf y}_{k+1}$ , the predicted bounding ellipsoid $ \\mathcal {E}_{k+1\|k}$ and the bounding ellipsoid $\\mathcal {E}_{h_{k+1}}$ , a bounding ellipsoid $ \\mathcal {E}_{k+1}=\\lbrace {\\bf x}:({\\bf x}-\\hat{{\\bf x}}_{k+1})^T({\\bf P}_{k+1})^{-1}({\\bf x}-\\hat{{\\bf x}}_{k+1})\\le 1\\rbrace $ can be obtained by solving the optimization problem in the variables ${\\bf P}_{k+1}$ , $\\hat{{\\bf x}}_{k+1}$ , nonnegative auxiliary variables $\\tau ^u\\ge 0,\\tau ^v\\ge 0, \\tau ^h\\ge 0$ , $ &&\\min ~~ f({\\bf P}_{k+1}) \\\\[5mm] &&~~\\mbox{subject to}~~ -\\tau ^u\\le 0,~-\\tau ^v\\le 0,~ -\\tau ^h\\le 0,\\\\[5mm] && -{\\bf P}_{k+1}\\prec 0,\\\\[5mm]&&\\left[\\begin{array}{cc}-{\\bf P}_{k+1}&\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }\\\\[3mm](\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot })^T& ~~-(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }^T\\Xi (\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }\\\\\\end{array}\\right]\\preceq 0,$ where $ \\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})&=&[\\hat{{\\bf x}}_{k+1\|k}-\\hat{{\\bf x}}_{k+1},{\\bf E}_{k+1\|k}, ~0, ~0], ~~0\\in \\mathcal {R}^{n,n_1}, \\\\[3mm] \\Psi _{k+1}({\\bf y}_{k+1})&=& [h_{k+1}(\\hat{{\\bf x}}_{k+1\|k})+{\\bf e}_{h_{k+1}}-{\\bf y}_{k+1},{\\bf J}_{h_{k+1\|k}}{\\bf E}_{k+1\|k}, ~{\\bf I}, ~{\\bf B}_{h_{k+1}}].$ $(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }$ is the orthogonal complement of $\\Psi _{k+1}({\\bf y}_{k+1})$ .", "${\\bf E}_{k+1\|k}$ is the Cholesky factorization of ${\\bf P}_{k+1\|k}$ , i.e, ${\\bf P}_{k+1\|k}={\\bf E}_{k+1\|k}({\\bf E}_{k+1\|k})^T$ .", "$\\hat{{\\bf x}}_{k+1\|k}$ is the center of the predicted bounding ellipsoid $ \\mathcal {E}_{k+1\|k}$ .", "${\\bf e}_{h_{k+1}}$ and ${\\bf B}_{h_{k+1}}$ are denoted by $(\\ref {Eqpre_153})$ at the time step $k+1$ .", "${\\bf J}_{h_{k+1\|k}}=\\frac{\\partial h_{k+1}({\\bf x}_k)}{\\partial {\\bf x}}\|_{\\hat{{\\bf x}}_{k+1\|k}}$ .", "$ \\Xi =\\qopname{}{o}{diag}(1-\\tau ^u-\\tau ^v-\\tau ^h,\\tau ^u{\\bf I},\\tau ^v{\\bf R}_{k+1}^{-1},\\tau ^hI).$ Proof: See Appendix.", "Remark 3.3 Notice that if $f(P)=tr(P)$ , the optimization problem in Theorems REF -REF is an SDP problem.", "If $f(P)=\\mbox{logdet(P)}$ , it is a MAXDET problem.", "Both of them can also be efficiently solved in polynomial-time by interior point methods for convex programming (see, e.g., [16], [26]) and related softwares [27], [28]." ], [ "Monte Carlo Set Membership Filtering", "In this section, we discuss the key problem that how to adaptively determine a bounding ellipsoid to cover the high-order remainders.", "In the first subsection, for the general case, the problem can be converted to solve a SDP problem via Monte Carlo sampling.", "Moreover, the Monte Carlo set membership filtering is presented.", "In the second subsection, for target tracking, we prove that the remainder can be bounded via Monte Carlo boundary sampling.", "Thus, the computation complexity Algorithm REF can be reduced much more." ], [ "Ellipsoid bounding of the remainder via Monte Carlo sampling", "By (REF )-(), the high-order remainders are $\\nonumber \\Delta f_k({\\bf u}_k)=f_k(\\hat{{\\bf x}}_k+{\\bf E}_k{\\bf u}_k)-f_k(\\hat{{\\bf x}}_k)-{\\bf J}_{f_k}{\\bf E}_k{\\bf u}_k,\\\\\\nonumber \\Delta h_k({\\bf u}_k)=h_k(\\hat{{\\bf x}}_k+{\\bf E}_k{\\bf u}_k)-h_k(\\hat{{\\bf x}}_k)-{\\bf J}_{h_k}{\\bf E}_k{\\bf u}_k,$ whenever $\\parallel {\\bf u}_k\\parallel \\le 1$ .", "Obviously, it is a hard problem to cover a remainder by an ellipsoid since $f_k$ and $h_k$ are generally nonlinear functions.", "The outer bounding ellipsoid for $\\Delta f_k({\\bf u}_k)$ is not uniquely defined, but which can be optimized by minimizing the size $f(P)$ of the bounding ellipsoid.", "Thus, the optimization problem for the bounding ellipsoid of $\\Delta f_k({\\bf u}_k)$ can be written as $ &&\\min ~~ f({\\bf P}_{f_k}) \\\\[5mm] && \\mbox{subject to}~ (\\Delta f_k({\\bf u}_k)-{\\bf e}_{f_k})^T({\\bf P}_{f_k})^{-1}(\\Delta f_k({\\bf u}_k)-{\\bf e}_{f_k})\\le 1, \\mbox{for~ all}~ \|\|{\\bf u}_k\|\|\\le 1.$ where ${\\bf P}_{f_k}={\\bf B}_{f_k}{\\bf B}_{f_k}^T$ , and ${\\bf e}_{f_k}$ , ${\\bf P}_{f_k}$ are decision variables.", "It is called a semi-infinite optimization problem by [29].", "For a general nonlinear dynamic system, to solve the problem (REF ), we may use Monte Carlo sampling by uniformly taking some samples from the boundary and interior-points of the sphere $\|\|{\\bf u}_k\|\|\\le 1$ so that we can get a finite set of ${\\bf u}_k^1,\\ldots ,{\\bf u}_k^N$ , then the infinite constraint () can be approximated by $N$ constraints based on ${\\bf u}_k^1,\\ldots ,{\\bf u}_k^N$ .", "Moreover, by Schur complement, an approximate bounding ellipsoid for $\\Delta f_k({\\bf u}_k)$ can be derived by solving the flowing SDP optimization problem: $ &&\\min ~~ f({\\bf P}_{f_k}) \\\\[5mm]&&\\mbox{subject to}~ \\left[\\begin{array}{cc}-1 & (\\Delta f_k({\\bf u}_k^i)-{\\bf e}_{f_k})^T \\\\\\Delta f_k({\\bf u}_k^i)-{\\bf e}_{f_k} & -{\\bf P}_{f_k} \\\\\\end{array}\\right] \\prec 0, i=1,\\ldots ,N.$ Similarly, the outer bounding ellipsoid for $h_k({\\bf u}_k)$ can be derived by solving $ &&\\min ~~ f({\\bf P}_{h_{k}}) \\\\[5mm] &&\\mbox{subject to}~ \\left[ \\begin{array}{cc}-1 & (\\Delta h_{k}({\\bf u}_k^i)-{\\bf e}_{h_{k}})^T \\\\\\Delta h_{k}({\\bf u}_k^i)-{\\bf e}_{h_{k}} & -{\\bf P}_{h_{k}} \\\\\\end{array}\\right] \\prec 0, i=1,\\ldots ,N.$ Remark 4.1 The problem (REF ) is an SDP problem that can be efficiently solved using modern interior-point methods, which have been developed by [26] and [30].", "When large number of samples are required to guarantee the bounding ellipsoid contain the remainder, the one-order optimizing algorithm [31] may be used for solving the problem (REF ) with a lower computation complexity.", "In addition, in the next subsection, we will develop boundary sampling technique for target tracking, where the samples on boundary are sufficient to derive the outer bounding ellipsoids of the remainder.", "Thus, computation complexity can be reduced much more.", "Numerical examples show that only 50 uniform samples on the boundary are enough to guarantee the bounding ellipsoid contain the remainder.", "Remark 4.2 Note that the bounding ellipsoid of [23] is derived by interval mathematics.", "We derive the bounding ellipsoid by solving a semi-infinite optimization problem.", "Figure REF illustrates the difference of two methods.", "It is obvious to see that the bounding ellipsoid derived by solving the SDP (REF ) is tighter than that obtained by interval mathematics.", "The cumulative effect of the conservative bounding ellipsoid at each time step may yield disconvergence of a filtering.", "Figure: (top) The bounding ellipsoid is derived by covering the solid points of the remainder which are obtained by Monte Carlo sampling; (bottom) The bounding ellipsoid is derived by covering the vertices of the rectangle obtained by interval mathematics .Based on Theorems REF –REF and the ellipsoids derived by solving the SDP optimization problems (REF )-(), (REF )-(), the filtering algorithm can be summarized as follows: Algorithm 4.3 (Monte Carlo Set Membership Filtering) Step 1: (Initialization step) Set $k=0$ and initial values $(\\hat{{\\bf x}}_0,{\\bf P}_0)$ such that ${\\bf x}_0\\in \\mathcal {E}_0$ .", "Step 2: (Bounding step) Take samples ${\\bf u}_k^1,\\ldots ,{\\bf u}_k^N$ from the sphere $\|\|{\\bf u}_k\|\|\\le 1$ , and then determine two bounding ellipsoids to cover the remainders $\\Delta f_k$ and $\\Delta h_k$ by (REF )-() and (REF )-(), respectively.", "Step 3: (Prediction step) Optimize the center and shape matrix of the state prediction ellipsoid $(\\hat{{\\bf x}}_{k+1\|k},{\\bf P}_{k+1\|k})$ such that ${\\bf x}_{k+1\|k}\\in \\mathcal {E}_{k+1\|k}$ by solving the optimization problem (REF )-().", "Step 4: (Bounding step) Take samples ${\\bf u}_{k+1\|k}^1,\\ldots ,{\\bf u}_{k+1\|k}^N$ from the sphere $\|\|{\\bf u}_{k+1\|k}\|\|\\le 1$ , and then determine one bounding ellipsoid to cover the remainder $\\Delta h_{k+1\|k}$ by (REF )-().", "Step 5: (Measurement update step) Optimize the center and shape matrix of the state estimation ellipsoid $(\\hat{{\\bf x}}_{k+1},{\\bf P}_{k+1})$ such that ${\\bf x}_{k+1}\\in \\mathcal {E}_{k+1}$ by solving the optimization problem (REF )-().", "Step 6: Set $k=k+1$ and go to step 2.", "A flowchart of the Algorithm REF is given in Figure REF .", "Figure: The flowchart of Algorithm ." ], [ "Monte Carlo set membership filtering for target tracking", "In this subsection, for a typical nonlinear dynamic system in target tracking, we discuss that the remainder can be bounded by an ellipsoid via Monte Carlo boundary sampling for target tracking.", "We prove that the boundary of the remainder set $\\lbrace \\Delta h_{k+1}({\\bf u}_k): \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ must be from the the boundary of the sphere $\\lbrace \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ when we linearize the nonlinear equations by Taylor's Theorem.", "Thus, when we take samples from the set $\\lbrace \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ , the samples on the boundary $\\lbrace \|\|{\\bf u}_k\|\|=1\\rbrace $ are sufficient to derive the outer bounding ellipsoids of the remainder set.", "Therefore, the computation complexity in the bounding steps of Algorithm REF can be reduced much more.", "Let us consider the following nonlinear measurement equation [2]: $ h({\\bf x})=\\left[\\begin{array}{c}\\sqrt{({\\bf x}(1)-a)^2+({\\bf x}(2)-b)^2} \\\\[3mm]arctan\\left(\\frac{{\\bf x}(2)-b}{{\\bf x}(1)-a}\\right) \\\\\\end{array}\\right], a, b \\in \\mathcal {R}$ where ${\\bf x}$ is a four-dimensional state variable that includes position and velocity $(x, y, \\dot{x}, \\dot{y})$ .", "Note that the $h({\\bf x})$ only depends on the first two dimensions ${\\bf x}(1)$ and ${\\bf x}(2)$ .", "We discuss the relationship between the set $\\lbrace \|\|{\\bf u}_k\|\|\\le 1, {\\bf u}_k= [{\\bf u}_k(1) ~{\\bf u}_k(2)]\\rbrace $ and the remainder set $\\lbrace \\Delta h_{k+1}({\\bf u}_k): \|\|{\\bf u}_k\|\|\\le 1\\rbrace $ .", "Theorem 4.4 If we let the remainder $g({\\bf u})=h({\\bf x}+{\\bf E}{\\bf u})-h({\\bf x})-{\\bf J}_h{\\bf E}{\\bf u}$ where h(x) is defined in (REF ), ${\\bf E}$ is a Cholesky factorization of a positive-definite ${\\bf P}$ such that $\\lbrace {\\bf x}+{\\bf E}{\\bf u}:\\parallel u\\parallel \\le 1\\rbrace $ is not intersect with the radial ${\\bf x}(1)<=a, {\\bf x}(2)=b$ , then the boundary of the remainder set ${\\bf S}=\\lbrace g({\\bf u}): \\parallel {\\bf u}\\parallel \\le 1\\rbrace $ belongs to the set $\\lbrace g({\\bf u}): \\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "The proof of Theorem REF relies on the following three lemmas.", "Lemma 4.5 (Remainder Lemma) The determinant of the derivative of the remainder $g({\\bf u})$ is not less than 0, and the equality holds if and only if $c{\\bf u}(1)+d{\\bf u}(2)=0$ , where $c={\\bf E}_{11}({\\bf x}(2)-b)-{\\bf E}_{21}({\\bf x}(1)-a)$ , $d={\\bf E}_{12}({\\bf x}(2)-b)-{\\bf E}_{22}({\\bf x}(1)-a)$ , and ${\\bf E}_{ij}$ is the entry of the $ith$ row and the $jth$ column of the matrix ${\\bf E}$ .", "Meanwhile, if $c{\\bf u}(1)+d{\\bf u}(2)=0$ , then $g({\\bf u})=0$ .", "Proof.", "From the definition of the function $g({\\bf u})$ , it is easy to see that $g({\\bf u})$ is a continuously differentiable function.", "By simple calculations, Jacobian matrix ${\\bf J}_g$ of $g({\\bf u})$ is $\\nonumber &&{\\bf J}_g=\\\\\\nonumber &&\\left[\\begin{array}{cc}\\frac{\\bigtriangleup _1}{\\sqrt{\\bigtriangleup _1^2+\\bigtriangleup _2^2}}- \\frac{\\bigtriangledown _1}{\\sqrt{\\bigtriangledown _1^2+\\bigtriangledown _2^2}}& \\frac{\\bigtriangleup _2}{\\sqrt{\\bigtriangleup _1^2+\\bigtriangleup _2^2}}- \\frac{\\bigtriangledown _2}{\\sqrt{\\bigtriangledown _1^2+\\bigtriangledown _2^2}} \\\\\\frac{-\\bigtriangleup _2}{\\bigtriangleup _1^2+\\bigtriangleup _2^2}- \\frac{-\\bigtriangledown _2}{\\bigtriangledown _1^2+\\bigtriangledown _2^2}& \\frac{\\bigtriangleup _1}{\\bigtriangleup _1^2+\\bigtriangleup _2^2}- \\frac{\\bigtriangledown _1}{\\bigtriangledown _1^2+\\bigtriangledown _2^2} \\\\\\end{array}\\right]{\\bf E}\\\\[3mm]\\nonumber && ={\\bf J}_h{\\bf E}.$ where $\\bigtriangleup _1&=&{\\bf x}(1)+{\\bf E}_{11}{\\bf u}(1)+{\\bf E}_{12}{\\bf u}(2)-a,\\\\\\bigtriangleup _2&=&{\\bf x}(2)+{\\bf E}_{21}{\\bf u}(1)+{\\bf E}_{22}{\\bf u}(2)-b,\\\\\\bigtriangledown _1&=&{\\bf x}(1)-a,\\\\\\bigtriangledown _2&=&{\\bf x}(2)-b,$ $\\nonumber &&{\\bf J}_h=\\\\\\nonumber &&\\left[\\begin{array}{cc}\\frac{\\bigtriangleup _1}{\\sqrt{\\bigtriangleup _1^2+\\bigtriangleup _2^2}}- \\frac{\\bigtriangledown _1}{\\sqrt{\\bigtriangledown _1^2+\\bigtriangledown _2^2}}& \\frac{\\bigtriangleup _2}{\\sqrt{\\bigtriangleup _1^2+\\bigtriangleup _2^2}}- \\frac{\\bigtriangledown _2}{\\sqrt{\\bigtriangledown _1^2+\\bigtriangledown _2^2}} \\\\\\frac{-\\bigtriangleup _2}{\\bigtriangleup _1^2+\\bigtriangleup _2^2}- \\frac{-\\bigtriangledown _2}{\\bigtriangledown _1^2+\\bigtriangledown _2^2}& \\frac{\\bigtriangleup _1}{\\bigtriangleup _1^2+\\bigtriangleup _2^2}- \\frac{\\bigtriangledown _1}{\\bigtriangledown _1^2+\\bigtriangledown _2^2} \\\\\\end{array}\\right].$ Simplifying the determinant of ${\\bf J}_h$ , $\\nonumber &&det({\\bf J}_h)=\\\\\\nonumber &&~~~\\frac{\\left(\\sqrt{(\\bigtriangleup _1^2+\\bigtriangleup _2^2)(\\bigtriangledown _1^2+\\bigtriangledown _2^2)}-(\\bigtriangleup _1\\bigtriangledown _1+\\bigtriangleup _2\\bigtriangledown _2)\\right)}{(\\bigtriangleup _1^2+\\bigtriangleup _2^2)(\\bigtriangledown _1^2+\\bigtriangledown _2^2)}\\\\\\nonumber &&~~~\\cdot \\left(\\sqrt{\\bigtriangleup _1^2+\\bigtriangleup _2^2}+\\sqrt{\\bigtriangledown _1^2+\\bigtriangledown _2^2}\\right).$ Thus, $det({\\bf J}_h)\\ge 0$ and the equality holds if and only if $\\bigtriangleup _2\\bigtriangledown _1=\\bigtriangleup _1\\bigtriangledown _2$ .", "Since $det({\\bf J}_g)=det({\\bf J}_h)det({\\bf E})$ and $det({\\bf E})>0$ , then $det({\\bf J}_g)\\ge 0$ and the equality holds if and only if $\\bigtriangleup _2\\bigtriangledown _1=\\bigtriangleup _1\\bigtriangledown _2$ , at the same time, we have $g({\\bf u})=0$ .", "Moreover, by (REF )-(), it is easy to see that $\\bigtriangleup _2\\bigtriangledown _1=\\bigtriangleup _1\\bigtriangledown _2$ is equivalent to $c{\\bf u}(1)+d{\\bf u}(2)=0$ , where $c={\\bf E}_{11}({\\bf x}(2)-b)-{\\bf E}_{21}({\\bf x}(1)-a)$ , $d={\\bf E}_{12}({\\bf x}(2)-b)-{\\bf E}_{22}({\\bf x}(1)-a)$ , and ${\\bf E}_{ij}$ is the entry of the $ith$ row and the $jth$ column of the matrix ${\\bf E}$ .", "Lemma 4.6 If the sets ${\\bf S}^1\\bigcup {\\bf S}^2={\\bf S}^3\\bigcup {\\bf S}^4$ , ${\\bf S}^3\\bigcap {\\bf S}^4=\\emptyset $ , ${\\bf S}^1\\subset {\\bf S}^3$ , then ${\\bf S}^4\\subset {\\bf S}^2$ .", "Proof.", "Since ${\\bf S}^1\\subset {\\bf S}^3$ , then ${\\bf S}^1\\bigcup {\\bf S}^2\\subset {\\bf S}^3\\bigcup {\\bf S}^2$ .", "Using ${\\bf S}^1\\bigcup {\\bf S}^2={\\bf S}^3\\bigcup {\\bf S}^4$ , we obtain ${\\bf S}^3\\bigcup {\\bf S}^4\\subset {\\bf S}^3\\bigcup {\\bf S}^2$ , then $({\\bf S}^3\\bigcup {\\bf S}^4)\\bigcap {\\bf S}^4\\subset ({\\bf S}^3\\bigcup {\\bf S}^2)\\bigcap {\\bf S}^4$ .", "By ${\\bf S}^3\\bigcap {\\bf S}^4=\\emptyset $ , we have ${\\bf S}^4\\subset {\\bf S}^4\\bigcap {\\bf S}^2$ .", "Moreover, ${\\bf S}^4\\subset {\\bf S}^2$ .", "Lemma 4.7 (Inverse Function Theorem by [32]) Suppose that $\\varphi :{\\bf R}^n\\rightarrow {\\bf R}^n$ is continuously differentiable in an open set containing ${\\bf u}$ , and $det(\\varphi ^{^{\\prime }}({\\bf u}))\\ne 0$ .", "Then there is an open set ${\\bf V}$ containing ${\\bf u}$ and open set ${\\bf W}$ containing $\\varphi ({\\bf u})$ such that $\\varphi : {\\bf V}\\rightarrow {\\bf W}$ has a continuous inverse $\\varphi ^{-1}: {\\bf W}\\rightarrow {\\bf V}$ which is differentiable and for all ${\\bf y}\\in {\\bf W}$ satisfies $\\nonumber (\\varphi ^{-1})^{^{\\prime }}({\\bf y})=[\\varphi ^{^{\\prime }}(\\varphi ^{-1}({\\bf y}))]^{-1}.$ Proof.", "[Proof of Theorem REF ] Since $g({\\bf u})$ is a continuous function in ${\\bf S}_1=\\lbrace \|\|{\\bf u}\|\|\\le 1\\rbrace $ and ${\\bf S}_1$ is compact, ${\\bf S}=\\lbrace g({\\bf u}): \\parallel {\\bf u}\\parallel \\le 1\\rbrace $ is compact [33].", "If we denote the interior and the boundary of the set ${\\bf S}$ by ${\\bf S}^i$ and ${\\bf S}^b$ respectively, then ${\\bf S}={\\bf S}^i\\bigcup {\\bf S}^b$ and ${\\bf S}^i\\bigcap {\\bf S}^b=\\emptyset $ .", "We need to prove ${\\bf S}^b\\subset \\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "By definition of the set ${\\bf S}_1$ , we can divide it into two parts, i.e., ${\\bf S}_1={\\bf S}_1^1\\bigcup {\\bf S}_1^2$ , where ${\\bf S}_1^1=\\lbrace {\\bf u}:\\parallel {\\bf u}\\parallel = 1~ or~ c{\\bf u}(1)+d{\\bf u}(2)=0\\rbrace $ , ${\\bf S}_1^2=\\lbrace {\\bf u}:\\parallel {\\bf u}\\parallel < 1, c{\\bf u}(1)+d{\\bf u}(2)\\ne 0\\rbrace $ and $c, d$ are defined in Lemma REF .", "According to the expression of the set ${\\bf S}$ , then, we can divide the set ${\\bf S}$ into the corresponding parts, i.e., ${\\bf S}={\\bf S}^1\\bigcup {\\bf S}^2$ , where ${\\bf S}^1=\\lbrace g({\\bf u}): {\\bf u}\\in {\\bf S}_1^1\\rbrace $ and ${\\bf S}^2=\\lbrace g({\\bf u}): {\\bf u}\\in {\\bf S}_1^2\\rbrace $ .", "Thus ${\\bf S}^1\\bigcup {\\bf S}^2={\\bf S}^i\\bigcup {\\bf S}^b$ .", "Next, we prove ${\\bf S}^2\\subset {\\bf S}^i$ .", "For $\\forall {\\bf z}\\in {\\bf S}^2$ , $\\exists {\\bf u}\\in {\\bf S}_1^2$ , s.t., ${\\bf z}=g({\\bf u})$ .", "From the definition of the set ${\\bf S}_1^2$ , we can see that $det({\\bf J}_g)>0$ with Lemma REF .", "Using Lemma REF , we can find an open set ${\\bf W}\\in {\\bf S}$ containing $g({\\bf u})$ , in other words, ${\\bf z}$ is the interior point of ${\\bf S}$ , i.e., ${\\bf z}\\in {\\bf S}^i$ , thus, ${\\bf S}^2\\subset {\\bf S}^i$ .", "According to Lemma REF , we can obtain ${\\bf S}^b\\subset {\\bf S}^1$ .", "Moreover, we prove that ${\\bf S}^1=\\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "Note that ${\\bf S}^1=\\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace \\bigcup \\lbrace g({\\bf u}):c{\\bf u}(1)+d{\\bf u}(2)=0\\rbrace $ .", "According to Lemma REF , it is obvious that $\\lbrace g({\\bf u}):c{\\bf u}(1)+d{\\bf u}(2)=0\\rbrace =\\lbrace 0\\rbrace $ .", "Let ${\\bf u}_0=[\\frac{-d}{\\sqrt{d^2+c^2}}~ \\frac{c}{\\sqrt{d^2+c^2}}]$ , then ${\\bf u}_0\\in \\lbrace {\\bf u}:c{\\bf u}(1)+d{\\bf u}(2)=0\\rbrace \\bigcap \\lbrace {\\bf u}:\\parallel {\\bf u}\\parallel = 1\\rbrace $ , we can also get $g({\\bf u}_0)\\in \\lbrace g({\\bf u}):c{\\bf u}(1)+d{\\bf u}(2)=0\\rbrace =\\lbrace 0\\rbrace $ and $g({\\bf u}_0)\\in \\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ , then $\\lbrace 0\\rbrace \\subset \\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "Thus, ${\\bf S}^1=\\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "Therefore, we have ${\\bf S}^b\\subset \\lbrace g({\\bf u}):\\parallel {\\bf u}\\parallel = 1\\rbrace $ , in other words, the boundary of ${\\bf S}=\\lbrace g({\\bf u}): \\parallel u\\parallel \\le 1\\rbrace $ belongs to the set $\\lbrace g({\\bf u}): \\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "Example 4.8 To illustrate Theorem REF , we give an example as follow: if $a=50$ , $b=100$ , $x=[80 ~130]^T$ , ${\\bf P}=diag(500,1000)$ , it is easy to check that $g({\\bf u})$ is continuously differentiable in set ${\\bf S}_1=\\lbrace {\\bf u}:\\parallel {\\bf u}\\parallel \\le 1\\rbrace $ .", "We divide ${\\bf S}_1$ into three parts, i.e., ${\\bf S}_1={\\bf A}^1\\cup {\\bf B}^1\\cup {\\bf C}^1$ , where ${\\bf A}^1=\\lbrace {\\bf u}: c{\\bf u}(1)+d{\\bf u}(2)<0, \\parallel {\\bf u}\\parallel \\le 1\\rbrace $ , ${\\bf B}^1=\\lbrace {\\bf u}: c{\\bf u}(1)+d{\\bf u}(2)>0, \\parallel {\\bf u}\\parallel \\le 1\\rbrace $ , and ${\\bf C}^1=\\lbrace {\\bf u}: c{\\bf u}(1)+d{\\bf u}(2)=0, \\parallel {\\bf u}\\parallel \\le 1\\rbrace $ .", "Meanwhile, we can also divide ${\\bf S}$ into the corresponding parts, such that ${\\bf A}=\\lbrace g({\\bf u}): {\\bf u}\\in {\\bf A}^1\\rbrace $ , ${\\bf B}=\\lbrace g({\\bf u}): {\\bf u}\\in {\\bf B}^1\\rbrace $ , ${\\bf C}=\\lbrace g({\\bf u}): {\\bf u}\\in {\\bf C}^1\\rbrace $ , then ${\\bf S}={\\bf A}\\cup {\\bf B}\\cup {\\bf C}$ .", "Figure REF shows that the separation area of the circle and their corresponding area of $g({\\bf u})$ .", "Three observations can be seen: The remainder set is the union of two sets.", "The (red) line ${\\bf C}^1$ is mapped to the point 0.", "The boundary of ${\\bf S}$ belongs to the set $\\lbrace g({\\bf u}): \\parallel {\\bf u}\\parallel = 1\\rbrace $ .", "Thus, when take samples by Monte Carlo methods, the samples on boundary are sufficient to derive the outer bounding ellipsoids of the remainder set.", "Therefore, based on Theorem REF , the computation complexity in the bounding steps of Algorithm REF can be reduced much more.", "Figure: (left) the separation of circle.", "(right) the corresponding area of of g(𝐮)g({\\bf u})Remark 4.9 Note that the assumption that ${\\bf E}$ is a Cholesky factorization of a positive-definite ${\\bf P}$ such that $\\lbrace {\\bf x}+{\\bf E}{\\bf u}:\\parallel u\\parallel \\le 1\\rbrace $ is not intersect with the radial ${\\bf x}(1)<=a, {\\bf x}(2)=b$ is a weak condition.", "If the true target is near it, we can transform the data to a new coordinate system where the target far way the the radial, then the assumption can be satisfied." ], [ "Numerical examples in target tracking", "In this section, we compare the performance between Monte Carlo set membership filter and particle filter when the underlying probability density functions of noises are known or unknown.", "Meanwhile, we also compare it with the extended set-membership filter (ESMF) in [23].", "Considering a two-dimensional Cartesian coordinate system, we track a moving target using measured range and angle from one sensor.", "The system equation is as follows [2]: ${\\bf x}_{k+1}&=&f_k({\\bf x}_k)+{\\bf w}_k,\\\\[3mm]{\\bf y}_k&=&h_k({\\bf x}_k)+{\\bf v}_k,$ where $\\nonumber f_k({\\bf x}_k)&=& \\left[\\begin{array}{cccc}1 & 0& T& 0\\\\0 & 1 & 0 & T \\\\0 & 0 & 1 & 1 \\\\0 & 0 & 0 & 1 \\\\\\end{array}\\right]{\\bf x}_k$ $\\nonumber h_k({\\bf x}_k)=\\left[\\begin{array}{c}\\sqrt{({\\bf x}_k(1))^2+({\\bf x}_k(2))^2} \\\\[3mm]arctan\\left(\\frac{{\\bf x}_k(2)}{{\\bf x}_k(1)}\\right) \\\\\\end{array}\\right].$ The ${\\bf x}$ is a four-dimensional state variable that includes position and velocity $(x, y, \\dot{x}, \\dot{y})$ , $T=0.2$ s is the time sampling interval.", "The process noise and measurement noise assumed to be confined to specified ellipsoidal sets $\\nonumber {\\bf W}_k&=&\\lbrace {\\bf w}_k: {\\bf w}_k^T{\\bf Q}_k^{-1}{\\bf w}_k\\le 1\\rbrace \\\\\\nonumber {\\bf V}_k&=&\\lbrace {\\bf v}_k: {\\bf v}_k^T{\\bf R}_k^{-1}{\\bf v}_k\\le 1\\rbrace .$ where $\\nonumber {\\bf Q}_k&=&\\sigma ^2\\left[\\begin{array}{cccc}\\frac{T^3}{3} & 0 & \\frac{T^2}{2} & 0 \\\\0 & \\frac{T^3}{3} & 0 & \\frac{T^2}{2} \\\\\\frac{T^2}{2} & 0 & T & 0 \\\\0 & \\frac{T^2}{2} & 0 & T \\\\\\end{array}\\right]\\\\\\nonumber {\\bf R}_k&=&\\left[\\begin{array}{cc}0.3^2 & 0\\\\0 & {0.1}^2\\\\\\end{array}\\right].$ The target acceleration is $\\sigma ^2=50$ .", "In the example, the target starts at the point $(50,30)$ with a velocity of $(5,5)$ .", "The center and the shape matrix of the initial bounding ellipsoid are $\\hat{{\\bf x}}_0=\\left[\\begin{array}{cccc}49.5& 29.5 & 5 &5\\\\\\end{array}\\right]^T$ , $\\nonumber {\\bf P}_0=\\left[\\begin{array}{cccc}5 & 0 & 0 & 0 \\\\0 & 5 & 0 & 0 \\\\0 & 0 & 2 & 0 \\\\0 & 0 & 0 & 2 \\\\\\end{array}\\right],$ respectively.", "Assume that the noises are confined to specified ellipsoidal sets, the state noise is truncated Gaussian with mean $[-0.2~-0.2~-1~-1]$ and covariance ${\\bf Q}_k/3^2$ and measurement noise is truncated Gaussian, with mean $[-0.4~0]^T$ , covariance ${\\bf R}_k/3^2$ on the ellipsoidal sets, respectively.", "From the description of the above, we can see that the condition of Algorithm REF is satisfied, then, using MCSMF to calculate the error bound, which is defined as follows: $error(k)=\\frac{1}{m}\\sum _{i=1}^m\|{\\bf x}_k^i-\\hat{{\\bf x}}_k^i\|,$ where ${\\bf x}_k^i$ and $\\hat{{\\bf x}}_k^i$ are the $ith$ true state and state estimate at time $k$ , respectively, and $m$ is the number of the Monte Carlo runs.", "When the underlying probability density functions of noises are known, we use the particle filter in [9], which is denoted by PF-T.", "When the underlying probability density functions of noises are unknown, we denote PF-G for the particle filter where the state noise and measurement noise are assumed the truncated Gaussian noise with zero mean.", "At the same time, we may assume that the noises are uniform density functions, then we still use particle filter, which is denoted by PF-U.", "The extended set-membership filter in [23] is denoted by ESMF.", "These four filters have the same initial bounding ellipsoid in this example.", "The following simulation results are under Matlab R2012a with YALMIP.", "Figures REF -REF present a comparison of the error bounds along position and velocity direction of MCSMF with those of PF-T, PF-G, PF-U and ESMF, respectively.", "Figures REF -REF show that when the probability density functions of noises are known, the performance of the particle filter is better than that of MCSMF and ESMF.", "The reason may be that more information of the probability density of noises is used.", "However, when it is unknown, the performance of the particle filter is worse than that of MCSMF.", "In addition, the figures also show that performance of ESMF is unstable.", "The reason may be that there are some uncertain parameters to be used in ESMF and the remainder is bounded by interval mathematics method, which is conservative and leads a bigger bounding ellipsoid than MCSMF.", "Figures REF -REF present the target tracking trajectories along ${\\bf x}$ direction by MCSMF and PF-T, respectively.", "The bounds of MCSMF and the $3\\sigma $ confidence bounds of PF-T are also plotted.", "Figures REF -REF show that the $3\\sigma $ confidence bounds of particle filter is indeed tighter than that of MCSMF, but it cannot contain the true state at some time step.", "It is an too optimistic bound.", "However, the bounds of MCSMF do guarantee the containment of the true state at each time step.", "This is useful in some applications.", "For example, in a civilian air traffic control system, the confidence bounds of trajectories can be used to check the standard separation between pairs of targets for maintenance of safety conditions (collision avoidance) and regularity of traffic flow in [34].", "The CPU times of MCSMF, PF-T, PF-U and PF-G are plotted as a function of number of samples and particles in Figure REF , respectively.", "It shows that CPU times of the three filters are increasing as the number of samples and particles is increasing.", "The magnitude of the CPU time of the three filters are similar.", "Figure: Comparison of the error bounds along position 𝐱{\\bf x} direction based on 200Monte Carlo runs.Figure: Comparison of the error bounds along position 𝐲{\\bf y} direction based on 200Monte Carlo runs.Figure: The target's trajectory along 𝐱{\\bf x} direction by MCSMFFigure: The target's trajectory along 𝐱{\\bf x} direction by PF-TFigure: (up) The CPU times for MCSMF with different sampling numbers from the boundary.", "(bottom) The CPU times for PF-T, PF-U and PF-G with different particle numbers." ], [ "Conclusion", "We have proposed a new class of filtering methods in bounded noise setting via set-membership theory and Monte Carlo (boundary) sampling technique to determine a state estimation ellipsoid.", "The set-membership prediction and measurement update are derived by recent convex optimization methods based on S-procedure and Schur complement.", "To guarantee the on-line usage, the nonlinear dynamics are linearized about the current estimate and the remainder terms are then bounded by an ellipsoid, which can be written as a semi-infinite optimization problem.", "For a typical nonlinear dynamic system in target tracking, based on the remainder properties and the Inverse Function Theorem, the semi-infinite optimization problem can be efficiently solved by Monte Carlo boundary sampling technique.", "Numerical example shows that when the probability density functions of noises are unknown, the performance of MCSMF is better than that of the particle filter, and which is more robust than particle filter.", "Future work will involve, in the setting of MCSMF, the multi-sensor fusion, multiple target tracking and various applications such as sensor management and placement for structures and different types of wireless networks." ], [ "APPENDIX", "Lemma 7.1 [35] Let ${\\bf F}_0(\\eta ), {\\bf F}_1(\\eta ),\\ldots , {\\bf F}_p(\\eta )$ , be quadratic functions in variable $\\eta \\in \\mathcal {R}^{n}$ ${\\bf F}_i(\\eta )=\\eta ^T{\\bf T}_i\\eta , ~~i=0,\\ldots , p$ with ${\\bf T}_i={\\bf T}_i^T$ .", "Then the implication ${\\bf F}_1(\\eta )\\le 0,\\ldots ,{\\bf F}_p(\\eta )\\le 0\\Rightarrow {\\bf F}_0(\\eta )\\le 0$ holds if there exist $\\tau _1,\\ldots ,\\tau _p\\ge 0$ such that ${\\bf T}_0-\\sum _{i=1}^p\\tau _i{\\bf T}_i\\preceq 0.$ Lemma 7.2 Schur Complements [35]: Given constant matrices ${\\bf A}$ , ${\\bf B}$ , ${\\bf C}$ , where ${\\bf C}={\\bf C}^T$ and ${\\bf A}={\\bf A}^T<0$ , then ${\\bf C}-{\\bf B}^T{\\bf A}^{-1}{\\bf B}\\preceq 0$ if and only if $\\left[\\begin{array}{cc}{\\bf A}& {\\bf B}\\\\{\\bf B}^T & {\\bf C}\\\\\\end{array}\\right]\\preceq 0$ or equivalently $\\left[\\begin{array}{cc}{\\bf C}& {\\bf B}^T \\\\{\\bf B}& {\\bf A}\\\\\\end{array}\\right]\\preceq 0$ Proof.", "[Proof of Theorem REF ]: Note that ${\\bf x}_k\\in \\mathcal {E}_{k}$ is equivalent to ${\\bf x}_k=\\hat{{\\bf x}}_{k}+{\\bf E}_{k}{\\bf u}_{k}$ , $\\parallel {\\bf u}_{k}\\parallel \\le 1$ , where ${\\bf E}_{k}$ is a Cholesky factorization of ${\\bf P}_{k}$ ; By the equations (REF ) and (REF ), $\\nonumber {\\bf x}_{k+1}-\\hat{{\\bf x}}_{k+1\|k}&=& f_k({\\bf x}_k)+{\\bf w}_k-\\hat{{\\bf x}}_{k+1\|k}\\\\[3mm]\\nonumber &=&f_k(\\hat{{\\bf x}}_{k}+{\\bf E}_{k}{\\bf u}_{k})+{\\bf w}_k-\\hat{{\\bf x}}_{k+1\|k}\\\\[3mm] &=&f_k(\\hat{{\\bf x}}_{k})+{\\bf J}_{f_k}{\\bf E}_{k}{\\bf u}_{k}+{\\bf e}_{f_k}+{\\bf B}_{f_k}\\Delta _{f_k}+{\\bf w}_k-\\hat{{\\bf x}}_{k+1\|k}$ and by the equations () and () $\\nonumber {\\bf y}_{k}&=& h_{k}({\\bf x}_{k})+{\\bf v}_{k}\\\\[3mm] &=&h_k(\\hat{{\\bf x}}_{k})+{\\bf J}_{h_k}{\\bf E}_{k}{\\bf u}_{k}+{\\bf e}_{h_k}+{\\bf B}_{h_k}\\Delta _{h_k}+{\\bf v}_k$ If we denote by $ \\xi =[1, ~{\\bf u}_{k}^T, ~{\\bf w}_k^T, ~{\\bf v}_{k}^T, ~\\Delta _{f_k}^T, ~\\Delta _{h_{k}}^T]^T,$ then (REF ) and (REF ) can be rewritten as $ {\\bf x}_{k+1}-\\hat{{\\bf x}}_{k+1\|k}&=& \\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})\\xi \\\\[3mm] 0&=& \\Psi _{k+1\|k}({\\bf y}_{k})\\xi ,$ where $\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})$ and $\\Psi _{k+1\|k}({\\bf y}_{k})$ are denoted by (REF ) and (), respectively.", "Moreover, the condition that ${\\bf x}_{k+1}\\in \\mathcal {E}_{k+1\|k}$ whenever I) ${\\bf x}_k$ is in $\\mathcal {E}_{k}$ II) the process and measurement noises ${\\bf w}_k, {\\bf v}_{k}$ are bounded in ellipsoidal sets, i.e., ${\\bf w}_k\\in {\\bf W}_k$ , ${\\bf v}_{k}\\in {\\bf V}_{k}$ is equivalent to $ \\xi ^T\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})^T({\\bf P}_{k+1\|k})^{-1}\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})\\xi \\le 1,$ whenever $ \\parallel {\\bf u}_{k}\\parallel &\\le & 1 ,\\\\[3mm] {\\bf w}_k^T{\\bf Q}_k^{-1}{\\bf w}_k&\\le & 1,\\\\[3mm] {\\bf v}_{k}^T{\\bf R}_k^{-1}{\\bf v}_{k}&\\le & 1,\\\\[3mm]\\parallel \\Delta _{f_k}\\parallel &\\le &1,\\\\[3mm]\\parallel \\Delta _{h_{k}}\\parallel &\\le &1.$ The equations (REF )–() is equivalent to $ \\xi ^T\\qopname{}{o}{diag}(-1,I,0,0,0,0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,{\\bf Q}_k^{-1},0,0,0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,0,{\\bf R}_k^{-1},0,0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,0,0,I,0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,0,0,0,I)\\xi &\\le & 0.$ where $I$ and 0 are matrices with compatible dimensions.", "By $\\mathcal {S}$ -procedure Lemma REF and Eq.", "(), a sufficient condition such that the inequalities (REF )-() imply (REF ) to hold is that there exist scalars $\\tau ^y$ and nonnegative scalars $\\tau ^u\\ge 0, \\tau ^w\\ge 0,\\tau ^v\\ge 0,\\tau ^f\\ge 0,\\tau ^h\\ge 0$ , such that $\\nonumber && \\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})^T({\\bf P}_{k+1\|k})^{-1}\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})\\\\[3mm]\\nonumber &&-\\qopname{}{o}{diag}(1,0,0,0,0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^u\\qopname{}{o}{diag}(-1,I,0,0,0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^w\\qopname{}{o}{diag}(-1,0,{\\bf Q}_k^{-1},0,0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^v\\qopname{}{o}{diag}(-1,0,0,{\\bf R}_k^{-1},0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^f\\qopname{}{o}{diag}(-1,0,0,0,I,0,0)\\\\[3mm]\\nonumber &&-\\tau ^h\\qopname{}{o}{diag}(-1,0,0,0,0,0,I)\\\\[3mm]&&-\\tau ^y \\Psi _{k+1\|k}({\\bf y}_{k})^T \\Psi _{k+1\|k}({\\bf y}_{k}) \\preceq 0$ Furthermore, (REF ) is written in the following compact form: $ \\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})^T({\\bf P}_{k+1\|k})^{-1}\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})-\\Xi -\\tau ^y \\Psi _{k+1\|k}({\\bf y}_{k})^T \\Psi _{k+1\|k}({\\bf y}_{k}) \\preceq 0$ where $\\Xi $ is denoted by (REF ).", "If we denote $(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }$ is the orthogonal complement of $\\Psi _{k+1\|k}({\\bf y}_{k})$ , then (REF ) is equivalent to $ &&((\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot })^T\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})^T({\\bf P}_{k+1\|k})^{-1}\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }\\\\\\nonumber &&~~~-((\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot })^T\\Xi (\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot } \\preceq 0$ Using Schur complements, (REF ) is equivalent to $&&\\left[\\begin{array}{cc}-{\\bf P}_{k+1\|k}&\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }\\\\[3mm](\\Phi _{k+1\|k}(\\hat{{\\bf x}}_{k+1\|k})(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot })^T& ~~-(\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }^T\\Xi (\\Psi _{k+1\|k}({\\bf y}_{k}))_{\\bot }\\\\\\end{array}\\right]\\preceq 0,\\\\[3mm]&&-{\\bf P}_{k+1\|k}\\prec 0.$ Therefore, if $\\hat{x}_{k+1\|k}$ , ${\\bf P}_{k+1\|k}$ satisfy (REF ) and (), then the state $x_{k+1}$ belongs to $\\mathcal {E}_{k+1\|k}$ , whenever I) ${\\bf x}_k$ is in $\\mathcal {E}_{k}$ , II) the process and measurement noises ${\\bf w}_k, {\\bf v}_{k}$ are bounded in ellipsoidal sets, i.e., ${\\bf w}_k\\in {\\bf W}_k$ , ${\\bf v}_{k}\\in {\\bf V}_{k}$ .", "Summarizing the above results, the computation of the predicted bounding ellipsoid by minimizing a size measure $f({\\bf P}_{k+1\|k})$ (REF ) is Theorem REF .", "Proof.", "[Proof of Theorem REF ]: Note that we have get ${\\bf x}_{k+1}\\in \\mathcal {E}_{k+1\|k}$ in prediction step, which is equivalent to ${\\bf x}_{k+1}=\\hat{{\\bf x}}_{k+1\|k}+{\\bf E}_{k+1\|k}{\\bf u}_{k+1\|k}$ , $\\parallel {\\bf u}_{k+1\|k}\\parallel \\le 1$ , where ${\\bf E}_{k+1\|k}$ is a Cholesky factorization of ${\\bf P}_{k+1\|k}$ , then, $ {\\bf x}_{k+1}-\\hat{{\\bf x}}_{k+1}&=& \\hat{{\\bf x}}_{k+1\|k}+{\\bf E}_{k+1\|k}{\\bf u}_{k+1\|k}-\\hat{{\\bf x}}_{k+1}$ and by the equations () and () $\\nonumber {\\bf y}_{k+1}&=& h_{k+1}({\\bf x}_{k+1})+{\\bf v}_{k+1}\\\\ &=&h_k(\\hat{{\\bf x}}_{k+1\|k})+{\\bf J}_{h_{k+1}}{\\bf E}_{k+1\|k}{\\bf u}_{k+1\|k}+{\\bf e}_{h_k+1}+{\\bf B}_{h_{k+1}}\\Delta _{h_{k+1}}+{\\bf v}_{k+1}$ If we denote by $ \\xi =[1, ~{\\bf u}_{k+1\|k}^T, ~{\\bf v}_{k+1}^T, ~\\Delta _{h_{k+1}}^T]^T,$ then (REF ) and (REF ) can be rewritten as $ {\\bf x}_{k+1}-\\hat{{\\bf x}}_{k+1}&=& \\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})\\xi \\\\[3mm] 0&=& \\Psi _{k+1}({\\bf y}_{k+1})\\xi ,$ where $\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})$ and $\\Psi _{k+1}({\\bf y}_{k+1})$ are denoted by (REF ) and (), respectively.", "Moreover, the condition that ${\\bf x}_{k+1}\\in \\mathcal {E}_{k+1}$ whenever I) ${\\bf x}_{k+1}$ is in $\\mathcal {E}_{k+1\|k}$ II) measurement noises ${\\bf v}_{k+1}$ are bounded in ellipsoidal sets, i.e., ${\\bf v}_{k+1}\\in {\\bf V}_{k+1}$ is equivalent to $ \\xi ^T\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})^T({\\bf P}_{k+1})^{-1}\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})\\xi \\le 1,$ whenever $ \\parallel {\\bf u}_{k}\\parallel &\\le & 1 ,\\\\[3mm] {\\bf v}_{k+1}^T{\\bf R}_{k+1}^{-1}{\\bf v}_{k+1}&\\le & 1,\\\\[3mm]\\parallel \\Delta _{h_{k+1}}\\parallel &\\le &1.$ The equations (REF )–() is equivalent to $ \\xi ^T\\qopname{}{o}{diag}(-1,I,0,0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,{\\bf R}_{k+1}^{-1},0)\\xi &\\le & 0,\\\\[3mm] \\xi ^T\\qopname{}{o}{diag}(-1,0,0,I)\\xi &\\le & 0,$ where $I$ and 0 are matrices with compatible dimensions.", "By $\\mathcal {S}$ -procedure Lemma REF and Eq.", "(), a sufficient condition such that the inequalities (REF )-() imply (REF ) to hold is that there exist scalars $\\tau ^y$ and nonnegative scalars $\\tau ^u\\ge 0, \\tau ^v\\ge 0, \\tau ^h\\ge 0$ , such that $\\nonumber && \\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})^T({\\bf P}_{k+1})^{-1}\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})\\\\[3mm]\\nonumber &&-\\qopname{}{o}{diag}(1,0,0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^u\\qopname{}{o}{diag}(-1,I,0,0,0)\\\\[3mm]\\nonumber &&-\\tau ^v\\qopname{}{o}{diag}(-1,0,0,{\\bf R}_{k+1}^{-1},0)\\\\[3mm]\\nonumber &&-\\tau ^f\\qopname{}{o}{diag}(-1,0,0,0,I)\\\\[3mm]&&-\\tau ^y \\Psi _{k+1}({\\bf y}_{k+1})^T \\Psi _{k+1}({\\bf y}_{k+1}) \\preceq 0$ Furthermore, (REF ) is written in the following compact form: $ \\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})^T({\\bf P}_{k+1})^{-1}\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})-\\Xi -\\tau ^y \\Psi _{k+1}({\\bf y}_{k+1})^T \\Psi _{k+1}({\\bf y}_{k+1}) \\preceq 0$ where $\\Xi $ is denoted by (REF ).", "If we denote $(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }$ is the orthogonal complement of $\\Psi _{k+1}({\\bf y}_{k+1})$ , then (REF ) is equivalent to $\\nonumber && ((\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot })^T\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})^T({\\bf P}_{k+1})^{-1}\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})(\\Psi _{k+1}({\\bf y}_{k}))_{\\bot }\\\\&&-((\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot })^T\\Xi (\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot } \\preceq 0$ Using Schur complements Lemma REF , (REF ) is equivalent to $&&\\left[\\begin{array}{cc}-{\\bf P}_{k+1}&\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }\\\\[3mm](\\Phi _{k+1}(\\hat{{\\bf x}}_{k+1})(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot })^T& ~~-(\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }^T\\Xi (\\Psi _{k+1}({\\bf y}_{k+1}))_{\\bot }\\\\\\end{array}\\right]\\preceq 0,\\\\[3mm]&&-{\\bf P}_{k+1}\\prec 0.$ Therefore, if $\\hat{x}_{k+1}$ , ${\\bf P}_{k+1}$ satisfy (REF ) and (), then the state $x_{k+1}$ belongs to $\\mathcal {E}_{k+1}$ , whenever I) ${\\bf x}_{k+1}$ is in $\\mathcal {E}_{k+1\|k}$ , II) measurement noises $ {\\bf v}_{k+1}$ are bounded in ellipsoidal sets, i.e., ${\\bf v}_{k+1}\\in {\\bf V}_{k+1}$ .", "Summarizing the above results, the computation of the measurement update bounding ellipsoid by minimizing a size measure $f({\\bf P}_{k+1})$ (REF ) is Theorem REF ." ] ]	1606.05046
[ [ "Intertwined Orders in Heavy-Fermion Superconductor CeCoIn$_5$" ], [ "Abstract The appearance of spin-density-wave (SDW) magnetic order in the low-temperature and high-field corner of the superconducting phase diagram of CeCoIn$_5$ is unique among unconventional superconductors.", "The nature of this magnetic $Q$ phase is a matter of current debate.", "Here, we present the thermal conductivity of CeCoIn$_5$ in a rotating magnetic field, which reveals the presence of an additional order inside the $Q$ phase that is intimately intertwined with the superconducting $d$-wave and SDW orders.", "A discontinuous change of the thermal conductivity within the $Q$ phase, when the magnetic field is rotated about antinodes of the superconducting $d$-wave order parameter, demands that the additional order must change abruptly together with the recently observed switching of the SDW.", "A combination of interactions, where spin-orbit coupling orients the SDW, which then selects the secondary $p$-wave pair-density-wave component (with an average amplitude of 20\\% of the primary $d$-wave order parameter), accounts for the observed behavior." ], [ "Introduction", "Magnetism is considered to be detrimental to conventional superconductivity, which is mediated by lattice vibrations—phonons [1].", "An external magnetic field, for example, destroys superconductivity via either orbital [2] or spin [3] (Pauli) limiting mechanisms.", "A growing number of cases, however, display the coexistence of magnetism and superconductivity and constitute a fascinating problem in condensed-matter physics [4], [5].", "CeCoIn$_5$ presents a unique case among all unconventional superconductors wherein a novel magnetic state, the so-called $Q$ phase, develops at high fields and requires superconductivity for its very existence.", "This $Q$ phase was originally suggested [6], [7], [8] to be a realization of spatially inhomogeneous superconductivity, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [9], [10].", "Subsequent NMR [11] and neutron scattering measurements [12], [13] revealed the presence of a magnetic spin-density-wave (SDW) order in the $Q$ phase.", "A number of theories were proposed for its origin [14], [15], [16], [17], [18], [19], [20], many of them involving additional orders, distinct from the $d$ -wave superconductivity [21], [22] and the SDW.", "The issue of intertwined orders (magnetic, multiple and inhomogeneous superconductivity, etc.)", "is increasingly common in correlated systems [23], [24].", "The $Q$ phase is a model system for studying such intertwined orders, with a uniquely tunable single-domain structure due to the high purity of CeCoIn$_5$ .", "Experimentally, neutron-scattering measurements suggest the condensation of a superconducting spin resonance as a possible origin [18], [25], [26] of the $Q$ phase.", "Recent neutron-scattering measurements reveal that its SDW order is single domain, with the ordering wave vector $\\mathbf {Q}_\\mathrm {SDW}$ either $\\mathbf {Q}_1 = (q, q, 0.5)$ or $\\mathbf {Q}_2 = (q, -q, 0.5)$ , with $q\\approx 0.44$ along the two nodal directions of the superconducting $d$ -wave order parameter [27].", "When the magnetic field is rotated within the crystallographic $ab$ plane about the [100] direction, $\\mathbf {Q}_\\mathrm {SDW}$ switches abruptly between $\\mathbf {Q}_1$ and $\\mathbf {Q}_2$ , choosing the one that is more perpendicular to the magnetic field [Figs.", "1(b) and 1(c)].", "It was suggested that a secondary $p$ -wave pair-density-wave (PDW) component drives the hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ to the direction of the magnetic field [27].", "This mechanism, however, at present lacks theoretical support (see Appendix A).", "One recently proposed scenario explains the hypersensitivity as being due to the magnetic field lifting the degeneracy of the direction of $\\mathbf {Q}_\\mathrm {SDW}$ via spin-orbit coupling [28], without requiring any additional order besides the existing superconducting $d$ -wave and SDW orders.", "Yet another scenario introduces the spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, which couples to the SDW state and lowers the energy of the $Q$ phase with $\\mathbf {Q}_\\mathrm {SDW}$ more perpendicular to ${q}_{\\mathrm {FFLO}}$ [29].", "The mechanism responsible for the switching of the direction of $\\mathbf {Q}_\\mathrm {SDW}$ is a matter of a current debate, and we experimentally establish a new microscopic scenario.", "Figure: (a) Phase diagram of CeCoIn 5 _5, showing the QQ phase, based on specific heat measurements .", "The red data points are obtained from the present measurements, and the details are explained in Fig.", "2.", "(b,c) Schematic diagrams that illustrate switching of the SDW magnetic domain (𝐐 SDW \\mathbf {Q}_\\mathrm {SDW}) as the magnetic field 𝐇\\mathbf {H} is rotated about [100].", "The heat current 𝐉\\mathbf {J} is in the nodal [110] direction.", "𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} switches to be more perpendicular to 𝐇\\mathbf {H}, while lying along the nodes of the dd-wave order parameter represented by the green curve.", "The blue circle represents the normal Fermi surface and the magnetization of the SDW points out of the plane.", "(d) The thermal conductivity of CeCoIn 5 _5 κ\\kappa divided by temperature TT in the QQ phase as a function of the angle θ\\theta between 𝐇\\mathbf {H} and the heat current 𝐉∥[110]\\mathbf {J}\\parallel [110], at 11 T and 108 mK.", "The magnetic field is rotated between -90 ∘ ^{\\circ } and +90 ∘ ^{\\circ } within the crystallographic abab plane.", "At 45 ∘ ^{\\circ } and -45 ∘ ^{\\circ }, the antiferromagnetic ordering vector 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} switches between (0.44, 0.44, 0.5) and (0.44, -0.44, 0.5), as in (b,c) .", "When 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} switches from 𝐐 SDW ⊥𝐉\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J} to 𝐐 SDW ∥𝐉\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}, the thermal conductivity increases by approximately 15%.", "(e) Hysteretic behavior of the thermal conductivity in the switching region around θ=-45 ∘ \\theta =-45^{\\circ }, showing a first-order-like anomaly, for several fields.", "(f) Hysteretic behavior around θ=45 ∘ \\theta =45^{\\circ }.", "The inset shows the width of the hysteresis as a function of magnetic field [from (e)].Though neutron scattering has been essential in identifying the nature of magnetism in the $Q$ phase, it does not probe the superconducting state with which magnetism couples.", "Thermal conductivity, however, is a powerful probe of superconductivity [30], [31] because it depends on the presence of normal quasiparticles (excitations), as the superconducting condensate itself does not carry heat.", "Thermal conductivity is particularly sensitive to the presence of states where the energy gap in an unconventional superconductor is zero, i.e., gap nodes.", "This sensitivity arises because normal quasiparticles are easily excited around the nodes, where the energy gap is small, and therefore dominate the heat transport.", "As we show, measurements on the thermal conductivity of CeCoIn$_5$ in a rotating magnetic field reveal the nature of the $Q$ phase.", "A needlelike single-crystal sample ($2.5\\times 0.5\\times 0.2~mm^3$ ) was prepared with the long axis along the [110] crystallographic direction that coincides with superconducting nodes.", "The heat current ($\\mathbf {J}$ ) was applied along the [110] direction, and the thermal conductivity was measured with the standard steady-state method with two thermometers that were calibrated in advance.", "The magnetic field was applied within the crystallographic $ab$ plane, and the crystal (equivalently, magnetic field) was rotated about the $c$ axis using an Attocube piezoelectric rotator [32].", "The alignment of the crystallographic axis was confirmed by Laue x-ray diffraction to be within 1$^\\circ $ .", "A total of eight sections (approximately 1 cm) of 50-$\\mu $ m-diameter platinum wire were spot welded to the sample, and small amounts of silver epoxy were applied over the welds for mechanical strength.", "The cold end of the sample was rigidly attached to a sample holder, a semicylindrical copper rod 2 mm in diameter.", "The sample was glued to the sample holder with varnish first; a pair of the platinum wires were wrapped around the sample and the sample holder; as the final step, silver paint was applied around the Pt wires, the sample holder, and the cold end of the sample, to enhance the electrical contacts between the bound wires and the sample holder and to ensure mechanical stability of the sample.", "The remaining three pairs of wires were used for thermal connections to two thermometers and a heater.", "The angle between the crystal and the magnetic field was monitored with two Hall sensors, parallel to the $ac$ and $bc$ planes, mounted on the sample stage." ], [ "Results", "As the magnetic field is rotated clockwise through [100] within the $ab$ plane, $\\mathbf {Q}_\\mathrm {SDW}$ flips from being perpendicular [Fig.", "1(b)] to being parallel [Fig.", "1(c)] to $\\mathbf {J}$ .", "Figure 1(d) shows the thermal conductivity as a function of the angle ($\\theta $ ) between the magnetic field ($\\mathbf {H}$ ) and the heat-current direction ($\\mathbf {J}\\parallel [110]$ ) at a temperature $T=108$ mK.", "The thermal conductivity exhibits sharp first-order jumps when the magnetic field is rotated around the [100] and [010] directions [Figs.", "1(d)-1(f)] with a narrow hysteresis region of approximately 0.2$^\\circ $ at 11 T. This response is identical to the switching of $\\mathbf {Q}_\\mathrm {SDW}$ observed by neutron diffraction [27] and, therefore, reflects the same hypersensitivity phenomenon.", "The width of the hysteresis is roughly linear with the magnetic field [Fig.", "1(e) and inset of Fig.", "1(f)] and tracks the development of the magnetic Bragg peak intensity [27].", "Figure: (a) Temperature dependence of the thermal conductivity over TT (κ/T\\kappa /T) for θ\\theta =44.7 ∘ ^\\circ when 𝐐 SDW ⊥𝐉\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J} (blue symbols), and for θ\\theta =45.7 ∘ ^\\circ when 𝐐 SDW ∥𝐉\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J} (red symbols) at a fixed field.", "The step at 0.6 K reflects the superconducting–normal transition.", "The data with higher thermal gradients (cyan and magenta triangles) show no difference from the other data (ΔT/T≈0.06\\Delta T/T\\approx 0.06); i.e., there are no indications of the sliding mode of the SDW (Appendix D).", "Inset: The difference (Δκ ∥,⊥ /T\\Delta \\kappa _{\\parallel , \\perp }/T) between κ/T\\kappa /T for the two orientations of 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} (green circles, left axis) and the width of the hysteresis Δθ\\Delta \\theta (from Fig.", "6) at two temperatures for μ 0 \\mu _0H = 11 T (orange triangles, right axis).", "The onset temperature of the QQ phase at μ 0 \\mu _0H=11 T is depicted as a diamond on the phase diagram in Fig.", "1(a).", "(b) Magnetic-field dependence of the thermal conductivity over TT (κ/T\\kappa /T) at several temperatures.", "The field directions for the two different 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} orientations are the same as in (a).", "The inset shows the difference between κ/T\\kappa /T for the two orientations.", "The onset of the rise in Δκ ∥,⊥ /T\\Delta \\kappa _{\\parallel , \\perp }/T is taken as a QQ-phase transition and is displayed as red circles in Fig.", "1(a).Figure: (a) Magnetic-field dependence of the thermal conductivity over TT (κ/T\\kappa /T) in and around the QQ phase.", "The field directions are the same as in Fig.", "2.", "The data for field sweeps down (circles) and up (diamonds) are shown.", "The difference (Δκ ∥,⊥ /T\\Delta \\kappa _{\\parallel , \\perp }/T) for the two orientations of 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} (green triangles, right axis) starts to grow above 9.9 T, and is well described by the (H-H c )/H c \\sqrt{(H-H_c)/H_c} fit shown (orange curve).", "Weak hysteresis within the QQ phase is likely due to a vortex-lattice transition for 𝐇∥[100]\\mathbf {H}\\parallel [100] at μ 0 \\mu _0H≈\\approx 11 T, observed recently with scanning tunneling microscopy .", "(b) κ/T\\kappa /T for θ\\theta =-90 ∘ ^\\circ when 𝐐 SDW ∥𝐉\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J} (red symbols), and θ\\theta =0 ∘ ^\\circ when 𝐐 SDW ⊥𝐉\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J} (blue symbols).", "The data are very similar to those for the same relative orientation of 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} and 𝐉\\mathbf {J}, correspondingly colored, in (a).", "The scales of the y axes in (a) and (b) are the same but with different offsets.", "The features for both θ\\theta =-90 ∘ ^\\circ and θ\\theta =0 ∘ ^\\circ between 8 and 9 tesla are likely due to a vortex-lattice transition for 𝐇∥[110]\\mathbf {H}\\parallel [110] between 7.5 and 8.7 T .Figure 2 displays the thermal conductivity over temperature ($\\kappa /T$ ) for two magnetic-field directions very close to the switching region at $\\theta $ =45$^\\circ $ , with $\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}$ ($\\theta $ =45.7$^\\circ $ , red) and $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ ($\\theta $ =44.7$^\\circ $ , blue).", "The temperature dependence at 11 T in Fig.", "2(a) shows that the difference between the thermal conductivities for the two directions (see the inset) develops below 0.3 K. A similar increase of thermal conductivity with decreasing temperature at high fields was recently reported for $\\mathbf {J}\\parallel [100]$ [35].", "The magnetic-field dependence in Fig.", "2(b) also shows that the difference in $\\kappa /T$ for the two directions of $\\mathbf {Q}_\\mathrm {SDW}$ develops at high magnetic field.", "The magnetic-field intensities of the onset of the increase of $\\Delta \\kappa /T$ from zero, with the corresponding measurement temperature, are displayed in the phase diagram of Fig.", "1(a).", "These points coincide with the $Q$ -phase boundary.", "The temperature and field dependence of CeCoIn$_5$ is complex, and has not been reproduced in detail theoretically.", "The task of including (1) $d$ -wave superconductivity, (2) a magnetic field which leads to both growth of the density of states due to the Doppler shift of the quasiparticle energies (the so-called Volovik effect) and a decrease of a quasiparticle mean-free path due to increased vortex scattering, (3) Pauli limiting, and (4) non-Fermi-liquid (NFL) behavior in the vicinity of H$_{c2}$ , even leaving out the SDW of the $Q$ phase, is a monumental one.", "We can, however, offer potential explanations of some of the observed trends based on the phenomena mentioned above.", "For example, the increase of thermal conductivity with reducing temperature within the superconducting state in magnetic fields close to H$_{c2}$ [Fig.", "2(a)] can be attributed to a similar NFL behavior in the normal state at or above H$_{c2}$ [36], [35].", "As shown by the data for the lowest temperature of 105 mK in Fig.", "2(b), the thermal conductivity is flat as a function of field between 4 and 9 tesla.", "Therefore, any deviation of $\\kappa /T$ from the flat behavior in the high-field regime (above 9 T) should be attributed to the formation of the $Q$ phase.", "Figure 3 displays the thermal conductivity data inside and around the $Q$ phase.", "The abrupt changes at H$_{c2}$ , 11.7 T for H$\\parallel $ [100] and 11.5 T for H$\\parallel $ [110] and H$\\parallel [\\bar{1}10]$ , agree well with the first-order transitions found in previous studies [7].", "There are a couple of salient features: (1) The difference between $\\kappa /T$ for the two orientations of $\\mathbf {Q}_\\mathrm {SDW}$ [right axis of Fig.", "3(a)] grows above 9.9 T and drops abruptly to zero above H$_{c2}$ , similar to the behavior of the SDW intensity measured by neutron scattering [27].", "The functional dependence, however, is different: While neutron intensity $I\\propto (H-H_c)$ , $\\Delta \\kappa /T \\propto \\sqrt{H-H_c}$ as shown in Fig.", "3(a) by the orange curve.", "The ordered magnetic moment $M\\propto \\sqrt{I} \\propto \\sqrt{H-H_c}$ [27].", "Therefore, $\\Delta \\kappa /T$ grows linearly with the magnetic moment.", "(2) For both directions of $\\mathbf {Q}_\\mathrm {SDW}$ , $\\kappa /T$ starts to decrease around 9 T, well before entering the $Q$ phase.", "This reduction of $\\kappa /T$ may be related to the results of NMR measurements [37] that were interpreted as an additional phase, but it may also arise from fluctuations due to the quantum-critical point ($T=0, H\\approx 9.8$ T) associated with the $Q$ phase.", "The key observations in current measurements are that, for both orientations of $\\mathbf {Q}_\\mathrm {SDW}$ , the thermal conductivity drops in the $Q$ phase, and the reduction is larger for $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ .", "The data in Fig.", "3 also show that $\\kappa /T$ for $\\theta $ =0$^\\circ $ and 44.7$^\\circ $ , with $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ , closely reproduce each other as do the data for $\\theta $ =45.7$^\\circ $ and -90$^\\circ $ with $\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}$ .", "Whatever changes in the $Q$ phase to affect thermal conductivity as the magnetic field rotates, those changes occur abruptly at $\\theta $ =45$^\\circ $ and then remain unchanged for the subsequent 90$^\\circ $ of the field rotation." ], [ "Discussion", "SDW order alone [28] cannot account for our data.", "SDW gaps quasiparticles along the $\\mathbf {Q}_\\mathrm {SDW}$ , and as a result, the thermal conductivity along the $\\mathbf {Q}_\\mathrm {SDW}$ must be smaller than the thermal conductivity perpendicular to it, contrary to our observation.", "There must be an additional component in the $Q$ phase that has an opposite and stronger effect on thermal conductivity compared to that of its SDW.", "A proposal for the origin of the hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ based on the formation of the FFLO state [29] is also incompatible with our result.", "Within this theory, ${q}_{\\mathrm {FFLO}}\\parallel \\mathbf {H}$ , which leads to a smooth change of the effect of the FFLO state on the thermal conductivity as both $\\mathbf {H}$ and ${q}_{\\mathrm {FFLO}}$ rotate together through [100].", "Therefore, the influence of the SDW will dominate, and $\\kappa /T(\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}) < \\kappa /T(\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ ) should be observed in the vicinity of $\\theta $ =45$^\\circ $ , in contrast to the experiment.", "To possibly reconcile this theory with the data, the requirement that ${q}_{\\mathrm {FFLO}}\\parallel \\mathbf {H}$ must be relaxed, with ${q}_{\\mathrm {FFLO}}$ pointing along the $d$ -wave nodes (see Appendix C).", "Figure: Schematics of the dd-wave (green) and the pp-wave PDW (orange, described by the vector 𝐝 1 \\mathbf {d}_1) superconducting order parameters, the 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} vectors and the corresponding gaps (magenta arrow and arcs, respectively), and the magnetic-field directions (cyan arrow).", "A combination of a SDW and a pp-wave PDW , 𝐝 1 \\mathbf {d}_1 component is consistent with the thermal conductivity data.", "(a) When 𝐐 SDW ⊥𝐉\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}, the pp-wave antinodes gap the nodes of the dd wave along 𝐉\\mathbf {J} (black arrow), sharply reducing the thermal conductivity.", "(b) The effect of the SDW gapping the nodes along 𝐉\\mathbf {J} must be smaller (by a factor of approximately 2) than the similar effect of the pp-wave PDW state in (a).A natural explanation that accounts for the observed reduction of the thermal conductivity in the $Q$ phase is the existence of a spatially inhomogeneous $p$ -wave PDW that couples the superconducting $d$ -wave and SDW order parameters [14], [27].", "One of the two $p$ -wave PDW components (Appendix A), compatible with $d$ -wave and SDW order parameters in CeCoIn$_5$ , is $\\mathbf {d}_1(k) = (0, 0, k_x-k_y)$ for $\\mathbf {Q}_\\mathrm {SDW} \\parallel [110]$ or, equivalently, $\\mathbf {d}_1(k) = (0, 0, k_x+k_y)$ for $\\mathbf {Q}_\\mathrm {SDW}\\parallel [1\\bar{1}0]$ , shown schematically in Figs.", "4(b) and 4(a), respectively.", "An interplay between the SDW and the PDW $\\mathbf {d}_1$ locks the node of the $p$ wave along the direction of $\\mathbf {Q}_\\mathrm {SDW}$ , leading to an additional anisotropy of the thermal conductivity.", "To allow the SDW with $\\mathbf {Q}_\\mathrm {SDW}\\parallel [1\\bar{1}0]$ to form, as illustrated in Fig.", "4(a), $\\mathbf {d}_1$ has to leave the quasiparticles along $\\mathbf {Q}_\\mathrm {SDW}$ ungapped by aligning its nodes along this direction, which is also a nodal direction of the $d$ wave.", "The $p$ -wave antinodes then gap the remaining $d$ -wave nodes along [110] and reduce the thermal conductivity for $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ [Fig.", "4(a)].", "We estimate that the average amplitude of the $p$ -wave gap required to suppress the thermal conductivity for $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ by 19%, as observed experimentally at 108 mK and 11 T [Fig.", "3(a)], is approximately 20% of the primary $d$ -wave gap, and the magnitude of the SDW gap is comparatively smaller and approximately 10% of the $d$ -wave gap (see Appendix B).", "A hierarchy of interactions between various orders shown in Fig.", "4 accounts for both the hypersensitivity and the thermal conductivity data.", "(1) The SDW must lie along one of the nodes of the superconducting $d$ -wave order parameter.", "(2) The spin-orbit coupling effect on the interaction between the SDW and the magnetic field drives the hypersensitivity [28] and orients $\\mathbf {Q}_\\mathrm {SDW}$ as perpendicular to $\\mathbf {H}$ as possible.", "(3) The selected $\\mathbf {Q}_\\mathrm {SDW}$ , in turn, orients the (allowed) $p$ -wave PDW $\\mathbf {d}_1$ component.", "Finally, (4) the PDW gaps the $d$ -wave nodes more effectively than the SDW, leading to the observed trend in thermal conductivity.", "Our measurements demonstrate a macroscopic realization of intertwined orders.", "As systems with multiple orders are becoming increasingly common in correlated electronic materials, we expect more examples of similar intertwined orders in which the manipulation of one order by the other is possible.", "Discussions with James A. Sauls, Anton B. Vorontsov, Ilya Vekhter, Stuart E. Brown, Alexander V. Balatsky, David M. Fobes, and Marc Janoschek are gratefully acknowledged.", "This work was conducted at the Los Alamos National Laboratory under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.", "We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD Program." ], [ "Scenario for the Hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ on the Direction of the Magnetic Field based on the ", "The switching of $\\mathbf {Q}_\\mathrm {SDW}$ , reported in Ref.", "[27], was suggested by the authors to be due to the formation of the spatially inhomogeneous $p$ -wave PDW, which, in a $d$ -wave superconductor, couples to the SDW [14].", "It was suggested that the anisotropic magnetic susceptibility of the $p$ -wave component orients it with respect to the magnetic field, and the interaction between the PDW and the SDW will then orient $\\mathbf {Q}_\\mathrm {SDW}$ .", "The two $p$ -wave components within the PDW scenario, compatible with $d$ -wave and SDW order parameters in CeCoIn$_5$ , are $\\mathbf {d}_1(\\mathbf {k})=(0,0,k_x-k_y)$ and $\\mathbf {d}_2(\\mathbf {k})=(k_z, -k_z, 0)$ for $\\mathbf {Q}_\\mathrm {SDW}\\parallel [110]$ [27], [14].", "Magnetic susceptibility of the $\\mathbf {d}_2$ component is indeed anisotropic.", "Its nodal plane, however, lies within the $ab$ plane, and $\\mathbf {d}_2$ , therefore, cannot preferentially select one of the two possible $\\mathbf {Q}_\\mathrm {SDW}$ .", "The $\\mathbf {d}_1$ component has nodes along [110] and would select the SDW domain with $\\mathbf {Q}_\\mathrm {SDW}$ along that direction.", "However, $\\mathbf {d}_1$ has isotropic $ab$ -plane susceptibility, and it cannot be the sole source of the hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ .", "The $p$ -wave PDW, therefore, can be the cause of the hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ only when the $\\mathbf {d}_1(\\mathbf {k})$ and $\\mathbf {d}_2(\\mathbf {k})$ are coupled.", "Currently, there is no theoretical support for the existence of such a coupling in CeCoIn$_5$ .", "Consequently, we exclude the possibility that the anisotropy of the magnetic susceptibility of $\\mathbf {d}_2$ is the origin of the field hypersensitivity.", "Nevertheless, the triplet $\\mathbf {d}_1$ component directly couples to the SDW and $d$ -wave orders, it is allowed to form in the $Q$ phase, and it does explain our thermal conductivity results." ], [ "Contribution of the Composite Order Parameter to the Thermal Conductivity", "To ascertain which nodes of the $d$ -wave order parameter contribute most to heat transport, we calculated the thermal conductivity using the theory in Ref.", "[38] for a superconductor with a composite order parameter, $\|\\Delta _d\|+i~a\|\\Delta _p\|$ , where $\|\\Delta _d\|$ and $\|\\Delta _p\|$ are the magnitudes of the $d$ -wave and the $p$ -wave ($\\mathbf {d}_1$ ) components, respectively.", "The heat current $\\mathbf {J}$ was taken to be along one of the nodes of the $d$ -wave gap.", "The nodes of the $p$ -wave order parameter were arranged to either coincide with $\\mathbf {J}$ (and one of the nodal directions of the $d$ wave) or to be perpendicular to it.", "Addition of the $p$ -wave component with imaginary phase guaranteed that the antinodes of the $p$ wave gap the nodes of the $d$ wave parallel to them and reduce the thermal conductivity of their quasiparticles.", "The calculations, shown in Fig.", "5, demonstrate that thermal conductivity is reduced much more, between a factor of 5 and 10, when the $p$ -wave nodes are perpendicular to $\\mathbf {J}$ ; i.e., the $p$ -wave antinodes gap the $d$ -wave nodes that are along the heat transport.", "This means that the $d$ -wave nodes along the heat flow dominate thermal transport because the velocity of the quasiparticles in these nodes has a large component parallel to the direction of the heat current.", "Figure: The calculated thermal conductivity as a function of the relative amplitude aa of the pp-wave component for \|Δ d \|+ia\|Δ p \|\|\\Delta _d\|+i~a\|\\Delta _p\| pairing symmetry without normalization for two orientations of the pp-wave component 𝐝 1 \\mathbf {d}_1 with respect to the heat current 𝐉\\mathbf {J} depicted in Figs.", "4(a) and 4(b) of the main text.", "The nodes of the pp-wave component are either perpendicular to 𝐉\\mathbf {J} [Fig.", "4(a)] or parallel to it [Fig.", "4(b)].", "The electron mean-free path l=10ξl=10\\xi , where ξ\\xi is the superconducting coherence length, T=0.05T c T=0.05T_c, and H=0.3H c2 H=0.3H_{c2}, where H c2 H_{c2} is the orbital upper critical field.", "The reduction of κ\\kappa is much stronger when the pp-wave antinode is along the heat current (pp-wave nodes ⊥𝐉\\mathrm {nodes}\\perp \\mathbf {J}) because the pp-wave antinode gaps the dd-wave nodal quasiparticles with momenta along the heat current 𝐉\\mathbf {J}.", "In contrast, when the pp-wave antinode is perpendicular to 𝐉\\mathbf {J} (pp-wave nodes ∥𝐉\\mathrm {nodes}\\parallel \\mathbf {J}), the pp-wave antinode only gaps quasiparticles with momenta perpendicular to the heat current, resulting in a much smaller effect.", "The thermal transport in a dd-wave superconductor is therefore dominated by the quasiparticles in the nodes that are along the heat current.These calculations also allow us to estimate the relative magnitude ($a$ ) of the $p$ -wave order parameter required to achieve the reduction of thermal conductivity by 19% observed experimentally for the case of $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ .", "The hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ is due to the spin-orbit coupling, whereas, the observed anisotropic thermal conductivity is due to the appearance of the allowed $\\mathbf {d}_1$ component of the $p$ -wave PDW.", "The reduction of thermal conductivity when $\\mathbf {Q}_\\mathrm {SDW}$ points along the heat current $\\mathbf {J}$ ($\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}$ , when the dominant nodes along $\\mathbf {J}$ are gapped by the SDW) is less than half of the reduction for the case of $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ , where the nodes along $\\mathbf {J}$ are gapped by the secondary $p$ -wave component.", "The contribution of the SDW to the reduction of thermal conductivity in the latter case is reduced even further, by a factor of 10, as seen in Fig.", "5.", "Therefore, when $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ , we can neglect the effect of SDW on the thermal conductivity for the purpose of making an estimate of the magnitude of the $p$ -wave order parameter.", "We then consider the case of the $p$ -wave nodes perpendicular to the heat current, shown in Fig.", "5.", "The horizontal dashed line represents the observed suppression of thermal conductivity for $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ , and we can read off the magnitude of the $p$ -wave $\\mathbf {d}_1$ component from their intersections with the blue ($p$ -wave $\\mathrm {nodes}\\perp \\mathbf {J}$ ) curve (vertical dashed lines).", "The resulting $a\\approx 0.2$ , a reasonable number for the amplitude of a secondary superconducting order parameter.", "We can roughly estimate the magnitude of the SDW gap (or the equivalent $p$ -wave gap) required to suppress the thermal conductivity by 8%, as observed experimentally for $\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}$ .", "We obtain the SDW gap to be approximately 10% of the primary $d$ -wave order parameter, half of the average $p$ -wave PDW gap.", "Figure: The hysteresis of the hypersensitive switching via thermal conductivity around θ=-45 ∘ \\theta =-45^{\\circ } at 11 T and two temperatures, 106 and 199 mK.", "The data for 106 mK are the same as in Fig.", "1(e) of the main text.", "The widths of the hysteresis at two different temperatures are plotted in the inset of Fig.", "2(a)." ], [ "FFLO state as the Origin of Hypersensitivity vis-à-vis Thermal Conductivity in the ${Q}$ phase", "Strong Pauli-limiting effects [39], evidenced by a first-order superconducting transition [7] above 10 T, and an extremely long electron mean-free path in the superconducting state [40] raise the possibility of the formation of a spatially inhomogeneous FFLO state.", "The FFLO state is characterized by a wave vector ${q}_{\\mathrm {FFLO}}$ , with the superconducting order parameter in the Larkin-Ovchinnikov (LO) scenario varying as $\\Delta =\|\\Delta \|\\mathrm {cos}({q}_{\\mathrm {FFLO}}\\cdot {r})$ and leading to a periodic array of nodal planes perpendicular to ${q}_{\\mathrm {FFLO}}$ where the superconducting gap ($\\Delta $ ) is zero.", "There is a number of experiments that are consistent with a FFLO state in CeCoIn$_5$ .", "One of the most notable works is the NMR investigation [41] that showed the resonance signal expected from the normal electrons in the FFLO nodal planes.", "The fragile nature of the $Q$ phase found in the doping experiment [42] also implies the existence of the FFLO state [43].", "Figure: Possible configuration of a FFLO state in CeCoIn 5 _5.", "Schematic of the dd-wave and the FFLO order parameters, the 𝐐 SDW \\mathbf {Q}_\\mathrm {SDW} vector, and the magnetic-field directions that are compatible with both hypersensitivity and thermal conductivity within a modified version of the FFLO scenario .", "This modification would require that q FFLO {q}_{\\mathrm {FFLO}} be forced to lie along the nodes of the dd wave, instead of always being parallel to the applied magnetic field.", "Dashed lines indicate the nodes of the FFLO state with normal quasiparticles.", "(a) FFLO nodal planes increase scattering and reduce thermal conductivity when they are perpendicular to 𝐉\\mathbf {J}.", "(b) The nodal planes contribute to thermal transport along 𝐉\\mathbf {J} when they are parallel to it.As stated in the main text, the proposal for the origin of the hypersensitivity of $\\mathbf {Q}_\\mathrm {SDW}$ based on the formation of a FFLO state [29] does not explain the thermal conductivity data.", "Within this theory, ${q}_{\\mathrm {FFLO}}$ is parallel to $\\mathbf {H}$ , and ${q}_{\\mathrm {FFLO}}$ rotates gradually through [100] together with the magnetic field.", "The FFLO state will therefore provide a smooth background to the thermal conductivity as the field rotates.", "With only a SDW present in addition to a FFLO state, the SDW will dominate the response in the vicinity of the switching region around $\\theta =45^{\\circ }$ and give $\\kappa /T(\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J})<\\kappa /T(\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J})$ , which is in contrast to the experimental result.", "To reconcile this theory with the data, the requirement that ${q}_{\\mathrm {FFLO}}\\parallel \\mathbf {H}$ must be relaxed.", "In fact, ${q}_{\\mathrm {FFLO}}$ was shown to lie along the nodes in the majority of the FFLO phase or along the antinodes of the $d$ -wave order parameter [44] when orbital (vortex) effects were not considered.", "The interaction between a FFLO and a SDW [29], [17], [45] also prefers $\\mathbf {Q}_\\mathrm {SDW}\\perp {q}_{\\mathrm {FFLO}}$ and would therefore tend to allign ${q}_{\\mathrm {FFLO}}$ along the $d$ -wave nodes in the $Q$ phase of CeCoIn$_5$ .", "This alignment occurs because Cooper pairs traveling in a direction perpendicular to $\\mathbf {Q}_\\mathrm {SDW}$ experience a uniform magnetization, and it is preferable for superconductivity to be modulated in this direction [45].", "If these requirements were allowed to be satisfied, i.e., if ${q}_{\\mathrm {FFLO}}$ is allowed to not follow $\\mathbf {H}$ and to instead lie along the $d$ -wave nodes and be perpendicular to $\\mathbf {Q}_\\mathrm {SDW}$ , the following will take place: For $\\mathbf {Q}_\\mathrm {SDW}\\perp \\mathbf {J}$ , ${q}_{\\mathrm {FFLO}}\\parallel \\mathbf {J}$ , and the FFLO nodal planes would be perpendicular to $\\mathbf {J}$ [Fig.", "7(a)] and increase quasiparticle scattering, decreasing $\\kappa $ .", "For $\\mathbf {Q}_\\mathrm {SDW}\\parallel \\mathbf {J}$ , ${q}_{\\mathrm {FFLO}}\\perp \\mathbf {J}$ , and the FFLO nodal planes would be parallel to $\\mathbf {J}$ [Fig.", "7(b)], increasing both the density of states of quasiparticles with momentum ${k}$ along $\\mathbf {J}$ and $\\kappa $ .", "The effect of the FFLO state described therefore has the right trend and, if larger than the effect of the SDW, could explain the thermal conductivity data.", "The requirement that ${q}_{\\mathrm {FFLO}}$ cannot be allowed to follow the magnetic field is necessitated by the fact that the changes must take place abruptly at $\\theta =45^{\\circ }$ , and after that the orders relevant to thermal conductivity must remain constant until the next antinodal plane of the $d$ wave (at $\\theta =-45^{\\circ }$ or $\\theta =135^{\\circ }$ ) is crossed by the applied magnetic field.", "In summary, to be compatible with our thermal conductivity data, the FFLO-based scenario for hypersensitivity [29] must be modified to allow ${q}_{\\mathrm {FFLO}}$ to be locked to the nodal direction of the primary $d$ -wave order parameter, with a possibility that needs to be tested theoretically and experimentally.", "In particular, small-angle neutron-scattering (SANS) measurements, with the neutron flux along the nodal [110] direction and the magnetic field applied to select ${q}_{\\mathrm {FFLO}}\\parallel [1\\bar{1}0]$ , may reveal the FFLO state if it is present." ], [ "Sliding Mode of a Spin-Density-Wave", "We rule out a contribution of the sliding mode of the SDW.", "A sliding mode along the ordering wave vector $\\mathbf {Q}_\\mathrm {SDW}$ can be expected in an incommensurate SDW state [46].", "Such effects depend heavily on pinning the SDW at impurity centers.", "When an incommensurate charge-density-wave (CDW) is depinned at a critical driving potential, the current it carries is a nonlinear function of the driving potential.", "We therefore expect a nonlinear response of thermal conductivity as a function of a sufficiently large thermal gradient in the sample.", "Our measurements for high thermal gradients (between 30% and 80% higher than normal) are also displayed in Fig.", "2(a).", "We did not observe any changes in thermal conductivity as a function of thermal gradient.", "Either the SDW remains pinned by impurities, or the contribution of the sliding mode to thermal conductivity is negligible." ] ]	1606.05015
[ [ "Machine Learning meets Data-Driven Journalism: Boosting International\n Understanding and Transparency in News Coverage" ], [ "Abstract Migration crisis, climate change or tax havens: Global challenges need global solutions.", "But agreeing on a joint approach is difficult without a common ground for discussion.", "Public spheres are highly segmented because news are mainly produced and received on a national level.", "Gain- ing a global view on international debates about important issues is hindered by the enormous quantity of news and by language barriers.", "Media analysis usually focuses only on qualitative re- search.", "In this position statement, we argue that it is imperative to pool methods from machine learning, journalism studies and statistics to help bridging the segmented data of the international public sphere, using the Transatlantic Trade and Investment Partnership (TTIP) as a case study." ], [ "The need for cross-national analysis", "The recently published news on the Panama Papers leak demonstrates firstly that tax fraud is an international phenomenon and secondly how cross-national cooperation can be beneficial to investigating and reporting.", "Admittedly, this is an exceptional case.", "Global events are still \"primarily covered in accordance with the traditional national outlook, i.e.", "national domestications and the 'domestic vs. foreign news” logic'\" [3].", "A global public sphere to address globally relevant issues has not been established yet and national biases impede possible international approaches.", "This way \"the global sociopolitical order becomes defined by the realpolitik of nation-states that cling to the illusion of sovereignty despite the realities wrought by globalization\" [5].", "Reciprocal knowledge about controversial issues across national borders is necessary to provide common ground for fruitful global discussions and proposals.", "In this position statement, we provide evidence that joining forces improves media transparency on a global scale: Combining machine learning with statistics and journalism studies contributes to bridging the segmented data of the international public sphere.", "Following an interdisciplinary approach we tackle the question of how methods from machine learning help to deepen our understanding of the discussion on cross-national issues.", "LDA, @TM, PDNs and word2vec are used to enhance transparency on international media coverage: Range, amount and framing of issues can be compared with fewer translation efforts.", "Differences in perception become obvious and evaluation divides can be interpreted.", "This will be demonstrated by an analysis of the coverage on the controversial Transatlantic Trade and Investment Partnership (TTIP) between the United States of America (U.S.) and the European Union (E.U.).", "TTIP was designed to facilitate trade between the U.S. and the E.U.", "However, TTIP's actual impact on economies and societies has been discussed controversially in both the U.S. and Europe.", "Media perception has differed in many aspects.", "The comparison of a U.S. newspaper (New York Times) and a German newspaper (Süddeutsche Zeitung) reveals that TTIP is more hotly debated in Germany than in the U.S., see Fig.", "REF .", "The New York Times highlighted the need of bank regulations and the threat that exporting nations pose to local markets.", "Whereas, Süddeutsche Zeitung focused largely on consumer protection.", "On the one hand TTIP was criticized for its implications on environmental and food standards, on the other for the negotiation proceedings that were characterized by democratic deficits and insufficient transparency.", "Intricate questions derive from the simple comparison of word frequencies: Why does the range of reported arguments differ so considerably?", "Is the German media reluctant to the Trade Partnership in general?", "And what are the reasons for the German obsession with the 'chlorinated chicken' as evidenced in the significant number of these words in articles broaching the TTIP issue?", "The TTIP example illustrates that media coverage can largely differ across nations.", "Public discussion is still strongly influenced by national media.", "Multiple languages add to the difficulties to frame a common global perspective.", "However, in a globalized world, crisis and political decisions have become far too complex to be dealt with on a national level only.", "Combining Machine Learning and data-driven journalism enables researchers to investigate large corpora of texts to reveal national patterns of argumentation, which in turn can promote international understanding.", "Figure: Attentional curves capture the development of topics in news articles over time, here illustrated for the war on Ukraine." ], [ "ML meets Data-Driven Journalism", "There is an arms race to `deeply' understand text data, and consequently a range of different techniques has been developed for media analysis.", "However, when using them for data-driven journalism, e.g.", "to gain a deeper understanding of the news reception of important political and societal issues, there are also challenges.", "Just to name few of the recent ML techniques, DeepDive [18] aims to extract structured data from texts, Metro Maps [21] extract easy to understand networks of news stories, and word2vec [17] computes Euclidean embeddings of words.", "It is trained on a corpus of documents and transforms each word into a vector by calculating word correlations.", "Similarity measures can be applied to the resulting vectors.", "Particularly, word2vec can be used to compute those words that are most likely to occur in the same context as a given word.", "It thus has the potential to reveal which words are linked closest to a given issue and hence provides a semantically enriched alternative to classical keyword searches, which is well used by journalists.", "Finally, topic models, have been used successfully in many scenarios, in particular to model discourses.", "Most prominent among them is Latent Dirichlet Allocation (LDA) [4] that characterizes each topic as a list of words and their respective probabilities to appear in the topic.", "Topics over Time (TOT) [25] follows this paradigm, but introduces a temporal component.", "In TOT, each document has a timestamp and the probability of a topic grows and declines over time.", "Thus, TOT can be employed to analyze trends in news.", "Due to this rich machine learning toolbox for analyzing news articles, it is tempting to put a stack of news articles on a data journalist's desk saying `Enjoy'.", "Unfortunately, data-driven journalism is not that simple.", "Reconsider topic models, the main focus of the present paper.", "TOT does not model attention of the crowd in a physically plausible way.", "Triggered by models from communication studies [15] and the observation that the Shifted Gompertz distribution models attentional curves [1], we developed a novel Attentional Topic Model (@TM) [19].", "It captures well the growth and decline of the popularity of topics in a physically plausible way.", "Moreover, multinomial word distributions, such as in LDA and TOT capture the most common words used in each topic.", "However, they often fail to give a deeper understanding of topics required when investigating media discourse.", "That is why APMs [13], which discover word dependencies in each topic, have been introduced; essentially, they encode topics as weighted undirected graphs.", "Often, however, word dependencies are asymmetric.", "If the word 'treaty' appears in a text, it is very likely that the text will refer to the museum's 'secretary of state', too.", "The phrase 'secretary of state', on the other hand, is a very general term and can be used in many different contexts.", "Thus, it does not make the word 'treaty' per se more likely.", "In [9], we therefore extended APMs to directed dependencies using Poisson Dependency Networks [12].", "Moreover, longer chains of directed dependencies may provide interesting clues to understand a topic.", "Finally, topic models have been traditionally evaluated using intrinsic measurements such as the likelihood and the perplexity of topics [24].", "As these measurements do not necessarily correspond to human judgment [6], we pool together the talents of journalists, machine learners and statisticians to obtain a better understanding of what makes a good topic.", "If we use topic models to create subcorpora e.g.", "for content analysis we have to ensure that the subcorpora are at least as good as the ones from other methods like keyword searches.", "The gold standard is the evaluation based on human judgment [22].", "The use of statistical methods helps to reduce the time requirement for human coders to come to a significant statement about the quality of a subcorpus.", "Moreover, our interdisciplinary research led to several interesting observations about the quality of topics: While researchers from a mathematical background tended to focus on topics linked to large quantities of documents, journalists oftentimes preferred those topics that were created from only few meaningful documents.", "Likewise words like 'can', 'need' and 'do', that were considered stopwords by machine learners, really caught the journalist's attention.", "We believe that these small and seemingly insignificant notices can help to improve the application and lead a way to new computational models.", "Can new topic models be developed to cater better to the specific needs of journalists?", "Are there different approaches to gain deeper insight into each topic?", "We will illustrate this using the case of TTIP." ], [ "Towards an International View on TTIP", "TTIP affects millions of people living in the U.S. and the E.U.", "and its negotiations have been controversial.", "However, content analysis of newspapers indicates that the issue is of diverging national importance.", "Both compared newspapers are high-circulation dailies from metropolises that exhibit a rather liberal orientation.", "Despite these similarities, coverage on TTIP varies significantly (see Fig.", "REF ).", "It is notable that coverage on TTIP increases considerably from 2014 in Süddeutsche Zeitung (SZ), whereas the number of articles in the New York Times (NYT) remains unaltered.", "In order to shed light on this disparity the general coverage on the U.S.A. and Europe, respectively, was analyzed.", "In SZ the sub-corpus with all articles including the pattern of the letters usa contained 59.637 articles (36 per cent of the corpus) whereas the europe-corpus in NYT contained 34.177 (11 per cent of the corpus).", "Figure: Topics in NYT found by LDA and labeled by journalists.", "(Topic Models) LDA was used to find 100 topics in each sub-corpus.", "The topics were labeled by journalists using top words and top articles.", "The topics in NYT are illustrated in Fig.", "REF .", "A glance at the results shows distinctly the different perspectives on TTIP in the public spheres.", "In SZ TTIP appears on the top word list of a topic covering articles on policies of the European Commission along with conjoined words like customs, arbitration and investor protection.", "In NYT TTIP is not among the top words of the European Commission topic which is dominated by stakeholders dealing with financial issues around the euro crisis.", "Interpreting the LDA topics further TTIP plays a less significant role in the U.S.-European relations represented by the LDA topics which are mainly various international conflicts, art, sports and general economy related topics.", "(Directed word dependencies) A PDN [12] trained on the articles of Süddeutsche Zeitung shows that the constitution of the E.U.", "as a politico-economic union of 28 states places the question of parliamentary participation in the foreground (see Fig.", "REF ).", "In the U.S., this question does not arise.", "(Attentional Topic Models) When analyzing the discourse on Europe in NYT through Attentional Topic Models [19], we found no attentional topic corresponding to the TTIP.", "Instead, the discussion focused on different topics such as the war in the Ukraine (see Fig.", "REF ).", "(word2vec) Analyzing word2vec results highlights diverse reciprocal perception: U.S. and German newspaper both coincide covering Germany mainly considering the recent migration.", "However, on the coverage on the U.S. SZ and NYT drift apart: In the SZ the importance of U.S. as an economic partner is demonstrated, whereas the NYT covers the U.S. in a broader range of topics including several sports (see Table REF ).", "Comparing the use of TTIP indicates that the NYT uses more matter-of-fact words in connection with TTIP while SZ seems to include more commenting and evaluating words including 'chlorinated chicken'.", "Word2vec solves the mystery of the 'chlorinated chicken': free trade agreement, genetically modified food and genetically modified corn are among the most similar words.", "For German TTIP opponents chicken meat disinfected with chlorine has become a symbol for lowering food safety standards and the disadvantages of TTIP in general.", "Applying machine learning to understand the cross-nationally diverse discourse on TTIP highlights the benefit which can be derived from an interdisciplinary approach.", "Yet, this interdisciplinary approach is still in its infancy.", "Table: Most similar words to 'USA', 'Germany' and 'TTIP' in NYT and SZ (translated from German) according to word2Vec." ], [ "Lesson Learned: Bridging Fields", "The absence of a common public sphere has already been constituted as an enduring obstacle to further political and economic integration in Europe [11], [14], [23], [20], a region where the difficulty of understanding between people speaking different languages becomes evident in spite of small distances.", "Agents in politics and business face a confusing multitude of partly conflicting national discourses.", "Therefore, finding common solutions in a democratic context is hindered.", "The defiance of finding common ground for discussion becomes even more challenging if international understanding is volitional.", "To amend the development of a global public sphere discussing and approaching international challenges it is imperative that computer scientists, information scientists, and experts in communication studies pool their talents and knowledge to help find efficient and effective ways of managing the news sources available.", "Research in this new field is necessarily interdisciplinary since developing new methods and applying established ones should eventually lead to instruments that enable not only researches and experienced data journalists but also practitioners in the media, in politics and business to compare debates internationally.", "So far using machine learning methods for content analysis is still uncommon in communication studies and best practices for algorithmic text analysis (ATA) are still being negotiated.", "In the TTIP case they were used as part of a hybrid approach \"that combines computational and manual methods throughout the process .", ".", ".", "[to] retain the strengths of traditional content analysis while maximizing the accuracy, efficiency, and largescale capacity of algorithms for examining Big Data.\"", "[16] Following this approach, patterns that so far have been hidden can be made visible.", "Transparency will be achieved on the prevailing debates, showing how they evolve and relate to existing narratives, identifying national frames and agenda setters and showing divergences and convergences across national debates.", "The focus of research should be on interlocking of long-term discourse patterns with current issues.", "Which arguments and frames have dominated the debate on the refugee crisis?", "Why is the opposition against TTIP so strong in the German speaking countries of Germany, Austria and Luxembourg while others embrace the deal?", "Why have Germany and France differed so fundamentally on how to handle the Greek debt crisis?", "Which persons and institutions are dominating the debates in the respective countries?", "In which areas do new topics or new frames emerge?", "Our TTIP study also motivates to revisit a considerable number of important theories of communication studies.", "The mechanisms of Agenda-setting [2] and issue attention cycles [7] can be visualized using clustering models in an entirely new dimension; the most important agents of the public discourse [10] can be analyzed with named-entity-recognition and network-visualizations, framing of news [8] can be illustrated using sentiment-analysis.", "In a nutshell, the potential of machine learning for analyzing international communication and discourses is high.", "However, the potential can only be achieved if new methods are developed and made available as easy-to-use applications.", "Overall, applying machine learning to broaden insight in international news coverage opens up fundamentally new intellectual territory with great potential to advance the state of the art of computer science and related disciplines and to provide unique societal benefits.", "Measures to achieve this potential involve intense interdisciplinary collaboration and the mutual objective to develop methods being easily usable for everyone interested in profound international understanding." ], [ "Acknowledgment", "This work was supported by the DFG Collaborative Research Center SFB 876 project A6 and A1 and the Dortmund Center for Media Analysis (DoCMA)." ] ]	1606.05110
[ [ "Scalable Partial Least Squares Regression on Grammar-Compressed Data\n Matrices" ], [ "Abstract With massive high-dimensional data now commonplace in research and industry, there is a strong and growing demand for more scalable computational techniques for data analysis and knowledge discovery.", "Key to turning these data into knowledge is the ability to learn statistical models with high interpretability.", "Current methods for learning statistical models either produce models that are not interpretable or have prohibitive computational costs when applied to massive data.", "In this paper we address this need by presenting a scalable algorithm for partial least squares regression (PLS), which we call compression-based PLS (cPLS), to learn predictive linear models with a high interpretability from massive high-dimensional data.", "We propose a novel grammar-compressed representation of data matrices that supports fast row and column access while the data matrix is in a compressed form.", "The original data matrix is grammar-compressed and then the linear model in PLS is learned on the compressed data matrix, which results in a significant reduction in working space, greatly improving scalability.", "We experimentally test cPLS on its ability to learn linear models for classification, regression and feature extraction with various massive high-dimensional data, and show that cPLS performs superiorly in terms of prediction accuracy, computational efficiency, and interpretability." ], [ "Introduction", "Massive data are now abundant throughout research and industry, in areas such as biology, chemistry, economics, digital libraries and data management systems.", "In most of these fields, extracting meaningful knowledge from a vast amount of data is now the key challenge.", "For example, to remain competitive, e-commerce companies need to constantly analyze huge data of user reviews and purchasing histories [24].", "In biology, detection of functional interactions of compounds and proteins is an important part in genomic drug discovery [29], [7] and requires analysis of a huge number of chemical compounds [3] and proteins coded in fully sequenced genomes [4].", "There is thus a strong and growing demand for developing new, more powerful methods to make better use of massive data and to discover meaningful knowledge on a large scale.", "Learning statistical models from data is an attractive approach for making use of massive high-dimensional data.", "However, due to high runtime and memory costs, learning of statistical models from massive data — especially models that have high interpretability — remains a challenge.", "Table: Summary of scalable learning methods of linear models.Partial least squares regression (PLS) is a linear statistical model with latent features behind high-dimensional data [26], [35], [36] that greedily finds the latent features by optimizing the objective function under the orthogonal constraint.", "PLS is suitable for data mining, because extracted latent features in PLS provide a low-dimensional feature representation of the original data, making it easier for practitioners to interpret the results.", "From a technical viewpoint, the optimization algorithm in PLS depends only on elementary matrix calculations of addition and multiplication.", "Thus, PLS is more attractive than other machine learning methods that are based on computationally burdensome mathematical programming and complex optimization solvers.", "In fact, PLS is the most common chemoinformatics method in pharmaceutical research.", "However, applying PLS to massive high-dimensional data is problematic.", "While the memory for the optimization algorithm in PLS depends only on the size of the corresponding data matrix, storing all high-dimensional feature vectors in the data matrix consumes a huge amount of memory, which limits large-scale applications of PLS in practice.", "One can use lossy compression (e.g., PCA [14], [9] and $b$ -bit minwise hashing [12], [20]) to compactly represent data matrices and then learn linear models on the compact data matrices [21].", "However, although these lossy compression-based methods effectively reduce memory usage [21], [30], their drawback is that they cannot extract informative features from the learned models, because the original data matrices cannot be recovered from the compressed ones.", "Grammar compression [2], [27], [16] is a method of lossless compression (i.e., the original data can be completely recovered from grammar-compressed data) that also has a wide variety of applications in string processing, such as pattern matching [37], edit-distance computation [13], and $q$ -gram mining [1].", "Grammar compression builds a small context-free grammar that generates only the input data and is very effective at compressing sequences that contain many repeats.", "In addition, the set of grammar rules has a convenient representation as a forest of small binary trees, which enables us to implement various string operations without decompression.", "To date, grammar compression has been applied only to string (or sequence) data; however, as we will see, there remains high potential for application to other data representations.", "A fingerprint (or bit vector) is a powerful representation of natural language texts [23], bio-molecules [32], and images [11].", "Grammar compression is expected to be effective for compressing a set of fingerprints as well, because fingerprints belonging to the same class share many identical features.", "Contribution.", "In this paper, we present a new scalable learning algorithm for PLS, which we call lossless compression-based PLS (cPLS), to learn highly-interpretable predictive linear models from massive high-dimensional data.", "A key idea is to convert high-dimensional data with fingerprint representations into a set of sequences and then build grammar rules for representing the sequences in order to compactly store data matrices in memory.", "To achieve this, we propose a novel grammar-compressed representation of a data matrix capable of supporting row and column access while the data matrix is in a compressed format.", "The original data matrix is grammar-compressed, and then a linear model is learned on the compressed data matrix, which allows us to significantly reduce working space.", "cPLS has the following desirable properties: Scalability: cPLS is applicable to massive high-dimensional data.", "Prediction Accuracy: cPLS can achieve high prediction accuracies for both classification and regression.", "Usability: cPLS has only one hyper parameter, which enhances the usability of cPLS.", "Interpretability: Unlike lossy compression-based methods, cPLS can extract features reflecting the correlation structure between data and class labels/response variables.", "We experimentally test cPLS on its ability to learn linear models for classification, regression and feature extraction with various massive high-dimensional data, and show that cPLS performs superiorly in terms of prediction accuracy, computational efficiency, and interpretability." ], [ "Literature Review", "Several efficient algorithms have been proposed for learning linear models on a large scale.", "We now briefly review the state of the art, which is also summarized in Table REF .", "Principal component analysis (PCA) [14] is a widely used machine learning tool, and is a method of lossy compression, i.e., the original data cannot be recovered from compressed data.", "There have been many attempts to extend PCA [31], [28] and present a scalable PCA in distributed settings for analyzing big data [9].", "For classification and regression tasks, a data matrix is compressed by PCA, and linear models are learned on the compressed data matrix by a supervised learning method (SL), which is referred to as PCA-SL.", "Despite these attempts, PCA and its variants do not look at the correlation structure between data and output variables (i.e., class labels/response variables), which results in not only the inability of feature extractions in PCA but also the inaccurate predictions by PCA-SL.", "Li et al.", "[21] proposed a compact representation of fingerprints for learning linear models by applying $b$ -bit minwise hashing (bMH).", "A $d$ -dimensional fingerprint is conceptually equivalent to the set $s_{i} \\subset \\lbrace 1,...,d\\rbrace $ that contains element $i$ if and only if the $i$ -th bit in the fingerprint is 1.", "Li et al.", "'s method works as follows.", "We first pick $h$ random permutations $\\pi _i$ , $i=1,..,h$ , each of which maps $[1,d]$ to $[1,d]$ .", "We then apply a random permutation $\\pi $ on a set $s_i$ , compute the minimum element as $\\min (\\pi (s_i))$ , and take as a hash value its lowest $b$ bits.", "Repeating this process $h$ times generates $h$ hash values of $b$ bits each.", "Expanding these $h$ values into a ($2^b \\times h$ )-dimensional fingerprint with exactly $h$ 1's builds a compact representation of the original fingerprint.", "Linear models are learned on the compact fingerprints by SL, which is referred to as bMH-SL.", "Although bMH-SL is applicable to large-scale learning of linear models, bMH is a method of lossy compression and cannot extract features from linear models learned by SL.", "Other hashing-based approaches have been proposed such as Count-Min sketch [5], Vowpal Wabbit [34], and Hash-SVM [25].", "However, like bMH-SL, these algorithms cannot extract features, which is a serious problem in practical applications.", "Stochastic gradient descent (SGD) [8], [33] is a computationally efficient algorithm for learning linear models on a large-scale.", "SGD samples $\\nu $ feature vectors from an input dataset and computes the gradient vector from the sampled feature vectors.", "The weight vector in linear models is updated using the gradient vector and the learning rate $\\mu $ , and this process is repeated until convergence.", "Unfortunately however, learning linear models using SGD is numerically unstable, resulting in low prediction accuracy.", "This is because SGD has three parameters ($\\nu $ , $\\mu $ , and $C$ ) that must be optimized if high classification accuracy is to be attained.", "Online learning is a specific version of SGD that loads an input dataset from the beginning and updates the weight vector in a linear model for each feature vector.", "AdaGrad [8] is an efficient online learning that automatically tunes parameters of $\\nu $ and $\\mu $ in SGD.", "Although online learning is space-efficient (owing to its online nature), it is also numerically unstable.", "Even worse, AdaGrad is applicable only to differentiable loss functions, which limits its applicability to simple linear models, e.g., SVM and logistic regression, making the learned model difficult to interpret.", "Despite the importance of scalable learning of interpretable linear models, no previous work has been able to achieve high prediction accuracy for classification/regression tasks and high interpretability of the learned models.", "We present a scalable learning algorithm that meets both these demands and is made possible by learning linear models on grammar-compressed data in the framework of PLS.", "Details of the proposed method are presented in the next section." ], [ "Grammar Compression", "Given a sequence of integers $S$ , a grammar-compressor generates a context-free grammar (CFG) that generates $S$ and only $S$ .", "The grammar consists of a set of rulesIn this paper we assume without loss of generality that the grammar is in Chomsky Normal Form.. Each rule is of the form $Z_i\\rightarrow ab$ .", "Symbols that appear on the left-hand side of any rule are called non-terminals.", "The remaining symbols are called terminals, all of which are present in the input sequence.", "Informally, a rule $Z_i\\rightarrow ab$ indicates that on the way to recovering the original sequence from its grammar-compressed representation, occurrences of the symbol $Z_i$ should be replaced by the symbol pair $ab$ (the resulting sequence may then be subject to yet more replacements).", "A data structure storing a set of grammar rules is called a dictionary and is denoted by $D$ .", "Given a non-terminal, the dictionary supports access to the symbol pair on the right-hand of the corresponding grammar rule, i.e., $D[Z_i]$ returns $ab$ for rule $Z_i\\rightarrow ab$ .", "The original sequence can be recovered from the compressed sequence and $D$ .", "The set of grammar rules in $D$ can be represented as a forest of (possibly small) binary trees called grammar trees, where each node and its left/right children correspond to a grammar rule.", "See Figure REF for an illustration.", "The size of a grammar is measured as the number of rules plus the size of compressed sequence.", "The problem of finding the minimal grammar producing a given string is known to be NP-complete [2], but several approximation algorithms exist that produce grammars that are small in practice (see, e.g., [27], [19], [16]).", "Among these is the simple and elegant Re-Pair [19] algorithm, which we review next.", "Figure: Illustration of grammar compression." ], [ "Re-Pair Algorithm", "The Re-Pair grammar compression algorithm by Larsson and Moffat [19] builds a grammar by repeatedly replacing the most frequent symbol pair in an integer sequence with a new non-terminal.", "Each iteration of the algorithm consists of the following two steps: (i) find the most frequent pair of symbols in the current sequence, and then (ii) replace the most frequent pair with a new non-terminal symbol, generating a new grammar rule and a new (and possibly much shorter) sequence.", "Steps (i) and (ii) are then applied to the new sequence and iterated until no pair of adjacent symbols appears twice.", "Apart from the dictionary $D$ that stores the rules as they are generated, Re-Pair maintains a hash table and a priority queue that together allow the most frequent pair to be found in each iteration.", "The hash table, denoted by $H$ , holds the frequency of each pair of adjacent symbols $ab$ in the current sequence, i.e., $H:ab\\rightarrow \\textbf {N}$ .", "The priority queue stores the symbol pairs keyed on frequency and allows the most frequent symbol to be found in step (i).", "In step (ii), a new grammar rule $Z_1\\rightarrow ab$ is generated where $ab$ is the most frequent symbol pair and $Z_1$ is a new non-terminal not appearing in a sequence.", "The rule is stored in the dictionary $D$ .", "Every occurrence of $ab$ in the sequence is then replaced by $Z_1$ , generating a new, shorter sequence.", "This replacement will cause the frequency of some symbol pairs to change, so the hash table and priority queue are then suitably updated.", "Let $s^c$ denote a sequence generated at $c$ -th iteration in the Re-Pair algorithm.", "For input sequence $s$ in Figure REF , the most frequent pair of symbols is 12.", "Thus, we generate rule $Z_1\\rightarrow 12$ to be added to the dictionary $D$ and replace all the occurrences of 12 by non-terminal $Z_1$ in $s$ .", "After four iterations, the current sequence $s^4$ has no repeated pairs, and thus the algorithm stops.", "Dictionary $D$ has four grammar rules that correspond to a forest of two small trees.", "As described by Larsson and Moffat [19], Re-Pair can be implemented to run in linear time in the length of the input sequence, but it requires the use of several heavyweight data structures to track and replace symbol pairs.", "The overhead of these data structures (at least 128 bits per position) prevents the algorithm from being applied to long sequences, such as the large data matrices.", "Another problem that arises when applying Re-Pair to long sequences is the memory required for storing the hash table: a considerable number of symbol pairs appear twice in a long sequence, and the hash table stores something for each of them, consuming large amounts of memory.", "In the next section, we present scalable Re-Pair algorithms that achieve both space-efficiency and fast compression time on large data matrices.", "Specifically, our algorithms need only constant working space." ], [ "Our Grammar-Compressed Data Matrix", "Our goal is to obtain a compressed representation of a data matrix ${\\textbf {X}}$ of $n$ rows and $d$ columns.", "Let $x_i$ denote the $i$ th row of the matrix represented as a fingerprint (i.e.", "binary vector).", "An alternative view of a row that will be useful to us is as a sequence of integers $s_i=(p_1,p_2,...,p_m)$ , $p_1<p_2<\\cdots < p_m$ , where $p_i \\in s_i$ if and only if $x_i[p_i] = 1$ .", "In other words the sequence $s_i$ indicates the positions of the 1 bits in $x_i$ .", "In what follows we will deal with a differentially encoded form of $s_i$ in which the difference for every pair of adjacent elements in $s_i$ is stored, i.e., $s_i=(p_1,p_2,...,p_m)$ is encoded as $s_{gi}=(p_1,p_2-p_1,p_3-p_2,...,p_m-p_{m-1})$ .", "This differential encoding tends to increase the number of repeated symbol pairs, which allows the sequences $s_{gi}$ to be more effectively compressed by the Re-Pair algorithm.", "A grammar compressor captures the underlying correlation structure of data matrices: by building the same grammar rules for the same (sequences of) integers, it effectively compresses data matrices with many repeated integers." ], [ "Re-Pair Algorithms in Constant Space", "We now present two ideas to make Re-Pair scalable without seriously deteriorating its compression performance.", "Our first idea is to modify the Re-Pair algorithm to identify top-$k$ frequent symbol pairs in all rows $s^c_{gi}$ in step (i) and replace all the occurrences of the top-$k$ symbol pairs in all rows $s^c_{gi}$ in step (ii), generating new $k$ grammar rules and new rows $s^{c+1}_{gi}$ .", "This new replacement process improves scalability by reducing the number of iterations required by roughly a factor of $k$ .", "Since we cannot replace both frequent symbol pairs $ab$ and $bc$ in triples $abc$ in step (ii), we replace the first appearing symbol pair $ab$ , preferentially.", "However, such preferential replacement can generate a replacement of a pair only once and can add redundant rules to a dictionary, adversely affecting compression performance.", "To overcome this problem, we replace the first and second appearances of each frequent pair at the same time and replace the next successive appearance of the frequent pair as usual, which guarantees generating grammar rules that appear at least twice.", "Our second idea is to reduce the memory of the hash table by removing infrequent symbol pairs.", "Since our modified Re-Pair algorithm can work storing compressed sequences $s_{gi}^c$ at each iteration $c$ in a secondary storage device, the hash table consumes most of the memory in execution.", "Our modified Re-Pair generates grammar rules from only top-$k$ frequent symbol pairs in the hash table, which means only frequent symbol pairs are expected to contribute to the compression.", "Thus, we remove infrequent symbol pairs from the hash table by leveraging the idea behind stream mining techniques originally proposed in [15], [6], [22] for finding frequent items in data stream.", "Our method is a counter-based algorithm that computes the frequency of each symbol pair and removes infrequent ones from the hash table at each interval in step (i).", "We present two Re-Pair algorithms using lossy counting and frequency counting for removing infrequent symbol pairs from the hash table.", "We shall refer to the Re-Pair algorithms using lossy counting and frequency counting as Lossy-Re-Pair and Freq-Re-Pair, respectively." ], [ "Lossy-Re-Pair", "The basic idea of lossy counting is to divide a sequence of symbols into intervals of fixed length and keep symbol pairs in successive intervals in accordance with their appearance frequencies in a hash table.", "Thus, if a symbol pair has appeared $h$ times in the previous intervals, it is going to be kept in the next $h$ successive intervals.", "Let us suppose a sequence of integers made by concatenating all rows $s_{gi}$ of ${\\bf X}$ and let $N$ be the length of the sequence.", "We divide the sequence into intervals of fixed-length $\\ell $ .", "Thus, the number of intervals is $N/\\ell $ .", "We use hash table $H$ for counting the appearance frequency of each symbol pair in the sequence.", "If symbol pair $ab$ has count $H(ab)$ , it is ensured that $ab$ is kept in hash table $H$ until the next $H(ab)$ -th interval.", "If symbol pair $ab$ first appears in the $q$ -th interval, $H(ab)$ is initialized as $qN/\\ell + 1$ , which ensures that $ab$ is kept until at least the next interval, i.e., the $(qN/\\ell +1)$ -th interval.", "Algorithm REF shows the pseudo-code of lossy counting.", "The estimated number of symbol pairs in the hash table is $O(\\ell )$ [22], resulting in $O(\\ell \\log \\ell )$ bits consumed by the hash table.", "Lossy counting.", "$H$ : hash table, $N$ : length of an input string at a time point, $\\ell $ : length of each interval.", "Note that lossy counting can be used in step (i) in the Re-Pair algorithm.", "[1] Initialize $N=0$ and $\\Delta =0$ LossyCounting$a,b$ $N=N+1$ $H(ab) \\ne 0$ $H(ab)=H(ab)+1$ $H(ab)=\\Delta + 1$ $\\lfloor \\frac{N}{\\ell } \\rfloor \\ne \\Delta $ $\\Delta =\\lfloor \\frac{N}{\\ell } \\rfloor $ each symbol pair $ab$ in $H$ $H(ab)<\\Delta $ Remove $ab$ from $H$" ], [ "Freq-Re-Pair", "The basic idea of frequency counting is to place a limit, $v$ , on the maximum number of symbol pairs in hash table $H$ and then keep only the most frequent $v$ symbol pairs in $H$ .", "Such frequently appearing symbol pairs are candidates to be replaced by new non-terminals, which generates a small number of rules.", "The hash table counts the appearance frequency for each symbol pair in step (i) of the Re-Pair algorithm.", "When the number of symbol pairs in the hash table reaches $v$ , Freq-Re-Pair removes the bottom $\\epsilon $ percent of symbol pairs with respect to frequency.", "We call $\\epsilon $ the vacancy rate.", "Algorithm REF shows the pseudo-code of frequency counting.", "The space consumption of the hash table is $O(v \\log {v})$ bits.", "Frequency counting.", "$H$ : hash table, $\|H\|$ : number of symbol pairs in $H$ , $v$ : the maximum number of symbol pairs in $H$ , $\\epsilon $ : vacancy rate.", "Note that frequency counting can be used in step (i) in the Re-Pair algorithm.", "[1] FrequencyCounting$a,b$ $H(ab) \\ne 0$ $H(ab)=H(ab)+1$ $\|H\| \\ge v$ $v(1-\\epsilon /100)<\|H\|$ each symbol pair $a^\\prime b^\\prime $ in H $H(a^\\prime b^\\prime )=H(a^\\prime b^\\prime )-1$ $H(a^\\prime b^\\prime )=0$ Remove $a^\\prime b^\\prime $ from $H$ $H(ab)=1$" ], [ "Direct Access to Row and Column", "In this section, we present algorithms for directly accessing rows and columns of a grammar-compressed data matrix, which is essential for us to be able to apply PLS on the compressed matrix in order to learn linear regression models." ], [ "Access to Row", "Accessing the $i$ -th row corresponds to recovering the original $s_i$ from grammar-compressed $s_{gi}^c$ .", "We compute this operation by traversing the grammar trees.", "For recovering the $i$ -th row $s_{i}$ , we start traversing the grammar tree having a node of the $q$ -th symbol $s_{gi}^c[q]$ as a root for each $q$ from 1 to $\|s_{gi}^c\|$ .", "Leaves encountered in the traversal must have integers in sequence $s_{gi}$ , which allows us to recover $s_{gi}$ via tree traversals, starting from the nodes with non-terminal $s^c_{gi}[q]$ for each $q\\in [1,\|s_{gi}^c\|]$ .", "We recover the original $i$ -th row $s_i$ from $s_{gi}$ by cumulatively adding integers in $s_{gi}$ from 1 to $\|s_{gi}\|$ , i.e, $s_i[1]=s_{gi}[1]$ , $s_i[2]=s_{gi}[2]+s_i[1]$ ,...,$s_i[\|s_{gi}\|]=s_{gi}[\|s_{gi}\|]+s_i[\|s_{gi}\|-1]$ ." ], [ "Access to Column", "Accessing the $j$ -th column of a grammar-compressed data matrix requires us to obtain a set of row identifiers $R$ such that $x_{ij}=1$ for $i\\in [1,n]$ , i.e., $R=\\lbrace i\\in [1,n]; x_{ij}=1 \\rbrace $ .", "This operation enables us to compute the transpose ${\\textbf {X}}^\\intercal $ from ${\\textbf {X}}$ in compressed format, which is used in the optimization algorithm of PLS.", "$P[Z_i]$ stores a summation of terminal symbols as integers at the leaves under the node corresponding to terminal symbol $Z_i$ in a grammar tree.", "For example, in Figure REF , $P[Z_1]=3$ , $P[Z_2]=6$ , $P[Z_3]=8$ and $P[Z_4]=4$ .", "$P$ can be implemented as an array that is randomly accessed from a given non-terminal symbol.", "We shall refer to $P$ as the weight array.", "The size of $P$ depends only on the grammar size.", "The $j$ -th column is accessed to check whether or not $x_{ij} = 1$ in compressed sequence $s^c_{gi}$ , for each $i \\in [1,n]$ .", "We efficiently solve this problem on grammar-compressed data matrix by using the weight array $P$ .", "Let $u_q$ store the summation of weights from the first symbol $s^c_{gi}[1]$ to the $q$ -th symbol $s^c_{gi}[q]$ , i.e., $u_q=P[s^c_{gi}[1]]+P[s^c_{gi}[2]]+\\cdots +P[s^c_{gi}[q]]$ , and let $u_0=0$ .", "If $u_{q}$ is not less than $j$ , the grammar tree with the node corresponding to a symbol $s^c_{gi}[q]$ as a root can encode $j$ at a leaf.", "Thus, we traverse the tree in depth-first order from the node corresponding to symbol $s^c_{gi}[q]$ as follows.", "Suppose $Z=s^c_{gi}[q]$ and $u=u_{q-1}$ .", "Let $Z_\\ell $ (resptively $Z_r$ ) be $a$ (respectively $b$ ) of $Z\\rightarrow ab$ in $D$ .", "(i) if $j < u$ , we go down to the left child in the tree; (ii) otherwise, i.e., $j \\ge u$ , we add $P[Z_\\ell ]$ to $u$ and go down to the right child.", "We continue the traversal until we reach a leaf.", "If $s=j$ at a leaf, this should be $x_{ij}=1$ at row $i$ ; thus we add $i$ to solution set $R$ .", "Algorithm REF shows the pseudo-code for column access.", "[t] Access to the $j$ -th column on grammar-compressed data matrix.", "$R$ : solution set of row identifiers $i$ at column $j$ s.t.", "$x_{ij}=1$ .", "[1] AccessColumn$j$ $i$ in $1..n$ $u_{0}=0$ $q$ in $1..\|s^c_{gi}\|$ $u_{q}=u_{q-1}+P[S^c_{gi}[q]]$ $j \\le u_{q}$ Recursion$(i,j,s^c_{gi}[q],u_{q-1})$ break [1] Recursion$i$ ,$j$ ,$Z$ ,$u$ $Z$ is a terminal symbol $u+Z=j$ Add $i$ to $R$ return Set $Z_l$ (resp.", "$Z_r$ ) as $a$ (resp.", "$b$ ) of $Z\\rightarrow ab$ in $D$ $u+P[Z_{l}] > j$ Recursion($i$ ,$j$ ,$Z_l$ ,$u$ ) Go to left child Recursion($i$ ,$j$ ,$Z_r$ ,$u+P[Z_l]$ ) Go to right child" ], [ "cPLS", "In this section we present our cPLS algorithm for learning PLS on grammar-compressed data matrices.", "We first review the PLS algorithm on uncompressed data matrices.", "NIPALS [35] is the conventional algorithm for learning PLS and requires the deflation of the data matrix involved.", "We thus present a non-deflation PLS algorithm for learning PLS on compressed data matrices." ], [ "NIPALS", "Let us assume a collection of $n$ data samples and their output variables $(x_1,y_1),$ $(x_2,y_2),...,(x_n,y_n)$ where $y_i \\in \\Re $ .", "The output variables are assumed to be centralized as $\\sum _{i=1}^{n}y_i=0$ .", "Denote by $y\\in \\Re ^n$ the vector of all the training output variables, i.e., $y=(y_1,y_2,...,y_n)^\\intercal $ .", "The regression function of PLS is represented by the following special form, $f(x) = \\sum _{i=1}^m \\alpha _i w_i^\\intercal x,$ where the $w_i$ are weight vectors reducing the dimensionality of $x$ ; they satisfy the following orthogonality condition: $ w_i^\\intercal \\textbf {X}^\\intercal \\textbf {X}w_j = \\left\\lbrace \\begin{array}{cl}1 & i = j \\\\0 & i \\ne j\\end{array}.\\right.$ We have two kinds of variables $w_i$ and $\\alpha _i$ to be optimized.", "Denote by ${\\textbf {W}} \\in \\Re ^{d\\times m}$ the weight matrix $i$ -th column of which is weight vector $w_i$ , i.e., ${\\textbf {W}}=(w_1,w_2,...,w_m)$ .", "Let $\\alpha \\in \\Re ^m$ be a vector whose $i$ -th element is $\\alpha _i$ , i.e., $\\alpha =(\\alpha _1,\\alpha _2,...,\\alpha _m)^\\intercal $ .", "Typically, ${\\textbf {W}}$ is first optimized and then $\\alpha $ is determined by minimizing the least squares error without regularization, $\\min _{\\alpha } \|\|y-\\textbf {X}\\textbf {W}\\alpha \|\|^2_2.$ By computing the derivative of equation (REF ) with respect to $\\alpha $ and setting it to zero, $\\alpha $ is obtained as follows: $\\alpha = (\\textbf {W}^\\intercal \\textbf {X}^\\intercal \\textbf {X}\\textbf {W})^{-1}\\textbf {W}^\\intercal \\textbf {X}^\\intercal y.$ The weight vectors are determined by the following greedy algorithm.", "The first vector $w_1$ is obtained by maximizing the squared covariance between the mapped feature $\\textbf {X}w$ and the output variable $y$ as follows: $w_1 = \\operatornamewithlimits{argmax}_{w} cov^2(\\textbf {X}w,y)$ , subject to $w^\\intercal \\textbf {X}^\\intercal \\textbf {X} w=1$ , where $cov(\\textbf {X}w,\\textbf {y})=y^\\intercal \\textbf {X}w$ .", "The problem can be analytically solved as $w_1=\\mathbf {X}^\\intercal y$ .", "For the $i$ -th weight vector, the same optimization problem is solved with additional constraints to maintain orthogonality, $w_i = \\operatornamewithlimits{argmax}_{w} cov^2(\\textbf {X}w,y),$ subject to $w^\\intercal \\textbf {X}^\\intercal \\textbf {X} w=1$ , $w^\\intercal \\textbf {X}^\\intercal \\textbf {X}^\\intercal w_j=0$ , $j=1,...,i-1$ .", "The optimal solution of this problem cannot be obtained analytically, but NIPALS solves it indirectly.", "Let us define the $i$ -th latent vector as $t_i=\\textbf {X}w_i$ .", "The optimal latent vectors $t_i$ are obtained first and the corresponding $w_i$ is obtained later.", "NIPALS performs the deflation of design matrix $\\textbf {X}$ to ensure the orthogonality between latent components $t_i$ as follows, $\\textbf {X} = \\textbf {X} - t_it_i^\\intercal \\textbf {X}$ .", "Then, the optimal solution has the form, $w_i = \\mathbf {X}^\\intercal y$ .", "Due to the deflation, ${\\mathbf {X}} = {\\mathbf {X}} - t_it_i^\\intercal {\\mathbf {X}}$ , NIPALS completely destroys the structure of ${\\mathbf {X}}$ .", "Thus, it cannot be used for learning PLS on grammar-compressed data matrices." ], [ "cPLS Algorithm", "We present a non-deflation PLS algorithm for learning PLS on grammar-compressed data matrices.", "Our main idea here is to avoid deflation by leveraging the connection between NIPALS [35] and the Lanczos method [18] which was originally proposed for recursive fitting of residuals without changing the structure of a data matrix.", "We define residual $r_{i+1}=(r_i - (y^\\intercal t_{i-1})t_{i-1})$ that is initialized as $r_1=y$ .", "The $i$ -th weight vector is updated as $w_i = {\\textbf {X}}^\\intercal (r_{i-1} - (y^\\intercal t_{i-1})t_{i-1})$ , which means $w_i$ can be computed without deflating the original data matrix $\\textbf {X}$ .", "The $i$ -th latent vector is computed as $t_i={\\textbf {X}}w_i$ and is orthogonalized by applying the Gram-Schmidt orthogonalization to the $i$ -th latent vector $t_i$ and previous latent vectors $t_1$ ,$t_2$ ,...,$t_{i-1}$ as follows, $t_i = (\\textbf {I} - \\textbf {T}_{i-1}\\textbf {T}_{i-1}^\\intercal )\\textbf {X}w_i$ , where $\\textbf {T}_{i-1}=(t_1,t_2,...,t_{i-1}) \\in \\Re ^{n\\times (i-1)}$ .", "The non-deflation PLS algorithm updates the residual $r_i$ instead of deflating ${\\textbf {X}}$ , thus enabling us to learn PLS on grammar-compressed data matrices.", "cPLS is the non-deflation PLS algorithm that learns PLS on grammar-compressed data matrices.", "The input data matrix is grammar-compressed and then the PLS is learned on the compressed data matrix by the non-deflation PLS algorithm.", "Our grammar-compressed data matrix supports row and column accesses directly on the compressed format for computing matrix calculations of addition and multiplication, which enables us to learn PLS by using the non-deflation PLS algorithm.", "Let ${\\textbf {X}}_G$ be the grammar-compressed data matrix of ${\\textbf {X}}$ .", "Algorithm REF shows the pseudo-code of cPLS.", "Since our grammar-compression is lossless, the cPLS algorithm on grammar-compressed data matrices learns the same model as the non-deflation PLS algorithm on uncompressed data matrices and so achieves the same prediction accuracy.", "[h] The cPLS algorithm.", "${\\textbf {X}}_G$ : the grammar-compressed data matrix of ${\\textbf {X}}$ .", "[1] $r_1=y$ $i=1,...,m$ $w_i = \\textbf {X}_G^\\intercal r_i$ access to column $i=1$ $t_1=\\textbf {X}_G w_i$ access to row $t_i=(\\textbf {I}-\\textbf {T}_{i-1}\\textbf {T}_{i-1}^\\intercal )\\textbf {X}_Gw_{i}$ access to row $t_i=t_i/\|\|t_i\|\|_2$ $r_{i+1}=r_i-(y^\\intercal t_i)t_i$ Compute the coefficients $\\alpha $ using equation (REF ).", "We perform feature extraction after line REF at each iteration in Algorithm REF .", "The features corresponding to the top-$u$ largest weights $w_i$ are extracted.", "Due to the orthogonality condition (REF ), the extracted features give users a novel insight for analyzing data, which is shown in Section .", "The cPLS algorithm has three kinds of variables to be optimized: $w_i$ , $r_i$ , and $t_i$ .", "The memory for $w_m$ is $O(md)$ and the memory for $t_m$ and $r_i$ is $O(mn)$ .", "Thus, the total memory for the variables in cPLS is $O(m\\min (n,d))$ highly depending on parameter $m$ .", "The parameter $m$ controls the amount of fitting of the model to the training data and is typically chosen to optimize the cross validation error.", "Since the cPLS algorithm learns the model parameters efficiently, $m$ can be set to a small value, which results in overall space-efficiency." ], [ "Experiments", "In this section, we demonstrate the effectiveness of cPLS with massive datasets.", "We used five datasets, as shown in Table REF .", "\"Book-review\" consists of 12,886,488 book reviews in English from Amazon [24].", "We eliminated stop-words from the reviews and then represented them as 9,253,464 dimensional fingerprints, where each dimension of the fingerprint represents the presence or absence of a word.", "\"Compound\" is a dataset of 42,682 chemical compounds that are represented as labeled graphs.", "We enumerated all the subgraphs of at most 10 vertices from the chemical graphs by using gSpan [38] and then converted each chemical graph into a 52,099 dimensional fingerprint, where each dimension of the fingerprint represents the presence or absence of a chemical substructure.", "\"Webspam\" is a dataset of 16,609,143 fingerprints of 350,000 dimensionsThe dataset is downloadable from http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html.. \"CP-interaction\" is a dataset of 216,121,626 compound-protein pairs, where each compound-protein pair is represented as a 3,621,623 dimensional fingerprint and 300,202 compound-protein pairs are interacting pairs according to the STITCH database [17].", "We used the above four datasets for testing the binary classification ability.", "\"CP-intensity\" consists of 1,329,100 compound-protein pairs represented as 682,475 dimensional fingerprints, where the information about compound-protein interaction intensity was obtained from several chemical databases (e.g., ChEMBL, BindingDB and PDSP Ki).", "The intensity was observed by IC50 (half maximal (50%) inhibitory concentration).", "We used the \"CP-intensity\" dataset for testing the regression ability.", "The number of all the nonzero dimensions in each dataset is summarized in the #nonzero column in Table REF , and the size for storing fingerprints in memory by using 32bits for each element is written in the memory column in Table REF .", "We implemented all the methods by C++ and performed all the experiments on one core of a quad-core Intel Xeon CPU E5-2680 (2.8GHz).", "We stopped the execution of each method if it had not finished within 24hours in the experiments.", "In the experiments, cPLS did not use a secondary storage device for compression, i.e., cPLS compressed data matrices by loading all data in memory." ], [ "Compression Ability and Scalability", "First, we investigated the influence on compression performance of the top-$k$ parameter in our Re-Pair algorithms.", "For this setting, we used the Lossy-Re-Pair algorithm, where parameter $\\ell $ is set to the total length of all rows in an input data matrix in order to keep all the symbols in the hash table.", "We examined $k=\\lbrace 1\\times 10^4, 2.5\\times 10^4, 5\\times 10^4, 7.5\\times 10^4, 10\\times 10^4 \\rbrace $ for the Book-review, Compound and Webspam datasets and examined $k=\\lbrace 1\\times 10^5, 2.5\\times 10^5, 5\\times 10^5, 7.5\\times 10^5, 10\\times 10^5\\rbrace $ for the CP-interaction and CP-intensity datasets.", "Figure REF shows compression size and compression time for various top-$k$ .", "We observed a trade-off between compressed size and compression time for all the datasets.", "The smaller the compressed size, the larger the compression time for larger values of $k$ .", "In particular, significantly faster compression time was possible at the cost of only slightly worse compression.", "For example, Lossy-Re-Pair took 57,290 seconds to compress the Book-review dataset and its size was 1,498 mega bytes (MB) for $k$ =10000.", "When $k$ =100000, compression time dropped to 20,004 seconds (less than half), while compressed size increased negligibly to 1,502MB.", "The same trends for the Book-review dataset were observed in the other datasets, which suggests that in practice a large value of $k$ can be chosen for fast compression, without adversely affecting compression performance.", "Notably, we observed our compression method to be particularly effective for the larger datasets: CP-interaction and CP-intensity.", "The original sizes of CP-interaction and CP-intensity were 125GB and 110GB, respectively, while the compressed sizes of CP-interaction and CP-intensity were at most 5GB and at 535MB, respectively.", "Our compression method thus achieved compression rates of 4% and less than 1% for CP-interaction and CP-intensity, respectively.", "Such significant reductions in data size enable the PLS algorithm to scale to massive data.", "Next, we evaluated the performance of Lossy-Re-Pair and Freq-Re-Pair, where parameters $\\ell $$=$$\\lbrace 1$ MB, 10MB, 100MB, 1000MB$\\rbrace $ were examined for Lossy-Re-Pair, and parameters $v$ $=$$\\lbrace 1$ MB, 10MB, 100MB, 1000MB$\\rbrace $ and $\\epsilon =\\lbrace 30\\rbrace $ were examined for Freq-Re-Pair.", "Table REF shows the compressed size, compression time and the working space used for the hash table in Lossy-Re-Pair and Freq-Re-Pair.", "We observed that both Lossy-Re-Pair and Freq-Re-Pair achieved high compression rates using small working space.", "Such efficiency is crucial when the goal is to compress huge data matrices that exceed the size of RAM; our Re-Pair algorithm can compress data matrices stored in external memory (disk).", "For compressing the CP-interaction dataset, Lossy-Re-Pair and Freq-Re-Pair consumed 16GB and 13GB, respectively, achieving a compressed size of 5GB.", "We observed the same tendency for the other datasets (See Table REF )." ], [ "Prediction Accuracy", "We evaluated the classification and regression capabilities of cPLS, PCA-SL, bMH-SL and SGD.", "Following the previous works [39], [21], we randomly selected 20% of samples for testing and used the remaining 80% of samples for training.", "cPLS has one parameter $m$ , so we selected the best parameter value among $m=\\lbrace 10, 20,...,100\\rbrace $ that achieved the highest accuracy for each dataset.", "The PCA phase of PCA-SL has one parameter deciding the number of principal components $m$ , which was chosen from $m=\\lbrace 10,25,50,75,100\\rbrace $ whose maximum value of 100 is the same as that of cPLS's parameter $m$ .", "Linear models were learned with LIBLINEAR [10], one of the most efficient implementations of linear classifiers, on PCA's compact feature vectors, where the hinge loss of linear SVM for classification and the squared error loss for regression were used with $L_2$ -regularization.", "The learning process of PCA-SL [14], [9] has one parameter $C$ for $L_2$ -regularization, which was chosen from $C=\\lbrace 10^{-5}, 10^{-4},$ $...,10^{5}\\rbrace $ .", "For PCA-SL [14], [9], we examined all possible combinations of two parameters ($m$ and $C$ ) and selected the best combination achieving the highest accuracy for each dataset.", "The hashing process of bMH-SL [21] has two parameters (the number of hashing values $h$ and the length of bits $b$ ), so we examined all possible combinations of $h=\\lbrace 10, 30, 100\\rbrace $ and $b=\\lbrace 8, 16\\rbrace $ .", "As in PCA-SL, linear models were learned with LIBLINEAR [10] on bMH's compact feature vectors, where the hinge loss of linear SVM for classification and the squared error loss for regression were used with $L_2$ -regularization.", "The learning process of bMH-SL [21] has one parameter $C$ for $L_2$ -regularization, which was chosen from $C=\\lbrace 10^{-5}, 10^{-4},$ $...,10^{5}\\rbrace $ .", "For bMH-SL, we examined all possible combinations of three parameters ($h$ , $b$ , and $C$ ) and selected the best combination achieving the highest accuracy for each dataset.", "We implemented SGD on the basis of the AdaGrad algorithm [8] using the logistic loss for classification and the squared error loss for regression with $L_2$ -regularization.", "SGD [8] has one parameter $C$ for $L_2$ -regularization, which was also chosen from $C=\\lbrace 10^{-5}, 10^{-4},...,10^{5}\\rbrace $ .", "We measured the prediction accuracy by the area under the ROC curve (AUC) for classification and Pearson correlation coefficient (PCC) for regression.", "Note that AUC and PCC return 1 for perfect inference in classification/regression, while AUC returns 0.5 for random inference and PCC returns 0 for random inference.", "We report the best test accuracy under the above experimental settings for each method below.", "Table REF shows the prediction accuracy, working space and training time of cPLS, PCA-SL, bMH-SL, and SGD.", "The working space for the storing data matrix and the working space needed for optimizations were separately evaluated.", "While PCA-SL and bMH-SL significantly reduced the working space for storing data matrices, the classification and regression accuracies were low.", "Since PCA-SL and bMH-SL compress data matrices without looking at the correlation structure between data and output variables for compressing data matrices, high classification and regression accuracies were difficult to achieve.", "SGD significantly reduced the working space, since it did not store data matrices in memory.", "Classification and regression accuracies of SGD were not high, because of the instability of the optimization algorithm.", "In addition, SGD is applicable only to simple linear models, making the learned model difficult to interpret.", "Our proposed cPLS outperformed the other methods (PCA-SL, bMH-SL, and SGD) in terms of AUC and PCC and significantly reduced the working space.", "The results showed cPLS's efficiency for learning PLS on compressed data matrices while looking at the correlation structure between data and output variables.", "Such a useful property enables us to extract informative features from the learned model." ], [ "Interpretability", "Figure REF shows the top-10 highly-weighted features that were extracted for each component in the application of cPLS to the Compound dataset, where one feature corresponds to a compound chemical substructure.", "It was observed that structurally similar chemical substructures were extracted together as important features in the same component, and the extracted chemical substructures differed between components.", "This observation corresponds to a unique property of cPLS.", "Analysing large-scale compound structure data is of importance in pharmaceutical applications, especially for rational drug design.", "For example, the extracted chemical substructures are beneficial for users who want to identify important chemical fragments involved in therapeutic drug activities or adverse drug reactions." ], [ "Conclusions and Future Work", "We presented a scalable algorithm for learning interpretable linear models — called cPLS — which is applicable to large-scale regression and classification tasks.", "Our method has the following appealing properties: Scalability: cPLS is applicable to large numbers of high-dimensional fingerprints (see Sections REF and REF ).", "Prediction Accuracy: The optimization of cPLS is numerically stable, which enables us to achieve high prediction accuracies (see Section REF ).", "Usability: cPLS has only one hyperparameter to be tuned in cross-validation experiments (see Section REF ).", "Interpretability: Unlike lossy compression-based methods, cPLS can extract informative features reflecting the correlation structure between data and class labels (or response variables), which makes the learned models easily interpretable (see Section REF ).", "In this study, we applied our proposed grammar compression algorithm to scaling up PLS, but in principle it can be used for scaling up other machine learning methods or data mining techniques.", "An important direction for future work is therefore the development of scalable learning methods and data mining techniques based on grammar-compression techniques.", "Such extensions will open the door for machine learning and data mining methods to be applied in various large-scale data problems in research and industry." ], [ "Acknowledgments", "This work was supported by MEXT/JSPS Kakenhi (24700140, 25700004 and 25700029), the JST PRESTO program, the Program to Disseminate Tenure Tracking System, MEXT and Kyushu University Interdisciplinary Programs in Education and Projects in Research Development, and the Academy of Finland via grant 294143." ] ]	1606.05031
[ [ "New nonbinary code bounds based on divisibility arguments" ], [ "Abstract For $q,n,d \\in \\mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \\subseteq [q]^n$ with minimum distance at least $d$.", "We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \\leq 65$, $A_4(11,8)\\leq 60$ and $A_3(16,11) \\leq 29$.", "These in turn imply the new upper bounds $A_5(9,6) \\leq 325$, $A_5(10,6) \\leq 1625$, $A_5(11,6) \\leq 8125$ and $A_4(12,8) \\leq 240$.", "Furthermore, we prove that for $\\mu,q \\in \\mathbb{N}$, there is a 1-1-correspondence between symmetric $(\\mu,q)$-nets (which are certain designs) and codes $C \\subseteq [q]^{\\mu q}$ of size $\\mu q^2$ with minimum distance at least $\\mu q - \\mu$.", "We derive the new upper bounds $A_4(9,6) \\leq 120$ and $A_4(10,6) \\leq 480$ from these `symmetric net' codes." ], [ "Introduction", "For any $m \\in \\mathbb {N}$ , we write $[m]:=\\lbrace 1,\\ldots ,m\\rbrace $ .", "Fix $n ,q \\in \\mathbb {N}$ .", "A word is an element $v \\in [q]^n$ .", "So $[q]$ serves as the alphabet.", "(If you prefer $\\lbrace 0,1,\\ldots ,q-1\\rbrace $ as alphabet, take the letters mod $q$ .)", "For two words $u,v \\in [q]^n$ , their (Hamming) distance $d_H(u,v)$ is the number of indices $i$ with $u_i \\ne v_i$ .", "A code is a subset of $ [q]^n$ .", "For any code $C \\subseteq [q]^n$ , the minimum distance $d_{\\text{min}}(C)$ of $C$ is the minimum distance between any two distinct code words in $C$ .", "For $d \\in \\mathbb {N}$ , an $(n,d)_q$-code is a set $C\\subseteq [q]^n$ that satisfies $d_{\\text{min}}(C)\\ge d$ .", "Define $A_q(n,d) := \\max \\lbrace \| C\| \\,\\, \| \\,\\, C \\text{ is an $(n,d)_q$-code} \\rbrace .$ Computing $A_q(n,d)$ and finding upper and lower bounds for it is a long-standing research interest in combinatorial coding theory (cf.", "MacWilliams and Sloane [12]).", "In this paper we find new upper bounds on $A_q(n,d)$ (for some $q,n,d$ ), based on a divisibility-argument.", "In some cases, it will sharpen a combination of the following two well-known upper bounds on $A_q(n,d)$ .", "Fix $q,n,d \\in \\mathbb {N}$ .", "Then $ qd >(q-1)n\\,\\,\\, \\Longrightarrow \\,\\,\\, A_q(n,d) \\le \\frac{qd}{qd-n(q-1)}.$ This is the $q$ -ary Plotkin bound.", "Moreover, $ A_q(n,d) \\le q \\cdot A_q(n-1,d).$ A proof of these statements can be found in $\\cite {sloane}$ .", "Plotkin's bound can be proved by comparing the leftmost and rightmost terms in $(\\ref {detruc})$ below.", "The second bound follows from the observation that in a $(n,d)_q$ -code any symbol can occur at most $A_q(n-1,d)$ times at the first position.", "We view an $(n,d)_q$ -code $C$ of size $M$ as an $M \\times n$ matrix with the words as rows.", "Two codes $C,D \\subseteq [q]^n$ are equivalent (or isomorphic) if $D$ can be obtained from $C$ by first permuting the $n$ columns of $C$ and subsequently applying to each column a permutation of the $q$ symbols in $[q]$ (we will write `renumbering a column' instead of `applying a permutation to the symbols in a column').", "Table: An overview of the results obtained and discussed in this paper.", "All previous lower and upper bounds are taken from references 4ary,5ary\\cite {4ary,5ary}, except for the lower bounds A 5 (8,6)≥50A_5(8,6)\\ge 50 and A 4 (11,8)≥48A_4(11,8) \\ge 48.", "These follow from the exact values A 5 (10,8)=50A_5(10,8)=50 and A 4 (12,9)=48A_4(12,9)=48 (plotkin\\cite {plotkin}).", "For updated tables with all most recent code bounds, we refer to brouwertable\\cite {brouwertable}.In $\\cite {4ary,5ary}$ , the lower bounds $A_5(8,6)\\ge 45$ and $A_4(11,8) \\ge 34$ are given.", "If an $(n,d)_q$ -code $C$ is given, then for $j=1,\\ldots ,n$ , let $c_{\\alpha ,j}$ denote the number of times symbol $\\alpha \\in [q]$ appears in column $j$ of $C$ .", "For any two words $u,v \\in [q]^n$ , we define $g(u,v):=n-d_H(u,v)$ .", "In our divisibility arguments, we will use the following observations (which are well known and often used in coding theory and combinatorics).", "Proposition 1.1 If $C$ is an $(n,d)_q$ -code of size $M$ , then $\\binom{M}{2}(n-d) \\ge \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subseteq C\\\\ u \\ne v\\end{array}} g(u,v) = \\sum _{j=1}^n \\sum _{\\alpha \\in [q]} \\binom{c_{\\alpha ,j}}{2} \\ge n \\cdot \\left( (q-r)\\binom{m}{2} +r \\binom{m-1}{2} \\right),$ where $m := {M/q}$ and $ r:=qm-M$ , so that $M=qm-r$ and $0 \\le r < q$ .", "Moreover, writing $L$ and $R$ for the leftmost term and the rightmost term in $(\\ref {detruc})$ , respectively, we have $ \| \\lbrace \\lbrace u,v\\rbrace \\subseteq C \\,\\, \| \\,\\, u \\ne v, \\,\\, d_H(u,v)\\ne d \\rbrace \| \\le L - R,$ i.e., the number of pairs of distinct words $\\lbrace u,v\\rbrace \\subseteq C$ with distance unequal to $d$ is at most the leftmost term minus the rightmost term in $(\\ref {detruc})$ .", "The first inequality in $(\\ref {detruc})$ holds because $n-d \\ge g(u,v)$ for all $u,v \\in C$ .", "The equality is obtained by counting the number of equal pairs of entries in the same columns of $C$ in two ways.", "The second inequality follows from the (strict) convexity of the binomial coefficient $F(x):=x(x-1)/2$ .", "Fixing a column $j$ , the quantity $\\sum _{\\alpha \\in [q]} F(c_{\\alpha ,j})$ , under the condition that $\\sum _{\\alpha \\in [q]} c_{\\alpha ,j} = M$ , is minimal if the $c_{\\alpha ,j}$ are as equally divided as possible, i.e., if $c_{\\alpha ,j} \\in \\lbrace {M/q}, {M/q}\\rbrace $ for all $\\alpha \\in [q]$ .", "The desired inequality follows.", "To prove the second assertion, note that it follows from $(\\ref {detruc})$ that $\\sum _{\\lbrace u,v\\rbrace \\subseteq C, \\, u \\ne v} g(u,v) \\ge R$ , so $\| \\lbrace \\lbrace u,v\\rbrace \\subseteq C \\,\\, \| \\,\\, u \\ne v, \\,\\, d_H(u,v)\\ne d \\rbrace \| &\\le \\sum _{\\begin{array}{c}\\lbrace u,v\\rbrace \\subseteq C\\\\ u \\ne v\\end{array}} (n-d-g(u,v))\\\\&\\le \\binom{M}{2}(n-d) - R = L-R.$ Corollary 1.2 If, for some $q, n, d$ and $M$ , the left hand side equals the right hand side in $(\\ref {detruc})$ , then for any $(n,d)_q$ -code $C$ of size $M$ , $g(u,v)=n-d$ for all $u,v \\in C$ with $u \\ne v$ , i.e., $C$ is equidistant, and for each column $C_j$ of $C$ , there are $q-r$ symbols in $[q]$ that occur $m$ times in $C_j$ and $r$ symbols in $[q]$ that occur $m-1$ times in $C_j$ .", "In the next sections we will use (REF ), (REF ) and the bound in $(\\ref {boundnd})$ to give (for some $q,n,d$ ) new upper bounds on $A_q(n,d)$ , based on divisibility arguments.", "Furthermore, in Section $\\ref {symsec}$ , we will prove that, for $\\mu ,q \\in \\mathbb {N}$ , there is a 1-1-correspondence between symmetric $(\\mu ,q)$ -nets (which are certain designs) and $(n,d)_q=(\\mu q, \\mu q - \\mu )_q$ -codes $C$ with $\|C\|=\\mu q^2 $ .", "We derive some new upper bounds from these `symmetric net' codes." ], [ "The divisibility argument", "In this section, we describe the divisibility argument and illustrate it by an example.", "Next, we show how the divisibility argument can be applied to obtain upper bounds on $A_q(n,d)$ for certain $q,n,d$ .", "In subsequent sections, we will see how we can improve upon these bounds for certain fixed $q,n,d$ .", "We will use the following notation.", "Definition 2.1 ($k$ -block) Let $C$ be an $(n,d)_q$ -code in which a symbol $\\alpha \\in [q]$ is contained exactly $k$ times in column $j$ .", "The $k \\times n$ matrix $B$ formed by the $k$ rows of $C$ that have symbol $\\alpha $ in column $j$ is called a ($k$ -)block (for column $j$ ).", "In that case, columns $[n]\\setminus \\lbrace j\\rbrace $ of $B$ form an $(n-1,d)_q$ -code of size $k$ .", "At the heart of the divisibility arguments that will be used throughout this paper lies the following observation.", "Proposition 2.1 (Divisibility argument) Suppose that $C$ is an $(n,d)_q$ -code and that $B$ is a block in $C$ (for some column $j$ ) containing every symbol exactly $m$ times in every column except for column $j$ .", "If $n-d$ does not divide $m(n-1)$ , then for each $u \\in C \\setminus B$ there is a word $v \\in B$ with $d_H(u,v) \\notin \\lbrace d,n\\rbrace $ .", "Let $u \\in C \\setminus B$ .", "We renumber the symbols in each column such that $u$ is $\\mathbf {1}:=1\\ldots 1$ , the all-ones word.", "The total number of 1's in $B$ is $m(n-1)$ (as the block $B$ does not contain 1's in column $j$ since $u\\notin B$ and $B$ consists of all words in $C$ that have the same symbol in column $j$ ).", "Since $n-d$ does not divide $m(n-1)$ , there must be a word $v \\in B$ that contains a number of 1's not divisible by $n-d$ .", "In particular, the number of 1's in $v$ is different from 0 and $n-d$ .", "So $d_H(u,v) \\notin \\lbrace d,n\\rbrace $ .", "Example 2.1 We apply Proposition $\\ref {parity}$ to the case $(n,d)_q=(8,6)_5$ .", "The best known upper boundThe Delsarte bound $\\cite {delsarte}$ on $A_5(8,6)$ , the bound based on Theorem $\\ref {elementarybounds}$ , and the semidefinite programming bound based on quadruples of code words [10] all are equal to 75. is $A_5(8,6) \\le 75$ , which can be derived from $(\\ref {elementarybounds})$ and $(\\ref {elementarybounds2})$ , as the Plotkin bound yields $A_5(7,6) \\le 15$ and hence $A_5(8,6) \\le 5 \\cdot 15 = 75$ .", "Since, for $(n,d)_q=(7,6)_5$ and $M=15$ , the left hand side equals the right hand side in $(\\ref {detruc})$ , any $(7,6)_5$ -code $D$ of size 15 is equidistant and each symbol appears exactly $m=3$ times in every column of $D$ .", "Note $2=n-d \\nmid m(n-1)=21$ .", "Suppose there exists a $(8,6)_5$ -code $C$ of size 75.", "As $A_5(7,6) \\le 15$ , for each column, $C$ is divided into five 15-blocks.", "Let $B$ be a 15-block for the $j$ th column and let $u \\in C \\setminus B$ .", "Note that the other columns of $B$ contain each symbol 3 times, and $3(n-1)=3\\cdot 7 =21$ is not divisible by $n-d=2$ .", "So by Proposition REF , there must be a word $v \\in B$ with $d_H(u,v) \\notin \\lbrace 6,8\\rbrace $ .", "However, since all $(7,6)_5$ -codes of size 15 are equidistant, all distances in $C$ belong to $\\lbrace 6,8\\rbrace $ : either two words are contained together in some 15-block (hence their distance is 6) or there is no column for which the two words are contained in a 15-block (hence their distance is 8).", "This implies that an $(8,6)_5$ -code $C$ of size 75 cannot exist.", "Hence $A_5(8,6) \\le 74$ .", "Theorem $\\ref {importantth}$ and Corollary $\\ref {1mod4}$ below will imply that $A_5(8,6)\\le 70$ and in Section $\\ref {sec586}$ we will show that, with some computer assistance, the bound can be pushed down to $A_5(8,6) \\le 65$ .", "To exploit the idea of Proposition $\\ref {parity}$ , we will count the number of so-called irregular pairs of words occuring in a code.", "Definition 2.2 (Irregular pair) Let $C$ be an $(n,d)_q$ -code and $u,v \\in C$ with $u\\ne v$ .", "If $d_H(u,v) \\notin \\lbrace d,n\\rbrace $ , we call $\\lbrace u,v\\rbrace $ an irregular pair.", "For any code $C \\subseteq [q]^n$ , we write $X:= \\text{ the set of irregular pairs~$\\lbrace u,v\\rbrace $ for~$u,v \\in C$}.$ Using Proposition $\\ref {parity}$ , we can for some cases derive a lower bound on $\|X\|$ .", "If we can also compute an upper bound on $\|X\|$ that is smaller than the lower bound, we derive that the code $C$ cannot exist.", "The proof of the next theorem uses this idea.", "For fixed $q,n,d,m \\in \\mathbb {N}$ with $q \\ge 2$ , define the following quadratic polynomial in $r$ : $\\phi (r) := n(n-1-d)(r-1)r - (q-r+1)(mq(q+r-2)-2r).$ Theorem 2.2 Suppose that $q\\ge 2$ , that $m:=d/(qd-(n-1)(q-1))$ is a positive integer, and that $n-d$ does not divide $m(n-1)$ .", "If $r \\in \\lbrace 1,\\ldots ,q-1\\rbrace $ with $\\phi (r)<0$ , then $A_q(n,d) < mq^2 -r$ .", "By Plotkin's bound $(\\ref {elementarybounds})$ we have $ A_q(n-1,d) \\le mq.$ Let $D$ be an $(n-1,d)_q$ -code of size $mq-t$ with $t<q$ .", "Note that $d=m(n-1)(q-1)/(mq-1)$ .", "Then the right-hand side in $(\\ref {boundnd})$ (taking $C:= D)$ is equal to $(n-1)(m-1)t(t-1)/(2mq-2) = (n-1-d)\\binom{t}{2}$ .", "Hence $ \\text{$D$ contains at most~$(n-1-d)\\binom{t}{2}$ pairs of words with distance~$\\ne d$}.$ Therefore, all $(n-1,d)_q$ -codes $D$ of size $mq$ are equidistant (then $t=0$ ) and each symbol occurs $m$ times in every column of $D$ .", "Now let $C$ be an $(n,d)_q$ -code of size $M:=mq^2-r$ with $r \\in \\lbrace 1,\\ldots ,q-1\\rbrace $ .", "Consider an $mq$ -block $B$ for some column of $C$ .", "As $n-d$ does not divide $m(n-1)$ , by Proposition $\\ref {parity}$ we know $\\text{if~$u \\in C\\setminus B$, then there exists~$v \\in B$ with~$d_H(u,v) \\notin \\lbrace d,n\\rbrace $.", "}$ Let $B_1,\\ldots , B_{s}$ be $mq$ -blocks in $C$ for some fixed column.", "Since $\|C\|=mq^2-r$ , the number of $mq$ -blocks for any fixed column is at least $q-r$ (so we can take $s=q-r$ ).", "Then, with $(\\ref {obs})$ , one obtains a lower bound on the number $\|X\|$ of irregular pairs in $C$ .", "Every pair $\\lbrace B_{i},B_{k}\\rbrace $ of $mq$ -blocks gives rise to $mq$ irregular pairs: for each word $u \\in B_{i}$ , there is a word $v \\in B_k$ such that $\\lbrace u,v\\rbrace \\in X$ .", "This implies that in $\\cup _{i=1}^s B_i \\subseteq C$ there are at least $\\binom{s}{2}mq$ irregular pairs.", "Moreover, for each word $u$ in $C\\setminus \\cup _{i=1}^s B_i$ (there are $M-mq\\cdot s$ of such words) there is, for each $i=1,\\ldots ,s$ , a word $v_i \\in B_i$ with $\\lbrace u,v_i\\rbrace \\in X$ .", "This gives an additional number of at least $(M-mqs)s$ irregular pairs in $C$ .", "Hence: $ \|X\| &\\ge \\binom{s}{2} mq + (M-mqs)s \\\\&= \\mbox{$\\frac{1}{2}$}s (mq(2q-s-1)-2r) =: l(s).$ On the other hand, note that the $i$ th block for the $j$ th column has size $mq-r_{i,j}$ for some integer $r_{i,j}\\ge 0$ by $(\\ref {part})$ , where $\\sum _{i=1}^q r_{i,j}=r\\le q-1$ (hence each $r_{i,j} <q$ ).", "So by $(\\ref {most})$ , the number of irregular pairs in $C$ that have the same entry in column $j$ is at most $(n-1-d)\\sum _{i=1}^q \\binom{r_{i,j}}{2}.$ As each irregular pair $\\lbrace u,v\\rbrace $ has $u_j=v_j$ for at least one column $j$ , we conclude $ \|X\| \\le (n-1-d)\\sum _{j=1}^n \\sum _{i=1}^q \\binom{r_{i,j}}{2} \\le n (n-1-d)\\binom{r}{2}.$ Here the last inequality follows by convexity of the binomial function, since (for fixed $j$ ) the sum $\\sum _{i=1}^q \\binom{r_{i,j}}{2}$ under the condition that $\\sum _{i=1}^q r_{i,j}=r$ is maximal if one of the $r_{i,j}$ is equal to $r$ and the others are equal to 0.", "If each $r_{i,j}\\in \\lbrace 0,1\\rbrace $ , then $\|X\|=0$ by $(\\ref {upperb})$ .", "As $q-r\\ge 1$ , there is at least one $mq$ -block for any fixed column, so $\|X\|\\ge 1$ by $(\\ref {obs})$ , which is not possible.", "Hence we can assume that $r_{i,j} \\ge 2$ for some $i,j$ (this also implies $A_q(n,d) \\le mq^2-2$ ).", "Then the number $s$ of $mq$ -blocks for column $j$ satisfies $s \\ge q-r+1$ .", "This gives by $(\\ref {lowerb})$ and $(\\ref {upperb})$ that $ l(q-r+1) \\le \|X\| \\le (n-1-d) \\binom{r}{2}.$ Subtracting the left hand side from the right hand side in $(\\ref {newineq2})$ yields $\\phi (r)/2 \\ge 0$ , i.e., $\\phi (r) \\ge 0$ .", "So if $\\phi (r)< 0$ , then $A_q(n,d) < mq^2-r$ , as was needed to prove.", "We give two interesting applications of Theorem $\\ref {importantth}$ .", "Corollary 2.3 If $q \\equiv 1 \\pmod {4}$ and $q\\ne 1$ , then $A_q(q+3,q+1) \\le \\mbox{$\\frac{1}{2}$}q^2(q+1)-q =\\mbox{$\\frac{1}{2}$}(q-1)q (q+2).$ Apply Theorem $\\ref {importantth}$ to $n=q+3$ , $d=q+1$ and $r=q-1$ .", "Then $m=(q+1)/2 \\in \\mathbb {N}$ and $n-d=2$ does not divide $m(n-1)=(q+1)(q+2)/2$ , as $q\\equiv 1 \\pmod {4}$ .", "Furthermore, $\\phi (q-1)=-(q^3-q^2-2)<0$ .", "Hence $A_q(q+3,q+1) < q^2(q+1)/2-(q-1)$ .", "Applying Corollary $\\ref {1mod4}$ to $q=5$ gives $A_5(8,6) \\le 70$ .", "In Section $\\ref {sec586}$ we will improve this to $A_5(8,6) \\le 65$ .", "Remark 2.1 Note that for bound $(\\ref {q+3})$ to hold it is necessary that $q\\equiv 1 \\pmod {4}$ .", "If $q \\equiv 3 \\pmod {4}$ the statement does not hold in general.", "For example, $A_3(6,4)= 18$ (see $\\cite {brouwertable}$ ), which is larger than bound $(\\ref {q+3})$ .", "Theorem $\\ref {importantth}$ also gives an upper bound on $A_q(n,d)=A_q(kq+k+q,kq)$ , where $q\\ge 2$ and $k$ does not divide $q(q+1)$ (which is useful for $k <q-1$ ; for $k \\ge q+1$ the Plotkin bound gives a better bound).", "One new upper bound for such $q,n,d$ is obtained: Proposition 2.4 $A_4(11,8)\\le 60$ .", "This follows from Theorem $\\ref {importantth}$ with $q=4$ , $n=11$ , $d=8$ and $r=3$ .", "Then $m=4 \\in \\mathbb {N}$ , and $n-d=3$ does not divide $m(n-1)=40$ .", "Moreover, $\\phi (3)=-16 <0$ .", "Therefore $A_4(11,8) < 61$ .", "This implies the following bound, which is also new: Corollary 2.5 $A_4(12,8) \\le 240$ .", "By Proposition $\\ref {60}$ and $(\\ref {elementarybounds2})$ ." ], [ "Kirkman triple systems and $A_5(8,6)$ .", "In this section we consider the case $(n,d)_q=(8,6)_5$ from Example $\\ref {586}$ .", "Corollary REF implies that $A_5(8,6) \\le 70$ .", "Using small computer experiments, we will obtain $A_5(8,6) \\le 65$ .", "As in the proof of Theorem $\\ref {importantth}$ , we will compare upper and lower bounds on $\|X\|$ .", "But since an $(8,6)_5$ -code $C$ of size at most 70 does not necessarily contain a 15-block (as $70=5 \\cdot 14$ ), we need information about 14-blocks.", "To this end we show, using an analogous approach as in $\\cite {bogzin}$ (based on occurrences of symbols in columns of an equidistant code): Proposition 3.1 Any $(7,6)_5$ -code $C$ of size 14 can be extended to a $(7,6)_5$ -code of size 15.", "For $M=14$ , the leftmost term in $(\\ref {detruc})$ equals the rightmost term.", "So $C$ is equidistant and for each $j \\in \\lbrace 1,\\ldots ,7\\rbrace $ there exists a unique $\\beta _j \\in [q]$ with $c_{\\beta _j,j} =2$ and $c_{\\alpha ,j}=3$ for all $\\alpha \\in [q] \\setminus \\lbrace \\beta _j \\rbrace $ .", "We can define a 15-th codeword $u$ by putting $u_j:= \\beta _j$ for all $j=1,\\ldots ,7$ .", "We claim that $C \\cup \\lbrace u \\rbrace $ is a $(7,6)_5$ -code of size 15.", "To establish the claim we must prove that $d_H(u,w) \\ge 6$ for all $w \\in C$ .", "Suppose that there is a word $w \\in C$ with $d_H(u,w) <6$ .", "We can renumber the symbols in each column of $C$ such that $w=\\mathbf {1}$ .", "Since $C$ is equidistant, each word in $C \\setminus \\lbrace w \\rbrace $ contains precisely one 1.", "On the other hand, there are two column indices $j_1$ and $j_2$ with $u_{j_1} = 1$ and $u_{j_2} = 1$ .", "Then $C\\setminus \\lbrace w\\rbrace $ contains at most $1+1+5\\cdot 2 = 12$ occurrences of the symbol 1 (since in columns $j_1$ and $j_2$ there is precisely one 1 in $C \\setminus \\lbrace w\\rbrace $ ).", "But in that case, since $\|C\\setminus \\lbrace w \\rbrace \|=13>12$ , there is a row in $C$ that contains zero occurrences of the symbol 1, contradicting the fact that $C$ is equidistant.", "Note that a code of size more than 65 must have at least one 15- or 14-block, and therefore it must have a subcode of size 65 containing at least one 15- or 14-block.", "We shall now prove that this is impossible because $ \\text{each~$(8,6)_5$-code of size~$65$ only admits~$13$-blocks.", "}$ It follows that $A_5(8,6) \\le 65$ .", "In order to prove $(\\ref {only13})$ , let $C$ be a $(8,6)_5$ -code of size 65.", "We first compute a lower bound on the number of irregular pairs in $C$ .", "Define, for $x,y \\in \\mathbb {Z}_{\\ge 0}$ , $f(x,y) &:= (3 x+ y)(65-15 x-14 y) + 3 \\cdot 15 \\binom{x}{2} + 14 \\binom{y}{2} + 3\\cdot 14 xy\\\\& \\phantom{={,}} - 2 \\cdot 21 x- 8 y + \\mathbf {1}_{\\lbrace y>0 \\text{ and } x=0\\rbrace } (65-14-39).", "$ Proposition 3.2 (Lower bound on $\|X\|$ ) Let $C$ be an $(n,d)_q=(8,6)_5$ -code of size 65 and let $j \\in [n]$ .", "Let $x$ and $y$ be the number of symbols that appear 15 and 14 times (respectively) in column $j$ .", "Then the number $\|X\|$ of irregular pairs in $C$ is at least $f(x,y)$ .", "First consider a $(7,6)_5$ -code $D$ of size 15 or size 14 and define $S := \\lbrace u \\in [5]^7 \\,\\, \| \\,\\,d_H(w,u) \\ge 5 \\,\\,\\,\\, \\forall \\, w \\in D \\rbrace .$ For any $u \\in S$ , define $\\alpha (u) := \|\\lbrace w \\in D\\,\\,: \\, \\, d_H(u,w)=6 \\rbrace \|.$ Then $ \\text{if $\|D\|=15$, then} \\phantom{aaai} & & \\text{if $\|D\|=14$, then}\\phantom{aaai}& \\\\\| \\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)=0\\rbrace \| &=0, & \| \\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)=0\\rbrace \| &\\le 8, \\\\\| \\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)=1\\rbrace \| &\\le 21, & \|\\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)\\le 1\\rbrace \| &\\le 39.", "\\\\\| \\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)=2\\rbrace \|&=0.", "& &$ This can be checked efficiently with a computerAll computer tests in this paper are small and can be executed within a minute on modern personal computers.", "by checking all possible $(7,6)_5$ -codes of size 15 and 14 up to equivalence.", "Here we note that a $(7,6)_5$ -code $D$ (which must be equidistant, see Example REF ) of size 15 corresponds to a solution to Kirkman's school girl problem $\\cite {zinoviev}$ .Kirkman's school girl problem asks to arrange 15 girls 7 days in a row in groups of 3 such that no two girls appear in the same group twice.", "The 1-1-correspondence between $(n,d)_q=(7,6)_5$ -codes $D$ of size 15 and solutions to Kirkman's school girl problem is given by the rule: $\\text{girls~$i_1$ and~$i_2$ walk in the same triple on day~$j$ } \\Longleftrightarrow D_{i_1,j} = D_{i_2,j}$ .", "So to establish $(\\ref {calign})$ , it suffices to checkBy `check' we mean that given a $(7,6)_5$ -code $D$ of size 14 or 15, we first compute $S$ , then $\\alpha (u)$ for all $u \\in S$ , and subsequently verify $(\\ref {calign})$ .", "all $(7,6)_5$ -codes of size 15, that is, Kirkman systems (there are 7 nonisomorphic Kirkman systems $\\cite {7sol}$ ), and all $(7,6)_5$ -codes of size 14, of which there are at most $7 \\cdot 15$ by Proposition $\\ref {14block}$ .", "Let $G=(C,X)$ be the graph with vertex set $V(G):=C$ and edge set $E(G):=X$ .", "Consider a 15-block $B$ determined by column $j$ .", "By $(\\ref {calign})$ , each $u \\in C \\setminus B$ has $\\ge 1$ neighbour in $B$ .", "We observed this also in Example $\\ref {586}$ : for any $u \\in C \\setminus B$ there exists at least one $v \\in B$ such that $d_H(u,v) \\notin \\lbrace 6,8\\rbrace $ , so $d_H(u,v)=7$ and $\\lbrace u,v\\rbrace \\in X$ .", "In $(\\ref {calign})$ this is represented as: if $\|D\|=15$ then $\|\\lbrace u \\in S\\,\\, \| \\,\\, \\alpha (u)=0\\rbrace \|=0$ , i.e., for any word $u^{\\prime }$ of length 7 that has distance $\\ge 5$ to all words in a $(7,6)_5$ -code $D$ of size 15, there is at least one $v^{\\prime }\\in D$ such that $d_H(u^{\\prime },v^{\\prime })=6$ .", "Furthermore, $(\\ref {calign})$ gives that all but $\\le 21$ elements $u\\in C\\setminus B$ have $\\ge 3$ neighbours in $B$ .", "So by adding $\\le 2 \\cdot 21$ new edges, we obtain that each $u \\in C \\setminus B$ has $\\ge 3$ neighbours in $B$ .", "Similarly, for any 14-block $B$ determined by column $j$ , by adding $\\le 8$ new edges we achieve that each $u \\in C \\setminus B$ has $\\ge 1$ neighbour in $B$ .", "Hence, by adding $\\le ( 2 \\cdot 21 \\cdot x+ 8\\cdot y )$ edges to $G$ , we obtain a graph $G^{\\prime }$ with $\|E(G^{\\prime })\| \\ge (3 x+ y)(65-15 x-14 y) + 3 \\cdot 15 \\binom{x}{2} + 14 \\binom{y}{2}+ 3\\cdot 14 xy.$ This results in the required bound, except for the term with the indicator function.", "That term can be added because $\|\\lbrace u \\in S\\,\\, \|\\,\\, \\alpha (u)\\le 1\\rbrace \| \\le 39$ if $\|D\|=14$ , by $(\\ref {calign})$ .", "It is also possible to give an upper bound on $\|X\|$ .", "If $D$ is a $(7,6)_5$ -code of size $k$ , an upper bound $h(k)=L-R$ on the number of pairs $\\lbrace u,v\\rbrace \\subseteq D$ with $u \\ne v$ and $d_H(u,v)\\ne 6$ (hence $d_H(u,v)=7$ ) is given by $(\\ref {boundnd})$ .", "The resulting values $h(k)$ are given in Table REF .", "Table: Upper bound h(k)h(k) on the number of pairs {u,v}⊆D\\lbrace u,v\\rbrace \\subseteq D with d H (u,v)=7d_H(u,v)=7 for a (7,6) 5 (7,6)_5-code DD with \|D\|=k\|D\|=k.Theorem 3.3 ($A_{5}(8,6) \\le 65$ ) Suppose that $C$ is an $(n,d)_q=(8,6)_5$ -code with $\|C\|=65$ .", "Then each symbol appears exactly 13 times in each column of $C$ .", "Hence, $A_{5}(8,6) \\le 65$ .", "Let $a^{(j)}_{k}$ be the number of symbols that appear exactly $k$ times in column $j$ of $C$ .", "Then the number of irregular pairs that have the same entry in column $j$ is at most $\\sum _{k =5}^{15} a^{(j)}_{k}h(k)$ .", "It follows that $\|X\| \\le U := \\sum _{j=1}^8 \\sum _{k =5}^{15} a^{(j)}_{k} h(k).$ One may check that if $\\mathbf {a}, \\mathbf {b} \\in \\mathbb {Z}_{\\ge 0}^{15}$ are 15-tuples of nonnegative integers, with $\\sum _k a_k k =65$ , $\\sum _k b_k k =65$ , $\\sum _k a_k=5$ , $\\sum _k b_k=5$ , and $f(a_{15},a_{14}) \\le f(b_{15},b_{14}) \\ne 0$ , then $ \\sum _{k=5}^{15} (7a_k +b_k) h(k) < f(b_{15},b_{14}).$ (There are 30 $\\mathbf {a} \\in \\mathbb {Z}_{\\ge 0}^{15}$ with $\\sum _k a_k k =65$ and $\\sum _{k}a_k = 5$ .", "So there are 900 possible pairs $\\mathbf {a},\\mathbf {b}$ .", "A computer now quickly verifies $(\\ref {tocheck})$ .)", "By permuting the columns of $C$ we may assume that $\\max _j f( a^{(j)}_{15},a^{(j)}_{14})=f(a^{(1)}_{15},a^{(1)}_{14})$ .", "Hence if $f(a^{(1)}_{15},a^{(1)}_{14})>0$ , then $U &= \\sum _{j=1}^8 \\sum _{k =5}^{15} a^{(j)}_{k} h(k) = \\frac{1}{7}\\sum _{j=2}^8\\left( \\sum _{k =5}^{15} \\left(7a^{(j)}_{k} + a^{(1)}_{k} \\right) h(k) \\right)\\\\&< f(a^{(1)}_{15},a^{(1)}_{14}) \\le \|X\| $ (where we used Proposition $\\ref {LBX}$ in the last inequality), contradicting $(\\ref {contra})$ .", "So $f( a^{(j)}_{15},a^{(j)}_{14})=0$ for all $j$ , which implies (for $\\mathbf {a}^{(j)} \\in \\mathbb {Z}_{\\ge 0}^{15}$ with $\\sum _k a^{(j)}_k k =65$ , $\\sum _k a^{(j)}_k=5$ ) that $a^{(j)}_{15}=a^{(j)}_{14}=0$ for all $j$ , hence each symbol appears exactly 13 times in each column of $C$ .", "Corollary 3.4 $A_5(9,6) \\le 325$ , $A_5(10,6) \\le 1625$ and $A_5(11,6) \\le 8125$ .", "By Theorem $\\ref {65th}$ and $(\\ref {elementarybounds2})$ ." ], [ "Improved bound on $A_3(16,11)$ .", "We show that $A_3(16,11) \\le 29$ using a surprisingly simple argument.", "Proposition 4.1 $A_3(16,11) \\le 29$ .", "Suppose that $C$ is an $(n,d)_q=(16,11)_3$ -code of size 30.", "We can assume that $\\mathbf {1} \\in C$ .", "It is known that $A_3(15,11)=10$ , so the symbol 1 is contained at most 10 times in every column of $C$ .", "Since $\|C\|=30$ , the symbol 1 appears exactly 10 times in every column of $C$ , so the number of 1's in $C$ is divisible by 5.", "On the other hand it is easy to check that a $(15,11)_3$ -code of size 10 is equidistant (using $(\\ref {boundnd})$ , as $L=R$ ).", "This implies that all distances in a $(16,11)_3$ -code of size 30 belong to $\\lbrace 11, 16\\rbrace $ .", "So the number of 1's in any code word $\\ne \\mathbf {1}$ is 0 or 5.", "As $\\mathbf {1}$ contains 16 1's, it follows that the total number of 1's is not divisible by 5, a contradiction." ], [ "Codes from symmetric nets", "In this section we will show that there is a 1-1-correspondence between symmetric $(\\mu ,q)$ -nets and $(n,d)_q=(\\mu q,\\mu q-\\mu )_q$ -codes of size $\\mu q^2$ .", "From this, we derive in Section $\\ref {496}$ the new upper bound $A_4(9,6) \\le 120$ , implying $A_4(10,6) \\le 480$ .", "Definition 5.1 (Symmetric net) Let $\\mu ,q \\in \\mathbb {N}$ .", "A symmetric $(\\mu , q)$ -net (also called symmetric transversal design $\\cite {beth}$ ) is a set $X$ of $\\mu q^2$ elements, called points, together with a collection $\\mathcal {B}$ of subsets of $X$ of size $\\mu q$ , called blocks, such that: $\\mathcal {B}$ can be partitioned into $\\mu q$ partitions (block parallel classes) of $X$ .", "Any two blocks that belong to different parallel classes intersect in exactly $\\mu $ points.", "$X$ can be partitioned into $\\mu q$ sets of $q$ points (point parallel classes), such that any two points from different classes occur together in exactly $\\mu $ blocks, while any two points from the same class do not occur together in any block.That is, a symmetric $(\\mu ,q)$ -net is a $1-(\\mu q^2, \\mu q, \\mu q)$ design $D$ , which is resolvable (s1), affine (s2), and the dual design $D^*$ of $D$ is affine resolvable (s3).", "Remark 5.1 From the 1-1-correspondence between symmetric $(\\mu ,q)$ -nets and $(n,d)_q=(\\mu q,\\mu q - \\mu )_q$ -codes $C$ of size $\\mu q^2$ in Theorem REF below it follows that (s2) and (s3) can be replaced by the single condition: (s') Each pair of points is contained in at most $\\mu $ blocks, since the only condition posed on such a code is that $g(u,v) \\le \\mu $ for all distinct $u,v \\in C$ .", "Example 5.1 Let $X=\\lbrace 1,2,3,4\\rbrace $ and $\\mathcal {B}=\\lbrace \\lbrace 1,3\\rbrace ,\\,\\lbrace 2,4\\rbrace ,\\,\\lbrace 1,4\\rbrace ,\\,\\lbrace 2,3\\rbrace \\rbrace $ .", "Then $(X,\\mathcal {B})$ is a symmetric $(1,2)$ -net.", "The block parallel classes are $\\lbrace \\lbrace 1,3\\rbrace ,\\,\\lbrace 2,4\\rbrace \\rbrace $ and $\\lbrace \\lbrace 1,4\\rbrace ,\\,\\lbrace 2,3\\rbrace \\rbrace $ .", "The point parallel classes are $\\lbrace 1,2\\rbrace $ and $\\lbrace 3,4\\rbrace $ .", "By labeling the points as $x_1,\\ldots ,x_{\\mu q^2}$ and the blocks as $B_1,\\ldots ,B_{\\mu q^2}$ , the $\\mu q^2 \\times \\mu q^2$ -incidence matrix $N$ of a symmetric $(\\mu ,q)$ -net is defined by $N_{i,j} := {\\left\\lbrace \\begin{array}{ll}1 &\\mbox{if } x_i \\in B_j, \\\\0 &\\mbox{else}.\\end{array}\\right.", "}$ An isomorphism of symmetric nets is a bijection from one symmetric net to another symmetric net that maps the blocks of the first net into the blocks of the second net.", "That is, two symmetric nets are isomorphic if and only if their incidence matrices are the same up to row and column permutations.", "Symmetric nets are, in some sense, a generalization of generalized Hadamard matrices.", "Definition 5.2 (Generalized Hadamard matrix) Let $M$ be an $n \\times n$ -matrix with entries from a finite group $G$ .", "Then $M$ is called a generalized Hadamard matrix GH$(n,G)$ (or GH$(n,\|G\|)$ ) if for any two different rows $i$ and $k$ , the $n$ -tuple $(M_{ij}M_{jk}^{-1})_{j=1}^n$ contains each element of $G$ exactly $n/\|G\|$ times.", "Figure: An incidence matrix of the unique (up to isomorphism) symmetric (2,4)(2,4)-net is obtained by writing the elements e,a,b,ce,a,b,c as 4×44 \\times 4-permutation matrices in the generalized Hadamard matrix GH(8,V 4 )(8,V_4) (with V 4 V_4 the Klein 4-group).", "See Al-Kenani 42net\\cite {42net}.Each generalized Hadamard matrix GH$(n,G)$ gives rise to a symmetric $(n/\|G\|,\|G\|)$ -net: by replacing $G$ by a set of $\|G\| \\times \|G\|$ -permutation matrices isomorphic to $G$ (as a group), one obtains the incidence matrix of a symmetric net.", "Not every symmetric $(n/q,q)$ -net gives rise to a generalized Hadamard matrix GH$(n,q)$ , see $\\cite {matrixnet}$ .", "But if the group of automorphisms (bitranslations) of a symmetric $(n/q,q)$ -net has order $q$ , then one can construct a generalized Hadamard matrix GH$(n,q)$ from it.", "See $\\cite {beth}$ for details.", "Assumption 5.1 In this section we consider triples $(n,d)_q$ of natural numbers for which $qd = (q-1)n,$ hence $n-d=n/q=:\\mu $ and $\\mu \\in \\mathbb {N}$ .", "So $(n,d)_q=(\\mu q, \\mu q - \\mu )_q$ .", "The fact that a generalized Hadamard matrix $\\text{GH}(n,q)$ gives rise to an $(n,d)_q$ -code of size $qn$ , was proved in $\\cite {plotkin}$ and for some parameters it can also be deduced from an earlier paper $\\cite {zinoviev2}$ .", "Using a result by Bassalygo, Dodunekov, Zinoviev and Helleseth $\\cite {granking}$ about the structure of $(n,d)_q$ -codes of size $qn$ ,Note that $A_q(n,d)\\le qn$ , since by Plotkin's bound $(\\ref {elementarybounds})$ , $A_q(n-1,d) \\le n$ , hence $A_q(n,d) \\le qn =\\mu q^2$ by $(\\ref {elementarybounds2})$ .", "we prove that such codes are in 1-1-relation with symmetric $(n/q,q)$ -nets.", "Theorem 5.1 Let $\\mu , q \\in \\mathbb {N}$ .", "There is a 1-1-relation between symmetric $(\\mu ,q)$ -nets (up to isomorphism) and $(n,d)_q=(\\mu q,\\mu q -\\mu )_q $ -codes $C$ of size $\\mu q^2$ (up to equivalence).", "Given an $(n,d)_q= (\\mu q,\\mu q -\\mu )_q$ -code $C$ of size $\\mu q^2$ , we construct a $(0,1)$ -matrix $M$ of order $\\mu q^2 \\times \\mu q^2$ with the following properties: $M$ is a $\\mu q^2 \\times \\mu q^2$ matrix that consists of $q \\times q$ blocks $\\sigma _{i,j}$ (so $M$ is a $\\mu q \\times \\mu q$ matrix of blocks $\\sigma _{i,j}$ ), where each $\\sigma _{i,j}$ is a permutation matrix.", "$MM^T= M^TM= A$ , where $A$ is a $\\mu q^2 \\times \\mu q^2$ matrix that consists of $q \\times q$ blocks $A_{i,j}$ (so $A$ is an $\\mu q \\times \\mu q$ matrix of blocks $A_{i,j}$ ), with $ A_{i,j} ={\\left\\lbrace \\begin{array}{ll} \\mu q \\cdot I_q &\\mbox{if } i =j, \\\\\\mu \\cdot J_q & \\mbox{if } i \\ne j.", "\\end{array}\\right.", "}$ Here $J_q$ denotes the $q \\times q$ all-ones matrix.", "By Proposition 4 of $\\cite {granking}$ , since $d=n(q-1)/q$ and $\|C\|=qn$ , $C$ can be partitioned as $ C = V_1 \\cup V_2 \\cup \\ldots \\cup V_{n},$ where the union is disjoint, $\|V_i\|=q$ for all $i=1,\\ldots ,n$ , and where $d_H(u,v)=n$ if $u,v \\in C$ are together in one of the $V_i$ , and $d_H(u,v)=d$ if $u\\in V_i$ and $v \\in V_j$ with $i\\ne j$ .", "Now we write each word $w \\in [q]^{n}$ as a $(0,1)$ -row vector of size $qn = \\mu q^2$ by putting a 1 on positions $(i,w_i) \\in [n] \\times [q]$ (for $i=1,\\ldots ,n$ ) and 0's elsewhere.", "The $q$ words in any of the $V_i$ then form a $q \\times qn$ matrix consisting of $n$ permutation matrices $\\sigma _{i,j}$ of size $q \\times q$ .", "By placing the matrices obtained in this way from all $n$ tuples $V_1,\\ldots ,V_{n}$ underneath each other, we obtain a $qn \\times qn$ matrix $M$ consisting of $n^2$ permutation matrices of order $q\\times q$ , so (REF ) is satisfied.", "Property (REF ) also holds, since for any $u,v \\in C$ written as row vectors of size $qn$ , with the $V_i$ as in $(\\ref {partition})$ , it holds that $\\sum _{k \\in [n] \\times [q]}u_kv_k= g(u,v)={\\left\\lbrace \\begin{array}{ll}n = \\mu q &\\mbox{if } u = v, \\\\0 &\\mbox{if } u \\ne v \\text{ and } u,v \\in V_i,\\\\n-d=\\mu &\\mbox{if } u \\ne v \\text{ and } u \\in V_i, v \\in V_j \\text{ with } i \\ne j.\\\\\\end{array}\\right.", "}$ So $MM^T=A$ .", "Moreover, if $j_1:=(j_1^{\\prime },a_1) \\in [n] \\times [q]$ and $j_2:=(j_2^{\\prime },a_2) \\in [n] \\times [q]$ , then $\\sum _{k \\in [qn]}M_{k,j_1} M_{k,j_2}={\\left\\lbrace \\begin{array}{ll}n = \\mu q &\\mbox{if } j_1^{\\prime }=j_2^{\\prime } \\text{ and } a_1=a_2, \\\\0 &\\mbox{if } j_1^{\\prime } = j_2^{\\prime } \\text{ and } a_1\\ne a_2,\\\\n/q = \\mu &\\mbox{if } j_1^{\\prime } \\ne j_2^{\\prime },\\\\\\end{array}\\right.", "}$ where the last statement follows by considering the words in $C$ that have $a_1$ at the $j_1^{\\prime }$ -th position.", "(The remaining columns form an $n$ -block for the $j_1^{\\prime }$ -th column.", "In this $n$ -block, each symbol occurs exactly $n/q$ times at each position, since the leftmost term equals the rightmost term in $(\\ref {detruc})$ for $(n-1,d)_q$ -codes of size $n$ .)", "We see that also $M^TM=A$ .", "Hence, $M$ is the incidence matrix of a symmetric $(\\mu ,q)$ -net (see $\\cite {beth}$ , Proposition I.7.6 for the net and its dual).", "Note that one can do the reverse construction as well: given a symmetric $(\\mu ,q)$ -net, the incidence matrix of $M$ can be written (after possible row and column permutations) as a matrix of permutation matrices such that $MM^T=M^TM=A$ , with $A$ as in $(\\ref {M})$ .", "From $M$ we obtain a code $C$ of size $\\mu q^2$ of the required minimum distance by mapping the rows $(i,w_i) \\in [\\mu q] \\times [q]$ to $w \\in [q]^{\\mu q}$ .", "Observe that equivalent codes yield isomorphic incidence matrices $M$ and vice versa.", "Figure: An (n,d) q =(3,2) 3 (n,d)_q=(3,2)_3-code C={w 1 ,...,w 9 }C=\\lbrace w_1,\\ldots ,w_9\\rbrace of size 9 (left table) gives rise to an incidence matrix of a symmetric (1,3)(1,3)-net (right table) and vice versa." ], [ "New upper bound on $A_4(9,6)$ .", "In this section we use the 1-1-correspondence between symmetric $(\\mu ,q)$ -nets and $(n,d)_q=(\\mu q, \\mu q - \\mu )_q$ -codes of size $\\mu q^2$ in combination with a known result about symmetric $(2,4)$ -nets $\\cite {42net}$ to derive that $A_4(9,6) \\le 120$ .", "As $A_4(8,6)=32$ , any $(9,6)_4$ -code of size more than 120 must contain at least one 31- or 32-block, and therefore it contains a subcode of size 120 containing at least one 31- or 32-block.", "We will show (using a small computer check) that this is impossible because a $(9,6)_4$ -code of size 120 does not contain any 31- or 32-blocks.", "Therefore $A_4(9,6) \\le 120$ .", "In order to do prove this, we need information about $(8,6)_4$ -codes of size 31.", "Proposition 6.1 Let $q,n,d \\in \\mathbb {N}$ satisfy $qd=(q-1)n$ .", "Any $(n,d)_q$ -code $C$ of size $qn-1$ can be extended to an $(n,d)_q$ -code of size $qn$ .", "Let $C$ be an $(n,d)_q$ -code of size $qn-1$ .", "By Plotkin's bound, $A_q(n-1,d) \\le n$ , so each symbol occurs at most $n$ times in each column of $C$ , hence there exists for each $j \\in [n]$ a unique $\\beta _j \\in [q]$ with $c_{\\beta _j,j} =n-1$ and $c_{\\alpha ,j}=n$ for all $\\alpha \\in [q] \\setminus \\lbrace \\beta _j \\rbrace $ .", "We can define a $qn$ -th codeword $u$ by putting $u_j:= \\beta _j$ for all $j=1,\\ldots ,n$ .", "We claim that $C \\cup \\lbrace u \\rbrace $ is an $(n,d)_q$ -code of size $qn$ .", "To establish the claim we must prove that $d_H(u,w) \\ge d$ for all $w \\in C$ .", "Let $w \\in C$ with $d_H(u,w) < n$ .", "We can renumber the symbols in each column of $C$ such that $w=\\mathbf {1}$ .", "Then $w$ is contained in an $(n-1)$ -block $B$ for some column in $C$ (otherwise $d_H(u,w)=n$ ).", "The number of 1's in $B$ is $n+(n-2)n/q$ (since any $(q,n-1,d)$ -code of size $n-1$ is equidistant, as $L-R=0$ in $(\\ref {boundnd})$ for $(n-1,d)_q$ -codes of size $n-1$ ) and the number of 1's in $C \\setminus B$ is $(q-1)(n-1)n/q$ (since in any $(n-1,d)_q$ -code of size $n$ , each symbol appears exactly $n/q$ times in each column, as the leftmost term equals the rightmost term in $(\\ref {detruc})$ for $(n-1,d)_q$ -codes of size $n$ ).", "Adding these two numbers we see that the number of 1's in $C$ is $n^2-n/q$ .", "Since $C \\cup \\lbrace u\\rbrace $ contains each symbol $n^2$ times by construction, $u$ contains symbol 1 exactly $n/q$ times, hence $d_H(u,w)=n-n/q=d$ , which gives the desired result.", "Proposition 6.2 $A_4(9,6) \\le 120$ .", "The $(n,d)_q=(8,6)_4$ -code of size 32 is unique up to equivalence, since the symmetric $(2,4)$ -net is unique up to equivalence (see Al-Kenani $\\cite {42net}$ ).", "By checking all $(8,6)_4$ -codes $D$ of size 31 (of which there are at most 32 up to equivalence since each $(8,6)_4$ -code of size 31 arises by removing one word from a $(8,6)_4$ -code of size 32 by Proposition REF ) we find that $\|\\lbrace u \\in [4]^{8} \\,\\, \| \\,\\,d_H(w,u) \\ge 5 \\,\\,\\,\\, \\forall \\, w \\in D \\rbrace \| \\le 25.$ This implies that an $(n,d)_q=(9,6)_4$ -code $C$ of size 120 cannot contain a 31- or 32-block.", "Therefore $A_4(9,6) \\le 120$ .", "Corollary 6.3 $A_4(10,6) \\le 480$ .", "By Proposition $\\ref {120}$ and $(\\ref {elementarybounds2})$ ." ], [ "Acknowledgements", "I am most grateful to Lex Schrijver for his supervision, for his help regarding both the content and the presentation of the paper and for all the conversations we had about the code bounds (in person and by e-mail).", "Thank you, Lex!", "Also I am grateful to Bart Litjens, Guus Regts and Jacob Turner for their comments.", "Furtermore I want to thank the editor and the anonymous referees for their very helpful comments concerning the presentation of the material.", "english" ] ]	1606.05144
[ [ "Boltzmann approach to high-order transport: the non-linear and non-local\n responses" ], [ "Abstract The phenomenological textbook equations for the charge and heat transport are extensively used in a number of fields ranging from semiconductor devices to thermoelectricity.", "We provide a rigorous derivation of transport equations by solving the Boltzmann equation in the relaxation time approximation and show that the currents can be rigorously represented by an expansion in terms of the 'driving forces'.", "Besides the linear and non-linear response to the electric field, the gradient of the chemical potential and temperature, there are also terms that give the response to the higher-order derivatives of the potentials.", "These new, non-local responses, which have not been discussed before, might play an important role for some materials and/or in certain conditions, like extreme miniaturization.", "Our solution provides the general solution of the Boltzmann equation in the relaxation time approximation (or equivalently the particular solution for the specific boundary conditions).", "It differs from the Hilbert expansion which provides only one of infinitely many solutions which may or may not satisfy the required boundary conditions." ], [ "Introduction", "The phenomenological transport equations for the charge and heat currents are at the core of the description of electric and electronic devices of any type.", "These equations relate the local charge and heat current densities, $\\mathbf {J}(\\mathbf {r})$ and $\\mathbf {J}_{\\mathcal {E}}(\\mathbf {r})$ , to the local thermodynamic forces given by the gradients of the electrical potential $\\phi $ , chemical potential $\\mu $ , and temperature $T$ .", "Often, they are written as $\\mathbf {J}(\\mathbf {r})&= \\sigma (\\mathbf {E}+ \\frac{\\nabla _{} \\mu }{e})-\\sigma \\alpha \\nabla _{} T+ \\sigma ^{\\scriptscriptstyle [\\mathbf {E}^2]} (\\mathbf {E}+ \\frac{\\nabla _{} \\mu }{e})^2 \\\\\\mathbf {J}_{\\mathcal {E}}(\\mathbf {r})& = \\sigma \\alpha T (\\mathbf {E}+ \\frac{\\nabla _{} \\mu }{e}) - (\\kappa +\\sigma \\alpha ^2T) \\nabla _{} T ~,$ where $\\mathbf {E}=-\\nabla \\phi $ is the local electric field, and the conductivity $\\sigma $ , the Seebeck coefficient $\\alpha $ , and the thermal conductivity $\\kappa $ are the position-dependent transport coefficients describing the linear part of the response.", "In some cases, the experiments indicate the presence of the non-linear response, but, of all the possible terms, we wrote down only the one that is proportional to the square of the electro-chemical force, with $\\sigma ^{\\scriptscriptstyle [\\mathbf {E}^2]}$ as the transport coefficient.", "Within the linear response theory, Onsager[1], Kubo,[2] and Luttinger[3] explained how to relate the coefficients of the driving fields to microscopic quantities but the microscopic content of $\\sigma ^{\\scriptscriptstyle [\\mathbf {E}^2]} $ is less clear.", "Our aim, within the semi-classical Boltzmann approach, is to relate $\\sigma ^{\\scriptscriptstyle [\\mathbf {E}^2]} $ and similar terms in the expansion of the current densities to the energy dispersion and the scattering matrix of the electrons.", "However, as shown in detail below, the expansion should contain all the powers of the thermodynamic forces and their derivatives, i.e., a consistent theory should include all the driving forces up to a given order.", "Beyond the linear order, the current response is usually very small but, under specific conditions, the non-linear driving forces can play a role; for instance, when the forces are large or when they exhibit large variations across the sample.", "In this article, we show how to obtain a systematic expansion of the current densities in terms of the driving forces.", "We find that Eqs.", "(REF ) and () are incomplete and that non-local terms proportional to the higher order derivatives of the potentials, such as $\\nabla \\mathbf {E}$ , $\\partial ^2 \\mu /\\partial \\mathbf {r}^2$ and $\\partial ^2 T/\\partial \\mathbf {r}^2$ , have to be included as well.", "This has an obvious fundamental relevance.", "While some non local effects have been studied so far within different approximations and approaches (as for instance the anomalous skin effect [4], [5]), there is no systematic and broad classification of non-local effects, consistent with the definition of other effects like conductivity, Seebeck, Hall, and so on.", "This consequently leads to neglecting effects which can be important in the description of devices, as for instance the charge current proportional to $\\nabla \\mathbf {E}$ in charged bulk regions.", "However, our results are also of a more direct and specific interest for the treatment of modern semiconductor devices.", "The miniaturization requires ever smaller components, with smaller and smaller active regions.", "[6], [7], [8], [9] In these regions, the thermodynamic potentials charge enormously over small distances and their higher-order derivatives gain in importance.", "Thus, even if the prefactors of the new terms are very small, their overall contribution to the total current could be significant and they can have an impact on the performance of the device.", "Similarly, modern thermoelectric devices,[10], [11], [12], [13], [14] designed so as to optimize the efficiency under given operating conditions,[15], [16], [17], [18] are often heterogeneous and have a non-linear distributions of temperature and chemical potential.", "In that case, additional terms are needed to describe the charge and energy current densities.", "An advance along these line requires a quantum mechanical engineering that is hardly possible without an insight from theory[19], [20], [21] The theoretical analysis is usually performed in several steps.", "First, the electronic structure is calculated by density functional theory and the transport coefficients are obtained by the linearized Boltzmann equation,[22] because the Kubo approach is most often too difficult to use for real materials.", "Even the Boltzmann approach requires a number of simplifications, as described in the classic textbooks.", "[23], [24] Once the transport coefficients are known, the currents given by Eqs.", "(REF ) and () can be substituted in the continuity equations for the charge and energy conservation.", "Given the appropriate boundary conditions, these equations provide, together with the Poisson equation for the electric field, the temperature and electrical and chemical potential at every macroscopic point of the sample.", "Thus, one can find the operating conditions, engineer the right composition of the material, and optimize the overall efficiency of the device.", "Using the semi-classical Boltzmann theory, and the relaxation time approximation (RTA), the present paper formalizes the procedure outlined above and shows how to obtain, in a systematic and rigorous way, the transport equations and transport coefficients of inhomogeneous samples.", "Expanding the solution of the Boltzmann equation (BE) in terms of the driving forces, we derive the terms beyond the linear response, reproduce the terms in Eqs.", "(REF ) and (), and show that additional, new terms arise.", "We also show that the effect of the microscopic boundary conditions can be neglected, when the size of the system is large with respect to the diffusion length $l=\\tau v_{\\mathbf {k}}$ .", "Here, $\\tau $ is the scattering relaxation time, $ v_{\\mathbf {k}}$ is the group velocity of the electron wave packet, and we are referring to the microscopic boundary conditions for the BE, which define the momentum space distribution of incoming electrons at the boundary.", "From the mathematical point of view, the proposed expansion has a major advantage over other expansions that are commonly used for the BE, like the Hilbert [25] and Chapman-Enskog expansion[26], [27], [28], [29].", "These expansions disregard the boundary conditions i.e., they expand just one of infinitely many possible solutions.", "In general, the solution generated by these expansions will not satisfy the boundary conditions of the real problem.", "The expansion proposed in this paper yields, within the RTA, the general solution of the BE.", "This is then used to find the particular solution satisfying the required boundary conditions.", "Thus, we can deal in a consistent way not only with the response to the higher-order driving forces but we can also describe the transport properties in the vicinity of the boundary (for instance, the anomalous skin effect [4], [5]).", "For simplicity, the present article is restricted to the static case and it addresses in more detail only the former effects.", "The paper is organized as follows.", "Section introduces the Boltzmann and Poisson equations, and the relaxation time approximation, which leads to two further equations required by the energy and particle conservation.", "For a given boundary condition, these four coupled equations determine the electron distribution function, the temperature, the chemical potential, and the electrical field everywhere in the sample.", "Supplementarily, we show in Appendix the equivalence of the equations for the charge and energy conservation to the continuity equations for the charge and heat current densities.", "In Section , rather than solving all these equations simultaneously, we focus on the Boltzmann equation treating the temperature, chemical and electrical potential as arbitrary known functions.", "The central part of the article is the expansion of these arbitrary functions in Taylor series, which generates an expansion of the non-equilibrium electron distribution function in terms of the driving forces.", "The coefficients in that expansion satisfy coupled differential equations, as shown in Section REF .", "The problem of setting up the proper boundary conditions and solving these equations is discussed in Section REF .", "The major advantage of our expansion over the Hilbert expansion is explained in Appendix .", "In Section we show that, for macroscopic samples, a further approximation can be done which leads to major simplifications (this approximation is equivalent to neglecting the effects that are relevant only close to the surface).", "In Section we derive the generalized transport equations for the heat and charge currents in the bulk, and find additional terms that have not been discussed hitherto.", "Sec.", "provides one example which shows the influence of the new thermodynamic forces on the behavior of materials: we analyze the depletion region in a metal-semiconductor junction.", "Section briefly discusses and summarizes our results." ], [ "Transport equations", "Bloch's quantum extension of the Boltzmann's theory derives the transport properties of a degenerate electron gas from the distribution function $g(t,\\mathbf {r},\\mathbf {k})$ , where $\\mathbf {r}$ and $\\mathbf {k}$ are the coordinates of an electron at time $t$ in the real and momentum space, respectively.", "For electrons moving in the presence of a scalar and vector potentials $V(\\mathbf {r},t)$ and $\\mathbf {A}(\\mathbf {r},t)$ , and confined to a single band, the distribution function satisfies the Boltzmann equation [22], [23] (we neglect, for simplicity, the anomalous contribution to the velocity, which can easily be included in the BE [30], [31]) $ &\\frac{\\partial g}{\\partial t}+\\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} g ~ + \\\\& \\frac{e}{\\hbar } \\left[ \\nabla _{\\mathbf {r}} V + \\frac{\\partial \\mathbf {A}}{\\partial t} - \\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\times \\left( \\nabla _{\\mathbf {r}} \\times \\mathbf {A}\\right)\\right] \\cdot \\nabla _{\\mathbf {k}} g =\\left(\\frac{dg}{dt}\\right)_{\\!\\!col} \\nonumber ~.$ Here, $\\mathcal {E}=\\mathcal {E}(\\mathbf {k})$ is the energy dispersion provided by the band structure calculations, $e<0$ is the electron charge, $\\hbar $ is the Planck constant, and $\\left(dg/dt\\right)_{\\!~\\!col}$ is the collision integral which describes the change of the distribution function due to the electron-electron (e-e) and, possibly, the electron-phonon (e-ph) scatterings.", "The extension to a multi-band system is straightforward and it amounts to a summation over a band index[23], [24].", "The charge and current densities defined by $g(t,\\mathbf {r},\\mathbf {k})$ have to be compatible with the electrodynamic potentials $V(\\mathbf {r},t)$ and $\\mathbf {A}(\\mathbf {r},t)$ , as required by the Maxwell equations.", "The exact solution of the Boltzmann-Maxwell system of equations satisfies all the conservation laws compatible with the invariance of the Hamiltonian with respect to the symmetry operations[32] and it determines completely the transport properties of the system.", "Unfortunately, except in the most simple cases, the presence of the collision integral makes the exact solution inaccessible.", "In many applications, for example, when engineering an optimal material for a thermoelectric device, one tries to infer the transport properties from the available band structure data, neglecting the details of the relaxation mechanisms.", "In that case, the standard approach is to assume that the scattering drives the system towards equilibrium and to replace the collision integral by a simple expression, $\\left(\\frac{dg}{dt}\\right)_{\\!\\!col} \\approx -\\frac{g-g_0}{\\tau } ,$ where $g_0 \\left( T, \\mu , \\mathcal {E}\\right)$ is the unperturbed local distribution function defined by local temperature $T(\\mathbf {r})$ and local chemical potential $\\mu (\\mathbf {r})$ .", "The assumption that the main effect of the scattering processes is the restoration of local thermodynamic equilibrium on the timescale given by $\\tau $ defines the RTA of the Boltzmann equation.", "[24] Since we are interested in the transport properties of an electron fluid, we choose, $ g_0 (\\mathbf {r},\\mathbf {k}) = f_{FD} \\left( T (\\mathbf {r}), \\mu (\\mathbf {r}), \\mathcal {E}(\\mathbf {k}) \\right)=\\frac{1}{1+e^{\\tfrac{\\mathcal {E}(\\mathbf {k})-\\mu (\\mathbf {r})}{k_B T(\\mathbf {r})}}} ~,$ where, $k_B$ is the Boltzmann constant and $f_{FD} $ the Fermi-Dirac distribution.", "In what follows, we consider the transport properties in a stationary state, such that $\\partial g/\\partial t=0$ , and solve rigorously the static Boltzmann equation using the RTA.", "We take into account the electric field $\\mathbf {E}(\\mathbf {r})= - \\nabla _{} V(\\mathbf {r})$ but neglect, for simplicity, the magnetic field.", "Thus, we replace the integro-differential equation Eq.", "(REF ) by a generalized drift-reaction (convection-reaction) equation $ \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g - \\frac{e}{\\hbar } \\mathbf {E}\\cdot \\nabla _{\\mathbf {k}} g=-\\frac{g-f_{FD} \\left( T\\left(\\mathbf {r}\\right), \\mu \\left(\\mathbf {r}\\right), \\mathcal {E}\\left( \\mathbf {k}\\right) \\right)}{\\tau } ~.$ The relaxation time $\\tau $ is treated either as a free parameter, which provides the best fit to the experimental data, or it is calculated in the perturbation theory [23].", "Unlike the exact solution of Eq.", "(REF ), the solution of Eq.", "(REF ) does not automatically satisfy the fundamental conservation laws, like the particle number and the energy conservation.", "To make the RTA physically acceptable we enforce the local particle and energy conservation by constraining the functions $T=T(\\mathbf {r})$ and $\\mu =\\mu (\\mathbf {r})$ .", "The conservation of the local particle density $n\\left(\\mathbf {r}\\right) =\\int g \\left(\\mathbf {r},\\mathbf {k}\\right) d^3k$ follows from the requirement $ \\int g \\left(\\mathbf {r},\\mathbf {k}\\right) d^3k = \\int f_{FD} \\left( T\\left(\\mathbf {r}\\right), \\mu \\left(\\mathbf {r}\\right), \\mathcal {E}\\left(\\mathbf {k}\\right) \\right) d^3k$ while the conservation of the total energy density of interacting electrons is enforced by the equation $\\begin{split} \\int \\!\\mathcal {E}\\left( \\mathbf {k}\\right)\\, g \\left(\\mathbf {r},\\mathbf {k}\\right) d^3k =\\!\\!\\int \\!", "\\mathcal {E}\\left( \\mathbf {k}\\right)\\, f_{FD} \\left( T\\left(\\mathbf {r}\\right)\\!, \\mu \\left(\\mathbf {r}\\right)\\!, \\mathcal {E}\\left( \\mathbf {k}\\right) \\right) d^3k.\\end{split}$ If the electrons scatter on some additional degrees of freedom, like phonons, the scattering process changes their energy by $ \\Delta \\epsilon _{e-ph} $ which has to be added to the right-hand-side of Eq.", "(REF ).", "The consistency of the charge density and the electrical field is enforced by the Poisson equation, $\\nabla _{} \\mathbf {E}(\\mathbf {r},t) = \\frac{e\\, n(\\mathbf {r}) + \\rho _{\\mbox{ion}}(\\mathbf {r})}{\\epsilon _0} ~,$ where $\\rho _{\\mbox{ion}}(\\mathbf {r})$ is the background charge that ensures the overall charge neutrality.", "The self-consistent solution of Eqs.", "(REF ) - (REF ) provides $ g \\left(\\mathbf {r},\\mathbf {k}\\right) $ , $\\mathbf {E}(\\mathbf {r})$ , $T(\\mathbf {r})$ and $\\mu (\\mathbf {r})$ at every point in the sample.", "As shown in Appendix , the conservation of charge and energy imply the continuity equations for the charge and energy current densities.", "In a stationary state (and in the absence of electron-phonon scatterings), the current densities satisfy $ & \\nabla _{}\\cdot \\mathbf {J}(\\mathbf {r})=0 \\\\& \\nabla _{}\\cdot \\mathbf {J}_{\\mathcal {E}}(\\mathbf {r})=W~,$ where the charge and energy current densities are $ \\mathbf {J}\\left(\\mathbf {r}\\right) &=& \\frac{e}{\\hbar } \\int \\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\; g\\left(\\mathbf {r},\\mathbf {k}\\right) d^3k ~, \\\\\\mathbf {J}_\\mathcal {E}\\left(\\mathbf {r}\\right) & =& \\ \\frac{e}{\\hbar } \\int {\\nabla _{\\mathbf {k}} \\mathcal {E}} \\; \\mathcal {E}\\; g\\left(\\mathbf {r},\\mathbf {k}\\right) d^3k ~,$ and $W$ is the work done by the applied electric field per unit time $ W\\left(\\mathbf {r}\\right)&=& \\frac{e}{\\hbar } \\int \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\mathbf {E}\\left(\\mathbf {r}\\right) \\; g\\left(\\mathbf {r},\\mathbf {k}\\right) \\,d^3k= \\mathbf {J}\\cdot \\mathbf {E}.", "$ Thus, finding the electron distribution function in the RTA implies solving the Boltzmann equation (REF ) for $g(\\mathbf {r},\\mathbf {k})$ , together with the continuity equations (REF ) and (), and the Poisson equation (REF ).", "Equivalently one can solve the Boltzmann equation (REF ) for $g(\\mathbf {r},\\mathbf {k})$ and Eqs.", "(REF ) – () for five unknown quantities $\\mathbf {J}\\left(\\mathbf {r}\\right), \\mathbf {J}_\\mathcal {E}\\left(\\mathbf {r}\\right), T \\left(\\mathbf {r}\\right), \\mu \\left(\\mathbf {r}\\right)$ and $\\mathbf {E}(\\mathbf {r})$ .", "In general, the above equations have infinitely many solutions and the physically relevant one is defined by the specific boundary conditions which provide $g (\\mathbf {r}_B,\\mathbf {k})=g_B(\\mathbf {r}_B,\\mathbf {k})$ at every point $\\mathbf {r}_B$ of the boundary.", "Note the difference between these detailed, microscopic boundary conditions and the one which specifies just the macroscopic quantities, like temperature and electrical and chemical potentials, at the interfaces.", "The microscopic boundary conditions for the Boltzmann equation specify the momentum distribution of the electrons coming from the neighboring material, and take also into account the reflection and scattering of the incoming electrons at the interface.", "In addition to the temperature of the injection, they should provide, for example, a detailed information on the band structure of the neighboring material, the $\\mathbf {k}$ -dependent injection probability or reflectivity.", "The construction of boundary conditions that determines a specific physical situation at the interface is a non-trivial problem which is not addressed further in this work.", "In the following, we simply assume the boundary conditions to be known." ], [ "Expansion of the Boltzmann distribution function in terms of the generalized forces", "Although the Boltzmann equation simplifies considerably within the RTA, solving Eq.", "(REF ) for $g(\\mathbf {r},\\mathbf {k})$ , together with Eqs.", "(REF ) – (REF ) for $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ , and $\\mathbf {E}(\\mathbf {r})$ , is still a formidable task.", "To solve it, we integrate Eq.", "(REF ) for completely arbitrary functions $\\mu (\\cdot )$ , $T(\\cdot )$ , and $E(\\cdot )$ , using an expansion of $g(\\mathbf {r},\\mathbf {k})$ in terms of the forces that arise out of equilibrium and to which the system responds by setting up the currents.", "Such approximations, based on physical arguments and irreversible thermodynamics, are often used but, here, we present a systematic expansion which clarifies the range of validity of the textbook solutions and identifies the new driving forces.", "For a given microscopic boundary condition ($BC$ ), the solution of Eq.", "(REF ) has a unique value at every point $\\lbrace \\mathbf {r},\\mathbf {k}\\rbrace $ of the phase space, so that $g(\\mathbf {r},\\mathbf {k})$ is a functional $g_F$ defined on functions $\\mu (\\cdot )$ , $T(\\cdot )$ and $\\mathbf {E}(\\cdot )$ , and the BC themselves.", "That is, $ g(\\mathbf {r},\\mathbf {k})=g_F \\left[\\mathbf {r}, \\mathbf {k}, T(\\cdot ) ,\\mu (\\cdot ),\\mathbf {E}(\\cdot ),BC\\right] ~.$ Assuming $\\mu (\\cdot )$ , $T(\\cdot )$ and $\\mathbf {E}(\\cdot )$ are analytic functions, we expand them into Taylor series around point $\\mathbf {r}$ and treat $g(\\mathbf {r},\\mathbf {k})$ , without any loss of information[33], not as a functional but as a function of infinitely many variables, $ g(\\mathbf {r},\\mathbf {k})=\\tilde{g} \\big (&\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}),\\mu (\\mathbf {r}), \\mathbf {E}(\\mathbf {r}) , \\nabla T(\\mathbf {r}),\\nabla \\mu (\\mathbf {r}),\\\\& \\nabla \\mathbf {E}(\\mathbf {r}), \\frac{\\partial ^2 T(\\mathbf {r})}{\\partial \\mathbf {r}^2},\\frac{\\partial ^2 \\mu (\\mathbf {r})}{\\partial \\mathbf {r}^2},\\frac{\\partial ^2 \\mathbf {E}(\\mathbf {r})}{\\partial \\mathbf {r}^2}, ..., BC\\big ) .\\nonumber $ (Operator $\\nabla $ denotes $\\nabla _{\\mathbf {r}}$ , whenever the function operated on depends solely on the position $\\mathbf {r}$ and no ambiguity can arise.)", "Obviously, analytic functions $T(\\cdot ) ,\\mu (\\cdot ),\\mathbf {E}(\\cdot )$ are completely defined by their values and the values of all their derivatives at any point of the sample.", "For example, $\\mu (\\cdot )$ is determined everywhere in its region of definition, if we provide, at point $ \\mathbf {r}$ , the values $\\mu (\\mathbf {r})$ , $\\nabla _{} \\mu (\\mathbf {r})$ , $\\partial ^2 \\mu (\\mathbf {r})/\\partial \\mathbf {r}^2$ , $\\partial ^3 \\mu (\\mathbf {r})/\\partial \\mathbf {r}^3$ , etc.", "If one is interested in the charge transport close to the interface, say, to model the Kapitza resistance [34], the interface defines a discontinuity and the Taylor extension cannot be used.", "To circumvent that problem, one can split the system into two halves, one to the left and one to the right of the boundary, and use separate Taylor expansions on each sides of the interface.", "However, solving for such boundary conditions, including the Kapitza resistance, becomes cumbersome.", "To proceed, we introduce the vector $\\vec{\\xi }=\\lbrace \\xi _1,\\xi _2,\\xi _3,...\\rbrace $ , where $\\xi _1=\\mathbf {r}$ , $\\xi _2=\\mathbf {k}$ , $\\xi _3=T$ , $\\xi _4=\\mu $ , $\\xi _5= \\mathbf {E}$ , $\\xi _6=\\nabla _{} T$ , etc.", "The components ${\\xi _i}$ , for $i\\ge 5$ , describe the driving forces to which the system responds.", "This notation indicates that even though $\\tilde{g}(\\vec{\\xi }\\,)$ is defined on an infinite dimensional vector space, only the values assumed by $\\tilde{g}(\\vec{\\xi }\\,)$ on a small subspace of the whole definition space are physically relevant.", "In particular, we are interested in $\\tilde{g}(\\vec{\\xi }\\,)$ on the hypersurface defined by $\\xi _1=\\mathbf {r}$ , $\\xi _2=\\mathbf {k}$ , $\\xi _3=T(\\mathbf {r})$ , $\\xi _4=\\mu (\\mathbf {r})$ , $\\xi _5=\\mathbf {E}(\\mathbf {r})$ , $\\xi _6=\\nabla _{} T(\\mathbf {r})$ , $\\xi _7=\\nabla _{} \\mu (\\mathbf {r})$ etc.", "Next, we expand $\\tilde{g}(\\vec{\\xi }\\,)$ in a Taylor series around the point $\\vec{\\xi }_0=\\lbrace \\xi _1,\\xi _2,\\xi _3,\\xi _4,0,0,0,\\ldots \\rbrace $ , i.e., we expand $\\tilde{g}(\\vec{\\xi }\\,)$ with respect to the variables $\\xi _i$ around $\\xi _i=0$ , for all $i\\ge 5$ .", "Thus, we write $ &\\tilde{g}(\\vec{\\xi }\\,)= g^{[0]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4] \\nonumber \\\\&+\\;\\;\\delta g^{[ \\xi _5]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _5 \\nonumber \\\\&+\\;\\; \\delta g^{[\\xi _6 ]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _6 \\nonumber \\\\&+\\;\\; \\delta g^{[\\xi _7 ]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _7 \\\\&+\\;\\; \\delta g^{[ \\xi _5^2]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _5^2 \\nonumber \\\\&+\\;\\; \\delta g^{[\\xi _8]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _8 \\nonumber \\\\&+\\;\\; \\delta g^{[\\xi _8^2 ]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4]\\; \\xi _8^2 \\nonumber \\\\&+\\;\\; \\delta g^{[\\xi _5 \\xi _{6}]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4] \\;\\xi _5 \\xi _{6}+ ... \\nonumber $ where the coefficients $ \\delta g^{[\\alpha ,\\beta ,\\gamma ,....]} [\\xi _1,\\xi _2, \\xi _3, \\xi _4] $ depend on the first four variables and the boundary conditions.", "(The explicit dependence on the boundary conditions has been omitted, for brevity.)", "In terms of the physically more transparent symbols, we have $ g(\\mathbf {r},\\mathbf {k}) &= g^{[0]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\nonumber \\\\&\\;\\;\\;\\; +\\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,\\mathbf {E}(\\mathbf {r})+ ... \\\\&\\;\\;\\;\\; +\\delta g^{[ \\nabla _{} T]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,\\nabla _{} T(\\mathbf {r}) \\nonumber \\\\&\\;\\;\\;\\;+ \\delta g^{[ \\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,\\nabla _{} \\mu (\\mathbf {r}) \\nonumber \\\\&\\;\\;\\;\\;+ \\delta g^{[ \\mathbf {E}^2]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})]\\, (\\mathbf {E}(\\mathbf {r}))^2+ ... \\nonumber \\\\&\\;\\;\\;\\;+ \\delta g^{[ \\partial ^2 T/\\partial \\mathbf {r}^2 ]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\, \\partial ^2 T(\\mathbf {r})/\\partial \\mathbf {r}^2+ ... \\nonumber \\\\&\\;\\;\\;\\;+\\delta g^{[ (\\partial ^2 T/\\partial \\mathbf {r}^2)^2 ]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,(\\partial ^2 T(\\mathbf {r})/\\partial \\mathbf {r}^2)^2 \\nonumber \\\\&\\;\\;\\;\\;+ \\delta g^{[ \\mathbf {E}, \\nabla _{} T ]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\, \\mathbf {E}(\\mathbf {r}) \\nabla _{} T(\\mathbf {r}) + ... ,\\nonumber $ where the (still unknown) coefficients $\\delta g^{[E]}$ , $\\delta g^{[ \\nabla _{} T]}$ , $\\delta g^{[ \\nabla _{} \\mu ]}$ , etc.", "describe the change in the distribution function due to the applied forces $\\mathbf {E}$ , $\\nabla _{} T$ , $\\nabla _{} \\mu $ , etc.", "These coefficients can be computed by substituting $\\tilde{g}(\\vec{\\xi }\\,)$ , given by Eq.", "(REF ), into the Boltzmann equation (REF ) and collecting the terms to order $\\mathbf {E}(\\mathbf {r})$ , $\\nabla _{} T(\\mathbf {r})$ , $\\nabla _{} \\mu (\\mathbf {r})$ , $ \\partial ^2 T(\\mathbf {r})/\\partial \\mathbf {r}^2$ , $\\mathbf {E}(\\mathbf {r})^2$ , $\\mathbf {E}(\\mathbf {r}) \\nabla _{} T(\\mathbf {r})$ , $\\mathbf {E}(\\mathbf {r}) \\nabla _{} \\mathbf {E}(\\mathbf {r})$ , etc.", "The first term in Eq.", "(REF ) yields (we only show the first few terms): $ &\\nabla _{\\mathbf {r}} g(\\mathbf {r},\\mathbf {k})= \\nabla _{\\mathbf {r}} \\left[ g^{[0]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})]+ \\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\, \\mathbf {E}(\\mathbf {r}) \\right.\\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+ \\delta g^{[ \\nabla _{} T]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,\\nabla _{} T(\\mathbf {r})+ \\delta g^{[ \\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,\\nabla _{} \\mu (\\mathbf {r}) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; + \\left.\\delta g^{[ \\mathbf {E}^2]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,(\\mathbf {E}(\\mathbf {r}))^2+ \\delta g^{[ \\mathbf {E},\\nabla _{} \\mu ]} [\\mathbf {r}, \\mathbf {k}, T(\\mathbf {r}), \\mu (\\mathbf {r})] \\,(\\mathbf {E}(\\mathbf {r}))\\nabla _{} \\mu + ... \\right] \\nonumber \\\\&=\\nabla _{\\mathbf {r}} g^{[0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] + \\frac{\\partial g^{[0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ]}{\\partial T} \\nabla _{} T + \\frac{\\partial g^{[0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ]}{\\partial \\mu }\\nabla _{} \\mu \\\\&\\;\\;\\;+\\nabla _{\\mathbf {r}} \\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\,\\mathbf {E}+ \\frac{\\partial \\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial T} \\mathbf {E}\\, \\nabla _{} T + \\frac{\\partial \\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial \\mu } \\mathbf {E}\\, \\nabla _{} \\mu + \\delta g^{[ \\mathbf {E}]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\, \\nabla _{} \\mathbf {E}\\nonumber \\\\&\\;\\;\\; +\\nabla _{\\mathbf {r}} \\delta g^{[ \\nabla _{} T]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\,\\nabla _{} T + \\frac{\\partial \\delta g^{[ \\nabla _{} T]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial T} (\\nabla _{} T)^2 + \\frac{\\partial \\delta g^{[ \\nabla _{} T]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial \\mu } \\nabla _{} T\\, \\nabla _{} \\mu + \\delta g^{[\\nabla _{} T ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\, \\frac{\\partial ^2 T}{\\partial \\mathbf {r}^2} \\nonumber \\\\&\\;\\;\\; +\\nabla _{\\mathbf {r}} \\delta g^{[ \\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\,\\nabla _{} \\mu + \\frac{\\partial \\delta g^{[ \\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial \\mu } (\\nabla _{} \\mu )^2 + \\frac{\\partial \\delta g^{[ \\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial T} \\nabla _{} T\\, \\nabla _{} \\mu + \\delta g^{[\\nabla _{} \\mu ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\, \\frac{\\partial ^2 \\mu }{\\partial \\mathbf {r}^2} \\nonumber \\\\&\\;\\;\\; +\\nabla _{\\mathbf {r}} \\delta g^{[ \\mathbf {E}^2]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\,\\mathbf {E}^2 + \\frac{\\partial \\delta g^{[ \\mathbf {E}^2]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial T} \\nabla _{} T\\, \\mathbf {E}^2 + \\frac{\\partial \\delta g^{[ \\mathbf {E}^2]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] }{\\partial \\mu } \\mathbf {E}^2\\, \\nabla _{} \\mu +2 \\delta g^{[\\mathbf {E}^2 ]} [\\mathbf {r},\\mathbf {k}, T, \\mu ] \\, \\mathbf {E}\\,\\nabla _{} \\mathbf {E}\\nonumber \\\\&\\;\\;\\;+ .... \\nonumber $ where the effect of $\\nabla _{\\mathbf {r}}$ , operating on composite functions in the first equation, is computed using the usual rules for the derivatives of the composite function, while $\\nabla _{\\mathbf {r}} g^{[ 0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ]$ , etc.", ", in the second equation, denotes the derivative of $g^{[0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ]$ with respect to the first variable only.", "The terms obtained by substituting Eq.", "(REF ) in the second and the third term of the Boltzmann Eq.", "(REF ) are easily written down and are not reported separately.", "The sum of all three terms can be written as $ \\begin{split}\\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g - \\frac{e}{\\hbar } \\mathbf {E}\\cdot \\nabla _{\\mathbf {k}} g&+\\frac{g-f_{FD} \\left( T\\left(\\mathbf {r}\\right), \\mu \\left(\\mathbf {r}\\right), \\mathcal {E}\\left( \\mathbf {k}\\right) \\right)}{\\tau } = \\\\=~~~~~~&\\left(\\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g^{[0]}+\\frac{g^{[0]}-f_{FD} \\left( T, \\mu , \\mathcal {E}\\right)}{\\tau } \\right) \\\\&+ \\left( \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\nabla _{} T]} +\\frac{1}{\\hbar } \\frac{\\partial g^{[0]} }{\\partial T} \\nabla _{\\mathbf {k}} \\mathcal {E}+\\frac{\\delta g^{[\\nabla _{} T]}}{\\tau } \\right) \\nabla _{} T \\\\&+ \\left( \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\nabla _{} \\mu ]}+\\frac{1}{\\hbar } \\frac{\\partial g^{[0]} }{\\partial \\mu } \\nabla _{\\mathbf {k}} \\mathcal {E}+\\frac{\\delta g^{[\\nabla _{} \\mu ]}}{\\tau } \\right) \\nabla _{} \\mu \\\\& +\\left( \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\mathbf {E}]}- \\frac{e}{\\hbar } \\mathbf {u}_{\\mathbf {E}} \\cdot \\nabla _{\\mathbf {k}} g^{[0]} +\\frac{\\delta g^{[\\mathbf {E}]}}{\\tau } \\right) \\mathbf {E}+ ... \\\\ =0 ~,\\end{split}$ where $\\mathbf {u}_{\\mathbf {E}}$ is the unit vector in the direction of the electric field and only the linear terms are shown, because the structure of the higher order terms is obvious.", "Since $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ and $\\mathbf {E}(\\mathbf {r})$ are arbitrary functions, the above equation can only be satisfied if all the brackets vanish.", "We therefore have to impose that all the expressions within brackets in Eq.", "REF have to vanish separately.", "Thus, we reduced the Boltzmann equation to an infinite sequence of coupled differential equations which describe the change in the distribution function in response to the driving forces.", "Each equation specifies a particular response function $\\delta g^{[\\alpha ]}[\\mathbf {r},\\mathbf {k}, T, \\mu ] $ (where $T$ and $\\mu $ are treated as variables) which corresponds to a particular driving force $\\alpha $ and the differential operator in these equation is operating on the first variable of $\\delta g^{\\alpha }[\\xi _1, \\xi _2, \\xi _3,\\xi _4]$ only.", "The solution can be constructed sequentially, starting from the lowest order and specifying, for every equation, a particular boundary condition regarding the variable $\\mathbf {r}$ .", "The construction has to ensure that the sum of all the contributions yields $g(\\mathbf {r},\\mathbf {k})$ which satisfies the boundary condition imposed on the solution of Eq.", "(REF ) (more on this in Sec.", "REF )." ], [ "Drift-reaction equations ", "We now discuss the zeroth and the first order distribution function defined by the expansion Eq.", "(REF ) and relate them to what is known from the literature.", "We also provide a few typical examples of the higher order terms.", "The zeroth-order distribution function is obtained by setting to zero the first bracket in Eq.", "(REF ), which gives $ \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g^{[0]} +\\frac{g^{[0]}-f_{FD} \\left( T, \\mu , \\mathcal {E}\\right)}{\\tau } =0~,$ where $\\nabla _{\\mathbf {r}} g^{[0]} $ is again the derivative of $g^{[0]} [\\mathbf {r},\\mathbf {k}, T, \\mu ]$ with respect to its first variable.", "The spatial part of Eq.", "(REF ) is a convection-reaction equation and the solution requires the value of $g^{[0]}[\\mathbf {r},\\mathbf {k},T,\\mu ]$ on the boundary.", "The change in the distribution function due to an applied electric field $\\mathbf {E}$ , is defined by the equation (see Eq.", "(REF )) $ \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\mathbf {E}]}- \\frac{e}{\\hbar } \\frac{\\mathbf {E}}{\\left\|\\mathbf {E}\\right\|} \\cdot \\nabla _{\\mathbf {k}} g^{[0]} + \\frac{\\delta g^{[\\mathbf {E}]}}{\\tau }=0~.$ Similarly, the response to a thermal force is obtained by collecting all the first order terms in $\\nabla _{} T$ , which gives $ \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\nabla _{} T]} +\\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\frac{\\partial g^{[0]}}{\\partial T} + \\frac{\\delta g^{[\\nabla _{} T]}}{\\tau } =0 ~;$ the response to a diffusion force $\\nabla _{} \\mu $ is given by $ \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\nabla _{} \\mu ]} + \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\frac{\\partial g^{[0]}}{\\partial \\mu }+ \\frac{\\delta g^{[\\nabla _{} \\mu ]}}{\\tau } =0.$ Equations (REF ) – (REF ), without the first term, yield the linear corrections to the Boltzmann distribution function, which is the same as in most textbooks [23], [24].", "These approximate expressions agree also with the results obtained, for instance, by the Hilbert expansion of Eq.", "(REF ) (see Appendix ).", "At this stage, it is not obvious that $\\nabla _{\\mathbf {r}} \\delta g^{[\\alpha ]}$ can be neglected but, in Section , we show that the rigorous solution of equations (REF ) – (REF ) indeed assumes the textbook form sufficiently far away from the boundaries.", "The higher order response follows straightforwardly from the expansion Eq.", "(REF ) and yields the terms of two basic types.", "The first type describes the non-linear response due to the higher powers of the gradients of potentials (like $ (\\nabla _{} \\mu )^2$ , $\\mathbf {E}^2$ , $(\\nabla _{} T)^2$ , etc.).", "Many such terms have previously been discussed in the literature[23], [24].", "The second order response to the diffusion force, $ (\\nabla _{} \\mu )^2$ , is defined by the equation $\\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} \\delta g^{[(\\nabla _{} \\mu )^2]} + \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\frac{\\partial \\delta g^{[\\nabla _{} \\mu ]}}{\\partial \\mu } + \\frac{\\delta g^{[(\\nabla _{} \\mu )^2]}}{\\tau } =0 ~$ and the response to $(\\nabla _{} T)^2$ is similar.", "The terms of the second type describe the response to the higher order derivatives of the potentials (like $ \\partial ^2 \\mu (\\mathbf {r})/\\partial \\mathbf {r}^2$ or $\\nabla _{} E$ , etc.)", "and all their powers.", "These terms have not been considered before, even though they can be comparable to the non-linear terms of the same (and higher) order.", "For example, the response to the second derivative of the chemical potential $ \\partial ^2 \\mu (\\mathbf {r})/\\partial \\mathbf {r}^2$ reads $ \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\cdot \\nabla _{\\mathbf {r}} \\delta g^{[\\frac{\\partial ^2 \\mu }{\\partial \\mathbf {r}^2}]} + \\frac{ \\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\delta g^{[\\nabla _{} \\mu ]} + \\frac{\\delta g^{[\\frac{\\partial ^2 \\mu }{\\partial \\mathbf {r}^2}]}}{\\tau } =0.$ A multidimensional Taylor expansion generates also a large number of mixed terms.", "The equations for other higher order responses (say, the one proportional to $ (\\nabla _{} \\mu )^3$ or $E^3$ ) or the equations for the coefficients depending on the higher derivatives of the fields (like $ \\partial ^3 \\mu (\\mathbf {r})/\\partial \\mathbf {r}^3$ ), or all the cross terms, are obtained by straightforward but tedious calculations.", "The distribution function is now obtained by solving successively equations (REF ), (REF ), (REF ), etc.", "The structure of these equations is always the same and we can write in the $n$ -th order $ \\nabla _{\\xi _1} \\delta g^{[\\xi _n]}[\\xi _1,\\xi _2,\\xi _3,\\xi _4] & +F_n(\\delta g^{[\\xi _0]},\\delta g^{[\\xi _6]},\\ldots ,\\delta g^{[\\xi _{n-1}]}) \\nonumber \\\\& +\\frac{ \\delta g^{[\\xi _n]}}{\\tau } =0,$ where $F_n(\\delta g^{[\\xi _0]},\\delta g^{[\\xi _6]},\\ldots ,\\delta g^{[\\xi _{n-1}]})$ is a known function obtained from the solution of the lower-order equations." ], [ "Expansion of the boundary conditions", "The boundary conditions for Eq.", "(REF ), written as $g(\\mathbf {r}_B,\\mathbf {k})=g_B(\\mathbf {r}_B,\\mathbf {k})$ , define how the electrons are injected in the region under consideration.", "For instance a surface subject to an electron flux from vacuum (as in the case of inverse photoemission) is subject to an electron current with an electronic distribution at the entrance that depends on the energy distribution of the injected electrons.", "Another interesting case is that of a current flowing from a semiconductor into a metal.", "The injected electrons do not have the same energy distribution as in the case when they are excited by an electric field inside the metal.", "In general, from a mathematical point of view.", "the population of electrons in the k-space can be described at the boundary $\\mathbf {r}=\\mathbf {r}_B$ by any function of momentum, not necessarily by the Fermi-Dirac distribution or the non-equilibrium distribution which is giving rise to stationary currents.", "A unique solution of Eqs.", "(REF ), (REF ), (REF ), etc.", "requires a boundary condition in every order.", "Since the differential operator in the drift-reaction equation for $\\delta g^{[\\xi _n]}$ is operating on the the first argument of $\\delta g^{[\\xi _n]}[\\mathbf {r},\\mathbf {k}, T, \\mu ]$ , the most general form of the boundary conditions for the drift-reaction equations is $\\delta g^{[\\xi _n]}[\\mathbf {r}_{B},\\mathbf {k}, T, \\mu ] =\\delta g_B^{[\\xi _n]}[\\mathbf {r}_{B},\\mathbf {k}, T, \\mu ],$ where $\\delta g_B^{[\\xi _n]}$ specifies the value of $\\delta g^{[\\xi _n]}$ at the boundary for any given $\\mathbf {k}$ , $T$ , and $\\mu $ .", "Note, any choice of boundary conditions for the drift-reaction equations is acceptable, as long as the sum of all terms in Eq.", "(REF ) yields the correct boundary condition for the solution of the Boltzmann equation (REF ).", "That is, the boundary conditions for the drift-reaction equations have to satisfy the supplementary condition $ g_B(\\mathbf {r}_B,\\mathbf {k}) &=g_B^{[0]}[\\mathbf {r}_B,\\mathbf {k},T(\\mathbf {r}_B),\\mu (\\mathbf {r}_B)] \\\\& \\!\\!\\!\\!\\!\\!\\!", "+ \\sum \\delta g_B^{[\\xi _n]}[\\mathbf {r}_B,\\mathbf {k},T(\\mathbf {r}_B),\\mu (\\mathbf {r}_B)] \\;\\xi _n(\\mathbf {r}_B).", "\\nonumber $ As long as Eq.", "(REF ) holds, the sum of all terms in Eq.", "(REF ) gives the particular solution of Boltzmann equation (REF ) which satisfies the required boundary condition (assuming the series expansion converges).", "This concludes the construction of the expansion of the solution of the BE in the RTA.", "We emphasize that our expansion of the distribution function takes into account the boundary conditions.", "This implies that our solution yields not just the response to the higher driving forces or the non-local effects in the bulk, but it can also describe the effects caused by the specific choice of the boundary conditions (for instance, our solution can be used to discuss the anomalous skin effect).", "In Appendix we show why this cannot be achieved by other, often used, expansions, as for instance the Hilbert expansion." ], [ "Solution of the drift-reaction equation in a macroscopic sample", "In the rest of this paper we focus on the high order responses and non-local effects in the bulk, leaving the description of surface effects to future work (by surface effects, we mean the features of the solution that depend on the specific form of the boundary conditions).", "We will proceed in two steps.", "First we show that due to the dissipative nature of the scattering term in the RTA, a specific shape of the BC modifies the solution only close to the boundary.", "This implies that, for a given functional form of $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ and $\\mathbf {E}(\\mathbf {r})$ , any BC imposed on the BE leads to the same solution in the bulk.", "For a very small device, the full set of the drift-reaction equations (REF ), (REF ), (REF ), etc., has to be solved for a given choice of the boundary conditions, subject to the supplementary condition Eq.", "(REF ).", "This rises the problem of the optimal distribution of the boundary conditions among various drift-reaction equations.", "Similar issues arise also in larger systems close to the physical boundary.", "However, these problems are not addressed here.", "To understand the extent in which the details of the BC affect the solution far away from the boundary, we study the lowest-order drift-reaction equation, Eq.", "(REF ), in one dimension.", "The solution can be written as $ \\begin{split}&g^{[0]} [x,\\mathbf {k}, T, \\mu ]= \\\\&\\;\\;\\;\\;\\;f_{FD} \\left( T, \\mu , \\mathcal {E}\\left( \\mathbf {k}\\right) \\right) +C(\\mathbf {k},T,\\mu ) e^{-\\tfrac{\\hbar \\, x}{\\tau \\partial _{k_x} \\mathcal {E}}}\\end{split}$ where $\\mathbf {k}, T, \\mu $ are arbitrary and the coefficient $C(\\mathbf {k},T,\\mu )$ is defined by the BC at $x=0$ .", "Obviously, the effect of the boundary on the solution $g^{[0]} [x,\\mathbf {k}, T, \\mu ]$ decreases exponentially with the distance from the boundary.", "The characteristic decay length is given by the mean free path, $l=\\tau v_k$ .", "For $x\\gg l$ , the effect of the boundary is obliterated by the scattering, so that the particular value of $g^{[0]} [x,\\mathbf {k}, T, \\mu ]$ at $x=0$ becomes irrelevant at distances which are much larger than the mean free path.", "This feature also holds in higher dimensions and for every term in the expansion.", "Therefore, if we are interested only in the bulk solution, which holds far from the boundary, we are free to choose the BC as we like; the difference between the true particular solution and the one obtained for a different BC vanishes far away from the boundary.", "This implies that the solution of the Boltzmann equation in the bulk of the sample, is completely determined by the local temperature, chemical potential and electric field, regardless of the microscopic BC.", "As regards the response of the entire sample, the larger the system, the less important the region close to the boundary.", "In a macroscopic device, an accurate treatment of microscopic boundary conditions gives only a very small correction to the response functions but it increases dramatically the computational complexity.", "Hence, we choose the boundary condition so as to minimize the computational efforts; for large enough systems, the error of using such a solution is insignificant.", "Note the difference between the microscopic boundary conditions for the BE and the macroscopic boundary conditions for the continuity equations providing the thermodynamic variables $T$ and $\\mu $ .", "The conservations laws given by Eqs.", "(REF ) and (REF ) require $T$ and $\\mu $ to assume the boundary values specified by the reservoirs.", "The microscopic boundary conditions, imposed on the Boltzmann equation, provide an information on the state of the electrons at the boundary.", "If we decide to disregard the surface effects, we can choose the BC for Eq.", "(REF ) as $g^{[0]}[0,\\mathbf {k},T,\\mu ]\\!=\\!f_{FD} \\left( T, \\mu , \\mathcal {E}\\left( \\mathbf {k}\\right) \\right)$ , which yields $C(\\mathbf {k},T,\\mu )=0$ and makes the function $g^{[0]}$ independent of the variable $x$ .", "The differential operator in Eq.", "(REF ) can now be dropped and the solution of the zeroth order drift-reaction equation becomes $ g^{[0]}[\\mathbf {r},\\mathbf {k},T,\\mu ]=f_{FD} \\left( T, \\mu , \\mathcal {E}\\left( \\mathbf {k}\\right) \\right) ~$ everywhere in the sample.", "This solution deviates from the exact one only very close to the boundary, where the exponential term cannot be neglected.", "Similarly, if we chose the BC for Eq.", "(REF ) as $\\delta g^{[\\xi _n]}[\\mathbf {r}_B,\\mathbf {k},T,\\mu ]=- \\tau F_n(\\delta g^{[\\xi _0]},\\delta g^{[\\xi _6]},\\ldots ,\\delta g^{[\\xi _{n-1}]})~,$ the $n$ -th order solution can be computed analytically and it will be independent on $\\xi _1$ .", "The set of approximate equations obtained in such a way coincides with the equations generated by the Hilbert expansion in the static approximation (see Appendix ).", "The approximate solution of the BE which works very well for bulk materials is obtained by summing up all the solutions of the drift-reaction equations.", "The ensuing distribution function reads $ \\begin{split}&g(\\mathbf {r},\\mathbf {k}) = f_{FD} \\left( T,\\mu , \\mathcal {E}\\left( \\mathbf {k}\\right) \\right) \\\\& - \\frac{\\tau \\,e}{\\hbar } \\cdot \\nabla _{\\mathbf {k}} f_{FD}\\left( T,\\mu ,\\mathcal {E}\\left(\\mathbf {k}\\right)\\right) \\cdot E \\\\& + \\frac{\\tau }{\\hbar } \\frac{\\mathcal {E}-\\mu }{T} \\nabla _{\\mathbf {k}} f_{FD}\\left(T,\\mu , \\mathcal {E}\\left(\\mathbf {k}\\right)\\right) \\cdot \\nabla _{} T \\\\& + \\frac{\\tau }{\\hbar } \\nabla _{\\mathbf {k}} f_{FD}\\left(T,\\mu , \\mathcal {E}\\left(\\mathbf {k}\\right)\\right)\\cdot \\nabla _{} \\mu \\\\& - \\frac{\\tau ^2}{\\hbar ^2} \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\frac{\\partial }{\\partial \\mu }\\nabla _{\\mathbf {k}} f_{FD}\\left(T,\\mu ,\\mathcal {E}\\left(\\mathbf {k}\\right)\\right)(\\nabla _{} \\mu )^2 \\\\& - \\frac{\\tau ^2}{\\hbar ^2} \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} f_{FD}\\left(T,\\mu ,\\mathcal {E}\\left(\\mathbf {k}\\right)\\right) \\frac{\\partial ^2 \\mu }{\\partial \\mathbf {r}^2} \\\\& - \\frac{\\tau ^2}{\\hbar ^2} \\frac{\\mathcal {E}-\\mu }{T} \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} f_{FD}\\left(T,\\mu ,\\mathcal {E}\\left(\\mathbf {k}\\right)\\right)\\frac{\\partial ^2 T}{\\partial \\mathbf {r}^2}+ ...\\end{split}$ and it satisfies Eq.", "(REF ) but it does not satisfy the original BC.", "However, the difference with respect to the full solution is exponentially small as soon as we move away from the boundary.", "Note, even though the terms $\\delta g^{[\\alpha ]}[\\mathbf {r},\\mathbf {k}, T, \\mu ]$ do not have an explicit $\\mathbf {r}$ -dependence far away from the boundary, they become position-dependent once we substitute for $T=T(\\mathbf {r})$ , $\\mu =\\mu (\\mathbf {r})$ , and $\\mathbf {E}=\\mathbf {E}(\\mathbf {r})$ the functions obtained by solving self-consistently the continuity and Poisson equations (see Sec.).", "The first few terms in Eq.", "(REF ) coincide with the expressions for the Boltzmann distribution function in the presence of the well known driving forces, i.e., we have reproduced, in a mathematically consistent way, the known textbook expressions[23], [24].", "However, we also have the terms, like the last two, that have not been reported before.", "These terms, together with similar, higher-order ones, are easy to overlook in the heuristic derivations that are often used to justify the first five terms.", "They only appear from the second order onwards and, therefore, usually give small corrections.", "However, the second order effects are not always negligible (for instance, the fifth term in Eq.", "(REF ) is sometimes very important).", "The last three terms can be important in inhomogeneous materials (say, multilayers) where the concentration and temperature vary rapidly across the sample.", "In that case, all the terms of the same order should be treated on the same footing, i.e., one should not neglect the non-local forces proportional to the higher order derivatives of the temperature, chemical potential and electric field.", "As shown in Appendix , the expansion obtained by neglecting the differential operator in Eq.", "(REF ) is the same as the one generated from the time-independent Boltzmann equation by the Hilbert expansion [25], [35], with the Knudsen number as the expansion parameter.", "(Knudsen number is given by the ratio of the mean free path, or the mean free time between collision, to some characteristic length (or time) of the system.)", "However, unlike the Hilbert expansion, our method retains its validity close to the boundary, provided we calculate the distribution function in each order from the differential drift-reaction equations (REF ).", "In that case, the transport coefficients are not simply defined by local thermodynamic variables but have an explicit position dependence." ], [ "Macroscopic transport equations ", "In the previous section, we derived an approximate solution of the Boltzmann equation (REF ) for arbitrary functions $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ and $E(\\mathbf {r})$ .", "Substituting that solution in equations for the charge and energy conservation, Eqs.", "(REF ) and (REF ), and using the Poisson equation (REF ), we can find the physical functions $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ and $E(\\mathbf {r})$ .", "Substituting the power series for $g(\\mathbf {r},\\mathbf {k})$ in Eqs.", "(REF ) and (), yields the transport equations $ \\mathbf {J}_{}^{}=&\\mathbf {J}^{[\\nabla _{} T]}+\\mathbf {J}^{[\\nabla _{} \\mu ]}+\\mathbf {J}^{[E]}+\\mathbf {J}^{[E^2]}+\\mathbf {J}^{[\\nabla _{} E]}+... \\\\\\mathbf {J}_{\\mathcal {E}}^{}=&\\mathbf {J}_{\\epsilon }^{[\\nabla _{} T]}+\\mathbf {J}_{\\epsilon }^{[\\nabla _{} \\mu ]}+\\mathbf {J}_{\\epsilon }^{[E]}+\\mathbf {J}_{\\epsilon }^{[E^2]}+\\mathbf {J}_{\\epsilon }^{[\\nabla _{} E]}+...$ where, $\\mathbf {J}^{[\\alpha ]}$ is the charge current density due to the driving force $\\xi _\\alpha $ , $ \\mathbf {J}^{[\\alpha ]}=e \\left(\\int \\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\delta g^{[\\alpha ]} \\left(\\mathbf {r},\\mathbf {k}\\right) d^3k \\right) \\xi _\\alpha = N_{\\alpha }^J \\xi _\\alpha ~,$ $\\mathbf {J}_{\\mathcal {E}}^{[\\alpha ]}$ is the corresponding energy current density, $ \\mathbf {J}_{\\mathcal {E}}^{[\\alpha ]}=\\left( \\int \\mathcal {E}\\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\delta g^{[\\alpha ]} \\left(\\mathbf {r},\\mathbf {k}\\right) d^3k \\right) \\xi _\\alpha = N_{\\alpha }^\\mathcal {E}\\xi _\\alpha ~,$ and $N_{\\alpha }^J$ and $N_{\\alpha }^\\mathcal {E}$ are the transport coefficients associated with the force $ \\xi _\\alpha $ .", "Since the charge and energy conservation imply the continuity equations for the charge and energy currents, we can equivalently obtain $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ and $E(\\mathbf {r})$ by solving the continuity equations, $\\nabla _{}\\cdot \\mathbf {J}(\\mathbf {r})=0$ and $\\nabla _{}\\cdot \\mathbf {J}_{\\mathcal {E}}(\\mathbf {r})=\\mathbf {J}(\\mathbf {r})\\cdot \\mathbf {E}(\\mathbf {r})$ , together with the Poisson equation, and the transport equations, Eqs.", "(REF ) and ().", "The above current densities and transport coefficients reproduce all the standard results for the response due to the known driving forces.", "For instance, the current which is first order in the electric field $\\mathbf {J}^{[E]}=\\sigma \\mathbf {E}$ has the conductivity coefficient $\\sigma = - \\frac{\\tau \\,e^2}{\\hbar ^2} \\int \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} f_{FD}\\;d^3k , $ which is the textbook result.", "The same agreement is found for the Seebeck coefficient $S$ and the thermal conductivity $\\kappa $ .", "We now report a few higher-order terms generated by the expansion of the distribution function given by Eq.", "(REF ).", "The current due to the second power of the electric field is $\\mathbf {J}^{[\\mathbf {E}^2]}=\\sigma ^{[\\mathbf {E}^2]}\\mathbf {E}^2$ , where the conductivity coefficient, $ \\sigma ^{[\\mathbf {E}^2]}= \\frac{e^3\\tau ^2}{\\hbar ^3}\\int \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\;\\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\frac{\\partial ^2 f_{FD}}{\\partial \\mu ^2} d^3k ~,$ describes the Boltzmann expression for the non-linear response to an electric field.", "The current driven by $ \\nabla _{}\\mathbf {E}(\\mathbf {r}) $ is $\\mathbf {J}^{[\\nabla _{}\\mathbf {E}]}=\\sigma ^{[\\nabla _{}\\mathbf {E}]}\\nabla _{}\\mathbf {E}$ , with the conductivity coefficient $ \\sigma ^{[\\nabla _{}\\mathbf {E}]}= \\frac{e^2\\tau ^2}{\\hbar ^3} \\int \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\frac{\\partial f_{FD}}{\\partial \\mu } d^3k ~,$ while the current driven by $ \\partial ^2 T/\\partial \\mathbf {r}^2 $ is $\\mathbf {J}^{[\\frac{\\partial ^2 T}{\\partial \\mathbf {r}^2}]}=\\alpha ^{[\\frac{\\partial ^2 T}{\\partial \\mathbf {r}^2}]}\\partial ^2 T/\\partial \\mathbf {r}^2$ , with the thermal conductivity coefficient $ \\alpha ^{[\\frac{\\partial ^2 T}{\\partial \\mathbf {r}^2}]}= \\frac{e\\tau ^2}{\\hbar ^3} \\int \\frac{\\mathcal {E}-\\mu }{T} \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\nabla _{\\mathbf {k}} \\mathcal {E}\\; \\frac{\\partial f_{FD}}{\\partial \\mu } d^3k.$ The other contributions to the currents, with the corresponding transport coefficients, can be calculated in the same way.", "The first few terms for the bulk current density read $\\mathbf {J}= \\sigma (\\mathbf {E}+\\frac{1}{e}\\nabla _{}\\mu ) + \\alpha \\nabla _{} T + \\sigma ^{[\\mathbf {E}^2]} \\mathbf {E}^2+ \\frac{\\sigma ^{[\\mathbf {E}^2]}}{e^2} \\left(\\nabla _{}\\mu \\right)^2 + \\sigma ^{[\\nabla _{}\\mathbf {E}]}\\nabla _{}\\mathbf {E}+ \\frac{\\sigma ^{[\\nabla _{}\\mathbf {E}]}}{e} \\frac{\\partial ^2 \\mu }{\\partial \\mathbf {r}^2} + .... ~,$ where the first term describes the response to a electro-chemical force, the second term describes to the Seebeck effect due to the thermal gradient, the third and the fourth term give the second-order response to the gradients of the electrical and chemical potentials, while the remaining terms describe the non-local response.", "A similar expression can be written for the heat current.", "Thus, our expansion supplements the well-known steady state macroscopic transport equations by additional terms which are due to the higher powers of the thermodynamic forces and their spatial derivatives.", "The difference with respect to the usual textbook equations is the appearance of new, higher-order terms.", "The current in Eq.", "(REF ) does not have the 0-th order term, since it can be proven to vanish because of the periodicity of the band structure and the fact that the integrand for the zeroth-order current is an exact differential.", "Let us also note that even-order contributions to the current densities are obtained by integrating the odd powers of the velocity.", "Such terms can be finite if inversion and time reversal symmetry is broken, e.g., for a ferromagnet and a lattice without inversion symmetry.", "Otherwise $ \\mathcal {E}_\\uparrow (-\\mathbf {k}) = \\mathcal {E}_\\downarrow (\\mathbf {k})$ so that there is no charge current, but possibly a spin current if inversion symmetry is broken." ], [ "Example: Depletion region in semiconductors", "To show the impact of the new terms on the behavior of real devices we now consider an example involving the terms proportional to $\\partial ^2 \\mu /\\partial \\mathbf {r}^2$ and $\\nabla _{} \\mathbf {E}$ .", "These terms are proportional to $\\tau ^2$ , so $\\sigma ^{[\\nabla _{}\\mathbf {E}]}$ is usually small, but if the scattering life time is long or the derivatives of the potential are large, their contribution can be important.", "Taking the case that is familiar to most readers, we examine the width and the shape of the depletion region in a metal-semicondutor (M-S) junction shown in Fig.", "REF .a.", "We are only interested in the qualitative features due to the new terms, revealed by our treatment, so we neglect several effects that are relevant for real junctions, like finite jumps in temperature and chemical potential or the formation of defects.", "Figure: [Colour online]Panel a) Geometry of the metal-semiconductor junction and the direction of positive current.", "The interface is at x=0x=0.Panel b) Modification of the bottom of the band structure as in Eq.", ".Panel c) Position-dependent chemical potential calculated for different currents at T=300KT=300K for the band structure with a finite δ\\delta , butusing the textbook transport equations, i.e.", "setting σ [∇ 𝐄] =0\\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}=0 in the complete set of equations.The results depend on the value of the current running through the device (see the inset, which also provides the color code for the currents).Panel d) Same as panel c) but calculated with the correct transport equations, including the new term σ [∇ 𝐄] \\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}.The insets in c) and d) provide the current vs voltage characteristic of the semiconductor layer.Let us, first, recapitulate the textbook approach to isothermal M-S junctions, ignoring the heat transport [36].", "The junction is described by two equations: $&J= \\sigma (\\mu (x))\\left( \\mathbf {E}(x)-\\frac{\\mu ^{\\prime }(x)}{e}\\right) = \\mbox{const.}", "\\\\&\\epsilon _0 \\mathbf {E}^{\\prime } (x)= \\rho (\\mu (x))$ where the first one is the trasnsport equation (REF ) and the second one is the Poisson equation (Eq.", "(REF )).", "As shown below, the conductivity $\\sigma $ and the charge $\\rho $ depend explicitly on the chemical potential, so that the solution of these coupled equations yields, for a constant current density $J$ , the spatial profile of the electric field and the chemical potential.", "The conductivity is calculated from Eq.", "(REF ) and to obtain an explicit expression for $\\sigma (x)$ and $\\rho (x)$ we consider an n-type semiconductor with a parabolic band $\\mathcal {E}(k)=\\hbar ^2 k^2/2m$ .", "We also assume that the chemical potential is below the bottom of the conduction band and approximate the Fermi-Dirac distribution by $f_{FD}\\approx \\exp [(\\mu -\\mathcal {E})/k_B T]$ .", "The conductivity and the charge $\\rho $ (see Eq.", "(REF )) assume the simple form $\\sigma &= \\sigma _0 e^{\\frac{\\mu }{k_B T}} \\\\\\rho &= \\rho _D -\\rho _0 e^{\\frac{\\mu }{k_B T}} $ where $\\sigma _0$ and $\\rho _0$ are given by the expressions, $& \\rho _0 = \\frac{\\sqrt{2 \\pi k_B T m}}{ \\hbar },\\\\& \\sigma _0 = \\frac{\\tau e^2 }{\\hbar } \\sqrt{\\frac{2\\pi k_B T}{ m}} ,$ and $\\rho _D$ is the dopant charge density.", "Thus, we obtain the textbook equations for the charge transport in semiconductors, $&J= \\sigma _0 e^{\\frac{\\mu (x)}{k_B T}}\\left( \\mathbf {E}(x)-\\frac{\\mu ^{\\prime }(x)}{e}\\right) ,\\\\&\\epsilon _0 \\mathbf {E}^{\\prime } (x)= \\rho _D -\\rho _0 e^{\\frac{\\mu (x)}{k_B T}} $ which have to be solved for the appropriate boundary conditions.", "For a complete description, we also need similar equations for the metal part of the junction (with the appropriate expressions for the conductivity and the charge) and, then, we have to link the two regions.", "The requirement is that the chemical potential and the electric field (or the electric displacement field, when the semiconductor and the metal have different dielectric constants) are continuous across the interface (again, we neglect the surface discontinuities in the potentials).", "However, if the metal is highly conductive and has a high density of states at the Fermi energy, it is sufficient to solve Eqs.", "(REF ) and () with the boundary conditions $\\mu (0)=\\mathcal {E}_{F,m}$ and $\\mu (+\\infty )=k_B T \\log (\\rho _D/\\rho _0)$ .", "That is, we require that the chemical potential at the interface is set by the Fermi level $\\mathcal {E}_{F,m}$ of the metal and that there is no net charge far away from the interface.", "We consider a low symmetry material with the dispersion (close to the band minimum) given by $ \\mathcal {E}(\\mathbf {k})=\\frac{\\hbar ^2 (k^2 +\\delta \\, k_x^3)}{2m}$ where $k$ is the magnitude of the crystal momentum $\\mathbf {k}$ , $k_x$ its the $x$ component, and $\\delta $ measures the asymmetry.", "The modification of band structure is depicted in Fig.", "REF b.", "The difference with respect to the parabolic dispersion is that $\\mathcal {E}(\\mathbf {k})$ now has a finite third derivative at the minimum.", "As an example, we take a semiconductor similar to silicon ($m=0.1 m_e$ , where $m_e$ is the electron mass, and $\\tau =10$ fs) but with an asymmetry $\\delta =5.25$ Å, doped with $3.6*10^{22}$ carriers$/m^3$ , and attach it to a metal with Fermi level at $-155 ~ meV$ below the semiconductor conduction band.", "The solution of Eqs.", "(REF ) and () provides the spatial profile of the electric field and the chemical potential.", "The results are shown in Fig.", "REF c for various operating conditions.", "The black line represents an open circuit (no current flowing) and shows the formation of the depletion region.", "As the current increases, the depletion region either expands, for negative currents, or shrinks, for positive ones.", "The inset shows a typical current-voltage characteristic of a metal-semiconductor junction, i.e.", "it shows the dependence of the electro-chemical potential in the semiconductor, $\\mu (+\\infty )/e-V(+\\infty ) - (\\mu (0))/e-V(0))$ , on the current running through the junction.", "We now compare this textbook solution with the case when the response of a semiconductor in the presence of additional driving forces $\\partial ^2 \\mu /\\partial \\mathbf {r}^2$ and $\\nabla _{} \\mathbf {E}$ .", "It is easy to prove that $\\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]} = \\sigma _0^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]} e^{\\frac{\\mu }{k_B T}}\\; ,$ where $\\sigma _0^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}$ is a constant, and that $\\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}$ vanishes for centrosymmetric materials ($\\delta =0$ ).", "That is, the shape of the conductivity and charge is still given by Eqs.", "(REF ) and () but the proportionality constants are now more complex.", "The macroscopic transport equation, including the new terms, reads $&J = \\sigma _0 e^{\\frac{\\mu }{k_B T}}\\left( \\mathbf {E}-\\frac{\\mu ^{\\prime }}{e}\\right) + \\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}_0e^{\\frac{\\mu }{k_B T}}\\left( \\mathbf {E}^{\\prime }-\\frac{\\mu ^{\\prime \\prime }}{e}\\right) \\;,$ while the Poisson equation is unchanged.", "As before, we also have to consider the transport equations in the metal and ensure that the electric field (or electric displacement field if the semiconductor and the metal have different dielectric constants), the chemical potential, and its first derivative are continuous across the interface.", "In the absence of the gradient of the electro-chemical potential we can set $\\sigma ^{\\scriptscriptstyle [\\nabla \\mathbf {E}]}_0=0$ and reduce Eq.", "(REF ) to Eq.", "(REF ).", "When $\\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}_0\\ne 0$ , the order of the differential equation rises and the boundary condition for the derivative is needed.", "We require that the first derivative has the same value as in the case $\\sigma ^{\\scriptscriptstyle [\\nabla _{} \\mathbf {E}]}_0=0$ and report the solution of Eqs.", "(REF ) and () in Fig.", "REF c. The comparison of Figs.", "REF c and REF d shows that in the absence of the current, the textbook equations and the complete transport equations lead to the same result (the black curves in Fig.", "REF c and REF d are the same), as can be deduced from Eq.", "(REF ) for $J=0$ .", "However, for $J\\ne 0$ , the dimension and shape of the depletion region are modified by the new terms, i.e., the linear response theory differs considerably from the full description of the device.", "For negative currents, the depletion region given by the non-linear description is wider than the textbook one, while for positive currents, the depletion region is reduced.", "The above concepts can have a straightforward experimental verification in the case of a perfectly symmetric metal-semiconductor-metal device, when the semiconductor has both inversion and time reversal symmetries broken.", "The standard transport equations predict that when the current is flowing in positive direction, the voltage drop is exactly the opposite to the one when the direction of the current is reversed.", "Thus, the standard equations predict that the two running conditions are perfectly symmetric.", "However, if the terms proportional to the derivative of the electric field and the second derivative of the chemical potential are taken into account we expect the asymmetric behaviour, as can be verified by applying the transformation $x \\rightarrow -x$ to Eq.", "(REF ) and (REF ).", "We provided just one example of possible effects due to the new terms obtained by the expansion of the Boltzmann distribution function.", "But the range of applicability and the relevance of that expansion is much wider and we expect that other terms will also become important." ], [ "Discussion and conclusions", "In summary, our starting point is the stationary Boltzmann equation, in its most general form, subject to specific microscopic boundary conditions.", "To obtain the solution, we carefully define the relaxation time approximation (RTA) and relate, for a given spatial profiles of temperature $T(\\mathbf {r})$ , chemical potential $\\mu (\\mathbf {r})$ , and electric field $\\mathbf {E}(\\mathbf {r})$ , the Boltzmann distribution function to the relaxation time.", "We then show that to obtain the physically meaningful results $\\mathbf {E}(\\mathbf {r})$ has to satisfy the Poisson equation, while $T(\\mathbf {r})$ and $\\mu (\\mathbf {r})$ have to satisfy the charge and energy conservations.", "Thus, together with the the Boltzmann equation, we now have to self-consistently solve three additional equations.", "The initial problem for $g(\\mathbf {r},\\mathbf {k})$ has, apparently, been turned into a more complicated one, where $g(\\mathbf {r},\\mathbf {k})$ is a functional defined on the functions $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ , and $\\mathbf {E}(\\mathbf {r})$ .", "We now focus on the strategy for solving the Boltzmann equation within the RTA, by assuming that $T(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ , and $\\mathbf {E}(\\mathbf {r})$ , are analytic functions.", "Representing these functions by their respective Taylor series[33], we can write $g(\\mathbf {r},\\mathbf {k})$ as a function of infinitely many variables, $ \\nonumber g(\\mathbf {r},\\mathbf {k})=\\tilde{g} \\big (&\\mathbf {r},\\mathbf {k}, T(\\mathbf {r}),\\mu (\\mathbf {r}), \\mathbf {E}(\\mathbf {r}) , \\nabla T(\\mathbf {r}),\\nabla \\mu (\\mathbf {r}),\\\\& \\nabla \\mathbf {E}(\\mathbf {r}), \\frac{\\partial ^2 T(\\mathbf {r})}{\\partial \\mathbf {r}^2},\\frac{\\partial ^2 \\mu (\\mathbf {r})}{\\partial \\mathbf {r}^2},\\frac{\\partial ^2 \\mathbf {E}(\\mathbf {r})}{\\partial \\mathbf {r}^2}, ..., BC\\big ) .\\nonumber $ The general solution of the Boltzmann equation is obtained by expanding the distribution function with respect to all its variables, except the first five, $\\mathbf {r},\\mathbf {k}, V(\\mathbf {r}), T(\\mathbf {r})$ , and $\\mu (\\mathbf {r})$ , and writing $g(\\mathbf {r},\\mathbf {k})$ as a multivariable power series.", "Since the expansion variables are completely arbitrary, substituting $g(\\mathbf {r},\\mathbf {k})$ in the Boltzmann equation yields an infinite number of coupled differential equations for the expansion coefficients.", "Integrating these equations for a particular set of microscopic boundary conditions yields $g(\\mathbf {r},\\mathbf {k})$ as a power series in terms of the expansion variables $\\mathbf {E}(\\mathbf {r}) , \\nabla _{} T(\\mathbf {r}),\\nabla _{} \\mu (\\mathbf {r}), \\nabla _{}^2 V(\\mathbf {r}), \\nabla _{}^2 T(\\mathbf {r}), \\nabla _{}^2 \\mu (\\mathbf {r}) \\ldots $ .", "Substituting $g(\\mathbf {r},\\mathbf {k})$ in the expressions for the charge and energy current densities we obtain an expansion of $\\mathbf {J}(\\mathbf {r})$ and $\\mathbf {J}_\\mathcal {E}(\\mathbf {r})$ in terms of their respective driving forces.", "The coefficients of the driving forces define the generalized transport coefficients.", "The physically relevant functions $\\mathbf {J}(\\mathbf {r})$ , $\\mathbf {J}_\\mathcal {E}(\\mathbf {r})$ , $\\mu (\\mathbf {r})$ , $T(\\mathbf {r})$ , and $\\mathbf {E}(\\mathbf {r})$ are obtained at every macroscopic point of the sample, by solving self-consistently the transport equations, the charge and energy continuity equations, and the Poisson equation.", "We also show, under which conditions the surface effects can be neglected and the distribution function of a macroscopic sample assumes a simple, textbook form.", "The above procedure elucidates the commonly used derivation of transport equations and exposes various approximations employed in that derivation.", "In addition, it reveals new contributions to the response functions which are proportional to the higher powers of the forces and their higher-order derivatives.", "The ensuing corrections to the charge and energy currents are usually small, which explains the success of the phenomenological transport theory in Eqs.", "(REF ) and ().", "However, in certain situations, the new terms lead to qualitatively new phenomena.", "For example, they can become important for heterogeneous devices, for materials in which the transport properties are strongly temperature- and potential-dependent, for systems driven out of equilibrium by large thermodynamic forces (e.g.", "large temperature differences), or when the thermodynamic potentials vary strongly over small distances.", "Unlike the second order response to the thermodynamic forces, like the one due to $\\mathbf {E}^2$ , $(\\nabla \\mu )^2$ or $(\\nabla T)^2$ , the response to the spatial derivatives of these forces, like $\\nabla _{} \\mathbf {E}$ , $\\nabla _{}^2 \\mu $ , or $\\nabla _{}^2 T$ , has not been discussed before.", "Since the magnitude of the new higher-order terms is comparable to the already known ones, all the terms of the same order should be treated on the same footing.", "In other words, a consistent semi-classical description of transport phenomena should not just consider the higher powers of the thermodynamic forces but should also take into account the driving forces which are proportional to the higher order derivatives of temperature, chemical potential and electric field.", "The expansion of the distribution function described in this paper respects the microscopic boundary conditions of the Boltzmann equation.", "Thus, it can be used to treat, on the same footing, not just the higher-order and non-local effects in the bulk but also the multitude of surface effects.", "Our expansion in terms of the driving forces provides a substantial improvements over the Hilbert expansion or similar expansions of the solution of the Boltzmann equation which do not take into account the microscopic boundary conditions, and therefore yield the solution which is valid only in the bulk.", "The authors wish to thank Jan Tomczak and Michael Wais for fruitful discussions, and acknowledge financial support by the European Research Council/ERC through grant agreement n. 306447 and by the Austrian Science Fund (FWF) through SFB ViCoM F41 and Lise Meitner position M1925-N28.", "V.Z acknowledges the support by the Ministry of Science of Croatia under the bilateral agreement with the USA on the scientific and technological cooperation, Project No.", "1/2016." ], [ "Continuity equations", "The particle continuity equation is obtained by integrating Eq.", "(REF ) over the whole $k$ -space.", "The first term of Eq.", "(REF ) becomes the time evolution of the local total number of particles, defined as $n(t,\\mathbf {r})=\\int g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k.", "$ In the second term of Eq.", "(REF ) the divergence with respect to spacial coordinates can be brought out of the $k$ -space integral, leading to the spatial divergence of the particle current written as $\\mathbf {J}(t,\\mathbf {r}) = e \\int \\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } \\, g\\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k.$ The third term of Eq.", "(REF ) can be proven to integrate to zero.", "The part multiplying the electric field $\\mathbf {E}=\\nabla _{} V + \\partial \\mathbf {A}/\\partial t$ vanishes due to a corollary of the divergence theorem in the $k$ -space and the periodicity of all the involved functions in $k$ -space.", "The part multiplying the magnetic field $\\mathbf {B}= \\nabla _{} \\times \\mathbf {A}$ requires the use of the identity $\\nabla _{\\mathbf {k}} \\cdot ( \\nabla _{\\mathbf {k}} \\mathcal {E}\\times \\mathbf {B}) =0$ and then the same considerations above.", "The integral of the scattering term on the right hand side of Eq.", "(REF ) depends on its precise expression.", "We assume the RTA in Eq.", "(REF ).", "As already mentioned in section , the value of the integral will depend on the local temperature $T(\\mathbf {r})$ and chemical potential $\\mu (\\mathbf {r})$ , as well as the local electron distribution $g\\left(t, \\mathbf {r},\\mathbf {k}\\right)$ and can be, in general, different from zero.", "This would imply that some particles are either destroyed or created during the scattering.", "It is indeed to prevent this unphysical effect that we imposed the constraint in Eq.", "(REF ), for the RTA to make sense.", "Using Eq.", "(REF ), the conservation equation reduces to $ \\frac{\\partial }{\\partial t} \\int g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k +\\nabla _{}\\cdot \\int \\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k =0.$ which states that the variation in the local number of particles $n$ , has to be equal to the divergence of the particle current density $\\mathbf {J}(\\mathbf {r})$ .", "Similarly the energy continuity equation can be obtained by multiplying Eq.", "(REF ) by the particle energy $\\mathcal {E}(\\mathbf {k})$ and integrating over the whole $k$ -space.", "The first term will give the change in the energy density $\\epsilon (t,\\mathbf {r})=\\int \\mathcal {E}(\\mathbf {k}) \\,g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k$ .", "The second yields the divergence of the energy current density $\\mathbf {J}_{\\epsilon }(t,\\mathbf {r}) = \\int \\mathcal {E}\\, g \\,\\nabla _{\\mathbf {k}} \\mathcal {E}/\\hbar \\, d^3k$ .", "The third term in this case does not vanish but is the work made by the electrical field on the system $W(t,\\mathbf {r})=e/\\hbar \\int \\mathcal {E}\\, \\mathbf {E}\\cdot \\nabla _{\\mathbf {k}} g \\,d^3k $ leading to the Joule heating.", "Again we use the RTA for the scattering term.", "Its integral is now constrained by Eq.", "(REF ) (eventually with the effect of phonons included as explained in the text below Eq.", "(REF )).We therefore obtain: $ \\begin{split}&\\frac{\\partial }{\\partial t} \\int \\mathcal {E}\\, g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k +\\nabla _{}\\cdot \\int \\mathcal {E}\\frac{\\nabla _{\\mathbf {k}} \\mathcal {E}}{\\hbar } g \\left(t, \\mathbf {r},\\mathbf {k}\\right) d^3k \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;+\\frac{e}{\\hbar } \\int \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{} V g \\,d^3k = \\frac{\\Delta \\epsilon _{e-ph} }{\\tau } .\\end{split}$ where we have integrated by parts the Joule heating term and used again the periodicity of all the involved functions in $k$ -space.", "The equation implies that any change in the local total energy $\\epsilon $ , is caused by the divergence of the energy current $\\mathbf {J}_{\\epsilon }$ and the work $W$ done on the charged particles by the electric field and the energy dissipated into phonons $\\Delta \\epsilon _{e-ph}$ ." ], [ "Comparison with Hilbert expansion", "Some of the results derived in this paper can also be obtained from the Hilbert expansion for the Boltzmann equation in the relaxation time approximation.", "However, as shown below, our approach overcomes one fundamental limitation of the Hilbert expansion, which critically limits its range of applicability (as well as its use in the longstanding mathematical problem of the proof of the existence of the solution of the Boltzmann equation in presence of the boundary conditions).", "By adapting the Hilbert expansion to the present case, we look for the solution of Eq.", "REF in the form: $g\\left(\\mathbf {r},\\mathbf {k}\\right) = \\sum _{i=0}^{\\infty } \\tau ^i g^{[i]}_H\\left(\\mathbf {r},\\mathbf {k}\\right)$ where here the relaxation time $\\tau $ plays the role of the Knudsen parameter.", "Substituting the above series in the Boltzmann equation and collecting all the terms of the same order in $\\tau $ gives the result $ \\begin{split}g^{[0]}_H\\left(\\mathbf {r},\\mathbf {k}\\right)&= f_{FD} \\left( T\\left(\\mathbf {r}\\right), \\mu \\left(\\mathbf {r}\\right), \\mathcal {E}\\left( \\mathbf {k}\\right) \\right) \\\\g^{[1]}_H\\left(\\mathbf {r},\\mathbf {k}\\right)&=- \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g^{[0]}_H +\\frac{e}{\\hbar } \\mathbf {E}\\cdot \\nabla _{\\mathbf {k}} g^{[0]}_H \\\\g^{[2]}_H\\left(\\mathbf {r},\\mathbf {k}\\right)&=- \\frac{1}{\\hbar } \\nabla _{\\mathbf {k}} \\mathcal {E}\\cdot \\nabla _{\\mathbf {r}} g^{[1]}_H +\\frac{e}{\\hbar } \\mathbf {E}\\cdot \\nabla _{\\mathbf {k}} g^{[1]}_H \\\\...&=...\\end{split}$ Substituting successively the lower-order corrections into the higher-order ones, we find that the distribution function defined by Eq.", "REF is equivalent to the one given by Eq.", "REF .", "Note, the expansion in Eq.", "(REF ) does not take into account the boundary conditions, so that it provides just one of the infinitely many solutions of the Boltzmann equation.", "In fact, the result given by Eqs.", "(REF ) or (REF ) corresponds to a specific choice of the boundary conditions, as discussed in Sec..", "Hence, the solution obtained by the Hilbert expansion does not, in general, satisfy the imposed boundary conditions This shows the most important difference between the approach taken in this work and the one taken by Hilbert expansion.", "Each term in the expansion defined by Eq.", "REF of this work satisfies one of the differential equations given by Eqs.", "(REF )-(REF ) and the sum of all these terms provides a particular solution of the Boltzmann equation that complies with the imposed boundary conditions.", "Thus, the expansion in terms of the driving forces presented in this work is much more powerful than the Hilbert expansion." ] ]	1606.05084
[ [ "Method for Determining AGN Accretion Phase in Field Galaxies" ], [ "Abstract Recent observations of AGN activity in massive galaxies (log Mstar / Msun > 10.4) show that: 1) at z < 1, AGN-hosting galaxies do not show enhanced merger signatures compared to normal galaxies, 2) also at z < 1, most AGNs are hosted by quiescent galaxies; and 3) at z > 1, percentage of AGNs in star forming galaxies increases and becomes comparable to AGN percentage in quiescent galaxies at z ~ 2.", "How can major mergers explain AGN activity in massive quiescent galaxies which have no merger features and no star formation to indicate recent galaxy merger?", "By matching merger events in a cosmological N-body simulation to the observed AGN incidence probability in the COSMOS survey, we show that major merger triggered AGN activity is consistent with the observations.", "By distinguishing between \"peak\" AGNs (recently merger triggered and hosted by star forming galaxies) and \"faded\" AGNs (merger triggered a long time ago and now residing in quiescent galaxies), we show that the AGN occupation fraction in star forming and quiescent galaxies simply follows the evolution of the galaxy merger rate.", "Since the galaxy merger rate drops dramatically at z < 1, the only AGNs left to be observed are the ones triggered by old mergers and are now in the declining phase of their nuclear activity, hosted by quiescent galaxies.", "As we go toward higher redshifts the galaxy merger rate increases and the percentages of \"peak\" AGNs and \"faded\" AGNs become comparable." ], [ "INTRODUCTION", "Galaxies residing outside of galaxy clusters are known as field galaxies.", "Their name implies a certain level of isolation; either in time between major interactions ($\\sim $ 3 Gyr, Verdes-Montenegro et al.", "2005), or through the surrounding environmental density (Dressler 1980).", "The topic of this paper is AGN activity in field elliptical galaxies.", "These are massive (log M$_$ /$\\:{\\rm M_{\\odot }}$ $>$ 10.4) galaxies, thought to be formed in gas rich major mergers of disk/spiral galaxies (Toomre 1977, White 1978, 1979, Gerhard 1981, Negroponte $\\&$ White 1983, Barnes 1988, Hernquist 1989, Barnes $\\&$ Hernquist 1996, Naab, Jesseit $\\&$ Burkert 2006, Novak et al.", "2012).", "We focus on field AGNs because AGNs in massive elliptical galaxies are a field phenomena.", "Hwang et al.", "2012 have studied a sample of almost a million SDSS galaxies.", "They found factor of three larger AGN fraction in the field compared to clusters.", "At higher redshift this increase is even more pronounced.", "Martini, Sivakoff $\\&$ Mulchaey 2009 found an increase by factor of 8 at redshift $z=1$ .", "Galaxy mergers are also a field phenomena.", "Low velocity dispersion in galaxy groups in the field leads to “slow encounters” (Binney $\\&$ Tremaine 1987) which are necessary for the merger to occur.", "“Fast encounters” are a characteristic of galaxy clusters.", "Energy input and dynamical friction scale as v$^{-2}$ (Binney $\\&$ Tremaine 1987) and do not lead to the merger but rather small perturbations which can fuel a low luminosity AGN (Lake, Katz $\\&$ Moore 1998).", "For a long time, major galaxy mergers have been a main mechanism for driving AGN activity (Sanders et al.", "1988, Barnes $\\&$ Hernquist 1996, Cavaliere $\\&$ Vittorini 2000, Menci et al.", "2004, Croton et al.", "2006, Hopkins et al.", "2006, Menci et al.", "2008), both supermassive black hole (SMBH) accretion and star formation (Sanchez et al.", "2004, Bohm et al.", "2007, Schawinski et al.", "2007, Silverman et al.", "2008, Rafferty et al.", "2011, Hopkins 2012).", "Observational evidence indicates postmerger features in galaxies hosting AGNs and quasars (Surace $\\&$ Sanders 1999, Surace, Sanders $\\&$ Evans 2000, Canalizo $\\&$ Stockton 2000, 2001) lending credence to the theoretical picture of mergers as drivers of AGN activity.", "Fiore et al.", "2012 found that theoretical models using galaxy interactions as AGN triggering mechanism are able to reproduce the high redshift ($z$ =[3, 7]) AGN luminosity functions.", "The AGN fraction is higher in galaxy pairs (Silverman et al.", "2011, Ellison et al.", "2011, 2013).", "This entire model was challenged recently (Gabor et al.", "2009, Darg et al.", "2010, Cisternas et al.", "2011, Kocevski et al.", "2012).", "Cisternas et al.", "2011 found that 85 $\\%$ of galaxies with AGNs do not show evidence of a previous merger at $z \\le $ 1, which is consistent with the merger fraction of non-active galaxies.", "Schawinski et al.", "2011, Kocevski et al.", "2012, and Simmons et al.", "2012 showed that at z$\\le $ 3 there is a high disk fraction in AGN hosts.", "Bohm et al.", "2013 found that morphologies of the AGN hosts are similar to undisturbed galaxies.", "These observations suggested that secular evolution is responsible for SMBH growth, at least at low $z$ .", "These could be internal processes such as bar-driven gas inflow (Kormendy and Kennicutt 2004), and stellar wind (Ciotti and Ostriker 2007, Ciotti, Ostriker $\\&$ Proga 2010, Cen 2012).", "At that moment it seemed that secular evolution is the dominant mechanism behind the activity of low luminosity AGNs, while major mergers of galaxies are responsible for luminous AGNs.", "Theoretical works also support this picture (Lapi et al.", "2006, Hopkins, Kocevski $\\&$ Bundy 2014).", "Hopkins, Kocevski $\\&$ Bundy 2014 combined both merger and non-merger triggering of AGNs in semi-empirical model and found that secular (stochastic) fuelling is dominant in low luminosity AGNs which host SMBHs with mass $\\le 10^7 \\:{\\rm M_{\\odot }}$ .", "For luminous AGNs hosting SMBHs with masses $\\ge 10^8 \\:{\\rm M_{\\odot }}$ it accounts for just $\\sim 10 \\%$ of black hole (BH) growth.", "This is consistent with the observations of post-starburst quasars (PSQs) which show that PSQs with lower luminosities reside in disk/spiral galaxies, while more luminous PSQs reside in early type galaxies (Cales et al.", "2013).", "However, Villforth et al.", "2014 analysed the morphological properties of AGN host galaxies as a function of AGN and host galaxy luminosity and compared them to a carefully matched sample of control galaxies in the redshift range $z$ = [0.5, 0.8] and luminosity range log L$_{\\rm X}$ [erg/s] = [41, 44.5].", "They found no increase in the prevalence of merger signatures with AGN luminosity and concluded that major mergers, even for higher luminosities, either play only a very minor role in the triggering of AGN or time delays are too long for merger features to remain visible.", "This conclusion questions galaxy mergers as drivers of any AGN activity.", "In this paper we apply Shen 2009 SMBH growth model to the dark matter halo (DMH) merger trees in cosmological N-body simulation, in order to test if merger driven AGN activity is consistent with the activity of the observed AGNs in massive galaxies of COSMOS survey (Bongiorno et al.", "2012).", "In the first part of the paper we determine initial BH mass, and final (true) BH mass in the merger trees and then we use Shen 2009 AGN light curve model to grow initial BHs into final BHs.", "Next, we find our best fit model by matching it to the observed AGN luminosity function, active BHs mass function, duty cycle, and bias factor.", "In the second part of the paper, we replace our best fit AGNs with COSMOS AGNs.", "We do this by using probability functions for galaxies to host AGNs in COSMOS, to determine probable AGN luminosities.", "We proceed with Monte Carlo procedure (40,000 realisations) where we replace peak luminosities in our best fit model with the COSMOS AGN luminosities.", "As the result, every postmerger halo has a 40,000 possible final BHs predicted by the model.", "Finally, we compare predicted BH masses to the true BH masses.", "We calculate the percentage of realisations when predicted SMBH mass is at least as large as the true SMBH mass.", "If that percentage is high, then the observed luminosity is the peak AGN luminosity.", "Otherwise, AGN is observed in the declining phase of its nuclear activity.", "This would place it in a massive, red, elliptical galaxy long after merger features can be detected, but its activity would still be consistent with merger driven model.", "AGN hosts in COSMOS survey are mainly massive, red galaxies.", "Hence, their AGNs could potentially be merger driven, passed their peak activity during Green Valley, and in the declining Red Sequence phase.", "This interpretation would be consistent with the merger driven scenario for AGN activity and with the recent Schawinski et al.", "2014 scheme for galaxy evolution.", "We describe our method in section 2 and introduce two models based on initial BH mass function.", "In section 3, we present our best fit model.", "In section 4, we determine the phase of AGNs activity is COSMOS survey.", "We discuss the implications of our results, in section 5.", "Figure: Graphical sketch showing the main steps of the methodology." ], [ "METHOD", "The three major components in our model are: dark matter halo (DMH) merger trees from cosmological N-body simulation; Shen 2009 fit-by-observations semi-analytical model for major merger driven growth of SMBH; and Bongiorno et al.", "2012 study of $\\sim $ 1700 AGNs and their host galaxies in COSMOS field survey.", "The main idea is to track “field DMHs” undergoing major mergers in N-body cosmological simulation.", "Use Shen 2009 SMBH growth model and match it to the observations.", "Then we overlay our field with COSMOS field, match simulated galaxies to the observed COSMOS galaxies and assign COSMOS AGNs to them.", "Find if the observed AGNs are at their peak activity or in the declining phase.", "Here is the outline of our model presented in figure 1. z$_{\\rm initial}$ is the redshift of DMH merger.", "Halos touch and the smaller halo is inside the larger halo at all later times.", "M$_{\\rm H, 1}$ and M$_{\\rm H, 2}$ are masses of merging halos at z$_{\\rm initial}$ .", "We consider major mergers only, when mass ratio of merging halos is $\\ge $ 0.3.", "If merging halo did not have major merger in its history, we assume that halo hosts a spiral galaxy.", "If halo had a major merger before, we assume it hosts an elliptical galaxy.", "We seed spiral galaxies with pristine BHs ($\\sim $ 10$^5$ - 10$^6$ $\\:{\\rm M_{\\odot }}$ ) and elliptical galaxies with BHs from Ferrarese 2002 M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation.", "Masses of BHs hosted by merging halos are M$_{\\rm BH, 1}$ and M$_{\\rm BH, 2}$ .", "Mergers of two elliptical galaxies do not trigger AGN activity (dry mergers).", "$z_{\\rm AGN}$ is the redshift when smaller halo can not be identified anymore inside the larger one which means that the merger of DMHs has finished.", "We assume that mergers of their galaxies and black holes have finished too, and that accretion onto the new SMBH starts and enters AGN phase which has pre-peak and peak activity.", "Even before central BHs (M$_{\\rm BH, 1}$ and M$_{\\rm BH, 2}$ ) form binary (BHB), they accrete at $\\sim $ 10$^{-3}$ - 10$^{-4}$ $\\:{\\rm M_{\\odot }}$ $\\rm yr^{-1}$ rate (Capelo et al.", "2015), from z$_{\\rm initial}$ at R$_{\\rm vir}$ separation until BHB forms at $\\sim $ kpc distance.", "This is the “pre-BHB accretion” phase.", "During the last $\\sim $ 100 Myr before BHs merge (at z$_{\\rm AGN}$ ), binary overcomes last couple of kpc and accretion increases to double the BH mass (Roskar et al.", "2015, Tamburello et al.", "2016).", "This is the “BHB accretion” phase.", "New masses of central BHs after these two accretion episodes are M$_{\\rm BH^{\\prime }, 1}$ and M$_{\\rm BH^{\\prime }, 2}$ .", "SMBH mass entering AGN phase at z$_{\\rm AGN}$ , is then M$_{\\rm BH,initial}$ = M$_{\\rm BH^{\\prime },1}$ + M$_{\\rm BH^{\\prime },2}$ .", "M$_{\\rm H, AGN}$ is the mass of DMH hosting an AGN at z$_{\\rm AGN}$ .", "We adopt Shen 2009 model for AGN activity in field galaxies.", "This model implies halo mass range 3 $\\times $ 10$^{11}$ h$^{-1}\\:{\\rm M_{\\odot }}$ $<$ M$_{\\rm H, AGN}$ $<$ 10$^{12}$ (1+z$_{\\rm AGN})^{3/2}$ h$^{-1}\\:{\\rm M_{\\odot }}$ .", "M$_{\\rm ,AGN}$ is the mass of the galaxy hosting an AGN at z$_{\\rm AGN}$ .", "We consider only AGNs in massive galaxies (log(M$_{,\\rm AGN}$ /$\\:{\\rm M_{\\odot }}$ ) $>$ 10.4).", "Galaxy mass is obtained by using Rodriguez-Puebla et al.", "2015 M$_$ - M$_{\\rm DMH}$ relation.", "P(L$_{\\rm X}$ ) is the probability of a galaxy to host an AGN of a given luminosity at z$_{\\rm AGN}$ (Bongiorno et al.", "2012).", "From it we obtain L$_{\\rm X}$ and calculate bolometric AGN luminosity L$_{\\rm COSMOS}$ .", "M$_{\\rm BH, predicted}$ is the SMBH mass predicted by Shen 2009 model, when M$_{\\rm BH,initial}$ is the input parameter given in point (vii) and the peak luminosity is replaced by L$_{\\rm COSMOS}$ .", "z$_{\\rm final}$ is the redshift of the postmerger halo M$_{\\rm H, final}$ .", "M$_{\\rm BH, final}$ is the “true” mass of the postmerger BH, derived from M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation, calibrated to the local M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation (Ferrarese 2002).", "If the observed COSMOS AGN is at the peak activity, then M$_{\\rm BH, predicted}$ has to be at least as large as M$_{\\rm BH, final}$ .", "Otherwise, AGN is in the declining phase of its nuclear activity.", "In the following sections we describe simulation, data, and modelling in more details." ], [ "Cosmological N-body Simulation", "Using GADGET2 (Springel, Yoshida $\\&$ White 2001, Springel et al.", "2005), we performed a high-resolution cosmological N-body simulation within a comoving periodic box with size of 130 Mpc$^3 $ .", "WMAP5-like (Komatsu et al.", "2009) cosmology was used ($\\Omega _{\\rm M}=0.25$ , $\\Omega _{\\Lambda }=0.75$ , $n_s=1$ , $\\sigma _8$ =0.8 and h=0.7) from $z=599$ to $ z=0$ (84 snapshots).", "Initial conditions were computed with the 2LPT code (Crocce, Pueblas $\\&$ Scoccimarro 2006).", "Simulation utilises 512$^3$ dark matter particles for a mass resolution of 1.14 x 10$^9$ $\\:{\\rm M_{\\odot }}$ .", "We generated halo catalogues using ROCKSTAR (Behroozi, Wechsler $\\&$ Wu 2013).", "ROCKSTAR combines friends-of-friends (FOF), phase-space and spherical overdensity analysis in locating halos.", "Please see Behroozi, Wechsler $\\&$ Wu 2013 for details on the ROCKSTAR algorithm.", "The merger tree was generated using Consistent Merger Tree (Behroozi et al.", "2013), a software package that is complementary with the ROCKSTAR halo finder." ], [ "AGNs and galaxies in COSMOS survey", "Bongiorno et al.", "(2012) have studied $\\sim $ 1700 AGNs in COSMOS field obtained by combining X-ray and optical spectroscopic selections.", "This is a highly homogeneous and representative sample of obscured and unobscured AGNs over a wide redshift range (0 $<$ z $<$ 4).", "By using Spectral Energy Distribution (SED) fitting procedure they have managed to separate host galaxies properties including the total stellar mass of galaxies hosting AGNs.", "One of their results is the probability of a galaxy to host an AGN of a given luminosity (P(L$_{\\rm X}$ )) as a function of stellar mass in three redshift bins: [0.3 - 0.8], [0.8 - 1.5], and [1.5 - 2.5] (Figure 14 in their paper, from here on F14).", "They grouped AGNs in four logarithmic X-ray (2 - 10 KeV) luminosity bins: [42.8 - 43.5], [43.5 - 44.0], [44.0 - 44.5], and [44.5 - 46.0] in erg/s units.", "They showed that for a fixed mass range, observed field galaxies are more likely to host less luminous AGNs.", "The probability that a field galaxy hosts an AGN decreases with increasing AGN luminosity." ], [ "SMBH growth model", "We adopt Shen 2009 model for the hierarchical growth and evolution of SMBHs assuming that AGN activity is triggered in major mergers.", "This model uses a general form of light curve where BH first grows exponentially at constant luminosity Eddington ratio of $\\lambda _0$ =3 (Salpeter 1964) to L$_{\\rm peak}$ at t = t$_{\\rm peak}$ , and then the luminosity decays monotonically as a power-law (Yu $\\&$ Lu 2008).", "Shen 2009 uses a variety of observations.", "Model adopts Hopkins et al.", "2007 compiled AGN bolometric luminosity function data for both unobscured and obscured SMBH growth, It also incorporates quasar clustering observations and the observed Eddington ratio distributions.", "Model successfully reproduces the observed AGN luminosity function and both the observed redshift evolution and luminosity dependence of the linear bias of AGN clustering.", "The input parameters for the Shen 2009 model are M$_{\\rm BH,initial}$ (mass of the BH entering AGN phase), L$_{\\rm peak}$ (peak bolometric AGN luminosity), and M$_{\\rm BH, relic}$ (BH mass in the postmerger halo M$_{\\rm DMH, post}$ immediately after the AGN phase).", "To match our nomenclature, we have renamed M$_{\\rm BH, relic}$ to M$_{\\rm BH, final}$ .", "In our model, values for the first parameter come from the numerical simulation combined with the semi-analytical modelling (details in section 2.4).", "We calculate M$_{\\rm BH, final}$ (details in section 2.6) from M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation (Kormendy $\\&$ Ho 2013) where $\\sigma _{\\rm sph}$ is the velocity dispersion of the stellar spheroid.", "$\\sigma _{\\rm sph}$ is correlated with V$_{\\rm vir}$ by a constant (Ferrarese 2002).", "We set this constant to a value which reproduces z = 0 Ferrarese 2002 M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation.", "Since M$_{\\rm BH}$ - $\\sigma $ relation is expected to be non-evolving, one can find M$_{\\rm BH}$ in M$_{\\rm DMH, post}$ at any redshift.", "The outcome of this procedure is that the BH mass in high redshift halos, right after AGN phase, is overestimated.", "This is expected to occur as BH grows first, followed by postmerger halo growth through minor mergers and diffuse matter accretion.", "As we go toward lower redshifts, DMH growth catches up to SMBH growth to reproduce local Ferrarese relation (Figure 6).", "Hence, we consider M$_{\\rm BH, final}$ to be the “true” final BH mass.", "L$_{\\rm peak}$ is the peak bolometric luminosity (details in section 2.7) in the Shen 2009 light curve model, set to the value which guarantees that the accretion onto M$_{\\rm BH,initial}$ produces M$_{\\rm BH, final}$ .", "This is our (L$_{\\rm peak, true}$ ) best fit model which reproduces the observed AGN activity, luminosity function, duty cycle and bias factor.", "After we demonstrate that our merger driven model reproduces observed AGN statistics, we test if the observed AGN activity in COSMOS survey corresponds to the peak or to the declining activity.", "Now, instead of L$_{\\rm peak, true}$ , values for the peak luminosity (L$_{\\rm COSMOS}$ ) are retrieved from the probability for a galaxy to host an AGN of a given luminosity (P(L$_{\\rm X}$ )) in COSMOS survey (details in section 2.8).", "We use this probability to seed galaxies with AGNs in 40,000 Monte Carlo realisations and grow SMBHs according to Shen 2009 model (details in section 2.9).", "The result is the probability that SMBHs grown in COSMOS AGNs match the true SMBHs grown in our best fit model." ], [ "Halos, galaxies, black holes: Initial values", "We start by identifying major merger events in the merger trees of our cosmological N-body simulation.", "We define masses of merging halos as M$_{\\rm H, 1}$ and M$_{\\rm H, 2}$ at the time of the merger z$_{\\rm initial}$ (figure 1).", "We also check if merging halos had major mergers previously.", "DMH without previous major merger is an ancient halo hosting disk/spiral galaxy with a large cold gas reservoir and the central BH that most likely formed through direct collapse of a gas cloud (Bromm $\\&$ Loeb 2003, Begelman, Volonteri $\\&$ Rees 2006, Begelman, Rossi $\\&$ Armitage 2008).", "Latest observations (Mortlock et al.", "2011) showed that BH seeds had to be massive ($\\sim $ 10$^5$ - 10$^6$$\\:{\\rm M_{\\odot }}$ ) in order to grow $\\sim $ 10$^9\\:{\\rm M_{\\odot }}$ BHs at redshift z $\\sim $ 7.", "The initial mass function (IMF) and the mass range of the seed BHs are unknown.", "These BHs settle at the centres of disk/spiral galaxies but their masses do not correlate with any of the galaxy properties.", "Growth of these initial BHs through accretion can occur even before they form a binary (BHB), during the early stages of the galaxy merger as galaxies go through subsequent pericentric passages (Capelo et al.", "2015).", "As the major merger of galaxies proceeds, gravitational torques generate large-scale gas inflows that drive the gas down to sub-pc scale where it can be accreted by the BH.", "Hence, BHs grow in a modest amount even before they form a binary (pre-BHB accretion).", "Modelling of this growth is a subject of numerous numerical studies.", "However, limited resolution and disparate subgrid physics recipes led to a very different estimates of the BH accretion rates.", "Latest results (Hayward et al.", "2014, Capelo et al.", "2015) show that BH accretes at the rate 10$^{-4}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ - 10$^{-3}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ for $\\sim $ 1 Gyr before AGN phase.", "After BH binary forms, accretion increases as BHs sink to overcome the last couple of kiloparsecs between them.", "During these last $\\sim $ 100 Myr before BH merger, BHs double their masses (Roskar et al.", "2015, Tamburello et al.", "2016).", "Assuming that Salpeter time is $\\sim $ 50 Myr, corresponding Eddington ratio is 0.35.", "This would mean that 10$^6\\:{\\rm M_{\\odot }}$ BH accretes at rate of 0.1 $\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ , while 10$^7\\:{\\rm M_{\\odot }}$ BH accretes at the rate of 1 $\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ (over 100 Myr).", "This “BHB accretion” phase finishes with BH binary coalescence into a new BH which enters an AGN phase.", "Unknown IMF for BH seeds, and rate of “pre-BHB accretion” are the main sources of uncertainty in our modelling.", "We overcome this issue by considering two models with the idea of constraining lower and upper end of possible initial BH mass.", "Our lower constraint model (M1) assumes log-normal IMF for BH seeds in the interval log (M$_{\\rm BH}/\\:{\\rm M_{\\odot }}$ ) = [4.5, 5.5] centered at 10$^5\\:{\\rm M_{\\odot }}$ , and pre-BHB accretion with a rate of $\\dot{\\rm m}$ = 10$^{-4}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ from z$_{\\rm initial}$ until z$_{\\rm AGN}$ minus 100 Myr.", "For a higher constraint model M2 we set log (M$_{\\rm BH}/\\:{\\rm M_{\\odot }}$ ) = [5.0, 6.0] centred at 10$^{5.5}\\:{\\rm M_{\\odot }}$ and $\\dot{\\rm m}$ = 10$^{-3}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ .", "We seed DMHs with BHs by randomly choosing BH masses from these IMFs in Monte Carlo realisations.", "In the last 100 Myr before z$_{\\rm AGN}$ , we double the BH mass (BHB accretion).", "If DMH already had a major merger in its history, we assume it hosts an elliptical galaxy.", "BH at the centre of an elliptical galaxy scales with the properties of the stellar spheroid but also with the mass of the host DMH.", "We use Ferrarese 2002 M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation with $\\pm $ 10$\\%$ scatter to seed elliptical galaxies with central BHs.", "Mergers of two elliptical galaxies do not trigger AGN activity (dry mergers).", "If an elliptical galaxy merges with a spiral, there is no pre-BHB or BHB accretion onto the BH at the centre of the elliptical galaxy.", "Initial mass of the BHs in both halos (M$_{\\rm BH, 1}$ and M$_{\\rm BH, 2}$ ) combined with pre-BHB and BHB accretion (if galaxy is spiral) produces M$_{\\rm BH^{\\prime }, 1}$ and M$_{\\rm BH^{\\prime }, 2}$ .", "Initial BH mass that enters AGN phase is then: M$_{\\rm BH,initial}$ = M$_{\\rm BH^{\\prime },1}$ + M$_{\\rm BH^{\\prime },2}$ ." ], [ "AGN phase", "At $z_{\\rm AGN}$ , initial BHs merge, form new BH (M$_{\\rm BH,initial}$ ).", "Mass of the DMH hosting an AGN is then M$_{\\rm H, AGN}$ .", "Accretion onto M$_{\\rm BH,initial}$ starts first with the pre-peak phase at super Eddington rate ($\\lambda $ = 3) followed by the declining phase best described by Figure 2 in Shen 2009.", "We assume that AGN reaches its peak activity at $z_{\\rm AGN}$ .", "Note that in Shen 2009 model, AGN activity starts at the time when halos merge (not galaxies).", "Hence, the AGN activity in their model is pushed toward slightly higher redshifts.", "We find that the typical delay between halo merger and consecutive galaxy merger is $\\Delta $ z = 0.2 in redshift space and it does not impact overall results.", "We adopt Shen 2009 model for AGN activity in field galaxies.", "We consider halos in mass range 3 $\\times $ 10$^{11}$ h$^{-1}\\:{\\rm M_{\\odot }}$ $<$ M$_{\\rm H, AGN}$ $<$ 10$^{12}$ (1+z$_{\\rm AGN})^{3/2}$ h$^{-1}\\:{\\rm M_{\\odot }}$ .", "If halo mass is too small mass, AGN activity can not be triggered, while overly massive halos can not cool gas efficiently and BH growth halts (especially at low redshift) (Shen 2009).", "This excludes high density environments (e.g galaxy clusters) from our model and we are left with the AGN activity in the field.", "We do find that increasing the upper limit on host halo mass overpredicts the AGN luminosity functions at low redshift (z $\\le $ 1).", "Mass of the galaxy hosting an AGN is M$_{\\rm ,AGN}$ .", "Since the topic of this paper is to examine merger driven AGN activity in massive galaxies in the field, we consider galaxies with log(M$_{,\\rm AGN}$ /$\\:{\\rm M_{\\odot }}$ ) $>$ 10.4.", "In lower mass galaxies, SMBHs are more likely to accrete through secular processes related to channeling of the gas through bars or disk instabilities.", "Galaxy mass is obtained by using Rodriguez-Puebla et al.", "2015 M$_$ - M$_{\\rm DMH}$ relation for early type (elliptical) galaxies (equations 17 and 18 and Figure 5 in their paper) with scatter $\\sigma _{\\rm r}$ = $\\pm $ 0.136 dex (equation 37 in Rodriguez-Puebla et al.", "2015).", "Scatter determines galaxy mass in every Monte Carlo realisation." ], [ "Halos, galaxies, black holes: Final values", "M$_{\\rm H, final}$ is the mass of the postmerger halo (immediately after the AGN phase) hosting final (relic) SMBH.", "We chose to define the time z$_{\\rm final}$ to be $\\sim $ 100 Myr after AGN phase z$_{\\rm AGN}$ , located in the first consecutive snapshot.", "However, mass of the postmerger halo changes insignificantly in more than one snapshot after z$_{\\rm AGN}$ .", "In fact, our results do not change even when we use M$_{\\rm H, AGN}$ instead of M$_{\\rm H, final}$ .", "This occurs because at the time of galaxy merger, new halo has already formed and for some time after the AGN phase it stays intact.", "Later, it continues growing by minor mergers and diffuse matter accretion.", "This implies that at first, mass of the final (relic) SMBH (M$_{\\rm BH, final}$ ) hosted by M$_{\\rm H, final}$ will be overestimated when compared to the local Ferrarese M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation.", "As M$_{\\rm H, final}$ grows in mass over time, M$_{\\rm BH, final}$ - M$_{\\rm DMH, final}$ relation approaches Ferrarese relation.", "Since M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation is expected to be non-evolving (Gaskell 2009, Shankar, Bernardi $\\&$ Haiman 2009, Salviander, Shields $\\&$ Bonning 2015, Shen et al.", "2015), one can find M$_{\\rm BH, final}$ in M$_{\\rm DMH, final}$ at any redshift.", "First, one can rewrite equation (3) in Ferrarese 2002 as: $\\frac{\\rm V_{{\\rm vir}}}{\\rm 200 km s^{-1}} = (\\frac{\\rm M_{H, final}}{2.7 \\times 10^{12} \\:{\\rm M_{\\odot }}})^{1/3} ,$ Next, $\\sigma _{\\rm sph}$ = C $\\times $ V$_{\\rm vir}$ .", "And from equation (7) in Kormendy $\\&$ Ho 2013: $\\frac{\\rm M_{{\\rm BH, final}}}{\\rm 10^9 \\:{\\rm M_{\\odot }}} = 0.309 (\\frac{\\rm \\sigma _{sph}}{200 {\\rm km s^{-1}}})^{4.38} ,$ with scatter $\\sigma $ = $\\pm $ 0.28 dex.", "We find that for C = 0.77, our M$_{\\rm BH, final}$ - M$_{\\rm DMH, final}$ relation at z=0 matches local Ferrarese relation.", "As we go toward higher redshifts, Ferrarese relation evolves (figure 6) and M$_{\\rm BH, final}$ is overestimated while M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ does not evolve." ], [ "Finding Best Fit Model", "Now that we have obtained M$_{\\rm BH, initial}$ and M$_{\\rm BH, final}$ , we can calculate L$_{\\rm peak}$ necessary to produce M$_{\\rm BH, final}$ .", "As already mentioned in section 2.5, we use Shen 2009 AGN light curve with pre-peak exponential growth phase followed by post peak power-law decline.", "To calculate L$_{\\rm peak}$ we rewrite equation (29) in Shen 2009: ${\\rm L_{peak}} = 3{\\rm M_{BH, final}}{l_{\\rm Edd}} (1-\\frac{2{\\rm lnf}}{3})^{-1} ,$ and $f = \\frac{3 l_{\\rm Edd} {\\rm M_{BH, initial}}}{{\\rm L_{peak}}}$ where $l_{\\rm Edd}$ =1.26 $\\times $ 10$^{38}{\\rm ergs}^{-1}\\:{\\rm M_{\\odot }}^{-1}$ .", "The descending phase is presented by equation 24 in Shen 2009: ${\\rm L(L_{peak}, t)} = {\\rm L_{peak}} (\\frac{\\rm t}{\\rm t_{peak}})^{-\\rm \\alpha } ,$ where $\\alpha $ = 2.5.", "Luminosities of all AGNs in all galaxies and at all redshifts decrease three orders of magnitude from their peak luminosity in $\\sim $ 2 Gyr.", "We use M$_{\\rm BH, initial}$ , M$_{\\rm BH, final}$ , and L$_{\\rm peak}$ to calculate AGN luminosity function, active SMBH mass function, duty cycle, and bias factor.", "We compare these to the observed values.", "We find that both M1 and M2 models reproduce observations without any additional modelling or parameter fixing.", "We continue with M1 and M2 as our best fit models and later replace L$_{\\rm peak}$ with L$_{\\rm COSMOS}$ to find the AGN activity phase in COSMOS survey." ], [ "Matching COSMOS AGNs to M$_{\\rm ,AGN}$", "For $\\sim $ 1700 AGNs in COSMOS field Bongiorno et al.", "2012 presented probability of a galaxy of a certain mass to host an AGN of a given luminosity as a function of stellar mass in three redshift bins: [0.3 - 0.8], [0.8 - 1.5], and [1.5 - 2.5] (F14).", "They group AGNs in four X-ray (2 - 10 KeV) luminosity bins in logarithm space: [42.8 - 43.5], [43.5 - 44.0], [44.0 - 44.5], and [44.5 - 46.0] (erg/s units).", "Masses of their AGN hosting galaxies are also separated in logarithmic bins: [9.0, 10.0], [10.0, 10.4], [10.4, 10.7], [10.7, 10.9], [10.9, 11.2] ($\\:{\\rm M_{\\odot }}$ units).", "How to pick a luminosity from F14 and assign it to our M$_{\\rm BH,initial}$ ?", "We do this by grouping our simulated galaxies at the moment their M$_{\\rm BH,initial}$ should start accreting.", "We determine z$_{\\rm AGN}$ , M$_{\\rm BH,initial}$ , and M$_{\\rm , AGN}$ for every merger in our simulation and group them into redshift bins: $\\Delta \\rm z = [0.3 - 0.8; 0.8 - 1.5; 1.5 - 2.5],$ and galaxy log-mass bins: $\\Delta \\rm M_{\\rm } = [10.4 - 10.7; 10.7 - 10.9; 10.9 - 11.2],$ since we study AGNs in massive galaxies only.", "Figure: Black hole mass function at three redshifts z = [2.00, 1.25, 0.75],for active black holes only, in AGNs with log L X _{\\rm X} [ erg /s]\\rm {[erg/s]} ≥\\ge 43,where X = [2 - 10] KeV.", "Horizontal and vertical bars show the full range ofmasses in our Monte Carlo realisations.", "Dotted, blue line shows our BH mass functionfor all BHs.", "Overplotted as thick black lineis active BH mass function for the same luminosity range from observations (HELLAS2XMM)of La Franca et al.", "2005 (presented in Fiore et al.", "2012).", "Also, in dashed redline is local BH mass function for all black holes (Merloni &\\& Heinz 2008).Our best fit model is a good match to the observations.In this manner we obtain nine $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ intervals.", "The number of galaxies belonging to each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval is N$_{\\rm , AGN}$ .", "Next we match these $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ intervals to the $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ intervals in F14.", "According to F14, galaxies can host AGNs with luminosities in intervals: $\\Delta \\rm L_{\\rm X} = [42.8 - 43.5; 43.5 - 44.0; 44.0 - 44.5; 44.5 - 46.0].$ So the BHs in N$_{\\rm , AGN}$ simulated galaxies can be assigned with any of the luminosities from $\\Delta $ L$_{\\rm X}$ intervals.", "How these luminosities should be assigned is determined by the probability P$_{\\rm AGN,i}$ (data points in F14) defined for every $\\Delta $ L$_{\\rm X, i}$ .", "P$_{\\rm AGN,i}$ in F14 tells us that every galaxy in a specific $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval is more likely to host low luminosity AGN.", "Since the number of galaxies in each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval is N$_{\\rm , AGN}$ , then the number of times a luminosity should be drawn from each luminosity interval $\\Delta $ L$_{\\rm X, i}$ is: $N_{\\rm L, i} = \\frac{P_{\\rm AGN,i}}{\\Sigma P_{\\rm AGN,i}} \\times N_{\\rm , AGN} ,$ Largest N$_{\\rm L,i}$ is for the interval $\\Delta $ L$_{\\rm X, i}$ with smallest luminosities.", "The sum of N$_{\\rm L,i}$ is equal to N$_{\\rm , AGN}$ .", "Next we randomly draw luminosities N$_{\\rm L,i}$ times from every corresponding $\\Delta $ L$_{\\rm X, i}$ and we randomly assign them to N$_{\\rm , AGN}$ galaxies.", "Figure: AGN luminosity function at three redshifts z = [2.00, 1.25, 0.75].Horizontal and vertical bars present the full range in our best fit model.Overplotted as thick black line is AGN luminosity function from observations(HELLAS2XMM) of La Franca et al.", "2005 (presented in Fiore et al.", "2012).In dashed red line is AGN luminosity function from a large combination ofX-ray surveys including XMM and Chandra COSMOS survey (Miyaji et al.", "2015).Our best fit model is a good match to the observations although we slightlyunderpredict luminosity function toward lower redshifts.This is the first out of 40,000 Monte Carlo realisations where we draw luminosity values to be assigned to AGNs in each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval.", "Thus, for each M$_{\\rm , AGN}$ we have a set of 40,000 COSMOS AGN luminosities.", "Since these are X-ray luminosities, we use equation 2) in Hopkins et al.", "2007 to calculate bolometric luminosities.", "We address these luminosities as L$_{\\rm COSMOS}$ .", "Next, we replace L$_{\\rm peak}$ in our best fit model with L$_{\\rm COSMOS}$ ." ], [ "Modelling SMBH growth in COSMOS", "With calculated M$_{\\rm BH,initial}$ (mass of the BH entering AGN phase) and L$_{\\rm COSMOS}$ (COSMOS bolometric AGN luminosity) we have two input parameters for Shen 2009 SMBH growth model.", "Evolution of AGN luminosities follows a universal general form of light curve with an initial exponential growth (pre-peak accretion) at constant Eddington ratio $\\lambda $ = 3 until it reaches L$_{\\rm peak}$ , followed by a power-law decay.", "We replace L$_{\\rm peak}$ with L$_{\\rm COSMOS}$ .", "Note that there are two sets of 40,000 M$_{\\rm BH,initial}$ and L$_{\\rm COSMOS}$ for each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval, obtained from Monte Carlo realisations in two models: M1 (lower range of seed BH masses) and M2 (upper range of seed BH masses).", "After applying best fit parameters of Shen 2009 to their equation 29, predicted SMBH mass (M$_{\\rm BH, predicted}$ ), after AGN phase, can be written as: ${\\rm M_{BH, predicted}} = \\frac{{\\rm L_{\\rm COSMOS}}}{3l_{\\rm Edd}} (1-\\frac{2{\\rm lnf}}{3}) ,$ and $f = \\frac{3 l_{\\rm Edd} {\\rm M_{BH, initial}}}{{\\rm L_{COSMOS}}}$ where $l_{\\rm Edd}$ =1.26 $\\times $ 10$^{38}{\\rm ergs}^{-1}\\:{\\rm M_{\\odot }}^{-1}$ .", "Figure: AGN duty cycle as a function of stellar mass at three redshiftsz = [2.00, 1.25, 0.75].", "We consider AGNs with log L X _{\\rm X} [erg/s] ≥\\ge 43,where X = [2 - 10] KeV.", "Horizontal and vertical bars show the full rangefor duty cycle in our Monte Carlo realisations.", "Overplotted as thick blackline is duty cycle for the same luminosity range from observations(HELLAS2XMM) of La Franca et al.", "2005 (presented in Fiore et al.", "2012).Our best fit model is a good match to the observations although weslightly underpredict duty cycle toward lower redshifts.Through Monte Carlo realisations we take into account: all possible seed values that could be assigned to the merging DMHs; all possible luminosities in each luminosity bin of Bongiorno et al.", "2012; scatter in Ferrarese 2002 M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation; scatter in Shen 2009 M$_{\\rm BH, final}$ - M$_{\\rm DMH, post}$ relation; and scatter in Rodriguez-Puebla et al.", "2015 M$_$ - M$_{\\rm DMH}$ relation.", "In the last mentioned scatter, same halo can host a galaxy below or above log(M$_{}$ /$\\:{\\rm M_{\\odot }}$ ) = 10.4.", "As the result, depending on the random draw from the scatter in each Monte Carlo realisation, some halos might drop from the analysis while others might join.", "At the end we have M$_{\\rm BH,final}$ from our best fit model and in 40,000 Monte Carlo realisations we produce M$_{\\rm BH,predicted}$ in each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval.", "Now we can compare these two masses.", "If the observed AGN luminosities are indeed the peak luminosities when most of the SMBH growth occurs, then the mass of the predicted SMBH should match the mass of the final SMBH.", "We calculate the percentage of realisations when this condition is met." ], [ "Best fit model", "We apply Shen 2009 major merger driven AGN activity model to the merger trees in cosmological N-body simulation.", "There are some differences between Shen 2009 and our model.", "While Shen 2009 assumes constant ratio of 10$^{-3}$ between initial and peak BH mass, we seed DMHs with BH seeds and follow their evolution before AGN phase.", "Hence, BH mass right before AGN phase is not necessarily a constant fraction of the peak BH mass.", "Also, we calibrate final SMBH mass at redshift z = 0 to the local Ferrarese relation.", "In this manner, SMBH mass is overestimated at high redshift to accommodate for the late DMH evolution.", "In our model, super-Eddington accretion starts when galaxies merge while in Shen 2009 model same occurs when DMHs merge.", "Our best fit model for the SMBH growth reproduces observed AGN luminosity function, SMBH mass function, duty cycle, and bias.", "Both M1 and M2 models can be considered as best fit models.", "M2 model provides a slightly better fit to the observations hence we show this match for M2 model only.", "Figure: AGN bias factor at three redshifts z = [2.0, 1.5, 1.0].Horizontal and vertical bars show the full range for AGN biasin our Monte Carlo realisations.", "Thick black line shows the best fit modelin Shen 2009.", "Points are measurements from Croom et al.", "(2005, green-crosses),Porciani &\\& Norberg (2006, green-star), Shen et al.", "(2009, green-open square),da Angela et al.", "(2008, green-circles), Myers et al.", "(2007, green-squares),and Allevato et al.", "2011 (blue triangle).", "Red squares are from the semianalytic model of galaxy formation in Gatti et al.", "2016.Considering the uncertainties in determining AGN bias, our best fit modelis a good match to the observations.Figure 2 shows BH mass function at three redshifts z = [2.00, 1.25, 0.75].", "Horizontal and vertical bars show the full range for BH mass function, in our Monte Carlo realisations, for active black holes only, in AGNs with log L$_{\\rm X}$ [erg/s] $\\ge $ 43, where X = [2 - 10] KeV.", "Dotted, blue line shows our BHs mass function for all BHs.", "Overplotted as thick black line is active BH mass function for the same luminosity range from observations (HELLAS2XMM) of La Franca et al.", "2005 (presented in Fiore et al.", "2012).", "Our best fit model follows the observed mass functions for active BHs.", "We slightly overestimate masses of active BHs at z = 2.", "This effect transfers to the lower redshift where our BH mass function for all BHs (dotted, blue) slightly overpredicts the local BH mass function at $\\sim $ 10$^8\\:{\\rm M_{\\odot }}$ (dashed, red, Merloni $\\&$ Heinz 2008).", "At larger BH masses our model underpredicts local BH mass function.", "We find that this occurs due to the arbitrary cut off at the higher mass end for halos capable of hosting AGNs.", "When this upper limit for halo mass is doubled, we get a perfect match to the local BH mass function for M$_{\\rm BH}$ $>$ 10$^8\\:{\\rm M_{\\odot }}$ .", "However, at the same time, we overpredict AGN luminosity function at z$<$ 1 and log L$_{\\rm X} \\rm [erg/s]$ $>$ 44 by a factor of 4.", "Similarly to Shen 2009, our model is incomplete at M$_{\\rm BH}$ $\\le $ 10$^{7.5} \\:{\\rm M_{\\odot }}$ because we did not include contributions from AGNs triggered by secular processes or minor mergers.", "Figure 3 shows AGN luminosity function with horizontal and vertical bars presenting the full range in our best fit model.", "Overplotted as thick black line is AGN luminosity function from same observations as in figure 2.", "Our best fit model deviates from the observations at z = 2 and z = 0.75.", "However, AGN luminosity functions reported in the literature deviate between various surveys.", "This can be seen when we overplot AGN luminosity function (dashed red line in Figure 3) from a large combination of X-ray surveys including XMM and Chandra COSMOS survey (Miyaji et al.", "2015).", "Discrepancy between Fiore et al.", "2012 and Miyaji et al.", "2015 is comparable to the discrepancy between our best fit model and these observations.", "Figure 4 shows AGN duty cycle as a function of stellar mass at three redshifts z = [2.00, 1.25, 0.75].", "We consider AGNs with log L$_{\\rm X}$ [erg/s] $\\ge $ 43, where X = [2 - 10] KeV.", "Horizontal and vertical bars show the full range for the duty cycle in our Monte Carlo realisations.", "Observations are again from La Franca et al.", "2005 and Fiore et al.", "2012.", "Our best fit model is a good match to the observations.", "Figure 5 shows AGN bias factor at three redshifts z = [2.0, 1.5, 1.0].", "We have calculated AGN bias factor by using equations (3), (4), and (5) in Cappelluti, Allevato $\\&$ Finoguenov 2012.", "From these equations, AGN bias in a luminosity and redshift range $\\Delta \\rm L, \\Delta z$ can be written as: $\\rm bias(\\Delta \\rm L, \\Delta z) = \\frac{\\Sigma \\rm b_{\\rm DMH}(\\Delta \\rm L, \\Delta z)}{N_{\\rm AGN}(\\Delta \\rm L, \\Delta z)} ,$ where b$_{\\rm DMH}$ ($\\Delta \\rm L, \\Delta z$ ) is the large scale bias of dark matter halos which host AGNs in the luminosity and redshift range $\\Delta \\rm L, \\Delta z$ , and N$_{\\rm AGN}$ ($\\Delta \\rm L,\\Delta z$ ) is the total number of AGNs hosted by DMHs in the luminosity and redshift range $\\Delta \\rm L, \\Delta z$ .", "We obtain b$_{\\rm DMH}$ from figure 11 in Allevato et al.", "2011.", "Figure: M BH _{\\rm BH} - M DMH _{\\rm DMH} relation in our bestfit model.", "Red lines show full range of Monte Carlorealisations at redshift z = 2; blue lines represent z = 1;and green lines z = 0.", "Thick black line shows local Ferrareserelation at z = 0.", "Figure shows how M BH _{\\rm BH} - M DMH _{\\rm DMH}relation evolves into local Ferrarese relation as dark matterhalos grow in mass.Horizontal and vertical bars in figure 5 show the full range for AGN bias in our Monte Carlo realisations.", "Thick black line shows the best fit model in Shen 2009.", "Points are measurements from Croom et al.", "(2005, green-crosses), Porciani $\\&$ Norberg (2006, green-star), Shen et al.", "(2009, green-open square), da Angela et al.", "(2008, green-circles), Myers et al.", "(2007, green-squares), and Allevato et al.", "2011 (blue triangle).", "Red squares are from the semi analytic model of galaxy formation in Gatti et al.", "2016.", "AGN bias substantially differs from bias in Shen 2009 at low luminosities.", "This flattening could be induced by the Monte Carlo approach as it includes the broad scatters in calculated parameters.", "Considering the uncertainties in determining AGN bias, our best fit model is a good match to the observations.", "Figure: Top panel: M BH _{\\rm BH} - M * _* relation in our best fit model.Horizontal and vertical bars show full range of Monte Carlo realisationsat redshift z = 0.", "Black line shows Kormendy &\\& Ho 2013 relation,and dashed green line shows Merloni &\\& Heinz 2008 relation.Bottom panel: evolution of scatter in M BH _{\\rm BH} - M * _* relation inour best fit model (dotted blue line for z = 2; dashed red line for z = 1;and thick black line for z=0).Figure 6 shows M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation in our best fit model.", "Red lines show full range of Monte Carlo realisations at redshift z = 2; blue lines represent the same at z = 1; and green lines at z = 0.", "Thick black line shows local Ferrarese relation at z = 0 and it matches our best fit model at z = 0 by the default since we calibrate our model to do exactly that.", "We find that this match occurs when $\\sigma _{\\rm sph}$ = 0.77 $\\times $ V$_{\\rm vir}$ .", "Our model incorporates no evolution in M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation.", "It overpredicts BH mass at high redshift as BHs grow faster than DMHs.", "Figure 6 shows how M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation evolves into local Ferrarese relation as dark matter halos grow in mass and “catch up” to the BH growth.", "Figure 7 shows M$_{\\rm BH}$ - M$_$ relation in our best fit model (top panel).", "Horizontal and vertical bars show full range of Monte Carlo realisations at redshift z = 0.", "Black line shows Kormendy $\\&$ Ho 2013 relation, and dashed green line shows Merloni $\\&$ Heinz 2008 relation.", "Our best fit model underpredicts BH masses when compared to Kormendy $\\&$ Ho 2013 relation.", "However, the match is better when compared to Merloni $\\&$ Heinz 2008 relation.", "We also find no evolution of scatter in M$_{\\rm BH}$ - M$_$ relation in our best fit model (bottom panel in figure 7).", "Despite the BH mass being determined by $\\sigma _{\\rm sph}$ via V$_{\\rm vir}$ , and the scatter in M$_$ at fixed M$_{\\rm DMH}$ is very small, the resulting M$_{\\rm BH}$ - M$_$ relation of figure 7 is very broad and even significantly below the Kormendy $\\&$ Ho 2013 relation.", "This might be in support of the biases in the local scaling relations of BHs and galaxies discussed recently in the literature (Reines $\\&$ Volonteri 2015, van den Bosch et al.", "2015, Shankar et al.", "2016, Greene et al.", "2016, van den Bosch 2016)." ], [ "Determining AGN activity phase", "Assuming that major mergers are driving AGN activity in massive galaxies, we have selected simulated mergers of field galaxies in the redshift range 0.3 $<$ z $<$ 2.5 and matched them to the observed samples of AGNs in redshift and galaxy-mass bins in F14.", "Matching procedure briefly consists of: As halo merger finishes, galaxy merger starts.", "We define that as a time of AGN peak activity corresponding to the AGN observed in COSMOS survey.", "The mass of simulated galaxy hosting the AGN is derived from halo-galaxy scaling relation.", "Once we find redshift bin and galaxy-mass bin of the simulated merger, we trace the merging halos before the merger, and we trace merger remnant after the merger.", "We determine initial SMBH mass (before accretion during AGN phase) and final (“true”) SMBH mass (after accretion in AGN phase).", "Figure 8 shows BHs mass function in model M1 (left panels) and M2 (right panels) for both initial BHs M$_{\\rm BH, initial}$ (thick, black line) and for the final BHs M$_{\\rm BH, final}$ (thin, red line), in the three redshift ranges: top panel: 0.3 $<$ z $<$ 0.8; middle panel: 0.8 $<$ z $<$ 1.5; bottom panel: 1.5 $<$ z $<$ 2.5.", "Bars represent full range of Monte Carlo realisations which cover possible M$_{\\rm BH, initial}$ from log-normal distribution defined for massive seed BHs in spiral galaxies; scatter in Ferrarese 2002 M$_{\\rm BH}$ - M$_{\\rm DMH}$ relation for BHs at the centres of elliptical galaxies; and scatter in Kormendy $\\&$ Ho 2013 M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation for the final “true” BH mass in postmerger halos.", "It also covers (not that obvious) scatter in Rodriguez-Puebla et al.", "2015 M$_$ - M$_{\\rm DMH}$ relation applied as a selection criterion for elliptical galaxies at z$_{\\rm AGN}$ .", "Because of this scatter, same halo can host a galaxy below or above log(M$_{}$ /$\\:{\\rm M_{\\odot }}$ ) = 10.4.", "As the result, depending on the random draw from the scatter in each Monte Carlo realisation, some halos might drop from the analysis while others might join.", "Premerger accretion occurs between z$_{\\rm initial}$ and z$_{\\rm AGN}$ and it consists of two phases: first, the pre-BHB phase before initial BHs form binary; and second, BHB phase which lasts for $\\sim $ 100 Myr before BHs merge as BHs in the binary overcome last couple of kiloparsecs.", "The typical pre-BHB timescale for BHs in spiral galaxies is $\\sim $ 2 Gyr.", "Since accretion rate in model M1 is set to 10$^{-4}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ the amount of mass accreted then during this phase is $\\sim $ few $\\times $ 10$^5\\:{\\rm M_{\\odot }}$ .", "During BHB phase BHs double their masses.", "After adding mass from both pre-BHB and BHB phases to the seed BHs in spiral galaxies their mass function peaks at 10$^{5.5}\\:{\\rm M_{\\odot }}$ - 10$^{6}\\:{\\rm M_{\\odot }}$ depending on the redshift (figure 8, left panels).", "On the other hand, mass function of the initial BHs in the elliptic galaxies peaks at 10$^{6.5}\\:{\\rm M_{\\odot }}$ - 10$^{7}\\:{\\rm M_{\\odot }}$ (figure 8, left panels).", "In model M2, the pre-BHB accretion is set to 10$^{-3}\\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ so the amount of mass accreted during this phase is $\\sim $ few $\\times $ 10$^6\\:{\\rm M_{\\odot }}$ .", "After both pre-BHB and BHB phases, and after adding the accreted mass to the seed BHs in spiral galaxies, their masses overlap with the masses of BHs in elliptical galaxies.", "Resulting mass function peaks at 10$^{6.5}\\:{\\rm M_{\\odot }}$ - 10$^{6.9}\\:{\\rm M_{\\odot }}$ depending on the redshift (figure 8, right panels).", "The difference in mass function between initial and final BHs in figure 8, is the accreted mass during AGN phase in our best fit model where L$_{\\rm peak}$ is calculated from the light curve model in Shen 2009.", "M$_{\\rm final}$ obtained in this manner (the “true” final BH mass) is then compared to M$_{\\rm predicted}$ which is obtained by replacing L$_{\\rm peak}$ with AGN luminosities from COSMOS survey L$_{\\rm COSMOS}$ .", "All galaxies in the specific mass range, host AGNs with the probability defined in F14.", "Probability functions presented in F14 show that galaxies are more likely to host less luminous AGNs.", "As the observed AGN luminosity increases, the probability for that particular AGN to be observed in the COSMOS galaxy decreases.", "We incorporate COSMOS AGN luminosities into Shen 2009 model for SMBH growth.", "We perform 40,000 Monte Carlo realisations for every M$_{\\rm initial}$ in each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval, through all possible COSMOS AGN luminosities.", "As the result, we obtain 40,000 predicted BH masses which we compare to the “true” final BH mass.", "When M$_{\\rm predicted}$ $\\ge $ M$_{\\rm final}$ , COSMOS AGN luminosity is the peak AGN luminosity, corresponding to the AGN luminosity in our best fit model.", "We calculate the percentage of realisations where M$_{\\rm predicted}$ is at least as large as M$_{\\rm final}$ and present it in figure 9.", "Figure: BH mass function for model M1 with bars representingfull range of Monte Carlo realisations.", "M BH , initial _{\\rm BH, initial}represented in thick, black.", "M BH , final _{\\rm BH, final} represented inthin, red.", "Three redshift ranges: top panel: 0.3 << z << 0.8;middle panel: 0.8 << z << 1.5; bottom panel: 1.5 << z << 2.5.BH mass function for model M2.Figure 9 shows the probability function (occupation fraction) that the observed AGNs are at their peak activity.", "Nine panels present three redshift ranges and three galaxy log-mass ranges.", "Thick (black) bars represent probability functions in our model M1.", "Thin (red) bars represent probability functions in our model M2.", "Bars show full range of Monte Carlo realisations.", "Probability for peak AGN activity at low redshift z=[0.3, 0.8] is small in all galaxy mass bins.", "For M$_$ = [10.4, 10.7] all AGNs have probability $<$ 20 $\\%$ in model M1, and $<$ 30 $\\%$ in model M2.", "In the same $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval, probability of $<$ 10 $\\%$ have 90 - 100 $\\%$ of AGNs in M1 and 80 - 90 $\\%$ of AGNs in M2.", "The occupation fraction of AGNs with low probability for peak activity increases toward larger galaxy mass.", "AGNs hosted by most massive galaxies (M$_$ = [10.9, 11.2]) are all in the declining phase of their activity since probability drops to $<$ 20 $\\%$ in both models (top, right panel in figure 9).", "Overall, at low redshift, almost all AGNs are in non-star-forming Red Sequence galaxies.", "Increase in fraction of AGNs with larger probability means more AGNs are in star forming Green Valley galaxies.", "We see this trend as we go from low to high redshift in figure 9.", "At the intermediate redshifts (z=[0.8, 1.5]) AGN fraction with larger probability increases in both models (middle panels in Figure 9).", "As expected, this increase is larger for M2 where M$_{\\rm BH, initial}$ is larger.", "Still, most AGNs have low probability for being observed at their peak.", "AGNs at high redshifts (z=[1.5, 2.5]), and in the lowest galaxy mass range (M$_$ = [10.4, 10.7], bottom, left panel in figure 9) are dominantly at the peak activity since 30 - 50 $\\%$ of them in M1 and 55 - 75 $\\%$ in M2 have $>$ 80 $\\%$ probability for being at the peak.", "Overall, distribution of occupation fractions shifts toward larger probabilities.", "Similarly to lower redshifts, occupation fraction with large probabilities decreases toward more massive galaxies.", "For M$_$ = [10.7, 10.9] (bottom, middle panel in figure 9), AGN fraction is evenly distributed.", "Here we would expect to see comparable numbers of AGNs in both quiescent and star forming galaxies.", "In the largest mass range (panels on the right of Figure 9) AGNs are predominantly in quiescent galaxies at all redshifts.", "So the trend that emerges in figure 9 is that quiescent galaxies host almost all AGNs at low redshift.", "As we go toward higher redshift there are more AGNs in star forming galaxies and the percentages of AGNs inhabiting quiescent galaxies and star forming galaxies become comparable.", "We also see the trend with increasing galaxy mass.", "At larger galaxy masses there are more AGNs in quiescent galaxies.", "This exact trend we see in AGNs in COSMOS survey (Figures 12, 13 and 18, Bongiorno et al.", "2012).", "Figure: Probability function for the predicted SMBH massto be at least as large as the true SMBH mass.", "In other words,probability that the observed AGN luminosity is large enough toaccount for the final SMBH mass.", "Probability functions are splitinto redshift bins and galaxy mass bins which correspond tonomenclature in Bongiorno et al.", "2012.", "Thick (black) histogramsrepresent probability functions in our model M1.", "Thin (red)histograms represent probability functions in our model M2." ], [ "DISCUSSION AND CONCLUSIONS", "We ran cosmological (130 Mpc box) N-body (dark matter only) simulation from which we located field DMHs at all redshifts.", "We also followed their evolution while they stay in the field.", "We found merger events and traced merger progenitors and merger remnants.", "Through scaling relations we calculated SMBH masses for progenitors and remnants.", "In this manner we obtain the SMBH mass at the centres of DMHs before (initial SMBH) and after (final SMBH) the merger.", "We assume that at the time when halo merger finishes, galaxy merger starts.", "At that time newly formed SMBH ignites as AGN and quickly reaches its peak activity.", "We focus on two models with different range for the initial BH mass since BH seeds in spiral galaxies and there pre-coalescence growth are the source of largest uncertainty in our modelling.", "Model M1 has a lower mass range $\\sim $ [10$^5$ - 10$^6$ ] $\\:{\\rm M_{\\odot }}$ and pre-coalescence accretion rate of 10$^{-4} \\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ .", "Model M2 has a larger initial mass range $\\sim $ [10$^{5.5}$ - 10$^{6.5}$ ] $\\:{\\rm M_{\\odot }}$ and accretion rate of 10$^{-3} \\:{\\rm M_{\\odot }}{\\rm yr^{-1}}$ .", "We determine “true” final BH mass by using non-evolving M$_{\\rm BH}$ - $\\sigma _{\\rm sph}$ relation where $\\sigma _{\\rm sph}$ = 0.77 $\\times $ V$_{\\rm vir}$ .", "In this manner, M$_{\\rm DMH}$ - M$_{\\rm BH}$ relation evolves from overestimating BHs masses at high redshift to matching local Ferrarese relation at z=0.", "Our best fit model for the SMBH growth reproduces observed AGN luminosity function, SMBH mass function, duty cycle and bias.", "Next, we replace peak AGN luminosities in our best fit model with COSMOS AGN luminosities from Bongiorno et al.", "2012.", "For every galaxy hosting an AGN we determine redshift and mass, and sort them into redshift ranges and mass ranges as in Bongiorno et al.", "2012, COSMOS survey.", "For each mass and redshift, we assign an observed probability function for a galaxy to host an AGN of a certain luminosity (Figure 14 in Bongiorno et al.", "2012).", "Next we ran 40,000 Monte Carlo realisations in each $\\Delta \\rm z$ -$\\Delta \\rm M_{}$ interval where we draw from the observed probability functions and we assign luminosities to the initial BH.", "We obtain 40,000 predicted BH masses which we compare to the “true” final BH mass.", "When M$_{\\rm predicted}$ $\\ge $ M$_{\\rm final}$ , COSMOS AGN luminosity is the peak AGN luminosity, corresponding to the peak AGN luminosity in our best fit model.", "We calculate the percentage of realisations where M$_{\\rm predicted}$ is at least as large as M$_{\\rm final}$ .", "Large percentage implies large probability for AGNs to be at their peak activity.", "Small percentage means that AGNs are not observed at the peak but in the declining phase of their nuclear activity.", "In this manner, we distinguish “peak” AGNs (recently merger triggered and hosted by star forming galaxies, Green Valley) and “faded” AGNs (merger triggered a long time ago and now residing in quiescent galaxies, Red Sequence).", "At low redshift range (z=[0.3, 0.8]) all observed AGNs are in the declining phase of their nuclear activity, fading away (figure 9).", "The probability for being at their peak activity is $<$ 10 $\\%$ for $>$ 90 $\\%$ of AGNs in the most massive galaxies.", "AGN luminosity would have to be very large to account for the SMBH growth.", "But even if this highest possible luminosity was large enough to produce final SMBH, it is also the least probable one.", "Since the entire range of luminosities can not produce final SMBH, then these luminosities do not correspond to the AGN peak activity.", "The time of maximum nuclear activity when most of the mass was accreted has occurred in the past at higher Eddington ratio when AGN luminosity was larger and when AGN was most likely hosted by star-forming galaxy (Green Valley).", "Logical conclusion is that the observed luminosities belong to the AGN in the declining phase of its nuclear activity which places this particular AGN in the quiescent galaxy (Red Sequence).", "Theoretical modelling of AGN populations in hosts of various morphologies, mass ranges and redshifts, support the merger driven scenario for luminous AGN activity.", "At the same time observations are split between existence of merger features (Schawinski et al.", "2010, Smirnova, Moiseev $\\&$ Afanasiev 2010, Koss et al.", "2010, Cotini et al.", "2013) and the lack of them (Gabor et al.", "2009, Darg et al.", "2010, Cisternas et al.", "2011, Kocevski et al.", "2012, Villforth et al.", "2014).", "Villforth et al.", "2014 found no increase in the prevalence of merger signatures with AGN luminosity (in the redshift range z = [0.5, 0.8]) and concluded that major mergers either play only a very minor role in the triggering of AGN in the luminosity range studied (log L$_{\\rm X}$ = [41, 44.5]) or time delays are too long for merger features to remain visible.", "Our model shows that the merger driven scenario is still consistent with the observations even though there are no merger features in massive galaxies hosting low redshift AGNs and almost all of the AGN hosts are quiescent galaxies.", "How can mergers explain AGN activity in massive galaxies which have no merger features and no star formation to indicate recent galaxy merger?", "Since at z = [0.3, 0.8] (figure 9) the observed luminosities can not correspond to AGNs at their peak activity (can not produce final SMBH mass in the simulation), then they must be observed much later in their evolution long after the merger features can be detected.", "And our confirmation of Bongiorno et al.", "2012 results that almost all low redshift AGNs are in quiescent galaxies is a simple consequence of the drop in galaxy merger rates at z $<$ 1.", "Since galaxy merger rates fall dramatically at low redshift, there are very few recently activated AGNs which would be hosted by star forming galaxies.", "So most of the observed AGNs are the fading AGNs activated in the old mergers which occurred at higher redshifts.", "Since there are no new galaxy mergers, almost all observed AGNs are in non-star-forming galaxies.", "As we go toward higher redshifts, the probability for the AGNs being at their peak activity increases.", "There are more AGNs in star-forming Green Valley galaxies.", "At z=[1.5, 2.5] the percentage of AGNs in star forming galaxies is comparable to the percentage of AGNs in quiescent galaxies.", "This can be seen in our figure 9, bottom panels, and in Figure 18 of Bongiorno et al.", "2012.", "Again, this is a simple consequence of the large merger rate in galaxies at high redshift.", "The explanation for comparable number of star forming and quiescent AGN hosts is that AGNs in star forming galaxies at high redshift have “just” been triggered by galaxy mergers while AGNs in quiescent galaxies at the same redshift have been merger triggered at some time in the past.", "Schawinski et al.", "2014 had proposed a split of Green Valley transition into two paths.", "Current understanding is that late type galaxies transition slowly from Blue Cloud to Red Sequence, while hosting low to intermediate luminosity AGNs driven by secular processes.", "Early type galaxies transition fast, while hosting high luminosity AGNs driven by major mergers.", "In the context of galaxy evolution, our model addresses the evolution of early type galaxies which are produced in major mergers of gas-rich disk/spirals.", "These galaxies correspond to the massive, red galaxies in COSMOS survey (Bongiorno et al.", "2012) where they represent the majority of AGN hosting galaxies.", "According to our model, AGNs in massive galaxies of the COSMOS survey, belong to the rapid transition channel (Schawinski et al.", "2014).", "We find that, right after the merger, AGNs reach their peak activity (Green Valley phase).", "This is a short phase ($\\sim $ 100 Myr) during which star formation is quenched.", "Then, galaxies enter Red Sequence phase with AGNs in the decline (or at the end) of their nuclear activity and low Eddington accretion rate observed in COSMOS survey.", "Figure 2 shows that we are sampling the growth of SMBHs $>$ 10$^7 \\:{\\rm M_{\\odot }}$ .", "For the most part, final SMBHs are $>$ 10$^8 \\:{\\rm M_{\\odot }}$ .", "This is consistent with Hopkins, Kocevski $\\&$ Bundy 2014 conclusion that at these masses merger driven AGN activity dominates.", "There are a number of recent papers discussing possible biases in the local scaling relations of BHs and galaxies (Reines $\\&$ Volonteri 2015, van den Bosch et al.", "2015, Shankar et al.", "2016, Greene et al.", "2016, van den Bosch 2016).", "In support of this, we find that when $\\sigma _{\\rm sph}$ is determined via V$_{\\rm vir}$ , the resulting M$_{\\rm BH}$ - M$_$ relation of figure 7 is very broad and even significantly below the Kormendy $\\&$ Ho 2013 relation.", "Shankar et al.", "2016 found that the normalisation of the M$_{\\rm BH}$ - $\\sigma $ relation might be decreased by a factor of 3.", "Revising M$_{\\rm BH}$ - $\\sigma $ relation in our best fit model leads to smaller final BH masses.", "This in turn decreases BH mass functions and AGN luminosity functions by a similar factor but still consistent with the observations.", "We have tested how this fact would influence our results and we found that the probability functions in figure 9 would shift toward higher probabilities but would not qualitatively change our results.", "We conclude that merger driven scenario for AGN activity is consistent with the observations and that the occupation fractions of the observed AGNs simply follow the evolution of galaxy merger rates.", "Our model reproduces the observed trend that quiescent (Red Sequence) galaxies host almost all AGNs at low redshift due to the dramatic drop in galaxy merger rates at z $<$ 1.", "There are just few recently activated AGNs in star forming galaxies.", "Instead, most AGNs are in their declining nuclear activity hosted by quiescent galaxies.", "As we go toward higher redshift (z $>$ 1), galaxy merger rates increase, and there are more peak activity AGNs observed in star forming galaxies.", "The percentage of peak AGNs inhabiting star forming and the percentage of faded AGNs hosted by quiescent galaxies becomes comparable.", "We also confirm the observed trend with increasing galaxy mass.", "At larger galaxy masses there are more AGNs in quiescent galaxies.", "Our method for matching simulated DMH merger events with observations of field AGNs will be more accurate as the statistics improves with the future surveys.", "At this point, limited statistics of the sample prevents more detailed investigations of the incidence of AGN in galaxies as a function of redshift, stellar mass, star-formation rate and nuclear luminosity (Bongiorno et al.", "2012).", "Wide area surveys will be necessary to probe volumes at z $>$ 1 comparable to that explored by SDSS.", "At the low end of the galaxy mass distribution log [10.4, 10.7] $\\:{\\rm M_{\\odot }}$ , and high redshift [1.5, 2.5], the probability functions in Bongiorno et al.", "2012 do not have AGNs with log L$_{\\rm X}$ [erg/s] $\\le $ 44.", "Most likely missed in COSMOS survey, since there are AGNs in this luminosity and galaxy mass range at lower redshifts.", "Including lower luminosities would change probability functions in Figure 14 of Bongiorno et al.", "2012, and would most likely shift AGN occupation fractions toward lower probabilities.", "Even though we enforce criterion on AGN hosts to be $>$ 10$^{10.4} \\:{\\rm M_{\\odot }}$ in stellar mass, we must have some mixing of populations.", "We expect that a large majority of AGNs in these galaxies are merger driven.", "However, some percentage of AGNs is probably driven by secular processes.", "That being said, we would like to point out that we are not trying to show that mergers are definitely responsible for AGN activity.", "We are arguing that observed AGN activity (at least in $>$ 10$^{10.4} \\:{\\rm M_{\\odot }}$ galaxies) is consistent with mergers as drivers.", "However, this does not exclude other mechanisms.", "In fact, one could imagine a scenario where all of the moderate to faint AGNs are secularly driven.", "Two major concerns for our method are: precision in determining the mass of the AGN host galaxy in observations, and detecting low luminosity AGNs.", "Both concerns impact the relations between mass of the host galaxy and probability functions for AGN incidence." ], [ "ACKNOWLEDGMENTS", "Authors would like to thank the anonymous reviewer for the tremendous help with shaping this paper.", "This work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through project no.", "176021, “Visible and Invisible Matter in Nearby Galaxies: Theory and Observations”.", "The author acknowledges the financial support provided by the European Commission through project BELISSIMA (BELgrade Initiative for Space Science, Instrumentation and Modelling in Astrophysics, call FP7-REGPOT-2010-5, contract no.", "256772).", "Numerical results were obtained on the PARADOX cluster at the Scientific Computing Laboratory of the Institute of Physics Belgrade, supported in part by the national research project ON171017, funded by the Serbian Ministry of Education, Science and Technological Development." ] ]	1606.05138
[ [ "On the initial boundary value problem of a Navier-Stokes/$Q$-tensor\n model for liquid crystals" ], [ "Abstract This work is concerned with the solvability of a Navier-Stokes/$Q$-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter.", "We prove the existence of local in time strong solution to the system with the anisotropic elastic energy.", "The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition.", "This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument.", "Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution." ], [ "Introduction", "Nematic liquid crystal is a sort of material which may flow as a conventional liquid while the molecules are oriented in a crystal-like way.", "One of the successful continuum theories modeling nematic liquid crystals is the $Q$ -tensor theory, also referred to as Landau-de Gennes theory, which uses a $3\\times 3$ traceless and symmetric matrix-valued function $Q(x)$ as order parameters to characterize the orientation of molecules near material point $x$ (cf.", "[11]).", "The matrix $Q$ , also called $Q$ -tensor, can be interpreted as the second momentum of a number density function $Q(x)=\\int _{\\mathbb {S}^2} (m\\otimes m-\\frac{1}{3}I_3)f(x,m)\\mathrm {d}m,$ where $f(x, m)$ corresponds to the number density of liquid crystal molecules which orient along the direction $m$ near material point $x$ .", "The configuration space for $Q$ -tensor will be denoted by $\\mathcal {Q}=\\lbrace Q\\in \\mathbb {R}^{3\\times 3}\\mid Q_{ij}=Q_{ji},~\\sum _{i=1}^3Q_{ii}=0\\rbrace .$ If $Q(x)$ has three equal eigenvalues, it must be zero and this corresponds to the isotropic phase.", "When $Q(x)$ has two equal eigenvalues, it can be written as $Q(x)=s(x)\\left(n(x)\\otimes n(x)-\\frac{1}{3}I_3\\right),$ for some $n(x)\\in \\mathbb {S}^2$ and $s(x)\\in \\mathbb {R}$ and it is said to be uniaxial.", "If all three eigenvalues of $Q(x)$ are distinct, it is called biaxial and can be written as $Q(x)=s(x)\\left(n(x)\\otimes n(x)-\\frac{1}{3}I_3\\right)+r(x)\\left(m(x)\\otimes m(x)-\\frac{1}{3}I_3\\right),$ where $n(x),m(x)\\in \\mathbb {S}^2$ and $s(x),r(x)\\in \\mathbb {R}$ .", "The classic Landau-de Gennes theory associates to each $Q(x)$ a free energy of the following form: $\\nonumber \\mathcal {F}( Q ,\\nabla Q )=&\\int _{\\Omega }\\left( \\frac{a}{2}\\operatorname{tr}Q ^2-\\frac{ b}{3}\\operatorname{tr}Q ^3+\\frac{c}{4}(\\mathrm {tr} Q ^2)^2 \\right)\\mathrm {d}x\\\\ \\nonumber &+\\frac{1}{2}\\int _{\\Omega } \\Big (L_1\|\\nabla Q \|^2+L_2Q_{ij,j}Q_{ik,k}+L_3Q_{ij,k}Q_{ik,j}+L_4Q_{\\ell k}Q_{ij,k}Q_{ij,\\ell } \\Big ) \\mathrm {d}x\\\\ =: &~ \\mathcal {F}_b( Q )+\\mathcal {F}_e(\\nabla Q,Q).$ Here and in the sequel, we shall adopt the Einstein's summation convention by summing over repeated greek letters.", "In (REF ), $ \\mathcal {F}_b(Q)$ is the bulk energy, describing the isotropic-nematic phase transition while $\\mathcal {F}_e(\\nabla Q,Q )$ is the elastic energy which characterizes the distortion effect.", "The parameters $a, b,c $ are temperature dependent constants with $b, c >0$ , and $L_1, L_2, L_3, L_4$ are elastic coefficients.", "In the sequel, we shall call $\\mathcal {F}(Q,\\nabla Q)$ isotropic if $L_2=L_3=L_4=0$ and anisotropic if at least one of $L_2, L_3, L_4$ does not vanish.", "This work is devoted to the later case.", "Note that the term in (REF ) corresponding to $L_4$ is cubic and will lead to severe analytic difficulties: it is shown in [4] that $\\mathcal {F}(Q,\\nabla Q)$ with $L_4\\ne 0$ is not bounded from below.", "In this work, we follow [5], [22] and assume: $L_1>0, \\quad L_1+L_2+L_3=: L_0>0, \\quad L_4=0.$ In order to introduce the sytem under consideration, we need some notation.", "For any $Q\\in \\mathcal {Q}$ defined by (REF ), $\\mathcal {S}_Q(M)$ will be a linear operator acting on any $3\\times 3$ matrix $M$ $\\mathcal {S}_ Q ( M)=\\xi \\Big ( \\frac{1}{2}(M+M^T)\\cdot ( Q +\\frac{1}{3}I_3)+\\frac{1}{2}( Q +\\frac{1}{3}I_3)\\cdot (M+ M^T)-2( Q +\\frac{1}{3}I_3) (Q+\\frac{1}{3}I_3) : M\\Big ).$ Note that $\\mathcal {S}_ Q ( M)$ is a traceless symmetric matrix and if $M$ is symmetric and traceless additionally, it reduces to $\\mathcal {S}_ Q ( M)=\\xi \\Big ( M\\cdot ( Q +\\frac{1}{3}I_3)+( Q +\\frac{1}{3}I_3)\\cdot M-2( Q +\\frac{1}{3}I_3) (Q : M)\\Big ).$ Here $A:B=\\operatorname{tr}AB^T$ and $A\\cdot B$ denotes the usual matrix product of $A,B$ and the `dot' will be sometimes omitted if it is clear from the context.", "The parameter $\\xi $ is a constant depending on the molecular details of a given liquid crystal and measures the ratio between the tumbling and the aligning effect that a shear flow would exert over the liquid crystal directors.", "Concerning the hydrodynamic part, for any vector field $u$ , its gradient $\\nabla u$ can be written as the sum of the symmetric and anti-symmetric parts: $\\nabla u=D(u)+W(u),$ where $D(u)=\\frac{1}{2}(\\nabla u+(\\nabla u)^T),\\quad W(u)=\\frac{1}{2}(\\nabla u-(\\nabla u)^T).$ With these notation, the Navier-Stokes/Q-tensor system, proposed in Beris-Edwards [6], can be written as: $&u_t+u\\cdot \\nabla u=\\nabla P+\\nabla \\cdot (\\sigma ^{s}+\\sigma ^{a}+\\sigma ^d),\\quad \\\\&\\nabla \\cdot u=0, \\\\&Q_t+u\\cdot \\nabla Q + Q \\cdot W(u)-W(u)\\cdot Q ={\\Gamma } \\mathcal {H}(Q) +\\mathcal {S}_Q (D(u)).$ In (), $\\Gamma $ is the rotational diffusion constant and without loss of generality, we shall assume $\\Gamma =1$ in the sequel.", "The unknowns $(u,P)$ correspond to the velocity/pressure of the hydrodynamics respectively.", "The stress terms $\\sigma ^{s}$ , $\\sigma ^{a}$ and $\\sigma ^d$ on the right hand side of (REF ) are symmetric viscous stress, anti-symmetric viscous stress and distortion stress respectively: $\\sigma ^{s}(u,Q)&:= \\nu D(u)-\\mathcal {S}_Q( \\mathcal {H}(Q) ),\\\\\\sigma ^{a}(Q)&:= Q \\cdot \\mathcal {H}(Q) - \\mathcal {H}(Q) \\cdot Q ,\\\\\\sigma ^d(Q)&:= \\left(\\sigma ^d_{ij}(Q)\\right)_{1\\le i,j\\le 3}=-\\left(\\frac{\\partial \\mathcal {F}(Q,\\nabla Q)}{\\partial Q_{k\\ell ,j}}Q_{k\\ell ,i}\\right)_{1\\le i,j\\le 3}.$ $ \\mathcal {H}(Q)$ is the molecular field, defined as the variational derivative of (REF ) and is written as the sum of the bulk part and the elastic part: $\\mathcal {H}(Q):= -\\frac{\\delta \\mathcal {F}(Q,\\nabla Q)}{\\delta Q}:=-\\mathcal {L}(Q)-\\mathcal {J}(Q).$ The operator $\\mathcal {L}$ and $\\mathcal {J}$ can be written explicitly by $\\begin{split}-\\mathcal {L}_{ij}(Q)=& L_1\\Delta {Q_{ij}} +\\frac{L_2+L_3}{2}(Q_{ik,kj}+Q_{jk,ki}-\\frac{2}{3}Q_{\\ell k,k\\ell }\\delta _i^j), \\end{split}$ $-\\mathcal {J}_{ij}(Q)= -aQ_{ij}+b(Q_{jk}Q_{ki}-\\frac{1}{3}\\operatorname{tr}(Q^2)\\delta _{ij})-c\\operatorname{tr}(Q^2)Q_{ij} .$ We note that the operator defined via (REF ) can not be considered as a perturbation of $L_1\\Delta $ as we only assumes (REF ).", "Actually, one of the key results in this work is Lemma REF below, showing that (REF ) fulfills the strong Legendre condition.", "The coupled system () has been recently studied by several authors.", "For the case $\\xi = 0$ , which corresponds to the situation when the molecules only tumble in a shear flow but are not aligned by the flow, the existence of global weak solutions to the Cauchy problem in $\\mathbb {R}^d$ with $d = 2, 3$ is proved in [18].", "Moreover, solutions with higher order regularity and the weak-strong uniqueness for $d = 2$ is discussed.", "Later, these results are generalized in [19] to the case when $\|\\xi \|$ is sufficiently small.", "Large time behavior of the solution to the Cauchy problem in $\\mathbb {R}^3$ with $\\xi =0$ is recently discussed in [10].", "The global well-posedness and long-time behavior of system with nonzero $\\xi $ in the two-dimensional periodic setting are studied in [7].", "In [16], the authors considered Beris-Edwards system with anisotropic elastic energy (REF ) (with $L_2+L_3>0$ and $L_4=0$ ).", "They proved the existence of global weak solutions as well as the existence of a unique global strong solution for the Cauchy problem in $\\mathbb {R}^3$ provided that the fluid viscosity is sufficiently large.", "In [8], [17], the weak solution of the gradient flow generated by the general Landau-de Gennes energy (REF ) with $L_4\\ne 0$ is established for small initial data.", "Some recent progresses have also been made on the analysis of certain modified versions of Beris-Edwards system.", "In [23], when $\\xi =0$ and the polynomial bulk energy is replaced by a singular potential derived from molecular Maier-Saupe theory, the author proved, under periodic boundary conditions, the existence of global weak solutions in space dimension two and three.", "Moreover, the existence and uniqueness of global regular solutions for dimension two is obtained.", "In [12], [13], the authors derived a nonisothermal variants of () and proved the existence of global weak solutions in the case of a singular potential under periodic boundary conditions for general $\\xi $ .", "We also mention that a rigorous derivation of the general Ericksen-Leslie system from the small elastic limit of Beris-Edwards system (with arbitrary $\\xi $ ) is recently given in [22] using the Hilbert expansion.", "In the aforementioned works, the domain under consideration is either the whole space or the tori.", "The initial-boundary value problems of () have been also investigated by several authors, see for instance [1], [2], [14], [15], in which the existences of weak solutions has been studied.", "In addition, in [1], [14], the authors proved the existence of local in time solution with higher order time regularity for () through different approaches.", "However, the higher order spacial regularity is not obtained due to the lack of effective energy estimate in the presence of inhomogenous boundary condition for $Q$ .", "The main goal of the presented work is to improve the results in [1], [14] to nature regularity in space variable.", "This gives a full answer to the construction of local in time strong solution of () in the presence of inhomogenous boundary condition for $Q$ .", "We shall consider the initial-boundary conditions $&(u,Q)\|_{t=0}=(u_0,Q_0),\\\\&(u,Q)\|_{\\partial \\Omega }=(0,Q_0\|_{\\partial \\Omega }),$ where $Q_0=Q_0(x)$ is time-independent.", "Note that such a result requires a compatibility condition on the initial data $Q_0$ .", "To see that, we write () in an abstract form $\\frac{d}{dt} (u,Q) =\\mathcal {E}(u,Q),$ where $\\mathcal {E}:H^1_{0,\\sigma }(\\Omega )\\times H^2(\\Omega )\\mapsto H^{-1}_{\\sigma }(\\Omega )\\times L^2(\\Omega )$ is defined by $\\begin{split}&\\bigl < \\mathcal {E}(u,Q), (\\varphi ,\\Psi )\\bigr >=-\\int _{\\Omega } (-u\\otimes u+ \\sigma ^{s}+\\sigma ^{a}+\\sigma ^d):\\nabla \\varphi \\, {\\rm d}x\\\\&+ \\int _{\\Omega }(-u\\cdot \\nabla Q - Q \\cdot W(u)+W(u)\\cdot Q + \\mathcal {H}(Q) +\\mathcal {S}_Q (D(u))): \\Psi \\, {\\rm d}x,\\end{split}$ for all $(\\varphi ,\\Psi )\\in H^1_{0,\\sigma }(\\Omega )\\times L^2(\\Omega ;\\mathcal {Q})$ .", "Note that the functional spaces used here are defined in Section .", "Since () specifies a time-independent boundary condition, it follows that $\\partial _t Q\|_{\\partial \\Omega }=0$ which leads to the compatibility condition that the trace of the second component on the right-hand side of (REF ) vanishes on $\\partial \\Omega $ .", "This motives to define the admissible class for the initial data ${I}=\\bigl \\lbrace (u_0,Q_0)\\in H^1_{0,\\sigma }(\\Omega )\\times H^2(\\Omega ;\\mathcal {Q})\\mid \\mathcal {E}(u_0,Q_0)\\in L^2_{\\sigma }(\\Omega )\\times H^1_{0}(\\Omega )\\bigr \\rbrace .$ It is not hard to see ${I}$ is not empty.", "For example, for any $Q$ solving $\\mathcal {H}(Q)=0$ and any $u\\in H^2_0(\\Omega )$ , we have $(u,Q)\\in {I}$ .", "We note that such a compatibility condition is nature in the sense that it can not be disregarded by changing the function spaces unless one considers very weak solution.", "The main result of this paper can be stated as follows.", "Theorem 1.1 Assume the coefficients of elastic energy satisfy (REF ).", "Then for any $(u_0,Q_0)\\in {I}$ with $Q_0\|_{\\partial \\Omega }\\in H^{5/2}(\\partial \\Omega ) $ , there exists some $T>0$ such that the system () and () has a unique solution $\\begin{split}u&\\, \\in H^2(0,T;H^{-1}(\\Omega ))\\cap H^1(0,T;H_\\sigma ^1(\\Omega ))\\cap L^\\infty (0,T;H^2(\\Omega ))\\,,\\\\Q&\\, \\in H^2(0,T;L^2(\\Omega ;\\mathcal {Q}))\\cap H^1(0,T;H^2(\\Omega ;\\mathcal {Q}))\\cap L^\\infty (0,T;H^3(\\Omega ;\\mathcal {Q})).\\,\\end{split}$ Theorem REF essentially improves the spatial regularity of solution obtained in [1] and generalizes their result to the case of anisotropic elastic energy.", "This is accomplished by the crucial observation that the terms containing third order derivatives on $Q$ in (REF ) can be eliminated and the system can be reduced into a Stokes-type system with positive definite viscosity coefficient.", "Moreover, under the general assumption (REF ), the operator $\\mathcal {L}$ defined by (REF ) is strongly elliptic.", "This fact leads to $W^{2,p}$ -estimates for the solution so that we can work with the general case of anisotropic energy rather than the isotropic energy case ($L_2=L_3=0$ ).", "The strong ellipticity of $\\mathcal {L}$ is proved in Lemma REF by an explicit construction of the coefficient matrix, which involves a fairly sophisticated anisotropic tensor of order six.", "We also mention that the local in time strong solution constructed here is valid for any $\\xi \\in \\mathbb {R}$ .", "The rest parts of the work is organized as follows.", "In Section , we introduce notation and analytic tools that will be used throughout the paper.", "The most important results involve the solvability theorem on the generalized Stokes system, due to Solonnikov [21], as well as Lemma REF on the analysis of the operator $\\mathcal {L}$ .", "In Section an abstract evolution equation that incorporates (), () and a compatibility condition is introduced and the functional analytic framework is established.", "The core part, Section , is devoted to the proof of Theorem REF by showing that the abstract evolution equation has a local in time solution.", "This is accomplished by proving the existence of a local in time solution that is regular in temporal variable in the first stage, following the method in [1], and then using the structure of () to eliminate the higher order terms in the additional stress tensors of (REF ) and recast it into a generalized Stokes system.", "Afterwards, the spatial regularity of () with initial-boundary condition () is improved using the $L^p$ -estimate of the generalized Stokes system together with bootstrap arguments." ], [ "Notations", "Throughout this paper, the Einstein's summation convention will be adopted.", "That is, we shall sum over repeated greek letters.", "For any $3\\times 3$ martrix $A,B\\in \\mathbb {R}^{3\\times 3}$ , their usual matrix product will be denoted by $A\\cdot B$ or even shortly by $AB$ if it is clear from the context.", "The Frobenius product of two matrices corresponds to $A:B= \\operatorname{tr}AB^T = A_{ij}B_{ij}$ and this induces a norm $\|A\|=\\sqrt{A_{ij}A_{ij}}$ .", "For any matrix-valued function $F=(F_{ij})_{1\\le i,j\\le 3}$ , we denote $F_{ij,k}=\\partial _kF_{ij}$ and $\\operatorname{div}F=\\nabla \\cdot F = (\\partial _j F_{ij})_{1\\le i\\le 3}$ .", "In tensor analysis, the Levi-Civita symbol $\\lbrace \\varepsilon ^{ijk}\\rbrace _{1\\le i,j,k\\le 3}$ and Kronecker symbol $\\lbrace \\delta ^i_j\\rbrace _{1\\le i,j \\le 3}$ are very useful to deal with operations involving inner and wedge product: for any $a,b\\in \\mathbb {R}^3$ , their inner and wedge products are given by $a\\cdot b= a_ib_j\\delta _i^j,\\qquad a\\wedge b=(a_jb_k\\varepsilon ^{ijk})_{1\\le i\\le 3}$ respectively.", "The following identity is well known: $\\varepsilon ^{ijk}\\varepsilon ^{imn}=(\\delta ^{j}_{m}\\delta ^{k}_{n}-\\delta ^{j}_{n}\\delta ^{k}_m).$ For any vector field $u$ , its divergence and curl can be calculated by $\\nabla \\cdot u=\\delta _i^j u_{i,j},\\qquad \\nabla \\wedge u=(\\varepsilon ^{ijk}u_{ k,j})_{1\\le i\\le 3}.$" ], [ "Function spaces and the generalized Stokes system", "Throughout this work, $\\Omega \\subset \\mathbb {R}^3$ will be a bounded domain with smooth boundary and $\\Omega _T=\\Omega \\times (0,T)$ will denote the parabolic cylinder.", "Standard notation for the Lebesgue and Sobolev spaces $L^p(\\Omega )$ and $W^{s,p}( \\Omega )$ as well as $L^p( \\Omega ;M)$ and $W^{s,p}( \\Omega ;M)$ for the corresponding spaces for $M$ -valued functions will be employed.", "Sometimes the domain and the range is omitted for simplicity if it is clear from the context.", "The $L^2$ -based Sobolev spaces are denoted by $H^s( \\Omega ;M)$ or simply by $H^s(\\Omega )$ .", "For any Banach space ${H}$ , $\\langle \\cdot ,\\cdot \\rangle _{{H},{H}^}$ denotes the dual product between ${H}$ and its dual space ${H}^$ and we shall simply write $\\langle \\cdot ,\\cdot \\rangle $ if the function spaces under consideration are clear from the context.", "An important result related to the incompressible Navier-Stokes equation is the Helmholtz decomposition $L^2(\\Omega ;\\mathbb {R}^3) = L^2_\\sigma (\\Omega )\\oplus {(L^2_\\sigma (\\Omega ))}^\\perp ,$ where $L^2_\\sigma (\\Omega )$ denotes the space of solenoidal vector field and its orthogonal space is given by ${(L^2_\\sigma (\\Omega ))}^\\perp = \\bigl \\lbrace u \\in L^2(\\Omega ;\\mathbb {R}^3),\\,u=\\nabla q\\text{ for some }q\\in H^1(\\Omega ) \\bigr \\rbrace \\,.$ The Helmholtz projection (also referred to as Leray projection), i.e., the orthogonal projection $L^2( \\Omega ;\\mathbb {R}^3) \\mapsto L^2_\\sigma (\\Omega )$ , is denoted by $P_\\sigma $ .", "The readers can refer to [20] for its basic properties.", "For any $f\\in H^{-1}( \\Omega ;\\mathbb {R}^3)$ , $P_\\sigma f \\in H^{-1}_{\\sigma }( \\Omega )$ is interpreted by $P_\\sigma f=f\|_{H^1_{0,\\sigma }( \\Omega )}$ .", "Moreover, for $F\\in L^2( \\Omega ;\\mathbb {R}^{3\\times 3})$ , $\\operatorname{div}F\\in H^{-1}( \\Omega ;\\mathbb {R}^3)$ is defined by $\\langle \\operatorname{div}F, \\Phi \\rangle _{H^{-1},H^1_0}= -\\int _{\\Omega } F: \\nabla \\Phi \\,{\\rm d}x,\\qquad \\text{for all }\\Phi \\in H^1_0( \\Omega ;\\mathbb {R}^3)\\, .$ We end this subsection by the following result due to Solonnikov [21], which is crucial in the discussion of the spatial regularity during the proof of Theorem REF .", "Proposition 2.1 Let $\\Omega \\subset \\mathbb {R}^d~(d=2,3)$ be a bounded domain with smooth boundary and $p> 3/2$ .", "Assume that the tensor-valued function $A^{k\\ell }_{ij}(x,t)\\in C(\\overline{\\Omega }_T)$ satisfies $A^{k\\ell }_{ij}(\\cdot , t)\\in W^{1,q}(\\Omega )$ for almost every $t\\in [0,T]$ , with $\\frac{1}{q}<\\frac{1}{d}+\\min (\\frac{1}{p},\\frac{p-1}{p},\\frac{1}{d})$ , and the strong Legendre condition, i.e.", "there exists two positive constants $\\Lambda >\\lambda >0$ such that $\\lambda \|\\xi \|^2\\le \\xi _{ki}A^{k\\ell }_{ij}(x,t)\\xi _{j\\ell }\\le \\Lambda \|\\xi \|^2,\\quad \\forall \\xi \\in \\mathbb {R}^{3\\times 3},\\quad \\forall (x,t)\\in \\overline{\\Omega }\\times (0,T),$ then for any $v^0\\in W^{2-2/p,p}(\\Omega ;\\mathbb {R}^3)$ with $\\operatorname{div}v^0=0$ and $f\\in L^p(\\Omega _T;\\mathbb {R}^3)$ , the system $\\left\\lbrace \\begin{array}{rl}\\partial _t v_k&= A^{k\\ell }_{ij}(x,t) \\partial _i\\partial _j v_\\ell +\\partial _kP+f_k,~\\text{with}~1\\le k\\le d,\\\\\\operatorname{div}v&=0,\\\\v\|_{t=0}&=v^0,\\\\v\|_{\\partial \\Omega }&=0\\end{array}\\right.$ has a unique solution $(v,P)$ such that $v\\in L^p(0,T;W^{2,p}(\\Omega )),~\\text{and}~ v_t,\\nabla P\\in L^p(\\Omega _T).$ Moreover, the following estimate holds for some constant $C$ that is independent of $v^0$ and $f$ : $\\Vert v_t\\Vert _{L^p(\\Omega _T)}+\\Vert v\\Vert _{L^p(0,T;W^{2,p}(\\Omega ))}+\\Vert \\nabla P\\Vert _{L^p(\\Omega _T)}\\le C(\\Vert v^0\\Vert _{W^{2-2/p,p}(\\Omega )}+\\Vert f\\Vert _{L^p(\\Omega _T)}).$ The result is still valid when $p=3/2$ and $v^0\\equiv 0$ .", "In the sequel, we shall also need the stationary version of the above result when $p=\\frac{3}{2}$ : Corollary 2.2 Assume $A^{k\\ell }_{ij}(x)\\in C(\\overline{\\Omega })\\cap W^{1,6}(\\Omega )$ and there exists two positive constants $\\Lambda >\\lambda >0$ such that $\\lambda \|\\xi \|^2\\le \\xi _{ki}A^{k\\ell }_{ij}(x)\\xi _{j\\ell }\\le \\Lambda \|\\xi \|^2,\\quad \\forall \\xi \\in \\mathbb {R}^{3\\times 3},\\quad \\forall x\\in \\overline{\\Omega } ,$ then for any $f\\in L^{3/2}(\\Omega ;\\mathbb {R}^3)$ , the system $\\left\\lbrace \\begin{array}{rl}\\partial _i( A^{k\\ell }_{ij}(x) \\partial _j v_\\ell )+\\partial _kP&=f_k,~(1\\le k\\le d)\\\\\\operatorname{div}v&=0,\\\\v\|_{\\partial \\Omega }&=0 \\end{array}\\right.$ has a unique solution $(v,P)$ with $v\\in W^{2,3/2}(\\Omega ),~\\nabla P\\in L^{3/2}(\\Omega )$ and the following estimate holds $\\Vert v\\Vert _{ W^{2,3/2}(\\Omega )}+\\Vert \\nabla P\\Vert _{L^{3/2}(\\Omega )}\\le C(1+\\Vert A\\Vert _{W^{1,6}(\\Omega )})\\Vert f\\Vert _{L^{3/2}(\\Omega )}.$ Moreover, if $f\\in L^2(\\Omega )$ , we have the improved estimate $\\Vert v\\Vert _{ W^{2,2}(\\Omega )}+\\Vert \\nabla P\\Vert _{L^{2}(\\Omega )}\\le C(1+\\Vert A\\Vert ^2_{W^{1,6}(\\Omega )})\\Vert f\\Vert _{L^{2}(\\Omega )}.$ In the above two inequalities, $C$ only depends on the continuous modulus of $A^{k\\ell }_{ij}(x)$ and geometric information of $\\Omega $ .", "Since $H^1(\\Omega )\\hookrightarrow L^6(\\Omega )$ , it follows from duality that $L^{6/5}(\\Omega )\\hookrightarrow H^{-1}(\\Omega )$ and this implies that $f\\in L^{3/2}(\\Omega )\\hookrightarrow H^{-1}(\\Omega )$ .", "The assumption on $A^{k\\ell }_{ij}(x)$ , especially (REF ), implies that the bilinear form $a(u,v):=\\int _\\Omega \\partial _i u_{k}A^{k\\ell }_{ij}(x)\\partial _j v_{\\ell }\\mathrm {d}x$ is coercive on $H^1_{0,\\sigma }(\\Omega )$ and the existence of solution $u\\in H^1_{0,\\sigma }(\\Omega )$ to (REF ) follows from Lax-Milgram theorem.", "In order to obtain the $L^p$ -estimate of (REF ), we set $v(x,t)=\\zeta (t) u(x)$ where $\\zeta $ is a non-negative smooth function such that $\\zeta (t)=0$ for $t\\le 0$ and $\\zeta (t)=1$ for $t\\ge 1$ .", "It can be verified that $v(x,t)$ satisfies the following equation on $\\Omega \\times [0,2]$ in the sense of distribution: $\\left\\lbrace \\begin{array}{rl}\\partial _t v_k- A^{k\\ell }_{ij}(x)\\partial _i\\partial _j v_\\ell -\\partial _k (\\zeta (t)P)&=\\zeta (t)\\partial _iA_{ij}^{k\\ell }(x)\\partial _j u_\\ell +\\zeta ^{\\prime }(t)u-\\zeta (t)f(x),\\\\\\nabla \\cdot v&=0,\\\\v\|_{\\left(\\partial \\Omega \\times [0,2]\\right)\\cup \\left(\\lbrace t=0\\rbrace \\times \\Omega \\right)}&=0.\\end{array}\\right.$ It follows from $A_{ij}^{k\\ell }\\in W^{1,6}(\\Omega )$ and $u\\in H^1_{0,\\sigma }(\\Omega )$ that $\\check{f}:=\\zeta (t)\\partial _iA_{ij}^{k\\ell }(x)\\partial _j u_\\ell +\\zeta ^{\\prime }(t)u-\\zeta (t)f(x)\\in L^{3/2}(\\Omega _T).", "$ Using Proposition REF $\\Vert v\\Vert _{L^{3/2}(0,2;W^{2,3/2}(\\Omega ))}+\\Vert \\nabla (\\zeta (t) P)\\Vert _{L^{3/2}(\\Omega \\times (0,2) )}\\le C\\Vert \\check{f} \\Vert _{L^{3/2}(\\Omega \\times (0,2))},$ and this leads to $\\Vert u\\Vert _{W^{2,3/2}(\\Omega )}+\\Vert \\nabla P\\Vert _{L^{3/2}(\\Omega )}\\le C \\left(1+\\Vert \\nabla A\\Vert _{L^{6}(\\Omega )}\\right)\\Vert f\\Vert _{L^{3/2}(\\Omega )}.$ To prove the second equality in the statement, note that $u\\in W^{2,3/2}(\\Omega )\\hookrightarrow W^{1,3}(\\Omega )$ and together with $f\\in L^2(\\Omega )$ improves the estimate for $\\check{f}$ $\\Vert \\check{f}\\Vert _{L^2(\\Omega \\times (0,2))}\\le C (1+\\Vert \\nabla A\\Vert ^2_{L^{6}(\\Omega )})\\Vert f\\Vert _{L^2(\\Omega )}.$ So applying Proposition REF to solve (REF ) again leads to the second inequality." ], [ "Abstract parabolic equation", "Following the method in [1], we shall prove the regular in time solution with the aid of the following result: Proposition 2.3 Suppose that $\\mathbb {V}$ and $\\mathbb {H}$ are two separable Hilbert spaces such that the embedding $\\mathbb {V}\\hookrightarrow \\mathbb {H}$ is injective, continuous, and dense.", "Fix $T\\in (0,\\infty )$ .", "Suppose that a bilinear form $a( \\cdot , \\cdot ): \\mathbb {V}\\times \\mathbb {V}\\mapsto \\mathbb {R}$ is given which satisfies for all $\\phi $ , $\\psi \\in \\mathbb {V}$ the following assumptions: (a) there exists a constant $c>0$ , independent of $ \\phi $ and $\\psi $ , with $\\bigl \| a(\\phi , \\psi ) \\bigr \| \\le c \\Vert \\phi \\Vert _\\mathbb {V}\\Vert \\psi \\Vert _\\mathbb {V}\\,;$ (b) there exist $k_0$ , $\\alpha >0$ independent of $\\phi $ , with $a( \\phi , \\phi ) + k_0 \\Vert \\phi \\Vert _\\mathbb {H}^2 \\ge \\alpha \\Vert \\phi \\Vert _\\mathbb {V}^2 \\,;$ Then there exists a representation operator $L : \\mathbb {V}\\mapsto \\mathbb {V}^\\prime $ with $a( \\phi , \\psi ) = \\langle L \\phi , \\psi \\rangle _{\\mathbb {V}^\\prime ,\\mathbb {V}}$ , which is continuous and linear.", "Moreover, for all $f\\in L^2((0,T);\\mathbb {V}^\\prime )$ and $y_0\\in \\mathbb {H}$ , there exists a unique solution $y\\in \\bigl \\lbrace v:[0,T]\\mapsto \\mathbb {V}\\text{ with }v\\in L^2(0,T;\\mathbb {V}),\\,\\partial _t v\\in L^2(0,T;\\mathbb {V}^\\prime ) \\bigr \\rbrace $ solving the equation $\\partial _t y+ L y = f \\quad \\text{ in } \\mathbb {V}^\\prime \\ \\text{for a.e.", "}t\\in (0,T)\\,,$ subject to the initial condition $y(0) =y_0$ .", "Finally, assume additionally that $y_0 \\in \\mathbb {V}$ .", "Then $L: H^{1}((0,T);\\mathbb {V}) \\mapsto H^{1}((0,T);\\mathbb {V}^\\prime )$ is continuous and for all $f\\in H^{1}((0,T);\\mathbb {V}^\\prime )$ which satisfy the compatibility condition $f(0) \\in \\mathbb {H}$ , the solution $y$ satisfies $y\\in H^{1}((0,T);\\mathbb {V})\\quad \\text{ and }\\quad \\partial _t^2 y\\in L^2\\bigl ((0,T);\\mathbb {V}^\\prime \\bigr )\\,.$ The proof of this theorem can be found in [24]." ], [ "Anisotropic Laplacian", "We consider the following bilinear form $a( \\Psi ,\\Phi )= \\int _{\\Omega }\\Big (L_1 \\Psi _{i j,k}\\Phi _{i j,k}+L_2\\Psi _{ij,j}\\Phi _{ik,k}+L_3\\Psi _{ij,k}\\Phi _{ik,j}\\Big ) \\mathrm {d}x.$ for $\\Psi ,\\Phi \\in H^1(\\Omega ;\\mathcal {Q})$ .", "Lemma 2.4 For any $f\\in H^{-1}(\\Omega ;\\mathcal {Q})$ , there exists a unique $ Q \\in H^1_0(\\Omega ;\\mathcal {Q})$ such that $a( Q ,\\Phi )=\\langle f,\\Phi \\rangle ~\\text{for any}~\\Phi \\in H^1_0(\\Omega ;\\mathcal {Q}),$ and there exists a constant $C$ depending on the geometry of $\\Omega $ such that $\\Vert Q\\Vert _{H^1_0(\\Omega )}\\le C \\Vert f\\Vert _{H^{-1}(\\Omega )}.$ This lemma can be proved using the construction in the proof of Lemma REF below.", "However, we present a simpler proof here : One can verify that $a( Q ,\\Phi )=\\langle \\mathcal {L}( Q ),\\Phi \\rangle _{H^{-1},H^1_0}$ where $\\mathcal {L}$ is the operator defined by (REF ).", "In order to apply Lax-Milgram theorem to deduce the existence of solution to (REF ), we need to show that $a(\\cdot ,\\cdot )$ is coercive in $H^1_0(\\Omega ;\\mathcal {Q})$ : $\\begin{split}a( Q , Q )\\ge &\\lambda \\int _{\\Omega } \|\\nabla Q \|^2\\mathrm {d}x,\\end{split}$ for some $\\lambda >0$ .", "Note that, it suffices to prove the above inequality for smooth functions that vanishes on $\\partial \\Omega $ .", "Actually, for any $ Q \\in H_0^1(\\Omega ;\\mathcal {Q})$ , we choose $ Q _n\\in C_0^\\infty (\\overline{\\Omega };\\mathcal {Q})$ such that $ Q _n\\rightarrow Q $ strongly in $H^1_0(\\Omega ;\\mathcal {Q})$ .", "Then the conclusion follows from the continuity of $a(\\cdot ,\\cdot )$ .", "Now we focus on the proof of (REF ) for $Q\\in C_0^\\infty (\\overline{\\Omega })$ .", "It follows from (REF ) and (REF ) that $\\begin{split}\|\\nabla \\wedge Q \|^2&= \|\\varepsilon ^{ijk}Q_{\\ell j,k}\|^2= \\varepsilon ^{ijk}Q_{\\ell j,k}\\varepsilon ^{imn}Q_{\\ell m,n}= (\\delta ^{j}_{m}\\delta ^{k}_{n}-\\delta ^{j}_{n}\\delta ^{k}_m)Q_{\\ell j,k}Q_{\\ell m,n}\\\\&= Q_{ij,k}Q_{ij,k}-Q_{ij,k}Q_{ik,j}= \|\\nabla Q \|^2-Q_{ij,k}Q_{ik,j}.\\end{split}$ Therefore by $Q\|_{\\partial \\Omega }=0$ , $\\nonumber \\int _{\\Omega } Q_{ij,j}Q_{ik,k}\\mathrm {d}x =&\\int _{\\Omega }\\left(Q_{ij,k}Q_{ik,j}+ \\partial _j(Q_{ij}Q_{ik,k}) -\\partial _k(Q_{ij}Q_{ik,j})\\right)\\mathrm {d}x\\\\ \\nonumber =&\\int _{\\Omega } \\left(\\nabla Q \|^2-\|\\nabla \\wedge Q \|^2\\right)\\mathrm {d}x+\\int _{\\partial \\Omega }\\left(Q_{ij}Q_{ik,j}\\nu _k-Q_{ij}Q_{ik,k}\\nu _j\\right)\\mathrm {d}S\\\\=&\\int _{\\Omega } \\left(\|\\nabla Q \|^2-\|\\nabla \\wedge Q \|^2\\right)\\mathrm {d}x.$ The above two formula together implies $a( Q , Q )=&\\int _\\Omega \\Big (L_1\|\\nabla Q \|^2+L_2Q_{ij,j}Q_{ik,k}+L_3Q_{ij,k}Q_{ik,j}\\Big ) \\mathrm {d}x\\nonumber \\\\ \\nonumber =&\\int _\\Omega L_1\|\\nabla Q \|^2\\mathrm {d}x+\\int _\\Omega (L_2+L_3)\\left(\|\\nabla Q \|^2-\|\\nabla \\wedge Q \|^2\\right)\\mathrm {d}x.$ Therefore, it is easy to see that $a(Q,Q)\\ge \\int _\\Omega L_1\|\\nabla Q \|^2 \\mathrm {d}x $ when $L_2+L_3\\ge 0,$ and $a(Q,Q)\\ge \\int _\\Omega (L_1+L_2+L_3)\|\\nabla Q \|^2 \\mathrm {d}x= L_0\\int _\\Omega \|\\nabla Q \|^2\\mathrm {d}x$ when $L_2+L_3\\le 0$ .", "The validity of the $L^p$ -estimate requires the verification of the strong Legendre condition for $\\mathcal {L}$ in (REF ).", "To this end, we consider the following second order operator defined for $Q\\in H^2(\\Omega ;\\mathbb {R}^{3\\times 3})$ : $\\begin{split}(\\widetilde{\\mathcal {L}} (Q) )_{ij}=&~L_1\\Delta Q_{ij}-\\frac{1}{4}(L_2+L_3)\\Big (\\partial _i\\partial _{k}Q_{jk}+\\partial _j\\partial _{ k}Q_{ik}+\\partial _i\\partial _{k}Q_{kj}+\\partial _j\\partial _{ k}Q_{ki}\\\\&\\quad -\\frac{4}{3}\\partial _{i}\\partial _jQ_{kk}-\\frac{4}{3}\\delta ^i_j\\partial _k\\partial _{\\ell }Q_{k\\ell }+\\frac{4}{9}\\delta ^i_j\\Delta Q_{kk}\\Big ).\\end{split}$ Lemma 2.5 Let $p>1$ be fixed.", "For any $F\\in L^p(\\Omega ; \\mathbb {R}^{3\\times 3})$ and $g\\in W^{2-1/p,p}(\\partial \\Omega ; \\mathbb {R}^{3\\times 3})$ , there exists a unique $ Q \\in W^{2,p}(\\Omega ; \\mathbb {R}^{3\\times 3})$ that solves $\\widetilde{\\mathcal {L}} Q =F$ with boundary condition $Q\|_{\\partial \\Omega }=g$ .", "Moreover, there exists $C>0$ depending only on $\\Omega $ such that $\\Vert Q \\Vert _{W^{2,p}(\\Omega )}\\le C\\left(\\Vert g\\Vert _{W^{2-1/p,p}(\\partial \\Omega )}+\\Vert F\\Vert _{L^p(\\Omega )}\\right).$ Especially, when $F,g\\in \\mathcal {Q}$ , we have $Q\\in W^{2,p}(\\Omega ;\\mathcal {Q})$ satisfying $\\mathcal {L}Q=F$ .", "It suffices to verify the strong Legendre condition (see (REF ) for instance) for $\\widetilde{\\mathcal {L}}$ and then the conclusion follows from standard theory of elliptic system (cf.", "[3] or [9] ).", "To this end, we first note that $\\widetilde{\\mathcal {L}}$ can be written as $\\begin{split}(\\widetilde{\\mathcal {L}}(Q) )_{ij}=&-L_1\\partial _\\ell \\Big (\\delta ^\\ell _k\\delta ^i_{i^{\\prime }}\\delta ^j_{j^{\\prime }}\\partial _kQ_{i^{\\prime }j^{\\prime }}\\Big )-\\frac{1}{4}(L_2+L_3)\\partial _\\ell \\Big ((\\delta ^j_{i^{\\prime }}\\delta ^k_{i}\\delta ^\\ell _{j^{\\prime }}+\\delta ^i_{i^{\\prime }}\\delta ^\\ell _{j^{\\prime }}\\delta ^k_j+\\delta _k^i\\delta _\\ell ^{i^{\\prime }}\\delta _j^{j^{\\prime }}+\\delta _k^j\\delta _\\ell ^{i^{\\prime }}\\delta _i^{j^{\\prime }})\\partial _kQ_{i^{\\prime }j^{\\prime }}\\\\&-\\frac{4}{3}\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^{i}_\\ell \\delta ^{j}_k\\partial _kQ_{i^{\\prime }j^{\\prime }}-\\frac{4}{3}\\delta ^i_j\\delta ^{i^{\\prime }}_\\ell \\delta ^{j^{\\prime }}_k\\partial _kQ_{i^{\\prime }j^{\\prime }}+\\frac{4}{9}\\delta ^i_j\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^k_\\ell \\partial _kQ_{i^{\\prime }j^{\\prime }}\\Big )\\\\=&-\\partial _\\ell \\Big (A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}\\partial _kQ_{i^{\\prime }j^{\\prime }}\\Big ),\\end{split}$ where $\\begin{split}A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}=&~L_1\\delta ^\\ell _k\\delta ^i_{i^{\\prime }}\\delta ^j_{j^{\\prime }}+\\frac{1}{4}(L_2+L_3)\\big (\\delta ^j_{i^{\\prime }}\\delta ^k_{i}\\delta ^\\ell _{j^{\\prime }}+\\delta ^i_{i^{\\prime }}\\delta ^\\ell _{j^{\\prime }}\\delta ^k_j+\\delta _k^i\\delta _\\ell ^{i^{\\prime }}\\delta _j^{j^{\\prime }}+\\delta _k^j\\delta _\\ell ^{i^{\\prime }}\\delta _i^{j^{\\prime }}\\\\&-\\frac{4}{3}\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^{i}_\\ell \\delta ^{j}_k-\\frac{4}{3}\\delta ^i_j\\delta ^{i^{\\prime }}_\\ell \\delta ^{j^{\\prime }}_k+\\frac{4}{9}\\delta ^i_j\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^k_\\ell \\big ).\\end{split}$ To verify the strong Legendre condition for $A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}$ , we need to compute $\\begin{split}A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}\\xi _\\ell ^{(ij)}\\xi _k^{(i^{\\prime }j^{\\prime })}=&L_1\\sum _{i,j,\\ell } (\\xi _\\ell ^{(ij)})^2+\\frac{1}{4}(L_2+L_3)\\Big (\\xi ^{(ij)}_{\\ell }\\xi ^{(j\\ell )}_i+\\xi ^{(ij)}_{\\ell }\\xi ^{(i\\ell )}_j+\\xi ^{(ij)}_{\\ell }\\xi ^{(\\ell j)}_i+\\xi ^{(ij)}_{\\ell }\\xi ^{(\\ell i)}_j\\\\&-\\frac{4}{3}\\xi ^{(ij)}_{i}\\xi ^{(\\ell \\ell )}_{j}-\\frac{4}{3}\\xi ^{(ii)}_j\\xi ^{(j\\ell )}_\\ell +\\frac{4}{9}\\sum _\\ell \\big \|\\xi ^{(ii)}_\\ell \\big \|^2\\Big ).\\end{split}$ To this end, we define a new tensor by $\\zeta ^{(ij)}_{\\ell }=\\xi ^{(ij)}_{\\ell }-\\frac{1}{3}\\delta ^i_j\\xi ^{(kk)}_\\ell $ .", "Then it is easy to verify that $\\sum _{i,j,\\ell } \|\\zeta ^{(ij)}_\\ell \|^2 \\le \\sum _{i,j,\\ell }\|\\xi ^{(ij)}_\\ell \|^2.$ Thus we have, for the case $L_2+L_3\\le 0$ , $\\begin{split}A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}\\xi _\\ell ^{(ij)}\\xi _k^{(i^{\\prime }j^{\\prime })}=&L_1 \\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2+\\frac{1}{4}(L_2+L_3)\\Big (\\zeta ^{(ij)}_{\\ell }\\zeta ^{(j\\ell )}_i+\\zeta ^{(ij)}_{\\ell }\\zeta ^{(i\\ell )}_j+\\zeta ^{(ij)}_{\\ell }\\zeta ^{(\\ell j)}_i+\\zeta ^{(ij)}_{\\ell }\\zeta ^{(\\ell i)}_j\\Big )\\\\\\ge &L_1 \\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2+\\frac{1}{4}\\sum _{i,j,\\ell }(L_2+L_3)\\Big (\\big \|\\zeta ^{(ij)}_{\\ell }\\big \|^2+\\big \|\\zeta ^{(i\\ell )}_j\\big \|^2+\\big \|\\zeta ^{(ij)}_{\\ell }\\big \|^2+\\big \|\\zeta ^{(j\\ell )}_i\\big \|^2\\Big )\\\\\\ge &L_1 \\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2+\\frac{1}{4}\\sum _{i,j,\\ell }(L_2+L_3)\\Big (\\big \|\\xi ^{(ij)}_{\\ell }\\big \|^2+\\big \|\\xi ^{(i\\ell )}_j\\big \|^2+\\big \|\\xi ^{(ij)}_{\\ell }\\big \|^2+\\big \|\\xi ^{(j\\ell )}_i\\big \|^2\\Big )\\\\=&(L_1+L_2+L_3)\\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2.\\end{split}$ In the case when $L_2+L_3\\ge 0$ , we can write $(\\widetilde{\\mathcal {L}} (Q) )_{ij}=-\\partial _\\ell \\Big (\\widetilde{A}^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}\\partial _kQ_{i^{\\prime }j^{\\prime }}\\Big ),$ where $ \\widetilde{A}^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}$ is defined by interchanging $k,\\ell $ in (REF ): $\\begin{split}\\widetilde{A}^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}=&~L_1\\delta ^\\ell _k\\delta ^i_{i^{\\prime }}\\delta ^j_{j^{\\prime }}+\\frac{1}{4}(L_2+L_3)\\big (\\delta ^j_{i^{\\prime }}\\delta ^\\ell _{i}\\delta ^k_{j^{\\prime }}+\\delta ^i_{i^{\\prime }}\\delta ^k_{j^{\\prime }}\\delta ^\\ell _j+\\delta _\\ell ^i\\delta _k^{i^{\\prime }}\\delta _j^{j^{\\prime }}+\\delta _\\ell ^j\\delta _k^{i^{\\prime }}\\delta _i^{j^{\\prime }}\\\\&-\\frac{4}{3}\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^{i}_k\\delta ^{j}_\\ell -\\frac{4}{3}\\delta ^i_j\\delta ^{i^{\\prime }}_k\\delta ^{j^{\\prime }}_\\ell +\\frac{4}{9}\\delta ^i_j\\delta ^{i^{\\prime }}_{j^{\\prime }}\\delta ^k_\\ell \\big ).\\end{split}$ This yields $\\begin{split}A^{\\ell k}_{(ij)(i^{\\prime }j^{\\prime })}\\xi _\\ell ^{(ij)}\\xi _k^{(i^{\\prime }j^{\\prime })}=&L_1 \\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2+\\frac{1}{4}(L_2+L_3)\\Big (\\xi ^{(ij)}_{i}\\xi ^{(j\\ell )}_\\ell +\\xi ^{(ij)}_{j}\\xi ^{(i\\ell )}_\\ell +\\xi ^{(ij)}_{i}\\xi ^{(\\ell j)}_\\ell +\\xi ^{(ij)}_{j}\\xi ^{(\\ell i)}_\\ell \\\\&-\\frac{4}{3}\\xi ^{(ij)}_{j}\\xi ^{(\\ell \\ell )}_{i}-\\frac{4}{3}\\xi ^{(ii)}_{\\ell }\\xi ^{(j\\ell )}_j+\\frac{4}{9}\\sum _\\ell \\big \|\\xi ^{(ii)}_\\ell \\big \|^2\\Big )\\\\=&L_1 \\sum _{i,j,\\ell }\\big \|\\xi _\\ell ^{(ij)}\\big \|^2+\\frac{1}{4}(L_2+L_3)\\sum _{j}\\Big \|\\xi ^{(ij)}_{i}+\\xi ^{(ji)}_i-\\frac{2}{3}\\xi ^{(ii)}_{j}\\Big \|^2\\ge L_1\\sum _{i,j,\\ell } \\big \|\\xi _\\ell ^{(ij)}\\big \|^2\\end{split}$ and we conclude that $\\widetilde{\\mathcal {L}}$ satisfies the strong Legendre condition in both cases.", "To prove the `especially' part, we first note that $\\widetilde{\\mathcal {L}} (Q)={\\mathcal {L}} (Q)$ for $Q\\in \\mathcal {Q}$ .", "Consequently, if $Q\\in \\mathcal {Q}$ solves $\\mathcal {L}(Q)= F $ for some $ F $ with image in $\\mathcal {Q}$ , then $\\widetilde{\\mathcal {L}}(Q)= F $ .", "On the other hand, if $ F \\in \\mathcal {Q}$ , and $Q\\in \\mathbb {R}^{3\\times 3}$ solves $\\widetilde{\\mathcal {L}}(Q)= F $ , then we have $L_1\\Delta (Q-Q^T)= \\widetilde{\\mathcal {L}}(Q)-(\\widetilde{\\mathcal {L}}(Q))^T=0,$ and $L_1\\Delta (\\operatorname{tr}Q)=\\operatorname{tr}\\widetilde{\\mathcal {L}}(Q) =\\operatorname{tr}F =0.$ Thus $Q\\in \\mathcal {Q}$ and $\\mathcal {L}(Q)= F $ .", "The above lemma implies the following: Corollary 2.6 The operator $\\mathcal {L}:H^2(\\Omega ;\\mathcal {Q})\\cap H^1_0(\\Omega ;\\mathcal {Q})\\mapsto L^2(\\Omega ;\\mathcal {Q}))$ defined by (REF ) is an isomorphism." ], [ "Abstract form of the system", "The task of this section is to setup the functional analytic framework for () and ().", "We first remark that the Beris-Edward system () obeys the basic energy dissipation law $\\frac{d}{dt}\\left(\\mathcal {F}( Q,\\nabla Q )+\\int _{\\Omega } \\frac{1}{2}\| u\|^2\\mathrm {d}x\\right)+\\int _{\\Omega }\\left( \|\\nabla u\|^2+\\left\| \\mathcal {H}(Q)\\right\|^2\\right)\\mathrm {d}x=0.$ This can be formally done by first testing equation (REF ) by the velocity field $u$ and testing () by $ \\mathcal {H}(Q)$ in (REF ), then simple integration by parts lead to: $\\begin{split}&\\frac{d}{dt}\\int _{\\Omega } \\frac{1}{2}\|u\|^2\\mathrm {d}x+\\int _{\\Omega } \|\\nabla u\|^2\\mathrm {d}x\\\\=&-\\int _{\\Omega } \\left(-\\mathcal {S}_Q ( \\mathcal {H} )+ Q \\cdot \\mathcal {H} - \\mathcal {H} \\cdot Q \\right):\\nabla u \\mathrm {d}x-\\int _{\\Omega } \\partial _j\\left(\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell ,j}}Q_{k\\ell ,i}\\right)u_i\\mathrm {d}x,\\end{split}$ and $\\begin{split}&-\\frac{d}{dt}\\mathcal {F}( Q,\\nabla Q )+\\int _{\\Omega } u\\cdot \\nabla Q : \\mathcal {H} (Q) \\mathrm {d}x\\\\=&\\int _{\\Omega }\\left(\| \\mathcal {H} (Q) \|^2+\\big (\\mathcal {S}_Q (D(u))+W(u)\\cdot Q - Q \\cdot W(u)\\big ): \\mathcal {H}\\right)\\mathrm {d}x .\\end{split}$ Since we have $\\int _{\\Omega } \\partial _j\\left(\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell ,j}}Q_{k\\ell ,i}\\right)u_i\\mathrm {d}x=&\\int _{\\Omega } \\left(\\partial _j\\left(\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell ,j}}\\right)Q_{k\\ell ,i}+\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell ,j}}Q_{k\\ell ,ij}\\right)u_i\\mathrm {d}x\\\\=&\\int _{\\Omega } \\left(\\mathcal {H}_{k\\ell }(Q)Q_{k\\ell ,i}+\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell }}Q_{k\\ell ,i}+\\frac{\\partial \\mathcal {F}}{\\partial Q_{k\\ell ,j}}Q_{k\\ell ,ij}\\right)u_i\\mathrm {d}x\\\\=&\\int _{\\Omega } \\left(\\mathcal {H}_{k\\ell }(Q)Q_{k\\ell ,i}+\\partial _i \\mathcal {F}(Q,\\nabla Q)\\right)u_i\\mathrm {d}x\\\\=&\\int _{\\Omega } \\mathcal {H}_{k\\ell }(Q)Q_{k\\ell ,i}u_i\\mathrm {d}x,$ thus subtracting (REF ) from (REF ) and using the following cancellation yields (REF ).", "The following lemma indicates the important cancellation law between nonlinear terms: Lemma 3.1 For any $Q\\in \\mathcal {Q}$ , the linear operator $\\mathcal {S}_Q ( M)$ defined by (REF ) is symmetric and traceless.", "Moreover, for any $3\\times 3$ matrices $P$ and $M$ , it holds $\\mathcal {S}_Q( M):P =\\mathcal {S}_Q(P):M.$ In addition, if $M$ is symmetric, then $\\left( -\\mathcal {S}_Q( M)+ Q \\cdot M- M\\cdot Q \\right):P=\\left(-\\mathcal {S}_Q(S)+ Q \\cdot A-A\\cdot Q \\right): M,$ where $S$ and $A$ are symmetric and anti-symmetric parts of $P$ respectively.", "Note that the space consisting of all $k$ by $k$ matrices under the Frobenius product $A:B= \\operatorname{tr}AB^T = A_{ij}B_{ij}$ is a Hilbert space and it allows the following direct product decomposition: $\\mathbb {R}^{n\\times n}=\\lbrace M\\in \\mathbb {R}^{n\\times n}, M_{ij}=M_{ji}\\rbrace \\oplus \\lbrace M\\in \\mathbb {R}^{n\\times n}, M_{ij}=-M_{ji}\\rbrace .$ Actually, any $M$ can be uniquely written as the sum of two orthogonal parts $M=\\frac{M+M^T}{2}+\\frac{M-M^T}{2}.$ Then direct calculations implies the identity (REF ).", "Formula (REF ) is trickier and can be proved using (REF ): $\\begin{split}&( -\\mathcal {S}_Q( M)+ Q \\cdot M- M^T\\cdot Q ): P\\\\=& ( -\\mathcal {S}_Q( M)+ Q \\cdot M- M^T\\cdot Q ):(S+A)\\\\=&-\\mathcal {S}_Q( M):S+( Q \\cdot M- M^T\\cdot Q ):A\\\\=&-\\mathcal {S}_Q(S): M+( Q \\cdot A-A \\cdot Q ): M.\\end{split}$ The typical situation for the application of (REF ) is when $P=\\nabla u=D(u)+W(u)$ for some vector field $u$ : $\\Big ( -\\mathcal {S}_Q( M)+ Q \\cdot M- M\\cdot Q \\Big ):\\nabla u=\\Big (-\\mathcal {S}_Q(D(u))+ Q \\cdot W(u)-W(u)\\cdot Q \\Big ): M,\\quad \\forall Q,M\\in \\mathcal {Q}.$ Let $(u_0,Q_0)\\in {I}$ , defined by (REF ).", "As usual, the first equation in () will be formulated by testing with divergence-free vector fields, or equivalently, by applying the Leray's projector $ \\left\\lbrace \\begin{array}{rl}&u_t=P_\\sigma \\operatorname{div}\\left( -u\\otimes u + \\sigma ^{s}(u,Q)+\\sigma ^{a}(Q)+\\sigma ^d(Q) \\right),\\quad \\\\&Q_t+u\\cdot \\nabla Q + Q \\cdot W(u)-W(u)\\cdot Q = \\mathcal {H}(Q) +\\mathcal {S}_Q (D(u)),\\end{array}\\right.$ where $P_\\sigma \\colon H^{-1}( \\Omega ;\\mathbb {R}^3)\\mapsto H^{-1}_\\sigma ( \\Omega )$ and $\\operatorname{div}\\colon L^2( \\Omega ;\\mathbb {R}^{3\\times 3})\\mapsto H^{-1}( \\Omega ;\\mathbb {R}^3)$ are defined in Section .", "The idea is to rewrite the nonlinear system (REF ) as an abstract evolutionary equation in a suitable Banach space.", "With the notation introduced in Section , we define the linearized operator at the initial director field $ Q_0 $ by ${L}_{ Q_0 }\\begin{pmatrix} u\\\\ Q \\end{pmatrix}:=\\frac{d}{dt}\\begin{pmatrix} u\\\\ Q \\end{pmatrix}-\\begin{pmatrix}P_{\\sigma }\\operatorname{div}\\left[ D( u)+\\mathcal {S}_{ Q _0}(\\mathcal {L}( Q ))- Q _0\\cdot \\mathcal {L}( Q )+\\mathcal {L}( Q )\\cdot Q _0\\right]\\\\-\\mathcal {L}( Q )+\\mathcal {S}_{ Q _0}( D( u))- Q _0\\cdot W(u)+W(u)\\cdot Q _0\\end{pmatrix}\\,$ and nonlinear part is given by $\\begin{split}\\widetilde{ {N}}_{ Q _0}\\begin{pmatrix} u\\\\ Q \\end{pmatrix}&:=\\begin{pmatrix}P_{\\sigma }\\operatorname{div}\\left[ \\mathcal {S}_Q(\\mathcal {J}( Q ))- Q \\cdot \\mathcal {J}( Q )+\\mathcal {J}( Q )\\cdot Q - u\\otimes u-\\frac{\\partial \\mathcal {F}}{\\partial \\nabla Q }:\\nabla Q \\right]\\\\-\\mathcal {J}( Q )- u\\cdot \\nabla Q\\end{pmatrix}\\\\&+\\begin{pmatrix}P_{\\sigma }\\operatorname{div}\\left[\\mathcal {S}_Q(\\mathcal {L}( Q ))-\\mathcal {S}_{ Q _0}(\\mathcal {L}( Q ))-( Q - Q _0)\\cdot \\mathcal {L}( Q )+\\mathcal {L}( Q )\\cdot ( Q - Q _0)\\right]\\\\\\mathcal {S}_Q( D( u))-\\mathcal {S}_{ Q _0}(D( u))+( Q _0- Q )\\cdot W(u)-W(u)\\cdot ( Q _0- Q )\\end{pmatrix}.\\end{split}$ So if $(u,Q)$ is a solution to () satisfying initial-boundary conditions (), then $(u_h,Q_h)=(u-u_0,Q-Q_0)$ satisfies the operator equation ${L}_{Q_0}\\begin{pmatrix} u_h+u_0\\\\ Q_h+Q_0 \\end{pmatrix}=\\widetilde{{N}}_{Q_0}\\begin{pmatrix} u_h+u_0\\\\ Q_h+Q_0 \\end{pmatrix}\\,,$ as well as the homogeneous initial-boundary conditions.", "Due to the inhomogenous boundary conditions, the operator $\\widetilde{{N}}_{Q_0}$ is defined on an affine space.", "For the purpose of applying classical result in functional analysis and operator theory, we shall rewrite it as an nonlinear operator between two Banach spaces.", "To this end, we denote the stationary version of (REF ) by ${S}_{ Q_0 }\\begin{pmatrix} u\\\\ Q \\end{pmatrix}:=\\begin{pmatrix}P_{\\sigma }\\operatorname{div}\\left[ D( u)+\\mathcal {S}_{ Q _0}(\\mathcal {L}( Q ))- Q _0\\cdot \\mathcal {L}( Q )+\\mathcal {L}( Q )\\cdot Q _0\\right]\\\\-\\mathcal {L}( Q )+\\mathcal {S}_{ Q _0}( D( u))- Q _0\\cdot W(u)+W(u)\\cdot Q _0\\end{pmatrix}\\,.$ Then it follows from the linearity of (REF ) and the assumption that $(u_0,Q_0)$ is time-independent that, equation (REF ) is equivalent to ${L}_{Q_0}\\begin{pmatrix} u_h\\\\ Q_h\\end{pmatrix}=\\widetilde{{N}}_{Q_0}\\begin{pmatrix} u_h+u_0\\\\ Q_h+Q_0 \\end{pmatrix}+{S}_{Q_0}\\begin{pmatrix} u_0\\\\ Q_0 \\end{pmatrix}.$ The right-hand side of (REF ) is a translated version of (REF ) and is a mapping between linear spaces rather than affine spaces: $ {N }_{(u_0,Q_0)}\\begin{pmatrix}u_h\\\\Q_h\\end{pmatrix}:=\\widetilde{{N}}_{Q_0}\\begin{pmatrix}u_h+u_0\\\\Q_h+Q_0\\end{pmatrix}+ {S}_{Q_0}\\begin{pmatrix}u_0\\\\Q_0\\end{pmatrix}.$ So we end up with the following abstract parabolic system that is equivalent to (REF ): ${L}_{Q_0}\\begin{pmatrix} u_h\\\\ Q_h\\end{pmatrix}={N }_{(u_0,Q_0)}\\begin{pmatrix}u_h\\\\Q_h\\end{pmatrix}\\,.$ To incorporate the initial-boundary condition (), we turn to the definition of functional spaces $X_0$ and $Y_0$ such that ${L}_{Q_0},\\,{N }_{(u_0,Q_0)}:X_0\\mapsto Y_0$ with ${L}_{Q_0}$ being an isomorphism.", "Motivated by the idea to construct solutions which are twice differentiable in time and the precise assertions in Theorem REF , we need to prove the existence of regular solutions of the linear equation ${L}_{Q_0}\\begin{pmatrix}u_h\\\\Q_h\\end{pmatrix} =\\begin{pmatrix}f\\\\g\\end{pmatrix}$ subject to homogeneous initial data with right-hand side $(f,g)\\in Y_0$ .", "The general linear theory requires a compatibility condition which is taken care of by the definition of $Y_0$ as $ Y_0&=\\left\\lbrace (f,g)\\in H^1(0,T;H^{-1}_{\\sigma }(\\Omega ))\\times H^1(0,T;L^2(\\Omega ;\\mathcal {Q})):( f, g)\|_{t=0}\\in L^2_\\sigma (\\Omega )\\times H^1_0(\\Omega ) \\right\\rbrace \\,.$ These spaces are equipped with the usual norms in product spaces and for spaces of functions of one variable with values in a Banach space together with the correct norm of the initial data.", "More precisely, the norm of $Y_0$ is given by $\\Vert (f,g)\\Vert ^2_{Y_0}= \\Vert f\\Vert _{H^1(0,T;H^{-1}_{\\sigma }(\\Omega )) }^2+\\Vert g\\Vert ^2_{H^1(0,T;L^2(\\Omega ))}+\\Vert (f,g)\|_{t=0}\\Vert _{L^2_\\sigma (\\Omega )\\times H^1(\\Omega )}^2 \\,.$ Note that the last part of the norm is not controlled by applying trace theorems to the first two parts.", "Now we turn to the domain of ${L}_{Q_0}$ : $\\begin{split}X_u&=H^2(0,T;H^{-1}_{\\sigma }(\\Omega )) \\cap H^1(0,T;H^1_{0,\\sigma }(\\Omega )),\\\\X_Q&=H^2(0,T;L^2(\\Omega ;\\mathcal {Q}))\\cap H^1(0,T;H^2(\\Omega ;\\mathcal {Q})),\\end{split}$ together with the norms $\\begin{split}\\Vert u\\Vert ^2_{X_u}&= \\Vert u\\Vert _{H^2(0,T;H^{-1}_{\\sigma }(\\Omega )) }^2+\\Vert u\\Vert _{H^1(0,T;H^1_{0,\\sigma }(\\Omega ))}^2+\\Vert u\|_{t=0}\\Vert _{H^1_{0,\\sigma }(\\Omega )}^2+\\Vert u_t\|_{t=0}\\Vert _{L^2(\\Omega )}^2 \\,,\\\\ \\Vert Q\\Vert _{X_Q}^2&= \\Vert Q\\Vert _{H^2(0,T;L^2(\\Omega ;\\mathcal {Q}))}^2+\\Vert Q\\Vert _{H^1(0,T;H^2(\\Omega ;\\mathcal {Q}))}^2+\\Vert Q\|_{t=0}\\Vert _{ H^2(\\Omega )}^2+\\Vert \\partial _t Q\|_{t=0}\\Vert _{ H^1(\\Omega )}^2 \\,.\\end{split}$ Note that the last two terms in the norms are important to obtain in the sequel constants that are uniformly bounded as $T\\rightarrow 0$ .", "The corresponding subspaces related to the homogeneous initial and boundary conditions in the formulation of the problem are defined by $X_0= \\left\\lbrace (u,Q)\\in X_u\\times X_Q\\mid Q\|_{\\partial \\Omega }=0,\\left(u,Q \\right)\|_{t=0}=\\left(0,0 \\right)\\right\\rbrace ,$ which is equipped with the product norm $\\Vert (u,Q)\\Vert _{X_0}=\\Vert (u,Q)\\Vert _{X_u\\times X_Q}\\,.$ Proposition 3.2 If $(u_h,Q_h)$ is a strong solution to (REF ), then $(u,Q)= (u_h,Q_h)+(u_0,Q_0)$ is a solution to () and (REF )." ], [ "Proof of Theorem ", "The first step towards the proof of the local in time existence of strong solutions is to construct a regular in time solution, following the method in [1].", "The following result establishes the invertibility of the linear operator equation.", "Note that we are seeking a solution of the linear equation in $X_0$ , i.e., a solution with homogeneous initial and boundary conditions.", "Proposition 4.1 For any fixed $T\\in (0,1]$ , ${L}_{Q_0}$ defined by (REF ) is a bounded linear operator $X_0\\mapsto Y_0$ and for every $(f,g)\\in Y_0$ , the operator equation ${L}_{Q_0} (u,Q) = (f,g)$ has a unique solution $(u,Q)\\in X_0$ satisfying $\\Vert {L}^{-1}_{Q_0}(f,g)\\Vert _{X_0}=\\Vert (u,Q)\\Vert _{X_0}\\le C_{{L}}\\Vert (f,g)\\Vert _{Y_0},$ where $C_{{L}}$ is independent of $T\\in (0,1]$ .", "In particular ${L}_{Q_0}:X_0\\mapsto Y_0$ is invertible and ${L}^{-1}_{Q_0}$ is a bounded linear operator with norm independent of $T\\in (0,1]$ .", "In order to apply Proposition REF , we define the Hilbert spaces $\\mathbb {H}= \\mathbb {H}_1 \\times \\mathbb {H}_2& =L^2_\\sigma (\\Omega )\\times H^1_{0}( \\Omega ;\\mathcal {Q}) \\,,\\\\\\mathbb {V}= \\mathbb {V}_1 \\times \\mathbb {V}_2& = H^1_{0,\\sigma }(\\Omega )\\times \\left( H^2( \\Omega ;\\mathcal {Q})\\cap H^1_{0}( \\Omega ;\\mathcal {Q}) \\right),$ and equip them with standard product Sobolev norm.", "The dual spaces of $\\mathbb {V}$ with respect to pivot space $\\mathbb {H}$ is $\\mathbb {V}^{\\prime }=\\mathbb {V}_1^{\\prime }\\times \\mathbb {V}_2^{\\prime }=H^{-1}_\\sigma (\\Omega )\\times L^2(\\Omega ;\\mathcal {Q}),$ and the dual product is given by $\\bigl < (u,Q), (\\varphi ,\\Phi )\\bigr >_{\\mathbb {V}^{\\prime },\\mathbb {V}}:=\\bigl < u, \\varphi \\bigr >_{\\mathbb {V}_1^{\\prime } \\times \\mathbb {V}_1 }+\\bigl < Q, \\Phi \\bigr >_{\\mathbb {V}_2^{\\prime } \\times \\mathbb {V}_2 }=\\bigl < u, \\varphi \\bigr >_{H^{-1}_\\sigma ,H^1_\\sigma }+\\int _{\\Omega } Q:\\mathcal {L}(\\Phi )\\,{\\rm d}x,$ according to Lemma REF .", "As a result, the space $Y_0$ defined by (REF ) can be written by $Y_0=\\lbrace (f,g) \\in H^1(0,T;\\mathbb {V}^{\\prime })\\mid (f,g)\|_{t=0}\\in \\mathbb {H}\\rbrace .$ We shall define the bilinear form $a(\\cdot ,\\cdot )$ on $\\mathbb {V}$ by $a((u,Q),(v,P))=& \\int _{\\Omega } \\Big (D(u)+\\mathcal {S}_{Q_0}(\\mathcal {L}(Q))-Q_0\\cdot \\mathcal {L}(Q)+\\mathcal {L}(Q) \\cdot Q_0\\Big ):D v \\,{\\rm d}x\\\\&+\\int _{\\Omega } \\Big (\\mathcal {L}(P)-\\mathcal {S}_{Q_0}(D(u))+Q_0 \\cdot W(u)-W(u) \\cdot Q_0\\Big ):\\mathcal {L}(P)\\,{\\rm d}x.$ One can verify that this bilinear form satisfies the hypothesis for applying Proposition REF .", "Especially, coerciveness follows from cancellation law (REF ), $a((u,Q),(u,Q))=& \\int _{\\Omega } \\Big (D(u)+\\mathcal {S}_{Q_0}(\\mathcal {L}(Q))-Q_0\\cdot \\mathcal {L}(Q)+\\mathcal {L}(Q)\\cdot Q_0\\Big ):D u \\,{\\rm d}x\\\\&+\\int _{\\Omega } \\Big (\\mathcal {L}(Q)-\\mathcal {S}_{Q_0}(D(u))+Q_0 \\cdot W(u)-W(u)\\cdot Q_0\\Big ):\\mathcal {L}(Q)\\,{\\rm d}x\\\\=&\\int _{\\Omega } (\|D(u)\|^2+\|\\mathcal {L}(Q)\|^2)\\,{\\rm d}x\\ge C\\Vert (u,Q)\\Vert ^2_{\\mathbb {V}}.$ In the last step, we employed Corollary REF .", "So there exists a bounded linear operator $L:\\mathbb {V}\\mapsto \\mathbb {V}^{\\prime }$ such that $\\bigl < L (u, Q), (\\varphi ,\\Phi )\\bigr >_{\\mathbb {V}^{\\prime },\\mathbb {V}}=a((u,Q),(\\varphi ,\\Phi )).$ Moreover, for any $(f,g)\\in Y_0$ , the abstract evolution equation $\\bigl < (\\partial _t u,\\partial _t Q), (\\varphi ,\\Phi )\\bigr >_{\\mathbb {V}^{\\prime },\\mathbb {V}}+\\bigl < L (u, Q), (\\varphi ,\\Phi )\\bigr >_{\\mathbb {V}^{\\prime },\\mathbb {V}}= \\langle (f,g), (\\varphi , \\Phi )\\rangle _{\\mathbb {V}^{\\prime },\\mathbb {V}}$ has a unique solution $(u,Q)$ satisfying $(u,Q)\\in H^{1}((0,T);\\mathbb {V})\\quad \\text{ and }\\quad (\\partial ^2_t u,\\partial _t^2 Q)\\in L^2\\bigl ((0,T);\\mathbb {V}^\\prime \\bigr )\\,.$ or shortly $(u,Q)\\in X_0$ .", "Now we need to show that (REF ) is equivalent to (REF ): it is evident that, choosing $(\\varphi ,0) \\in \\mathbb {V}$ in (REF ) implies the first equation in (REF ).", "To identify the equation for $Q$ , we choose $(0,\\Phi ) \\in \\mathbb {V}$ as test function and deduce from (REF ) that $&\\int _{\\Omega } g(t,x):\\mathcal {L}( \\Phi (x))\\,{\\rm d}x\\\\&=\\langle \\partial _t Q,\\Phi \\rangle _{\\mathbb {V}_2^{\\prime },\\mathbb {V}_2}+\\int _{\\Omega } \\Big (\\mathcal {L}(Q)-\\mathcal {S}_{Q_0}(D(u))+Q_0\\cdot W(u)-W(u)\\cdot Q_0\\Big ):\\mathcal {L}(\\Phi )\\,{\\rm d}x\\\\ &= \\int _{\\Omega } \\left(\\partial _t Q +\\mathcal {L}(Q)-\\mathcal {S}_{Q_0}(D(u))+Q_0 \\cdot W(u)-W(u) \\cdot Q_0\\right):\\mathcal {L}(\\Phi ) \\,{\\rm d}x\\,.$ In view of Lemma REF , $\\mathcal {L}\\colon \\mathbb {V}_2\\mapsto L^2( \\Omega ;\\mathcal {Q})$ is bijective and thus $\\partial _t Q +\\mathcal {L}(Q)-\\mathcal {S}_{Q_0}(D(u))+Q_0 \\cdot W(u)-W(u) \\cdot Q_0 = g, \\qquad \\text{a.e.", "in } \\Omega \\times (0,T).$ Altogether, we have proven that ${L}_{Q_0}\\colon X_0\\mapsto Y_0$ is an isomorphism.", "Since ${L}_{Q_0}$ is also a bounded linear operator and the operator norm only depends on $Q_0$ and geometry of $\\Omega $ , the boundedness of its inverse operator ${L}_{Q_0}^{-1}\\colon Y_0\\mapsto X_0$ follows from inverse mapping theorem.", "The assertion that $C_{{L}}$ is independent of $T$ follows from standard energy estimate and the cancellation law (REF ).", "Here we omit the details.", "Proposition 4.2 Fix $0<T\\le 1$ , $R>0$ , $(u_0, Q_0)\\in {I}$ .", "Let ${N}_{(u_0,Q_0)}$ be the nonlinear operator defined in (REF ) and ${B}_{X_0}(0,R)=\\lbrace (v,P)\\in X_0,\\, \\Vert (v,P)\\Vert _{X_0}\\le R\\rbrace $ .", "Then the following assertions hold for all $(u_i,Q_i)\\in {B}_{X_0}(0,R)$ , $i=1,\\,2$ : (i) ${N}_{(u_0,Q_0)}$ maps $X_0$ to $Y_0$ .", "(ii) Local Lipschitz continuity: there exists a constant $C_{{N}}(T,R,Q_0,u_0)>0$ such that $\\begin{aligned}&\\Vert {N}_{(u_0,Q_0)}(u_1,Q_1)-{N}_{(u_0,Q_0)}(u_2,Q_2)\\Vert _{Y_0} \\\\&\\,\\qquad \\qquad \\le C_{{N}}(T,R,Q_0,u_0)\\Vert (u_1-u_2,Q_1-Q_2)\\Vert _{X_0}\\,.\\end{aligned}$ (iii) Local boundedness: $\\Vert {N}_{(u_0,Q_0)}(u_1,Q_1)\\Vert _{Y_0}\\le C_{{N}}(T,R,Q_0,u_0)\\Vert (u_1,Q_1)\\Vert _{X_0}+\\Vert \\mathcal {E}(u_0, Q_0)\\Vert _{Y_0}\\,.$ where $\\mathcal {E}$ is given by (REF ).", "(iv) For any fixed $R>0$ , $\\lim _{T\\rightarrow 0}C_{{N}}(T,R,Q_0,u_0)= 0$ .", "The proof of this result can be adapted line by line from [1].", "Actually, the proof in [1] is slightly more general since they work with variable viscosity in the fluid equation and mixed boundary condition for the $Q$ -tensor field.", "The proof will be divided into two steps.", "First, we shall use Proposition REF and REF to prove the existence and uniqueness of a regular in time solution.", "Based on this, in the second step, owning to a special structure of the system, we improve the spatial regularity of $u$ and also $Q$ and this leads to the strong solution of ().", "Step 1: Regularity in time.", "We first show that ${A} := {L}^{-1}_{Q_0}{N}_{(u_0,Q_0)}:X_0\\mapsto X_0$ has a unique fixed-point.", "By (REF ) and (REF ) we find for all $(u_{hi},Q_{hi})\\in {B}_{X_0}(0,R)$ that $&\\bigl \\Vert {L}_{Q_0}^{-1}{N}_{(u_0,Q_0)} (u_{h1},Q_{h1})- {L}^{-1}_{Q_0}{N}_{(u_0,Q_0)}(u_{h2},Q_{h2}) \\bigr \\Vert _{X_0}\\\\ &\\, \\qquad \\le C_{{L}}\\Vert {N}_{(u_0,Q_0)}(u_{h1},Q_{h1})-{N}_{(u_0,Q_0)}(u_{h2},Q_{h2})\\Vert _{Y_0}\\\\ &\\, \\qquad \\le C_{{L}}C_{{N} }(T,R,u_0,Q_0)\\Vert (u_{h1}-u_{h2},Q_{h1}-Q_{h2})\\Vert _{X_0}\\,.$ Therefore ${A} $ is a contraction mapping for $T\\ll 1$ .", "A similar argument shows that ${A}$ maps ${B}_{X_0}(0,R)$ into itself.", "In fact, by (REF ), we deduce that $\\bigl \\Vert {A} (u_{h1}, Q_{h1}) \\bigr \\Vert _{X_0}&\\le C_{{L}} \\bigl \\Vert {N}_{(u_0,Q_0)}(u_{h1},Q_{h1}) \\bigr \\Vert _{Y_0}\\\\&\\le C_{{L}}\\bigl ( C_{{N}}(T,R,u_0,Q_0)\\Vert (u_{h1},Q_{h1})\\Vert _{X_0}+\\Vert \\mathcal {E}(u_0,Q_0)\\Vert _{Y_0}\\bigr ).$ So we can fix $R \\gg 1$ large enough and then choose $T \\ll 1$ small enough in such a way that $\\bigl \\Vert {A}(u_{h1},Q_{h1}) \\bigr \\Vert _{X_0}&\\le C_{{L}}C_{{N}}(T,R,u_0,Q_0)\\Vert (u_{h1},Q_{h1})\\Vert _{X_0} + \\frac{R}{2} \\le R\\,.$ We conclude from Banach's fixed-point theorem that ${A}$ possess a unique fixed-point $(u_h,Q_h)\\in X_0$ and it is a solution of the system (), according to (REF ) and Proposition REF .", "The argument implies the uniqueness as well.", "Suppose that there was another solution $(\\hat{u}_h,\\hat{Q}_h)$ in ${B}_{X_0}(0,R_1)$ with $R_1>R$ .", "Choose $\\hat{T}\\le T$ and repeat the above argument to show the uniqueness of fixed-points of ${A}$ , which implies $(u_h,Q_h)=(\\hat{u}_h,\\hat{Q}_h)$ on $(0,\\hat{T})\\times \\Omega $ .", "Then the uniqueness follows by the continuity argument.", "So $(u,Q)=(u_h,Q_h)+(u_0,Q_0)$ is a solution of () with $\\begin{split}u&\\, \\in H^2(0,T;H^{-1}_{\\sigma }(\\Omega ))\\cap H^1(0,T;H^1_{0,\\sigma }(\\Omega ))\\,,\\\\Q&\\, \\in H^2(0,T;L^2(\\Omega ;\\mathcal {Q}))\\cap H^1(0,T;H^2(\\Omega ;\\mathcal {Q}))\\,.\\end{split}$ and it follows from standard interpolation result that $u\\in C([0,T];H^1_{0,\\sigma }(\\Omega )),\\quad Q\\in C([0,T];H^2(\\Omega ;\\mathcal {Q}))$ and $\\begin{split}u_t \\in C([0,T];L^2_\\sigma (\\Omega )),\\quad Q_t \\in C([0,T];H^1(\\Omega ;\\mathcal {Q})).\\end{split}$ These also imply, together with Sobolev embedding that $\\begin{split}Q_t+u\\cdot \\nabla Q\\in C([0,T];W^{1,3/2}(\\Omega )).\\end{split}$ Step 2: Spatial regularity.", "For any vector field $v\\in H^1(\\Omega ;\\mathbb {R}^{3})$ , we denote the symmetric matrix $\\mathcal {T}( Q ,\\nabla v):=\\mathcal {S}_Q ( D(v))- Q \\cdot W(v)+ W(v)\\cdot Q,$ where the operator $\\mathcal {S}_Q$ is defined by (REF ).", "We also denote $\\sigma ( Q ,\\nabla v):= \\mathcal {S}_Q (\\mathcal {T}( Q ,\\nabla v))- Q \\cdot \\mathcal {T}( Q ,\\nabla v)+\\mathcal {T}( Q ,\\nabla v)\\cdot Q .$ Note that, for any vector field $v\\in H^1(\\Omega ;\\mathbb {R}^{3})$ , not necessarily divergence-free, $\\mathcal {T}( Q ,\\nabla v)$ is traceless and symmetric according to (REF ).", "Then we have from $Q$ -tensor equation () that $\\mathcal {H}(Q) =- \\mathcal {T}( Q , \\nabla u)+ (\\partial _t Q + u\\cdot \\nabla Q ).$ In addition, if we define $f:=-\\mathcal {S}_Q(Q_t+u\\cdot \\nabla Q)+Q\\cdot (Q_t+u\\cdot \\nabla Q)-(Q_t+u\\cdot \\nabla Q)\\cdot Q,$ then the following identity holds: $\\begin{split}& D(u)+ \\sigma ( Q ,\\nabla u)+f\\\\=& D(u)+\\mathcal {S}_Q(\\mathcal {T}(Q,\\nabla u)-(Q_t+u\\cdot \\nabla Q))\\\\&+Q\\cdot (Q_t+u\\cdot \\nabla Q-\\mathcal {T}(Q,\\nabla u))-(Q_t+u\\cdot \\nabla Q-\\mathcal {T}(Q,\\nabla u))\\cdot Q\\\\=& D(u)-\\mathcal {S}_Q(\\mathcal {H}(Q))+Q\\cdot \\mathcal {H}(Q)-\\mathcal {H}(Q)\\cdot Q\\\\=&\\sigma ^s+\\sigma ^a,\\end{split}$ or equivalently $D(u)+ \\sigma ( Q ,\\nabla u)+f=\\sigma ^s+\\sigma ^a.$ Substituting (REF ) into (REF ) leads to $\\nabla \\cdot ( D(u)+\\sigma (Q,\\nabla u))+\\nabla P=-\\nabla \\cdot (f+\\sigma ^d)+u\\cdot \\nabla u+u_t.$ If we denote $\\widetilde{f}:= -\\nabla \\cdot (f+\\sigma ^d) +u\\cdot \\nabla u,$ where $f$ is defined by (REF ), then due to the regularity result (REF ), we can show that $\\widetilde{f}\\in C([0,T];L^{3/2}(\\Omega )).$ Actually, it follows from (REF ), (REF ) and Sobolev embedding that $\\mathcal {S}_Q(Q_t+u\\cdot \\nabla Q)\\in C([0,T];W^{1,3/2}(\\Omega )\\cap L^3(\\Omega ))$ and also $f\\in C([0,T];W^{1,3/2}(\\Omega ))$ .", "These together with (REF ) imply (REF ).", "The crucial observation is that, (REF ) is a Stokes system with variable coefficient.", "To show this, we claim that, the bilinear form $a(u,v):= \\langle D(u)+ \\sigma (Q,\\nabla u),\\nabla v\\rangle $ defines a symmetric positive definite bilinear form on $H^1(\\Omega ;\\mathbb {R}^3)$ .", "Actually, note that $\\mathcal {S}_Q (\\mathcal {T})$ is symmetric while $ Q \\cdot \\mathcal {T}-\\mathcal {T}\\cdot Q $ is antisymmetric, we infer from (REF ) that $\\begin{split}\\sigma ( Q ,\\nabla v):\\nabla u &= \\Big ( \\mathcal {S}_Q (\\mathcal {T}( Q ,\\nabla v))- Q \\cdot \\mathcal {T}( Q ,\\nabla v)+\\mathcal {T}( Q ,\\nabla v)\\cdot Q \\Big ):\\nabla u \\\\\\nonumber &=\\Big ( \\mathcal {S}_Q(D(u))- Q \\cdot W(u)+W(u)\\cdot Q \\Big ): \\mathcal {T}(Q,\\nabla v) \\\\&= \\mathcal {T}(Q,\\nabla u) : \\mathcal {T}(Q,\\nabla v).\\end{split}$ This formula together with the definition of $\\mathcal {T}( Q ,\\nabla v)$ implies that, there exists a smooth tensor-valued function $\\left\\lbrace \\hat{A}^{k\\ell }_{ij}(z): \\mathcal {Q}\\mapsto \\mathbb {R}\\right\\rbrace _{1\\le i,j,k,\\ell \\le 3}$ with $\\xi _i^k\\hat{A}^{k\\ell }_{ij}(z)\\xi _j^\\ell \\ge 0,~\\forall \\xi \\in \\mathbb {R}^{3\\times 3},\\quad \\forall z\\in \\mathcal {Q}$ such that the following identity holds almost everywhere for $(x,t)\\in \\Omega _T$ : $\\sigma (Q(x,t),\\nabla u):\\nabla v = \\partial _k u_i\\hat{A}^{k\\ell }_{ij}(Q(x,t))\\partial _\\ell v_j,\\quad \\forall u,v\\in H^1(\\Omega ;\\mathbb {R}^3).$ Consequently, the system (REF ) can be reduced to $\\left\\lbrace \\begin{array}{rl}\\partial _i \\left((\\delta _{i}^j\\delta _{k}^\\ell +\\hat{A}^{k\\ell }_{ij}(Q(x,t)))\\partial _j u_\\ell \\right)+\\partial _k P&=\\widetilde{f}_k+\\partial _t u_k\\\\\\nabla \\cdot u&=0,\\\\u\|_{\\partial \\Omega }&=0.\\end{array}\\right.$ Then it follows from Corollary REF as well as (REF ) that $\\Vert u\\Vert _{W^{2,3/2}(\\Omega )}+\\Vert \\nabla P\\Vert _{L^{3/2}(\\Omega )}\\le C \\left(1+\\Vert Q\\Vert _{C([0,T];H^2(\\Omega ))}\\right)\\Vert \\widetilde{f}+\\partial _t u\\Vert _{L^{3/2}(\\Omega )}, ~a.e.~t\\in [0,T],$ and this yields the second order derivative estimate for the velocity field: $\\begin{split}&\\Vert u\\Vert _{L^\\infty (0,T;W^{2,3/2}(\\Omega ))} +\\Vert \\nabla P\\Vert _{L^\\infty (0,T;L^{3/2}(\\Omega ))}\\\\\\le &C \\left(1+\\Vert Q\\Vert _{C([0,T];H^2(\\Omega ))}\\right)\\Vert \\widetilde{f}+\\partial _t u\\Vert _{C([0,T];L^{3/2}(\\Omega ))}.\\end{split}$ On the other hand, we can write () as $\\nonumber Q_t-\\mathcal {L}(Q)&=-\\mathcal {J}(Q)-u\\cdot \\nabla Q - Q \\cdot W(u)+W(u)\\cdot Q +\\mathcal {S}_Q (D(u))\\\\ &=: \\mathcal {N}(u,\\nabla u, Q, \\nabla Q).$ We claim that $\\mathcal {N}(u,\\nabla u,Q,\\nabla Q)\\in L^\\infty (0,T;W^{1,3/2}(\\Omega )).$ Indeed, it follows from (REF ) that $\\nabla u\\in L^\\infty (0,T;W^{1,3/2}\\cap L^3(\\Omega ))$ and this together with (REF ), Sobolev embedding and Hölder's inequality implies (REF ).", "Consequently, we can apply Lemma REF to deduce the higher order regularity of $Q$ : $\\begin{split}&\\Vert Q\\Vert _{L^\\infty (0,T;W^{3,3/2}(\\Omega ))}\\\\\\le &C \\left(\\Vert \\mathcal {N}(u,\\nabla u,Q,\\nabla Q)\\Vert _{L^\\infty (0,T;W^{1,3/2}(\\Omega ))}+\\Vert Q_t\\Vert _{L^\\infty (0,T;W^{1,3/2}(\\Omega ))}+\\Vert Q\\Vert _{W^{3-2/3,3/2}(\\partial \\Omega )}\\right).\\end{split}$ Combining (REF ), (REF ) and (REF ), one can verify as previously that (REF ) can be improved to be $Q_t+u\\cdot \\nabla Q\\in L^\\infty ( 0,T ;H^{1}(\\Omega )),$ and this will in turn improve (REF ) to be $\\mathcal {S}_Q(Q_t+u\\cdot \\nabla Q)\\in L^\\infty ( 0,T ;H^1(\\Omega )).$ So we end up with an improved estimate $\\widetilde{f}\\in L^{\\infty }( 0,T ;L^2(\\Omega ))$ , in contrast to (REF ).", "So one can argue in the same manner as in the previous step by employing the second part of Corollary REF : $\\Vert u\\Vert _{L^{\\infty }(0,T;H^{2}(\\Omega ))} \\le C \\Vert (\\widetilde{f},u_t)\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))}.$ This implies $\\mathcal {S}_Q(D(u))\\in L^\\infty (0,T;H^1(\\Omega ))$ and thus $ \\mathcal {N}(u,\\nabla u,Q,\\nabla Q)\\in L^\\infty (0,T;H^1(\\Omega ))$ .", "Together with (REF ) and the boundary condition $Q\|_{\\partial \\Omega }=Q_0\|_{\\partial \\Omega }\\in H^{5/2}(\\partial \\Omega )$ , we can employ Lemma REF to enhance the regularity of (REF ) by $Q\\in L^\\infty (0,T;H^3(\\Omega ))$ .", "This completes the proof of the main theorem." ], [ "Acknowledgment", "Y. Liu is supported by the Affiliated Faculty Research Grant of NYU shanghai.", "W. Wang is supported by NSF of China under Grant 11501502 and “the Fundamental Research Funds for the Central Universities\" 2016QNA3004." ] ]	1606.05216
[ [ "Particle models with self sustained current" ], [ "Abstract We present some computer simulations run on a stochastic CA (cellular automaton).", "The CA simulates a gas of particles which are in a channel, the interval $[1,L]$ in $\\mathbb Z$, but also in \"reservoirs\" $\\mathcal R_1$ and $\\mathcal R_2$.", "The evolution in the channel simulates a lattice gas with Kawasaki dynamics with attractive Kac interactions, the temperature is chosen smaller than the mean field critical one.", "There are also exchanges of particles between the channel and the reservoirs and among reservoirs.", "When the rate of exchanges among reservoirs is in a suitable interval the CA reaches an apparently stationary state with a non zero current, for different choices of the initial condition the current changes sign.", "We have a quite satisfactory theory of the phenomenon but we miss a full mathematical proof." ], [ "Introduction", "In this paper we introduce models of macroscopic dissipative systems made of interacting particles which move stochastically in a circuit and exhibit a very surprising behavior.", "Despite the fact that there is no external bias we see, after a transient, an apparently stationary state with a non zero current, with suitably different initial conditions we may select another state with the opposite value of the current.", "We speculate that on much longer times there is a “dynamical phase transition” with the two states alternating one after the other.", "To make an analogy with equilibrium phase transitions, consider the 2D Ising model in a large but finite box with nearest neighbor ferromagnetic interactions.", "Running the Glauber dynamics at a temperature below the critical value we typically see long time intervals where the magnetization density has approximately the plus equilibrium value alternating via tunneling with those where it is close to the minus equilibrium value.", "The analogue of the equilibrium magnetization in our model is the current as we have two states with opposite values of the current.", "However we observe our circuit for times long but much smaller than those for tunneling so that we only see one of the two currents (selected by the initial condition) which then looks stationary.", "Our analysis relies mostly on computer simulations, we have theoretical explanations but we miss a mathematical proof.", "There is a huge literature on the more general question of existence of periodic motions or oscillations especially in the context of biological systems and chemical reactions, the classical reference is the book by Kuramoto, [11].", "We just quote here a few examples selected with the purpose of introducing what we will be doing in this paper.", "In [15] P. Tass discusses a simple system of rotators which interact attractively with each other and are subject to white noise forces.", "For small interactions the stationary state is homogeneous and even though each particle rotates there is no macroscopic change.", "However if the interaction increases the rotators form a macroscopic cluster which then moves periodically.", "This is a simplified model for neural activities, the angle of the rotator is related to the neuron potential and the crossing from $2\\pi $ to 0 is interpreted as the neuron discharging its potential (“firing”): the appearance of a cluster causes a great potential change when the cluster crosses $2\\pi $ which could explain some diseases related to anomalous neuron firing.", "A quantum analogue of the rotator model has been studied by Wilczek in [16] where it is shown that there are ground states with a localized cluster which rotates, this phenomenon called a “time crystal”.", "Comments on time crystals can be found in [1].", "Experimental evidence of “time crystals” are presented in [18].", "Time crystals in a classical (i.e.", "non quantum) context have been considered in [17].", "A rotators model is also considered in [9] where an additional external force is present.", "The main point in the paper is to show that for a critical set of values of the parameters there is a cluster which is however blocked (by the external force).", "However if the white noise strength is increased then the cluster starts moving and performs a periodic motion, this being a nice example of noise-induced periodicity.", "Also in our models noise is the fuel which makes the system run.", "In the above models each particle by itself rotates: the macroscopic rotations arise from a “phase synchronization” of the rotators.", "Instead in the FitzHugh Nagumo class of models for the firing cycles of a neuron, the appearance of periodic motions is due to a different, more intrinsic mechanism.", "For what follows it is convenient to consider a particular model in the class which can and will be read in a statistical mechanics language.", "In such a context the model is defined by two (macroscopic) variables, the magnetization $m$ and the magnetic field $h$ .", "$m$ is the “fast” and $h$ the “slow variable” as the evolution is defined by the equations: ${ \\epsilon }\\frac{dm}{dt} = -m + \\tanh \\lbrace \\beta (m + h)\\rbrace ,\\quad \\frac{dh}{dt} =-m$ where ${ \\epsilon }>0$ is the “small parameter” and $\\beta >1$ the inverse temperature.", "It can be seen that (REF ) has a (stable) periodic solution which in the limit ${ \\epsilon }\\rightarrow 0$ becomes the hysteresis cycle: $m= m_\\pm (h)$ , $m_{+}(h)$ the positive solution of $m = \\tanh \\lbrace \\beta (m + h)\\rbrace $ which exists for $h>-h_c$ , $h_c>0$ ; $m_{-}(h)=-m_{+}(-h)$ , $h<h_c$ , see Fig.", "REF .", "The transition from the upper curve $m_{+}(\\cdot )$ to the lower one $m_{-}(\\cdot )$ (and viceversa) is discontinuous and hence very sharp for ${ \\epsilon }>0$ small, a fact which catches the main feature of the neuron voltage cycle namely that at the firing the potential changes very abruptly.", "Observe that $m_+(h)$ is metastable for $h<0$ as well as $m_-(h)$ for $h>0$ , the metastable values of the magnetization will play a fundamental role also in this paper.", "Figure: Hysteresis cycle, with β=2.5\\beta =2.5.Dai Pra et al., [4], derived similar patterns in a macroscopic limit from a Ising spin model with mean field interactions giving nice examples of “intrinsic” periodic oscillations in the stochastic Ising model.", "In this paper we will consider the relaxed version of mean field as defined by Kac potentials.", "All the above examples can be interpreted in terms of a current in a circuit but in all of them there is a more or less hidden bias because the current can flow only in one direction and not in the opposite one, so that they do not fit in what we are looking for.", "However they have all a common feature with our models, namely the presence of a phase transition, responsible in the rotator models for the formation of a cluster and in the FitzHugh Nagumo models for the presence of a hysteresis cycle.", "The way phase transitions appear in our analysis is the following.", "In a first order phase transition there is a spontaneous separation of phases which gives rise to gradients of the order parameter without currents being present.", "The Fourier law associates to a gradient a current (in the opposite direction) so that the phase transition generates “effective forces” which prevent the gradients to give rise to currents.", "Our idea is to exploit such forces to construct a “battery” which allows for a non zero current in a circuit.", "Our battery is a cellular automaton which simulates the Kawasaki dynamics in a lattice gas with interactions given by an attractive Kac potential which in the Lebowitz-Penrose limit has a van der Waals phase transition.", "Therefore we can distinguish between stable, metastable and unstable values of the density.", "The main and somehow unexpected feature of the system is that if we connect the endpoints of the channel to “infinite” (i.e.", "true) reservoirs which fix the density at values $\\rho _- $ and $\\rho _+= 1-\\rho _->\\rho _-$ with $\\rho _{\\pm }$ metastable densities we observe numerically a current which goes through the channel from the reservoir with smaller density $\\rho _-$ to the one with the larger density $\\rho _+$ .", "We have a theoretical explanation of the phenomenon in terms of properties of the solution of an integral equation obtained from the process in the “mesoscopic limit” where the scaling parameter ${ \\gamma }$ of the Kac potential vanishes, but we could verify these properties only numerically.", "In [3] we have presented numerical evidence that the current in the CA flows from the reservoir with smaller density to the one with larger density.", "In this paper we present a more complete set of simulations from where a very complex structure emerges for which we have a theoretical explanation, but we miss a complete mathematical proof.", "The other main point in this paper is that we can exploit the above to construct a circuit with a self sustained current without an external bias, as claimed in the first sentence of this Introduction.", "This is obtained by making the reservoirs finite and allowing also particles exchanges among the reservoirs.", "We show (via the simulations) that for suitable values of the parameters there are initial conditions which give rise to a steady non zero current (stationary for the times of our simulations); there are also other initial conditions where the current flows in the opposite direction and still others where there is no current at all.", "The state with zero current seems unstable while those with a non zero current seem locally stable.", "As suggested by a referee, similar phenomena have also been studied in other models, e.g.", "the Bunimovich's mushroom billiard model [2] in which the presence of peculiar transport regimes can be traced back to the lack of ergodicity of the microscopic dynamics; but we have not yet explored this issue.", "The paper is organized as follows.", "In Section we define two different versions of the CA used in the simulations, namely: one describing a single (open) channel in contact with two reservoirs (hereafter called OS-CA), and another mimicking the particle dynamics in a closed circuit (called CC-CA).", "In Section we present the results of the simulations obtained by running the OS-CA and also explain how to run the CC-CA by exploiting the results first obtained with the OS-CA.", "In Section we illustrate the behavior of the particle current in the CC-CA and comment on the dependence of this quantity on the parameters of the model.", "In Section we study the continuum (mesoscopic) limits of both the OS-CA and the CC-CA, which are described by an integro-differential equation; proofs are deferred to the Appendix .", "In Section we discuss the adiabatic limit of the model, and check the consistency of our simulations of the CC-CA with the predicted adiabatic behavior.", "In Section we consider the case where the reservoirs have stable densities and in Section where the densities are not stable.", "In Section we study the stability of a stationary density profile, referred to below as the “bump” solution, close to the boundary.", "Concluding remarks are finally drawn in Section ." ], [ "The cellular automata", "In this section we define two cellular automata: the first one, called “open system cellular automaton”, OS-CA in short, has been first introduced in [12] and then used in [3] to simulate a system in contact with reservoirs.", "The second one, simply called “closed circuit cellular automaton”, CC-CA, is a modification of the first one obtained by making finite the reservoirs and adding direct exchanges between them, so that it simulates a closed circuit.", "The OS-CA.", "The OS-CA describes the evolution of particles in a “channel\" $ \\lbrace 1,2,..,L\\rbrace $ , $L>1$ a positive integer.", "Besides moving in the channel particles may also leave from or enter into the channel through $L$ and 1 (we then say that they are absorbed or released from the reservoir $\\mathcal {R}_2$ if this happens at $L$ and from reservoir $\\mathcal {R}_1$ if it happens at 1).", "The two reservoirs are “infinite\" in the sense that they do not have memory of the particles which are absorbed or released.", "The CA in the channel is a parallel updating version of a weakly asymmetric simple exclusion process, designed for computer simulations.", "The $d=1$ symmetric simple exclusion process is a system of random walks jumping to the right and left with equal probability, the jump being suppressed if the arrival site is occupied.", "The weak asymmetry that we add is a small bias to jump in the direction where the density is higher.", "If the channel was a torus this would produce a phase separation into a region where the density is higher and another where it is smaller.", "But our channel is open as particles may leave or enter into the channel in a setup typical of the Fourier law but in a context where phase transitions are present.", "Let us now go back to the definition of the CA.", "The phase space is $\\mathcal {S}=\\lbrace (x,v), x\\in \\lbrace 1,..,L\\rbrace , v\\in \\lbrace -1,1\\rbrace \\rbrace $ , particles configurations are functions $\\eta :\\mathcal {S}\\rightarrow \\lbrace 0,1\\rbrace $ , $\\eta (x,v) \\in \\lbrace 0,1\\rbrace $ denotes the occupation variable at $(x,v)$ and $v$ will be interpreted as a velocity.", "$\\eta (x)=\\eta (x,-1)+\\eta (x,1)\\in \\lbrace 0,1,2\\rbrace $ denotes the total number of particles at $x$ .", "We may add a suffix $t$ when the occupation variables are computed at time $t$ .", "The definition of the OS-CA involves four more parameters: ${ \\gamma }^{-1}\\in \\mathbb {N}$ , $C>0$ and $\\rho _{\\pm }\\in [0,1]$ .", "In the simulations presented in this paper we have fixed ${ \\gamma }^{-1}=30$ , $C=1.25$ , while the length of the channel is set equal to $L=600$ .", "$\\rho _{\\pm }$ are referred to as the density of reservoir $\\mathcal {R}_2$ , respectively $\\mathcal {R}_1$ , they are fixed during a simulation but they may be changed in different simulations.", "In the definition of the CA we will use the notation $N_{+,x,{ \\gamma }}= \\sum _{y=x+1}^{ x+{ \\gamma }^{-1}}\\eta ^{(+)}(y),\\;N_{-,x,{ \\gamma }}= \\sum _{y= x-{ \\gamma }^{-1}}^{x-1}\\eta ^{(-)}(y),\\quad x \\in [1,L]$ where $\\eta ^{(+)}(y)= \\eta (y)$ if $y \\in [1,L]$ and $\\eta ^{(+)}(y)= 2\\rho _{+}$ if $y >L$ ; similarly $\\eta ^{(-)}(y)= \\eta (y)$ if $y \\in [1,L]$ and $\\eta ^{(-)}(y)= 2\\rho _{-}$ if $y <1$ .", "We want $N_{+,x,{ \\gamma }}$ to be the total number of particles to the right of $x$ within distance ${ \\gamma }^{-1}$ from $x$ , however it may happen that if $x$ is close to the right boundary then there are not ${ \\gamma }^{-1}$ sites in the channel to the right of $x$ .", "Suppose that there are only ${ \\gamma }^{-1}-m$ such sites, we then add fictitiously $2m$ phase points $(y,v)$ , $v=\\pm 1$ and $y$ takes $m$ values to be thought as $m$ physical sites to the right of the channel.", "The occupation number $\\eta (y,v)$ is then set equal to $\\rho _+$ so that the contribution to $N_{+,x,{ \\gamma }}$ of the extra $m$ sites is $2\\rho _+ m$ , which explains the factor 2 in the definition of $\\eta ^{(+)}(y)$ .", "Analogous interpretation applies to $\\eta ^{(-)}(y)$ .", "We are now ready to define how the OS-CA operates: the unit time step updating (from $t$ to $t+1$ ) is obtained as the result of three successive operations, we denote by $\\eta $ the configuration at time $t$ , by $\\eta ^{\\prime }$ and $\\eta ^{\\prime \\prime }$ two consecutive updates starting from $\\eta $ and by $\\eta ^{\\prime \\prime \\prime }$ the final update which gives the configuration at time $t+1$ .", "velocity flip.", "At all sites $x\\in \\lbrace 1,..,L\\rbrace $ where there is only one particle we update its velocity to become $+1$ with probability $\\frac{1}{2}+ { \\epsilon }_{x,{ \\gamma }}$ and $-1$ with probability $\\frac{1}{2}- { \\epsilon }_{x,{ \\gamma }}$ , ${ \\epsilon }_{x,{ \\gamma }}= C{ \\gamma }^2[N_{+,x,{ \\gamma }}-N_{-,x,{ \\gamma }}]$ (the definition is well posed because $(2{ \\gamma }^{-1}) C{ \\gamma }^2= 2.5/30<\\frac{1}{2}$ , $(2{ \\gamma }^{-1}) $ being an upper bound for $\|N_{+,x,{ \\gamma }}-N_{-,x,{ \\gamma }}\|$ ).", "At all other sites the occupation numbers are left unchanged.", "We denote by $\\eta ^{\\prime }$ the occupation numbers after the flip.", "advection.", "After deleting the particles in the channel at $(1,-1)$ and $(L,1)$ (if present) we let each one of the remaining particles in the channel move by one lattice step in the direction of its velocity.", "We denote by $\\eta ^{\\prime \\prime }$ the occupation numbers after this advection step.", "exchanges with the reservoirs.", "With probability $\\rho _+$ we put a particle at $(L,-1)$ and with probability $1-\\rho _+$ we leave $(L,-1)$ empty.", "We do independently the same operations at $(1,1)$ but with $\\rho _-$ instead of $\\rho _+$ .", "The final configuration is then denoted by $\\eta ^{\\prime \\prime \\prime }$ .", "The CC-CA.", "We now turn to the second CA which describes the evolution of particles in a “closed circuit\".", "The phase space is the disjoint union $\\mathcal {S} \\cup \\mathcal {R}_1\\cup \\mathcal {R}_2$ , where $\\mathcal {S}$ is as before while the two reservoirs $\\mathcal {R}_1$ and $\\mathcal {R}_2$ are finite sets both with cardinality $R$ , $R$ a positive, even integer.", "$R$ is interpreted as the number of phase points in the reservoir, thus there will be $R/2$ sites with velocity 1 and $R/2$ sites with velocity $-1$ : the velocities in the reservoirs however do not play any role in the evolution, they are used only to have a symmetric description of the channel and the reservoirs.", "Unlike in the OS-CA now the total number of particles (i.e.", "those in the channel and in the reservoirs) is constant in time.", "In the CC-CA the densities $\\rho _{\\pm }$ in the two reservoirs are no longer constant but given by $N_{\\mathcal {R}_1}/R$ and $N_{\\mathcal {R}_2}/R$ where $N_{\\mathcal {R}_1} = \\sum _{(x,v)\\in \\mathcal {R}_1}\\eta (x,v),\\quad N_{\\mathcal {R}_2} = \\sum _{(x,v)\\in \\mathcal {R}_2}\\eta (x,v)$ Accordingly we define $N_{\\pm ,x,{ \\gamma }}$ in the CA as in (REF ) but with $\\rho _{\\pm }$ replaced by the instantaneous values $N_{\\mathcal {R}_1}/R$ and $N_{\\mathcal {R}_2}/R$ of the density in $\\mathcal {R}_1$ and $\\mathcal {R}_2$ .", "With these notation the first two steps of the evolution in the CA are the same as in the OS-CA.", "We call again $\\eta ^{\\prime }$ and $\\eta ^{\\prime \\prime }$ the configurations in the system after the first and the second step, with $\\eta ^{\\prime \\prime }=\\eta ^{\\prime }=\\eta $ in $\\mathcal {R}_1 \\cup \\mathcal {R}_2$ (i.e.", "the occupation numbers in the reservoirs are unchanged in the first two steps).", "In the third step instead they may change as we are going to see.", "3.", "The new third step, (reservoirs exchanges).", "Its definition involves a new, suitably small parameter ${ \\gamma }p > 0$ .", "We first select with uniform probability a phase point $(x_1,v_1)\\in \\mathcal {R}_1$ and $(x_2,v_2)\\in \\mathcal {R}_2$ : if $\\eta (x_1,v_1)=0$ we set $\\eta ^{\\prime \\prime \\prime }(1,1)=0$ , if instead $\\eta (x_1,v_1)=1$ we set $\\eta ^{\\prime \\prime \\prime }(1,1)=1$ .", "Analogously $\\eta ^{\\prime \\prime \\prime }(L,-1)=0,1$ if $\\eta (x_2,v_2)=0,1$ .", "This concludes the definition of $\\eta ^{\\prime \\prime \\prime }$ in the channel while in the reservoirs $\\eta ^{\\prime \\prime \\prime } = \\theta ^{\\prime \\prime \\prime }$ , with $\\theta ^{\\prime \\prime \\prime }$ defined as follows.", "We first define $\\theta ^{\\prime }$ by setting $\\theta ^{\\prime }(x,v)=\\eta (x,v)$ for $(x,v)$ in $\\mathcal {R}_1$ with $(x,v) \\ne (x_1,v_1)$ and $\\theta ^{\\prime }(x_1,v_1)=0$ .", "$\\theta ^{\\prime }(x,v)$ is defined analogously in $\\mathcal {R}_2$ .", "$\\theta ^{\\prime \\prime }(x,v)$ is obtained from $\\theta ^{\\prime }(x,v)$ by adding a particle in the first empty point of $\\mathcal {R}_1$ (according to a fixed but arbitrary order) if $\\eta ^{\\prime }(1,-1)=1$ , otherwise $\\theta ^{\\prime \\prime }=\\theta ^{\\prime }$ in $\\mathcal {R}_1$ .", "$\\theta ^{\\prime \\prime }$ is defined analogously in $\\mathcal {R}_2$ .", "Finally $\\theta ^{\\prime \\prime \\prime }$ is obtained from $\\theta ^{\\prime \\prime }$ in the following way.", "With probability $1-{ \\gamma }p $ we let $\\theta ^{\\prime \\prime \\prime }=\\theta ^{\\prime \\prime }$ while with probability ${ \\gamma }p $ we do the following: we choose with uniform probability $(y_1,v_1)\\in \\mathcal {R}_1$ and $(y_2,v_2)\\in \\mathcal {R}_2$ and exchange $\\theta ^{\\prime \\prime }(y_1,v_1)$ with $\\theta ^{\\prime \\prime }(y_2,v_2)$ .", "To be well defined we have tacitly supposed that $ { \\gamma }p \\le 1$ , actually $ { \\gamma }p \\ll 1$ in the simulations.", "Heuristically ${ \\gamma }p$ is the rate at which particles jump directly from a reservoir to the other.", "Without the channel these exchanges would eventually make the densities of the two reservoirs equal to each other.", "Magnetization variables.", "To exploit the symmetries in the system it is convenient to introduce spin variables.", "We set in the CC-CA: ${\\sigma }(x) = \\eta (x,1)+\\eta (x,-1)-1$ both in the channel and in the reservoirs.", "(possibly adding $t$ when the variables are computed at time $t$ ).", "We call $\\displaystyle {S_{\\rm ch}= \\sum _{x=1}^L{\\sigma }(x)}$ the total spin in the channel, thus $S_{\\rm ch} = N_{\\rm ch}-L,\\qquad N_{\\rm ch}:=\\sum _{x=1}^L\\eta (x)$ Recalling (REF ), we define analogously to (REF ) $S_{\\mathcal {R}_1}= N_{\\mathcal {R}_1} - \\frac{R}{2},\\quad S_{\\mathcal {R}_2}=N_{\\mathcal {R}_2}- \\frac{R}{2}$ We define also the magnetization density in the two reservoirs $m^{CC}_{-} =\\frac{ S_{\\mathcal {R}_1}}{R/2},\\qquad m^{CC}_{+} =\\frac{ S_{\\mathcal {R}_2}}{R/2}$ In the OS-CA the magnetization density in the “reservoirs” is $m_{\\pm }= 2\\rho _{\\pm } -1$ Currents.", "For the OS-CA we define $j_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)$ and $j_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(t)$ as the number of particles which go from $\\mathcal {R}_1$ to the channel minus those which go from the channel to $\\mathcal {R}_1$ in the time step $t\\rightarrow t+1$ and respectively, the number of particles which go from the channel to $\\mathcal {R}_2$ minus those which go from $\\mathcal {R}_2$ to the channel in the time step $t\\rightarrow t+1$ .", "Thus $&&j_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)= \\eta ^{\\prime \\prime \\prime }(1,1;t) - \\eta ^{\\prime }(1,-1;t)\\nonumber \\\\ \\\\&&\\nonumber j_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(t)= \\eta ^{\\prime }(L,1;t)-\\eta ^{\\prime \\prime \\prime }(L,-1;t)$ with $\\eta ^{\\prime }$ , $\\eta ^{\\prime \\prime }$ and $\\eta ^{\\prime \\prime \\prime }$ the occupation numbers after the three updates which lead from $t$ to $t+1$ .", "In the CC-CA the currents $j^{CC}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)$ and $j^{CC}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(t) $ are defined by the same expression as in (REF ) with the new $\\eta $ 's.", "The current between the reservoirs is defined as the number of particles which go from $\\mathcal {R}_2$ to $\\mathcal {R}_1$ minus those which go from $\\mathcal {R}_1$ to $\\mathcal {R}_2$ in the time step $t\\rightarrow t+1$ , thus: $j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)= -\\sum _{(x,v)\\in \\mathcal {R}_2} [\\theta ^{\\prime \\prime \\prime }(x,v;t)-\\theta ^{\\prime \\prime }(x,v;t)]$ Conservation laws.", "In the OS-CA we have $N_{\\rm ch}(t+1)- N_{\\rm ch}(t)= j_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)-j_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(t)$ In the CC-CA the analogue of (REF ) holds as well: $N_{\\mathcal {R}_1} (t+1)- N_{\\mathcal {R}_1} (t)= j^{CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)-j^{CC}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t) $ with analogous formula for $N_{\\mathcal {R}_2}$ .", "As a consequence, in the CC-CA, the total number of particles $N_{\\mathcal {R}_1} +N_{\\mathcal {R}_1} + N_{\\rm ch}$ is conserved as well as the total spin $S_{\\mathcal {R}_1} +S_{\\mathcal {R}_1} + S_{\\rm ch}$ .", "Initial conditions.", "In the OS-CA we impose $\\rho _+> \\rho _-$ , $\\rho _++\\rho _-=1$ .", "Observe that this implies $m_+>0$ and $m_++m_-=0$ .", "Analogously in the CC-CA we initially impose that $N_{\\mathcal {R}_1} + N_{\\mathcal {R}_2} = R, \\quad N_{\\mathcal {R}_2} > \\frac{R}{2}$ The initial state in the channel will be specified in the sequel.", "Parameters of the simulations.", "We conclude the section by recalling the values of the parameters that will be used in the simulations: ${ \\gamma }^{-1}=30, \\;C=1.25, \\;\\beta =2.5,\\;L=600,\\;R= 10^5,\\quad R=:{ \\gamma }^{-1}a,\\;L =: { \\gamma }^{-1} \\ell $" ], [ "The OS-CA", "In this section we present the simulations obtained by running the OS-CA, recall that this CA has been defined in terms of two fixed densities $\\rho _+$ and $\\rho _-$ , $\\rho _++\\rho _-=1$ , which in the magnetization variables, see (REF ), amounts to fix $m_+>0$ , $m_-=-m_+$ .", "As already mentioned the OS-CA simulates the typical Fourier law experiments therefore the physically most relevant quantity is the stationary current $j(m_+)$ : $j=j(m_+)$ plays the role of the equation of state in a non equilibrium context (due to the presence of the reservoirs) and defines the “non equilibrium thermodynamics” of the system.", "Thus our first task is to consider the currents in the CA, since the instantaneous currents defined in (REF ) are strongly fluctuating, we take averages: $&&j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}= \\frac{1}{T}\\sum _{t=0}^{T-1}j_{ \\mathcal {R}_1\\rightarrow {\\rm ch}}(t)$ In general with $f^T$ we will denote the average of $f$ , thus $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}$ is the averaged current from the channel to $\\mathcal {R}_2$ .", "Strictly speaking stationarity is reached as $T\\rightarrow \\infty $ , existence of the limit should follow (almost everywhere) from the Birkhoff theorem.", "Of course in the simulations we cannot take such a limit and the value of $T$ is chosen empirically in such a way that $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}\\approx j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}$ looks independent of $T$ .", "The initial condition in the channel is with all phase points empty, we have checked that with other conditions the final current does not change appreciably.", "The stationarity condition $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)- j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(m_+)\\approx 0$ is also satisfied, typical values are $10^{-8}$ while the currents have order $10^{-5}$ .", "$10^{-8}$ is also considerably smaller than the a-priori bound $\|j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}- j^T_{ {\\rm ch}\\rightarrow \\mathcal {R}_2}\| \\le \\frac{2L}{T} = 1.2 \\cdot 10^{-7}, \\text{when $T= 10^{10}$}$ Figure: We plot j:=j ch →ℛ 2 T j:=j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2} as a function of m + m_+ (black dots).", "The continuous line is j(m + )j(m_+).", "Shown are the values m ' =0.500m^{\\prime }= 0.500, m '' =0.825m^{\\prime \\prime }=0.825, m ''' =0.912m^{\\prime \\prime \\prime }=0.912, m iv =0.985m^{iv}= 0.985.The black dots in Fig.", "REF are the values of $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)$ in the simulations done with $T= 3 \\cdot 10^9$ for $m_+ \\in (0,m^{\\prime })$ and $m_+>m^{\\prime \\prime \\prime }$ and $T= 10^{10}$ elsewhere.", "The continuous line in Fig.", "REF , denoted by $j(m_+)$ , is a continuous interpolation of $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)$ which we presume to be a good approximation of simulations done with the other values of $m_+$ , it is therefore the “experimental” value for the non equilibrium equation of state $j=j(m_+)$ .", "The main features in Fig.", "REF (where $m^{\\prime }= 0.500$ , $m^{\\prime \\prime }= 0.825$ , $m^{\\prime \\prime \\prime }=0.912$ and $m^{iv}= 0.985$ )) are: For $m_+\\in (m^{iv},1]$ the current $j(m_+)$ is negative in agreement with the Fourier law, while for $m_+<m^{iv}$ the current is positive going from smaller to larger values of the magnetization (i.e.", "from $m_-$ to $m_+$ ).", "$j(m_+)$ is first increasing till $m^{\\prime }$ , then decreasing till $m^{\\prime \\prime }$ , again increasing till $m^{\\prime \\prime \\prime }$ and finally decreasing then after.", "In Fig.", "REF we plot $j^{t}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(m_+)t$ and $j^t_{ {\\rm ch}\\rightarrow \\mathcal {R}_2}(m_+)t$ , $t \\le T$ with $m_+ \\in [m^{\\prime },m^{\\prime \\prime \\prime }]$ .", "We see significant fluctuations around the linear slope $j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(m_+) t$ , while for $m_+ \\notin [m^{\\prime },m^{\\prime \\prime \\prime }]$ the fluctuations are “negligible”.", "Figure: We plot j ℛ 1 → ch t (m + )tj^{t}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(m_+)t (black circles) and j ch →ℛ 2 t (m + )tj^t_{ {\\rm ch}\\rightarrow \\mathcal {R}_2}(m_+)t (empty circles) as functions of time tt , with m + ∈[m ' ,m ''' ]m_+ \\in [m^{\\prime },m^{\\prime \\prime \\prime }].The most striking feature in the simulations is undoubtedly the fact that the current is positive when $m_+<m^{iv}$ so that it flows along the gradient going from the reservoir with smaller magnetization to the one with larger magnetization.", "If we dropped the interaction among particles in the channel, namely put ${ \\epsilon }_{x,{ \\gamma }}\\equiv 0$ , then the current would flow according to the Fourier law opposite to the gradient, namely from $\\mathcal {R}_2$ to $\\mathcal {R}_1$ .", "A heuristic argument.", "Let us now imagine to have two channels connected to $\\mathcal {R}_1$ and $\\mathcal {R}_2$ , channel 1 is the channel considered so far while channel 2 is some other channel where the Fourier law is satisfied (for instance the OS-CA with no bias, ${ \\epsilon }_{x,{ \\gamma }}\\equiv 0$ , or some simpler connection as the one discussed later).", "When $m_+<m^{iv}$ , in channel 1 there is a current $j(m_+)$ going from $\\mathcal {R}_1$ to $\\mathcal {R}_2$ , while in channel 2 the current is $j_2 = \\kappa m_+$ , $\\kappa >0$ , going from $\\mathcal {R}_2$ to $\\mathcal {R}_1$ (recall $m_-=-m_+$ ).", "Thus in a time $t$ the reservoir $\\mathcal {R}_1$ will loose a magnetization $j(m_+)t$ through channel 1 and gain a magnetization $\\kappa m_+t$ through channel 2; the opposite happens to $\\mathcal {R}_2$ .", "This will go forever because the reservoirs in the OS-CA are not changed by what comes and goes; if instead the reservoirs were realized by large but finite systems (as in CC-CA) then after a time which depends on the size of the reservoirs and the difference $j(m_+)- \\kappa m_+$ the magnetization in the reservoirs would change and stationarity would be lost.", "However if we choose channel 2 so that $\\kappa m_+ = j(m_+)$ there is a perfect balance so that what $\\mathcal {R}_1$ gives to $\\mathcal {R}_2$ through channel 1 comes back from channel 2.", "We may thus hope that even if the reservoirs are finite (yet sufficiently large) this is again approximately true and that there is a non zero current which looks stationary for long times.", "The simplest choice for channel 2 leads to the CC-CA of Section where channel 2 is made by just allowing direct exchanges between the two reservoirs.", "Then, as we shall see later, the average current in the CC-CA from $\\mathcal {R}_2$ to $\\mathcal {R}_1$ is equal to ${ \\gamma }p m_+$ , hence the conjecture that for such a particular value of ${ \\gamma }p$ there is a non zero stationary current in the circuit which is close to $j(m_+)$ .", "To check this we have defined for each $m_+<m^{iv}$ in Fig.", "REF ${ \\gamma }p= j(m_+)/m_+$ as a function of $m_+$ , see Fig.", "REF .", "Figure: We plot j/m + :=j(m + )/m + j/m_+:=j(m_+)/m_+ as a function of m + m_+ (black dots).", "The continuous line is a black dots interpolation.We have then run the CC-CA with such values of ${ \\gamma }p$ , putting $m_{+}(0) = m_+$ in $\\mathcal {R}_2$ , $m_{-}(0) = -m_+$ in $\\mathcal {R}_1$ and choosing the initial state in the channel equal to the configuration in the OS-CA simulation at the final time $T$ .", "For all the values of $m_+$ considered in Fig.", "REF we have run the CC-CA for a same time $T= 3\\cdot 10^9$ and computed the averaged currents $j^{T,CC}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}$ , $j^{T,CC}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}$ and $j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}$ defined as in (REF ).", "Recalling (REF ) we have also defined the averaged magnetization $m^{T,CC}_{\\pm }$ in the two reservoirs writing $j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(m_+), j^{T,CC}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(m_+)$ , $j^{T,CC}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+), m^{T,CC}_{\\pm }(m_+)$ when we want to underline that the values are obtained starting from $m_+$ .", "The previous heuristic argument suggests that the three currents above are all close to each other and thus approximately equal to $j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)$ and moreover that $m^{T,CC}_{\\pm }(m_+)\\approx \\pm m_+$ .", "In the next section we will see what the simulations say." ], [ "Self sustained currents", "Fig.", "REF is obtained by running the CC-CA in the setup described at the end of the previous section.", "It reports the values of the differences $10^5[j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(m_+)-j^{T,CC}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)]$ and $m_+^{T,CC}(m_+)-m_+$ as a function of $m_+$ , recall from Fig.", "REF that the typical values of the current have order $10^{-5}$ .", "Figure: We plot Δj:=10 5 ×[j ℛ 2 →ℛ 1 T,CC (m + )-j ch →ℛ 2 T,CC (m + )]\\Delta j:=10^5\\times [j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(m_+)-j^{T,CC}_{{\\rm ch} \\rightarrow \\mathcal {R}_2}(m_+)] (left panel) and Δm + :=m + T,CC -m + \\Delta m_+:=m_+^{T,CC}-m_+ (right panel) as a function of m + m_+.", "Note that the large fluctuations occur in the interval (m '' ,m ''' )(m^{\\prime \\prime },m^{\\prime \\prime \\prime }), with m '' =0.825m^{\\prime \\prime }=0.825, m ''' =0.912m^{\\prime \\prime \\prime }=0.912.We have also reported for each $m_+$ in Fig.", "REF the values of the pair $({ \\gamma }p,j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1})$ , see Fig.", "REF left, the continuous line is obtained by interpolating between such values.", "Analogously in Fig.", "REF right the dots are the values of $({ \\gamma }p,m_+^{T,CC})$ and the continuous line is obtained by interpolation.", "The continuous lines are multi-valued functions denoted respectively by $j^{CC}({ \\gamma }p)$ and $m_+^{CC}({ \\gamma }p)$ , we presume they are a good approximation of what would be obtained by following the same procedure for other values of $m_+$ in Fig.", "REF .", "Let us point out the main features of our simulations.", "Figure: We plot the values of the pairs (γp,j T :=j ℛ 2 →ℛ 1 T,CC )({ \\gamma }p,j^T:=j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}) (left panel) and (γp,m + T :=m + T,CC )({ \\gamma }p,m_+^T:=m_+^{T,CC}) (right panel).", "The black circles in the panels above denote, respectively, the stationary values of j ℛ 2 →ℛ 1 T,CC j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1} and m + T,CC m_+^{T,CC} obtained with m + ∈(m '' ,m ''' )m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime }).", "Shown are also the values of q ' =1.98×10 -5 q^{\\prime }=1.98\\times 10^{-5} and q '' =5.26×10 -5 q^{\\prime \\prime }=5.26\\times 10^{-5}.", "Fig REF shows that the simulations are in good agreement with the conjectures stated at the end of the previous section except in the interval $m_+ \\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ .", "The values of $ j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(m_+)$ and $m_+^{T,CC}(m_+)$ when $m_+ \\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ , are however approximately the same as those obtained for different values of $m_+$ , see the black circles in Fig.", "REF ).", "The values of ${ \\gamma }p$ are all in the interval $(0,q_c)$ , $q_c= 11.25\\times 10^{-5}$ , and $j^{CC}({ \\gamma }p)$ is positive for all such values of ${ \\gamma }p$ .", "We have also done simulations with ${ \\gamma }p > q_c$ with several choices of the initial condition and we have always seen zero current (not reported here).", "$j^{CC}({ \\gamma }p)$ is multi-valued, it has two distinct branches (separated from each other), the upper one in the interval $(0,q^{\\prime \\prime })$ , $q^{\\prime \\prime }=5.26\\times 10^{-5}$ , the lower one in the interval $(q^{\\prime },q_c)$ ; $q^{\\prime \\prime }>q^{\\prime }$ , $q^{\\prime }=1.98\\times 10^{-5}$ .", "In the interval $(q^{\\prime },q^{\\prime \\prime })$ there are two positive currents different from each other.", "$m_+^{CC}({ \\gamma }p)$ has the analogous structure, being two valued in $(q^{\\prime },q^{\\prime \\prime })$ .", "Both branches are decreasing, $m_+^{CC}({ \\gamma }p)\\rightarrow m^{iv} = 0.985$ , as ${ \\gamma }p \\rightarrow 0$ , and to 0 as ${ \\gamma }p \\rightarrow q_c$ .", "There is a gap in the range of $m_+^{CC}({ \\gamma }p)$ , namely the interval $(m^{\\prime },m^{\\prime \\prime })$ .", "Conclusions.", "The simulations in Fig.", "REF show good agreement with the conjectures of Section except when $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ .", "Thus, with such exception, we may say that the stationary state found in the OS-CA evolution persists in the CC-CA provided that ${ \\gamma }p=j(m_+)/m_{+}$ .", "There is no mystery about the current between the two reservoirs being ${ \\gamma }p(m_{+}-m_-)/2\\approx { \\gamma }p m_+$ because we can prove (see Appendix ) that $E\\Big [ \\lbrace j^{T,CC}_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1} - { \\gamma }p \\frac{1}{2}[ m_+^{T,CC}-m_-^{T,CC}]\\rbrace ^2 \\Big ] \\le \\frac{{ \\gamma }p}{T} +16 \\frac{({ \\gamma }p)^2}{R}+ \\; \\text{\\rm corrections}$ Since ${ \\gamma }p \\approx 10^{-5}$ , $R = 10^{5}$ and $T \\approx 10^{9}$ , the corrections have order $10^{-19}$ , see (REF ).", "Fig.", "REF can be obtained from Fig.", "REF : in fact according to the above statements $j^{CC}({ \\gamma }p)$ is (approximately) equal to $j(m_+)$ with $j(m_+)= { \\gamma }p m_+$ .", "Since this may have multiple roots, $j^{CC}({ \\gamma }p)$ will be correspondingly multi-valued.", "However the roots with $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ are absent in the simulations (see the black circles in Fig.", "REF ) but their values are the same as those obtained with other values of $m_+$ .", "Same if we look at $m^{CC}({ \\gamma }p)$ and compare with Fig.", "REF .", "As a conclusion we have a consistent explanation of what seen in the OS-CA and the CC-CA, but we still need to explain (i) what happens when $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ ; (ii) why the typical values of $j(m_+)$ have order $10^{-5}$ which is much smaller than $1/L \\approx 10^{-3}$ which is what expected from Fourier law experiments; (iii) why the true reservoir current has the behavior shown in Fig.", "REF .", "We can gain a theoretical insight on what is going on by looking at what happens in the mesoscopic limit ${ \\gamma }\\rightarrow 0$ which we study in the next section." ], [ "The mesoscopic limit", "This is defined by letting ${ \\gamma }\\rightarrow 0$ with $L= { \\gamma }^{-1} \\ell ,\\quad R = { \\gamma }^{-1}a,\\quad \\ell ,a >0\\;{\\rm fixed}$ In the channel space and time are scaled diffusively, thus $x \\rightarrow r ={ \\gamma }x$ and $t \\rightarrow \\tau = { \\gamma }^{2}t$ .", "In mesoscopic units the channel after the limit ${ \\gamma }\\rightarrow 0$ becomes the real interval $[0,\\ell ]$ .", "We will prove existence of the limit (for the relevant quantities) under the assumption of a strong form of propagation of chaos, the details are given in an appendix.", "We denote by $E_{ \\gamma }$ the expectation in the CA processes (randomness coming from the initial datum and from the updating rules of the CA's).", "Assumptions.", "We suppose that In both CA the limit below (denoted in the same way for both CA) exists and is smooth $\\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } E_{ \\gamma }[ \\eta (x,v,t)] =\\frac{m(r,\\tau )+1}{2}, \\quad r\\in [0,\\ell ], v\\in \\lbrace -1,1\\rbrace , \\tau \\ge 0$ In the CC-CA $\\lim _{{ \\gamma }\\rightarrow 0, { \\gamma }^{2}t\\rightarrow \\tau } m_{\\pm ,{ \\gamma }}({ \\gamma }^2 t)=m_{\\pm }(\\tau )$ where, recalling (REF ), we have set $m_{\\pm ,{ \\gamma }}({ \\gamma }^2 t):= E_{ \\gamma }[m^{CC}_{\\pm } (t)]$ In both CA for all $r,r_1,r_2 \\in (0,\\ell )$ , $r_1\\ne r_2$ , $v\\in \\lbrace -1,1\\rbrace $ and $\\tau \\ge 0$ $\\nonumber && \\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } \| E_{ \\gamma }[ \\eta (x,v,t)\\eta (x,-v,t)]- E_{ \\gamma }[ \\eta (x,v,t)]E_{ \\gamma }[\\eta (x,-v,t)] \| =0 \\\\\\\\&&\\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r_1, { \\gamma }y\\rightarrow r_2, { \\gamma }^{2}t\\rightarrow \\tau } \| E_{ \\gamma }[ \\eta (x,t)\\eta (y,t)]- E_{ \\gamma }[ \\eta (x,t)]E_{ \\gamma }[\\eta (y,t)] \| =0\\nonumber $ In the CC-CA for all $\\tau \\ge 0$ $\\lim _{{ \\gamma }\\rightarrow 0, { \\gamma }^{2}t\\rightarrow \\tau } R^{-1} E_{ \\gamma }\\Big [ \\big \|N_{\\mathcal {R}_i}(t)-E_{ \\gamma }[N_{\\mathcal {R}_i}(t)]\\big \|\\Big ]=0,\\;\\; i=1,2$ In Appendix we will prove the following two Theorems.", "Theorem 1 (Mesoscopic limit) Under the above assumptions, in both CA, the limit magnetization $m(r,t)$ satisfies: $&&\\frac{\\partial }{\\partial t}m(r,t) = -\\frac{\\partial }{\\partial r} I(r,t),\\quad r \\in (0,\\ell )\\\\&& \\hspace{-8.5359pt}I(r,t)= -\\frac{1}{2}\\big \\lbrace \\frac{\\partial m(r,t)}{\\partial r}- 2C[1-m(r,t)^2]\\int _{r}^{r+1}[m(r+\\xi ,t)- m(r-\\xi ,t)] d\\xi \\big \\rbrace \\nonumber $ with $m(r+\\xi ,t)= m_+(t)$ if $r+\\xi \\ge \\ell $ and $m(r-\\xi ,t)= m_-(t)$ if $r-\\xi \\le 0$ in the CC-CA; same expression holds in the OS-CA but with $m_\\pm (t)$ replaced by $m_\\pm $ .", "Moreover $m(0,t)=m_{-},\\quad m(\\ell ,t)=m_{+},\\qquad \\text{in the OS-CA}$ while in the CC-CA $m(0,t)=m_{-}(t),\\quad m(\\ell ,t)=m_{+}(t) $ $\\frac{d}{dt}m_{+}(t) = \\frac{1}{a} \\Big (2I(\\ell , t)+ p [m_{-}(t)-m_{+}(t)]\\Big )$ $\\frac{d}{d t}m_{-}(t) = \\frac{1}{a} \\Big ( -2I(0,t)+ p [m_{+}(t)-m_{-}(t)]\\Big )$ A proof which avoids our assumptions of propagation of chaos has been obtained in [10] for a lattice gas with Kac potential and Kawasaki dynamics in a torus.", "In magnetization variables the system becomes the Ising model with Kac potential and the limit equation is (REF ).", "In [12] it has been studied the macroscopic scaling limit of this system with space scaled by ${ \\gamma }^{-\\alpha }$ and time by ${ \\gamma }^{-2\\alpha }$ , $\\alpha >1$ ( $\\alpha =1$ is the mesoscopic limit considered above).", "Theorem 2 (Currents) Denote by $j_{x,x+1}(t)$ the number of particles which in the time step $t, t+1$ cross the bond $(x,x+1)$ , $x\\in \\lbrace 1,..,L-1\\rbrace $ (counting as positive those which jump from $x$ to $x+1$ and as negative those from $x+1$ to $x$ ).", "Then, under the above assumptions, in both CA, for all $r\\in (0,\\ell )$ and $\\tau >0$ $\\lim _{{ \\gamma }\\rightarrow 0: { \\gamma }x\\rightarrow r} { \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[j_{x,x+1}(t)]= \\int _0^\\tau I(r,s)ds,\\qquad T=[{ \\gamma }^{-2}\\tau ]$ $&&\\lim _{{ \\gamma }\\rightarrow 0}{ \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[ j_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)]= -\\int _{0}^{\\tau } I(0,s)ds,\\nonumber \\\\&&\\lim _{{ \\gamma }\\rightarrow 0}{ \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)]=\\int _{0}^{\\tau } I(\\ell ,s)ds$ where $I(r,s)$ is given in (REF ).", "In the CC-CA the current between reservoirs converges by (REF ) to: $&&\\lim _{{ \\gamma }\\rightarrow 0}{ \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[ j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)]=\\int _{0}^{\\tau } p m_+(s)ds$ In the simulations we have plotted the quantity $ j^{T}_{{\\rm ch}\\rightarrow \\mathcal {R}_2}$ .", "This is related by (REF ) to the mesoscopic current $I$ by $&&E_{ \\gamma }[ j^{T}_{\\mathcal {R}_1\\rightarrow {\\rm ch}}]=\\frac{{ \\gamma }}{T} \\sum _{t=0}^{T-1} E_{ \\gamma }[ j_{\\mathcal {R}_1\\rightarrow {\\rm ch}}(t)] \\approx - \\frac{{ \\gamma }}{\\tau }\\int _{0}^{\\tau } I(0,s)ds ,\\nonumber \\\\&&E_{ \\gamma }[ j^{T}_{{\\rm ch}\\rightarrow \\mathcal {R}_2}]=\\frac{{ \\gamma }}{T} \\sum _{t=0}^{T-1} E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)] \\approx \\frac{{ \\gamma }}{\\tau }\\int _{0}^{\\tau } I(\\ell ,s)ds \\nonumber \\\\&&E_{ \\gamma }[ j^T_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}]=\\frac{{ \\gamma }}{T} \\sum _{t=0}^{T-1} E_{ \\gamma }[ j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)]\\approx \\frac{{ \\gamma }}{\\tau }\\int _{0}^{\\tau } p m_+(s)ds$ so that the experimental values of the three currents scale all as ${ \\gamma }$ when ${ \\gamma }\\rightarrow 0$ .", "We next show that there is a natural interpretation of the solutions of the system (REF )–(REF ) in terms of statistical mechanics, which then allows to relate what seen in the simulations to phase transitions and metastable–unstable magnetization values.", "Free energy functional and thermodynamic potentials.", "The evolution equation (REF ) in $[0,\\ell ]$ with periodic boundary conditions is the gradient flow relative to a non local free energy functional $F(m)$ , in fact $&&I(r)= - \\chi \\frac{\\partial }{\\partial r}\\frac{\\delta F(m)}{\\delta m(r)},\\quad \\chi =\\frac{\\beta }{2}(1-m^2), \\quad \\beta =2C\\\\&& F(m) = \\int \\Big (-\\frac{m^2}{2} - \\frac{S}{\\beta }\\Big ) + \\frac{1}{4}\\int \\int J (r,r^{\\prime }) [m(r)-m(r^{\\prime })]^2\\nonumber \\\\&&S(m)= -\\frac{1-m}{2}\\log \\frac{1-m}{2} - \\frac{1+m}{2}\\log \\frac{1+m}{2}\\nonumber \\\\&& J(r,r^{\\prime })=1-\|r-r^{\\prime }\|,\\qquad \\text{for } \|r-r^{\\prime }\|\\le 1\\quad \\text{ and }=0 \\quad \\text{elsewhere}\\nonumber $ $F(m)$ is “the mesoscopic free energy functional”, the Ginzburg-Landau functional is a local approximation of $F(m)$ where the non local term becomes a gradient squared.", "The corresponding gradient flow evolution is the Cahn-Hilliard equation, which can then be viewed as a local approximation of (REF ).", "The important point for us is that $F(m)$ specifies the thermodynamics of the system.", "In fact $&&f_\\beta (m)= -\\frac{m^2}{2} - \\frac{S(m)}{\\beta }$ is the van der Waals mean field free energy; its convex envelope $f^{}_\\beta (m)$ is the thermodynamic free energy.", "$f^{}_\\beta (s)$ is obtained by minimizing $F(m)/\\ell $ under the constraint $\\int m(r) = \\ell s$ and then taking the limit $\\ell \\rightarrow \\infty $ , see for instance [14], Ch.", "6.", "The equilibrium magnetization density when there is a magnetic field $h$ is the solution of the mean field equation $&&m = \\tanh \\lbrace \\beta (m+h)\\rbrace $ When $\\beta >1$ there is $h_c(\\beta )>0$ so that for any $\|h\|<h_c(\\beta )$ , $f_\\beta (m,h)=f_\\beta (m)-hm $ is a double well function of $m$ .", "The local minima are $m_+(h)$ and $m_-(h)$ and their graph is the hysteresis cycle, see Fig.", "REF .", "In particular at $h=-h_c(\\beta )$ , $m_+(h)= m^$ $&&m^>0 : \\beta [1-(m^)^2] =1$ so that the magnetization in $(m^,m_\\beta )$ and in $(-m_\\beta ,- m^)$ is metastable.", "At $h=0$ the double well is symmetric and the local minima are global minima, they are attained at $m=\\pm m_\\beta $ , $m_\\beta $ the positive solution of (REF ) with $h=0$ .", "$\\pm m_\\beta $ are the equilibrium magnetization at the phase transition with $h=0$ and $\\beta >1$ .", "$m_+(h)$ and $m_-(h)$ are the unique equilibrium magnetization at $h> 0$ and respectively $h<0$ ." ], [ "The adiabatic limit.", "Some of the characteristic parameters of the simulations are related to the thermodynamics associated to the mesoscopic equations, see the end of Section .", "Indeed in fig REF which refers to simulations with the OS-CA, the value $ 0.985$ is very close to $m_\\beta $ so that the simulation shows that the current is negative when $m_+$ is stable, namely $m_+ > m_\\beta $ and positive when $m_+ < m_\\beta $ (metastable or unstable).", "Correspondingly when there is a current in the CC-CA then $m_+ < m_\\beta $ , see fig.", "REF right.", "The above validates the considerations in the Introduction about the relation between the appearance of a current in the circuit and the occurrence of phase transitions.", "Also the metastable region $(m^,m_\\beta )$ has a role in the simulations as the interval $(m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ is a subset of $(m^,m_\\beta )$ (because $m^{\\prime \\prime }=0.825$ and $m^{\\prime \\prime \\prime }=0.912$ while $m^\\approx 0.775$ and $m_\\beta \\approx 0.985$ ); thus the gap phenomenon (i.e.", "that some values of the magnetization in $\\mathcal {R}_2$ are never seen for all ${ \\gamma }p$ ) occurs only inside the metastable region.", "We turn now to the heuristic argument at the end of Section by observing that it becomes rigorous in the context of the mesoscopic equations.", "In fact if $m$ is a stationary solution of the mesoscopic equation for the OS-CA when the reservoirs magnetizations are $m_\\pm $ , and the corresponding current $I$ is positive, then $u=m$ in the channel and $u=m_\\pm $ in $\\mathcal {R}_i$ $i=1,2$ is a stationary solution of the CC-CA mesoscopic equations with $p= \\frac{I}{\\frac{1}{2} (m_+-m_-)}$ , recall that the ratio between the mesoscopic current and the current in the CA scales as ${ \\gamma }^{-1}$ .", "Thus for sufficiently small ${ \\gamma }$ we may expect to see what conjectured at the end of Section .", "We can also give an explanation of the gap phenomenon (i.e.", "that for all ${ \\gamma }p$ the magnetization in the reservoirs $\\mathcal {R}_2$ is never in the interval $(m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ ) by assuming that the evolution of the OS-CA is well approximated by the mesoscopic equations in the adiabatic limit that we are going to define.", "We first observe that the OS-CA can be regarded as the “infinite reservoirs limit” of the CC-CA, in fact in the limit $R\\rightarrow \\infty $ the updating rules of the CC-CA become those of the OS-CA.", "This is true also at the mesoscopic level: when $a\\rightarrow \\infty $ the magnetizations $m_\\pm (t)$ converge to their initial value $m_\\pm (0)$ and the evolution becomes that of the OS-CA.", "The above is true when we let $a\\rightarrow \\infty $ keeping the time finite, more interesting behaviour is seen if we scale time proportionally to $a$ , which is the so called adiabatic scaling limit.", "Suppose (in agreement with the simulations in fig.", "REF ) that for each value of $m_+$ (and with $m_-=-m_+$ ) there is a unique stationary solution of the mesoscopic equations for the OS-CA, $I_{\\rm {stat}}(m_+)$ being the corresponding current.", "We then say that the CC-CA mesoscopic equations have a “good adiabatic behavior” if in the adiabatic limit the magnetizations $m_\\pm (t)$ satisfy the equations $\\frac{d m_{+}(t)}{d t} = 2\\Big ( I_{\\rm {stat}}(m_{+}(t)) - pm_{+}(t))\\Big ) \\qquad m_-(t)=-m_+(t)$ Suppose now that $ I_{\\rm {stat}}$ is positive with a graph like $j(m_+)$ , see fig.", "REF .", "Then the stationary solutions of $p m_+= I_{\\rm {stat}}(m_+)$ with $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ are linearly unstable because $I_{\\rm {stat}}(m_+)$ is decreasing while $p m_+$ is increasing.", "Thus a small perturbation will lead the magnetization away from the stationary value $p m_+= I_{\\rm {stat}}(m_+)$ , $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ , and presumably it will converge to one of the two other solutions of $p m_+= I_{\\rm {stat}}(m_+)$ .", "This may therefore explain why in the simulations we do not see the magnetization $m_+\\in (m^{\\prime \\prime },m^{\\prime \\prime \\prime })$ and instead find another solution of $p m_+= I_{\\rm {stat}}(m_+)$ .", "We can check experimentally whether the CC-CA has a good adiabatic behavior by doing simulations with non stationary initial data.", "In fig.", "REF we report the experimental values and those obtained by solving numerically the adiabatic equations.", "Figure: We plot m ± m_{\\pm } as functions of time tt (empty circles), obtained by running the CC-CA, as well as the predicted behavior in the adiabatic limit (dashed line).", "The initial values of the magnetization are, respectively, m + (0)=-m - (0)=1m_+(0)=-m_-(0)=1 (left panel) and m + (0)=-m - (0)=0.5m_+(0)=-m_-(0)=0.5 (right panel).We do not have an analytic proof of good adiabatic behavior which instead can be rigorously proved for another particle model.", "This is the simple symmetric exclusion process in an interval with boundary processes at the endpoints which simulate reservoirs with densities $\\rho _{\\pm }(t)$ dependent on time.", "In [6] it is proved that in a scaling limit where $\\rho _{\\pm }(t)$ are “slowly varying” the current in the system becomes at each time $t$ the same as the stationary current when the densities at the endpoints are kept fixed at the values $\\rho _{\\pm }(t)$ .", "Summarizing, we have a reasonable explanation of the simulations in the CC-CA once we accept the behavior of the current $j(m_+)$ in the OS-CA as given in Fig.", "REF .", "To explain the latter we need to go deeper in the analysis of the simulations discussing the magnetization profile in the channel, which will be the argument of the remaining sections." ], [ "The instanton and the Stefan problem", "We have a good understanding of what happens when $m_+\\in (m_\\beta ,1]$ .", "In Fig.", "REF we plot the time evolution of the magnetization pattern when $m_+ = 1$ , but a similar picture is observed for the other values of $m\\in (m_\\beta ,1]$ .", "The simulation shows convergence as time increases to a profile which is therefore stationary (in the times of the simulation) and it agrees with what found studying the mesoscopic equations.", "The existence of stationary solutions $m_{\\rm st}(r;\\ell ;m_{\\pm })$ of (REF ) with boundary conditions (REF ) when $m_+>m_\\beta $ has been proved in [5] for $\\ell $ large enough.", "It is also shown that $\\lim _{\\ell \\rightarrow \\infty }m_{\\rm st}(r\\ell ;\\ell ;m_{\\pm })=m_{\\rm st}(r;m_{\\pm }),\\quad r\\in (0,1)$ where the limit $m_{\\rm st}(r;m_{\\pm })$ is antisymmetric around $r= 1/2$ and satisfies the equation $-\\frac{1}{2} [1-\\beta (1-m_{\\rm st}^2)]\\frac{dm_{\\rm st}}{dr} =I_{\\rm st}(m_+),\\quad r\\in [\\frac{1}{2},1]$ where $I_{\\rm st}(m_+)$ is determined by requiring that $m_{\\rm st}(1/2)=m_\\beta $ and $m_{\\rm st}(1)=m_+$ .", "For $m_+=1$ , $\\beta =2.5$ and $m_\\beta =0.985$ from (REF ) we get $I_{\\rm st}(1)\\simeq -7.2\\times 10^{-3}$ .", "To compare with the simulations we have to divide by $L={ \\gamma }^{-1}\\ell = 600$ getting $-1.2 \\times 10^{-5}$ the current in the simulations of the cellular automata OS -CA is instead $\\simeq -2.2\\times 10^{-5}$ .", "The discrepancy is possibly due to $\\ell $ not being large enough.", "In [5] it is also proved that $\\lim _{\\ell \\rightarrow \\infty }m_{\\rm st}(\\frac{1}{2} + x;\\ell )=\\bar{m} (x),\\quad x\\in \\mathbb {R}$ where $\\bar{m} (x)$ is the instanton solution of $\\bar{m} (x) = \\tanh \\lbrace J\\bar{m} (x)\\rbrace $ namely the antisymmetric function solution of (REF ) which converges to $m_\\beta $ as $x \\rightarrow \\infty $ .", "See for instance [14] for existence and properties of the instanton.", "Figure: Magnetization profile with m + =1m_{+}=1.", "The different curves in the plot correspond to the averaged magnetization computed at different times: t=10 5 t=10^5 (empty squares), t=10 6 t=10^6 (black squares), t=10 7 t=10^7 (empty circles) and t=10 8 t=10^8 (black circles).", "The black thin line denotes the initial configuration, corresponding to a step function centered at r=5r=5.In Fig.", "REF it is also plotted the time evolution of the magnetization pattern when starting away from the stationary one.", "The approach to the latter occurs on the time scale $L^2$ .", "Conjecture.", "Let $m(r,t;\\ell ;m_{\\pm })$ be the solution of (REF ) with boundary conditions (REF ) and with initial datum $m_0(r\\ell )$ , $r\\in [0,1]$ , such that: $m_0(r) < -m_\\beta $ is smooth in $r<r_0$ , $r_0 \\in (0,1)$ with limits $m_-$ and $-m_\\beta $ as $r\\rightarrow 0$ and $r\\rightarrow r_0$ $m_0(r) >m_\\beta $ is smooth in $r>r_0$ , with limits $m_\\beta $ and $m_+$ as $r\\rightarrow r_0$ and $r\\rightarrow 1$ .", "Then $\\lim _{\\ell \\rightarrow \\infty } m(r\\ell ,t\\ell ^2;\\ell )= m(r,t)$ where $m(r,t)$ is the solution of the Stefan problem with initial datum $m_0(r)$ : $\\frac{\\partial }{\\partial t}m(r,t) = -\\frac{\\partial }{\\partial r} I(r,t),\\quad I(r,t)= -\\frac{1}{2} [1- \\beta (1- m(r,t)^2)]\\frac{\\partial }{\\partial r}m(r,t)$ where (REF ) holds in $\\lbrace r<r_t\\rbrace $ and in $\\lbrace r>r_t\\rbrace $ with Dirichlet boundary conditions $m_-$ and $-m_\\beta $ in $\\lbrace r<r_t\\rbrace $ and $m_\\beta $ and $m_+$ in $\\lbrace r>r_t\\rbrace $ .", "The free boundary $r_t$ is also an unknown and it is determined by (REF ) and the condition $2m_\\beta \\frac{dr_t}{dt} = I(r_t^-,t) - I(r_t^+,t)$ We do not have a proof that $m(r,t)\\rightarrow m_{\\rm st}(r;m_{\\pm })$ as $t\\rightarrow \\infty $ ($m_{\\rm st}(r;m_{\\pm })$ as in (REF )).", "However if the pattern looks like the one in Fig.", "REF , i.e.", "essentially linear away from $\\pm m_\\beta $ , then the current (being proportionally to the slope) when $r> r_t > 1/2$ is larger (in absolute value) than the one when $r< r_t$ .", "Thus the magnetization increases and therefore $r_t$ moves to the left.", "(REF ) has been derived in [12] from the spin dynamics on a torus when the initial profile $m_0(r)$ has values in $(m^,1)$ for all $r$ or when it has values in $(-1,-m^)$ .", "The result does not apply in the case of the Stefan problem where there are both positive and negative values of the magnetization: the derivation of the Stefan problem for Ising spins with Kawasaki dynamics and Kac potential is still an open problem." ], [ "Boundary layers, the bump", "When $m_+<m_\\beta $ we get a completely different picture.", "Compare in fact the simulations in Fig.", "REF and Fig.", "REF where the initial state is the same but $m_+$ is stable in the former ($m_+=1$ ) and metastable ($m_+=0.93$ ) in the latter.", "In both cases, after a transient, we see a profile with a sharp (instanton-like) transition from $-m_\\beta $ to $+m_\\beta $ and then approximately linear profiles which connect $m_-$ to $-m_\\beta $ and $m_\\beta $ to $m_+$ .", "But, in the stable case the instanton-like region moves towards the center, while in the metastable case it moves towards 0 which is eventually reached.", "The same [heuristic] argument which explained in the case of Fig.", "REF the motion of the instanton towards the center, now explains its motion away from the center: since $m_+<m_\\beta $ the slope of the pattern from the endpoint to the instanton is negative in the case of Fig.", "REF (as it connects $m_-$ to $-m_\\beta $ and $m_\\beta $ to $m_+$ ); consequently the current in the interval from $m_-$ to $-m_\\beta $ is positive and larger than the one from $m_\\beta $ to $m_+$ (as the instanton in Fig.", "REF is closer to 0 than to $\\ell $ ), thus the total magnetization increases and the instanton moves further towards 0.", "Figure: Magnetization profile with m + =0.93m_{+}=0.93.", "The different curves in the plot correspond to the averaged magnetization computed at different times: t=10 5 t=10^5 (empty squares), t=10 6 t=10^6 (black squares) and t=10 8 t=10^8 (empty circles).", "The black thin line denotes the initial configuration, corresponding to a step function centered at r=5r=5.The transition region in the stable case is approximated by an instanton which is a stationary solution of the evolution equations on the whole line.", "Analogously, when $m_+<m_\\beta $ we speculate that the transition region is approximated by a bump which is again a stationary solution $m(r)$ , $r\\ge 0$ , of (REF ) on the half line with zero current and given boundary condition at 0, say $\\mu $ , namely: $&& m(r) = \\tanh \\Big \\lbrace \\beta [ Jm(r)+h]\\Big \\rbrace ,\\quad r \\ge 0\\\\&& h= -Jm(0) + \\frac{1}{\\beta }\\tanh ^{-1} \\mu , \\qquad m(r)=\\mu \\,\\,\\text{for $r<0$} $ Indeed it can be easily seen that a stationary solution of (REF ) with zero current is necessarily a solution of (REF ).", "The Gibbsian formula (in the mesoscopic limit) would give (REF ) with $h=0$ , thus the problem (REF ) is not in the framework of the equilibrium theory.", "This is reflected by the appearance of an auxiliary magnetic field which has to be determined consistently with the magnetization pattern (as in the FitzHugh Nagumo models of the introduction where however by a mean field assumption the magnetization was simply a real number).", "Observe that if $m(r)$ solves (REF ) with boundary condition $\\mu $ then $-m(r)$ solves (REF ) with boundary condition $-\\mu $ , this symmetry will play an important role in the sequel.", "Besides the trivial solution $m(r)\\equiv \\mu $ , existence of other solutions of (REF ) is an open problem.", "The simulations indicate the existence of increasing solutions, we thus define: Definition.", "The bump $B_{\\mu }(r)$ , $\\mu \\in (-m_\\beta , m^{})$ , is a non constant solution of (REF ) which is monotone non decreasing, We call $b(\\mu )$ its asymptotic value: $\\lim _{r \\rightarrow \\infty } B_{\\mu }(r)=: b(\\mu )$ Analogously we call $B^-_\\mu $ , $m\\in (-m^,m_\\beta )$ a non constant solution which is monotonic non increasing and denote by $b^-_\\mu $ its asymptotic value.", "The existence $B_\\mu $ implies the existence of $B^-_\\mu $ , in fact by simmetry $B^-_\\mu =-B_{-\\mu }$ .", "Thus what we will say for $B_\\mu $ extends to $B^-_\\mu $ and in the sequel we will consider only $B_\\mu $ .", "As mentioned above the existence of bumps is an open problem, the simulations indicate that bumps do indeed exist.", "The relation between bump and instanton can be understood in the following way.", "Call $\\bar{x} (\\mu )$ the value of $r$ such that $\\bar{m}(r)=\\mu $ .", "Replace the boundary condition $m(r)=\\mu $ , $r<0$ , in the definition of the bump by $m(r)= \\bar{m}(r+\\bar{x} (\\mu ))$ , $r<0$ .", "Then the solution of (REF ) would be $m(r)= \\bar{m}(r +\\bar{x} (\\mu ))$ , $r>0$ , with $h=0$ , the asymptotic value at $r=+\\infty $ being $m_\\beta $ .", "Replacing $m(r)=\\mu $ by $m(r)= \\bar{m}(r+\\bar{x} (\\mu ))$ for $r<0$ is a small error if $\\mu $ is close to $-m_\\beta $ (because the instanton converges exponentially to its asymptotic values).", "One may then hope to prove in such a case the existence of the bump using perturbative techniques as in [7]–[8].", "This has been done successfully in [8] for the equation $m(r) = \\tanh \\lbrace \\beta [ J^{\\rm neum}m(r)+h]\\rbrace ,\\quad r \\ge 0$ where $J^{\\rm neum}$ is defined with Neumann conditions; $h$ above is fixed and sufficiently small.", "We have numerical evidence of the existence of bumps.", "We have simulated (REF ) by looking at its discrete version with ${ \\gamma }^{-1}=120$ , $\\ell =5$ and Neumann conditions at the right boundary.", "We have solved such an equation by iteration: we start with $m\\equiv 1$ , compute $h$ via () with such $m$ and then define the first iterate $m_1$ as $m_1= \\tanh {\\beta (Jm+h)}$ .", "We then repeat the procedure till we find a fixed point.", "This is indeed reached (approximately) after a few iterations (in fact, three iterations already suffice to obtain good numerical convergence), see Fig.", "REF .", "The numerical values of $b(\\mu )$ are reported in Fig.", "REF , the main features are: the values of $b(\\mu )$ are all in the metastable region, $b(m_+)=m_+$ if $m_+\\in (m^,m_\\beta )$ , i.e.", "in the plus metastable region (left panel) $b(m_-)>b(m_+)$ for $m_-\\in (-m_\\beta , 0)$ and $m_+=-m_-$ (right panel).", "Figure: Iterations of Eq.", "(), with β=2.5\\beta =2.5, γ -1 =120\\gamma ^{-1}=120, ℓ=5\\ell =5 and μ=-0.7\\mu =-0.7.", "The different points denote, respectively, the initial condition (empty squares), the first iteration (black squares), the second iteration (empty circles) and the third iteration (black circles).Figure: Left panel: Behavior of b(μ)b(\\mu ), with β=2.5\\beta =2.5 and γ -1 =120\\gamma ^{-1}=120.", "The black dashed line denotes the curve b(μ)=μb(\\mu )=\\mu .Right panel: For any μ∈(-m β ,0)\\mu \\in (-m_\\beta ,0) we report with an empty circle the value of b(m - )b (m_-) and with a black circles the value of b(m + )b(m_+), m + =-m - m_+=-m_-.Conjecture The bump $B_\\mu $ exists for all $\\mu < m^$ , when $\\mu \\in [m^,m_\\beta )$ there is no bump and we call $b(\\mu )=\\mu $ .", "When $m_+ < m_\\beta $ for all $\\ell $ large enough there is a stationary solution $m_{\\rm st}(r;\\ell ;m_{\\pm })$ of (REF ), such that $\\lim _{\\ell \\rightarrow \\infty }m_{\\rm st}(r\\ell ;\\ell ;m_{\\pm })=m_{\\rm st}(r;m_{\\pm }),\\quad r\\in (0,1)$ $m_{\\rm st}(0;m_{\\pm })= b(m_-),\\quad m_{\\rm st}(1;m_{\\pm })= b(m_+)$ and $-\\frac{1}{2} [1-\\beta (1-m_{\\rm st}^2)]\\frac{dm_{\\rm st}}{dr} =I_{\\rm st}(m_+),\\quad r\\in [0,1]$ Remark.", "Under the above Conjecture the channel has a positive current if $b(\\mu )>-\\mu , \\; \\mu \\in (-m_\\beta ,-m^);\\quad b(\\mu )>b(-\\mu ) , \\; \\mu \\in (-m^,0)$ As shown in the right panel of Fig.", "REF there is clear numerical evidence of the validity of (REF ).", "The equation (REF ) with boundary conditions (REF ) can be easily solved analytically thus determining $ I_{\\rm st}(m_+)$ .", "By using the numerical values obtained for $b(m_-)$ and $b(m_+)$ we get the graph shown with empty circles in Fig.", "REF , where however $ I_{\\rm st}$ is divided by $L$ in order to compare it with the experimental value $j(m_+)$ (black circle in Fig.", "REF ) as given in Fig.", "REF .", "The agreement is good except in the interval $m_+\\in (m^{\\prime },m^{\\prime \\prime \\prime })$ , such a discrepancy will be discussed in the next section.", "Figure: We plot I st /LI_{\\rm st}/L (empty circles) and j:=j ch →ℛ 2 T j:=j^{T}_{{\\rm ch} \\rightarrow \\mathcal {R}_2} (black circles) as functions of m + m_+ ." ], [ "Stability of the bump", "The numerical analysis of (REF ) suggests the following: there exists a bump solution $B_\\mu $ for all $\\mu \\in (-m_\\beta ,m^)$ , when $\\mu \\in (-m_\\beta ,-m^)$ there are two solutions: $m(x)\\equiv \\mu $ and $m(x)=B_\\mu (x)$ , when $\\mu \\in (-m^*,0)$ there are three solutions: $m(x)\\equiv \\mu $ , $m(x)=B_\\mu (x)$ and $m(x)=-B_{-\\mu }(x)$ , An alternative way to study the existence of the bump is by running the OS-CA with boundary conditions $\\mu $ on the left and Neumann on the right.", "We take the same parameters ${ \\gamma }^{-1}=120$ and $\\ell =5$ used for the numerical analysis of the solutions of (REF ) and start with an initial condition where all sites in the channel are occupied.", "Referring to Fig.", "REF , when $\\mu \\in (-m_\\beta ,m^{\\prime })$ we see, after a transient, a steady pattern close to $B_\\mu $ .", "When $\\mu \\in (m^{\\text{iv}},m_\\beta )$ we see, after a transient, a steady pattern close to $m(x)\\equiv \\mu $ .", "When $\\mu \\in (m^{\\prime },m^{\\text{iv}})$ the final pattern is close to $-B_{-\\mu }$ , see Fig.", "REF .", "Figure: β=2.5\\beta =2.5, γ -1 =120\\gamma ^{-1}=120, ℓ=5\\ell =5 and μ=0.7∈(m iv ,m '' )\\mu =0.7\\in (m^{\\text{iv}},m^{\\prime \\prime }).", "The black dashed line represents B μ B_\\mu , the black thin line represents the asymptotic pattern of the CA which is close to -B -μ -B_{-\\mu }.", "Observe that the OS-CA does not select the bump solution when $\\mu \\in (m^{\\prime },m^{\\text{iv}})$ which is approximately the region where there is discrepancy between the theoretical and the experimental curves in Fig REF .", "We conjecture that this is due to ${ \\gamma }$ being not small enough so we are far from the mesoscopic regime and stochastic fluctuations are relevant.", "Stochastic fluctuations may then determine tunnelling from the bump to patterns where there is a bump on the left and a minus bump on the right with an instanton in between them and patterns where the two bumps are both up.", "Indeed we have numerical evidence of all that, in the times of the simulations we see in fact the magnetization patterns oscillate as described above, see Fig.REF .", "Figure: Typical (average) magnetization profile obtained at the beginning of a cycle (left panel) and at the end of a cycle (right panel)." ], [ "Conclusions", "We have presented two sets of simulations: the first one, see Fig.", "REF , shows that in the CC-CA there is a non zero current $j^{CC}({ \\gamma }p)$ (provided the rate ${ \\gamma }p$ of exchanges between reservoirs is in some non zero interval); the second set of simulations, see Fig.", "REF , refers to the OS-CA at magnetization $m_+ = -m_->0$ and shows that when $m_+ \\in (0,m_\\beta )$ then the current $j(m_+)$ goes “in the wrong direction”, namely from the reservoir with $m_-$ to that with $m_+$ .", "We have a heuristic proof that what seen in Fig.", "REF follows from the behavior of the channel in the OS-CA, as shown in Fig.", "REF ; the proof relies on the validity of the mesoscopic and the adiabatic limits.", "In the case of Fig.", "REF the current is negative when $m_+>m_\\beta $ and positive when $m_+<m_\\beta $ , in the former case the magnetization pattern in the channel shows the coexistence of the plus and minus phases while in the latter case only one phase appears (the statement in both cases refers to what happens in most of the volume).", "When the current is negative the values of the magnetization in the plus phase are larger than $m_\\beta $ and smaller than $-m_\\beta $ in the negative one.", "Instead when the current is positive we are in the one phase regime and the values of the magnetization are metastable (thus there is a state with positive current which in the bulk takes positive metastable values and another state also with positive current which in the bulk takes negative metastable values).", "When the current is negative the plus and minus phases are connected via an instanton-like profile around the center of the channel, when the current is negative the unstable values of the magnetization are localized in a small region close to the endpoints.", "We thus have a boundary layer which leads quite abruptly from the imposed values of the magnetization at the boundaries to some metastable value after which the magnetization pattern is smooth and the current flows opposite to the magnetization gradient in agreement with the Fourier law.", "The strange phenomenon of the current going “in the wrong direction” depends on the fact that the magnetization-jump in the boundary layer is more pronounced if it starts from lower values of the magnetization.", "Such a property, see (REF ), follows from the solution (the bump) of a non local equation describing the boundary layer, but its solution is obtained only numerically and we do not have a mathematical proof or even a heuristic explanation of why (REF ) should hold.", "We expect that also in the Cahn-Hilliard equation the graph of $j(m_+)$ has a qualitatively similar shape as in Fig.", "REF , but we miss a proof.", "We imagine that our results extend to more general systems with Kac potentials and maybe to physical systems where a van der Waals type of phase transition is present.", "In such cases a metastable interval is well defined and the relevant density (or magnetization) patterns in the bulk of the channel should have metastable values.", "Also for short range interactions, as in the n.n.", "Ising model with ferromagnetic interactions there are metastable values but the metastable region depends on the size of the system and shrinks to 0 as the volume diverges.", "Take the 2D Ising model in a squared box of side $L$ : in the periodic case for $\\beta $ large it is proved that if $\\pm m_\\beta $ are the equilibrium magnetizations then for the canonical Gibbs measure with average magnetization $m \\in (-m_\\beta , -m_\\beta + c L^{-2/3})$ and $c$ small enough the phenomenon of phase separation is absent.", "Consider the Kawasaki dynamics at such values of $\\beta $ with periodic conditions on the horizontal sides of the box and exchanges of the spins in the vertical ones with infinite reservoirs at magnetization $m_-$ and $m_+$ on the left and right.", "If what we have observed extends to this 2D Ising model we should see in the bulk magnetization patterns in the metastable phase, hence with values in an interval of size $L^{-2/3}$ .", "The current should therefore scale as $L^{-1}L^{-2/3}= L^{-5/3}$ and if the boundary layer goes like in our case then the current would go from the small to the large values of the reservoirs magnetization." ], [ "Estimates on the current between reservoirs", "Recalling (REF ) we have $j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)=j_t := \\zeta _t \\sum _{i_+,i_-}\\mathbf {1}_{\\xi _t=(i_+,i_-)}[\\theta ^{\\prime \\prime }_t(i_+)- \\theta ^{\\prime \\prime }_t(i_-)]$ where $ \\zeta _t$ and $\\xi _t$ are random variables independent of the process till time $t$ and of $\\theta ^{\\prime \\prime }_t$ , they are also independent of each other.", "$ \\zeta _t$ takes value 1 with probability ${ \\gamma }p$ and value 0 with probability $1-{ \\gamma }p$ ; the values of $\\xi _t$ are pairs $(i_+,i_-)$ , $i_+\\in \\mathcal {R}_2$ , $i_-\\in \\mathcal {R}_1$ and $P(\\xi _t=(i_+,i_-))=\\frac{1}{R^2}$ .", "The sum $ \\sum _{i_+,i_-}$ is over $i_+\\in \\mathcal {R}_2$ and $i_-\\in \\mathcal {R}_1$ .", "We first estimate the expected value of $ j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)$ : $E_{ \\gamma }[ j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)]=E_{ \\gamma }[\\frac{N^{\\prime \\prime }_{\\mathcal {R}_2}(t)-N^{\\prime \\prime }_{\\mathcal {R}_1}(t)}{R}]{ \\gamma }p$ where $N^{\\prime \\prime }_{\\mathcal {R}_i}(t)=\\sum _{i\\in \\mathcal {R}_i}\\theta ^{\\prime \\prime }_t(i),\\quad i=1,2$ Since $\|N^{\\prime \\prime }_{\\mathcal {R}_i}(t)-N_{\\mathcal {R}_i}(t)\|\\le 2$ for all $t$ we have $\\Big \| E_{ \\gamma }[ j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)]-{ \\gamma }pE_{ \\gamma }[\\frac{N_{\\mathcal {R}_2}(t)-N_{\\mathcal {R}_1}(t)}{R}] \\Big \|\\le { \\gamma }p\\frac{4}{R} $ We will next prove (REF ).", "Since $\\theta ^{\\prime \\prime }$ has values 0, 1 we have from (REF ) $E[ j_t ]\\le { \\gamma }p$ By (REF ) the left hand site of (REF ), can be written as $A_T:=E\\Big [\\lbrace \\frac{1}{T} \\sum _{t=0}^{T-1} [j_t- { \\gamma }p R^{-1}(N_{+,t}-N_{-,t}) ] \\rbrace ^2\\Big ]$ where $N_{+,t}= \\sum _{i_+}\\eta _t(i_+)=N_{\\mathcal {R}_2}(t),\\quad N_{-,t}= \\sum _{i_-}\\eta _t(i_-)=N_{\\mathcal {R}_1} (t)$ Define $N^{\\prime \\prime }_{\\pm ,t}$ as in (REF ) but with $\\theta ^{\\prime \\prime }_t$ instead of $\\eta _t$ and $A^{\\prime \\prime }_T$ as in (REF ) but with $N^{\\prime \\prime }_{\\pm ,t}$ .", "Lemma 1 $A_{T} \\le A^{\\prime \\prime }_T + \\frac{16}{R} ({ \\gamma }p)^2 + \\frac{16}{R^2} ({ \\gamma }p)^2$ Proof.", "Call $a_{t} =j_t - { \\gamma }p R^{-1}(N^{\\prime \\prime }_{+,t}-N^{\\prime \\prime }_{-,t})$ $b_{t} = { \\gamma }p R^{-1}\\lbrace (N^{\\prime \\prime }_{+,t}-N^{\\prime \\prime }_{-,t}) - (N_{+,t}-N_{-,t})\\rbrace $ Then $A_{T} = E\\Big [\\frac{1}{T^2} \\sum _{s,t} (a_{t}-b_{t})(a_{s}-b_{s})\\Big ]$ Hence $A_{T} \\le A^{\\prime \\prime }_{T} +2 E\\Big [\\frac{1}{T^2} \\sum _{s,t}\|a_{t}\|\|b_{s}\|\\Big ]+ E\\Big [\\frac{1}{T^2} \\sum _{s,t}\|b_{s}\| \|b_t\|\\Big ]$ $\|b_t\|\\le { \\gamma }p \\frac{4}{R}$ because $\|N^{\\prime \\prime }_{+,t} - N^{\\prime \\prime }_{-,t}\| \\le R$ and $\|N^{\\prime \\prime }_{\\pm ,t} - N_{\\pm ,t}\| \\le 2$ .", "By (REF ) and $\|N^{\\prime \\prime }_{+,t} - N^{\\prime \\prime }_{-,t}\| \\le R$ we get $E_{ \\gamma }[\|a_t\|] \\le 2 { \\gamma }p$ , therefore $A_{T} \\le A^{\\prime \\prime }_{T} +2 { \\gamma }p \\frac{8}{R} { \\gamma }p + [{ \\gamma }p \\frac{4}{R}]^2$ $\\Box $ Lemma 2 Let $s<t$ and $a_t$ as in (REF ) then $E\\Big [ a_s a_t\\Big ] =0$ Proof.", "By the independence properties of $\\zeta _t$ and $\\xi _t$ : $E [ a_s j_t ] = E \\Big [ a_s { \\gamma }p \\sum _{i_+,i_-} R^{-2}[\\theta ^{\\prime \\prime }_t(i_+)- \\theta ^{\\prime \\prime }_t(i_-)]\\Big ] = E \\Big [ a_s { \\gamma }p R^{-1}[N^{\\prime \\prime }_{+,t}- N^{\\prime \\prime }_{-,t}]\\Big ]$ $\\Box $ As a consequence $A^{\\prime \\prime }_T = \\frac{1}{T^2} \\sum _{t=0}^{T-1} E[a_t^2]$ We expand the square in $E[a_t^2]$ , the first term is $E\\Big [ \\zeta _t \\sum _{i_+,i_-}\\sum _{i^{\\prime }_+,i^{\\prime }_-}\\mathbf {1}_{\\xi _t=(i_+,i_-)}\\mathbf {1}_{\\xi _t=(i^{\\prime }_+,i^{\\prime }_-)}[\\theta ^{\\prime \\prime }_t(i_+)- \\theta ^{\\prime \\prime }_t(i_-)][\\theta ^{\\prime \\prime }_t(i^{\\prime }_+)- \\theta ^{\\prime \\prime }_t(i^{\\prime }_-)]\\Big ]$ Due to the characteristic functions $i_{\\pm }=i^{\\prime }_{\\pm }$ so that the above is bounded by ${ \\gamma }p$ .", "The double product in the expansion of $E[a_t^2]$ is bounded by $2({ \\gamma }p)^2$ and the third term by $({ \\gamma }p)^2$ , so that $A^{\\prime \\prime }_T \\le \\frac{1}{T} \\lbrace { \\gamma }p + 3({ \\gamma }p)^2\\rbrace $ Going back to (REF ) we get $A_{T} \\le \\frac{1}{T} \\lbrace { \\gamma }p + 3({ \\gamma }p)^2\\rbrace +16 \\frac{({ \\gamma }p)^2}{R} + 16[ \\frac{{ \\gamma }p}{R}]^2$ which concludes the proof of (REF )." ], [ "Proof of Theorems ", "Proof of (REF ).", "Here we prove that $m(r,t)$ satisfies (REF ) both in the CC-CA and in the OS-CA.", "Let $u(r,t)=m(r,t)+1$ then $m$ satisfies (REF ) if and only if $u$ satisfies $&&\\frac{\\partial }{\\partial t}u(r,t) = \\frac{1}{2}\\frac{\\partial ^2 u}{\\partial r^2} - C \\frac{\\partial }{\\partial r} \\Big \\lbrace [u(2-u]\\int _{r}^{r+1}[u(r+\\xi ,t)- u(r-\\xi ,t)] d\\xi \\Big \\rbrace $ with $u(r+\\xi ,t)= u_+(t)=m_+(t)+1$ if $r+\\xi \\ge \\ell $ and $u(r-\\xi ,t)= u_-(t)=m_-(t)+1$ if $r-\\xi \\le 0$ .", "In the OS-CA $m_\\pm (t)\\equiv m_\\pm $ .", "By (REF ) $\\nonumber && \\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } E_{ \\gamma }[ \\eta (x,v,t)] =\\frac{1}{2} u(x,t), \\qquad v\\in \\lbrace -1,1\\rbrace \\\\&& \\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } u_{ \\gamma }(x,t)=u(x,t),\\qquad u_{ \\gamma }(x,t)=E_{ \\gamma }[ \\eta (x,t)] \\\\\\nonumber $ So that we need to prove that the limit of $u_{ \\gamma }$ satisfies (REF ).", "By assumpotion $u(r,t)$ is smooth so that it is enough to prove weak convergence namely that for any smooth test function $f(r,t)$ with compact support in $(0,\\ell )\\times (0,\\infty )$ , $&&\\hspace{-28.45274pt}\\int u(r,t)\\frac{\\partial f(r,t)}{\\partial t} dr dt= - \\frac{1}{2} \\int u(r,t)\\frac{\\partial ^2 f(r,t)}{\\partial r^2} dr dt \\nonumber \\\\&&\\hspace{2.84544pt}- \\int \\frac{\\partial f(r,t)}{\\partial r} C \\Big \\lbrace [u(2-u]\\int _{r}^{r+1}[u(r+\\xi ,t)- u(r-\\xi ,t)] \\Big \\rbrace dr dt$ By an integration by parts $&&\\int u(r,t)\\frac{\\partial f(r,t)}{\\partial t} dr dt=-\\lim _{{ \\gamma }\\rightarrow 0} { \\gamma }^3\\sum _{x,t} f({ \\gamma }x,{ \\gamma }^2t){ \\gamma }^{-2}\\lbrace u_{ \\gamma }(x;t+1)- u_{ \\gamma }(x;t)\\rbrace $ We will next consider $u_{ \\gamma }(x;t+1)- u_{ \\gamma }(x;t)$ .", "Recalling that $j_{x,x+1}(t)$ is the number of particles which in the time step $t, t+1$ cross the bond $(x,x+1)$ , $x\\in \\lbrace 1,..,L-1\\rbrace $ (counting as positive those which jump from $x$ to $x+1$ and as negative those from $x+1$ to $x$ ), we have $u_{ \\gamma }(x;t+1)- u_{ \\gamma }(x;t)=E_{ \\gamma }[j_{x-1,x}(t)]-E_{ \\gamma }[j_{x,x+1}(t)]$ We then have denoting by $\\nabla _{ \\gamma }$ the discrete derivative ($\\nabla _{ \\gamma }\\varphi (x)=\\varphi (x+1)-\\varphi (x)$ ), $\\int u(r,t)\\frac{\\partial f(r,t)}{\\partial t} dr dt=-\\lim _{{ \\gamma }\\rightarrow 0} { \\gamma }^3\\sum _{x,t} { \\gamma }^{-1} \\nabla _{ \\gamma }f({ \\gamma }x,{ \\gamma }^2t){ \\gamma }^{-1}E_{ \\gamma }[j_{x,x+1}(t)]$ Lemma 3 $E_{ \\gamma }[j_{x,x+1}(t)]= \\frac{1}{2}[u_{ \\gamma }(x;t)- u_{ \\gamma }(x+1;t)+E_{ \\gamma }\\Big [\\chi _{x,{ \\gamma };t}{ \\epsilon }_{x,{ \\gamma };t}+\\chi _{x+1,{ \\gamma };s}{ \\epsilon }_{x+1,{ \\gamma };t}\\Big ]$ where ${ \\epsilon }_{x,{ \\gamma };t}$ is ${ \\epsilon }_{x,{ \\gamma }}$ computed at time $t$ and $\\chi _{x,{ \\gamma };t} = \\eta (x,1;t)\\Big (1- \\eta (x,-1;t)\\Big )+\\eta (x,-1;t)\\Big (1- \\eta (x,1;t)\\Big )$ Proof.", "Observe that the expected number of particles that goes from $x$ to $x+1$ is $E_{ \\gamma }\\Big [\\eta (x,1;t)\\eta (x,-1,t)+\\chi _{x,{ \\gamma };t}(\\frac{1}{2}+{ \\epsilon }_{x,{ \\gamma };t})\\Big ]=\\frac{1}{2} u_{ \\gamma }(x,t)+ E_{ \\gamma }\\Big [\\chi _{x,{ \\gamma };t}{ \\epsilon }_{x,{ \\gamma };t}\\Big ]$ The expected number of particles that goes from $x+1$ to $x$ is $&&E_{ \\gamma }\\big [\\eta (x+1,1;t)\\eta (x+1,-1,t)+\\chi _{x+1,{ \\gamma };t}(\\frac{1}{2}-{ \\epsilon }_{x+1,{ \\gamma };t})\\big ]=\\frac{1}{2} u_{ \\gamma }(x+1,t)\\\\&& \\hspace{199.16928pt}- E_{ \\gamma }\\big [\\chi _{x+1,{ \\gamma };t}{ \\epsilon }_{x+1,{ \\gamma };t}\\big ]$ so that we get (REF ).$\\Box $ We insert (REF ) in (REF ) and, denoting by $\\Delta _{ \\gamma }$ the discrete laplacian, we get $\\nonumber &&{ \\gamma }^3\\sum _{x,t} { \\gamma }^{-1} \\nabla _{ \\gamma }f({ \\gamma }x,{ \\gamma }^2t){ \\gamma }^{-1}j_{ \\gamma }(x,x+1,t)= \\frac{1}{2} { \\gamma }^3\\sum _{x,t}{ \\gamma }^{-2} \\Delta _{ \\gamma }f({ \\gamma }x,{ \\gamma }^2t)u_{ \\gamma }(x,t)\\\\&&\\hspace{28.45274pt}+ { \\gamma }^3\\sum _{x,t} { \\gamma }^{-1} 2 f^{\\prime }({ \\gamma }x,{ \\gamma }^2t)E_{ \\gamma }[\\chi _{x,{ \\gamma };t}]E_{ \\gamma }[{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}] +R_{ \\gamma }\\nonumber \\\\$ where $2 f^{\\prime }({ \\gamma }x,{ \\gamma }^2t)= [\\nabla _{ \\gamma }f({ \\gamma }x,{ \\gamma }^2t)+ \\nabla _{ \\gamma }f({ \\gamma }(x-1),{ \\gamma }^2t)]$ and $R_{ \\gamma }:=2 { \\gamma }^3\\sum _{x,t} { \\gamma }^{-1} 2 f^{\\prime }({ \\gamma }x,{ \\gamma }^2t) E_{ \\gamma }[\\chi _{x,{ \\gamma };t}\\Big ({ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}-E_{ \\gamma }[{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}]\\Big )]$ By (REF ) and (REF ) $\\nonumber && \\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } E_{ \\gamma }[\\chi _{x,{ \\gamma };t}]=\\frac{1}{2} u(r,t)[2-u(r,t)]\\\\ \\\\&& \\lim _{{ \\gamma }\\rightarrow 0} \\lim _{{ \\gamma }x\\rightarrow r, { \\gamma }^{2}t\\rightarrow \\tau } E_{ \\gamma }[{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}] = \\int _{r}^{r+1} C[u(r+\\xi ,t)- u(r-\\xi ,t)]d\\xi \\nonumber $ We postpone the proof of $\\lim _{{ \\gamma }\\rightarrow 0} { \\gamma }^3\\sum _{x=2}^{{ \\gamma }^{-1}\\ell -1} \\sum _{t=1}^{{ \\gamma }^{-2}T} E_{ \\gamma }\\Big [\\big \|{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}-E_{ \\gamma }[{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}]\\big \| \\Big ] = 0$ where $(0,\\ell )\\times (0,T)$ contains the support of $f(r,t)$ .", "Observe that (REF ), (REF ), (REF ) and (REF ) yield (REF ) concluding the proof of $\\Box $ Proof of (REF ).", "By Cauchy-Schwartz it is enough to prove that $\\lim _{{ \\gamma }\\rightarrow 0} { \\gamma }^3\\sum _{x=2}^{{ \\gamma }^{-1}\\ell -1} \\sum _{t=1}^{{ \\gamma }^{-2}T} E_{ \\gamma }\\Big [\\big \|{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}-E_{ \\gamma }[{ \\gamma }^{-1}{ \\epsilon }_{x,{ \\gamma };t}]\\big \|^2 \\Big ] = 0$ We thus need to compute the limit of ${ \\gamma }^5\\sum _{r,r^{\\prime },r^{\\prime \\prime }\\in { \\gamma }\\mathbb {Z}} \\sum _{\\tau \\in { \\gamma }^2 \\mathbb {Z}}g_{ \\gamma }(r,r^{\\prime },r^{\\prime \\prime },\\tau )$ where ${ \\gamma }^{-1}r \\in [2,{ \\gamma }^{-1}\\ell -1]$ , $\|r^{\\prime }-r\| \\le 1$ , $\|r^{\\prime \\prime }-r\| \\le 1$ , ${ \\gamma }^{-2}\\tau \\in [1,{ \\gamma }^{-2}T]$ and $g_{ \\gamma }(r,r^{\\prime },r^{\\prime \\prime },\\tau ) = C^2E_{ \\gamma }[\\tilde{\\eta }_{{ \\gamma }^{-2}\\tau } ({ \\gamma }^{-1} (r^{\\prime }-r))\\tilde{\\eta }_{{ \\gamma }^{-2}\\tau } ({ \\gamma }^{-1} (r^{\\prime \\prime }-r)) ]$ where $\\tilde{\\eta }_t(x) = \\eta (x,t)-E_{ \\gamma }[\\eta (x,t)]$ if $x\\in [1,L]$ , otherwise it is $=\\frac{2 N_{\\mathcal {R}_i}}{R}- E_{ \\gamma }[\\frac{2 N_{\\mathcal {R}_i}}{R}]$ where $i=2$ if $x>L$ and $i=1$ if $x<1$ otherwise in the OS-CA is equal to $m_\\pm $ respectively.", "By (REF ) and (REF ), (REF ) vanishes as ${ \\gamma }\\rightarrow 0$ .", "$\\Box $ Proof of (REF ).", "We call $I_{x,{ \\gamma }}^T={ \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[j_{x,x+1}(t)]$ Lemma 4 There are $c$ and $c^{\\prime }$ so that for all $r^{\\prime }<r^{\\prime \\prime }$ in $(0,\\ell )$ $\\Big \|\\frac{1}{x^{\\prime \\prime }-x^{\\prime }}\\sum _{y=x^{\\prime }}^{x^{\\prime \\prime }}I_{y,{ \\gamma }}^T\\Big \|\\le c,\\qquad x^{\\prime }=[{ \\gamma }^{-1}r^{\\prime }],\\quad x^{\\prime \\prime }=[{ \\gamma }^{-1}r^{\\prime \\prime }]$ $\\Big \|I_{x^{\\prime \\prime },{ \\gamma }}^T-I_{x^{\\prime },{ \\gamma }}^T\\Big \|\\le c^{\\prime } \|r^{\\prime \\prime }-r^{\\prime }\|$ Proof.", "By (REF ), using that $\|\\chi _{x,{ \\gamma };t}\|\\le 2$ and $\|{ \\epsilon }_{x,{ \\gamma };t}\| \\le 2C{ \\gamma }$ for all $x$ and $t$ and after telescopic cancellations we get $\\Big \|\\frac{1}{x^{\\prime \\prime }-x^{\\prime }}\\sum _{y=x^{\\prime }}^{x^{\\prime \\prime }}I_{y,{ \\gamma }}^T\\Big \|\\le \\big \| { \\gamma }\\sum _{s=0}^{T-1}\\frac{1}{x^{\\prime \\prime }-x^{\\prime }} E_{ \\gamma }[\\frac{1}{2}(\\eta (x^{\\prime },s)-\\eta (x^{\\prime \\prime }+1,s))]\\big \|+ 8 C { \\gamma }^2T$ The right hand side converges to $\\frac{1}{r^{\\prime \\prime }-r^{\\prime }}\\int _0^\\tau \\frac{1}{2} [m(r^{\\prime }.s)-m(r^{\\prime \\prime },s)] ds+8C^2\\tau $ which, by the smoothness of $m$ , proves (REF ).", "We have that $\\Big \| { \\gamma }\\sum _{t=0}^{T-1} j_{x^{\\prime },x^{\\prime }+1}(t)]-{ \\gamma }\\sum _{t=0}^{T-1} j_{x^{\\prime \\prime },x^{\\prime \\prime }+1}(t)\\Big \|\\le c^{\\prime } { \\gamma }\|x^{\\prime \\prime }-x^{\\prime }\|$ because the particles which contribute to the left hand site are: (1) those which reach for the first time $x^{\\prime }+1$ jumping from $x^{\\prime }$ and at the final time are in $[x^{\\prime }+1,x^{\\prime \\prime }]$ ; (2) those which reach for the first time $x^{\\prime \\prime }$ jumping from $x^{\\prime \\prime }+1$ and at the final time are in $[x^{\\prime }+1,x^{\\prime \\prime }]$ ; (3) those initially in $[x^{\\prime }+1,x^{\\prime \\prime }]$ and which leave this interval for the last time jumping to $x^{\\prime \\prime }+1$ ; (4) those initially in $[x^{\\prime }+1,x^{\\prime \\prime }]$ and which leave this interval for the last time jumping to $x^{\\prime }$ .", "$\\Box $ The family $\\lbrace I_{x,{ \\gamma }}^T\\rbrace $ thought as functions of $r={ \\gamma }x$ are equibounded and equicontinuous in any compact of $(0,\\ell )$ , thus they converge pointwise by subsequences.", "We will then prove (REF ) by identifying the limit.", "By continuity it will be enough to prove $\\lim _{{ \\gamma }\\rightarrow 0}\\frac{1}{x^{\\prime \\prime }-x^{\\prime }}\\sum _{y=x^{\\prime }}^{x^{\\prime \\prime }} I_{x,{ \\gamma }}^T=\\frac{1}{r^{\\prime \\prime }-r^{\\prime }} \\int _{r^{\\prime }}^{r^{\\prime \\prime }}dr\\int _0^\\tau I(r,s)ds$ By (REF ) $\\frac{1}{x^{\\prime \\prime }-x^{\\prime }}\\sum _{y=x^{\\prime }}^{x^{\\prime \\prime }} I_{x,{ \\gamma }}^T&=&{ \\gamma }\\sum _{s=0}^{T-1}\\Big \\lbrace \\frac{1}{x^{\\prime \\prime }-x^{\\prime }} E_{ \\gamma }[\\frac{1}{2}(\\eta (x^{\\prime },s)-\\eta (x^{\\prime \\prime }+1,s))]\\\\&+& \\frac{1}{x^{\\prime \\prime }-x^{\\prime }}\\sum _{y=x^{\\prime }}^{x^{\\prime \\prime }}E_{ \\gamma }[\\chi _{x,{ \\gamma };s}{ \\epsilon }_{x,{ \\gamma };s}+\\chi _{x+1,{ \\gamma };s}{ \\epsilon }_{x+1,{ \\gamma };s}]\\Big \\rbrace $ The first term converges to $\\frac{1}{r^{\\prime \\prime }-r^{\\prime }} \\frac{1}{2}\\int _0^\\tau [m(r^{\\prime },s)-m(r^{\\prime \\prime },s)]ds= \\frac{1}{r^{\\prime \\prime }-r^{\\prime }} \\frac{1}{2}\\int _0^\\tau ds\\int _{r^{\\prime }}^{r^{\\prime \\prime }}dr\\frac{\\partial m(r,s)}{\\partial s}$ By (REF ) and (REF ) the second one converges to $-\\frac{1}{r^{\\prime \\prime }-r^{\\prime }} \\int _0^\\tau \\int _{r^{\\prime }}^{r^{\\prime \\prime }} C [1-m^2]\\int _{r}^{r+1}[m(r+\\xi ,s)- m(r-\\xi ,s)] d\\xi dr ds$ Proof of (REF ).", "As the two are similar, we just prove the second equality in (REF ).", "The same proof as the one for (REF ) shows that $\\Big \|I_{x,{ \\gamma }}^T-I_{{\\rm ch}\\rightarrow \\mathcal {R}_2,{ \\gamma }}^T\\Big \|\\le c^{\\prime } \|\\ell -r\|,\\qquad x=[{ \\gamma }^{-1}r]$ where $I_{{\\rm ch}\\rightarrow \\mathcal {R}_2,{ \\gamma }}^T= { \\gamma }\\sum _{t=0}^{T-1} E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)]$ Let $\\tilde{I}$ be a limit point of $I_{{\\rm ch}\\rightarrow \\mathcal {R}_2,{ \\gamma }}^T$ as ${ \\gamma }\\rightarrow 0$ then $\\Big \|\\int _0^\\tau I(r,s)ds -\\tilde{I}\\Big \|\\le c^{\\prime } \|\\ell -r\|$ Using the expression (REF ) for $I(r,t)$ and the continuity of $m$ , we get in the limit $r\\rightarrow \\ell $ that $\\tilde{I}=\\int _0^\\tau I(\\ell ,s)ds$ .", "Proof of (REF ).", "As the proofs are similar, we just prove the second equality in (REF ) for the CC-CA.", "Suppose by contradiction that there is $t>0$ such that $m(\\ell ,t)\\ne m_{+}(t)$ and for the sake of definiteness $m(\\ell ,t)< m_{+}(t)$ .", "Then there is $\\delta >0$ and an interval $[t^{\\prime },t^{\\prime \\prime }]$ so that for $s\\in [t^{\\prime },t^{\\prime \\prime }]$ , $m_{+}(t)> m(\\ell ,t) +\\delta $ .", "Recalling the proof of Lemma REF $E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(s)]&=&E_{ \\gamma }[ \\frac{N_{\\mathcal {R}_2}(s)}{R}]- \\frac{1}{2} u_{ \\gamma }(L,s)- E_{ \\gamma }\\Big [\\chi _{L,{ \\gamma };s}{ \\epsilon }_{L,{ \\gamma };s}\\Big ] \\\\&\\ge &E_{ \\gamma }[ \\frac{N_{\\mathcal {R}_2}(s)}{R}]-\\frac{1}{2} u_{ \\gamma }(L,s) -c{ \\gamma }$ $c$ a suitable constant, $c{ \\gamma }$ bounding the term with ${ \\epsilon }_{x,{ \\gamma }}$ .", "Then, recalling (REF ), (REF ), (REF ) and using the assumptions in Theorem REF we get $\\liminf _{{ \\gamma }\\rightarrow 0}{ \\gamma }^2\\sum _{s\\in \\mathbb {Z} \\cap { \\gamma }^{-2}[t^{\\prime },t^{\\prime \\prime }]} E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(s)]&\\ge &\\frac{1}{2} \\int _{t^{\\prime }}^{t^{\\prime \\prime }}[m_+(s)- m(\\ell ,s)]ds\\ge \\frac{\\delta }{2} [t^{\\prime \\prime }-t^{\\prime }]$ which contradicts (REF ).", "The dynamics of the reservoirs.", "We just prove (REF ).", "Let $\\tau _0\\ge 0$ , $\\tau >0$ , $t_0=[{ \\gamma }^{-2}\\tau _0]$ , $T=[{ \\gamma }^{-2}\\tau ]$ , then $N_{\\mathcal {R}_2}(t_0+T)-N_{\\mathcal {R}_2}(t_0)=\\sum _{t=t_0}^{t_0+T-1}\\Big [j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)-j_{\\mathcal {R}_2\\rightarrow \\mathcal {R}_1}(t)\\big ]$ We take the expectation and we use (REF ) to get $\\nonumber &&\\Big \|E_{ \\gamma }[N_{\\mathcal {R}_2}(t_0+T)-N_{\\mathcal {R}_2}(t_0)] -\\sum _{t=t_0}^{t_0+T-1}\\Big [E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)]-E_{ \\gamma }[\\frac{N_{\\mathcal {R}_2}(t)-N_{\\mathcal {R}_1}(t)}{R}]{ \\gamma }p\\Big ]\\Big \|\\\\&&\\hspace{85.35826pt}\\le \\frac{4{ \\gamma }p}{R}T$ We then get $\\nonumber \\frac{a}{2} [m_+(\\tau _0+\\tau )-m_+(\\tau _0)]=\\int _{\\tau _0}^{\\tau _0+\\tau } I(\\ell ,s)ds-p\\int _{\\tau _0}^{\\tau _0+\\tau } \\frac{1}{2}[m_+(s)-m_-(s)]ds\\\\ $ which is obtained from (REF ) by multiplying by ${ \\gamma }$ and taking the limit ${ \\gamma }\\rightarrow 0$ after using that (1) $R=a{ \\gamma }^{-1}$ , (2) by (REF ) $\\lim _{{ \\gamma }\\rightarrow 0}E_{ \\gamma }[\\frac{N_{\\mathcal {R}_2}(t)-N_{\\mathcal {R}_1}(t)}{R}]=\\frac{m_+(\\tau )-m_-(\\tau )}{2}, \\quad t =[{ \\gamma }^{-2}\\tau ]$ (3) by (REF ) $\\lim _{{ \\gamma }\\rightarrow 0}{ \\gamma }\\sum _{t=t_0}^{t_0+T-1} E_{ \\gamma }[j_{{\\rm ch}\\rightarrow \\mathcal {R}_2}(t)]=\\int _{\\tau _0}^{\\tau _0+\\tau } I(\\ell ,s)ds$ Then (REF ) is obtained from (REF ) by dividing by $\\tau $ and taking the limit $\\tau \\rightarrow 0$ .", "Acknowledgements The authors acknowledge very useful discussions with Dima Ioffe." ] ]	1606.04920
[ [ "The entropy emission properties of near-extremal Reissner-Nordstr\\\"om\n black holes" ], [ "Abstract Bekenstein and Mayo have revealed an interesting property of evaporating $(3+1)$-dimensional Schwarzschild black holes: their entropy emission rates $\\dot S_{\\text{Sch}}$ are related to their energy emission rates $P$ by the simple relation $\\dot S_{\\text{Sch}}=C_{\\text{Sch}}\\times (P/\\hbar)^{1/2}$.", "Remembering that $(1+1)$-dimensional perfect black-body emitters are characterized by the same functional relation, $\\dot S^{1+1}=C^{1+1}\\times(P/\\hbar)^{1/2}$, Bekenstein and Mayo have concluded that, in their entropy emission properties, $(3+1)$-dimensional Schwarzschild black holes behave effectively as $(1+1)$-dimensional entropy emitters.", "One naturally wonders whether all black holes behave as simple $(1+1)$-dimensional entropy emitters?", "In order to address this interesting question, we shall study in this paper the entropy emission properties of Reissner-Nordstr\\\"om black holes.", "We shall show, in particular, that the physical properties which characterize the neutral sector of the Hawking emission spectra of these black holes can be studied {\\it analytically} in the near-extremal $T_{\\text{BH}}\\to0$ regime.", "We find that the Hawking radiation spectra of massless neutral scalar fields and coupled electromagnetic-gravitational fields are characterized by the non-trivial entropy-energy relations $\\dot S^{\\text{Scalar}}_{\\text{RN}} = -C^{\\text{Scalar}}_{\\text{RN}} \\times (AP^3/\\hbar^3)^{1/4} \\ln(AP/\\hbar)$ and $\\dot S^{\\text{Elec-Grav}}_{\\text{RN}} = -C^{\\text{Elec-Grav}}_{\\text{RN}} \\times (A^4P^9/\\hbar^9)^{1/10} \\ln(AP/\\hbar)$ in the near-extremal $T_{\\text{BH}}\\to0$ limit (here $A$ is the surface area of the Reissner-Nordstr\\\"om black hole).", "Our analytical results therefore indicate that {\\it not} all black holes behave as simple $(1+1)$-dimensional entropy emitters." ], [ "Introduction", "The entropy emission rate $\\dot{S}$ and the energy emission rate (power) $P$ of a perfect black-body (BB) emitter in a flat $(3+1)$ -dimensional spacetime are related by the well known Stefan-Boltzmann radiation law [1] (we use gravitational units in which $G=c=k_{\\text{B}}=1$ ) $\\dot{S}^{3+1}_{\\text{BB}}=C^{3+1}\\times \\Big ({{AP^3}\\over {\\hbar ^3}}\\Big )^{1/4}\\ ,$ where $A$ is the surface area of the $(3+1)$ -dimensional radiating black-body and $C^{3+1}=({{32\\pi ^2}/{1215}})^{1/4}$ is a dimensionless proportionality coefficient.", "However, in a very interesting work, Bekenstein and Mayo [2] have revealed the remarkable fact that the Hawking radiation spectra [3] of $(3+1)$ -dimensional Schwarzschild black holes are characterized by the qualitatively different (and mathematically much simpler) entropy-energy relation [2] $ \\dot{S}^{3+1}_{\\text{Sch}}=C^{3+1}_{\\text{Sch}}\\times \\Big ({{P}\\over {\\hbar }}\\Big )^{1/2}\\ ,$ where $C^{3+1}_{\\text{Sch}}$ is a numerically computed coefficient [2] which depends on the characteristic greybody factors [4] of the $(3+1)$ -dimensional Schwarzschild black-hole spacetime.", "Bekenstein and Mayo [2] have emphasized the interesting fact that the entropy-energy relation (REF ), which characterizes the Hawking emission spectra of $(3+1)$ -dimensional Schwarzschild black holes, has the same functional form as the entropy-energy relation [2] $\\dot{S}^{1+1}_{\\text{BB}}=C^{1+1}_{\\text{BB}}\\times \\Big ({{P}\\over {\\hbar }}\\Big )^{1/2}$ which characterizes the emission spectra of $(1+1)$ -dimensional perfect black-body emitters [one finds $C^{1+1}_{\\text{BB}}=(\\pi /3)^{1/2}$ for a $(1+1)$ -dimensional perfect black-body emitter [2]].", "Hence, Bekenstein and Mayo [2] have reached the intriguing conclusion that, in their entropy emission properties, $(3+1)$ -dimensional Schwarzschild black holes behave effectively as $(1+1)$ -dimensional [and not as $(3+1)$ -dimensional] thermal entropy emitters [see Eqs.", "(REF ) and (REF )].", "It is worth noting that it was later proved [5], [6] that this intriguing property of the $(3+1)$ -dimensional Schwarzschild black holes is actually a generic characteristic of all radiating $(D+1)$ -dimensional Schwarzschild black holes.", "One naturally wonders whether this intriguing physical property of the Schwarzschild black holes is shared by all black holes?", "In particular, we raise here the following question: do all radiating black holes behave as simple $(1+1)$ -dimensional entropy emitters?", "In order to address this interesting question, we shall analyze in this paper the entropy emission properties of Reissner-Nordström black holes.", "As we shall show below, the physical properties which characterize the neutral sector of the Hawking radiation spectra of these black holes can be studied analytically in the near-extremal $T_{\\text{BH}}\\rightarrow 0$ regime [here $T_{\\text{BH}}$ is the Bekenstein-Hawking temperature of the Reissner-Nordström black hole, see Eq.", "(REF ) below].", "Our analytical results (to be presented below) indicate that not all radiating black holes behave as simple $(1+1)$ -dimensional entropy emitters." ], [ "The Hawking radiation spectra of near-extremal Reissner-Nordström black\nholes", "In the present section we shall study the Hawking emission of massless neutral fields from near-extremal Reissner-Nordström black holes.", "The semi-classical Hawking radiation power $P_{\\text{RN}}$ and the semi-classical entropy emission rate $\\dot{S}_{\\text{RN}}$ for one bosonic degree of freedom are given respectively by the integral relations [4], [7], [8] $P_{\\text{RN}}={{\\hbar }\\over {2\\pi }}\\sum _{l,m}\\int _0^{\\infty } {{\\Gamma \\omega }\\over {e^{\\hbar \\omega /T_{\\text{BH}}}-1}}d\\omega \\ ,$ and $\\dot{S}_{\\text{RN}}={{1}\\over {2\\pi }}\\sum _{l,m}\\int _0^{\\infty }\\Big [{{\\Gamma }\\over {e^{\\hbar \\omega /T_{\\text{BH}}}-1}}\\ln \\Big ({{e^{\\hbar \\omega /T_{\\text{BH}}}-1}\\over {\\Gamma }}+1\\Big )+\\ln \\Big (1+{{\\Gamma }\\over {e^{\\hbar \\omega /T_{\\text{BH}}}-1}}\\Big )\\Big ]d\\omega \\ ,$ where $\\lbrace l,m\\rbrace $ are the angular harmonic indices of the emitted field mode, $\\Gamma =\\Gamma _{lm}(\\omega )$ are the black-hole-field greybody factors [4], and $T_{\\text{BH}}={{\\hbar (r_+-r_-)}\\over {4\\pi r^2_+}}\\ $ with $r_{\\pm }=M+(M^2-Q^2)^{1/2}$ is the semi-classical Bekenstein-Hawking temperature of the Reissner-Nordström black hole [here $\\lbrace r_+,r_-\\rbrace $ are the horizon radii of the Reissner-Nordström black hole, and $\\lbrace M,Q\\rbrace $ are the black-hole mass and charge, respectively].", "The characteristic thermal factor $\\omega /(e^{\\hbar \\omega /T_{\\text{BH}}}-1)$ that appears in the expression (REF ) for the semi-classical black-hole radiation power implies that the Hawking emission spectra peak at the characteristic frequency ${{\\hbar \\omega ^{\\text{peak}}}\\over {T_{\\text{BH}}}}=O(1)\\ .$ Furthermore, taking cognizance of the fact that the Bekenstein-Hawking temperature (REF ) of a near-extremal [9] Reissner-Nordström black hole is characterized by the strong inequality ${{MT_{\\text{BH}}}\\over {\\hbar }}\\ll 1\\ ,$ one finds the closely related strong inequality $M\\omega ^{\\text{peak}}\\ll 1\\ $ for the characteristic emission frequencies that constitute the Hawking black-hole radiation spectra in the near-extremal regime (REF ).", "It is well known [4], [10] that the dimensionless greybody factors $\\Gamma _{lm}(\\omega )$ , which quantify the interaction of the emitted field modes with the effective curvature barrier in the exterior region of the black-hole spacetime, can be calculated analytically in the low-frequency regime (REF ) [it is worth emphasizing again that the low-frequency regime (REF ) dominates the neutral sector of the Hawking radiation spectra in the low-temperature (near-extremal) regime (REF )].", "We shall now use this fact in order to study analytically the physical properties which characterize the neutral sector of the Hawking black-hole radiation spectra in the near-extremal (low-temperature) limit (REF )." ], [ "The Hawking emission of massless scalar quanta", "Following the analysis presented in [4], one finds the leading-order behavior $\\Gamma _{lm}=\\Big [{{(l!", ")^2}\\over {(2l)!(2l+1)!!", "}}\\Big ]^2\\prod _{n=1}^{l}\\Big [1+\\Big ({{\\hbar \\omega }\\over {2\\pi T_{\\text{BH}}\\cdot n}}\\Big )^2\\Big ]\\Big ({{AT_{\\text{BH}}\\omega }\\over {\\hbar }}\\Big )^{2l}{{A\\omega ^2}\\over {\\pi }}\\cdot [1+O(AT_{\\text{BH}}\\omega /\\hbar )^{2l+1}]\\ $ for the greybody factors which characterize the emission of scalar quanta in the low-frequency regime (REF ).", "Here $A=4\\pi r^2_+\\ $ is the surface area of the Reissner-Nordström black hole.", "From (REF ) one finds that, in the low-frequency regime (REF ), the scalar Hawking black-hole radiation spectrum is dominated by the fundamental $l=m=0$ mode [it is worth emphasizing the fact that the Hawking emission of scalar modes with $l>0$ is suppressed as compared to the Hawking emission of the fundamental $l=m=0$ scalar mode.", "This characteristic property of the Hawking black-hole emission spectra stems from the fact that the greybody factors of the higher scalar modes (that is, scalar modes which are characterized by $l>0$ ) contain higher powers (as compared to the fundamental $l=m=0$ scalar mode) of the small quantity $\\omega (r_+-r_-)\\ll 1$ [see Eq.", "(REF )]].", "In particular, one finds [see Eq.", "(REF )] $\\Gamma _{00}={{A\\omega ^2}\\over {\\pi }}\\cdot [1+O(AT_{\\text{BH}}\\omega /\\hbar )]$ in the small frequency regime (REF ) which characterizes the neutral sector of the Hawking emission spectra in the low-temperature (near-extremal) regime (REF ).", "Substituting (REF ) into Eqs.", "(REF ) and (REF ), one finds after some algebra [11] $P_{\\text{RN}}={{\\pi ^2}\\over {30}}{{AT^4_{\\text{BH}}}\\over {\\hbar ^3}}$ and $\\dot{S}_{\\text{RN}}=-{{\\zeta (3)}\\over {\\pi ^2}}{{AT^3_{\\text{BH}}}\\over {\\hbar ^{3}}}\\cdot \\Big [\\ln \\Big ({{AT^2_{\\text{BH}}}\\over {\\hbar ^2}}\\Big )+O(1)\\Big ]$ for the scalar Hawking radiation power and the scalar Hawking entropy emission rate of the near-extremal Reissner-Nordström black holes.", "Finally, substituting (REF ) into (REF ), one can express the black-hole entropy emission rate in terms of the Hawking radiation power: $\\dot{S}^{\\text{Scalar}}_{\\text{RN}}=-C^{\\text{Scalar}}_{\\text{RN}}\\times \\Big ({{AP^3_{\\text{RN}}}\\over {\\hbar ^3}}\\Big )^{1/4}\\ln \\Big ({{AP_{\\text{RN}}}\\over {\\hbar }}\\Big )\\ ,$ where the analytically calculated coefficient $C^{\\text{Scalar}}_{\\text{RN}}$ is given by $C^{\\text{Scalar}}_{\\text{RN}}={{30^{3/4}\\zeta (3)}\\over {2\\pi ^{7/2}}}\\ .$ It is worth emphasizing the fact that the entropy emission rate (REF ), which characterizes the scalar Hawking emission spectra of the near-extremal Reissner-Nordström black holes, does not have the simple $(1+1)$ -dimensional entropy-energy functional relation (REF ) which characterizes the Hawking emission spectra of the Schwarzschild black holes.", "It is important to emphasize that Mirza, Oboudiat, and Zare [6] have reached similar conclusions for 3-dimensional rotating BTZ black holes and for D-dimensional Lovelock black holes in odd and even dimensions (Refs.", "[5] and [6] have also considered the case of D-dimensional general relativistic black holes)." ], [ "The Hawking emission of coupled electromagnetic-gravitational\nquanta", "For the case of coupled electromagnetic-gravitational quanta, one finds the leading-order behavior [10] $\\Gamma _{11}=\\Gamma _{2m}={4\\over 9}\\Big ({{A\\omega ^2}\\over {4\\pi }}\\Big )^4$ for the greybody factors in the low-frequency regime (REF ) which characterizes the neutral sector of the Hawking black-hole emission spectra in the near-extremal (low-temperature) regime (REF ).", "[It is worth emphasizing the fact that the Hawking emission of coupled electromagnetic-gravitational modes with $l>2$ is suppressed as compared to the Hawking emission of coupled electromagnetic-gravitational modes with $l=1$ and $l=2$ .", "This characteristic property of the Hawking black-hole emission spectra stems from the fact that the greybody factors of coupled electromagnetic-gravitational modes with higher-$l$ values (that is, electromagnetic-gravitational modes which are characterized by $l>2$ ) contain higher powers (as compared to the $l=1$ and $l=2$ modes) of the small quantity $\\omega r_+\\ll 1$ [10]].", "Substituting (REF ) into Eqs.", "(REF ) and (REF ), one finds after some algebra [12] $P_{\\text{RN}}={{4\\pi ^5}\\over {297}}{{A^4T^{10}_{\\text{BH}}}\\over {\\hbar ^9}}$ and $\\dot{S}_{\\text{RN}}=-{{560\\zeta (9)}\\over {\\pi ^5}}{{A^4T^9_{\\text{BH}}}\\over {\\hbar ^{9}}}\\cdot \\Big [\\ln \\Big ({{AT^2_{\\text{BH}}}\\over {\\hbar ^2}}\\Big )+O(1)\\Big ]$ for the electromagnetic-gravitational Hawking radiation power and the electromagnetic-gravitational Hawking entropy emission rate of the near-extremal Reissner-Nordström black holes.", "Finally, substituting (REF ) into (REF ), one can express the black-hole entropy emission rate in terms of the Hawking radiation power: $\\dot{S}^{\\text{Elec-Grav}}_{\\text{RN}}=-C^{\\text{Elec-Grav}}_{\\text{RN}}\\times \\Big ({{A^4P^9_{\\text{RN}}}\\over {\\hbar ^9}}\\Big )^{1/10}\\ln \\Big ({{AP_{\\text{RN}}}\\over {\\hbar }}\\Big )\\ ,$ where the analytically calculated coefficient $C^{\\text{Elec-Grav}}_{\\text{RN}}$ is given by $C^{\\text{Elec-Grav}}_{\\text{RN}}={{112\\zeta (9)}\\over {\\pi ^{19/2}}}\\Big ({{297}\\over {4}}\\Big )^{9/10}\\ .$ It is worth emphasizing again that the entropy emission rate (REF ), which characterizes the electromagnetic-gravitational Hawking emission spectra of the near-extremal Reissner-Nordström black holes, does not have the simple $(1+1)$ -dimensional entropy-energy functional relation (REF ) which characterizes the Hawking radiation spectra of the Schwarzschild black holes." ], [ "Summary", "In a very interesting paper [2], Bekenstein and Mayo have revealed that, in their entropy emission properties, $(3+1)$ -dimensional Schwarzschild black holes behave effectively as $(1+1)$ -dimensional [and not as $(3+1)$ -dimensional] thermal entropy emitters [see Eqs.", "(REF ) and (REF )].", "Later studies [5], [6] have extended the analysis of [2] to higher dimensional black holes and proved that all radiating $(D+1)$ -dimensional Schwarzschild black holes are characterized by this intriguing physical property.", "One naturally wonders whether this interesting property of the Schwarzschild black holes is shared by all black holes?", "In particular, motivated by the results of [2], [5], [6], we have raised here the following question: do all radiating black holes behave as simple $(1+1)$ -dimensional entropy emitters?", "In order to address this intriguing question, we have explored in this paper the entropy emission properties of Reissner-Nordström black holes.", "In particular, we have shown that the physical properties which characterize the neutral sector of the Hawking emission spectra of these black holes can be studied analytically in the low-temperature (near-extremal) $T_{\\text{BH}}\\rightarrow 0$ regime.", "We have explicitly shown that the analytically derived expressions for the Hawking entropy emission rates of massless scalar fields and coupled electromagnetic-gravitational fields by near-extremal Reissner-Nordström black holes [see Eqs.", "(REF ) and (REF )] do not have the simple $(1+1)$ -dimensional entropy-energy functional relation (REF ) which characterizes the Hawking emission spectra of the Schwarzschild black holes.", "Our analytical results therefore indicate that not all black holes behave as simple $(1+1)$ -dimensional entropy emitters.", "ACKNOWLEDGMENTS This research is supported by the Carmel Science Foundation.", "I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B.", "Tea for stimulating discussions." ] ]	1606.04944
[ [ "On a class of left metacyclic codes" ], [ "Abstract Let $G_{(m,3,r)}=\\langle x,y\\mid x^m=1, y^3=1,yx=x^ry\\rangle$ be a metacyclic group of order $3m$, where ${\\rm gcd}(m,r)=1$, $1<r<m$ and $r^3\\equiv 1$ (mod $m$).", "Then left ideals of the group algebra $\\mathbb{F}_q[G_{(m,3,r)}]$ are called left metacyclic codes over $\\mathbb{F}_q$ of length $3m$, and abbreviated as left $G_{(m,3,r)}$-codes.", "A system theory for left $G_{(m,3,r)}$-codes is developed for the case of ${\\rm gcd}(m,q)=1$ and $r\\equiv q^\\epsilon$ for some positive integer $\\epsilon$, only using finite field theory and basic theory of cyclic codes and skew cyclic codes.", "The fact that any left $G_{(m,3,r)}$-code is a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$ is proved, where ${\\cal A}_i$ is a minimal cyclic code over $\\mathbb{F}_q$ of length $m$ and $C_i$ is a skew cyclic code of length $3$ over an extension field of $\\mathbb{F}_q$.", "Then an explicit expression for each outer code in any concatenated code is provided.", "Moreover, the dual code of each left $G_{(m,3,r)}$-code is given and self-orthogonal left $G_{(m,3,r)}$-codes are determined." ], [ "Introduction", "Let $\\mathbb {F}_q$ be a finite field of cardinality $q$ and $G$ a group of order $n$ .", "The group algebra $\\mathbb {F}_q[G]$ is a vector space over $\\mathbb {F}_q$ with basis $G$ .", "Addition, multiplication with scalars $c\\in \\mathbb {F}_q$ and multiplication are defined by: for any $a_g,b_g\\in \\mathbb {F}_q$ , $\\sum _{g\\in G}a_g g+\\sum _{g\\in G}b_g g=\\sum _{g\\in G}(a_g+b_g) g$ , $c(\\sum _{g\\in G}a_g g)=\\sum _{g\\in G}ca_g g$ , $(\\sum _{g\\in G}a_g g)(\\sum _{g\\in G}b_g g)=\\sum _{g\\in G}(\\sum _{uv=g}a_ub_v)g$ .", "Then $\\mathbb {F}_q[G]$ is an associative $\\mathbb {F}_q$ -algebra with identity $1=1_{\\mathbb {F}_q}1_{G}$ where $1_{\\mathbb {F}_q}$ and $1_{G}$ are the identity elements of $\\mathbb {F}_q$ and $G$ respectively.", "Readers are referred to for more details on group algebra.", "Let $G=G_{(m,s,r)}=\\langle x,y\\mid x^m=1, y^s=1,yx=x^ry\\rangle $ where ${\\rm gcd}(m,r)=1$ and $r^s\\equiv 1$ (mod $m$ ).", "Then $G$ is called a metacyclic group of order $sm$ .", "Sabin and Lomonaco provided a unique direct decomposition of the $\\mathbb {F}_2$ -algebra $\\mathbb {F}_2[G_{(m,n,r)}]$ to minimal two-sided ideals (central codes) and described a technique to decompose minimal central codes to a direct sum of $n$ minimal left ideals (left codes) and gave an algorithm to determine these minima left codes.", "They discovered several good metacyclic codes and they expressed the hope that more “good\" and perhaps even “best\" codes may be discovered among the ideals of non-abelian group rings.", "Recently, Olteanu et al provided algorithms to construct minimal left group codes and rediscovered some best codes.", "These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra $\\mathbb {F}_q[G]$ for a large class of groups $G$ .", "For example, by use of the computer algebra system GAP and the packages GUAVA and Wedderga some optimal codes and non-abelian group codes were obtained: $\\diamond $ A linear $[27,18,2]$ -code constructed by a left ideal in $\\mathbb {F}_2[G]$ , where $G=\\langle x,y\\mid x^9=1, y^3=1,yx=x^4y\\rangle $ ( ).", "$\\diamond $ A best linear $[20,4,8]$ -code constructed by a left ideal in $\\mathbb {F}_3[G]$ , where $G=\\langle x,y\\mid x^5=1, y^4=1,yx=x^2y\\rangle $ ().", "$\\diamond $ A non-abelian group code $[55,10,22]$ -code constructed by a left ideal in $\\mathbb {F}_2[G]$ , where $G=\\langle x,y\\mid x^{11}=1, y^5=1,yx=x^3y\\rangle $ ().", "For any $\\textbf {a}=(a_{0,0}, a_{1,0},\\ldots , a_{m-1,0},a_{0,1}, a_{1,1},\\ldots , a_{m-1,1}$ , $a_{0,s-1}, a_{1,s-1},\\ldots ,a_{m-1,s-1})\\in \\mathbb {F}_q^{sm}$ , define $\\Psi (\\textbf {a})=(1,x,\\ldots ,x^{m-1})M_{\\textbf {a}}\\left(\\begin{array}{c} 1 \\cr y\\cr \\ldots \\cr y^{s-1}\\end{array}\\right),$ where $M_{\\textbf {a}}=\\left(\\begin{array}{cccc} a_{0,0} & a_{0,1}& \\ldots &a_{0,s-1} \\cr a_{1,0} & a_{1,1} & \\ldots & a_{1,s-1}\\cr \\ldots & \\ldots \\cr a_{m-1,0} & a_{m-1,1}&\\ldots & a_{m-1,s-1}\\end{array}\\right).$ Then $\\Psi $ is an $\\mathbb {F}_q$ -linear isomorphism from $\\mathbb {F}_q^{sm}$ onto $\\mathbb {F}_q[G_{(m,s,r)}]$ .", "As in and a nonempty subset $C$ of $\\mathbb {F}_q^{sm}$ is called a left metacyclic code (or left $G_{(m,s,r)}$ -code for more precisely) over $\\mathbb {F}_q$ if $\\Psi (C)$ is a left ideal of the $\\mathbb {F}_q$ -algebra $\\mathbb {F}_q[G_{(m,s,r)}]$ .", "From now on, we will identify $C$ with $\\Psi (C)$ for convenience.", "In this paper, we focus our attention on the case of $s=3$ in the metacyclic group $G_{(m,s,r)}$ and $r=q^\\epsilon $ for some positive integer $\\epsilon $ .", "Compared with the known theory for cyclic codes over finite fields, literatures related with metacyclic codes were involved too much group algebra language and techniques.", "A system and elementary theory for left metacyclic codes over finite fields have not been developed fully to the best of our knowledge.", "In this paper, we try to achieve the following goals: $\\bullet $ Develop a system theory for left $G_{(m,3,r)}$ -codes using an elementary method.", "Specifically, only finite field theory and basic theory of cyclic codes and skew cyclic codes are involved, and it does not involve any group algebra language and technique.", "$\\bullet $ Provide a clear expression for each left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ and give a formula to count the number of all such codes.", "$\\bullet $ Give an explicit expression of the dual code for each left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ and determine its self-orthogonality.", "Using the expression provided, one can list all distinct left $G_{(m,3,r)}$ -codes for specific $m$ and $q$ (not too big) conveniently and easily, design left $G_{(m,3,r)}$ -codes for their requirements and encode the presented codes directly.", "The present paper is organized as follows.", "In section 2, we prove that any left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ is a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$ for the case of ${\\rm gcd}(n,q)=1$ and $r\\equiv q^\\epsilon $ (mod $m$ ) for some positive integer $\\epsilon $ , where ${\\cal A}_i$ is a minimal cyclic code over $\\mathbb {F}_q$ of length $m$ and $C_i$ is a skew $\\theta _i$ -cyclic code over $K_i$ of length 3, i.e., left ideals of the ring $K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ , where $K_i$ is an extension field of $\\mathbb {F}_q$ and $\\theta _i\\in {\\rm Aut}_{\\mathbb {F}_q}(K_i)$ satisfying $\\theta _i^3={\\rm id}_{K_i}$ .", "In Section 3, we give a precise description for skew $\\theta _i$ -cyclic codes over $K_i$ of length 3.", "Hence all distinct left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ can be determined by their concatenated structure.", "In Section 4, we give the dual code of each left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ precisely and determine all self-orthogonal left-$G_{(m,3,r)}$ -codes.", "Finally, we list all distinct 541696 left $G_{(14,3,9)}$ -codes and all 3364 self-orthogonal left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ in Section 5." ], [ "The concatenated structure of left $G_{(m,3,r)}$ -codes over {{formula:4c315ecf-ae9b-4c4e-8e64-c52597914ee1}}", "In this section, we overview properties for concatenated structure of linear codes and determine the concatenated structure of left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_{q}$ .", "Let $B$ be a linear $[n_B,k_B,d_B]$ -code over $\\mathbb {F}_{q}$ , $\\mathbb {F}_{q^{k_B}}$ an extension field of $\\mathbb {F}_{q}$ with degree $k_B$ , $\\varphi $ an $\\mathbb {F}_{q}$ -linear isomorphism from $\\mathbb {F}_{q^{k_B}}$ onto $B$ and $E$ a linear $[n_E,k_E,d_E]$ -code over $\\mathbb {F}_{q^{k_B}}$ .", "The concatenated code of the inner code $B$ and the outer code $E$ is defined as $B\\Box _{\\varphi }E=\\lbrace (\\varphi (c_1),\\ldots , \\varphi (c_{n_E}))\\mid (c_1,\\ldots ,c_{n_E})\\in E\\rbrace $ (cf.", ").", "It is known that $B\\Box _{\\varphi }E$ is a linear code over $\\mathbb {F}_{q}$ with parameters $[n_Bn_E,k_Bk_E,\\ge d_Bd_E]$ .", "From now on, let $m$ be a positive integer satisfying ${\\rm gcd}(m,q)=1$ and $m\\ge 3$ , $(\\mathbb {Z}_m,+)$ the addition group of integer residue classes modulo $m$ where $\\mathbb {Z}_m=\\lbrace 0,1,\\ldots ,m-1\\rbrace $ , and denote by ${\\cal A}=\\mathbb {F}_q[x]/\\langle x^m-1\\rangle =\\lbrace \\sum _{i=0}^{m-1}a_ix^i\\mid a_0,a_1,\\ldots ,a_{m-1}\\in \\mathbb {F}_q\\rbrace $ the residue class ring of $\\mathbb {F}_q[x]$ modulo its ideal generated by $x^m-1$ with operations defined by the usual polynomial operations modulo $x^m-1$ .", "We will identify cyclic codes over $\\mathbb {F}_q$ of length $m$ with ideals of the ring ${\\cal A}$ under the identification map: $(a_0,a_1,\\ldots ,a_{m-1})\\mapsto \\sum _{i=0}^{m-1}a_ix^i$ .", "First, we define $\\theta : (\\mathbb {Z}_m,+)\\rightarrow (\\mathbb {Z}_m,+) \\ {\\rm by} \\ \\theta (s)\\equiv rs \\ ({\\rm mod} \\ m).$ As ${\\rm gcd}(m,r)=1$ , we see that $\\theta $ is a group automorphism on $(\\mathbb {Z}_m,+)$ .", "Moreover, from $1<r<n$ and $r^3\\equiv 1$ (mod $m$ ) we deduce that the multiplicative order of $\\theta $ is a factor of 3.", "Next, we define a map ${\\cal A}\\rightarrow {\\cal A}$ by the rule that $a(x)\\mapsto \\sum _{j=0}^{m-1}a_jx^{\\theta (j)}\\equiv a(x^r) \\ ({\\rm mod} \\ x^m-1),$ for any $a(x)=\\sum _{j=0}^{m-1}a_jx^j\\in {\\cal A}$ .", "In order to simplify notations, we also use $\\theta $ to denote this map on ${\\cal A}$ , i.e., $\\theta (a(x))=a(x^r) \\ {\\rm in} \\ {\\cal A}.$ Then $\\theta $ is an $\\mathbb {F}_q$ -algebra automorphism on ${\\cal A}$ satisfying $\\theta ^3={\\rm id}_{{\\cal A}}$ .", "In addition, $\\theta $ is a permutation of the coordinate positions $\\lbrace 0,1,\\ldots , m-1\\rbrace $ of a cyclic codes of length $m$ over $\\mathbb {F}_q$ and is called a multiplier.", "Readers are referred to for more details on basic properties of multipliers.", "Wether $\\theta $ denotes this automorphism of ${\\cal A}$ or the group automorphism on $(\\mathbb {Z}_m,+)$ is determined by the context.", "Let ${\\cal A}[y;\\theta ]=\\lbrace \\sum _{j=0}^ka_j(x)y^j\\mid a_0(x),\\ldots ,a_k(x)\\in {\\cal A}, \\ k\\ge 0\\rbrace $ be the skew polynomial ring over the commutative ring ${\\cal A}$ with coefficients written on the left side, where the multiplication is defined by the rule $y^ja(x)=\\theta ^{j}(a(x))y^j=a(x^{r^j})y^j, \\ \\forall a(x)\\in {\\cal A}$ and by the natural ${\\cal A}$ -linear extension to all polynomials in ${\\cal A}[y;\\theta ]$ .", "As $\\theta ^3={\\rm id}_{\\mathcal {A}}$ , we have $y^3a(x)=a(x)y^3$ for all $a(x)\\in {\\cal A}$ .", "So $y^3-1$ generates a two-sided ideal $\\langle y^3-1\\rangle $ of ${\\cal A}[y;\\theta ]$ .", "Let ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle =\\lbrace \\alpha +\\beta y+\\gamma y^2\\mid \\alpha ,\\beta ,\\gamma \\in {\\cal A}\\rbrace $ be the residue class ring of $A[y;\\theta ]$ modulo its two-sided ideal $\\langle y^3-1\\rangle $ .", "For any $\\xi =a_0(x)+a_1(x)y+a_2(x)y^2\\in {\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ , where $a_j(x)=\\sum _{i=0}^{m-1}a_{i,j}x^i\\in {\\cal A}$ with $a_{i,j}\\in \\mathbb {F}_q$ for $j=0,1,2$ , we define a natural map: $\\Phi : \\xi \\mapsto a(x,y)=\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}x^iy^j.$ Then it can be easily proved that $\\Phi $ is a ring isomorphism from ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ onto $\\mathbb {F}_q[G_{(m,3,r)}]$ .", "In the rest of this paper, we will identify $\\mathbb {F}_q[G_{(m,3,r)}]$ with ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ under this ring isomorphism $\\Phi $ .", "Theorem 2.1 Using the notations above, ${\\cal C}$ is a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ if and only if ${\\cal C}$ is a left ideal of the ring ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ .", "By the identification of $\\mathbb {F}_q[G_{(m,3,r)}]$ with ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ , we see that ${\\cal C}$ is a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ , i.e., ${\\cal C}$ is a left ideal of the ring $\\mathbb {F}_q[G_{(m,3,r)}]$ , if and only if ${\\cal C}$ is a left ideal of ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ .", "In order to determine all left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ , by Theorem REF it is sufficient to give all left ideals of the ring ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ .", "To do this, we need investigate the structure and properties of ${\\cal A}$ first.", "For any integer $s$ , $0\\le s\\le m-1$ , let $J_{s}^{(q)}$ be the $q$ -cyclotomic coset modulo $m$ , i.e., $J_{s}^{(q)}=\\lbrace s,sq,\\ldots ,sq^{l_s-1}\\rbrace $ (mod $m$ ) where $l_s={\\rm min}\\lbrace k\\in \\mathbb {Z}^{+}\\mid s(q^k-1)\\equiv 0 \\ ({\\rm mod} \\ m)\\rbrace $ .", "Then $\|J_{s}^{(q)}\|=l_s$ .", "It is obvious that $J_{0}^{(q)}=\\lbrace 0\\rbrace $ and $\\theta (J_{0}^{(q)})=J_{0}^{(q)}=\\lbrace 0\\rbrace $ .", "In this paper, we assume that $r\\equiv q^\\epsilon \\ ({\\rm mod} \\ m)$ for some positive integer $\\epsilon $ .", "Lemma 2.2 Let $0\\le s\\le m-1$ .", "Then $J_{s}^{(q)}$ satisfies one and only one of the following two conditions: (I) $\\theta (s)\\equiv s$ $({\\rm mod} \\ m)$ .", "In this case, $\\theta (k)\\equiv k$ $({\\rm mod} \\ m)$ for all $k\\in J_{s}^{(q)}$ .", "(II) $\\theta (s)\\in J_{s}^{(q)}$ and $\\theta (s)\\lnot \\equiv s$ $({\\rm mod} \\ m)$ .", "In this case, $l_s=\|J_{s}^{(q)}\|$ is a multiple of 3, and $\\theta (J_{s}^{(q)})=J_{s}^{(q)}$ .", "By Condition (1), it follows that $\\theta (s)\\in J_{s}^{(q)}$ and $\\theta (J_{s}^{(q)})=J_{s}^{(q)}$ .", "Then we have one of the following two cases.", "(I) $\\theta (s)=rs\\equiv s$ (mod $m$ ).", "In this case, for any $k\\in J_{s}^{(q)}$ we have $k\\equiv sq^j$ for some $0\\le j\\le l_s-1$ , and hence $\\theta (k)=rsq^j\\equiv sq^j\\equiv k$ (mod $m$ ).", "(II) $\\theta (s)\\in J_{s}^{(q)}$ and $\\theta (s)\\lnot \\equiv s$ (mod $m$ ).", "In this case, it is obvious that $l_s=\|J_{s}^{(q)}\|\\ge 2$ .", "By $\\theta (s)\\in J_{s}^{(q)}$ , there exits integer $v$ , $1\\le v\\le l_s-1$ , such that $\\theta (s)=rs\\equiv sq^v$ (mod $m$ ).", "By $r^3\\equiv 1$ (mod $m$ ) and $sq^v\\equiv rs$ (mod $m$ ), we have $sq^{3v}=s(q^v)^3\\equiv r^3s\\equiv s$ (mod $m$ ).", "From this and by the minimality of $l_s$ , we deduce that $l_s\|3v$ .", "Suppose that $l_s$ is not a multiple of 3.", "Then ${\\rm gcd}(l_s,3)=1$ and hence $l_s\|v$ .", "By $sq^{l_s}\\equiv s$ (mod $m$ ), we deduce that $sq^v\\equiv s$ (mod $m$ ), i.e., $\\theta (s)\\equiv s$ (mod $m$ ), which contradicts that $\\theta (s)\\lnot \\equiv s$ (mod $m$ ).", "Hence $l_s$ is a multiple of 3.", "In this paper, let $\\zeta $ be a primitive $m$ th root of unity in an extension field of $\\mathbb {F}_{q}$ .", "Then $x^m-1=\\sum _{s=0}^{m-1}(x-\\zeta ^s)$ .", "We will adopt the following notations: $\\bullet $ Let $J_{k_0}^{(q)}, J_{k_1}^{(q)}, \\ldots , J_{k_s}^{(q)}$ , where $k_0=0$ , be all distinct $q$ -cyclotomic cosets modulo $m$ satisfying Condition (I) in Lemma REF .", "$\\bullet $ Let $J_{k_{s+1}}^{(q)}, \\ldots , J_{k_{s+t}}^{(q)}$ be all distinct $q$ -cyclotomic cosets modulo $m$ satisfying Condition (II) in Lemma REF .", "$\\bullet $ Denote $J(i)=J_{k_i}^{(q)}$ , $f_i(x)=\\prod _{j\\in J(i)}(x-\\zeta ^j)$ , $K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ and assume $d_i={\\rm deg}(f_i(x))=\|J_{k_i}^{(q)}\|=\|J(i)\|$ for all $i=0,1,\\ldots ,s+t$ .", "Then $d_i$ is a multiple of 3 for all $s+1\\le i\\le s+t$ by Lemma REF (II).", "It is clear that $f_0(x),f_1(x),\\ldots , f_{s+t}(x)$ are pairwise coprime irreducible polynomials in $\\mathbb {F}_q[x]$ satisfying $x^m-1=f_0(x)f_1(x)\\ldots f_{s+t}(x).$ Hence $K_i$ is an extension field of $\\mathbb {F}_q$ with cardinality $q^{d_i}$ for $i=0,1,\\ldots ,s+t$ , and $m=\\sum _{i=0}^{s+t}d_i$ .", "For each integer $i$ , $0\\le i\\le s+t$ , denote $F_i(x)=\\frac{x^m-1}{f_i(x)}\\in \\mathbb {F}_q[x]$ .", "Then ${\\rm gcd}(F_i(x),f_i(x))$ $=1$ .", "By Extended Euclidian Algorithm, we find polynomials $u_i(x)$ , $v_i(x)\\in \\mathbb {F}_q[x]$ such that $u_i(x)F_i(x)+v_i(x)f_i(x)=1.$ In this paper, we denote $\\varepsilon _i(x)\\equiv u_i(x)F_i(x)=1-v_i(x)f_i(x) \\ ({\\rm mod} \\ x^m-1).$ By (REF ) and (REF ), it follows that $\\varepsilon _i(\\zeta ^j)=1$ for all $j\\in J(i)$ and $\\varepsilon _i(\\zeta ^j)=0$ for all $j\\in \\mathbb {Z}_m\\setminus J(i)$ , which implies $\\varepsilon _i(x)=\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-jl})x^l, \\ 0\\le i\\le s+t.$ Then we have the following conclusions.", "Lemma 2.3 Using the notations above, the following hold.", "(i) $\\sum _{i=0}^{s+t}\\varepsilon _i(x)=1$ , $\\varepsilon _i(x)^2=\\varepsilon _i(x)$ and $\\varepsilon _i(x)\\varepsilon _j(x)=0$ in $\\mathcal {A}$ for all $0\\le i\\ne j\\le s+t$ .", "(ii) ${\\cal A}={\\cal A}_0\\oplus {\\cal A}_1\\oplus \\ldots {\\cal A}_{s+t}$ , where ${\\cal A}_i={\\cal A}\\varepsilon _i(x)$ is the ideal of ${\\cal A}$ generated by $\\varepsilon _i(x)$ , and ${\\cal A}_i$ is a commutative ring with $\\varepsilon _i(x)$ as its multiplicative identity for all $i=0,1,\\ldots , s+t$ .", "(iii) For each integer $i$ , $0\\le i\\le s+t$ , define $\\varphi _i: a(x)\\mapsto \\varepsilon _i(x)a(x) \\ {\\rm mod} \\ x^m-1 \\ (\\forall a(x)\\in K_i).$ Then $\\varphi _i$ is a field isomorphism from $K_i$ onto ${\\cal A}_i$ .", "(iv) $\\theta (\\varepsilon _i(x))=\\varepsilon _i(x)$ and $\\theta ({\\cal A}_i)={\\cal A}_i$ , for all $i=0,1$ , $\\ldots , s+t$ .", "(v) For each integer $i$ , $0\\le i\\le s+t$ , define $\\theta _i: K_i\\rightarrow K_i \\ {\\rm via} \\ a(x)\\mapsto a(x^r) \\ (\\forall a(x)\\in K_i).$ Then $\\theta _i$ is an $\\mathbb {F}_q$ -algebra automorphism on $K_i$ satisfying $\\theta _i^3={\\rm id}_{K_i}$ , where ${\\rm id}_{K_i}$ is the identity automorphism on $K_i$ .", "Precisely, $\\theta _i={\\rm id}_{K_i}$ for all $i=0,1,\\ldots ,s$ , and the multiplicative order of $\\theta _i$ is equal to 3 for all $i=s+1,\\ldots ,s+t$ .", "(vi) The restriction $\\theta \|_{{\\cal A}_i}$ of $\\theta $ on ${\\cal A}_i$ is an $\\mathbb {F}_q$ -algebra automorphism on ${\\cal A}_i$ satisfying $(\\theta \|_{{\\cal A}_i})^3={\\rm id}_{{\\cal A}_i}$ and $\\theta \|_{{\\cal A}_i}=\\varphi _i\\theta _i\\varphi _i^{-1}$ .", "Hence the following diagram for $\\mathbb {F}_q$ -algebra isomorphisms commutes: $\\left.\\begin{array}{ccc} \\ \\ \\ \\ K_i & \\stackrel{\\theta _i}{\\longrightarrow } & K_i\\cr \\varphi _i\\downarrow & & \\ \\ \\ \\ \\downarrow \\varphi _i\\cr \\ \\ \\ \\ \\mathcal {A}_i & \\stackrel{\\theta \|_{\\mathcal {A}_i}}{\\longrightarrow } & \\mathcal {A}_i\\end{array}\\right..$ Then $\\theta \|_{{\\cal A}_i}={\\rm id}_{{\\cal A}_i}$ for all $i=0,1,\\ldots ,s$ , and the multiplicative order of $\\theta \|_{{\\cal A}_i}$ is equal to 3 for all $i=s+1,\\ldots ,s+t$ .", "(i)–(iii) follow from classical ring theory and Equations (REF ) and (REF ).", "(iv) By the definition of the automorphism $\\theta $ on $\\mathcal {A}$ and (REF ), it follows that $\\theta (\\varepsilon _i(x))=\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-jl})x^{rl \\ ({\\rm mod} \\ m)}.$ As $J(i)=\\theta (J(i))=\\lbrace rj \\ ({\\rm mod} \\ m)\\mid j\\in J(i)\\rbrace $ , we have $\\theta (\\varepsilon _i(x))&=&\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-(rj)l})x^{rl \\ ({\\rm mod} \\ m)}\\\\&=&\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-j(rl)})x^{rl \\ ({\\rm mod} \\ m)}.$ Moreover, since $\\theta $ is an automorphism of the group $(\\mathbb {Z}_m,+)$ , for each $k\\in \\mathbb {Z}_m$ there is a unique $l\\in \\mathbb {Z}_m$ such that $k=rl$ (mod $m$ ), and hence $\\theta (\\varepsilon _i(x))=\\frac{1}{m}\\sum _{k=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-jk})x^{k}=\\varepsilon _i(x).$ Therefore, $\\theta ({\\cal A}_i)=\\theta ({\\cal A})\\theta (\\varepsilon _i(x))={\\cal A}\\varepsilon _i(x)={\\cal A}_i.$ (v) It is clear that $\\theta _i$ is an $\\mathbb {F}_q$ -algebra endomorphism of $K_i$ .", "By $r^3\\equiv 1$ (mod $m$ ) and $f_i(x)=\\prod _{j\\in J(i)}(x-\\zeta ^j)$ , we see that $(\\zeta ^j)^{r^3}=\\zeta ^j$ for all $j\\in J(i)$ , which implies that $x^{r^3}\\equiv x$ (mod $f_i(x)$ ), i.e., $x^{r^3}=x$ in $K_i$ .", "Hence for any $a(x)\\in K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ we have $\\theta _i^3(a(x))=a(x^{r^3})=a(x)$ in $K_i$ .", "Therefore, $\\theta _i^3={\\rm id}_{K_i}$ and so $\\theta _i$ is an $\\mathbb {F}_q$ -algebra automorphism of $K_i$ .", "Then we consider the following two cases: (v-1) Let $0\\le i\\le s$ .", "For any $j\\in J(i)$ , by Condition (I) in Lemma 2.2 we have $(\\zeta ^j)^r=\\zeta ^j$ , which implies $x^r\\equiv x$ (mod $x-\\zeta ^j$ ), and hence $x^r\\equiv x$ (mod $f_i(x)$ ).", "Therefore, $\\theta _i(a(x))=a(x^r)=a(x)$ for any $a(x)\\in K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ , and so $\\theta _i={\\rm id}_{K_i}$ .", "(v-2) Let $s+1\\le i\\le s+t$ .", "By Condition (II) in Lemma 2.2, there exists $j\\in J(i)$ such that $rj\\lnot \\equiv j$ (mod $m$ ), which implies $(\\zeta ^j)^r\\ne \\zeta ^j$ .", "Then by the proof of (v-1), we conclude $\\theta _i\\ne {\\rm id}_{K_i}$ .", "From this and by $\\theta _i^3={\\rm id}_{K_i}$ , we deduce that the multiplicative order of $\\theta _i$ is equal to 3.", "(vi) By (iv) and $\\theta ^3={\\rm id}_{{\\cal A}}$ , it follows that $\\theta \|_{{\\cal A}_i}$ is an $\\mathbb {F}_q$ -algebra automorphism on ${\\cal A}_i$ satisfying $(\\theta \|_{{\\cal A}_i})^3={\\rm id}_{{\\cal A}_i}$ .", "Then the equation $\\theta \|_{{\\cal A}_i}=\\varphi _i\\theta _i\\varphi _i^{-1}$ follows from (v) and the definitions of $\\varphi _i$ , $\\theta _i$ and $\\theta \|_{{\\cal A}_i}$ immediately.", "For any integer $i$ , $0\\le i\\le s+t$ , it is known that $\\mathcal {A}_i$ is a minimal cyclic code of length $m$ over $\\mathbb {F}_q$ .", "Precisely, $f_i(x)$ is the parity check polynomial and $\\varepsilon _i(x)$ is the idempotent generator of $\\mathcal {A}_i$ .", "Hence ${\\rm dim}_{\\mathbb {F}_q}(\\mathcal {A}_i)={\\rm deg}(f_i(x))=d_i$ .", "Frow now on, we adopt the following notations.", "$\\bullet $ Let $K_i[y;\\theta _i]=\\lbrace \\sum _{j=0}^kb_j(x)y^j\\mid b_0(x),\\ldots ,b_k(x)\\in K_i, \\ k\\ge 0\\rbrace $ be the skew polynomial ring over $K_i$ with coefficients written on the left side, where the multiplication is defined by the rule $y^ja(x)=\\theta _i^{j}(a(x))y^j=a(x^{r^j})y^j, \\ \\forall a(x)\\in K_i$ and by the natural $K_i$ -linear extension to all polynomials in $K_i[y;\\theta _i]$ .", "Since $\\theta _i^3={\\rm id}_{K_i}$ by Lemma REF (v), we see that $y^3-1$ generates a two-sided ideal $\\langle y^3-1\\rangle $ of $K_i[y;\\theta _i]$ .", "$\\bullet $ Let $K_i[y;\\theta _i]/\\langle y^3-1\\rangle =\\lbrace a(x)+b(x)y+c(x)y^2\\mid a(x), b(x),c(x)\\in K_i\\rbrace $ be the residue class ring of $K_i[y;\\theta _i]$ modulo its two-sided ideal $\\langle y^3-1\\rangle $ .", "Recall that left ideals of $K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ are called skew $\\theta _i$ -cyclic codes over $K_i$ of length 3 (See ).", "For more details on skew cyclic codes, readers are referred to , , , , .", "Now, we can decompose any left $G_{(m,3,r)}$ -code into a direct sum of concatenated codes by the following theorem.", "Theorem 2.4 Using the notations above, we have the following conclusions.", "(i) ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle =\\oplus _{i=0}^{s+t}({\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]/\\langle \\varepsilon _i(x)y^3-\\varepsilon _i(x)\\rangle )$ .", "(ii) For each integer $i$ , $0\\le i\\le s+t$ , $\\varphi _i:K_i\\rightarrow {\\cal A}_i$ can be extended to a ring isomorphism from $K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ onto ${\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]/\\langle \\varepsilon _i(x)y^3-\\varepsilon _i(x)\\rangle $ by $\\varphi _i:\\xi (y)\\mapsto \\varphi _i(\\xi _0)+\\varphi _i(\\xi _1)y+\\varphi _i(\\xi _2)y^2=\\varepsilon _i(x)\\xi (y)$ for any $\\xi (y)=\\xi _0+\\xi _1+\\xi _2y^2$ with $\\xi _0,\\xi _1,\\xi _2\\in K_i$ .", "(iii) ${\\cal C}$ is a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ if and only if for each $0\\le i\\le s+t$ there is a unique skew $\\theta _i$ -cyclic code $C_i$ over $K_i$ of length 3 such that ${\\cal C}=\\bigoplus _{i=0}^{s+t}{\\cal A}_i\\Box _{\\varphi _i}C_i,$ where ${\\cal A}_i\\Box _{\\varphi _i}C_i=\\lbrace \\varepsilon _i(x)\\xi \\ ({\\rm mod} \\ x^m-1)\\mid \\xi \\in C_i\\rbrace $ .", "Moreover, we have $\|{\\cal C}\|=\\prod _{i=0}^{s+t}\|C_i\|$ .", "(i) By Lemma REF (ii),(iv) and (vi), we have ${\\cal A}[y;\\theta ]=\\oplus _{i=0}^{s+t}{\\cal A}_i[y;\\theta \|_{{\\cal A}_i}].$ Moreover, by ${\\cal A}_i={\\cal A}\\varepsilon _i(x)$ we know that the projection from ${\\cal A}[y;\\theta ]$ onto ${\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]$ is determined by $\\alpha (y)\\mapsto \\varepsilon _i(x)\\alpha (y)$ ($\\forall \\alpha (y)\\in {\\cal A}[y;\\theta ]$ ).", "Especially, we have $y^3-1\\mapsto \\varepsilon _i(x)y^3-\\varepsilon _i(x)$ under this projection.", "As $(\\theta \|_{{\\cal A}_i})^3={\\rm id}_{{\\cal A}_i}$ , $\\varepsilon _i(x)y^3-\\varepsilon _i(x)$ generates a two-sided ideal $\\langle \\varepsilon _i(x)y^3-\\varepsilon _i(x)\\rangle $ of ${\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]$ for all $i=0,1,\\ldots ,s+t$ .", "By Lemma REF (i) it follows that $y^3-1=\\sum _{i=0}^{s+t}(\\varepsilon _i(x)y^3-\\varepsilon _i(x))$ and $(\\varepsilon _i(x)y^3-\\varepsilon _i(x))(\\varepsilon _j(x)y^3-\\varepsilon _j(x))=0$ for all $0\\le i\\ne j\\le s+t$ .", "Hence ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle =\\oplus _{i=0}^{s+t}({\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]/\\langle \\varepsilon _i(x)y^3-\\varepsilon _i(x)\\rangle )$ .", "(ii) Since $\\varphi _i:K_i\\rightarrow {\\cal A}_i$ is a ring isomorphism by Lemma 2.3(iii), the conclusion follows from Lemma REF (vi) and a direct calculation.", "(iii) By Theorem REF and (i), we see that ${\\cal C}$ is a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ if and only if for each integer $i$ , $0\\le i\\le s+t$ , there is a unique left ideal ${\\cal C}_i$ of the ring ${\\cal A}_i[y;\\theta \|_{{\\cal A}_i}]/\\langle \\varepsilon _i(x)y^3-\\varepsilon _i(x)\\rangle $ such that ${\\cal C}=\\oplus _{i=0}^{s+t}{\\cal C}_i$ .", "By (ii), the latter condition is equivalent to that for each integer $i$ , $0\\le i\\le s+t$ , there is a unique left ideal $C_i$ of $K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ such that ${\\cal C}_i=\\varphi _i(C_i)=\\lbrace \\varepsilon _i(x)\\xi \\mid \\xi \\in C_i\\rbrace ={\\cal A}_i\\Box _{\\varphi _i}C_i$ .", "Finally, it is clear that the codewords contained in ${\\cal C}$ is equal to $\|{\\cal C}\|=\\prod _{i=0}^{s+t}\|{\\cal C}_i\|=\\prod _{i=0}^{s+t}\|C_i\|$ .", "By Theorem REF , in order to give all distinct left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ it is sufficient to determine all distinct skew $\\theta _i$ -cyclic codes over $K_i$ of length 3, for all $i=0,1,\\ldots ,s+t$ .", "For convenience and notations simplicity in the following sections, we introduce the following notations.", "Notation 2.5 For any integer $i$ , $0\\le i\\le s+t$ , denote $\\bullet $ ${\\cal R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ .", "For any $\\alpha =a_0+a_1y+a_2y^2\\in {\\cal R}_i$ with $a_0,a_1,a_2\\in K_i$ , the Hamming weight ${\\rm wt}_H^{(i)}(\\alpha )$ of $\\alpha $ over $K_i$ is defined as the number of nonzero coefficients of the polynomial $a_0+a_1y+a_2y^2\\in K_i[y;\\theta _i]$ , i.e., $\\bullet $ ${\\rm wt}_H^{(i)}(\\alpha )=\|\\lbrace j\\mid a_j\\ne 0, \\ j=0,1,2\\rbrace \|$ .", "For any nonzero left ideal $J$ of ${\\cal R}_i$ , the minimum Hamming weight ${\\rm wt}_H^{(i)}(J)$ of $J$ over $K_i$ is defined as $\\bullet $ ${\\rm wt}_H^{(i)}(J)={\\rm min}\\lbrace {\\rm wt}_H^{(i)}(\\alpha )\\mid \\alpha \\ne 0, \\ \\alpha \\in J\\rbrace $ ." ], [ "Skew $\\theta _i$ -cyclic codes\nover {{formula:4c2fce09-9c34-4494-bf99-a00b00d69e61}} of length 3", "In this section, we give all skew $\\theta _i$ -cyclic codes over $K_i$ of length 3, i.e., left ideals of the ring ${\\cal R}_i$ , where $0\\le i\\le s+t$ .", "$\\diamondsuit $ Let $0\\le i\\le s$ .", "Then $K_i$ is a finite field of cardinality $q^{d_i}$ and $\\theta _i$ is the identity automorphism of $K_i$ by Lemma REF (v).", "Hence ${\\cal R}_i=K_i[y]/\\langle y^3-1\\rangle $ which is a commutative ring.", "In this case, left ideals of ${\\cal R}_i$ are in fact ideals of ${\\cal R}_i$ .", "By the basic theory of cyclic codes over finite fields, we know that $C_i$ is an ideal of ${\\cal R}_i$ if and only if $C_i$ is a cyclic code over $K_i$ of length 3.", "The latter is equivalent to that there is a unique monic divisor $g(y)$ of $y^3-1$ in $K_i[y]$ such that $C_i={\\cal R}_ig(y)$ .", "Then $g(y)$ is called the generator polynomial of $C_i$ and ${\\rm dim}_{K_i}(C_i)=3-{\\rm deg}_y(g(y))$ where ${\\rm deg}_y(g(y))$ is the degree of $g(y)$ as a polynomial with indeterminate $y$ .", "Obviously, ${\\cal R}_i(y^3-1)=\\lbrace 0\\rbrace $ .", "Theorem 3.1 Let $0\\le i\\le s$ .", "Then ${\\cal R}_i=K_i[y]/\\langle y^3-1\\rangle $ which is a commutative ring, and the following hold.", "(i) If $q\\equiv 0$ $({\\rm mod} \\ 3)$ , there are 4 distinct ideals in ${\\cal R}_i$ : ${\\cal R}_ig(y), \\ {\\rm where} \\ g(y)\\in \\lbrace 1, y-1,(y-1)^2,y^3-1\\rbrace .$ (ii) If $q^{d_i}\\equiv 2$ $({\\rm mod} \\ 3)$ , there are 4 distinct ideals in ${\\cal R}_i$ : ${\\cal R}_ig(y), \\ {\\rm where} \\ g(y)\\in \\lbrace 1, y-1,y^2+y+1,y^3-1\\rbrace .$ (iii) Let $q^{d_i}\\equiv 1$ $({\\rm mod} \\ 3)$ , $\\zeta _i(x)$ a primitive element of $K_i$ and denote $\\omega _i(x)=\\zeta _i(x)^{\\frac{q^{d_i}-1}{3}}$ $({\\rm mod} \\ f_i(x))$ .", "Then there are 8 distinct ideals in ${\\cal R}_i$ : ${\\cal R}_ig(y)$ , where $&&g(y)\\in \\lbrace 1, y-1,y-\\omega _i(x),y-\\omega _i(x)^2,y^2+y+1,\\\\&& \\ \\ \\ \\ \\ (y-1)(y-\\omega _i(x)^2), (y-1)(y-\\omega _i(x)), y^3-1\\rbrace .$ (i) Since $q\\equiv 0$ (mod 3), we have $y^3-1=(y-1)^3$ in $K_i[y]$ .", "In this case, $y^3-1$ has 4 monic divisors in $K_i[y]$ : 1, $y-1$ , $(y-1)^2$ and $y^3-1$ ,.", "(ii) Since $q^{d_i}\\equiv 2$ (mod 3), we have that $y^3-1=(y-1)(y^2+y+1)$ where $y^2+y+1$ is irreducible in $K_i[y]$ .", "In this case, $y^3-1$ has 4 monic divisors in $K_i[y]$ : 1, $y-1$ , $y^2+y+1$ and $y^3-1$ .", "(iii) Since $q^{d_i}\\equiv 1$ (mod 3), we have $3\|(q^{d_i}-1)$ and $\\omega _i(x)$ is a primitive 3th root of unity in $K_i$ , which implies $y^3-1=(y-1)(y-\\omega _i(x))(y-\\omega _i(x)^2)$ .", "In this case, $y^3-1$ has 8 monic divisors in $K_i[y]$ : 1, $y-1$ , $y-\\omega _i(x)$ , $y-\\omega _i(x)^2$ , $y^2+y+1$ , $(y-1)(y-\\omega _i(x)^2)$ , $(y-1)(y-\\omega _i(x))$ and $y^3-1$ .", "$\\diamondsuit $ Let $s+1\\le i\\le s+t$ .", "Then $K_i$ is a finite field of cardinality $q^{d_i}$ , $d_i$ is a multiple of 3 by Lemma REF (II), and $\\theta _i$ is an $\\mathbb {F}_q$ -algebra automorphism of $K_i$ with multiplicative order 3 by Lemma REF (v).", "In this case, by we know that $C_i$ is a left ideal of ${\\cal R}_i$ if and only if $C_i$ is a skew $\\theta _i$ -cyclic code over $K_i$ of length 3, and the latter is equivalent to that there is a unique monic right divisor $g(y)$ of $y^3-1$ in the skew polynomial ring $K_i[y;\\theta _i]$ such that $C_i={\\cal R}_ig(y)$ .", "If the latter condition is satisfied, $g(y)$ is called the generator polynomial of $C_i$ , ${\\rm dim}_{K_i}(C_i)=3-{\\rm deg}_y(g(y))$ and the number of codewords in $C_i$ is equal to $\|C_i\|=(q^{d_i})^{3-{\\rm deg}_y(g(y))}$ .", "Precisely, a generator matrix of $C_i$ over $K_i$ is given by: $G_{C_i}=(a(x),b(x),1) \\ {\\rm if} \\ g(x)=a(x)+b(x)y+y^2;$ and $G_{C_i}=\\left(\\begin{array}{ccc}a(x) & 1 & 0 \\cr 0 &\\theta _i(a(x)) & 1\\end{array}\\right) \\ {\\rm if} \\ g(x)=a(x)+y.$ In order to describe monic right divisors of $y^3-1$ in the skew polynomial ring $K_i[y;\\theta _i]$ and the relationships between two nontrivial monic right divisors, we adopt the following notations in the rest of this paper: $\\bullet $ Let $\\zeta _i(x)$ be a primitive element of $K_i$ and denote $\\varrho _i(x)=\\zeta _i(x)^{q^{\\frac{d_i}{3}-1}}$ .", "Then the multiplicative order of $\\varrho _i(x)$ is equal to ${\\rm ord}(\\varrho _i(x))=\\frac{q^{d_i}-1}{q^{\\frac{d_i}{3}}-1}=1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}.$ Denote $\\mathcal {G}_i=\\lbrace \\varrho _i(x)^k\\mid k=0,1,\\ldots ,q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}\\rbrace $ which is the multiplicative cyclic subgroup of $K_i^\\times $ generated by $\\varrho _i(x)$ .", "Then $\|\\mathcal {G}_i\|=1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}$ .", "Lemma 3.2 Let $s+1\\le i\\le s+t$ .", "Then we have the following: (i) $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ or $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ .", "(ii) For any $\\alpha (x)\\in K_i^\\times $ , $\\alpha (x)\\in \\mathcal {G}_i$ if and only if $\\alpha (x)$ satisfies the following equation $\\alpha (x)\\theta _i(\\alpha (x))\\theta _i^2(\\alpha (x))=1.$ (iii) All distinct monic right divisors of $y^3-1$ with degree 1 in the skew polynomial ring $K_i[y;\\theta _i]$ are given by: $-\\alpha (x)+y, \\ \\alpha (x)\\in \\mathcal {G}_i.$ (iv) All distinct monic right divisors of $y^3-1$ with degree 2 in the skew polynomial ring $K_i[y;\\theta _i]$ are given by: $\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2, \\ \\alpha (x)\\in \\mathcal {G}_i.$ (v) For any $\\alpha (x)\\in \\mathcal {G}_i$ , we have $y^3-1&=&\\left(-\\alpha (x)+y\\right)\\cdot \\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2\\right)\\\\&=&\\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2\\right)\\cdot \\left(-\\alpha (x)+y\\right).$ Therefore, both the number of right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ with degree 1 and the number of right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ with degree 2 are equal to $1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}$ .", "(i) Let $\\sigma : \\alpha \\mapsto \\alpha ^q$ ($\\forall \\alpha \\in K_i$ ) be the Frobenius automorphism of $K_i$ over $\\mathbb {F}_q$ .", "Then the multiplicative order of $\\sigma $ is $d_i$ and every automorphism of $K_i$ over $\\mathbb {F}_q$ is of the form: $\\sigma ^k$ , $0\\le k\\le d_i-1$ .", "By Lemma REF , $d_i$ is a multiple of 3.", "Hence ${\\rm ord}(\\sigma ^k)=3$ if and only if ${\\rm gcd}(k,d_i)=\\frac{d_i}{3}$ , i.e., $k=\\frac{d_i}{3}$ or $k=\\frac{2d_i}{3}$ .", "By Lemma REF (v), $\\theta _i$ is an automorphism of $K_i$ over $\\mathbb {F}_q$ with multiplicative order 3, which implies that $\\theta _i=\\sigma ^{\\frac{d_i}{3}}$ or $\\theta _i=\\sigma ^{\\frac{2d_i}{3}}$ .", "Hence $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ or $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ .", "(ii) Denote $\\alpha =\\alpha (x)$ in order to simplify the notation.", "When $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ , it is clear that $\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=\\alpha ^{1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}}.$ When $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ , by $\\alpha ^{q^{\\frac{4d_i}{3}}}=(\\alpha ^{q^{d_i}})^{q^{\\frac{d_i}{3}}}=\\alpha ^{q^{\\frac{d_i}{3}}}$ we have $\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=\\alpha ^{1+q^{\\frac{2d_i}{3}}+q^{\\frac{4d_i}{3}}}=\\alpha ^{1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}}$ as well.", "Since $K_i^\\times $ is a multiplicative cyclic group with order $q^{d_i}-1$ and $\\mathcal {G}_i$ is a subgroup of $K_i^\\times $ with order $1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}$ , by basic group theory we conclude that $\\alpha \\in \\mathcal {G}_i$ if and only if $\\alpha ^{1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}}=1$ , i.e., $\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=1$ .", "(iii) Let $\\alpha \\in K_i$ .", "Dividing $y^3-1$ by $y-\\alpha $ from right hand in the skew polynomial ring $K_i[y;\\theta _i]$ , we $y^3-1&=&\\left(y^2+\\theta _i^2(\\alpha )y+\\theta _i^2(\\alpha )\\theta _i(\\alpha )\\right)(y-\\alpha )\\\\&&+1-\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha ).$ Hence $-\\alpha +y$ is a right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ if and only if $1-\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=0$ , i.e., $\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=1$ .", "(iv) Let $\\beta ,\\gamma \\in K_i$ .", "Then $\\gamma +\\beta y+y^2$ is a right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ if and only if there exists $\\alpha \\in K_i$ such that $y^3-1&=&(-\\alpha +y)(\\gamma +\\beta y+y^2)\\\\&=&y^3+(\\theta _i(\\beta )-\\alpha )y^2+(\\theta _i(\\gamma )-\\beta \\alpha )y-\\gamma \\alpha ,$ which is equivalent that $\\alpha ,\\beta ,\\gamma $ satisfy $\\theta _i(\\beta )=\\alpha $ , $\\theta _i(\\gamma )=\\beta \\alpha $ and $\\gamma \\alpha =1$ .", "From these and by $\\theta _i^3={\\rm id}_{K_i}$ , we deduce that $\\beta =\\theta _i^2(\\theta _i(\\beta ))=\\theta _i^2(\\alpha )$ and $\\gamma =\\theta _i^2(\\theta _i(\\gamma ))=\\theta _i^2(\\beta \\alpha )=\\theta _i(\\alpha )\\theta _i^2(\\alpha )$ , which implies $\\alpha \\theta _i(\\alpha )\\theta _i^2(\\alpha )=1$ .", "Then by the latter equation and (ii), we conclude that $\\alpha \\in \\mathcal {G}_i$ , and hence $\\gamma =\\alpha ^{-1} \\ {\\rm and} \\ \\beta =\\theta _i^2(\\alpha ).$ Therefore, both the number of right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ with degree 1 and the number of right divisors of $y^3-1$ in $K_i[y;\\theta _i]$ with degree 2 are equal to $1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}$ .", "(v) It follow from a direct calculation.", "Now, by Lemma REF and basic theory of skew cyclic codes we list all distinct left ideals of ${\\cal R}_i$ , i.e., skew $\\theta _i$ -cyclic codes over $K_i$ of length 3, by the following theorem.", "Theorem 3.3 Let $s+1\\le i\\le s+t$ .", "Then all distinct left ideals of ${\\cal R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ are given by one of the following three cases: (i) $C_{i,0}=\\lbrace 0\\rbrace $ ; $C_{i,3}={\\cal R}_i$ with $\|C_{i,3}\|=q^{3d_i}$ , ${\\rm wt}_H^{(i)}(C_{i,3})=1$ and $G_{C_{i,3}}=I_3$ is a generator matrix of $C_{i,3}$ , where $I_3$ is the identity matrix of size $3\\times 3$ .", "(ii) $C_{i,2,\\alpha }={\\cal R}_i(-\\alpha (x)+y)$ , where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ .", "Precisely, we have ${\\rm dim}_{K_i}(C_{i,2,\\alpha })=2$ , $\|C_{i,2,\\alpha }\|=q^{2d_i}$ and a generator matrix of $C_{i,2,\\alpha }$ is given by $G_{C_{i,2,\\alpha }}=\\left(\\begin{array}{ccc}-\\alpha (x) & 1 & 0\\cr 0 & -\\theta _i(\\alpha (x))& 1\\end{array}\\right).$ Therefore, $C_{i,2,\\alpha }=\\lbrace (a,b)G_{C_{i,2,\\alpha }}\\mid a,b\\in K_i\\rbrace $ and ${\\rm wt}_H^{(i)}(C_{i,2,\\alpha })=2$ .", "Hence $C_{i,2,\\alpha }$ is a MDS linear $[3,2,2]_{q^{d_i}}$ -code over $K_i$ .", "(iii) $C_{i,1,\\alpha }={\\cal R}_i(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2)$ , where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ .", "Precisely, we have ${\\rm dim}_{K_i}(C_{i,1,\\alpha })=1$ , $\|C_{i,1,\\alpha }\|=q^{d_i}$ and a generator matrix of $C_{i,1,\\alpha }$ is given by $G_{C_{i,1,\\alpha }}=(\\alpha (x)^{-1},\\theta _i^2(\\alpha (x)),1).$ Hence $C_{i,1,\\alpha }=\\lbrace a G_{C_{i,1,\\alpha }}\\mid a \\in K_i\\rbrace $ and ${\\rm wt}_H^{(i)}(C_{i,1,\\alpha })=3$ .", "Then $C_{i,1,\\alpha }$ is a MDS linear $[3,1,3]_{q^{d_i}}$ -code over $K_i$ .", "Therefore, the number of left ideals of ${\\cal R}_i$ is equal to $4+2q^{\\frac{d_i}{3}}+2q^{\\frac{2d_i}{3}}$ .", "In the rest of this paper, for any $\\alpha ,\\beta \\in \\mathcal {G}_i$ , we denote $\\bullet $ $C_{i,2,\\alpha }=\\mathcal {R}_i(-\\alpha +y)$ , $C_{i,1,\\beta }={\\cal R}_i(\\beta ^{-1}+\\theta _i^2(\\beta )y+y^2)$ which are skew $\\theta _i$ -cyclic codes over $K_i$ of length 3.", "By Theorems REF , REF and REF , we deduce the following corollary.", "Corollary 3.4 Using the notations in Section 2, denote $\\delta =\|\\lbrace d_i \\mid q^{d_i}\\equiv 1 \\ ({\\rm mod} \\ 3), \\ 0\\le i\\le s\\rbrace \|.$ Then the number $N_{(m,3,r;q)}$ of left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ is given by one of the following two cases: (i) When $q\\equiv 0$ $($ mod $3)$ , $N_{(m,3,r;q)}=4^{s+1}\\prod _{i=s+1}^{s+t}(4+2q^{\\frac{d_i}{3}}+2q^{\\frac{2d_i}{3}}).$ (ii) When $q\\lnot \\equiv 0$ $($ mod $3)$ , $N_{(m,3,r;q)}=2^{\\delta }4^{s+1}\\prod _{i=s+1}^{s+t}(4+2q^{\\frac{d_i}{3}}+2q^{\\frac{2d_i}{3}}).$ As the end of this section, we investigate the relationship between two left ideals of the ring $\\mathcal {R}_i$ for $i=s+1,\\ldots ,s+t$ .", "To do this, we consider the relationship between two nontrivial monic right divisors of $y^3-1$ in $K_i[y,\\theta _i]$ with different degrees.", "Lemma 3.5 For any $\\gamma \\in \\mathcal {G}_i$ , denote $\\phi _{i,\\gamma }(X)=X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{q^{\\frac{2d_i}{3}}}X+\\gamma ^{-1},$ $\\psi _{i,\\gamma }(X)=\\gamma ^{q^{\\frac{d_i}{3}}}X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{-1}X^{q^{\\frac{d_i}{3}}}+1.$ Then both $\\phi _{i,\\gamma }(X)$ and $\\psi _{i,\\gamma }(X)$ have exactly $q^{\\frac{d_i}{3}}+1$ roots and these roots are contained in $\\mathcal {G}_i$ .", "As ${\\rm gcd}(q,q^{\\frac{d_i}{3}}+1)=1$ , $\\phi _{i,\\gamma }(X)$ and $\\psi _{i,\\gamma }(X)$ have exactly $q^{\\frac{d_i}{3}}+1$ roots in some extended field of $K_i$ .", "We only need to prove that all these roots are contained in $\\mathcal {G}_i$ .", "By $\\gamma \\in \\mathcal {G}_i\\subseteq K_i^\\times $ , we have $\\gamma ^{1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}}=1$ and $\\gamma ^{q^{d_i}}=\\gamma $ , which implies $\\gamma ^{q^{\\frac{2d_i}{3}}+1}=\\gamma ^{q^{-\\frac{d_i}{3}}}$ .", "Then from $X^{q^{\\frac{d_i}{3}}+1}\\equiv -(\\gamma ^{q^{\\frac{2d_i}{3}}}X+\\gamma ^{-1})\\ ({\\rm mod} \\ \\phi _{i,\\gamma }(X)),$ we deduce that $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}&\\equiv &(X^{q^{\\frac{d_i}{3}}+1})^{q^{\\frac{d_i}{3}}}X\\\\&\\equiv & -(\\gamma ^{q^{\\frac{2d_i}{3}}}X+\\gamma ^{-1})^{q^{\\frac{d_i}{3}}}X\\\\&\\equiv & -(\\gamma ^{q^{d_i}}X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{q^{-\\frac{d_i}{3}}}X)\\\\&\\equiv & -\\gamma (X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{q^{\\frac{2d_i}{3}}}X)\\\\&\\equiv & -\\gamma \\gamma ^{-1}=1 \\ ({\\rm mod} \\ \\phi _{i,\\gamma }(X)),$ which implies that $\\phi _{i,\\gamma }(X)$ is a factor of $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}-1$ .", "Hence $\\alpha ^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}=1$ for any root $\\alpha $ of $\\phi _{i,\\gamma }(X)$ .", "Therefore, all roots of $\\phi _{i,\\gamma }(X)$ are contained in $\\mathcal {G}_i$ .", "The reciprocal polynomial of $\\psi _{i,\\gamma }(X)$ is equal to $\\psi _{i,\\gamma }^\\ast (X)=X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{-1}X+\\gamma ^{q^{\\frac{d_i}{3}}}.$ Then by $X^{q^{\\frac{d_i}{3}}+1}\\equiv -(\\gamma ^{-1}X+\\gamma ^{q^{\\frac{d_i}{3}}})$ (mod $\\psi _{i,\\gamma }^\\ast (X)$ ) and $\\gamma ^{-q^{\\frac{d_i}{3}}-1}=\\gamma ^{q^{\\frac{2d_i}{3}}}$ , it follows that $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}&\\equiv &(X^{q^{\\frac{d_i}{3}}+1})^{q^{\\frac{d_i}{3}}}X\\\\&\\equiv & -(\\gamma ^{-1}X+\\gamma ^{q^{\\frac{d_i}{3}}})^{q^{\\frac{d_i}{3}}}X\\\\&\\equiv &-(\\gamma ^{-q^{\\frac{d_i}{3}}}X^{q^{\\frac{d_i}{3}}+1}+\\gamma ^{q^{\\frac{2d_i}{3}}}X)\\\\&\\equiv &-(-\\gamma ^{-q^{\\frac{d_i}{3}}}(\\gamma ^{-1}X+\\gamma ^{q^{\\frac{d_i}{3}}})+\\gamma ^{q^{\\frac{2d_i}{3}}}X)\\\\&\\equiv & 1 \\ ({\\rm mod} \\ \\psi _{i,\\gamma }^\\ast (X)),$ which implies that $\\psi _{i,\\gamma }^\\ast (x)$ is a factor of $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}-1$ .", "Since $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}-1$ is self-reciprocal, we conclude that $\\psi _{i,\\gamma }(X)$ is also a factor of $X^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}-1$ .", "Therefore, all roots of $\\psi _{i,\\gamma }(X)$ are contained in $\\mathcal {G}_i$ .", "Theorem 3.6 (i) Let $\\alpha ,\\beta \\in \\mathcal {G}_i$ .", "Then $C_{i,1,\\beta }\\subset C_{i,2,\\alpha }$ if and only of $\\alpha $ and $\\beta $ satisfying the following equation $\\alpha \\theta _i(\\alpha )\\beta +\\alpha \\beta \\theta _i^2(\\beta )+1=0.$ (ii) For any $\\beta \\in \\mathcal {G}_i$ , there are $q^{\\frac{d_i}{3}}+1$ codes $C_{i,2,\\alpha }$ containing $C_{i,1,\\beta }$ where $\\alpha $ is given by one of the following two case: $\\alpha $ is a root of the polynomial $\\phi _{i,\\beta }(X)$ if $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ ; $\\alpha $ is a root of the polynomial $\\psi _{i,\\beta }(X)$ if $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ .", "(iii) For any $\\alpha \\in \\mathcal {G}_i$ , there are $q^{\\frac{d_i}{3}}+1$ codes $C_{i,1,\\beta }$ contained in $C_{i,2,\\alpha }$ where $\\beta $ is given by one of the following two case: $\\beta $ is a root of the polynomial $\\psi _{i,\\alpha }(X)$ if $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ ; $\\beta $ is a root of the polynomial $\\phi _{i,\\alpha }(X)$ if $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ .", "(i) It is clear that $C_{i,1,\\beta }\\subset C_{i,2,\\alpha }$ if and only if $-\\alpha +y$ is a right divisor of $\\beta ^{-1}+\\theta _i^2(\\beta )y+y^2$ .", "Dividing $\\beta ^{-1}+\\theta _i^2(\\beta )y+y^2$ by $y-\\alpha $ from right hand in the skew polynomial ring $K_i[y;\\theta _i]$ , we have $&&\\beta ^{-1}+\\theta _i^2(\\beta )y+y^2\\\\&=&(\\theta _i(\\alpha )+\\theta _i^2(\\beta ))(-\\alpha +y)+\\beta ^{-1}+\\alpha \\theta _i(\\alpha )+\\alpha \\theta _i^2(\\beta ).$ Hence $-\\alpha +y$ is a right divisor of $\\beta ^{-1}+\\theta _i^2(\\beta )y+y^2$ if and only if $\\beta ^{-1}+\\alpha \\theta _i(\\alpha )+\\alpha \\theta _i^2(\\beta )=0$ , which is equivalent to that $\\alpha $ and $\\beta $ satisfy (REF ).", "(ii) Let $\\alpha \\in \\mathcal {G}_i$ .", "We have one of the following two cases: When $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ , it is clear that $\\alpha \\theta _i(\\alpha )\\beta +\\alpha \\beta \\theta _i^2(\\beta )+1=\\beta \\phi _{i,\\beta }(\\alpha ).$ From this and by (i), we deduce that $C_{i,1,\\beta }\\subset C_{i,2,\\alpha }$ if and only if $\\phi _{i,\\beta }(\\alpha )=0$ .", "Furthermore, by Lemma REF we know that the $\\phi _{i,\\beta }(x)$ has exactly $q^{\\frac{d_i}{3}}+1$ roots in $\\mathcal {G}_i$ .", "When $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ , by $\\alpha \\theta _i(\\alpha )=\\alpha ^{1+q^{\\frac{2d_i}{3}}}=q^{-q^{\\frac{d_i}{3}}}$ we have $\\alpha \\theta _i(\\alpha )\\beta +\\alpha \\beta \\theta _i^2(\\beta )+1=\\alpha ^{1+q^{\\frac{2d_i}{3}}}\\beta \\psi _{i,\\beta }(\\alpha ).$ From this and by (i), we deduce that $C_{i,1,\\beta }\\subset C_{i,2,\\alpha }$ if and only if $\\psi _{i,\\beta }(\\alpha )=0$ .", "Furthermore, by Lemma REF we know that the polynomial $\\psi _{i,\\beta }(x)$ has exactly $q^{\\frac{d_i}{3}}+1$ roots in $\\mathcal {G}_i$ .", "(iii) By (REF ) and Lemma REF , it can be proved similarly as that of (ii).", "Here, we omit the proof." ], [ "The dual code of any left $G_{(m,3,r)}$ -code", "In this section, we give the dual code of any left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ and determine all self-orthogonal left $G_{(m,3,r)}$ -codes.", "As in , the Euclidian inner product in $\\mathbb {F}_{q}[G_{(m,3,r)}]$ is defined as follows.", "For $\\xi =\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}x^iy^j$ and $\\eta =\\sum _{i=0}^{m-1}\\sum _{j=0}^2b_{i,j}x^iy^j$ in $\\mathbb {F}_{q}[G_{(m,3,r)}]$ , we set $[\\xi ,\\eta ]_E=\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}b_{i,j}\\in \\mathbb {F}_q.$ The Euclidian dual code of a left $G_{(m,3,r)}$ -code $\\mathcal {C}$ over $\\mathbb {F}_{q}$ is defined by $\\mathcal {C}^{\\bot _E}=\\lbrace \\xi \\in \\mathbb {F}_{q}[G_{(m,3,r)}]\\mid [\\xi ,\\eta ]_E=0, \\ \\forall \\eta \\in \\mathcal {C}\\rbrace .$ $\\mathcal {C}$ is said to be self-orthogonal if $\\mathcal {C}\\subseteq \\mathcal {C}^{\\bot _E}$ .", "For any $\\xi (y)=\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}x^iy^j\\in \\mathbb {F}_{q}[G_{(m,3,r)}]$ , as in we define the conjugation $\\mu $ on $\\mathbb {F}_{q}[G_{(m,3,r)}]$ by $\\mu (\\xi (y))=\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}y^{-j}x^{-i} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $ $=a_0(x^{-1})+y^2a_1(x^{-1})+ya_2(x^{-1}),$ where $a_j(x)=\\sum _{i=0}^{m-1}a_{i,j}x^{i}$ and $a_j(x^{-1})=a_{0,j}+\\sum _{i=1}^{m-1}a_{i,j}x^{m-i}$ for $j=0,1,2$ .", "It can be verify easily that $\\mu (\\xi +\\eta )=\\mu (\\xi )+\\mu (\\eta ) \\ {\\rm and} \\ \\mu (\\xi \\eta )=\\mu (\\eta )\\mu (\\xi )$ for all $\\xi ,\\eta \\in \\mathbb {F}_{q}[G_{(m,3,r)}]$ .", "Moreover, we have the following Lemma 4.1 (i) The map $\\mu $ defined by $(6)$ is an $\\mathbb {F}_{q}$ -algebra anti-automorphism of $\\mathbb {F}_{q}[G_{(m,3,r)}]$ satisfying $\\mu ^{-1}=\\mu $ .", "(ii) For any $\\xi ,\\eta \\in \\mathbb {F}_{q}[G_{(m,3,r)}]$ , we have $[\\xi ,\\eta ]_E=0$ if $\\xi \\cdot \\mu (\\eta )=0$ in the ring $\\mathbb {F}_{q}[G_{(m,3,r)}]$ .", "(iii) Let $\\mathcal {C}$ be a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ and $\\mathcal {B}$ a right ideal of $\\mathbb {F}_{q}[G_{(m,3,r)}]$ .", "Then $\\mu (\\mathcal {B})\\subseteq \\mathcal {C}^{\\perp _E} \\ {\\rm if} \\ \\mathcal {C}\\cdot \\mathcal {B}=\\lbrace 0\\rbrace \\ {\\rm in} \\ \\mathbb {F}_{q}[G_{(m,3,r)}].$ (ii) For $\\xi =\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}x^iy^j$ and $\\eta =\\sum _{i=0}^{m-1}\\sum _{j=0}^2b_{i,j}x^iy^j$ in $\\mathbb {F}_{q}[G_{(m,3,r)}]$ , by ${\\rm ord}(x)=m$ and ${\\rm ord}(y)=3$ we deduce that $\\xi \\cdot \\mu (\\eta )&=&(\\sum _{i=0}^{m-1}\\sum _{j=0}^2a_{i,j}x^iy^j)(\\sum _{k=0}^{m-1}\\sum _{l=0}^2b_{k,l}y^{3-l}x^{m-k})\\\\&=&[\\xi ,\\eta ]_E+\\sum _{0\\le i\\le m-1, 0\\le j\\le 2,i+j\\ne 0}c_{i,j}x^iy^j$ for some $c_{i,j}\\in \\mathbb {F}_q$ .", "Hence $[\\xi ,\\eta ]_E=0$ if $\\xi \\cdot \\mu (\\eta )=0$ in the ring $\\mathbb {F}_{q}[G_{(m,3,r)}]$ .", "(iii) For any $\\beta \\in \\mathcal {B}$ and $\\eta \\in \\mathcal {C}$ , by $\\mathcal {C}\\cdot \\mathcal {B}=\\lbrace 0\\rbrace $ we have $\\eta \\beta =0$ , which implies $\\mu (\\beta )\\cdot \\mu (\\eta )=0$ by (i), and so $[\\mu (\\beta ),\\eta ]_E=0$ by (ii).", "From this we deduce that $\\mu (\\beta )\\in \\mathcal {C}^{\\bot _E}$ .", "Therefore, $\\mu (\\mathcal {B})\\subseteq \\mathcal {C}^{\\bot _E}$ .", "By the identification of $\\mathbb {F}_q[G_{(m,3,r)}]$ with ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ , we see that $\\mathcal {A}=\\mathbb {F}_q[x]/\\langle x^m-1\\rangle $ is a subring of $\\mathbb {F}_{q}[G_{(m,3,r)}]$ .", "In the following, we consider the restriction of $\\mu $ on $\\mathcal {A}$ .", "In order to simplify the notation, we still denote this restriction by $\\mu $ .", "Obviously, we have $\\mu (a(x))=a(x^{-1})=\\sum _{i=0}^{m-1}a_ix^{-i}=a_0+\\sum _{i=1}^{m-1}a_ix^{m-i}$ for all $a(x)=\\sum _{i=0}^{m-1}a_ix^{i}\\in \\mathcal {A}$ .", "It is clear that $\\mu $ is an $\\mathbb {F}_{q}$ -algebra automorphism of $\\mathcal {A}$ satisfying $\\mu ^{-1}=\\mu $ .", "Using the notations of Section 2, we know that $J(i)=J_{k_i}^{(q)}$ , $0\\le i\\le s+t$ , are the all distinct $q$ -cyclotomic cosets modulo $m$ .", "By Lemma REF , we have one of the following two cases: $\\diamondsuit $ $0\\le i\\le s$ .", "In this case, we have $\\theta (j)=rj\\equiv j$ (mod $m$ ) for all $j\\in J(i)=J_{k_i}^{(q)}$ .", "Then it is clear that $-J(i)=\\lbrace -j\\mid j\\in J(i)\\rbrace $ (mod $m$ ) is a $q$ -cyclotomic coset modulo $m$ satisfying $\\theta (-j)=-rj\\equiv -j$ (mod $m$ ) for all $j\\in J(i)$ .", "Hence $-J(i)$ is also a $q$ -cyclotomic coset modulo $m$ satisfying Condition (I) in Lemma REF .", "Therefore, there is a unique integer $i^\\prime $ , $0\\le i^\\prime \\le s$ , such that $-J(i)=J(i^\\prime )$ .", "$\\diamondsuit $ $s+1\\le i\\le s+t$ .", "In this case, we have $\\theta (j)\\in J(i)$ and $\\theta (j)\\lnot \\equiv j$ (mod $m$ ) for all $j\\in J(i)=J_{k_i}^{(q)}$ .", "Then it is clear that $-J(i)$ is a $q$ -cyclotomic coset modulo $m$ satisfying $\\theta (-j)=-\\theta (j)\\in -J(i)$ and $\\theta (-j)\\lnot \\equiv -j$ (mod $m$ ) for all $j\\in J(i)$ .", "Hence $-J(i)$ is also a $q$ -cyclotomic coset modulo $m$ satisfying Condition (II) in Lemma REF .", "Therefore, there is a unique integer $i^\\prime $ , $s+1\\le i^\\prime \\le s+t$ , such that $-J(i)=J(i^\\prime )$ .", "We also use $\\mu $ to denote this map $i\\mapsto i^\\prime $ , i.e., $\\mu (i)=i^\\prime $ .", "Whether $\\mu $ denotes the automorphism of $\\mathcal {A}$ or this map on the set $\\lbrace 0,1,\\ldots ,s+t\\rbrace $ is determined by context.", "The next lemma shows the compatibility of the two uses of $\\mu $ .", "Lemma 4.2 Using the notations above, the following assertions hold.", "(i) $\\mu $ is a permutation on $\\lbrace 0,1,\\ldots ,s+t\\rbrace $ satisfying $\\mu ^{-1}=\\mu $ , $\\mu (0)=0$ , $1\\le \\mu (i)\\le s$ for all $1\\le i\\le s$ and $s+1\\le \\mu (i)\\le s+t$ for all $s+1\\le i\\le s+t$ .", "(ii) After a rearrangement of $J(0),J(1)\\ldots ,J(s+t)$ , there are nonnegative integers $s_1,s_2,t_1,t_2$ satisfying the following conditions: $\\bullet $ $s=s_1+2s_2$ , $\\mu (i)=i$ for all $1\\le i\\le s_1$ , $\\mu (i)=i+s_2$ and $\\mu (i+s_2)=i$ for all $s_1+1\\le i\\le s_1+s_2$ ; $\\bullet $ $t=t_1+2t_2$ , $\\mu (i)=i$ for all $s+1\\le i\\le s+t_1$ , $\\mu (i)=i+t_2$ and $\\mu (i+t_2)=i$ for all $s+t_1+1\\le i\\le s+t_1+t_2$ .", "(iii) $\\mu (\\varepsilon _i(x))=\\varepsilon _{\\mu (i)}(x)$ and $\\mu (\\mathcal {A}_i)=\\mathcal {A}_{\\mu (i)}$ for all $i=0,1,\\ldots ,s+t$ .", "(iv) Let $\\mu $ be the map defined by $\\mu (a(x))=a(x^{-1})=a(x^{m-1})$ $({\\rm mod} \\ f_{\\mu (i)}(x))$ for all $a(x)\\in K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ .", "Then $\\mu $ is an $\\mathbb {F}_q$ -algebra isomorphism from $K_i$ onto $K_{\\mu (i)}=\\mathbb {F}_q[x]/\\langle f_{\\mu (i)}(x)\\rangle $ satisfying $\\mu \\theta _i=\\theta _{\\mu (i)}\\mu $ .", "(v) Let $0\\le i\\le s+t$ .", "Using the notations of Theorem REF (ii), the $\\mathbb {F}_{q}$ -algebra anti-automorphism $\\mu $ of $\\mathbb {F}_{q}[G_{(m,3,r)}]$ induces an $\\mathbb {F}_q$ -algebra anti-isomorphism $\\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i$ from $\\mathcal {R}_i$ onto $\\mathcal {R}_{\\mu (i)}$ .", "We denote this anti-isomorphism by $\\mu $ as well.", "Then for any $\\alpha (y)=a(x)+b(x)y+c(x)y^2\\in \\mathcal {R}_i$ where $a(x),b(x),c(x)\\in K_i$ , we have $\\mu (\\alpha (y))=a(x^{-1})+y^2b(x^{-1})+yc(x^{-1}).$ (i) follows from the definition of the map $\\mu $ , and (ii) follows from (i).", "(iii) It is clear that $\\mu (\\varepsilon _i(x))=\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-jl})x^{-l}$ by (REF ) in Section 2.", "From this, by $\\mathbb {Z}_m=-\\mathbb {Z}_m$ and $J(\\mu (i))=-J(i)=\\lbrace -j\\mid j\\in J(i)\\rbrace $ we deduce that $\\mu (\\varepsilon _i(x))&=&\\frac{1}{m}\\sum _{l=0}^{m-1}(\\sum _{j\\in J(i)}\\zeta ^{-(-j)(-l)})x^{-l}\\\\&=&\\frac{1}{m}\\sum _{k=0}^{m-1}(\\sum _{j^\\prime \\in J(\\mu (i))}\\zeta ^{-j^\\prime k})x^{k}\\\\&=&\\varepsilon _{\\mu (i)}(x).$ Hence $\\mu (\\mathcal {A}_i)=\\mu (\\mathcal {A}\\varepsilon _i(x))=\\mu (\\mathcal {A})\\mu (\\varepsilon _i(x))=\\mathcal {A}\\varepsilon _{\\mu (i)}(x)=\\mathcal {A}_{\\mu (i)}$ by Lemma REF (ii).", "(iv) By (iii), we know that $\\mu $ induces an $\\mathbb {F}_q$ -algebra isomorphism from $\\mathcal {A}_i$ onto $\\mathcal {A}_{\\mu (i)}$ .", "Then by Lemma REF (iii), we see that $\\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i$ is an $\\mathbb {F}_q$ -algebra isomorphism from $K_i$ onto $K_{\\mu (i)}$ .", "For any $a(x)\\in K_i$ , by Equation (REF ) in Section 2 we have $\\varepsilon _{\\mu (i)}(x)\\equiv 1$ (mod $f_{\\mu (i)}(x)$ ), which implies $(\\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i)(a(x))&=&\\varphi _{\\mu (i)}^{-1}\\mu (\\varepsilon _i(x)a(x))\\\\&=&\\varphi _{\\mu (i)}^{-1}(\\varepsilon _{\\mu (i)}(x)a(x^{-1}))\\\\&=&a(x^{-1}) \\ ({\\rm mod} \\ f_{\\mu (i)}(x)).$ Since we denote $\\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i$ by $\\mu $ as well, the map $\\mu : a(x)\\mapsto a(x^{-1})$ (mod $f_{\\mu (i)}(x)$ ) is an $\\mathbb {F}_q$ -algebra isomorphism from $K_i$ onto $K_{\\mu (i)}$ .", "Moreover, for any $a(x)\\in K_i$ by Lemma REF (v) and $a(x^{-1})\\in K_{\\mu (i)}$ it follows that $(\\mu \\theta _i)(a(x))&=&\\mu (a(x^r))=a(x^{-r})=\\theta _{\\mu (i)}(a(x^{-1}))\\\\&=&(\\theta _{\\mu (i)}\\mu )(a(x)).$ Hence $\\mu \\theta _i=\\theta _{\\mu (i)}\\mu $ .", "(v) By (iii) and Theorem REF (ii), we have the following commutative diagram form ring isomorphisms: $\\left.\\begin{array}{ccc}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal {R}_i & \\stackrel{\\varphi _i}{\\longrightarrow } &\\mathcal {A}_i[y;\\theta \|_{\\mathcal {A}_i}]/\\langle \\varepsilon _i(x)(y^3-1)\\rangle \\cr {\\small \\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i}\\downarrow & & \\ \\ \\ \\ \\downarrow \\mu \\cr \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal {R}_{\\mu (i)} & \\stackrel{\\varphi _{\\mu (i)}}{\\longrightarrow } & \\mathcal {A}_{\\mu (i)}[y;\\theta \|_{\\mathcal {A}_{\\mu (i)}}]/\\langle \\varepsilon _{\\mu (i)}(x)(y^3-1)\\rangle \\end{array}\\right.$ As we write $\\varphi _{\\mu (i)}^{-1}\\mu \\varphi _i$ by $\\mu $ , for any $a(x),b(x),c(x)\\in K_i$ by the identification of $\\mathbb {F}_q[G_{(m,3,r)}]$ with ${\\cal A}[y;\\theta ]/\\langle y^3-1\\rangle $ , $\\varepsilon _i(x^{-1})=\\mu (\\varepsilon _i(x))=\\varepsilon _{\\mu (i)}(x)$ , Equation (REF ) and $y\\varepsilon _{\\mu (i)}(x)=\\theta (\\varepsilon _{\\mu (i)}(x))y=\\varepsilon _{\\mu (i)}(x)y$ , we deduce that $&&\\mu (a(x)+b(x)y+c(x)y^2)\\\\&=&(\\varphi _{\\mu (i)}^{-1}\\mu )(\\varphi _i(a(x)+b(x)y+c(x)y^2))\\\\&=&\\varphi _{\\mu (i)}^{-1}\\left(\\mu (\\varepsilon _i(x)a(x)+\\varepsilon _i(x)b(x)y+\\varepsilon _i(x)c(x)y^2)\\right)\\\\&=&\\varphi _{\\mu (i)}^{-1}(\\varepsilon _i(x^{-1})a(x^{-1})+y^2\\varepsilon _i(x^{-1})b(x^{-1})\\\\&&+y\\varepsilon _i(x^{-1})c(x^{-1}))\\\\&=&\\varphi _{\\mu (i)}^{-1}\\left(\\varepsilon _{\\mu (i)}(x)\\left(a(x^{-1})+y^2b(x^{-1})+yc(x^{-1})\\right)\\right)\\\\&=&a(x^{-1})+y^2b(x^{-1})+yc(x^{-1})$ by (iv).", "Corollary 4.3 For any $\\alpha (x)\\in \\mathcal {G}_i$ , we denote $\\widehat{\\alpha }(x)=(\\alpha (x^{m-1}))^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}} \\ ({\\rm mod} \\ f_{\\mu (i)}(x)).$ Using the notations of Lemma REF (iv), we have that $\\widehat{\\alpha }(x)=(\\alpha (x^{-1}))^{-1}=(\\mu (\\alpha (x)))^{-1}\\in \\mathcal {G}_{\\mu (i)}$ , $\\alpha (x^{-1})\\widehat{\\alpha }(x)=1$ and $\\alpha (x)=(\\widehat{\\alpha }(x^{-1}))^{-1}$ .", "As $\\alpha (x)\\in \\mathcal {G}_i$ , we see that $\\alpha (x)$ is an element of $K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ satisfying $(\\alpha (x))^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}=1$ .", "By Lemma REF (iv), we know that $\\mu $ is an $\\mathbb {F}_q$ -algebra isomorphism from $K_i$ onto $K_{\\mu (i)}=\\mathbb {F}_q[x]/\\langle f_{\\mu (i)}(x)\\rangle $ .", "Hence $(\\mu (\\alpha (x)))^{-1}\\in K_{\\mu (i)}$ and $(\\mu (\\alpha (x)))^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}+1}=1$ in $K_{\\mu (i)}$ , which implies $\\mu (\\alpha (x))\\in \\mathcal {G}_{\\mu (i)}$ , and so $(\\mu (\\alpha (x))^{-1}\\in \\mathcal {G}_{\\mu (i)}$ .", "Finally, by $f_{\\mu (i)}(x)\|(x^m-1)$ it follows that $\\widehat{\\alpha }(x)&=&(\\alpha (x^{-1}))^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}}=(\\mu (\\alpha (x)))^{q^{\\frac{2d_i}{3}}+q^{\\frac{d_i}{3}}}\\\\&=&(\\mu (\\alpha (x)))^{-1}$ in $K_{\\mu (i)}$ .", "Then $\\alpha (x)=(\\mu (\\widehat{\\alpha }(x)))^{-1}=(\\widehat{\\alpha }(x^{-1}))^{-1}$ .", "For any integer $i$ , $0\\le i\\le s+t$ , and $g(y),h(y)\\in \\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ , in the following we define $g(y)\\sim _l h(y) \\ {\\rm if} \\ g(y)=\\alpha h(y) \\ {\\rm for} \\ {\\rm some} \\ \\alpha \\in \\mathcal {R}_i^\\times ,$ where $\\mathcal {R}_i^\\times $ is the set of invertible elements in $\\mathcal {R}_i$ .", "It is clear that $\\mathcal {R}_ig(y)=\\mathcal {R}_ih(y)$ if $g(y)=\\alpha h(y)$ .", "Lemma 4.4 For any integer $i$ , $0\\le i\\le s+t$ , we have the following conclusions: (i) $\\mu (y^2+y+1)\\sim _l y^2+y+1$ , $\\mu (y^3-1)\\sim _l y^3-1$ and $\\mu ((y-1)^j)\\sim _l (y-1)^j$ for all $j=0,1,2$ .", "(ii) Let $0\\le i\\le s$ and $q^{d_i}\\equiv 1$ (mod 3).", "Then $\\mu (y-\\omega _i(x))\\sim _l y-\\omega _i(x^{-1})^2$ and $\\mu (y-\\omega _i(x)^2)\\sim _l y-\\omega _i(x^{-1})$ in $\\mathcal {R}_{\\mu (i)}$ .", "(iii) Let $s+1\\le i\\le s+t$ and $\\alpha (x)\\in \\mathcal {G}_i$ .", "Then $\\mu (-\\alpha (x)+y)\\sim _l -\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))+y,$ $&&\\mu \\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x)) y+y^2\\right)\\\\&\\sim _l&\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right)^{-1}+\\theta _{\\mu (i)}^2\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right)y+y^2.$ (i) By Equation (REF ) and $y^3=1$ , it follows that $\\mu (y-1)=y^{2}-1=(-y^2)(y-1)$ where $-y^2\\in \\mathcal {R}_i^\\times $ .", "The other conclusion can be verified similarly.", "(ii) Since $\\omega _i(x)^3=1$ and $\\mu $ is a ring isomorphism from $\\mathcal {R}_i$ onto $\\mathcal {R}_{\\mu (i)}$ by Lemma REF (v), it follows that $\\omega _i(x^{-1})^3=(\\mu (\\omega _i(x)))^3=1$ .", "As $-\\omega _i(x^{-1})y^2\\in \\mathcal {R}_{\\mu (i)}^\\times $ , we have $\\mu (y-\\omega _i(x))&=&-\\omega _i(x^{-1})+y^2\\\\&=&(-\\omega _i(x^{-1})y^2)(y-\\omega _i(x^{-1})^2)\\\\&\\sim _l&y-\\omega _i(x^{-1})^2.$ Similarly, one can verify that $\\mu (y-\\omega _i(x)^2)\\sim _l y-\\omega _i(x^{-1})$ .", "(iii) By (REF ), Lemma REF (v) and Corollary REF we have $\\mu (-\\alpha (x)+y)&=&-\\alpha (x^{-1})+y^2\\\\&=&-\\mu (\\alpha (x))\\left(1-\\widehat{\\alpha }(x)y^2\\right)\\\\&=&-\\mu (\\alpha (x))\\left(1-y^2\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))\\right)\\\\&=&-\\mu (\\alpha (x))y^2\\cdot y\\left(1-y^2\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))\\right)\\\\&=&-\\mu (\\alpha (x))y^2\\left(-\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))+y\\right),$ where $-\\mu (\\alpha (x))y^2\\in \\mathcal {R}_{\\mu (i)}^\\times $ and $\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))\\in \\mathcal {G}_{\\mu (i)}$ by Corollary REF .", "Similarly, we have $&&\\mu \\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x)) y+y^2\\right)\\\\&=&\\alpha (x^{-1})^{-1}+y^2\\theta _{\\mu (i)}^2(\\alpha (x^{-1})) +y\\\\&=&y^2\\cdot y\\left(\\widehat{\\alpha }(x)+y^2\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)^{-1})+y\\right)\\\\&=&y^2\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)^{-1}) +\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))y+y^2\\right)$ where $y^2\\in \\mathcal {R}_{\\mu (i)}^\\times $ , $\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)^{-1})=\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right)^{-1}\\in \\mathcal {G}_{\\mu (i)}$ by Corollary 4.3 and $\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))=\\theta _{\\mu (i)}^2\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right)$ .", "Now, we give the dual code of any left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ by the following theorem.", "Theorem 4.5 Let ${\\cal C}=\\oplus _{i=0}^{s+t}({\\cal A}_i\\Box _{\\varphi _i}C_i)$ be a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ , where $C_i$ is a left ideal of the ring ${\\cal R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ given by Theorems REF and REF .", "Then the dual code ${\\cal C}^{\\bot _E}$ of ${\\cal C}$ is also a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ .", "Precisely, we have ${\\cal C}^{\\bot _E}=\\bigoplus _{i=0}^{s+t}({\\cal A}_i\\Box _{\\varphi _i}D_i),$ where $D_i$ is a left ideal of ${\\cal R}_i$ given by one of the following cases: (i) Let $0\\le i\\le s$ .", "Then $\\mathcal {R}_i=K_i[y]/\\langle y^3-1\\rangle $ and $D_i$ is given by one of the following subcases: (i-1) Let $q\\equiv 0$ $({\\rm mod} \\ 3)$ .", "Then $D_{\\mu (i)}=\\mathcal {R}_{\\mu (i)}\\frac{(y-1)^3}{g(y)}$ , if $C_i=\\mathcal {R}_{i}g(y)$ where $g(y)\\in \\lbrace 1,y-1,(y-1)^2,(y-1)^3\\rbrace $ .", "(i-2) Let $q^{d_i}\\equiv 2$ $({\\rm mod} \\ 3)$ .", "Then $D_{\\mu (i)}=\\mathcal {R}_{\\mu (i)}\\frac{y^3-1}{g(y)}$ , if $C_i=\\mathcal {R}_{i}g(y)$ where $g(y)\\in \\lbrace 1,y-1,y^2+y+1,y^3-1\\rbrace $ .", "(i-3) Let $q^{d_i}\\equiv 1$ $({\\rm mod} \\ 3)$ .", "Then $D_{\\mu (i)}=\\mathcal {R}_{\\mu (i)}\\vartheta (y)$ if $C_i=\\mathcal {R}_{i}g(y)$ , where the pair $(g(y),\\vartheta (y))$ of polynomials is given by the following table: Table: NO_CAPTION(ii) Let $s+1\\le i\\le s+t$ .", "Then $\\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ and $D_i$ is given by one of the following subcases: (ii-1) $D_{\\mu (i)}={\\cal R}_{\\mu (i)}\\left(-\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))+y\\right)=C_{\\mu (i),2,\\theta _{\\mu (i)}(\\widehat{\\alpha })}$ and a generator matrix of $D_{\\mu (i)}$ is given by $G_{D_{\\mu (i)}}=\\left(\\begin{array}{ccc}-\\theta _{\\mu (i)}(\\widehat{\\alpha }(x)) & 1 & 0 \\cr 0 & -\\theta _{\\mu (i)}\\left(\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))\\right) & 1 \\end{array}\\right)$ as a linear code over $K_{\\mu (i)}$ of length 3, if $C_i=C_{i,1,\\alpha }=\\mathcal {R}_{i}\\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2\\right)$ where $\\alpha (x)\\in \\mathcal {G}_i$ .", "(ii-2) $D_{\\mu (i)}={\\cal R}_{\\mu (i)}(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)^{-1})+\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))y+y^2 )=C_{\\mu (i),1,\\theta _{\\mu (i)}^2(\\widehat{\\alpha })}$ and a generator matrix of $D_{\\mu (i)}$ is given by $G_{D_{\\mu (i)}}=\\left(\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right)^{-1},\\theta _{\\mu (i)}^2\\left(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x))\\right),1\\right)$ as a linear code over $K_{\\mu (i)}$ of length 3, if $C_i=C_{i,2,\\alpha }=\\mathcal {R}_{i}(-\\alpha (x)+y)$ where $\\alpha (x)\\in \\mathcal {G}_i$ .", "(ii-3) $D_{\\mu (i)}=\\lbrace 0\\rbrace $ if $C_i={\\cal R}_i$ ; $D_{\\mu (i)}={\\cal R}_{\\mu (i)}$ if $C_i=\\lbrace 0\\rbrace $ .", "Let $K_i=\\mathbb {F}_q[x]/\\langle f_i(x)\\rangle $ , $\\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ and $B_i$ be an right ideal of the ring $\\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ given by one of the following two cases: (A) Let $0\\le i\\le s$ .", "Then $\\theta _i={\\rm id}_{K_i}$ , $\\mathcal {R}_i=K_i[y]/\\langle y^3-1\\rangle $ and $B_i$ is given by one of the following three subcases.", "(A-1) Let $q\\equiv 0$ (mod 3).", "Then $B_i=\\frac{y^3-1}{g(y)}\\mathcal {R}_i$ if $C_i=\\mathcal {R}_ig(y)$ where $g(y)\\in \\lbrace 1, y-1, (y-1)^2, y^3-1\\rbrace $ .", "(A-2) Let $q^{d_i}\\equiv 2$ (mod 3).", "Then $B_i=\\frac{y^3-1}{g(y)}\\mathcal {R}_i$ if $C_i=\\mathcal {R}_ig(y)$ where $g(y)\\in \\lbrace 1, y-1, y^2+y+1, y^3-1\\rbrace $ .", "(A-3) Let $q^{d_i}\\equiv 1$ (mod 3).", "Then $B_i=\\frac{y^3-1}{g(y)}\\mathcal {R}_i$ if $C_i=\\mathcal {R}_ig(y)$ where $g(y)\\in \\lbrace 1, y-1, y-\\omega _i(x), y-\\omega _i(x)^2, y^2+y+1, ( y-1)(y-\\omega _i(x)^2), ( y-1)(y-\\omega _i(x)),y^3-1\\rbrace $ .", "(B) Let $s+1\\le i\\le s+t$ .", "Then $B_i$ is given by one of the following four subcases.", "(B-1) $B_i=(-\\alpha (x)+y)\\mathcal {R}_i$ , if $C_i=\\mathcal {R}_i(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2)$ where $\\alpha (x)\\in \\mathcal {G}_i$ .", "(B-2) $B_i=\\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2\\right)\\mathcal {R}_i$ , if $C_i=\\mathcal {R}_i(-\\alpha (x)+y)$ where $\\alpha (x)\\in \\mathcal {G}_i$ .", "(B-3) $B_i=\\lbrace 0\\rbrace $ , if $C_i=\\mathcal {R}_i$ ; $B_i=\\mathcal {R}_i$ , if $C_i=\\lbrace 0\\rbrace $ .", "By Theorems REF and REF , Lemma REF (v) and direct calculations, one can easily verify that $C_i\\cdot B_i=\\lbrace 0\\rbrace \\ {\\rm in} \\ \\mathcal {R}_i, \\ i=0,1,\\ldots ,s+t.$ For any integer $0\\le i\\le s+t$ , let $D_{\\mu (i)}=\\mu (B_i)$ .", "By Lemma REF (v) we see that $D_{\\mu (i)}$ is a left ideal of $\\mathcal {R}_{\\mu (i)}$ .", "Let $\\mathcal {D}=\\sum _{i=0}^{s+t}\\mathcal {A}_i\\Box _{\\varphi _i}D_i=\\bigoplus _{i=0}^{s+t}\\mathcal {A}_{\\mu (i)}\\Box _{\\varphi _{\\mu (i)}}D_{\\mu (i)}.$ Then by Theorem REF (iii), we conclude that $\\mathcal {D}$ is a left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ .", "$\\diamondsuit $ First, we give the clear expression of $D_{\\mu (i)}=\\mu (B_i)$ .", "For the trivial case: $B_i=\\mathcal {R}_i$ or $B_i=\\lbrace 0\\rbrace $ , the conclusion follows from Lemma 4.2(v) immediately.", "Then we only need to consider the nontrivial cases in (A) and (B).", "In the case of (A-1), $B_i=\\frac{y^3-1}{g(y)}\\mathcal {R}_i$ .", "If $g(y)=(y-1)^2$ , then $B_i=(y-1)\\mathcal {R}_i$ .", "By Lemma REF (v) and Lemma REF (i), we have $D_{\\mu (i)}=\\mu (B_i)=\\mathcal {R}_{\\mu (i)}\\mu (y-1)=\\mathcal {R}_{\\mu (i)}(y-1)$ .", "Similarly, one can easily prove that the other conclusions in (i-1) and all conclusions in (i-2) hold from (A-1) and (A-2).", "In the case of (A-3), $B_i=\\frac{y^3-1}{g(y)}\\mathcal {R}_i$ .", "If $g(y)=(y-1)(y-\\omega (x)^2)$ , by Lemma REF (v) and Lemma REF (ii), we have $D_{\\mu (i)}&=&\\mu ((y-\\omega _i(x))\\mathcal {R}_i)=\\mu (\\mathcal {R}_i)\\mu ((y-\\omega _i(x))\\\\&=&\\mathcal {R}_{\\mu (i)}(y-\\omega _i(x^{-1})^2).$ Similarly, one can easily prove that the other conclusions in (i-3) hold from (A-3).", "In the case of (B-1), by Lemma REF (v) and Lemma REF (iii) we have $D_{\\mu (i)}&=&\\mu \\left((-\\alpha (x)+y)\\mathcal {R}_i\\right)=\\mu (\\mathcal {R}_i)\\mu (-\\alpha (x)+y)\\\\&=&{\\cal R}_{\\mu (i)}\\left(-\\theta _{\\mu (i)}(\\widehat{\\alpha }(x))+y\\right)\\\\&=&C_{\\mu (i),2,\\theta _{\\mu (i)}(\\widehat{\\alpha })}$ Hence the conclusion (ii-1) holds by Theorem REF (ii).", "Similarly, in the case of (B-2) we have $&&D_{\\mu (i)}=\\mu (\\mathcal {R}_i)\\mu \\left(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x)) y+y^2\\right)\\\\&=&{\\cal R}_{\\mu (i)}\\left((\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)))^{-1}+\\theta _{\\mu (i)}^2(\\theta _{\\mu (i)}^2(\\widehat{\\alpha }(x)))y+y^2\\right)\\\\&=&C_{\\mu (i),1,\\theta _{\\mu (i)}^2(\\widehat{\\alpha })}.$ Hence the conclusion (ii-2) holds by Theorem REF (iii).", "$\\diamondsuit $ Then we prove that $\|\\mathcal {C}\|\|\\mathcal {D}\|=\|\\mathbb {F}_q\|^{3m}$ .", "For any $0\\le i\\le s+t$ , by Theorems REF , REF and direct calculations we deduce that $\|C_i\|\|D_{\\mu (i)}\|=\|K_i\|^3=\|\\mathcal {R}_i\|$ .", "From this and by Theorem REF (i)–(iii), we obtain $\|\\mathcal {C}\|\|\\mathcal {D}\|&=&(\\prod _{i=0}^{s+t}\|C_i\|)(\\prod _{i=0}^{s+t}\|D_{\\mu (i)}\|)=\\prod _{i=0}^{s+t}\|C_i\|\|D_{\\mu (i)}\|\\\\&=&\\prod _{i=0}^{s+t}\|\\mathcal {R}_i\|=\\prod _{i=0}^{s+t}\|\\mathcal {A}_i[y;\\theta \|_{\\mathcal {A}_i}]/\\langle \\varepsilon _i(x)(y^3-1)\\rangle \|\\\\&=&\|\\mathcal {A}[y;\\theta ]/\\langle y^3-1\\rangle \|=\|\\mathbb {F}_q[G_{(m,3,r)}]\|\\\\&=&\|\\mathbb {F}_q\|^{3m}.$ $\\diamondsuit $ We claim that $\\mathcal {D}\\subseteq \\mathcal {C}^{\\bot _E}$ .", "In fact, let $\\xi \\in \\mathcal {D}$ and $\\eta \\in \\mathcal {C}$ .", "Then for each integer $i$ , $0\\le i\\le s+t$ , there exist $\\alpha _i\\in C_i$ and $\\beta _i\\in D_i$ such that $\\xi =\\sum _{i=0}^{s+t}\\varepsilon _i(x)\\alpha _i$ and $\\eta =\\sum _{i=0}^{s+t}\\varepsilon _i(x)\\beta _i$ , where $C_i$ and $D_i$ are left ideals of $\\mathcal {R}_i$ given by (i)–(ii) and $\\varepsilon _i(x)\\alpha _i, \\varepsilon _i(x)\\beta _i\\in \\mathcal {A}_i[y;\\theta \|_{\\mathcal {A}_i}]/\\langle \\varepsilon _i(x)(y^3-1)\\rangle $ .", "By Lemma REF (iv), we see that $\\varepsilon _i(x)$ is the multiplicative identity of $\\mathcal {A}_i[y;\\theta \|_{\\mathcal {A}_i}]/\\langle \\varepsilon _i(x)(y^3-1)\\rangle $ .", "Since $\\varepsilon _i(x)\\varepsilon _j(x)=0$ for all $0\\le i\\ne j\\le s+t$ , we have $\\varepsilon _i(x)\\varepsilon _{\\mu (j)}(x)=0$ if $i\\ne \\mu (j)$ , i.e., $j\\ne \\mu (i)$ .", "Hence $\\xi \\cdot \\mu (\\eta )&=&(\\sum _{i=0}^{s+t}\\varepsilon _i(x)\\alpha _i)(\\sum _{i=0}^{s+t}\\mu (\\varepsilon _i(x)\\beta _i)\\\\&=&\\left(\\sum _{i=0}^{s+t}(\\varepsilon _i(x)\\alpha _i)\\varepsilon _i(x)\\right)\\\\&&\\cdot \\left(\\sum _{i=0}^{s+t}\\varepsilon _{\\mu (i)}(x)(\\mu (\\beta _i)\\varepsilon _{\\mu (i)}(x))\\right)\\\\&=&\\sum _{i,j=0}^{s+t}(\\varepsilon _i(x)\\alpha _i)\\varepsilon _i(x)\\varepsilon _{\\mu (j)}(x)(\\mu (\\beta _j)\\varepsilon _{\\mu (j)}(x))\\\\&=&\\sum _{i=0}^{s+t}(\\varepsilon _i(x)\\alpha _i)\\varepsilon _i(x)(\\mu (\\beta _{\\mu (i)})\\varepsilon _{i}(x))\\\\&=&\\sum _{i=0}^{s+t}\\varepsilon _i(x)(\\alpha _i\\mu (\\beta _{\\mu (i)})).$ Since $\\beta _{\\mu (i)}\\in D_{\\mu (i)}$ , by Lemma REF (i) we see that $\\mu (\\beta _{\\mu (i)})\\in \\mu (D_{\\mu (i)})=\\mu (\\mu (B_i))=B_i.$ From this and by (REF ), we deduce that $\\alpha _i\\mu (\\beta _{\\mu (i)})=0$ for all $i=0,1,\\ldots ,s+t$ , which implies $\\xi \\cdot \\mu (\\eta )=0$ , and so $[\\xi ,\\eta ]_E=0$ by Lemma REF (ii).", "Therefore, $\\mathcal {D}\\subseteq \\mathcal {C}^{\\bot _E}$ .", "As stated above, we conclude that $\\mathcal {D}=\\mathcal {C}^{\\bot _E}$ since both $\\mathcal {C}$ and $\\mathcal {D}$ are linear codes over $\\mathbb {F}_q$ of length $3m$ .", "Finally, we determine self-orthogonal left $G_{(m,3,r)}$ -codes.", "Theorem 4.6 All distinct self-orthogonal left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ are given by the following $\\mathcal {C}=\\bigoplus _{i=0}^{s+t}\\mathcal {A}_i\\Box _{\\varphi _i}C_i=\\bigoplus _{i=0}^{s+t}\\lbrace \\varepsilon _i(x)\\xi \\mid \\xi \\in C_i\\rbrace \\ ({\\rm mod} \\ x^m-1),$ where $C_i$ is an left ideal of $\\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ given by one of the following four cases: (i) $0\\le i\\le s_1$ .", "In this case, $C_i$ is given by one of the following three subcases.", "(i-1) If $q\\equiv 0$ (mod 3), $C_i=\\lbrace 0\\rbrace $ or $C_i=\\mathcal {R}_i(y-1)^2$ .", "(i-2) If $q^{d_i}\\equiv 2$ (mod 3), $C_i=\\lbrace 0\\rbrace $ .", "(i-3) Let $q^{d_i}\\equiv 1$ (mod 3).", "If $\\omega _i(x^{-1})\\equiv \\omega _i(x)$ (mod $f_i(x)$ ), then $C_i=\\lbrace 0\\rbrace $ , $C_i=\\mathcal {R}_i(y-1)(y-\\omega _i(x))$ or $C_i=\\mathcal {R}_i(y-1)(y-\\omega _i(x)^2)$ .", "Otherwise, $C_i=\\lbrace 0\\rbrace $ .", "(ii) $s_1+1\\le i\\le s_1+s_2$ .", "In this case, $C_i=\\mathcal {R}_ig(y)$ , $C_{i+s_2}=\\mathcal {R}_{i+s_2}\\vartheta (y)$ and the pair $(g(y),\\vartheta (y))$ of polynomials is given by one of the following three subcases.", "(ii-1) Let $q\\equiv 0$ (mod 3).", "There are 10 pairs $(g(y),\\vartheta (y))$ : Table: NO_CAPTION(ii-2) Let $q^{d_i}\\equiv 2$ (mod 3).", "There are 9 pairs $(g(y),\\vartheta (y))$ : Table: NO_CAPTION(ii-3) Let $q^{d_i}\\equiv 1$ (mod 3).", "There are 27 pairs $(g(y),\\vartheta (y))$ : Table: NO_CAPTION(iii) $s+1\\le i\\le s+t_1$ .", "In this case, $C_i=\\lbrace 0\\rbrace $ or $C_i=C_{i,1,\\alpha }=\\mathcal {R}_i(\\alpha (x)^{-1}+\\theta _i^2(\\alpha (x))y+y^2)$ where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ satisfying $\\theta _{i}(\\widehat{\\alpha })\\theta _{i}^2(\\widehat{\\alpha })\\alpha +\\theta _{i}(\\widehat{\\alpha })\\alpha \\theta _i^2(\\alpha )+1=0.$ (iv) $s+t_1+1\\le i\\le s+t_1+t_2$ .", "In this case, there are exactly $10+8q^{\\frac{d_i}{3}}+8q^{\\frac{2d_i}{3}}+q^{d_i}$ pairs $(C_i,C_{i+t_2})$ given by one of the following four subcases: (iv-1) $4+2q^{\\frac{d_i}{3}}+2q^{\\frac{2d_i}{3}}$ pairs: $(\\lbrace 0\\rbrace ,C_{i+t_2})$ , where $C_{i+t_2}$ is any left ideal of $\\mathcal {R}_{i+t_2}$ listed by Theorem REF .", "(iv-2) $(1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}})(3+q^{\\frac{d_i}{3}})$ pairs: $(C_{i,1,\\alpha },C_{i+t_2})$ , where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ and $C_{i+t_2}$ is one of the following $3+q^{\\frac{d_i}{3}}$ left ideals of $\\mathcal {R}_{i+t_2}$ : $\\bullet $ $C_{i+t_2}=\\lbrace 0\\rbrace $ .", "$\\bullet $ $C_{i+t_2}=C_{i+t_2,1,\\beta }$ where $\\beta =\\theta _{i+t_2}^2(\\gamma (x^{-1})^{-1})$ (mod $f_{i+t_2}(x)$ ) and $\\gamma (x)\\in \\mathcal {G}_i$ satisfying the following conditions: $\\phi _{i,\\alpha }(\\gamma (x))=0$ , if $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ ; $\\psi _{i,\\alpha }(\\gamma (x))=0$ , if $\\theta _i(a(x))=a(x)^{q^{2\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ .", "$\\bullet $ $C_{i+t_2}=C_{i+t_2,2,\\beta }$ where $\\beta =\\theta _{i+t_2}(\\alpha (x^{-1})^{-1})$ (mod $f_{i+t_2}(x)$ ).", "(iv-3) $2(1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}})$ pairs: $(C_{i,2,\\alpha },C_{i+t_2})$ , where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ and $C_{i+t_2}$ is one of the following 2 left ideals of $\\mathcal {R}_{i+t_2}$ : $\\bullet $ $C_{i+t_2}=\\lbrace 0\\rbrace $ .", "$\\bullet $ $C_{i+t_2}=C_{i+t_2,1,\\beta }$ where $\\beta =\\theta _{i+t_2}^2(\\alpha (x^{-1})^{-1})$ (mod $f_{i+t_2}(x)$ ).", "(iv-1) 1 pair: $(\\mathcal {R}_i,\\lbrace 0\\rbrace )$ .", "By Theorem REF and its proof we have $\\mathcal {C}^{\\bot _E}=\\oplus _{i=0}^{s+t}\\mathcal {A}_i\\Box _{\\varphi _i}D_i$ , where $D_i=\\mu (B_{\\mu (i)})$ .", "From this and by Theorem REF , we deduce that $\\mathcal {C}$ is a self-orthogonal left $G_{(m,3,r)}$ -code over $\\mathbb {F}_q$ if and only if $C_i\\subseteq D_i=\\mu (B_{\\mu (i)})$ for all $i=0,1,\\ldots ,s+t$ .", "By Equation (REF ) and the proof of Theorem REF , it follows that $C_{\\mu (i)}\\cdot B_{\\mu (i)}=\\lbrace 0\\rbrace $ and $\|C_{\\mu (i)}\|\|B_{\\mu (i)}\|=\|C_{\\mu (i)}\|\|\\mu (B_{\\mu (i)})\|=\|C_{\\mu (i)}\|\|D_{i}\|=\|\\mathcal {R}_{\\mu (i)}\|,$ which implies that $C_{\\mu (i)}$ is the annihilating left ideal of $B_{\\mu (i)}$ in $\\mathcal {R}_{\\mu (i)}$ , i.e., $C_{\\mu (i)}={\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(B_{\\mu (i)})$ where ${\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(B_{\\mu (i)})=\\lbrace \\xi \\in \\mathcal {R}_{\\mu (i)}\\mid \\xi \\eta =0, \\ \\forall \\eta \\in B_{\\mu (i)}\\rbrace $ .", "Since $\\mu $ is an $\\mathbb {F}_q$ -algebra anti-isomorphism from $\\mathcal {R}_i$ onto $\\mathcal {R}_{\\mu (i)}$ , by $C_i\\cdot B_i=\\lbrace 0\\rbrace $ and $\|C_i\|\|B_i\|=\|\\mathcal {R}_i\|$ we have $\\mu (B_i)\\cdot \\mu (C_i)=\\lbrace 0\\rbrace $ and $\|\\mu (B_i)\|\|\\mu (C_i)\|=\|\\mathcal {R}_{\\mu (i)}\|$ , which implies $ {\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(\\mu (C_i))=\\mu (B_i)=D_{\\mu (i)}.$ From this, by $D_i=\\mu (B_{\\mu (i)})$ and (REF ), we deduce that $C_i\\subseteq D_i &\\Longleftrightarrow & B_{\\mu (i)}\\supseteq \\mu (C_i) \\\\&\\Longleftrightarrow & {\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(B_{\\mu (i)})\\subseteq {\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(\\mu (C_i))\\\\&\\Longleftrightarrow & C_{\\mu (i)}\\subseteq D_{\\mu (i)}$ for all $i=0,1,\\ldots ,s+t$ .", "$\\diamondsuit $ Let $0\\le i\\le s$ .", "Then $\\theta _i={\\rm id}_{K_i}$ and that $\\mathcal {R}_i=K_i[y]/\\langle y^3-1\\rangle $ is a commutative ring.", "By Theorem REF and its proof there is a unique pair $(g(y),h(y))$ of monic factors $g(y),h(y)$ of $y^3-1$ in $K_i[y]$ such that $C_i=\\mathcal {R}_ig(y)$ , $D_i=\\mathcal {R}_i\\frac{y^3-1}{h(y)}=\\mu (B_{\\mu (i)})$ with $B_{\\mu (i)}=\\mathcal {R}_{\\mu (i)}\\mu (\\frac{y^3-1}{h(y)})$ , and $C_{\\mu (i)}={\\rm Ann}^{(L)}_{\\mathcal {R}_{\\mu (i)}}(B_{\\mu (i)})=\\mathcal {R}_{\\mu (i)}\\mu (h(y)).$ Hence $C_i\\subseteq D_i \\Longleftrightarrow \\frac{y^3-1}{h(y)}\\mid g(y)\\Longleftrightarrow (y^3-1)\\mid g(y)h(y).$ By Lemma REF (ii) and Theorems REF , we have one of the following two cases.", "(i) When $0\\le i\\le s_1$ , $\\mu (i)=i$ , which implies $\\mu (h(y))\\sim _l g(y)$ by (REF ), and so $\\mu (g(y))\\sim _l h(y)$ .", "Form this and by (REF ) we deduce that $C_i\\subseteq D_i$ if and only if $(y^3-1)\\mid g(y)\\mu (g(x))$ .", "Then the conclusions follow from Lemma REF (i) and (ii).", "(ii) Let $s_1+1\\le i\\le s_1+s_2$ , $\\mu (i)=i+s_2$ .", "By Lemma REF (i) and (ii), we see that for each monic factor $h(y)$ of $y^3-1$ in $K_i[y]$ there is a unique monic factor $\\vartheta (y)$ of $y^3-1$ in $K_{\\mu (i)}[y]$ such that $\\mu (h(y))\\sim _l \\vartheta (y)$ , which implies $C_{i+s_2}=\\mathcal {R}_{\\mu (i)}\\vartheta (y)$ by (REF ).", "Then the conclusions follow from (REF ), Lemma REF (i) and (ii) immediately.", "$\\diamondsuit $ Let $s+1\\le i\\le s+t$ .", "Then $\\mathcal {R}_i=K_i[y;\\theta _i]/\\langle y^3-1\\rangle $ is a noncommutative ring.", "By Theorem REF (ii), the pair $(C_i,D_{\\mu (i)})$ is given by one of the following cases: $\\diamond $ $C_i=\\lbrace 0\\rbrace $ and $D_{\\mu (i)}=\\mathcal {R}_{\\mu (i)}$ , or $C_i=\\mathcal {R}_{i}$ and $D_{\\mu (i)}=\\lbrace 0\\rbrace $ ; $\\diamond $ $C_i=C_{i,1,\\alpha }$ and $D_{\\mu (i)}=C_{\\mu (i),2,\\theta _{\\mu (i)}(\\widehat{\\alpha })}$ , where $\\alpha \\in \\mathcal {G}_i$ ; $\\diamond $ $C_i=C_{i,2,\\alpha }$ and $D_{\\mu (i)}=C_{\\mu (i),1,\\theta _{\\mu (i)}^2(\\widehat{\\alpha })}$ , where $\\alpha \\in \\mathcal {G}_i$ .", "From these, we deduce that ${\\rm dim}_{K_i}(C_i)+{\\rm dim}_{K_{\\mu (i)}}(C_{\\mu (i)})=3.$ Then by Lemma REF (ii), we have one and only one of the following two cases.", "(iii) Let $s+1\\le i\\le s+t_1$ .", "Then $\\mu (i)=i$ .", "In this case, we deduce that the condition $C_i\\subseteq D_i$ if and only if $C_i=\\lbrace 0\\rbrace $ or $C_i=C_{i,1,\\alpha }$ where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ satisfying $C_{i,1,\\alpha }\\subset C_{i,2,\\theta _{i}(\\widehat{\\alpha })}$ .", "By Theorem REF (i), the condition $C_{i,1,\\alpha }\\subset C_{i,2,\\theta _{i}(\\widehat{\\alpha })}$ is equivalent to that $\\theta _{i}(\\widehat{\\alpha })\\theta _{i}(\\theta _{i}(\\widehat{\\alpha }))\\alpha +\\theta _{i}(\\widehat{\\alpha })\\alpha \\theta _i^2(\\alpha )+1=0$ , i.e., $\\theta _{i}(\\widehat{\\alpha })\\theta _{i}^2(\\widehat{\\alpha })\\alpha +\\theta _{i}(\\widehat{\\alpha })\\alpha \\theta _i^2(\\alpha )+1=0.$ (iv) Let $s+t_1+1\\le i\\le s+t_1+t_2$ .", "Then $\\mu (i)=i+t_2$ .", "By Theorem REF (ii), we have one of the following four situations: (iv-1) Let $C_i=\\lbrace 0\\rbrace $ .", "Then $D_{i+t_2}=\\mathcal {R}_{\\mu (i)}$ .", "In this case, $C_{i+t_2}\\subseteq D_{i+t_2}$ for any left ideal $C_{i+t_2}$ of $\\mathcal {R}_{\\mu (i)}$ .", "By Theorem REF , the number of pairs $(\\lbrace 0\\rbrace , C_{i+t_2})$ is equal to $4+2q^{\\frac{d_i}{3}}+2q^{\\frac{2d_i}{3}}$ .", "(iv-2) Let $C_i=C_{i,1,\\alpha }$ where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ .", "Then ${\\rm dim}_{K_{i+t_i}}(D_{i+t_2})=2$ by (REF ) and $D_{i+t_2}=C_{i+t_2,2,\\theta _{i+t_2}(\\widehat{\\alpha })}$ by Theorem REF (ii-1).", "Hence ${\\rm dim}_{K_{i+t_i}}(C_{i+t_2})\\le 2$ if $C_{i+t_2}\\subseteq D_{i+t_2}$ .", "Then we have one of the following three cases.", "$\\triangleright $ It is obvious that $C_{i+t_2}=\\lbrace 0\\rbrace $ satisfying $C_{i+t_2}\\subseteq D_{i+t_2}$ .", "$\\triangleright $ Let $C_{i+t_2}=C_{i+t_2,1,\\beta }$ where $\\beta =\\beta (x)\\in \\mathcal {G}_{i+t_2}$ .", "By Lemma REF (ii), we have $\\mu (i+t_2)=i$ , which implies $D_i=D_{\\mu (i+t_2)}=C_{i,2,\\theta _i(\\widehat{\\beta })}$ by Theorem REF (ii-1).", "From this and by Theorem REF (ii), we deduce that $C_i\\subset D_i$ if and only if $\\theta _i(\\widehat{\\beta })$ satisfies the following conditions: $\\phi _{i,\\alpha }(\\theta _i(\\widehat{\\beta }))=0$ , if $\\theta _i(a(x))=a(x)^{q^{\\frac{d_i}{3}}}$ for all $a(x)\\in K_i$ .", "$\\psi _{i,\\alpha }(\\theta _i(\\widehat{\\beta }))=0$ , if $\\theta _i(a(x))=a(x)^{q^{\\frac{2d_i}{3}}}$ for all $a(x)\\in K_i$ .", "We denote $\\gamma =\\gamma (x)=\\theta _i(\\widehat{\\beta }(x))$ .", "Then $\\gamma =\\theta _i(\\mu (\\beta (x))^{-1})\\in \\mathcal {G}_i$ by Corollary REF , which implies $\\mu (\\beta (x))^{-1}=\\theta _i^2(\\gamma (x))$ , and hence $\\beta (x)=\\theta _{i+t_2}^2(\\gamma (x^{-1})^{-1})$ by Lemma REF (iv).", "Moreover, by Lemma REF we know that both $\\phi _{i,\\alpha }(x)$ and $\\psi _{i,\\alpha }(x)$ have exactly $q^{\\frac{d_i}{3}}+1$ roots in $\\mathcal {G}_i$ .", "$\\triangleright $ Let $C_{i+t_2}=C_{i+t_2,2,\\beta }$ where $\\beta =\\beta (x)\\in \\mathcal {G}_{i+t_2}$ .", "As $\\mu (i+t_2)=i$ , we have $D_i=D_{\\mu (i+t_2)}=C_{i,1,\\theta _i^2(\\widehat{\\beta })}$ by Theorem REF (ii-2).", "Hence $C_i=C_{i,1,\\alpha }\\subseteq D_i$ if and only if $\\alpha =\\theta _i^2(\\widehat{\\beta })=\\theta _i^2(\\beta (x^{-1})^{-1})$ , which is equivalent to that $\\beta (x)=\\theta _{i+t_2}(\\alpha (x^{-1})^{-1})$ by Lemma REF (iv).", "Therefore, the number of pairs $(C_{i,1,\\alpha },C_{i+t_2})$ is equal to $(1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}})(3+q^{\\frac{d_i}{3}}).$ (iv-3) Let $C_i=C_{i,2,\\alpha }$ where $\\alpha =\\alpha (x)\\in \\mathcal {G}_i$ .", "Then ${\\rm dim}_{K_{i+t_i}}(D_{i+t_2})=1$ by (REF ) and $D_{i+t_2}=C_{i+t_2,1,\\theta _{i+t_2}^2(\\widehat{\\alpha })}$ by Theorem REF (ii-2).", "Hence ${\\rm dim}_{K_{i+t_i}}(C_{i+t_2})\\le 1$ if $C_{i+t_2}\\subseteq D_{i+t_2}$ .", "Then we have one of the following two cases.", "$\\triangleright $ $C_{i+t_2}=\\lbrace 0\\rbrace $ .", "$\\triangleright $ Let $C_{i+t_2}=C_{i+t_2,1,\\beta }$ where $\\beta =\\beta (x)\\in \\mathcal {G}_{i+t_2}$ .", "Then As $\\mu (i+t_2)=i$ , we have $D_i=D_{\\mu (i+t_2)}=C_{i,2,\\theta _i(\\widehat{\\beta })}$ by Theorem REF (ii-1).", "Hence $C_i=C_{i,2,\\alpha }\\subseteq D_i$ if and only if $\\alpha =\\theta _i(\\widehat{\\beta })=\\theta _i^2(\\beta (x^{-1})^{-1})$ , which is equivalent to that $\\beta (x)=\\theta _{i+t_2}^2(\\alpha (x^{-1})^{-1})$ by Lemma REF (iv).", "Therefore, the number of pairs $(C_{i,1,\\alpha },C_{i+t_2})$ is equal to $2(1+q^{\\frac{d_i}{3}}+q^{\\frac{2d_i}{3}}).$ (iv-4) Let $C_i=\\mathcal {R}_i$ .", "Then $D_{i+t_2}=\\lbrace 0\\rbrace $ .", "From this and by $C_{i+t_2}\\subseteq D_{i+t_2}$ , we deduce $C_{i+t_2}=\\lbrace 0\\rbrace $ .", "As stated above, we conclude that the number of pairs $(C_i,C_{i+t_2})$ , where $s+t_1+1\\le i\\le s+t_1+t_2$ , is equal to $10+8q^{\\frac{d_i}{3}}+8q^{\\frac{2d_i}{3}}+q^{d_i}.$" ], [ "An Example", "We consider left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ .", "Obviously, $9^3=729\\equiv 1$ (mod 14).", "All distinct 3-cyclotomic cosets modulo 14 are the following: $J^{(3)}_0=\\lbrace 0\\rbrace $ , $J^{(3)}_7=\\lbrace 7\\rbrace $ , $J^{(3)}_2=\\lbrace 2,6,4,12,8,10\\rbrace $ , $J^{(3)}_1=\\lbrace 1,3,9,13,11,5\\rbrace $ .", "It is clear that $\\theta (7)=9\\cdot 7\\equiv 7, \\ \\theta (2)=9\\cdot 2\\equiv 4, \\ \\theta (1)=9 \\ ({\\rm mod 14}).$ Using the notations is Section 2, we have that $s=1$ , $t=2$ , $J(0)=J^{(3)}_0$ , $J(1)=J^{(3)}_7$ , $J(2)=J^{(3)}_2$ and $J(3)=J^{(3)}_1$ .", "Hence $d_0=d_1=1$ , $d_i=6$ and $\\frac{d_i}{3}=2$ for $i=2,3$ .", "Obviously, $3\\equiv 0$ (mod 3).", "By Corollary REF (i), the number of left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ is equal to $4^2\\prod _{i=2,3}(4+2\\cdot 3^{\\frac{d_i}{3}}+2\\cdot 3^{\\frac{2d_i}{3}})=16\\cdot 184^2=541,696.$ We have $x^{14}-1=f_0(x)f_1(x)f_2(x)f_3(x)$ , where $f_0(x)=x-1$ , $f_1(x)=x+1$ , $f_2(x)=x^6+x^5+x^4+x^3+x^2+x+1$ and $f_3(x)=x^6+2x^5+x^4+2x^3+x^2+2x+1$ .", "Then $\\bullet $ $K_0=\\mathbb {F}_3[x]/\\langle x-1\\rangle =\\mathbb {F}_3$ and $\\mathcal {R}_0=K_0[y]/\\langle y^3-1\\rangle =\\mathbb {F}_3[y]/\\langle (y-1)^3\\rangle $ .", "By Theorem REF (ii), all distinct ideals of $\\mathcal {R}_0$ are given by: $C_0=\\mathcal {R}_0g(y)$ , where $g(y)\\in \\lbrace 1,y-1,(y-1)^2,y^3-1\\rbrace .$ $\\bullet $ $K_1=\\mathbb {F}_3[x]/\\langle x+1\\rangle =\\mathbb {F}_3$ and $\\mathcal {R}_1=K_1[y]/\\langle y^3-1\\rangle =\\mathbb {F}_3[y]/\\langle (y-1)^3\\rangle $ .", "By Theorem REF (ii), all distinct ideals of $\\mathcal {R}_1$ are given by: $C_1=\\mathcal {R}_1g(y)$ , where $g(y)\\in \\lbrace 1,y-1,(y-1)^2,y^3-1\\rbrace .$ Moreover, $\|\\langle 0\\rangle \|=1$ , $\|\\mathcal {R}_0\|=\|\\mathcal {R}_1\|=3^3=27$ , $\|\\mathcal {R}_0(y-1)\|=\|\\mathcal {R}_1(y-1)\|=3^2=9$ and $\|\\mathcal {R}_0(y-1)^2\|=\|\\mathcal {R}_1(y-1)^2\|=3$ .", "$\\bullet $ $K_2=\\mathbb {F}_2[x]/\\langle f_2(x)\\rangle =\\lbrace \\sum _{j=0}^{5}a_jx^j\\mid a_j\\in \\mathbb {F}_3, \\ j=0,1,\\ldots ,5\\rbrace $ and $\\varrho _2(x)=1+x$ is an element of multiplicative order $1+3^2+3^4=91$ in $K_2$ .", "Hence $\\mathcal {G}_2=\\lbrace (1+x)^\\lambda \\mid \\lambda =0,1,\\ldots ,90\\rbrace \\ ({\\rm mod} \\ f_2(x))$ and $\\mathcal {R}_2=K_2[y;\\theta _2]/\\langle y^3-1\\rangle $ where $\\theta _2$ is an $\\mathbb {F}_3$ -algebra automorphism of $K_2$ defined by: $\\theta _2(a(x))=a(x^r)=a(x^9)=a(x)^9 \\ ({\\rm mod} \\ f_2(x))$ for all $a(x)\\in K_2$ .", "Then $\\theta _2^2(\\varrho _2(x))=(1+x)^{9^2}=1+x^4.$ By Theorem REF , all distinct left ideals $C_2$ of $\\mathcal {R}_2$ are given by the following three cases: (i) $C_2=\\lbrace 0\\rbrace $ with $\|\\lbrace 0\\rbrace \|=1$ , and $C_2=\\mathcal {R}_2$ with $\|\\mathcal {R}_2\|=\|K_2\|^3=(3^6)^3=3^{18}=387420489$ .", "(ii) $C_2=C_{2,2,(1+x)^\\lambda }=\\mathcal {R}_2(-(1+x)^\\lambda +y)$ with $\|C_2\|=(3^6)^2=3^{12}=531441$ , $\\lambda =0,1,2,\\ldots ,90$ .", "(iii) $C_2=C_{2,1,(1+x)^\\lambda }=\\mathcal {R}_2((1+x)^{91-\\lambda }+(1+x^4)^\\lambda y+y^2)$ with $\|C_2\|=3^6=729$ , $\\lambda =0,1,2,\\ldots ,90$ .", "$\\bullet $ $K_3=\\mathbb {F}_2[x]/\\langle f_3(x)\\rangle =\\lbrace \\sum _{j=0}^5a_jx^j\\mid a_j\\in \\mathbb {F}_3\\rbrace $ .", "We find that $\\varrho _3(x)=1+2x$ is an element of multiplicative order $1+3^2+3^4=91$ in $K_3$ .", "Hence $\\mathcal {G}_3=\\lbrace (1+2x)^\\lambda \\mid \\lambda =0,1,\\ldots ,90\\rbrace \\ ({\\rm mod} \\ f_3(x))$ and $\\mathcal {R}_3=K_3[y;\\theta _2]/\\langle y^3-1\\rangle $ where $\\theta _3$ is an $\\mathbb {F}_3$ -algebra automorphism of $K_3$ defined by: $\\theta _3(a(x))=a(x^r)=a(x^9)=a(x)^9 \\ ({\\rm mod} \\ f_3(x))$ for all $a(x)\\in K_3$ .", "In particular, we have $\\theta _3^2(\\varrho _2(x))=(1+x)^{9^2}=1+x^4.$ By Theorem REF , all distinct left ideals $C_3$ of $\\mathcal {R}_3$ are given by the following three cases: (i) $\\lbrace 0\\rbrace $ and $\\mathcal {R}_3$ , where $\|\\lbrace 0\\rbrace \|=1$ and $\|\\mathcal {R}_3\|=\|K_3\|^3=(3^6)^3=3^{18}=387420489$ .", "(ii) $C_3=C_{3,2,(1+2x)^\\lambda }=\\mathcal {R}_3(-(1+2x)^\\lambda +y)$ with $\|C_3\|=(3^6)^2=3^{12}=531441$ , $\\lambda =0,1,2,\\ldots ,90$ .", "(iii) $C_3=C_{3,1,(1+x)^\\lambda }=\\mathcal {R}_3((1+2x)^{91-\\lambda }+(1+x^4)^\\lambda y+y^2)$ with $\|C_3\|=3^6=729$ , $\\lambda =0,1,2,\\ldots ,90$ .", "$\\bullet $ All distinct 541696 left $G$ -codes over $\\mathbb {F}_3$ are given by $\\mathcal {C}=\\bigoplus _{i=0}^3\\mathcal {A}_i\\Box _{\\varphi _i}C_i=\\sum _{i=0}^3\\lbrace \\varepsilon _i(x)\\xi _i\\mid \\xi _i\\in C_i\\rbrace $ (mod $ x^{14}-1$ ) by Theorem REF , where $\\varepsilon _0(x)=2+2x+2x^2+2x^3+2x^4+2x^5+2x^6+2x^7+2x^8+2x^9+2x^{10}+2x^{11}+2x^{12}+2x^{13}$ , $\\varepsilon _1(x)=2+x+2x^2+x^3+2x^4+x^5+2x^6+x^7+2x^8+x^9+2x^{10}+x^{11}+2x^{12}+x^{13}$ , $\\varepsilon _2(x)=x+x^2+x^3+x^4+x^5+x^6+x^8+x^9+x^{10}+x^{11}+x^{12}+x^{13}$ , $\\varepsilon _3(x)=2x+x^2+2x^3+x^4+2x^5+x^6+x^8+2x^9+x^{10}+2x^{11}+x^{12}+2x^{13}$ , and the number of codewords in $\\mathcal {C}$ is equal to $\|\\mathcal {C}\|=\|C_0\|\|C_1\|\|C_2\|\|C_3\|$ .", "As $-J(i)=J(i)$ (mod 14), we have $\\mu (i)=i$ for all $i=0,1,2,3$ .", "Using the notations of Lemma REF , we have $s=s_1=1$ , $s_2=0$ , $t=t_1=2$ and $t_2=0$ .", "Hence $\\mu (\\varepsilon _i(x))=\\varepsilon _i(x)$ for all $i=0,1,2,3$ .", "$\\bullet $ By Theorem REF , all self-orthogonal left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ are given by: $\\mathcal {C}=\\bigoplus _{i=0}^3\\mathcal {A}_i\\Box _{\\varphi _i}C_i$ , where $\\diamond $ $C_0=\\lbrace 0\\rbrace $ or $C_0=\\mathcal {R}_0(y-1)^2$ .", "$\\diamond $ $C_1=\\lbrace 0\\rbrace $ or $C_1=\\mathcal {R}_1(y-1)^2$ .", "$\\diamond $ $C_2=\\lbrace 0\\rbrace $ or $C_2=C_{2,1,\\alpha (x)}$ where $\\alpha (x)=(1+x)^\\lambda $ satisfying $\\theta _{i}(\\widehat{\\alpha })\\theta _{i}^2(\\widehat{\\alpha })\\alpha +\\theta _{i}(\\widehat{\\alpha })\\alpha \\theta _i^2(\\alpha )+1=0$ , i.e., $0&=&1+(1+x^{-1})^{-9\\lambda -81\\lambda }(1+x)^\\lambda \\\\&&+(1+x^{-1})^{-9\\lambda }(1+x)^{\\lambda +81\\lambda }\\\\&=&1+x^{90\\lambda }(1+x)^{-89\\lambda }+x^{9\\lambda }(1+x)^{73\\lambda }$ in $K_2$ ($0\\le \\lambda \\le 90$ ).", "Since $x^{14}=1$ and $(1+x)^{91}=1$ in $K_2$ , the above condition is equivalent to $1+x^{6\\lambda }(1+x)^{2\\lambda }+x^{9\\lambda }(1+x)^{73\\lambda }\\equiv 0 \\ ({\\rm mod} \\ f_2(x)).$ Precisely, we have $\\lambda =0,7,8,11,13,20,21,24,26,33,34,37,39,46,47,$ $\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 50,52,59,60,63,65,72,73,76,78,85,86,89.$ $\\diamond $ $C_3=\\lbrace 0\\rbrace $ or $C_3=C_{3,1,(1+2x)^\\lambda }$ where $\\lambda $ is given by (REF ).", "Therefore, the number of self-orthogonal left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ is equal to $2\\cdot 2\\cdot 29\\cdot 29=3364$ .", "For example, we have 21 self-orthogonal left $G_{(14,3,9)}$ -codes over $\\mathbb {F}_3$ : $\\mathcal {C}=\\lbrace \\varepsilon _2(x)\\xi \\mid \\xi \\in C_{2,1,(1+x)^\\lambda }\\rbrace $ where $\\lambda =7,8,11,20,21,24,33,34,37,46,47, 50, 59,60,63,72,73,76$ , $85,86,89$ , which are self-orthogonal linear $[42,6,18]$ -codes over $\\mathbb {F}_3$ with the following Hamming weight enumerator: $W_{\\mathcal {C}}(Y)=1+14Y^{18}+294Y^{24}+336Y^{30}+84Y^{36}.$" ], [ "Conclusion", "Let $G_{(m,3,r)}$ be a metacyclic group of order $3m$ , $r\\equiv q^\\epsilon $ (mod $m$ ) for some positive integer $\\epsilon $ and ${\\rm gcd}(m,q)=1$ .", "We present a system theory of left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ , only using finite field theory and basic theory of cyclic codes and skew cyclic codes.", "We prove that any left $G_{(m,3,r)}$ -code is a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$ , where ${\\cal A}_i$ is a minimal cyclic code over $\\mathbb {F}_q$ of length $m$ and $C_i$ is a skew cyclic code of length 3 over an extension field of $\\mathbb {F}_q$ , and provide an explicit expression for each outer code in every concatenated code.", "Moreover, we give the dual code of each left $G_{(m,3,r)}$ -code and determine all self-orthogonal left $G_{(m,3,r)}$ -codes over $\\mathbb {F}_q$ ." ], [ "Acknowledgment", "Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China.", "Yonglin Cao would like to thank the institution for the kind hospitality.", "This research is supported in part by the National Natural Science Foundation of China (Grant Nos.", "11471255, 61171082, 61571243) and the National Key Basic Research Program of China (Grant No.", "2013CB834204)." ] ]	1606.05019
[ [ "Simulations of solitonic core mergers in ultra-light axion dark matter\n cosmologies" ], [ "Abstract Using three-dimensional simulations, we study the dynamics and final structure of merging solitonic cores predicted to form in ultra-light axion dark matter halos.", "The classical, Newtonian equations of motion of a self-gravitating scalar field are described by the Schr\\\"odinger-Poisson equations.", "We investigate mergers of ground state (boson star) configurations with varying mass ratios, relative phases, orbital angular momenta and initial separation with the primary goal to understand the mass loss of the emerging core by gravitational cooling.", "Previous results showing that the final density profiles have solitonic cores and NFW-like tails are confirmed.", "In binary mergers, the final core mass does not depend on initial phase difference or angular momentum and only depends on mass ratio, total initial mass, and total energy of the system.", "For non-zero angular momenta, the otherwise spherical cores become rotating ellipsoids.", "The results for mergers of multiple cores are qualitatively identical." ], [ " Introduction", "If dark matter consists of a cold, ultra-light (pseudo)scalar field, the statistical distribution and structural properties of collapsed halos may be modified with respect to the predictions of standard cold dark matter (CDM) [1], [2], [3], [4].", "String theory compactifications provide a class of well-motivated candidate particles with axion-like properties which are naturally ultra-light, so-called ultra-light axions (ULAs) [5].", "Observations that are sensitive to the small-scale structure of dark matter halos thus open a unique window onto fundamental physics.", "Roughly speaking, Heisenberg's uncertainty relation suppresses gravitational collapse on scales below the de Broglie wavelength of particles with virial velocities [6], [7].", "For halo masses of $10^8$ $M_\\odot $ at $z \\sim 5$ , this is of the order of kpc if the particle mass is $m \\sim 10^{-22}$ eV [8].", "The strongest constraint in this mass range to date follows from the predicted suppression of early galaxy formation and the measured optical depth to reionization, yielding $m > 10^{-22}$ eV [3].", "While ULAs provide a well-motivated class of candidate particles, the same phenomenology applies more generally to any massive scalar field with negligible self-interactions in a coherent non-thermal state, e.g.", "as a consequence of being produced by vacuum realignment, which include subclasses of scalar field dark matter (SFDM) [9], [10], [11] and Bose-Einstein condensate (BEC) dark matter [12], [13], [14], [15].", "The class of scalars with negligible self-interactions in the mass range relevant for constraints from structure formation is often referred to as fuzzy dark matter (FDM) [6].", "We will follow this convention here.", "To clearly distinguish the effects of FDM from other modifications of CDM with suppressed small-scale growth such as warm dark matter, it will be necessary to search for generic signatures of FDM on halo density profiles and substructure.", "One such signature may be the existence of compact solitonic cores embedded in a halo with NFW-like density profile, found in cosmological simulations that used the comoving Schrödinger-Poisson (SP) equations to model FDM [16] (who used the notation $\\Psi $ DM to emphasize the wavelike nature of dark matter).", "Their density profiles are governed by well-known equilibrium solutions for self-gravitating scalar fields in the nonrelativistic regime whose properties have been studied intensively in the context of Newtonian boson stars [17], [18], [19] and BEC dark matter [20], [21], [22], [23].", "The core mass obeys a scaling relation with the mass of the host halo, $M_{\\rm c}\\sim M_{\\rm h}^{1/3}$ , which can be motivated by identifying the characteristic scale height of the solitonic core with the virial velocity de Broglie wavelength [24].", "The presence of solitonic cores was used to fit the profiles of dwarf galaxies and suggested as a possible solution of the cusp-core problem in CDM cosmologies [4], [25].", "In collisions of solitonic cores with exact phase opposition, destructive interference gives rise to a short-range repulsive force between the cores [23].", "As the authors of [23] pointed out, in the context of galaxy cluster observations with indications of an offset between dark and stellar matter [26], [27], this effect can provide an alternative explanation to self-interacting dark matter.", "These results suggest a rich phenomenology of hierarchical structure formation in the presence of solitonic halo cores.", "They motivate an investigation of their potential impact on halo substructure, baryonic physics, and the properties of the earliest generation of galaxies.", "Ideally, this would be achieved by direct simulations of the SP equations in large cosmological boxes.", "Requirements on the spatial resolution of the SP equations, however, limit the box size of currently affordable simulations to $\\sim 1$ Mpc$^3$ in the interesting range of $m \\sim 10^{-22}$ eV [16], making a detailed study of halo and subhalo core mergers infeasible.", "On the other hand, since the time scales for major mergers and subhalo evolution (determined by dynamical friction) are large compared to the gravitational time scales of the cores, much can be learned from the simplified problem of isolated mergers of two cores with different characteristic properties.", "One of the key questions is the efficiency of gravitational cooling of the newly formed core to shed mass and angular momentum as a function of the binary parameters [28], [29].", "The results can be used, for instance, in semi-analytic models for galaxy formation in FDM cosmologies.", "In this work, we therefore address the simplified setup of merging solitonic cores in three-dimensional SP simulations.", "This allows us to perform parameter studies of total energy, mass ratio, angular momenta, and relative phase in order to map out their impact on the merging time, final core mass, and final angular momentum.", "Collisions and mergers of Newtonian boson stars have been studied extensively in 2D [30], [31] but we are unaware of fully three-dimensional simulations with no imposed symmetries.", "Furthermore, a systematic investigation of final core masses after relaxation by gravitational cooling has been lacking so far.", "In addition to studying binary mergers, we also relax the assumption of isolated events by considering mergers of multiple cores in fast succession in order to compare our results with those presented in [24].", "The remainder of this paper is structured as follows.", "In sec:theory we briefly outline the underlying theory.", "Our numerical methods are described in sec:numerics.", "In sec:binary we summarize results from an in depth analysis of binary mergers of two solitonic cores.", "In sec:multimerge we extend this investigation to mergers of multiple cores.", "We conclude in sec:conclusion." ], [ "Solitonic halo cores", "The nonrelativistic dynamics of a coherent massive scalar field can be described by a function $\\psi $ which is governed by the Schrödinger-Poisson (SP) equations it = -22m2+mU 2U = 4G , where the density is defined as $\\rho =\|\\psi \|^{2}$ .", "Simulations of the (comoving) SP equations with cosmological initial conditions show the formation of halo cores with solitonic profiles [16] that coincide with spherically symmetric, stationary solutions of the SP equations otherwise known as (nonrelativistic) boson starsTo emphasize that this work is focused on galactic rather than stellar scales, we will refer to these solutions as (solitonic) cores instead of boson stars.", "All of our results, however, are independent of this interpretation.", "[18], [19].", "Their radial density profile is well approximated by [24] c(r) 0[1 + 0.091(r/rc)2]-8 where $r_{c}$ is the radius at which the density drops to one-half its peak value and the central density is given by 03.11015(2.510-22eVm)2(kpcrc)4MMpc3.", "As in [24], we define the core mass $M_{c}$ as the mass enclosed by $r_{c}$ and note that in the case of a core with total mass $M$ it is McM 0.237MM 8.64106(2.510-22eVm)2(kpcrc) 1.81106(2.510-22eVm)(EM)1/2skm .", "[24] show evidence for the same scaling relation between $M_c, E,$ and $M$ for the final state of multiple core mergers where $E$ and $M$ refer to the total energy and mass of the system instead of just the core.", "We revisit this claim in sec:multimerge below.", "The SP system and consequently the stationary solutions obey a scaling symmetry of the form [33]: {t,x,U,,}{-2t,-1x,2U,2,4}, where $\\lambda $ is an arbitrary parameter.", "Note that $x\\propto \\rho ^{-1/4}$ consistent with the relation between the average density of the core and its Jeans length [6].", "Throughout this paper we use an axion mass $m=2.5\\times 10^{-22}$ eV." ], [ "Numerical methods", "The Schrödinger equation in comoving coordinates [7] was implemented into the cosmological hydro code Nyx [34] in order to facilitate its later use for combined simulations of dark matter and baryons.", "The field $\\psi $ is discretized on a grid as an additional dark matter component and integrated using a 4th order Runge-Kutta solver.", "We employed the multigrid Gauss-Seidel red-black Poisson solver provided by Nyx to compute the gravitational potential.", "The cosmological scale factor was set to $a = 1$ in all of the simulations reported here.", "All simulations used a grid size of $512^3$ cells.", "In all runs, the total mass M[] = Vd3x and energy E[] = V[22m2\|\|2+12U\|\|2]d3x = V22m2()2d3x+ V2v2d3x+ V2Ud3x = K+Kv+W of the system was monitored.", "In the second line of eq:7, we used the Madelung representation [35] = eiS/ , v=S/m and in the last line we divided the total energy into gradient energy $K_{\\rho }$ , kinetic energy $K_{v}$ , and potential energy $W$ .", "Besides total energy, each contribution was measured separately in order to follow the dynamics of a particular system more closely.", "We use units $[M]=M_{\\odot }$ and $[E]=M_{\\odot }$ km$^{2}$ s$^{-2}$ .", "In addition, conservation of total angular momentum L[] =1mV*[r(-i)]d3x = 1mV[rS+i2 r]d3x =Vrvd3x was verified assuming that the density falls off sufficiently rapidly that boundary terms vanish.", "Owing to eq:5, the quantities defined above obey the scaling relations {M,K,Kv,W,LM,3K,3Kv,3W,L}.", "During the relaxation of the system, waves emitted by the merger carry mass and energy toward the numerical boundaries.", "In order to avoid spurious reheating from reflected waves, we follow [18] and place a 'sponge' in the outer regions of the grid by adding an imaginary potential V(r) = -i2V0{2+[(r-rs)/]-(rs/)} [r-rp] , to the Schrödinger equation which efficiently absorbs matter.", "Here $r$ is the distance from the center of the numerical domain.", "The Heaviside function $\\Theta $ ensures that the non-physical sponge is only added in the outer regions $r>r_{p}$ .", "Let $r_{N}$ be half the box size.", "We then set $r_{p}=7/8 r_{N}$ , $r_{s}=(r_{N}+r_{p})/2$ , $\\delta =(r_{N}-r_{p})$ and $V_{0}=0.6$ .", "Although our numerical domains are always cubic, we use a spherical sponge since the final states of our simulations are approximately spherically symmetric.", "In all runs, the time steps were chosen such that they fulfill the Courant-Friedrichs-Lewy (CFL) condition [36] t [m6x2,m\|V\|max] where in all conducted runs the first argument is more stringent than the second.", "We tested our code by considering a single solitonic core.", "It was shown in [18], [30] that it is a virialized attractor solution of a broad class of initial conditions.", "Hence, we expect the core to be stable with low-amplitude excitations caused by numerical errors.", "The excitation manifests itself in a periodic variation in the central density.", "Its amplitude decreases faster than quadratically with resolution implying fast convergence of our code.", "The central density varies at most on the percent level if $r_c$ is resolved by at least 3 cells.", "For the simulations described below, the typical resolution is greater than 4 cells for all binary mergers and most multiple mergers.", "The oscillation frequency matches the one found in [18].", "While kinetic and potential energy oscillate with opposite phase, total mass and energy are conserved to better than $10^{-3}$ .", "The oscillation of $\\psi $ in the complex plane has the expected frequency [18].", "We checked convergences of our code also for binary mergers.", "Increasing the resolution by a factor of two alters the results only negligibly.", "In all runs conserved quantities stay constant to better than $10^{-3}$ until matter is absorbed by the sponge.", "We use the yt toolkit [37] for our analysis of numerical data and for the volume rendering of fig:3b and fig:13.", "Core profiles were fitted employing the radial density profile routines around the density maxima.", "Although cores with non-vanishing angular momentum are not expected to be perfectly spherical, we find that they can be well fitted by eq:2.", "Below, we therefore always assume spherical symmetry of the final state." ], [ "Binary core mergers", "One of the distinctive features of hierarchical structure formation in FDM cosmologies is the presence of halo and subhalo cores evolving under a sequence of binary mergers which, to very good approximation, can be considered as isolated events.", "As a consequence of the scaling relations in eq:5, the initial conditions for an arbitrary binary collision are fully parametrized by few defining parameters, i.e.", "the relative velocity $v_{\|\|}$ and distance $d$ between the cores, the mass ratio $\\mu $ and total mass $M$ , the phase difference $\\Phi $ , and the angular momentum $L_{z}$ perpendicular to the orbital plane chosen to be in the x-y-plane.", "There are two distinct regimes.", "If the two cores are unbound ($E>0$ ) they superpose and pass through each other almost undisturbed [30], [31], [38], [39], behaving like solitons in this regime.", "If instead the cores are bound ($E<0$ ), they merge rapidly forming a new core [30].", "Our main result is that the mass of the emerging core is largely independent of the initial angular momentum, distance and relative phase, but depends on the ratio of initial core masses and total energy.", "Figure: Head-on collision of two cores with mass ratio μ=2\\mu =2 and high relative velocity.", "Upper panels: density profiles at different times for relative phases Φ=0\\Phi =0 (left) and Φ=π\\Phi =\\pi (right) along the symmetry axis.", "Numerical results are shown for the initial and final state as well as for the time of maximal interference.", "For comparison, we plot the interference pattern predicted from eq:17 at the same time.", "Deviations can be attributed mostly to a small offset in the time of maximal interference.", "Lower panels: mass and energy contributions.", "Total energy and mass are conserved, while kinetic energy associated to the cores' relative motion (K v K_{v}) is transferred into the interference pattern yielding large values of K ρ K_{\\rho } during the interaction.", "The equality of the lower panels shows the independence of the evolution with respect to the inital phase shift Φ\\Phi .In order to analyse the unbound case, we consider two solitonic cores with $\\mu \\equiv M_{1}/M_{2}=M_{c,1}/M_{c,2}=2$ , $L_{z}=0\\,M_{\\odot }$ Mpc km/s and $v_{\|\|}=4\\,$ km/s.", "The cores are scaled such that the heavier one has a central density (0) = 1.361011 MMpc-3, roughly corresponding to the present cosmic critical density, giving a core radius $r_{c}\\simeq 11.6\\,\\text{kpc}$ .", "We emphasize that all results are independent of this overall scaling of the problem.", "The two cores are placed centrally in a 512 kpc cubic box with $d=256\\,$ kpc yielding $E\\simeq 8.2\\times 10^{6}\\,M_{\\odot }$ km$^{2}$ s$^{-2}$ .", "fig:2 shows the density profiles along the symmetry axis and the evolution of global quantities (mass and energy components) for two runs with relative phases $\\Phi =0$ and $\\Phi =\\pi $ .", "The final density distribution as well as the evolution of the global quantities are practically indistinguishable in both cases.", "Only the interference pattern at the time of superposition depends on the relative phase.", "The observed interference pattern follows directly from a superposition of the two solitonic cores.", "Initially, the cores are placed at $\\pm \\hat{x}(t=0)=\\pm d/2$ .", "The corresponding wavefunction $\\psi (t,x)$ is given by (t,x) = A1(\|x+x\|)ei(kx/2+t+/2) +A2(\|x-x\|)ei(-kx/2+t-/2) where $(A_{1})^{2}$ and $(A_{2})^{2}$ are the density profiles of the two cores and $k=mv_{\|\|}/\\hbar $ is the wavenumber corresponding to their relative velocity.", "The time $t_{\\text{int}}$ of maximal interference is defined by $\\hat{x}(t_{\\text{int}})=0$ .", "At that time, \|(tint,x)\|2 = A1(\|x\|)2 + A2(\|x\|)2 + 2A1(\|x\|)A2(\|x\|)(kx+)) .", "We thus expect that the period of the interference pattern is given by the de Broglie wave length = 2k = 2mv\|\| corresponding to the relative velocity.", "Here, $\\lambda \\simeq 12$ kpc.", "It is therefore well resolved by 12 cells.", "Figure: Mass, energy and angular momentum evolution of two representative binary collisions with initial values μ=1\\mu =1, v \|\| =0v_{\|\|}=0 km/s, and L z =2.4×10 4 M ⊙ L_{z}=2.4\\times 10^{4}\\,M_{\\odot } Mpc km/s (rescaled by 10 2 10^2).", "Cores with equal phase (Φ=0\\Phi =0) immediately merge (left).", "In perfect phase opposition (Φ=π\\Phi =\\pi ), the two cores first mutually repel each other multiple times before merging (right).", "The bounces are indicated by black arrows.", "The emerging cores are excited as seen by the oscillations of gradient and gravitational energy, K ρ K_{\\rho } and WW.", "The loss of total mass, energy, and angular momentum results from matter absorption inside the sponge.The interference pattern predicted by eq:17 matches the numerical results as seen in fig:2.", "During the interaction, gravity slightly contracts the density profiles.", "Neglecting this small effect, we see that they remain in a superposition state of two solitonic cores even during their interaction.", "As expected, the potential energy mildly increases during the collision, while mass and total energy are conserved.", "During the collision, the kinetic energy from the cores' relative motion is stored in the interference pattern, strongly boosting the gradient energy contribution $K_{\\rho }$ .", "At later times, the energy is transferred back to the cores' motion.", "There is no significant decrease in velocity or deformation of the density profiles due to the collision.", "The cores thus indeed behave like solitons in this regime.", "The evolution of a bound binary system with negative total energy is very different.", "In this case, the cores rapidly merge and relax to a new solitonic core by gravitational cooling [40].", "One interesting exception is the case of binary collisions with perfect phase opposition $\\Phi =\\pi $ and equal masses $\\mu =1$ during which the destructive interference gives rise to a repulsive effect, causing the cores to bounce off each other [23].", "Figure: Evolution of the core (solid lines) and total (dashed lines) mass for binary mergers.", "The triplets identify the point (μ,Φ,L z )(\\mu ,\\Phi ,L_{z}) in parameter space.", "Angular momentum is given in units of L z =10 4 M ⊙ \\left[L_{z}\\right] = 10^{4}\\,M_{\\odot } Mpc km/s.For our study of bound binary collisions, we placed two halos along the central axis in a 1024 kpc cubic box with $d=256\\,$ kpc.", "As before, the cores are scaled such that the central density of the heavier core obeys $\\rho (0)=\\rho _{\\text{cr}}$ .", "We need the larger box compared to the previous runs since the two halos emit mass while merging.", "We require this mass to be able to propagate sufficiently far away from the merger before being absorbed inside the sponge.", "Figure: Volume rendered images of two representative binary mergers in phase (top) and with opposite phase (bottom) showing the central region of the computational domain at t=0.7t= 0.7, t=0.94t=0.94, t=2.0t=2.0 and t=7.0t=7.0 in Mpc/km s.In fig:3, we show the mass, energy and angular momentum evolution of two representative runs with $\\mu =1$ , $v_{\|\|}=0$ km/s and $L_{z}=2.4\\times 10^{4}\\,M_{\\odot }$ Mpc km/s.", "We again emphasize that the system can be arbitrarily rescaled using eq:5 without changing the results.", "On the left, the two cores are in phase.", "They merge after approximately one free fall time, $t_{\\text{ff}}\\simeq 0.94$ Mpc/km s, and form a new excited solitonic core within roughly one oscillation period.", "The core's frequency $f\\simeq 8\\,$ km/Mpc/s, implies that it consists of only $70\\%$ of the initial mass [18] whereas approximately $30\\,$ % of the initial total mass was radiated off by gravitational cooling.", "This estimate is confirmed by the evolution of the total core mass $M_{c}=M_{c,1}+M_{c,2}$ and the total mass $M$ shown in fig:8.", "Initially, $M_{c}\\simeq \\frac{1}{4}M$ as expected, decreasing roughly by 30% during the merger.", "After a while, the ejected mass reaches the sponge and is absorbed.", "This does not alter the results, since in all conducted runs, the ejected mass is roughly an order of magnitude above the escape velocity $v_{\\text{esc}}=\\sqrt{2GM/r}$ and will not fall back onto the core.", "Figure: Slice through the symmetry plane of a representative ellipsoid.", "Its density is color-coded while arrows denote the strength and direction of its velocity field.", "It roughly forms closed elliptical orbits.In the case of solitonic cores with equal mass ($\\mu = 1$ ) but opposite phase ($\\Phi =\\pi $ ), the destructive interference gives rise to a repulsive interaction, causing the cores to bounce off each other several times before merging (cf.", "right panel of fig:3).", "This behaviour was also observed in [23].", "The arrows indicate the bounces which result in a noticeable compression of the individual cores.", "Radiation produced by each encouter results in a damping of the bounces and a decreasing amplitude of the compression.", "Eventually, the symmetry is broken by the accumulation of small numerical errors producing a slight phase shift, causing the cores to merge in the end.", "At later times, the evolution is qualitatively identical to the case with $\\Phi =0$ as can be seen by comparing the core and halo mass evolution in fig:8.", "Volume rendered images of both runs are shown in fig:3b.", "Especially in the upper panels, a noticeable eccentricity of the newly formed core can be recognized.", "These rotating ellipsoids are qualitatively those investigated in [41], [42], [22], [43], [44].", "In particular, their internal velocity fields roughly confine density distributions on elliptical orbits.", "A slice through a representative ellipsoid is shown in fig:3c.", "Further analysis will be the subject of future work.", "We tested the sensitivity of the repulsive interaction to small deviations from exact phase opposition by considering a phase difference $\\Phi =7/8\\pi $ .", "In this case, only a single bounce occurs before the cores merge.", "Similarly, for a mass ratio $\\mu =2$ and $\\Phi =\\pi $ the cores merge without any observable repulsion.", "These results suggest that in any realistic scenario absent finely tuned phase opposition and mass equality, repulsive behavior of colliding solitonic cores can be ignored for all practical purposes.", "Figure: Binary mergers with different mass ratios μ\\mu .", "Left: evolution of the core mass of the more massive core.", "Right: final radial density profiles.", "Solid lines represent fitted core profiles as defined in eq:2.", "The black line corresponds to r -3 r^{-3} as expected for the outer parts of an NFW profile.We conducted a series of binary mergers spanning the parameter space $(\\mu ,\\Phi ,L_{z})$ .", "For all runs, we set $v_{\|\|}=0$ km/s, $\\mu \\le 2$ , and $L_{z} \\le 7.2 \\times 10^{4}\\,M_{\\odot }$ Mpc km/s so that the cores are bounded and overlap when reaching the semi-minor axis.", "Our main result is that the core mass evolution is nearly independent of these parameters within the considered ranges.", "In all cases, the mass of the emerging core is approximately $70\\%$ of the sum of the progenitors' core masses.", "The core and total mass evolution of eight representative runs are shown in fig:8.", "The ratio between final core and total masses is approximately one fifth implying that $80\\%$ of the remaining bound mass resides in the solitonic core while the remainder has formed a diffuse halo around it.", "Note that due to the restriction to small angular momenta and mass ratios, the total energy varies only very little for all runs.", "The energy dependent final core masses $M_{c}(E)$ of the above runs are shown in fig:10 (run 1).", "Figure: Core mass as a function of the total energy and mass.", "The star indicates the relation for a single solitonic core.", "Run 1 denotes the simulations with almost equal total energy for different angular momenta and phases.", "Runs 2 and 3 show the dependence on mass ratio μ\\mu and total energy EE, respectively.", "Multiple core mergers are shown as run 4 (cf.", "sec:multimerge).", "See main text for details.Assuming a constant fraction of final to initial core masses of $\\sim 70\\%$ even for $\\mu \\ne 1$ implies that the final core is less massive than the more massive progenitor if $\\mu \\gtrsim 7/3$ .", "We therefore expect the change of $M_c$ of the more massive core to saturate at roughly this mass ratio.", "This is qualitatively confirmed by our simulations.", "For $\\mu \\gtrsim 2$ , the less massive core is completely disrupted and forms a diffuse halo.", "fig:9 shows the core mass evolution for different mass ratios (left).", "Here, the initial core mass corresponds to the more massive core.", "On the right, the final radial density profiles can be seen.", "They consist of a solitonic core well fitted by eq:2 and a shallow outer tail.", "Interestingly, the tails in all cases approximately follow a power law decline with a logarithmic slope of roughly $-3$ as expected for the outer parts of a Navarro-Frenk-White (NFW) halo profile.", "This behavior is consistent with the results of [24] but finding NFW-like halos already in the case of binary mergers suggests that it may be more robust than previously expected.", "Figure: Density distribution of a multimerger simulation with 13 halos at different times.The fitted core masses are mildly energy dependent as can be seen in fig:10 (run 2).", "They very broadly follow a power law with McM = 656(\|E\|M3)1/4M1/2km-1/2s1/2.", "In fig:10, the final core mass $M_{c}$ is normalized to the initial total mass $M$ in order to obtain an invariant relation with respect to the scaling properties given in eq:9.", "For a single solitonic core, $M_{c}/M\\simeq 0.237$ and $\|E\|/M^{3}\\simeq 1.7\\times 10^{-14}M_{\\odot }^{-2}\\text{km}^{2}\\text{s}^{-2}$ as indicated by the black star in the upper right corner.", "This point is consistent with eq:20 since a single core is the limit of infinite mass ratio.", "A single core is the ground state solution of the SP system.", "It is therefore the point of minimum energy and maximum core mass per total mass.", "Finally, we conducted a series of runs with $\\Phi =0$ , $\\mu =1$ , $L_{z}=0$ and varying $d$ and $v_{\|\|}$ over a wide range of energies.", "The fitted final core masses are collectively shown in fig:10 (run 3).", "The dashed line corresponds to McM = 46.7(\|E\|M3)1/6M1/3km-1/3s1/3 , indicating a weaker energy dependence for $\\mu =1$ than for larger mass ratios.", "In conclusion, our results for binary mergers show consistently that the final core mass does not depend on initial phase difference but only on mass ratio, total initial mass, and total energy of the system.", "It depends on angular momentum, relative distance and velocity only via the total energy.", "Figure: Final radial density profiles for all conducted multimerger runs.", "Solid lines represent fitted core profiles as defined in eq:2.", "The black line corresponds to r -3 r^{-3} as expected for the outer parts of an NFW profile." ], [ "Mergers of Multiple Cores", "In order to study more complex, non-equlibrium problems we follow [24] and investigate mergers of multiple cores.", "From our previous analysis we know that the merging time of binaries is negligible with respect to the typical free-fall time.", "We can therefore safely assume that a multimerger consists of a series of binary mergers within a deeper gravitational well.", "For all runs, we draw halo masses from a Gaussian distribution within the $2\\sigma $ -band around a chosen average halo mass.", "We then place the halos uniformly inside the central numerical domain, rejecting positions that would result in an overlap of halos or close proximity to the outer sponge.", "Rejected halo positions are redrawn until acceptable.", "Halos are initialized with random phases.", "We simulated multimergers of up to 13 halos.", "As a typical example, fig:13 shows the volume rendered images of a multimerger with 13 halos at three different times.", "The final radial density profiles for all runs are presented in fig:12.", "As in the case of binary mergers and in full agreement with [24], their central regions can be fitted with a solitonic core profile, eq:2, while the tails fall off like $r^{-3}$ consistent with the outer profile of an NFW halo.", "The final core masses are summarized in fig:10 (run 4).", "We cannot confirm the $M_{c}\\sim (E/M)^{1/2}$ scaling shown in [24] which may in part be a consequence of the fact that, in contrast with their analysis, all results in fig:12 are normalized to the initial total mass $M$ .", "This eliminates any scaling with energy originating only from the scale invariance of the SP system, making the results more sensitive to the intrinsic energy dependence of multimergers.", "We verified that this discrepancy is unrelated to the initial phase shifts of individual halos." ], [ "Conclusions", "We presented an investigation of merging solitonic halo cores in full three-dimensional simulations of the Schrödinger-Poisson (SP) equations without assuming any symmetries.", "These cores have been predicted to form in the center of ultra-light axion dark matter halos.", "Their structure is identical to Newtonian oscillaton solutions also known as boson stars.", "Our results demonstrate a number of robust features of binary core mergers.", "Qualitatively, bound systems rapidly merge within roughly one oscillation period of the emerging core after approaching to a distance at which the characteristic core radii overlap.", "It was shown in [45] that luminous matter cannot follow these extreme dynamics and is expelled from the gravitational potential.", "During this dynamical phase, gravitational cooling is most efficient and essentially determines the loss of mass and angular momentum of the merged core, while continuing to dampen its excitations during the ensuing several oscillation periods.", "One exception is the case of perfect phase opposition and equal masses in which case the cores initially repel each other, leading to a bouncing behavior until small accumulated phase differences again cause a rapid merger on a dynamical time scale.", "Owing to the fine tuning required for this situation, we do not consider it relevant in the context of cosmology.", "The mass of the emerging core does not directly depend on the binary angular momentum, initial distance, and phase shift between the solitonic cores.", "It does depend weakly on their mass ratio and total energy.", "The mass of the more massive core can only be enhanced by binary mergers with mass ratio $\\mu <7/3$ .", "Otherwise, the smaller core is completely disrupted and forms an NFW-like halo around the more massive one.", "Neither for the binary mergers nor for the sample of multiple core mergers we were able to reproduce the scaling of core mass with total energy and mass, $M_{c}\\sim (E/M)^{1/2}$ , reported in [24].", "After normalizing our results to equal total mass using the scale invariance of the SP equations in order to eliminate spurious scaling behavior, we find no convincing evidence for a universal scaling of core mass with total energy.", "More detailed analysis with larger ranges of $M_c$ and $E$ and comparison to cosmological simulations of the SP system are needed to further elucidate this discrepancy.", "The final states of both the binary and multimergers are roughly spherical symmetric.", "We confirm that their radial density profiles consist of a solitonic core well modeled by eq:2 and an NFW-like outer region falling off as $r^{-3}$ [24].", "If the system is initialized with non-zero total angular momentum, we qualitatively recover the rotating ellipsoidal cores studied in [41], [42], [22], [43], [44].", "Our results are useful for a refined modeling of the properties of halo cores in FDM cosmologies, for instance in stochastic merger tree realizations of the halo and subhalo population [46].", "This approach complements other simplified structure formation models that include the effects of the linear transfer function [47], [48] and mass-dependent collapse barrier [2], [3], [49] but neglect the presence of solitonic cores.", "They might also help to understand the relation of core and halo masses in cosmological FDM simulations [24].", "Eventually, the consequences of solitonic cores for galaxy evolution will have to be better understood in order to tighten the constraints on ultra-light axion masses from reionization, the UV luminosity function, or halo substructure.", "Simulations of more realistic cosmological setups including baryonic physics are in preparation.", "We thank C. Behrens, X.", "Du, D.J.E.", "Marsh, and J. Veltmaat for helpful discussions.", "The simulations were performed with resources provided by the North-German Supercomputing Alliance (HLRN)." ] ]	1606.05151
[ [ "On certain product of Banach modules" ], [ "Abstract Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A-$bimodule.", "Then the $\\ell^1-$direct sum $A\\times B$ equipped with the multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\\cdot b_2+b_1\\cdot a_2+b_1b_2),~~ (a_1, a_2\\in A, b_1, b_2\\in B)$$ is a Banach algebra denoted by $A\\bowtie B$.", "Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework.", "We obtain characterizations of bounded approximate identities, spectrum, and topological center of this product.", "This provides a unified approach for obtaining some known results of both module extensions and Lau product of Banach algebras." ], [ "Introduction", "Let $A$ and $B$ be Banach algebra and $B$ is a Banach $A-$ bimodule, we say that $B$ is an algebraic Banach $A-$ bimodule if $a\\cdot (b_1b_2)=(a\\cdot b_1)b_2,\\quad (b_1b_2)\\cdot a=b_1(b_2\\cdot a),\\quad (b_1\\cdot a)b_2=b_1(a\\cdot b_2),$ for each $b_1, b_2\\in B$ and $a\\in A$ .", "Then the Cartesian product $A \\times B$ with the algebra multiplication $(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\\cdot b_2+b_1\\cdot a_2+b_1b_2),$ and with the norm $\\Vert (a,b)\\Vert =\\Vert a\\Vert +\\Vert b\\Vert $ becomes a Banach algebra provided $\\Vert a\\cdot b\\Vert \\le \\Vert a\\Vert \\Vert b\\Vert $ , which we denote it by $A\\bowtie B$ .", "We note that if we identify $A\\times \\lbrace 0\\rbrace $ with $A$ , and $\\lbrace 0\\rbrace \\times B$ with $B$ , in $A\\bowtie B$ , then $B$ is a closed ideal while $A$ is a closed subalgebra of $A\\bowtie B$ , and $(A\\bowtie B)/B$ is isometrically isomorphic to $A$ .", "In other words, $A\\bowtie B$ is a strongly splitting Banach algebra extension of $A$ by $B$ .", "Besides giving a new method of constructing Banach algebras, the product $\\bowtie $ has relevance with the following known products.", "(Direct product of two Banach algebras) Let $A$ and $B$ be Banach algebra.", "If we define $a\\cdot b=b\\cdot a=0$ , then $B$ is an algebraic Banach $A$ -bimodule and $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2).$ Therefore, $A\\bowtie B$ is the direct product $A\\times _1 B$ .", "(The module extensions) Let $X$ be a Banach $A$ -bimodule.", "Define $x_1x_2=0$ , then $X$ is an algebraic Banach $A$ -bimodule, and $(a_1,x_1)(a_2,x_2)=(a_1a_2,a_1\\cdot x_2+x_1\\cdot a_2).$ Therefore, $A\\bowtie X$ is the module extension $A\\oplus _1 X$ .", "Module extensions are known as a rich source of (counter-)examples in various situations in abstract harmonic analysis and functional analysis, [5].", "($\\theta -$ Lau product of Banach algebras) Let $A$ and $B$ be Banach algebra and $\\theta \\in \\Delta (A)$ , the set of all non zero multiplicative linear functional on $A$ .", "Then $B$ with a module actions given by $a\\cdot b=b\\cdot a=\\theta (a)b$ is an algebraic Banach $A$ -bimodule and $(a_1,b_1)(a_2,b_2)=(a_1a_2, \\theta (a_1)b_2+\\theta (a_2)b_1+b_1b_2).$ Thus $A\\bowtie B$ is the $\\theta $ -Lau product $A{~}_{\\theta }\\!\\!\\times B$ .", "This product was introduced by Lau [2] for certain class of Banach algebras and followed by Sangani Monfared [4] for the general case.", "An elementary very familiar example is the case that $A=\\mathbb {C}$ with $\\theta $ as the identity character $i$ that we get the unitization $B^\\sharp =\\mathbb {C}{~}_{\\theta \\!\\!", "}\\times B$ of $B$ .", "($T-$ Lau product of Banach algebras) Let $A$ and $B$ be Banach algebra and $T:A\\rightarrow B$ be an algebra homomorphism with $\\Vert T\\Vert \\le 1$ .", "Define $a\\cdot b=T(a)b=b\\cdot a$ .", "Then $B$ is an algebraic Banach $A$ -bimodule and $(a_1,b_1)(a_2,b_2)=(a_1a_2,T(a_1)b_2+b_1T(a_2)+b_1b_2).$ Thus $A\\bowtie B$ is the $T-$ Lau product $A{~}_T\\!\\!\\times B$ .", "This type of product was first introduced by Bhatt and Dabhi for the case where $B$ is commutative and was extended by Javanshiri and Nemati for the general case [1].", "The purpose of the present note is to determine the Gelfand space of $A\\bowtie B$ which turns out to be non trivial even though $A\\bowtie B$ need not be commutative and to discuss the topological center of $A\\bowtie B$ .", "These topics are central to the general theory of Banach algebras." ], [ "Main results", "Let $B$ be a Banach $A-$ bimodule, we recall that $B$ is called symmetric if $a\\cdot b=b\\cdot a$ for all $a\\in A$ and $b\\in B$ .", "We start with the following propositions which characterize the basic properties of $A\\bowtie B$ in terms of $A$ and $B$ .", "These results extend related results in [1], [4].", "Proposition 2.1 [3] Let $B$ be an algebraic Banach $A$ -bimodule.", "Then $A\\bowtie B$ is commutative if and only if $B$ is a symmetric Banach $A$ -bimodule and both $A$ and $B$ are commutative.", "Proposition 2.2 [3] Let $B$ be an algebraic Banach $A$ -bimodule.", "Then $(a_0, b_0)$ is an identity for $A\\bowtie B$ if and only if $a_0$ is an identity for $A$ , $b_0\\cdot a=a\\cdot b_0=0$ for all $a\\in A$ and $a_0\\cdot b+b_0b=b\\cdot a_0+bb_0=b$ for all $b\\in B$ .", "Proposition 2.3 [3] Let $B$ be an algebraic Banach $A$ -bimodule.", "Then $\\lbrace (a_\\alpha , b_\\alpha )\\rbrace $ is a bounded left approximate identity for $A\\bowtie B$ if and only if $\\lbrace a_\\alpha \\rbrace $ is a bounded left approximate identity for $A$ , $\\Vert b_\\alpha \\cdot a\\Vert \\rightarrow 0$ for all $a\\in A$ and $a_\\alpha \\cdot b+b_\\alpha b\\rightarrow b$ for all $b\\in B$ .", "The dual of the space $A\\bowtie B$ can be identified with $A^\\times B^$ in the natural way $(\\varphi , \\psi )(a,b)=\\varphi (a)+\\psi (b)$ .", "The dual norm on $A^\\times B^$ is of course the maximum norm $\\Vert (\\varphi ,\\psi )\\Vert =\\max \\lbrace \\Vert \\varphi \\Vert , \\Vert \\psi \\Vert \\rbrace $ .", "The following result identifies the Gelfand space of $A\\bowtie B$ .", "This is a generalization of [1] and [4].", "Proposition 2.4 [3] Let $B$ be an algebraic Banach $A$ -bimodule.", "If $E:=\\lbrace (\\varphi ,0) : \\varphi \\in \\Delta (A)\\rbrace $ and $F:=\\lbrace (\\varphi , \\psi ) : \\varphi \\in \\Delta (A)\\cup \\lbrace 0\\rbrace ,~ \\psi \\in \\Delta (B),\\text{ and} ~a\\cdot \\psi =\\psi \\cdot a=\\varphi (a)\\psi ~~ \\forall a\\in A \\rbrace ,$ then $\\Delta (A\\bowtie B)=E\\cup F$ .", "Corollary 2.5 Let $A$ and $B$ be commutative Banach algebras and $B$ is an algebraic Banach $A$ -bimodule which is also symmetric.", "Then $A\\bowtie B$ is semisimple if and only if both $A$ and $B$ are semisimple.", "Let $X$ be a Banach $A$ -bimodule, for $a\\in A, x\\in X, x^\\in X^, a^{}\\in A^{}$ and $x^{}\\in X^{}$ we define $\\begin{array}{lcl}(x^{}\\circ a^{})(x^)=x^{}(a^{}\\circ x^),&\\quad \\qquad &(a^{}\\circ x^{})(x^)=a^{}(x^{}\\circ x^)\\\\(a^{*}\\circ x^)(x)=a^{*}(x^\\circ x),&&(x^{*}\\circ x^)(a)=x^{*}(x^\\circ a)\\\\(x^\\circ x)(a)=x^(x\\cdot a),& &(x^\\circ a)(x)=x^(a\\cdot x).\\end{array}$ Clearly, for each $a^{}\\in A^{}$ and $x^{}\\in X^{}$ the mappings $b^{}\\rightarrow b^{}\\circ x^{}:A^{}\\rightarrow X^{}$ and $y^{}\\rightarrow y^{}\\circ a^{}:X^{}\\rightarrow X^{}$ are w$^$ -w$^$ -continuous.", "The first topological centres of module actions of $A$ on $X$ may therefore be defined as $&Z^1_A(X^{})=\\lbrace x^{}\\in X^{} : a^{}\\rightarrow x^{}\\circ a^{}\\mbox{ is w$^$-w$^$-continuous}\\rbrace ,\\\\&Z^1_X(A^{})=\\lbrace a^{}\\in A^{}~:~x^{}\\rightarrow a^{}\\circ x^{}\\mbox{ is w$^$-w$^$-continuous}\\rbrace .$ If we consider $X=A$ with the natural $A$ -bimodule structure then we obtain the first Arens product on $A^{}$ .", "In this case we write $Z^1(A^{})$ instead $Z^1_A(A^{})$ .", "The Banach algebra $A$ is called Arens regular if $Z^1(A^{})=A^{}$ .", "To state our next result we note that if $B$ is an algebraic Banach $A$ -bimodule then $B^{}$ is an algebraic Banach $A^{}$ -bimodule, when $A^{}$ and $B^{}$ are equipped with their first Arens products.", "Theorem 2.6 [3] Let $B$ be an algebraic Banach $A$ -bimodule.", "Suppose that $A^{}, B^{}$ , and $(A\\bowtie B)^{}$ are equipped with their first Arens products.", "Then $(A\\bowtie B)^{}\\cong A^{}\\bowtie B^{}\\qquad (\\text{isometric~isomorphism}).$ If $B$ is Arens regular then $&Z^1((A\\bowtie B)^{})=\\left(Z^1(A^{})\\cap Z^1_B(A^{})\\right)\\times Z^1_A(B^{**})$" ] ]	1606.04874
[ [ "Quivers with subadditive labelings: classification and integrability" ], [ "Abstract Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg's definition for undirected graphs.", "In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity.", "In this paper, we classify all quivers with subadditive labelings.", "We conjecture them to exhibit a certain form of integrability, namely, as the $T$-system dynamics proceeds, the values at each vertex satisfy a linear recurrence.", "Conversely, we show that every quiver integrable in this sense is necessarily one of the $19$ items in our classification.", "For the quivers of type $\\hat A \\otimes A$ we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called \\emph{Goncharov-Kenyon Hamiltonians}.", "We also consider tropical $T$-systems of type $\\hat A \\otimes A$ and explain how affine slices exhibit solitonic behavior, i.e.", "soliton resolution and speed conservation.", "Throughout, we conjecture how the results in the paper are expected to generalize from $\\hat A \\otimes A$ to all other quivers in our classification." ], [ "Introduction", "A quiver $Q$ is a directed graph without 1-cycles (i.e.", "loops) and directed 2-cycles.", "For a vertex $v$ of a quiver, one can define a certain operation called a mutation, which produces a new quiver denoted $\\mu _v(Q)$ (see Definition REF ).", "We say that a quiver is bipartite if its underlying graph is bipartite, in which case we say that a map $ {\\epsilon }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\lbrace 0,1\\rbrace ,\\ v\\mapsto {\\epsilon }_v$ is a bipartition if for every edge $u\\rightarrow v$ of $Q$ we have ${\\epsilon }_u\\ne {\\epsilon }_v$ .", "Here ${ { \\operatorname{Vert}}(Q)}$ is the set of vertices of $Q$ .", "It is clear from Definition REF that $\\mu _u$ and $\\mu _v$ commute if $u,v$ are not connected by an edge in $Q$ .", "Therefore, we can define $\\mu _\\circ =\\prod _{u:{\\epsilon }_u=0} \\mu _u;\\quad \\mu _\\bullet =\\prod _{v:{\\epsilon }_v=1} \\mu _v.$ We say that $Q$ is recurrent if $\\mu _\\circ (Q)=\\mu _\\bullet (Q)=Q^{{ \\operatorname{op}}}$ where $Q^{ \\operatorname{op}}$ is the same quiver as $Q$ but with all the arrows reversed.", "Let $Q$ be a bipartite recurrent quiver.", "Denote ${ \\mathbf {x}}:=\\lbrace x_v\\rbrace _{v\\in { { \\operatorname{Vert}}(Q)}}$ to be the set of indeterminates, one for each vertex of $Q$ , and let $\\mathbb {Q}({ \\mathbf {x}})$ be the field of rational functions in these variables.", "The $T$ -system associated with $Q$ is a family $T_v(t)$ of elements of $\\mathbb {Q}({ \\mathbf {x}})$ satisfying the following relations for all $v\\in { { \\operatorname{Vert}}(Q)}$ and all $t\\in \\mathbb {Z}$ : $T_v(t+1)T_v(t-1)=\\prod _{u\\rightarrow v} T_u(t)+\\prod _{v\\rightarrow w} T_w(t).$ Here the products are taken over all arrows connecting the two vertices.", "It is clear that the parity of $t+{\\epsilon }_v$ in all of the terms is the same, so the $T$ -system associated with $Q$ splits into two completely independent ones.", "Without loss of generality we may consider only one of them.", "From now on we assume that the $T$ -system is defined only for $t\\in \\mathbb {Z}$ and $v\\in { { \\operatorname{Vert}}(Q)}$ satisfying $t+{\\epsilon }_v\\equiv 0\\pmod {2}.$ The $T$ -system is set to the following initial conditions: $T_v({\\epsilon }_v)=x_v$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "Let us say that the $T$ -system associated with a recurrent quiver $Q$ is integrable if for every vertex $v\\in { { \\operatorname{Vert}}(Q)}$ , there exists an integer $N$ and elements $J_0,J_1,\\dots ,J_{N}\\in \\mathbb {Q}({ \\mathbf {x}})$ satisfying $J_0,J_N\\ne 0$ and $\\sum _{j=0}^{N} J_j T_v(t+2j)=0$ for all $t\\in \\mathbb {Z}$ with $t+{\\epsilon }_v$ even.", "We also refer to a recurrent quiver $Q$ as Zamolodchikov integrable if the associated $T$ -system is integrable.", "If the recurrence has the form $T_v(t+2N)=T_v(t)$ for all $t\\in \\mathbb {Z}$ and $v\\in { { \\operatorname{Vert}}(Q)}$ , then we call $Q$ Zamolodchikov periodic.", "Just as in the periodic case, we call a quiver Zamolodchikov integrable when the bipartite $T$ -system is integrable.", "More general notions of $T$ -systems can be found in [27].", "Zamolodchikov periodicity for the case when $Q$ is a tensor product of two finite $ADE$ Dynkin diagrams has been studied extensively (see [41], [30], [23], [24], [9], [11], [13], [40], [36]) and was proven in full generality in [19] and later in [16], [17], where tropical $Y$ -systems played a major role.", "By analyzing tropical $T$ -systems (see Part ), we have classified Zamolodchikov periodic quivers in [12], where we showed that these are exactly the quivers admitting a strictly subadditive labeling (Definition REF ).", "Besides thermodynamic Bethe ansatz [41], $T$ -systems and $Y$ -systems arise naturally in a lot of different contexts in physics and representation theory, e.g.", "[24], [21], [31], [28], [10], [22], [26], see [25] for a survey.", "Assem, Reutenauer and Smith [1] showed that the affine Dynkin diagrams of types ${\\hat{A}}$ and ${\\hat{D}}$ are Zamolodchikov integrable, and later Keller and Scherotzke [20] extended this result to all affine Dynkin diagrams.", "Conversely, it was shown in [1] that if every vertex of a Zamolodchikov integrable quiver $Q$ is either a source or a sink then $Q$ is necessarily an affine Dynkin diagram.", "In Sections and we prove (Theorem REF ) that if a bipartite recurrent quiver is Zamolodchikov integrable then it admits a subadditive labeling (see Definition REF ).", "We then classify (Theorem REF ) quivers that admit subadditive labelings in Part .", "We conjecture all of them to be Zamolodchikov integrable (Conjecture REF ).", "When $Q$ is a tensor product (see Definition REF ) of type ${\\hat{A}}\\otimes A$ , it was shown in [29] that $Q$ is Zamolodchikov integrable.", "In Sections -, we express the recurrence coefficients $J_1,\\dots ,J_N$ for the vertices of such $Q$ in terms of partition functions of domino tilings on the cylinder, called Goncharov-Kenyon Hamiltonians.", "See Theorem REF and Corollary REF .", "In Section we show that the above Goncharov-Kenyon Hamiltonians belong to the upper cluster algebra.", "In Part , we analyze the tropical $T$ -system associated with quivers admitting a subadditive labeling.", "We show (Corollary REF ) that when $t\\gg 0$ or $t\\ll 0$ , every affine slice of the tropical $T$ -system moves with some constant speed.", "We explain how this can be seen as soliton resolution in Section , and then we proceed to show speed conservation in Section : for the quiver of type ${\\hat{A}}_{2n-1}\\otimes A_m$ , the speeds of the solitons at $t\\gg 0$ are equal to the speeds of solitons at $t\\ll 0$ after one applies a diagram automorphism to $A_m$ .", "See Example REF for an illustration of these solitonic phenomena.", "Finally, in Sections and we conjecture most of our results for all other quivers in our classification." ], [ "Bigraphs", "In [35] Stembridge studies admissible $W$ -graphs for the case when $W=I(p)\\times I(q)$ is a direct product of two dihedral groups.", "These $W$ -graphs encode the structure of representations of Iwahori-Hecke algebras, and were first introduced by Kazhdan and Lusztig in [18].", "The following definitions are adapted from [35] with slight modifications.", "A bigraph is an ordered pair of simple (undirected) graphs $(\\Gamma , \\Delta )$ which share a common set of vertices $V:={ \\operatorname{Vert}}(\\Gamma )={ \\operatorname{Vert}}(\\Delta )$ and do not share edges.", "A bigraph is called bipartite if there is a map ${\\epsilon }:V\\rightarrow \\lbrace 0,1\\rbrace $ such that for every edge $(u,v)$ of $\\Gamma $ or of $\\Delta $ we have ${\\epsilon }_u\\ne {\\epsilon }_v$ .", "There is a simple one-to-one correspondence between bipartite quivers and bipartite bigraphs.", "Namely, to each bipartite quiver $Q$ with a bipartition ${\\epsilon }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\lbrace 0,1\\rbrace $ we associate a bigraph $G(Q)=(\\Gamma (Q),\\Delta (Q))$ on the same set of vertices defined as follows: $\\Gamma (Q)$ contains an (undirected) edge $(u,v)$ if and only if $Q$ contains a directed edge $u\\rightarrow v$ with ${\\epsilon }_u=0,{\\epsilon }_v=1$ ; $\\Delta (Q)$ contains an (undirected) edge $(u,v)$ if and only if $Q$ contains a directed edge $u\\rightarrow v$ with ${\\epsilon }_u=1,{\\epsilon }_v=0$ .", "Similarly, we can direct the edges of any given bipartite bigaph $G$ to get a bipartite quiver $Q(G)$ .", "It is convenient to think of $(\\Gamma , \\Delta )$ as of a single graph with edges of two colors: red for the edges of $\\Gamma $ and blue for the edges of $\\Delta $ .", "Definition 1.1 Let $S$ and $T$ be two bipartite undirected graphs.", "Then their tensor product $S\\otimes T$ is a bipartite bigraph $G=(\\Gamma ,\\Delta )$ with vertex set ${ \\operatorname{Vert}}(S)\\times { \\operatorname{Vert}}(T)$ and the following edge sets: for each edge $\\lbrace u,u^{\\prime }\\rbrace \\in S$ and each vertex $v\\in T$ there is an edge between $(u,v)$ and $(u^{\\prime },v)$ in $\\Gamma $ ; for each vertex $u\\in S$ and each edge $\\lbrace v,v^{\\prime }\\rbrace \\in T$ there is an edge between $(u,v)$ and $(u,v^{\\prime })$ in $\\Delta $ ; An example of a tensor product is given in Figure REF .", "Figure: A tensor product of a square (type A ^ 3 {\\hat{A}}_3) and a single edge (type A 2 A_2)." ], [ "Reformulation of the dynamics in terms of bigraphs", "Let $G=(\\Gamma ,\\Delta )$ be a bipartite bigraph with a vertex set $V$ .", "Then the associated $T$ -system for $G$ is defined as follows: $T_v(t+1)T_v(t-1)&=&\\prod _{(u,v)\\in \\Gamma } T_u(t)+\\prod _{(v,w)\\in \\Delta } T_w(t);\\\\T_v({\\epsilon }_v) &=& x_v.$ It is easy to see that this system is equivalent to the corresponding system defined for $Q(G)$ in the Introduction." ], [ "Finite and affine $ADE$ Dynkin diagrams and their Coxeter numbers", "By a finite $ADE$ Dynkin diagram we mean a Dynkin diagram of type $A_n,D_n,E_6,E_7,$ or $E_8$ .", "An affine $ADE$ Dynkin diagram is a Dynkin diagram of type ${\\hat{A}}_n,{\\hat{D}}_n,{\\hat{E}}_6,{\\hat{E}}_7,$ or ${\\hat{E}}_8$ , see Figure REF .", "The following characterization of finite and affine $ADE$ Dynkin diagrams is due to Vinberg [39]: Theorem 1.2 Let $G=(V,E)$ be an undirected graph with possibly multiple edges.", "Then: $G$ is a finite $ADE$ Dynkin diagram if and only if there exists a map $\\nu :V\\rightarrow \\mathbb {R}_{>0}$ such that for all $v\\in V$ , $2\\nu (v)>\\sum _{(u,v)\\in E} \\nu (u).$ $G$ is an affine $ADE$ Dynkin diagram if and only if there exists a map $\\nu :V\\rightarrow \\mathbb {R}_{>0}$ such that for all $v\\in V$ , $2\\nu (v)=\\sum _{(u,v)\\in E} \\nu (u).$ The values of $\\nu $ satisfying (REF ) are given in Figure REF .", "Figure: Affine ADEADE Dynkin diagrams and their additive labelings.For each finite $ADE$ Dynkin diagram $\\Lambda $ there is an associated integer $h(\\Lambda )$ called Coxeter number.", "We list Coxeter numbers of finite $ADE$ Dynkin diagrams in Figure REF .", "If ${\\hat{\\Lambda }}$ is an affine Dynkin diagram, we set $h({\\hat{\\Lambda }})=\\infty $ .", "Table: Coxeter numbers of finite ADEADE Dynkin diagramsIt is well-known that the Coxeter number has a nice interpretation in terms of eigenvalues of the adjacency matrix: Proposition 1.3 If $\\Lambda $ is a finite $ADE$ Dynkin diagram then the dominant eigenvalue of its adjacency matrix equals $2\\cos (\\pi /h(\\Lambda ))$ ; if ${\\hat{\\Lambda }}$ is an affine $ADE$ Dynkin diagram then the dominant eigenvalue of its adjacency matrix equals 2.", "$\\Box $ In particular, the second claim justifies setting $h({\\hat{\\Lambda }}):=\\infty $ ." ], [ "Subadditive labelings", "Let $G=(\\Gamma ,\\Delta )$ be a bipartite bigraph on vertex set $V$ .", "A labeling of its vertices is a function $\\nu : V\\rightarrow \\mathbb {R}_{>0}, \\;$ which assigns to each vertex $v$ of $G$ a positive real label $\\nu (v)$ .", "Definition 1.4 A labeling $\\nu :V\\rightarrow \\mathbb {R}_{>0}$ is called strictly subadditive if for any vertex $v\\in V$ , $2 \\nu (v) > \\sum _{(u,v)\\in \\Gamma } \\nu (u), \\; \\text{ and } \\; 2 \\nu (v) > \\sum _{(v,w)\\in \\Delta } \\nu (w).$ subadditive if for any vertex $v\\in V$ , $2 \\nu (v) \\ge \\sum _{(u,v)\\in \\Gamma } \\nu (u), \\; \\text{ and } \\; 2 \\nu (v) > \\sum _{(v,w)\\in \\Delta } \\nu (w).$ weakly subadditive if for any vertex $v\\in V$ , $2 \\nu (v) \\ge \\sum _{(u,v)\\in \\Gamma } \\nu (u), \\; \\text{ and } \\; 2 \\nu (v) \\ge \\sum _{(v,w)\\in \\Delta } \\nu (w).$ Examples of each type can be found in Figure REF .", "Figure: Different kinds of labelingsStrictly subadditive, subadditive and weakly subadditive labelings of quivers have been introduced in [29].", "The terminology is motivated by Vinberg's subadditive labelings [39] for non-directed graphs (see Theorem REF )." ], [ "Quivers", "Definition 1.5 For a vertex $v$ of $Q$ one can define the quiver mutation $\\mu _v$ at $v$ as follows: for each pair of edges $u \\rightarrow v$ and $v \\rightarrow w$ create an edge $u \\rightarrow w$ ; reverse the direction of all edges adjacent to $v$ ; if some directed 2-cycle is present, remove both of its edges; repeat until there are no more directed 2-cycles.", "Let us denote the resulting quiver $\\mu _v(Q)$ .", "See Figure REF for an example of each step.", "Figure: Mutating a quiver at vertex aa.Now, let $Q$ be a bipartite quiver.", "Recall that $\\mu _\\circ $ (resp., $\\mu _\\bullet $ ) is the simultaneous mutation at all white (resp., all black) vertices of $Q$ , and that $Q$ is recurrent if $\\mu _\\circ (Q)=\\mu _\\bullet (Q)=Q^{\\operatorname{op}}$ .", "As we have observed in [12], this property translates nicely into the language of bigraphs: Corollary 1.6 A bipartite quiver $Q$ is recurrent if and only if the associated bipartite bigraph $G(Q)$ has commuting adjacency matrices $A_\\Gamma ,A_\\Delta $ .", "We define three variations of Stembridge's admissible $ADE$ bigraphs (see [35]): Definition 1.7 Let $G=(\\Gamma ,\\Delta )$ be a bipartite bigraph, and assume that the adjacency $\|V\|\\times \|V\|$ matrices $A_\\Gamma $ and $A_\\Delta $ of $\\Gamma $ and $\\Delta $ commute.", "In this case we encode the three definitions in Table REF .", "For instance, $G$ is an affine $\\boxtimes $ finite $ADE$ bigraph if each connected component of $\\Gamma $ is an affine $ADE$ Dynkin diagram and each connected component of $\\Delta $ is a finite $ADE$ Dynkin diagram.", "We similarly define the notions of admissible and affine $\\boxtimes $ affine $ADE$ bigraphs.", "Table: Three types of bigraphsThe following fact is an easy consequence of [35]: Lemma 1.8 Let $G=(\\Gamma ,\\Delta )$ be a bigraph and assume that the adjacency matrices $A_\\Gamma ,A_\\Delta $ commute.", "Then the dominant eigenvalues of all components of $\\Gamma $ are equal to the same value $\\lambda _\\Gamma $ , and the dominant eigenvalues of all components of $\\Delta $ are equal to the same value $\\lambda _\\Delta $ .", "Matrices $A_\\Gamma $ and $A_\\Delta $ have a common dominant eigenvector $\\mathbf {v}$ such that $A_\\Gamma \\mathbf {v}={ \\lambda }_\\Gamma \\mathbf {v};\\quad A_\\Delta \\mathbf {v}={ \\lambda }_\\Delta \\mathbf {v}.$ $\\Box $ Corollary 1.9 Let $G=(\\Gamma ,\\Delta )$ be a bigraph and assume that the adjacency matrices $A_\\Gamma ,A_\\Delta $ commute, and assume that all connected components of $\\Gamma $ and of $\\Delta $ are either affine or finite $ADE$ Dynkin diagrams.", "Then all connected components of $\\Gamma $ have the same Coxeter number denoted $h(\\Gamma )$ , and all connected components of $\\Delta $ have the same Coxeter number denoted $h(\\Delta )$ .", "Combining Lemma REF , Definition REF , Definition REF , Vinberg's characterization (Theorem REF ), and Proposition REF , we get the following proposition, whose part (REF ) was shown in [12].", "The proof for parts (REF ) and (REF ) is completely analogous and we refer the reader to [12] for details.", "Proposition 1.10 Let $Q$ be a bipartite recurrent quiver $Q$ and $G(Q)=(\\Gamma ,\\Delta )$ be the corresponding bipartite bigraph.", "Then $Q$ admits a strictly subadditive labeling if and only if $G(Q)$ is an admissible $ADE$ bigraph; $Q$ admits a subadditive labeling which is not strictly subadditive if and only if $G(Q)$ is an affine $\\boxtimes $ finite $ADE$ bigraph; $Q$ admits a weakly subadditive labeling which is not subadditive if and only if $G(Q)$ is an affine $\\boxtimes $ affine $ADE$ bigraph.", "$\\Box $" ], [ "Zamolodchikov integrable quivers admit weakly subadditive labelings", "Recall that a bipartite recurrent quiver $Q$ is called Zamolodchikov integrable if for every vertex $v\\in { { \\operatorname{Vert}}(Q)}$ , there exists an integer $N$ and rational functions $J_0,\\dots ,J_N\\in \\mathbb {Q}({ \\mathbf {x}})$ such that $J_0,J_N\\ne 0$ and $\\sum _{j=0}^{N} J_j T_v(t+2j)=0$ for all $t\\in \\mathbb {Z}$ with $t+{\\epsilon }_v$ even.", "The following lemma is the first step towards the proof of Theorem REF Lemma 2.1 If a bipartite recurrent quiver $Q$ is Zamolodchikov integrable then $Q$ admits a weakly subadditive labeling.", "For $v\\in { { \\operatorname{Vert}}(Q)},t\\in \\mathbb {Z}$ , define a positive number $a(v,t):=T_v(2t+{\\epsilon }_v)\\mid _{{ \\mathbf {x}}:=1}$ to be the value of $T_v(2t+{\\epsilon }_v)$ if one substitutes $x_u:=1$ for all $u\\in { { \\operatorname{Vert}}(Q)}$ .", "By the Laurent Phenomenon (see [8]), the numbers $a(v,t)$ are integers.", "Note that, unlike $T_v(t)$ , the numbers $a(v,t)$ are defined for all $v,t$ , regardless of parity.", "Since $a(v,t)$ is always a positive integer, it is easy to see that the sequences $a(v,):=\\left(a(v,t)\\right)_{t\\in \\mathbb {Z}}$ are either simultaneously bounded or simultaneously unbounded (for all $v$ ).", "Assume for the sake of contradiction that for some vertex $v$ , the sequence $a(v,)$ is unbounded, but there is another vertex $u$ for which the sequence $a(u,)$ is bounded, say, $\|a(u,t)\|<C$ for all $t\\in \\mathbb {Z}$ .", "Since $Q$ is connected, we may assume that $u$ and $v$ are neighbors in $Q$ .", "Let $t$ be such that $a(v,t)>C^2$ .", "Then by the definition of the $T$ -system, we have $a(u,t+1)> \\frac{a(v,t)}{a(u,t)}>C,$ where the first inequality uses the fact that all the numbers involved are positive integers, hence each of them is at least 1.", "This leads to an immediate contradiction.", "If all the sequences are simultaneously bounded then they are periodic with the same period.", "This implies that the $T$ -system associated with $Q$ is periodic for any initial data, see [12].", "In particular, such $Q$ admits a strictly subadditive labeling by [12].", "Thus the only case left for us to consider is when the sequence $a(v,)$ is unbounded for every $v$ .", "We need to show that if $Q$ is Zamolodchikov integrable then $Q$ admits a weakly subadditive labeling.", "The way to find such a labeling is going to be very similar to the proof of [1].", "The fact that $Q$ is Zamolodchikov integrable implies that for each $v$ , the sequences $a(v,)$ satisfy a linear recurrence.", "Knowing that each of them is unbounded suggests using [1] that describes the asymptotic behavior of sequences $a(v,)$ .", "Before we state it, let us denote $A(k)\\approx B(k)$ for two functions of $k$ if their ratio tends to a positive constant as $k\\rightarrow \\infty $ .", "Lemma 2.2 (see [1]) Let $a(v,)$ be an unbounded sequence of positive integers satisfying a linear recurrence for each $v\\in { { \\operatorname{Vert}}(Q)}$ .", "Then there exist: an integer $p\\ge 1$ ; real numbers ${ \\lambda }(v,l)\\ge 1$ for each $v\\in { { \\operatorname{Vert}}(Q)},\\ l=0,\\dots ,p$ ; integers $d(v,l)\\ge 0$ for each $v\\in { { \\operatorname{Vert}}(Q)},\\ l=0,\\dots ,p$ ; a strictly increasing sequence $(n_k)_{k\\in \\mathbb {Z}_{\\ge 0}}$ of nonnegative integers such that the following things hold: for every $v\\in { { \\operatorname{Vert}}(Q)}$ and every $l=0,\\dots ,p$ , $a(v,pn_k+l)\\approx { \\lambda }(v,l)^{n_k} n_k^{d(v,l)}$ ; for every $v\\in { { \\operatorname{Vert}}(Q)}$ there exists $l=0,\\dots ,p$ such that ${ \\lambda }(v,l)>1$ or $d(v,l)\\ge 1$ ; for every $v\\in { { \\operatorname{Vert}}(Q)}$ we have ${ \\lambda }(v,0)={ \\lambda }(v,p)$ and $d(v,0)=d(v,p)$ .", "$\\Box $ Clearly, the sequences $a(v,)$ satisfy all the requirements of Lemma REF .", "For each $v\\in { { \\operatorname{Vert}}(Q)}$ , define ${ \\lambda }(v):=\\prod _{l=0}^{p-1} { \\lambda }(v,l)\\in \\mathbb {R}_{\\ge 1};\\quad d(v):=\\sum _{l=0}^{p-1} d(v,l)\\in \\mathbb {Z}_{\\ge 0}.$ For all $v\\in { { \\operatorname{Vert}}(Q)}$ and $t\\in \\mathbb {Z}$ , define $b(v,t):=\\prod _{l=0}^{p-1} a(v,t+l).$ Applying Lemma REF yields $b(v,pn_k)\\approx { \\lambda }(v)^{n_k} n_k^{d(v)}.$ By property (REF ) of Lemma REF , we have $a(v,pn_k)\\approx a(v,pn_k+p)$ for every $v\\in { { \\operatorname{Vert}}(Q)}$ and thus we can write $b(v,pn_k)^2 &\\approx & \\prod _{l=0}^{p-1} a(v,pn_k+l)a(v,pn_k+l+1) =\\prod _{l=0}^{p-1} T_v(2(pn_k+l)+{\\epsilon }_v)T_v(2(pn_k+l+1)+{\\epsilon }_v)\\\\&=&\\prod _{l=0}^{p-1} \\left(\\prod _{u\\rightarrow v}T_u(2(pn_k+l)+{\\epsilon }_v+1)+\\prod _{v\\rightarrow w}T_w(2(pn_k+l)+{\\epsilon }_v+1)\\right)\\\\&\\ge &\\left( \\prod _{u\\rightarrow v}\\prod _{l=0}^{p-1}T_u(2(pn_k+l)+{\\epsilon }_v+1)\\right)+\\left(\\prod _{v\\rightarrow w}\\prod _{l=0}^{p-1} T_w(2(pn_k+l)+{\\epsilon }_v+1)\\right)\\\\&\\approx & \\left( \\prod _{u\\rightarrow v}b(u,pn_k)\\right)+\\left(\\prod _{v\\rightarrow w}b(w,pn_k)\\right).$ The last equality is justified as follows: if ${\\epsilon }_v=0$ then $T_u(2(pn_k+l)+{\\epsilon }_v+1)$ is indeed equal to $a(u,pn_k+l)$ .", "If ${\\epsilon }_v=1$ then $T_u(2(pn_k+l)+{\\epsilon }_v+1)=a(u,pn_k+l+1)$ and then we again use $a(u,pn_k)\\approx a(u,pn_k+p)$ in order to get to the last line.", "By analyzing the asymptotics (REF ) of $b(v,pn_k)$ , we see that for all $v\\in { { \\operatorname{Vert}}(Q)}$ , ${ \\lambda }(v)^2\\ge \\max \\left(\\prod _{u\\rightarrow v}{ \\lambda }(u),\\prod _{v\\rightarrow w}{ \\lambda }(w)\\right);\\quad d(v)\\ge \\max \\left(\\sum _{u\\rightarrow v} d(u),\\sum _{v\\rightarrow w} d(w)\\right).$ Note that $\\log { \\lambda }(v)\\ge 0$ and $d(v)\\ge 0$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "Define $\\nu (v):=\\log { \\lambda }(v)+d(v)$ .", "By property (REF ) of Lemma REF , $\\nu (v)>0$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "By (REF ), $\\nu $ is a weakly subadditive labeling of ${ { \\operatorname{Vert}}(Q)}$ .", "Applying Proposition REF , we get Corollary 2.3 If a bipartite recurrent quiver $Q$ is Zamolodchikov integrable then $G(Q)$ is either an admissible $ADE$ bigraph, or an affine $\\boxtimes $ finite $ADE$ bigraph, or an affine $\\boxtimes $ affine $ADE$ bigraph.", "$\\Box $ Remark 2.4 We have shown in [12] that case (REF ) of the above corollary holds if and only if $Q$ is Zamolodchikov periodic, that is, the $T$ -system associated with $Q$ is periodic.", "Obviously, this is a special case of Zamolodchikov integrability.", "In fact, only cases (REF ) and (REF ) of Corollary REF are possible when $Q$ is Zamolodchikov integrable: Theorem 2.5 If a bipartite recurrent quiver $Q$ is Zamolodchikov integrable then $G(Q)$ is either an admissible $ADE$ bigraph, or an affine $\\boxtimes $ finite $ADE$ bigraph.", "We postpone the proof of this theorem until Section .", "Each affine $ADE$ Dynkin diagram ${\\hat{\\Lambda }}$ has the associated dominant eigenvector $\\mathbf {v}_{\\hat{\\Lambda }}:{ \\operatorname{Vert}}({\\hat{\\Lambda }})\\rightarrow \\mathbb {R}$ corresponding to the eigenvalue 2.", "In other words, for every $v\\in {\\hat{\\Lambda }}$ we have $2 \\mathbf {v}_{\\hat{\\Lambda }}(v)=\\sum _{(v,w)\\in \\operatorname{Edges}({\\hat{\\Lambda }})} \\mathbf {v}_{\\hat{\\Lambda }}(w).$ We normalize $\\mathbf {v}_{\\hat{\\Lambda }}$ so that its entries are positive integers with the smallest entry equal to 1.", "The values of $\\mathbf {v}_{\\hat{\\Lambda }}$ are given in Figure REF ." ], [ "Self and double bindings", "In this section, we classify all the bipartite affine $\\boxtimes $ finite $ADE$ bigraphs $G=(\\Gamma ,\\Delta )$ such that $\\Gamma $ has either one or two connected components.", "If $\\Gamma $ has just one connected component then $G$ is called a self binding, and if $\\Gamma $ has two connected components then $G$ is called a double binding.", "We start with self bindings.", "Throughout this section we assume that $h(\\Delta )>2$ , i.e.", "that $\\Delta $ has at least one edge (because if $h(\\Delta )=2$ then all connected components of $\\Delta $ are of type $A_1$ )." ], [ "Self bindings", "Lemma 3.1 If $G=(\\Gamma ,\\Delta )$ is a self binding then all the connected components of $\\Delta $ are of type $A_2$ .", "Let $\\mathbf {v}:{ \\operatorname{Vert}}(G)\\rightarrow \\mathbb {R}$ be the common eigenvector for $A_\\Gamma $ and $A_\\Delta $ from Lemma REF .", "Thus $A_\\Gamma \\mathbf {v}=2\\mathbf {v}$ .", "Since $\\Gamma $ has just one connected component, we may rescale $\\mathbf {v}$ so that it is equal to $\\mathbf {v}_\\Gamma $ .", "Now, let ${ \\lambda }_\\Delta :=2\\cos (\\pi /h(\\Delta ))$ be the dominant eigenvalue for $A_\\Delta $ .", "We have that for every $v\\in { \\operatorname{Vert}}(G)$ , $\\sum _{(v,w)\\in \\Delta } \\mathbf {v}(w)={ \\lambda }_\\Delta \\mathbf {v}(v).$ Since there exists a vertex $v$ for which $\\mathbf {v}(v)=1$ , it follows that ${ \\lambda }_\\Delta $ is an integer.", "This can only happen when $h(\\Delta )=3$ , that is, when all the connected components of $\\Delta $ have Coxeter number 3.", "The only finite $ADE$ Dynkin diagram with Coxeter number 3 is $A_2$ .", "Figure: Self binding 𝒮 5 \\mathcal {S}_{5}.Proposition 3.2 For every $n\\ge 1$ , there is a self binding $\\mathcal {S}_{4n+1}=(\\Gamma _n,\\Delta _n)$ where $\\Gamma _n$ is an affine $ADE$ Dynkin diagram of type ${\\hat{A}}_{4n+1}$ , that is, a single cycle with $4n+2$ vertices, and two vertices of $\\Gamma _n$ are connected by an edge of $\\Delta _n$ iff they are the opposite vertices of that cycle (see Figure REF ); There are no other self bindings.", "Let $G=(\\Gamma ,\\Delta )$ be a self binding.", "By Lemma REF , all the components of $\\Delta $ are just isolated single edges.", "Let us define an involution ${\\mathbf {i}}:{ \\operatorname{Vert}}(G)\\rightarrow { \\operatorname{Vert}}(G)$ such that $v$ and ${\\mathbf {i}}(v)$ are exactly the vertices connected by the edges of $\\Delta $ .", "This is a fixed point free involution, otherwise $\\Delta $ would have a connected component of type $A_1$ .", "Moreover, since $G$ is bipartite, ${\\mathbf {i}}$ should reverse the colors of vertices.", "Finally, if $(u,v)\\in \\Gamma $ then one must also have $({\\mathbf {i}}(u),{\\mathbf {i}}(v))\\in \\Gamma $ because otherwise the adjacency matrices $A_\\Gamma $ and $A_\\Delta $ would not commute.", "Thus ${\\mathbf {i}}$ is a color-reversing involutive automorphism of $G$ without fixed points.", "The only affine $ADE$ Dynkin diagram admitting such an automorphism is ${\\hat{A}}_{4n+1}$ for $n\\ge 1$ , where the automorphism is just a rotation by $180^\\circ $ ." ], [ "Double bindings: scaling factor", "The classification of double bindings is going to be much richer than that of self bindings.", "Throughout the rest of this section, we assume that $G=(\\Gamma ,\\Delta )$ is a double binding, and that ${ \\operatorname{Vert}}(G)=X\\sqcup Y$ , where $X$ and $Y$ are the two connected components of $\\Gamma $ , and recall that they are affine $ADE$ Dynkin diagrams.", "A parallel binding is a bigraph of type ${\\hat{\\Lambda }}\\otimes A_2$ and, following [35], is denoted ${\\hat{\\Lambda }}\\equiv {\\hat{\\Lambda }}$ .", "Definition 3.3 The scaling factor of $G$ (denoted $ \\operatorname{scf}(G)$ ) is the number ${ \\lambda }_\\Delta ^2$ where ${ \\lambda }_\\Delta =2\\cos (\\pi /h(\\Delta ))$ is the dominant eigenvalue for $A_\\Delta $ .", "Proposition 3.4 The scaling factor $ \\operatorname{scf}(G)$ is an integer equal to either $1,2,$ or 3.", "Moreover, if $ \\operatorname{scf}(G)=1$ then all connected components of $\\Delta $ are of type $A_2$ ; if $ \\operatorname{scf}(G)=2$ then all connected components of $\\Delta $ are of type $A_3$ ; if $ \\operatorname{scf}(G)=3$ then all connected components of $\\Delta $ are either of type $A_5$ or of type $D_4$ .", "We view maps $\\tau :{ \\operatorname{Vert}}(G)\\rightarrow \\mathbb {R}$ as pairs $\\begin{pmatrix}\\tau _X\\\\\\tau _Y\\end{pmatrix}$ where $\\tau _X:{ \\operatorname{Vert}}(X)\\rightarrow \\mathbb {R}$ and $\\tau _Y:{ \\operatorname{Vert}}(Y)\\rightarrow \\mathbb {R}$ are restrictions of $\\tau $ to the corresponding subsets.", "Let $\\tau =\\begin{pmatrix}\\tau _X\\\\\\tau _Y\\end{pmatrix}$ be the common dominant eigenvector for $A_\\Gamma $ and $A_\\Delta $ from Lemma REF .", "We may rescale it so that $\\tau _X=\\alpha \\mathbf {v}_X$ and $\\tau _Y=\\mathbf {v}_Y$ for some $\\alpha \\in \\mathbb {R}$ .", "Since the entries of the dominant eigenvector are positive, we may assume $\\alpha >0$ .", "Since $A_\\Delta \\tau ={ \\lambda }_\\Delta \\tau $ , we have $\\sum _{(v,w)\\in \\Delta } \\mathbf {v}_Y(w)&=&{ \\lambda }_\\Delta \\alpha \\mathbf {v}_X(v),\\quad \\forall \\,v\\in X;\\\\\\sum _{(v,w)\\in \\Delta }\\alpha \\mathbf {v}_X(v)&=&{ \\lambda }_\\Delta \\mathbf {v}_Y(w),\\quad \\forall \\,w\\in Y.$ If we substitute $v\\in X$ such that $\\mathbf {v}_X(v)=1$ in (REF ), we will get that ${ \\lambda }_\\Delta \\alpha \\in \\mathbb {Z}_{>0}$ .", "Similarly, if we substitute $w\\in X$ such that $\\mathbf {v}_Y(w)=1$ in (), we will get that ${ \\lambda }_\\Delta /\\alpha \\in \\mathbb {Z}_{>0}$ .", "Therefore their product ${ \\lambda }_\\Delta ^2$ belongs to $\\mathbb {Z}_{>0}$ as well.", "A straightforward case analysis shows that this can only happen when $h(\\Delta )=3,4,$ or 6, and the result follows.", "A simple consequence of the proof is the following observation: Corollary 3.5 Up to switching $X$ and $Y$ , we have: $\\sum _{(v,w)\\in \\Delta } \\mathbf {v}_Y(w)&=& \\operatorname{scf}(G) \\mathbf {v}_X(v),\\quad \\forall \\,v\\in X;\\\\\\sum _{(v,w)\\in \\Delta }\\mathbf {v}_X(v)&=&\\mathbf {v}_Y(w),\\quad \\forall \\,w\\in Y.$ We know that ${ \\lambda }_\\Delta ^2\\in \\lbrace 1,2,3\\rbrace $ and thus ${ \\lambda }_\\Delta \\in \\lbrace 1,\\sqrt{2},\\sqrt{3}\\rbrace $ .", "Thus the only $\\alpha \\in \\mathbb {R}$ satisfying ${ \\lambda }_\\Delta /\\alpha \\in \\mathbb {Z}_{>0}$ and ${ \\lambda }_\\Delta \\alpha \\in \\mathbb {Z}_{>0}$ is either $\\alpha ={ \\lambda }_\\Delta $ or $\\alpha =1/{ \\lambda }_\\Delta $ .", "By the same reasoning as in the proof of Proposition REF , if $ \\operatorname{scf}(G)=1$ then $G$ is a parallel binding.", "It remains to classify double bindings with scaling factor 2 and 3.", "We say that a double binding is nontrivial if it is not a parallel binding, i.e.", "if the scaling factor is 2 or 3.", "Definition 3.6 When $X$ is an affine $ADE$ Dynkin diagram of type ${\\hat{\\Lambda }}$ and $Y$ is an affine $ADE$ Dynkin diagram of type ${\\hat{\\Lambda }}^{\\prime }$ then we say that $G$ is a double binding of type ${\\hat{\\Lambda }}\\ast {\\hat{\\Lambda }}^{\\prime }$.", "Note that Corollary REF is not symmetric in $X$ and $Y$ , so if $G$ is a double binding of type ${\\hat{\\Lambda }}\\ast {\\hat{\\Lambda }}^{\\prime }$ then necessarily $X$ has type ${\\hat{\\Lambda }}$ , $Y$ has type ${\\hat{\\Lambda }}^{\\prime }$ and (REF ) and () hold.", "In other words, we treat double bindings of types ${\\hat{\\Lambda }}\\ast {\\hat{\\Lambda }}^{\\prime }$ and ${\\hat{\\Lambda }}^{\\prime }\\ast {\\hat{\\Lambda }}$ differently.", "A simple consequence of () is Corollary 3.7 For any double binding $G$ , the maximal value of $\\mathbf {v}_X$ is less than or equal to the maximal value of $\\mathbf {v}_Y$ .", "Let $\\mathbf {v}_X(u)$ be the maximal value of $\\mathbf {v}_X$ , then clearly $\\mathbf {v}_Y(w)\\ge \\mathbf {v}_X(u)$ for any $(u,w)\\in \\Delta $ by ().", "Denote by $\\mathbf {v}_Y^{-1}(1)$ the set of vertices $u$ of $Y$ with $\\mathbf {v}_Y(u)=1$ .", "Proposition 3.8 There are no non-trivial double bindings of type ${\\hat{\\Lambda }}\\ast {\\hat{\\Lambda }}$ (i.e.", "when $X$ and $Y$ have the same type).", "Let $M$ be the maximal value of $\\mathbf {v}_X$ and $\\mathbf {v}_Y$ , and let $W=\\mathbf {v}_X^{-1}(M),\\ U=\\mathbf {v}_Y^{-1}(M)$ be the sets of vertices where $\\mathbf {v}_X$ (resp., $\\mathbf {v}_Y$ ) takes the maximal value.", "It is clear from () that every vertex from $U$ is $\\Delta $ -connected to at most one vertex from $W$ .", "By the same reason, every vertex from ${ \\operatorname{Vert}}(Y)\\setminus U$ is not $\\Delta $ -connected to any vertex from $W$ .", "Thus every vertex from $W$ is allowed to be $\\Delta $ -connected only to vertices from $U$ , and by (REF ), each of them should be connected to at least two vertices in $U$ .", "We get a contradiction since the sizes of $W$ and $U$ are supposed to be the same." ], [ "Double bindings involving type ${\\hat{E}}$", "We say that $Y$ is one-two-bipartite if for every $u,w\\in { \\operatorname{Vert}}(Y)$ with $\\mathbf {v}_Y(u)=1$ and $\\mathbf {v}_Y(w)=2$ , we have ${\\epsilon }_u\\ne {\\epsilon }_w$ (that is, all ones in $\\mathbf {v}_Y$ are white and all twos in $\\mathbf {v}_Y$ are black, or vice versa).", "Note that if $Y$ is of type ${\\hat{E}}_6$ or ${\\hat{E}}_8$ then $Y$ is one-two-bipartite, see Figure REF .", "Lemma 3.9 Let $G$ be a double binding, and assume that $Y$ is one-two-bipartite.", "Then $ \\operatorname{scf}(G)$ divides $\\#\\mathbf {v}_Y^{-1}(1)$ .", "Let $w\\in \\mathbf {v}_Y^{-1}(1)$ .", "By , there is exactly one vertex $v\\in X$ with $(w,v)\\in \\Delta $ , and moreover, $\\mathbf {v}_X(v)=1$ .", "By (REF ), $\\sum _{(v,u)\\in \\Delta } \\mathbf {v}_Y(u)= \\operatorname{scf}(G).$ Since $ \\operatorname{scf}(G)\\le 3$ and $\\mathbf {v}_Y(w)=1$ is one of the terms in the left hand side, all the other terms in the left hand side are equal to either 1 or 2.", "But all vertices $u$ with $(v,u)\\in \\Delta $ must be of the same color, since the graph is bipartite.", "The set of $\\Delta $ -neighbors of $v$ consists of exactly $ \\operatorname{scf}(G)$ vertices $u$ with $\\mathbf {v}_Y(u)=1$ .", "By (), $v$ is the only $\\Delta $ -neighbor of each such $u$ .", "Therefore, the set $\\mathbf {v}_Y^{-1}(1)$ is partitioned into classes, and each class has $ \\operatorname{scf}(G)$ members that have the same $\\Delta $ -neighbor.", "Corollary 3.10 If $G$ is a non-trivial double binding of type ${\\hat{\\Lambda }}\\ast {\\hat{E}}_6$ then $ \\operatorname{scf}(G)=3$ .", "Proposition 3.11 There are no non-trivial double bindings of type ${\\hat{\\Lambda }}\\ast {\\hat{E}}_8$ ; the only non-trivial double binding of type ${\\hat{E}}_n\\ast {\\hat{\\Lambda }}$ is the double binding ${\\hat{E}}_6\\ast {\\hat{E}}_7$ depicted in Figure REF .", "To prove (REF ), just observe that if $Y$ is of type ${\\hat{E}}_8$ then $\\#\\mathbf {v}_Y^{-1}(1)=1$ and apply Lemma REF .", "To prove (REF ), we can first eliminate all the cases except for ${\\hat{E}}_6\\ast {\\hat{E}}_7$ : by (REF ), there are no bindings of type ${\\hat{E}}_n\\ast {\\hat{E}}_8$ ; by Proposition REF , there are no bindings of types ${\\hat{E}}_6\\ast {\\hat{E}}_6$ or ${\\hat{E}}_7\\ast {\\hat{E}}_7$ ; by Corollary REF , there are no bindings of types ${\\hat{E}}_7\\ast {\\hat{E}}_6$ , $E_n\\ast A_m$ , or $E_n\\ast D_m$ .", "Now we need to prove that there is only one double binding of type ${\\hat{E}}_6\\ast {\\hat{E}}_7$ .", "Let $\\lbrace w_1,w_2,w_3\\rbrace $ be all the vertices of $X$ (which is of type ${\\hat{E}}_6$ ) with $\\mathbf {v}_X(w_i)=2$ for $i=1,2,3$ .", "Since $Y$ is of type ${\\hat{E}}_7$ , it has 5, say, white vertices and 3 black vertices.", "Let $\\lbrace u_1,u_2,u_3\\rbrace $ be these three black vertices.", "Since $w_1,w_2,w_3$ are all of the same color, it is clear from () that they are white (because if the left hand side of () is even then the right hand side should be also even), and thus the other 4 vertices of $X$ are black.", "To sum up, the edges of $\\Delta $ connect the vertices $u_1,u_2,u_3$ to the vertices $w_1,w_2,w_3$ , and we have $\\mathbf {v}_X(w_1)=\\mathbf {v}_X(w_2)=\\mathbf {v}_X(w_3)=\\mathbf {v}_Y(u_1)=\\mathbf {v}_Y(u_3)=2,\\quad \\mathbf {v}_Y(u_2)=4.$ A simple case analysis shows that $u_2$ is $\\Delta $ -connected to two vertices, say, to $w_1$ and $w_2$ while $u_1$ and $u_3$ are then both connected to $w_3$ .", "Now, using the fact that the adjacency matrices $A_\\Gamma $ and $A_\\Delta $ commute, there is only one way to recover the rest of the double binding, and we get exactly ${\\hat{E}}_6\\ast {\\hat{E}}_7$ from Figure REF .", "We conclude the analysis of double bindings for which one of the components is of type ${\\hat{E}}_n$ with the following proposition: Proposition 3.12 There are no non-trivial double bindings of type ${\\hat{A}}_m\\ast {\\hat{E}}_n$ ; there is exactly one non-trivial double binding of type ${\\hat{D}}_m\\ast {\\hat{E}}_6$ , namely, the binding ${\\hat{D}}_4\\ast {\\hat{E}}_6$ depicted in Figure REF ; there is exactly one non-trivial double binding of type ${\\hat{D}}_m\\ast {\\hat{E}}_7$ , namely, the binding ${\\hat{D}}_6\\ast {\\hat{E}}_7$ depicted in Figure REF .", "First, we show (REF ).", "If $X$ is of type ${\\hat{A}}_m$ and $Y$ is of type ${\\hat{E}}_n$ , then $\\mathbf {v}_X(w)=1$ for all $w\\in X$ .", "If $Y$ has type ${\\hat{E}}_7$ then there is a vertex $u\\in Y$ with $\\mathbf {v}_Y(u)=4$ which is impossible since $u$ has at most three neighbors, so by (), $\\mathbf {v}_Y(u)\\le 3$ .", "By Proposition REF , $Y$ cannot be of type ${\\hat{E}}_8$ , so assume now that $Y$ is of type ${\\hat{E}}_6$ .", "Let $\\mathbf {v}_Y^{-1}(1)=\\lbrace u_1,u_2,u_3\\rbrace $ .", "Then there is a vertex $w_1\\in X$ connected by $\\Delta $ to all of them.", "Let $w_2$ be such that $(w_2,w_1)\\in \\Gamma $ .", "Since $w_2$ is of different color, it can only be $\\Delta $ -connected to vertices $u\\in Y$ with $\\mathbf {v}_Y(u)=2$ .", "But the sum $\\sum _{(u,w_2)\\in \\Delta } \\mathbf {v}_Y(u)$ should be equal to 3 which is impossible because it is even.", "Thus (REF ) follows.", "Next, we prove (REF ), so assume $X$ has type ${\\hat{D}}_m$ and $Y$ has type ${\\hat{E}}_6$ .", "By Corollary REF , the scaling factor in this case equals to 3.", "Let $\\mathbf {v}_Y^{-1}(1)=\\lbrace u_1,u_2,u_3\\rbrace $ .", "Then all of them are connected to some vertex $w_1\\in X$ with $\\mathbf {v}_X(w_1)=1$ .", "Therefore $w_1$ has a unique $\\Gamma $ -neighbor $w_2\\in X$ , and $\\mathbf {v}_X(w_2)=2$ .", "Since the adjacency matrices $A_\\Gamma $ and $A_\\Delta $ commute, $w_2$ should be connected to all three vertices $u_4,u_5,u_6$ of $Y$ satisfying $\\mathbf {v}_Y(u_i)=2$ for $i=4,5,6$ .", "Since $X$ has three more vertices $w_3,w_4,w_5$ with $\\mathbf {v}_X(w_i)=1$ for $i=3,4,5$ , each of them has to be connected to the remaining vertex $u_7$ of $Y$ with $\\mathbf {v}_Y(u_7)=3$ .", "It follows that there are no more vertices in $X$ , so we are done with (REF ).", "Finally, we show (REF ), so let $X$ have type ${\\hat{D}}_m$ and let $Y$ have type ${\\hat{E}}_7$ .", "Assume first that the scaling factor is 2, and let $u\\in { \\operatorname{Vert}}(Y)$ be a vertex with $\\mathbf {v}_Y(u)=3$ .", "Then by (REF ), if $(u,w)\\in \\Delta $ for some $w\\in { \\operatorname{Vert}}(X)$ , then $\\mathbf {v}_X(w)\\ge 2$ , but since $X$ is of type ${\\hat{D}}_m$ , $\\mathbf {v}_X(w)$ must be equal to 2.", "Since $\\mathbf {v}_Y(u)$ is odd, this contradicts ().", "Thus the scaling factor has to be equal to 3.", "Because $Y$ is of type ${\\hat{E}}_7$ , $Y$ has 3, say, black vertices $u_1,u_2,u_3$ , and 5 white vertices, and we have $\\mathbf {v}_Y(u_1)=\\mathbf {v}_Y(u_3)=2,\\mathbf {v}_Y(u_2)=4$ .", "It follows now that: $X$ has exactly 2 white vertices $w_1$ and $w_2$ ; one of the components of $\\Delta $ has type $A_5$ and connects the vertices $u_1 - w_1 - u_2 - w_2 - u_3$ .", "Again, using commuting adjacency matrices, one can reconstruct the rest of the double binding and see that it is in fact ${\\hat{D}}_6\\ast {\\hat{E}}_7$ in Figure REF ." ], [ "Double bindings involving type ${\\hat{A}}$", "One can identify the vertices of the cycle $A_{2m-1}$ with $\\mathbb {Z}_m:=\\mathbb {Z}/m\\mathbb {Z}$ .", "We define double and triple coverings to be the following double bindings: in a double covering ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}$ , a vertex $j\\in \\mathbb {Z}_{4m}$ of $Y$ is connected by a blue edge to a vertex $i\\in \\mathbb {Z}_{2m}$ of $X$ iff $i\\equiv j\\pmod {2m}$ .", "Similarly, in a triple covering ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{6n-1}$ , a vertex $j\\in \\mathbb {Z}_{6m}$ of $Y$ is connected by a blue edge to a vertex $i\\in \\mathbb {Z}_{2m}$ of $X$ iff $i\\equiv j\\pmod {2m}$ .", "These are obviously affine $\\boxtimes $ finite $ADE$ bigraphs.", "Proposition 3.13 The only possible double bindings of type ${\\hat{A}}_{m-1}\\ast {\\hat{A}}_{k-1}$ are: parallel bindings ${\\hat{A}}_{2n-1}\\equiv {\\hat{A}}_{2n-1}$ ; double coverings ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}$ ; triple coverings ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{6n-1}$ .", "By (), each vertex of $Y$ has exactly one blue neighbor, and each vertex of $X$ has exactly $ \\operatorname{scf}(G)$ blue neighbors.", "Let $(w_i)_{i\\in \\mathbb {Z}_k}$ be the vertices of $Y$ listed in cyclic order, and let $(v_i)_{i\\in \\mathbb {Z}_m}$ be the vertices of $X$ in cyclic order.", "Let $f:\\mathbb {Z}_k\\rightarrow \\mathbb {Z}_m$ be the map such that $v_{f(i)}$ is the unique blue neighbor of $w_i$ for all $i\\in \\mathbb {Z}_k$ .", "Since the adjacency matrices have to commute, we get that $\\lbrace f(i+1),f(i-1)\\rbrace =\\lbrace f(i)+1,f(i)-1\\rbrace $ which immediately yields the result of the proposition.", "By Corollary REF , there are no double bindings of type ${\\hat{D}}_m\\ast {\\hat{A}}_n$ so the only case left in this section is ${\\hat{A}}_n\\ast {\\hat{D}}_m$ .", "Proposition 3.14 The only possible double bindings of type ${\\hat{A}}_n\\ast {\\hat{D}}_m$ are the double bindings of type ${\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}$ in Figure REF and the exceptional double binding of type ${\\hat{A}}_3\\ast {\\hat{D}}_5$ in Figure REF .", "We have two options: either $ \\operatorname{scf}(G)=2$ or $ \\operatorname{scf}(G)=3$ .", "If $ \\operatorname{scf}(G)=2$ then we know that each non-leaf vertex of $Y$ is connected to exactly two vertices of $X$ , and is the only blue neighbor of each of them.", "On the other hand, there are two more vertices $v_1,v_2$ in $X$ , and each of them has two blue neighbors which are leaves in $Y$ .", "Now using commuting adjacency matrices condition one can easily recover that $G$ is the double binding of type ${\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}$ from Figure REF .", "Now assume that $ \\operatorname{scf}(G)=3$ .", "This means that each vertex of $X$ is connected to an odd number of leaves of $Y$ .", "Since $Y$ has exactly four leaves, it follows that $X$ has either two or four vertices.", "If $X$ has two vertices then the sum of values of $\\mathbf {v}_Y$ is six so $Y$ has type ${\\hat{D}}_4$ but then all the leaves of $Y$ have the same color so one of the vertices of $X$ is not going to be connected to any of them.", "We are left with the case when $X$ has four vertices and each of them is connected to a leaf of $Y$ and to a non-leaf of $Y$ .", "Therefore $Y$ has type ${\\hat{D}}_5$ from which one can quickly see that $G$ is the unique double binding of type ${\\hat{A}}_3\\ast {\\hat{D}}_5$ from Figure REF ." ], [ "Double bindings of type ${\\hat{D}}_{m+1}\\ast {\\hat{D}}_{k+1}$", "Proposition 3.15 The only possible double bindings of type ${\\hat{D}}_{m+1}\\ast {\\hat{D}}_{k+1}$ are the double bindings of type ${\\hat{D}}_{n}\\ast {\\hat{D}}_{2n-2}$ and the double bindings of type ${\\hat{D}}_{n+1}\\ast {\\hat{D}}_{3n-1}$ constructed in the proof of this proposition and depicted for small $n$ in Figure REF .", "Let $v_1^+,v_1^-,v_2,\\dots ,v_{m-1},v_{m}^+,v_{m}^-$ be the vertices of $X$ , the component of type ${\\hat{D}}_{m+1}$ , and $w_1^+,w_1^-,w_2,\\dots ,w_{k-1},w_k^+,w_k^-$ be the vertices of $Y$ which has type ${\\hat{D}}_{k+1}$ .", "Here we assume that $v_1^+,v_1^-$ are the leaves attached to $v_2$ and so on.", "By (), each of $w_1^+,w_1^-,w_k^+,w_k^-$ is connected to exactly one leaf of $X$ .", "Without loss of generality assume that $w_1^+$ is connected to $v_1^+$ by a blue edge.", "Since the adjacency matrices commute, $w_2$ has to be connected to $v_2$ by blue edges.", "We claim that $w_1^-$ cannot be connected to $v_1^+$ .", "Indeed, otherwise there would be at least two blue-red paths and at most one red-blue path from $v_1^+$ to $w_2$ , so the matrices would not commute.", "On the other hand, $w_1^-$ is connected to a leaf, and this leaf has to be a neighbor of $v_2$ .", "So without loss of generality we may assume that $w_1^-$ is connected to $v_1^-$ (we only make a choice here when $Y$ has type ${\\hat{D}}_4$ in which case all the four leaves of $Y$ are connected to $w_2$ ).", "By Proposition REF , we have $m\\ne k$ and by (REF )-() we actually have $m<k$ .", "We claim that for each $i=2,\\dots ,m-1$ , $w_i$ is connected to $v_i$ , and thus to nothing else by ().", "We show it by induction on $i$ , where the base $i=2$ has already been shown.", "Assume that $w_{i}$ is connected to $v_{i}$ .", "Then there is a red-blue path from $v_{i+1}$ to $w_{i}$ , and $v_{i+1}$ is not connected to $w_{i-1}$ so it has to be connected to $w_{i+1}$ , and the claim follows for $i=2,\\dots ,m-1$ .", "Now there is a red-blue path from $v_m^+$ to $w_{m-1}$ so by the same reasoning $v_m^+$ and $v_m^-$ are connected to $w_m$ .", "Now there is a red-blue path from $w_{m+1}$ to $v_m^+$ and to $v_m^-$ so $w_{m+1}$ is connected to $v_{m-1}$ .", "Again using induction we can show that for $i=1,2,\\dots ,m-2$ , $w_{m+i}$ is connected to $v_{m-i}$ .", "This includes the fact that $2m-2<k$ which is true since before we stop we need to add another blue edge to $v_1^+$ in order to satisfy (REF ).", "If $ \\operatorname{scf}(G)=2$ then (REF ) is satisfied for all vertices of $X$ except for $v_1^+$ and $v_1^-$ so we complete the construction of the graph by joining $v_1^+$ to $w_k^+$ and $v_1^-$ to $w_k^-$ , where necessarily $k=2m-1$ .", "This can considered to be the definition of ${\\hat{D}}_n\\ast {\\hat{D}}_{2n-2}$ .", "If $ \\operatorname{scf}(G)=3$ then (REF ) is not satisfied for $v_m^+$ yet so we note that $2m-1<k$ and thus have to connect both $v_1^+$ and $v_1^-$ to $w_{2m-1}$ .", "But then there is a red-blue path from $w_{2m}$ to $v_1^+$ so $w_{2m}$ has to be connected to $v_2$ .", "Now for $i=1,2,\\dots ,m-2$ it follows that $w_{2m+i-1}$ is connected to $v_{i+1}$ .", "After that (REF ) fails only for $v_m^+$ and $v_m^-$ which we connect to $w_k^+$ and $w_k^-$ respectively.", "Here $k$ is necessarily equal to $3m-1$ yielding the double binding of type ${\\hat{D}}_{n+1}\\ast {\\hat{D}}_{3n-1}$ ." ], [ "The classification of self and double bindings", "We summarize the results of Sections REF -REF in the following theorem: Theorem 3.16 The only possible self bindings are $\\mathcal {S}_{4n+1}$ for $n\\ge 1$ .", "all the double bindings with scaling factor 2 are listed in Figure REF ; all the double bindings with scaling factor 3 are listed in Figure REF ; the only other double bindings are parallel bindings ${\\hat{\\Lambda }}\\equiv {\\hat{\\Lambda }}$ .", "$\\Box $ Figure: Three infinite and one exceptional family of double bindings with scaling factor 2.", "All blue components have type A 3 A_3.Figure: Two infinite and three exceptional families of double bindings with scaling factor 3.", "All blue components have types A 5 A_5 or D 4 D_4.Remark 3.17 In [12], we introduced duality of symmetric bigraphs (not to be confused with Stembridge's dual bigraphs in [35]).", "Here we briefly list some pairs of dual symmetric bigraphs for certain choices of the auxiliary datai.e.", "${\\mathbf {i}},V_+,V_0,V_-,X$ in the notation of [12] which we omit: ${\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}$ is dual to ${\\hat{A}}_{2n-1}\\otimes A_3$ ; ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}$ is dual to ${\\hat{A}}_{4n-1}\\otimes A_3$ ; ${\\hat{D}}_{n}\\ast {\\hat{D}}_{2n-2}$ is dual to ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}$ ; ${\\hat{E}}_{6}\\ast {\\hat{E}}_{7}$ is dual to itself; ${\\hat{D}}_{n+1}\\ast {\\hat{D}}_{3n-1}$ is dual to ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{6n-1}$ ; ${\\hat{D}}_6\\ast {\\hat{E}}_7$ is dual to ${\\hat{D}}_4\\ast {\\hat{E}}_6$ ; ${\\hat{A}}_3\\ast {\\hat{D}}_5$ is dual to the triple covering ${\\hat{A}}_1\\ast {\\hat{A}}_5$ ." ], [ "The classification", "To classify affine $\\boxtimes $ finite $ADE$ bigraphs, we mostly follow the strategy of [35]: we are going to show that the component graph $\\mathcal {C}$ of $\\Gamma $ defined below is a path with either at most one loop (in case there is a self binding) or at most one non-parallel double binding.", "Definition 4.1 Let $G=(\\Gamma ,\\Delta )$ be a bigraph.", "Let $C_1,C_2,\\dots ,C_m$ be the connected components of $\\Gamma $ .", "Define the graph $\\mathcal {C}=\\mathcal {C}(G)$ with vertex set $[m]:=\\lbrace 1,2,\\dots ,m\\rbrace $ such that $(i,j)$ is an edge of $\\mathcal {C}$ iff there is a blue edge $(u,v)\\in \\Delta $ with $u\\in C_i$ and $v\\in C_j$ .", "Let $G$ be an affine $\\boxtimes $ finite $ADE$ bigraph.", "We define its reduced version $\\widetilde{G}$ to be the same as $G$ but with all the blue edges removed from each self binding in $G$ .", "Clearly, $\\mathcal {C}(\\widetilde{G})$ is $\\mathcal {C}(G)$ with all the loops removed.", "It is also clear that $\\widetilde{G}$ is going to be an affine $\\boxtimes $ finite $ADE$ bigraph as well.", "Several properties of affine $\\boxtimes $ finite $ADE$ bigraphs have literally the same statements as their analogs for admissible $ADE$ bigraphs of [35], so we list them with the corresponding references to the parts of [35] where they are proved: Lemma 4.2 Let $G$ be an affine $\\boxtimes $ finite $ADE$ bigraph.", "Then: the component graph $\\mathcal {C}(\\widetilde{G})$ is acyclic (see [35]; in fact, $\\mathcal {C}(\\widetilde{G})$ is a path (see [35]); $G$ contains at most one non-parallel double binding (see [35]).", "$\\Box $ These properties allow us to describe every affine $\\boxtimes $ finite $ADE$ bigraph by a string of symbols ${\\hat{A}}_{n},{\\hat{D}}_n,{\\hat{E}}_n,\\mathcal {S}_{n}$ with symbols $\\ast ,\\equiv $ inserted between them, for example, ${\\hat{\\Lambda }}_1\\equiv {\\hat{\\Lambda }}_1\\ast {\\hat{\\Lambda }}_2$ has three red connected components (i.e.", "$m=3$ ) and $C_1$ and $C_2$ form a parallel binding while $C_2$ and $C_3$ form a double binding of type ${\\hat{\\Lambda }}_1\\ast {\\hat{\\Lambda }}_2$ .", "Lemma 4.3 Assume that $G$ is an affine $\\boxtimes $ finite $ADE$ bigraph containing a self binding.", "Then it contains exactly one self binding and all the double bindings in $G$ are parallel.", "Assume for the sake of contradiction that $G$ has at least two self bindings.", "We may remove everything else so that they occur at the ends of $\\mathcal {C}(\\widetilde{G})$ (which is a path on $[m]$ ).", "After some relabeling, the edges of $\\mathcal {C}(\\widetilde{G})$ become exactly $\\lbrace (i,i+1)\\rbrace _{i\\in [m-1]}$ .", "We are going to construct a blue cycle in $G$ as follows: let $v^1_1\\in C_1$ be any vertex, then there is a blue path $v^1_1,v^1_2,\\dots ,v^1_m$ with $v^1_i\\in C_i$ .", "Since $C_m$ is a self binding, $v^1_m$ is connected by a blue edge to some other vertex $v^2_m\\in C_m$ , from which we can construct a blue path $v^2_m,v^2_{m-1},\\dots ,v^2_1$ with $v^2_i\\in C_i$ again.", "But now $v_2^1$ is connected by a blue edge to some other vertex $v_3^1\\in C_1$ , so we may continue our path until it crosses itself yielding a blue cycle in $G$ which is a contradiction since all the finite $ADE$ Dynkin diagrams are acyclic.", "Assume now that there is a self binding and a non-parallel double binding in $G$ .", "Again, we may assume that the self binding occurs in $C_1$ and the double binding occurs between $C_{m-1}$ and $C_m$ with $\\mathcal {C}(\\widetilde{G})$ being a path on $[m]$ .", "Take the maximal blue path $P$ in $G$ .", "Since all the vertices in $C_1,\\dots ,C_{m-1}$ have blue degree at least 2, both endpoints of $P$ belong to $C_m$ and have blue degree 1.", "But since the blue components of the double binding $C_{m-1}\\ast C_m$ are either $A_3,A_5,$ or $D_4$ (see Proposition REF ), the vertices of $P$ adjacent to the endpoints have blue degrees at least 3.", "Therefore they coincide because every finite $ADE$ Dynkin diagram contains at most one vertex of degree 3.", "So $P$ has length at most 3, and therefore $m=2$ .", "It is clear that adding a self binding to any of the double bindings involving type ${\\hat{A}}$ yields either a cycle or a blue component with at least two vertices of degree 3.", "Proposition 4.4 The only affine $\\boxtimes $ finite bigraphs involving self-bindings are $({\\hat{A}}_{4n+1})^m:=\\mathcal {S}_{4n+1}\\equiv {\\hat{A}}_{4n+1}\\equiv \\cdots \\equiv {\\hat{A}}_{4n+1} \\quad \\text{($m$ factors, $m\\ge 1,n\\ge 1$)}.$ If $G$ is an affine $\\boxtimes $ finite bigraph with a self binding then we know that $\\mathcal {C}(\\widetilde{G})$ is a path by Lemma REF , so let $C_1,C_2,\\dots ,C_m$ be its connected components with the self binding happening in $C_l$ for some $l\\in [m]$ .", "If $l\\ne 1,m$ then we immediately get two vertices of degree 3 in every blue component, so we may assume that $l=1$ .", "By Lemma REF , all the double bindings are parallel and the result follows.", "Figure: Items (), (), (), and () of our classification.Theorem 4.5 Let $G$ be an affine $\\boxtimes $ finite $ADE$ bigraph.", "Then $G$ is isomorphic to exactly one of the following bigraphs: ${\\hat{\\Lambda }}\\otimes \\Lambda ^{\\prime }$ where ${\\hat{\\Lambda }}$ and $\\Lambda ^{\\prime }$ are an affine and a finite $ADE$ Dynkin diagram respectively; $({\\hat{A}}_{4n+1})^m$ ($m\\ge 1,n\\ge 1$ ); $({\\hat{A}}{\\hat{D}}^{m-1})_n:={\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}\\equiv \\cdots \\equiv {\\hat{D}}_{n+2}$ ($m$ factors, $m\\ge 2, n\\ge 2$ ); $({\\hat{A}}^{m-1}{\\hat{D}})_n:={\\hat{A}}_{2n-1}\\equiv \\cdots \\equiv {\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}$ ($m$ factors, $m\\ge 3, n\\ge 2$ ); $({\\hat{A}}{\\hat{A}}^{m-1})_n:={\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}\\equiv \\cdots \\equiv {\\hat{A}}_{4n-1}$ ($m$ factors, $m\\ge 2, n\\ge 1$ ); $({\\hat{A}}^{m-1}{\\hat{A}})_n:={\\hat{A}}_{2n-1}\\equiv \\cdots \\equiv {\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}$ ($m$ factors, $m\\ge 3, n\\ge 1$ ); $({\\hat{D}}{\\hat{D}}^{m-1})_n:={\\hat{D}}_{n}\\ast {\\hat{D}}_{2n-2}\\equiv \\cdots \\equiv {\\hat{D}}_{2n-2}$ ($m$ factors, $m\\ge 2, n\\ge 4$ ); $({\\hat{D}}^{m-1}{\\hat{D}})_n:={\\hat{D}}_{n}\\equiv \\cdots \\equiv {\\hat{D}}_{n}\\ast {\\hat{D}}_{2n-2}$ ($m$ factors, $m\\ge 3, n\\ge 4$ ); ${\\hat{E}}{\\hat{E}}^{m-1}:={\\hat{E}}_{6}\\ast {\\hat{E}}_{7}\\equiv \\cdots \\equiv {\\hat{E}}_{7}$ ($m$ factors, $m\\ge 2$ ); ${\\hat{E}}^{m-1}{\\hat{E}}:={\\hat{E}}_{6}\\equiv \\cdots \\equiv {\\hat{E}}_{6}\\ast {\\hat{E}}_{7}$ ($m$ factors, $m\\ge 3$ ); ${\\hat{A}}_{2n-1}\\equiv {\\hat{A}}_{2n-1}\\ast {\\hat{D}}_{n+2}\\equiv {\\hat{D}}_{n+2}$ ($n\\ge 2$ ); ${\\hat{A}}_{2n-1}\\equiv {\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{4n-1}\\equiv {\\hat{A}}_{4n-1}$ ($n\\ge 1$ ); ${\\hat{D}}_n\\equiv {\\hat{D}}_{n}\\ast {\\hat{D}}_{2n-2}\\equiv {\\hat{D}}_{2n-2}$ ($n\\ge 4$ ); ${\\hat{A}}_{2n-1}\\ast {\\hat{A}}_{6n-1}$ ; ${\\hat{D}}_{n+1}\\ast {\\hat{D}}_{3n-1}$ ; ${\\hat{E}}_6\\equiv {\\hat{E}}_{6}\\ast {\\hat{E}}_{7}\\equiv {\\hat{E}}_{7}$ ; ${\\hat{A}}_3\\ast {\\hat{D}}_5$ ; ${\\hat{D}}_6\\ast {\\hat{E}}_7$ ; ${\\hat{D}}_4\\ast {\\hat{E}}_6$ .", "Note that the infinite families are (REF )-(REF ), so there are 15 infinite families and 4 exceptional bigraphs.", "Please see Figure REF for examples.", "By Proposition REF , we may assume that $G$ has no self bindings.", "If all the double bindings in $G$ are parallel then $G$ is a tensor product.", "Otherwise consider the unique double binding $C_l\\ast C_{l+1}$ of $G$ .", "If it has scaling factor 3 then all of its components are of type either $A_5$ or $D_4$ by Proposition REF , so it is clear that adding an edge to all vertices of the same color in $A_5$ or in $D_4$ does not produce a finite $ADE$ Dynkin diagram (in fact, it always produces an affine $ADE$ Dynkin diagram).", "Therefore if the scaling factor is 3 then $m=2$ and $G$ is just the double binding itself.", "If the scaling factor is 2 then all the blue components are of type $A_3$ so obviously either $l=1$ or $l=m-1$ and $m$ is arbitrary, or $l=2$ and $m=4$ , and the theorem follows." ], [ "Speyer's formula", "In [33] Speyer gives the following formula for variables of the octahedron recurrence.", "One can think of it as the $T$ -system associated with the tensor product of type $A_{\\infty } \\otimes A_{\\infty }$ , where $A_\\infty $ is the infinite path graph and the tensor product is in the sense of Definition REF , which works the same way for infinite graphs.", "Let $\\mathcal {Z}_v(t)$ be the Aztec diamond of radius $t$ centered at vertex $v$ .", "By abuse of notation, denote $\\mathcal {Z}_v(t)$ also the set of vertices of $\\mathcal {Z}_v(t)$ that are not its outer corners.", "Let $\\mathcal {D}$ be a domino tiling of $\\mathcal {Z}_v(t)$ .", "Each domino has a cut edge that separates its two halves.", "Let $G_{\\mathcal {D}}$ be the graph obtained by taking $\\mathcal {Z}_v(t)$ as the set of vertices, and the set of all cut edges in $\\mathcal {D}$ as the set of edges.", "For a vertex $u \\in \\mathcal {Z}_v(t)$ let $d_{\\mathcal {D}}(u)$ be the degree of vertex $u$ in graph $G_{\\mathcal {D}}$ .", "It is easy to see that each $d_{\\mathcal {D}}(u)$ can take values $0,1,2$ only.", "Theorem 5.1 [33] The formula for the variable $T_v(t)$ in an $A_{\\infty } \\otimes A_{\\infty }$ $T$ -system is as follows: $T_v(2t+1) = T_v(2t+2) = \\sum _{\\mathcal {D}} \\prod _{u \\in \\mathcal {Z}_v(2t+1)} u^{1 - d_{\\mathcal {D}}(u)},$ $T_v(-2t) = T_v(-2t-1) = \\sum _{\\mathcal {D}} \\prod _{u \\in \\mathcal {Z}_v(2t)} u^{1 - d_{\\mathcal {D}}(u)},$ where $t \\ge 0$ and the sum is taken over all domino tilings $\\mathcal {D}$ of $\\mathcal {Z}_v(2t+1)$ and of $\\mathcal {Z}_v(2t)$ , respectively.", "Example 5.2 In Figure REF we see an example of an Aztec diamond $\\mathcal {Z}_v(2)$ , its domino tiling $\\mathcal {D}$ , and the associated graph $G_{\\mathcal {D}}$ .", "The Laurent monomial this tiling contributes is $\\frac{a l n}{c m}$ , which is easily seen to be one of the monomials in $T_v(-2) = \\frac{\\frac{ev+bl}{f} \\frac{vk+dn}{h} + \\frac{av+bd}{c} \\frac{ln+vo}{m}}{v} = \\frac{evk}{fh}+\\frac{edn}{fh}+\\frac{blk}{fh}+\\frac{bldn}{fhv}+\\frac{aln}{cm}+\\frac{avo}{cm}+\\frac{bdo}{cm}+\\frac{bdln}{cmv}.$ Figure: An Aztec diamond 𝒵 v (2)\\mathcal {Z}_v(2), its domino tiling 𝒟\\mathcal {D}, and the associated graph G 𝒟 G_{\\mathcal {D}}.Remark 5.3 Alternative approaches to giving explicit formulas for the octahedron recurrence can be found in the works of Di Francesco and Kedem [6], [4], [7], [5] and Henriques [15].", "We shall use Speyer's language as the most convenient for our purposes." ], [ "Formula with cylindric boundary conditions", "Consider now the case of $T$ -system of type $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "The quiver is naturally embedded on a cylinder.", "Consider the lifting of the quiver to the universal cover of the cylinder, where the vertex variables are periodic.", "We claim that the following variation of Speyer's theorem holds.", "Let $\\mathcal {Z}_v(t)$ now be the intersection of the Aztec diamond of radius $t$ centered at vertex $v$ with the universal cover of the cylinder, where we include two layers of frozen variables with values 1 on both boundaries.", "An example for $n=2$ and $m=3$ is shown in Figure REF .", "For each domino tiling $\\mathcal {D}$ of $\\mathcal {Z}_v(t)$ define $G_{\\mathcal {D}}$ and $d_{\\mathcal {D}}(u)$ as before, but now using the periodicity of variables on the universal cover.", "Figure: An example of region 𝒵 v (4)\\mathcal {Z}_v(4) on the universal cover of a cylinder with n=2n=2 and m=3m=3.Theorem 5.4 The formula for the variable $T_v(t)$ in an $A_m \\otimes {\\hat{A}}_{2n-1}$ $T$ -system is as follows: $T_v(2t+1) = T_v(2t+2) = \\sum _{\\mathcal {D}} \\prod _{u \\in \\mathcal {Z}_v(2t+1)} u^{1 - d_{\\mathcal {D}}(u)},$ $T_v(-2t) = T_v(-2t-1) = \\sum _{\\mathcal {D}} \\prod _{u \\in \\mathcal {Z}_v(2t)} u^{1 - d_{\\mathcal {D}}(u)},$ where $t \\ge 0$ and the sum is taken over all domino tilings $\\mathcal {D}$ of $\\mathcal {Z}_v(2t+1)$ and of $\\mathcal {Z}_v(2t)$ , respectively.", "We are going to apply Speyer's theorem to the $A_{\\infty } \\otimes A_{\\infty }$ case with variables as shown in Figure REF .", "Figure: An assignment of variables outside of the universal cover.There are three logical steps to the proof.", "First, we claim that as we run the $T$ -system dynamics, the Laurent monomials with the minimal power of ${\\epsilon }$ remain the same in the vertices which carry variables $1,{\\epsilon },{\\epsilon }^2,\\ldots $ at the beginning, while at the same time the minimal degree of ${\\epsilon }$ in Laurent monomials in the rest of the vertices (i.e.", "the ones in the middle of the universal cover) is 0.", "Indeed, let us argue this by induction.", "Applying a mutation at a vertex with minimal Laurent monomial 1 (i.e.", "with value $1+O({\\epsilon })$ ), we see that the new value is $\\frac{(1+O({\\epsilon }))(1+O({\\epsilon }))+({\\epsilon }+O({\\epsilon }^2))(O(1)+O({\\epsilon }))}{1+O({\\epsilon })},$ where $O({\\epsilon }^k)$ denotes terms with ${\\epsilon }$ -degree at least $k$ .", "It is clear then that specializing at ${\\epsilon }= 0$ we get 1, which must be then the Laurent monomial with the smallest degree of ${\\epsilon }$ in the result.", "A similar argument applies in other locations carrying a power of ${\\epsilon }$ at the beginning.", "Next, we claim that plugging in ${\\epsilon }=0$ into the formulas for the $T$ -system of type $A_{\\infty } \\otimes A_{\\infty }$ constructed as above returns exactly the formulas for $T$ -system of type $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "Again, we can argue this by induction.", "At the very beginning the claim is obvious.", "The step is also easy to see from the first claim above.", "This is because by induction assumption the exchange relations for $A_{\\infty } \\otimes A_{\\infty }$ $T$ -system specialize to exchange relations for $A_m \\otimes {\\hat{A}}_{2n-1}$ $T$ -system, and all the powers of ${\\epsilon }$ involved are non-negative.", "Finally, we want to argue that Speyer's formula applied to the above $A_{\\infty } \\otimes A_{\\infty }$ case and specialized at ${\\epsilon }= 0$ indeed returns the formula stated in the theorem.", "For that, we claim that in order for a domino tiling $\\mathcal {D}$ to contribute a term with degree of ${\\epsilon }$ equal 0 (i.e.", "a term which will not die after specializing) the chunks of Aztec diamonds $\\mathcal {Z}_v(t)$ that are outside of the universal cover need to be tiled with horizontal tiles only.", "Such $\\mathcal {D}$ -s are then in bijection with the tilings of the part of $\\mathcal {Z}_v(t)$ that is inside the universal cover strip, as desired.", "Figure: An assignment of variables outside of the universal cover.Let us look at a chunk of $\\mathcal {Z}_v(t)$ that falls outside of the universal cover.", "Give each potential domino square weight ${\\epsilon }^r$ equal to the larger weight a vertex adjacent to this square has.", "Considering both ways a domino can be positioned, it is clear that the weight picked up by the corresponding edge in $G_{\\mathcal {D}}$ is equal to the weight of its squares minus one (see FigureREF ): $r + (r-1) = {\\color {red}{r} + {r}} -1 \\text{ and } r + r = {\\color {red}{(r+1)} + {r}} -1.$ From this it is easy to see that the dominos lying in this chunk can pick up maximal weight of at most the weight of all squares minus potential number of dominos, which is $2 \\cdot R + 4 \\cdot (R-1) + \\cdots + 2R \\cdot 1 - R(R+1)/2,$ where ${\\epsilon }^R$ is the maximal power of ${\\epsilon }$ in the chunk.", "On the other hand, the total weight to burn in the chunk is $1 \\cdot R + 3 \\cdot (R-1) + \\cdots + (2R-1) \\cdot 1,$ which is easily seen to be the same.", "Thus, in order for the ${\\epsilon }$ to not enter the resulting overall weight picked up by $G_{\\mathcal {D}}$ inside the chunk, we need the equality to hold, which happens only if every square in the chunk is covered by a domino that lies in this chunk.", "This happens only when the chunk is tiled by the horizontal dominoes." ], [ "Boundary affine slices and Goncharov-Kenyon Hamiltonians", "Let us refer to copies of ${\\hat{A}}_{2n-1}$ in $A_m \\otimes {\\hat{A}}_{2n-1}$ as affine slices.", "We will distinguish boundary affine slices which correspond to the two boundary vertices of the Dynkin diagram $A_m$ , and internal affine slices which correspond to the internal vertices of the Dynkin diagram $A_m$ .", "In this section, we identify the recurrence coefficients of boundary affine slices as Goncharov-Kenyon Hamiltonians introduced in [14].", "We shall see in Section that the recurrence coefficients of the internal affine slices can be expressed through the Goncharov-Kenyon Hamiltonians using plethysm of symmetric functions.", "While we leave the question of an explicit formula for internal affine slices coefficients open, we will be able to deduce some of their properties in Sections and ." ], [ "Thurston height", "Recall from [38] the following definition of Thurston height function associated to a domino tiling.", "Consider a cylinder $\\mathfrak {C}_{m,2n}$ which we can think of as $(m+1) \\times 2n$ rectangle with sides of length $m+1$ glued.", "We can identify the $m \\times 2n$ non-boundary nodes with vertices of the quiver $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "Fix a chessboard coloring of the cylinder, and fix a node $O$ at its bottom boundary.", "Let $\\mathcal {D}$ be a domino tiling of $\\mathfrak {C}_{m,2n}$ .", "Define the function $h: \\text{ nodes of } \\mathfrak {C}_{m,2n} \\longrightarrow \\mathbb {Z}$ as follows: $h(O)=0$ ; if $a \\rightarrow b$ is a directed edge of a domino in $\\mathcal {D}$ and the cell to the right of it is black, then $h(b) - h(a) = 1$ ; if $a \\rightarrow b$ is a directed edge of a domino in $\\mathcal {D}$ and the cell to the right of it is white, then $h(b) - h(a) = -1$ .", "If there is no cell to the right, we can still decide between the two options by looking at the cell to the left and assuming the cell to the right has an opposite color.", "Theorem 6.1 Thurston height $h$ is a well-defined function on the nodes of $\\mathfrak {C}_{m,2n}$ .", "An example of a domino tiling of $\\mathfrak {C}_{m,2n}$ and the associated Thurston height function can be seen in Figure REF .", "Figure: Two cases to consider when constructing hh and an example on ℭ 3,4 \\mathfrak {C}_{3,4}.It is known [38] that Thurston height function is well-defined for regions in the plane without holes.", "Thus, it is well-defined on the infinite periodic tiling obtained by lifting $\\mathcal {D}$ to the universal cover of $\\mathfrak {C}_{m,2n}$ .", "It remains to argue that this height function is also periodic, and thus can be folded back onto the cylinder.", "Assume it is not periodic, then it must steadily grow or steadily decline as we circle around the cylinder.", "However, then it would reach arbitrary high or arbitrary low values, which is impossible since any node is within distance $m$ from the lower boundary, which is filled with 0-s and $-1$ -s. The contradiction implies the desired property.", "We can now define the height of a tiling $\\mathcal {D}$ as $h(\\mathcal {D}) = h(O^{\\prime }) - h(O),$ where $O^{\\prime }$ is the node on the top boundary component opposite of $O$ .", "Proposition 6.2 The function $h(\\mathcal {D})$ takes values $4k$ , $k= -(m+1)/2, \\ldots , (m+1)/2$ if $m$ is odd, and takes values $4k+1$ , $k = -m/2-1, \\ldots , m/2$ if $m$ is even.", "There is a unique tiling having minimal height and a unique tiling having maximal height.", "As we walk from $O$ to $O^{\\prime }$ straight up, at each step the height changes either by $\\pm 1$ or $\\pm 3$ , depending on whether the step cuts a domino and what the colors on the sides are.", "The claims of the proposition then easily follow.", "Let us refer to the tiling with the minimal height as the see, and denote it $\\mathfrak {S}$ .", "One can give an alternative definition of the height of a tiling $h(\\mathcal {D})$ as follows.", "For any tiling $\\mathcal {D}$ , put $\\mathfrak {S}$ and $\\mathcal {D}$ on the same picture.", "What we get is a double dimer model, where all dominos will split into closed cycles.", "An example of such superposition for the tiling in Figure REF is given in Figure REF .", "The dominos of the sea $\\mathfrak {S}$ are shown in blue.", "Figure: The superposition of 𝔖\\mathfrak {S} and 𝒟\\mathcal {D}.The cycles created in the process may include contractible cycles and non-contractible cycles.", "Let us refer to the latter as hula hoops.", "Note that the contractible cycles may be just double edges, if $\\mathfrak {S}$ and $\\mathcal {D}$ share dominos.", "The example in Figure REF has zero contractible cycles and three hula hoops.", "Denote $\\mathfrak {h}(\\mathcal {D})$ the number of hula hoops created by superposing $\\mathcal {D}$ and $\\mathfrak {S}$ .", "Proposition 6.3 We have $h(\\mathcal {D}) = h(\\mathfrak {S}) + 4 \\mathfrak {h}(\\mathcal {D}).$ One can always walk from $O$ to $O^{\\prime }$ so that the only steps that cross dominos, rather than follow their boundaries, are the ones crossing the hula hoops.", "It is easy to see that each such crossing is responsible for a difference of 4 between the accumulated parts of $h(\\mathfrak {S})$ and $h(\\mathcal {D})$ .", "Furthermore, it is easy to see that since $\\mathfrak {S}$ is the tiling with minimal height, each such crossing must make $h(\\mathcal {D})$ larger by 4 than $h(\\mathfrak {S})$ , as opposed to smaller.", "Otherwise we could change $\\mathfrak {S}$ by using the dominos of $\\mathcal {D}$ from the hula hoop, and decrease its height even further, which is impossible.", "The proposition claim follows." ], [ "The recurrence", "Define Goncharov-Kenyon Hamiltonians to be the sums $H_r = \\sum _{\\mathfrak {h}(\\mathcal {D}) = r} \\prod _{u \\in \\mathfrak {C}_{m,2n}} u^{1 - d_{\\mathcal {D}}(u)},$ where $d_{\\mathcal {D}}(u)$ is as before the degree of $u$ in the associated graph $G_{\\mathcal {D}}$ on the cylinder, and the sum is taken over all tilings $\\mathcal {D}$ of height $4r + h(\\mathfrak {S})$ .", "Here on the boundary we always have $u=1$ , which makes $H_r$ -s into functions of variables at the vertices of the quiver $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "Example 6.4 Take $m=3$ and $n=1$ .", "We have six variables $a,b,c,d,e,f$ at the vertices of the quiver $A_3 \\otimes {\\hat{A}}_{1}$ .", "Figure REF shows the domino tilings contributing to $H_1$ and the monomials they contribute.", "Figure: The tilings 𝒟\\mathcal {D} with height h(𝒟)=-4h(\\mathcal {D})=-4.As a result, we find $H_1 = \\frac{ab}{de}+\\frac{a}{be}+\\frac{b}{ad}+\\frac{c}{f}+\\frac{d}{a}+\\frac{ef}{bc}+\\frac{e}{cf}.$ Similarly we find $H_2 = \\frac{abc}{def}+\\frac{ab}{dcf}+\\frac{bc}{adf}+\\frac{ac}{be}+\\frac{be}{acdf}+\\frac{ef}{ad}+\\frac{af}{cd}+\\frac{e}{b}+\\frac{cd}{af}+\\frac{de}{acf}+\\frac{b}{e}+\\frac{df}{be}+\\frac{def}{abc}, \\text{ and }$ $H_3 = \\frac{bc}{ef}+\\frac{c}{be}+\\frac{b}{cf}+\\frac{a}{d}+\\frac{f}{c}+\\frac{de}{ab}+\\frac{e}{ad}.$ The only tiling contributing to $H_0$ is the sea $\\mathfrak {S}$ , and it is easy to see that $H_0 = H_4 =1.$ Let $v^{\\prime }$ be the vertex diametrically opposite to $v$ on the same affine slice.", "Let $v(j) ={\\left\\lbrace \\begin{array}{ll}v & \\text{if $j$ is even;}\\\\v^{\\prime } & \\text{if $j$ is odd.}\\end{array}\\right.", "}$ We are ready to state the main theorem of the section.", "Theorem 6.5 For any vertex $v$ on the top boundary affine slice of the quiver $A_m \\otimes {\\hat{A}}_{2n-1}$ the $T$ -system satisfies for any $t$ the following recursion $T_{v(0)}(t+(m+1)n) - H_1 T_{v(1)}(t+mn) + \\cdots \\pm H_{m} T_{v(m)}(t+n) \\mp T_{v(m+1)}(t) = 0.$ Similarly, for any vertex $v$ on the bottom boundary affine slice and any $t$ we have $T_{v(0)}(t+(m+1)n) - H_m T_{v(1)}(t+mn) + \\cdots \\pm H_{1} T_{v(m)}(t+n) \\mp T_{v(m+1)}(t) = 0.$" ], [ "Proof of the recurrence", "In this section we prove Theorem REF .", "We consider the case of $v$ lying in the top affine slice of the quiver $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "The case of the bottom affine slice is similar.", "As we have seen, the Laurent monomials entering both the $H_i$ -s and the $T_v$ -s have an interpretation in terms of weights of domino tilings.", "We are going to construct an involution which associates each Laurent monomial in the expansion by linearity of $H_i T_{v(i)}(t+(m+1-i)n)$ to an equal Laurent monomial in either $H_{i-1} T_{v(i-1)}(t+(m+2-i)n)$ or $H_{i+1} T_{v(i+1)}(t+(m-i)n)$ .", "This implies that all of the terms cancel out as desired, since thus created pairs of Laurent monomials are equal but have opposite signs.", "Let $\\mathcal {D}_{\\mathfrak {C}}$ be a domino tiling of the cylinder, contributing a term into $H_i$ .", "Let $H_{\\mathfrak {C}}$ be the topmost among $i$ hula hoops created by superposing $\\mathcal {D}_{\\mathfrak {C}}$ with the sea $\\mathfrak {S}$ .", "Let $\\mathcal {Z}_{v(i)}(t+(m+1-i)n)$ be the fragment of an Aztec diamond lying inside the universal cover, as defined above.", "Let $\\mathcal {D}_{\\mathcal {Z}}$ be a domino tiling of $\\mathcal {Z}_{v(i)}(t+(m+1-i)n)$ , contributing a term into $T_{v(i)}(t+(m+1-i)n)$ .", "Superpose $\\mathcal {D}_{\\mathcal {Z}}$ with the universal cover of the sea $\\mathfrak {S}$ , which is a tiling of the universal cover of the cylinder.", "Consider the part of the result that intersects $\\mathcal {Z}_{v(i)}(t+(m+1-i)n)$ .", "Lemma 7.1 The resulting double dimer contains a single chain of dominos, called the hose, connecting the top left to the top right cells of $\\mathcal {Z}_{v(v)}(t+(m+1-i)n)$ .", "The rest is filled with pairs of dominos that are shared by $\\mathcal {D}_{\\mathcal {Z}}$ and the universal cover of $\\mathfrak {S}$ .", "Example 7.2 In Figure REF an example is presented of a superposition of $\\mathcal {D}_{\\mathcal {Z}}$ , shown in red, with the universal cover of $\\mathfrak {S}$ , shown in blue.", "Figure: An example of superposition of 𝒟 𝒵 \\mathcal {D}_{\\mathcal {Z}} (red) and the universal cover of 𝔖\\mathfrak {S} (blue).Here $m=3$ , $n=2$ , the vertex $v(2)=v$ is circled and $\\mathcal {Z}_v(4)$ is shown.", "One can clearly see the hose, while the rest of the dominos form 2-cycles.", "It is easy to see that the resulting double dimer in $\\mathcal {Z}_{v(i)}(t+(m+1-i)n)$ must consist of exactly one path and several cycles.", "This is because there are only two places where it crosses the boundary of $\\mathcal {Z}_{v(i)}(t+(m+1-i)n)$ , thus those two places must be the ends of the path, i.e.", "the hose.", "To see why all cycles must have length 2 observe that the sea always flows in the same direction once you start crossing between its dominos, and thus you can never really turn around to form a long cycle.", "Now we are ready to define the involution.", "Assume we are given a pair $(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}})$ with corresponding Laurent monomials contributing to the product $H_i T_{v(i)}(t+(m+1-i)n)$ .", "Take the hose associated with $\\mathcal {D}_{\\mathcal {Z}}$ and start following its edges on the cylinder $\\mathfrak {C}_{m,2n}$ .", "One of the two events is going to occur: either the hose wrapping around $\\mathfrak {C}_{m,2n}$ will intersect itself first, without intersecting the hula hoops of $\\mathcal {D}_{\\mathfrak {C}}$ ; or the hose will intersect the top hula hoop $H$ of $\\mathcal {D}_{\\mathfrak {C}}$ before intersecting itself.", "In the first case, take the first such self-intersection, and extract from it the corresponding hula hoop.", "By this we mean cut out from the hose the dominos of the part between endpoints of self-intersection, and add the corresponding red dominos to $\\mathcal {D}_{\\mathfrak {C}}$ instead of the blue ones it is currently using.", "In the second case, take the first such intersection with $H$ and insert $H$ to extend the hose, by pasting it at this first point of intersection.", "We then remove the red edges of $H$ from $\\mathcal {D}_{\\mathfrak {C}}$ , substituting the blue sea edges instead.", "In either case we get a new pair $(\\mathcal {D}_{\\mathfrak {C}}^{\\prime }, \\mathcal {D}_{\\mathcal {Z}}^{\\prime })$ .", "Lemma 7.3 The resulting pair is a well-defined pair of domino tilings that contributes either to $H_{i-1} T_{v(i-1)}(t+(m+2-i)n)$ or to $H_{i+1} T_{v(i+1)}(t+(m-i)n)$ , depending on which of the two events occurred.", "Example 7.4 An example of a pair $(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}})$ contributing to $H_1 T_{v(1)}(4)$ for which the first event occurs is shown in Figure REF .", "The new pair $(\\mathcal {D}_{\\mathfrak {C}}^{\\prime }, \\mathcal {D}_{\\mathcal {Z}})^{\\prime }$ Figure: The effect of the first event on a pair (𝒟 ℭ ,𝒟 𝒵 )(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}}).in this case contributes to $H_{i+1} T_{v(i+1)}(t+(m-i)n) = H_2 T_{v(2)}(2)$ .", "An example of a pair $(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}})$ contributing to $H_1 T_{v(1)}(4)$ for which the second event occurs is shown in Figure REF .", "The new pair $(\\mathcal {D}_{\\mathfrak {C}}^{\\prime }, \\mathcal {D}_{\\mathcal {Z}})^{\\prime }$ Figure: The effect of the second event on a pair (𝒟 ℭ ,𝒟 𝒵 )(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}}).in this case contributes to $H_{i-1} T_{v(i-1)}(t+(m+2-i)n) = H_0 T_{v(0)}(6)$ .", "In both cases, the fragments that get either extracted or inserted are circled by a green dashed line.", "The only somewhat non-trivial part of the claim is why after a hula hoop is extracted from a hose, what remains is still a proper hose.", "The reason is that all blue dominos in the hose flow East, which means that the red dominos must flow North, East or South, but not West.", "This means that the red and the blue dominos that we need to connect after the extraction are compatible.", "The final claim we need to conclude the theorem is the following.", "Lemma 7.5 This map is a weight-preserving involution on Laurent monomials.", "If the first event occurred in $(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}})$ and $(\\mathcal {D}_{\\mathfrak {C}}^{\\prime }, \\mathcal {D}_{\\mathcal {Z}}^{\\prime })$ was created, then the second event occurs in $(\\mathcal {D}_{\\mathfrak {C}}^{\\prime }, \\mathcal {D}_{\\mathcal {Z}}^{\\prime })$ at exactly the same place, and $(\\mathcal {D}_{\\mathfrak {C}}, \\mathcal {D}_{\\mathcal {Z}})$ is created.", "Same holds vice versa.", "Thus, the map is an involution.", "The fact that it is weight preserving is easy to see from the way we assign weights to domino tilings." ], [ "Affine slices and plethysm", "In this section, we explain how to express the recurrence coefficients of the affine slices in $A_m \\otimes {\\hat{A}}_{2n-1}$ through the Goncharov-Kenyon Hamiltonians $H_i$ .", "Note that Theorem REF did not quite answer that yet for the boundary slices, since it involved variables at two vertices $v$ and $v^{\\prime }$ , rather than a single $v$ .", "We rely here on results of [29], as well as on the language of tensors introduced there.", "Recall that in [29] the $T$ -system variables are interpreted as certain polynomial $SL_{m+1}$ -invariants of a collection of $2n$ vectors in $\\mathbb {C}^{m+1}$ and one matrix $A \\in SL_{m+1}$ .", "The key theorem is the following strengthening of [29].", "Theorem 8.1 The variables on the $r$ -th slice of $A_m \\otimes {\\hat{A}}_{2n-1}$ , $r=1, \\ldots , m$ satisfy the same recurrence as the exterior powers $\\wedge ^r(\\hat{A}^q)$ , $q \\in \\mathbb {Z}$ , where $\\hat{A} = A^2$ .", "In particular, according to the Cayley-Hamilton theorem, the recurrence for $r=1$ is given by the characteristic polynomial of $\\hat{A}$ .", "As $q$ grows, we keep repeating the Dehn twists, which inserts $\\hat{A} \\otimes \\cdots \\otimes \\hat{A}$ into the tensor.", "Thus, we obtain the tensor $\\hat{A}^q \\otimes \\cdots \\otimes \\hat{A}^q$ in the middle.", "Furthermore, this tensor is attached to the anti-symmetrizing Levi-Cevita tensor, which results in the anti-symmetrization of $\\otimes ^r(\\hat{A}^q)$ , which is $\\wedge ^r(\\hat{A}^q)$ .", "Corollary 8.2 The recurrence coefficients of the affine slices $r=1, \\ldots , m$ are expressed in terms of the Goncharov-Kenyon Hamiltonians $H_i$ as plethysms of elementary symmetric functions and the power sum symmetric function $e_j[e_r[p_2]]$ are expressed through the original elementary symmetric functions $e_i=H_i$ .", "Theorem REF tells us the recurrence satisfied by the sequence of $T_v$ -s and $T_{v^{\\prime }}$ -s. To obtain the recurrence satisfied by $T_v$ -s only, we need to take every second term of the sequence.", "In terms of the recurrence, this means we just need to square the roots of the recurrence polynomial.", "This means that if $H_i = e_i(\\lambda )$ , then the coefficients on boundary levels are just the plethysms $e_i[p_2]$ , which of course can be expressed as polynomials in the $H_i$ -s. Since in the construction of the ring of invariants in [29] the dimension count forces the vectors and the matrix $A$ to be generic, the $r=1$ affine slice cannot satisfy any linear recurrence of length shorter than $2n(m+1)$ .", "This means that any two such linear recurrences must coincide, and thus the plethysms $e_i[p_2]$ of Goncharov-Kenyon Hamiltonians $H_i$ are the coefficients of the characteristic polynomial of $\\hat{A}$ .", "Now, if $\\lambda _i$ , $i=1, \\ldots , m+1$ are the eigenvalues of $\\hat{A}$ , then products $\\lambda _{i_1} \\cdots \\lambda _{i_r}, \\;\\;1 \\le i_1 < \\cdots < i_r \\le m+1$ are the eigenvalues of $\\wedge ^r(\\hat{A})$ .", "Then the coefficients of the corresponding characteristic polynomial of $\\wedge ^r(\\hat{A}^q)$ are exactly the plethysms $e_j[e_r[p_2(\\lambda )]].$ Corollary 8.3 The Goncharov-Kenyon Hamiltonians $H_i$ are conserved quantities of the $T$ -system.", "As in the previous proof, the minimal recurrence satisfied by the boundary affine slice is unique, and thus its coefficients are the same no matter which moment we pick as $t=0$ .", "Corollary 8.4 The recurrence for the $r$ -th affine slice has the form $T_v \\left(t+2n{{m+1} \\atopwithdelims ()r}\\right) - \\cdots \\pm T_v(t)=0$ with exactly ${{m+1} \\atopwithdelims ()r}+1$ terms on the left.", "This is clear since the size of $\\wedge ^r(\\hat{A})$ is ${m+1} \\atopwithdelims ()r$ .", "Corollary 8.5 The recurrence coefficients of $r$ -th and $(m+1-r)$ -th affine slices are the same up to the reversal of the order.", "If $m$ is odd, then the coefficients of the slice $r=(m+1)/2$ are palindromic.", "Since $\\hat{A} \\in SL_{m+1}$ , we know that the constant term of the characteristic polynomial is 1.", "Alternatively, we have already seen that $H_{m+1}=1$ .", "Either way, we see that $\\prod _{i=1}^{m+1} \\lambda _i =1$ .", "This means that the eigenvalues of $\\wedge ^r(\\hat{A}^q)$ and of $\\wedge ^{m+1-r}(\\hat{A}^q)$ are inverses of each other, and the claim follows.", "Example 8.6 Consider the case $m=3$ .", "In this case we have 3 affine slices, two boundary and one internal.", "The recurrence relations satisfied by the $T$ -system are as follows.", "If $v$ lies on the $r=1$ affine slice, $T_v(t+8n) - (H_1^2-2H_2) T_v(t+6n) + (H_2^2-2H_1H_3+2) T_v(t+4n) - (H_3^2-2H_2) T_v(t+2n) + T_v(t) = 0.$ If $v$ lies on the $r=2$ affine slice, $T_v(t+12n) - (H_2^2-2H_1H_3+2) T_v(t+10n) + ((H_1^2-2H_2) (H_3^2-2H_2) - 1) T_v(t+8n) -$ $((H_1^2-2H_2)^2+(H_3^2-2H_2)^2-2(H_2^2-2H_1H_3+2)) T_v(t+6n)$ $+ ((H_1^2-2H_2) (H_3^2-2H_2) - 1) T_v(t+4n) - (H_2^2-2H_1H_3+2) T_v(t+2n) + T_v(t)= 0.$ If $v$ lies on the $r=3$ affine slice, $T_v(t+8n) - (H_3^2-2H_2) T_v(t+6n) + (H_2^2-2H_1H_3+2) T_v(t+4n) - (H_1^2-2H_2) T_v(t+2n) + T_v(t) = 0.$ Here for example $(H_1^2-2H_2)^2+(H_3^2-2H_2)^2-2(H_2^2-2H_1H_3+2)$ is determined by the plethysm $e_3[e_2(\\lambda _1, \\lambda _2, \\lambda _3, \\lambda _4)] = e_3(\\lambda _1 \\lambda _2, \\lambda _1 \\lambda _3, \\lambda _1 \\lambda _4, \\lambda _2 \\lambda _3, \\lambda _2 \\lambda _4, \\lambda _3 \\lambda _4) =$ $=(\\lambda _1^3\\lambda _2\\lambda _3\\lambda _4+\\cdots ) + (\\lambda _1^2+\\lambda _2^2+\\lambda _3^2+\\cdots ) + 2 (\\lambda _1^2\\lambda _2^2\\lambda _3\\lambda _4+\\cdots ) = e_4e_1^2+e_3^2-2e_2e_4 = e_1^2+e_3^2-2e_2,$ followed by the plethysm $e_1[p_2] = e_1^2-2e_2$ , $e_2[p_2] = e_2^2-2e_1e_3+2$ , $e_3[p_2]=e_3^2-2e_2$ .", "Throughout we use $e_4=1$ ." ], [ "Laurent property and positivity", "Recall that the upper cluster algebra $\\mathfrak {U}_{A}$ associated with a cluster algebra $A$ is the algebra of all elements of the fraction field of $A$ that can be expressed as Laurent polynomials in any cluster of $A$ .", "Due to Laurent property of cluster algebras [8] we know that $\\mathfrak {U}_{A} \\subseteq A$ .", "The equality holds in some cases, while in other cases $\\mathfrak {U}_{A}$ is strictly larger.", "We refer the reader to [2] for a rigorous definition and properties of upper cluster algebras.", "Theorem 9.1 Goncharov-Kenyon Hamiltonians $H_i$ are elements of the upper cluster algebra associated with the quiver $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "Of course, the $H_i$ -s are Laurent expressions in terms of the initial cluster of this $T$ -system by definition.", "Since we know they are conserved quantities, the same holds for any cluster in the $T$ -system.", "However, the claim of the theorem is much stronger, since the $T$ -system represents only one way to mutate the quiver, while the Laurentness is true for any such way.", "Corollary 9.2 The coefficients of recurrence polynomials of all vertices of $A_m \\otimes {\\hat{A}}_{2n-1}$ lie in the upper cluster algebra.", "Since those coefficients are polynomials in the $H_i$ -s by Corollary REF , the statement follows.", "Sherman and Zelevinsky [32] have defined a positive cone inside the upper cluster algebra $\\mathfrak {U}_A$ to be the subset of all elements of $\\mathfrak {U}_A$ that are expressible as positive Laurent expression in any cluster of $A$ .", "Conjecture 9.3 Goncharov-Kenyon Hamiltonians $H_i$ are elements of the positive cone of the corresponding upper cluster algebras.", "Again, by definition the $H_i$ -s are positive in terms of the clusters along time evolution of the $T$ -system, but the claim of the conjecture is much stronger." ], [ "Proof of Theorem ", "We are going to use the standard technique invented in [2].", "Specifically, to know that certain $H_i$ lies in the upper cluster algebra, it suffices to check the Laurent condition with respect to some seed together with all the seeds obtained from it by a single mutation.", "The fact that the $H_i$ -s are positive in the initial seed of the $T$ -system is true by definition.", "Thus, it remains to check positivity in all seeds obtained by mutating just a single variable in the initial seed.", "Let $v$ be the variable that is mutated, and assume the surrounding variables are as in Figure REF .", "Figure: The two possible ways to get vv in the denominator of the corresponding monomial.Note that some of the variables may be equal to 1 if $v$ is close to the boundary.", "When we mutate at $v$ , we make a substitution $v \\longleftarrow \\frac{bg+de}{v^{\\prime }}.$ Let us consider the effect of this substitution on the Laurent monomials entering $H_i$ , which as we know correspond to domino tilings $\\mathcal {D}$ : $H_i = \\sum _{\\mathfrak {h}(\\mathcal {D}) = i} \\prod _{u \\in \\mathfrak {C}_{m,2n}} u^{1 - d_{\\mathcal {D}}(u)}.$ For each tiling $\\mathcal {D}$ where $v$ does not appear at all in the monomial, i.e.", "where $d_{\\mathcal {D}}(u)=1$ , or where $v$ appears in the numerator, i.e.", "$d_{\\mathcal {D}}(u)=0$ , the Laurentness is not violated by the substitution $v \\longleftarrow \\frac{bg+de}{v^{\\prime }}.$ Thus, it remains to consider the terms where $v$ appears in the denominator, i.e.", "$d_{\\mathcal {D}}(u)=2$ .", "The key observation is that such tilings $\\mathcal {D}$ come in pairs.", "This is because locally around vertex $v$ they need to look in one of the two ways shown in Figure REF .", "Furthermore, the local move swapping between those two ways to tile the surrounding $2 \\times 2$ square does not change the height of the tiling.", "Thus, all tilings $\\mathcal {D}$ contributing to the terms of $H_i$ with $v$ in the denominator indeed come in pairs, differing by the application of this local $2 \\times 2$ square swap.", "Let $\\mathcal {D}$ and $\\mathcal {D}^{\\prime }$ be such a pair.", "Then $\\prod _{u \\in \\mathfrak {C}_{m,2n}} u^{1 - d_{\\mathcal {D}}(u)} + \\prod _{u \\in \\mathfrak {C}_{m,2n}} u^{1 - d_{\\mathcal {D}^{\\prime }}(u)}= \\left(\\frac{bg}{v} + \\frac{de}{v} \\right) b^{-d_{\\mathcal {D}}(b)} g^{-d_{\\mathcal {D}}(g)} \\prod _{u \\in \\mathfrak {C}_{m,2n}, u \\ne b,v,g} u^{1 - d_{\\mathcal {D}}(u)}.$ We see that after the substitution this becomes $\\left[\\left(\\frac{bg}{v} + \\frac{de}{v} \\right) b^{-d_{\\mathcal {D}}(b)} g^{-d_{\\mathcal {D}}(g)} \\prod _{u \\in \\mathfrak {C}_{m,2n}, u \\ne b,v,g} u^{1 - d_{\\mathcal {D}}(u)}\\right]_{v \\longleftarrow \\frac{bg+de}{v^{\\prime }}} =$ $= v^{\\prime } b^{-d_{\\mathcal {D}}(b)} g^{-d_{\\mathcal {D}}(g)} \\prod _{u \\in \\mathfrak {C}_{m,2n}, u \\ne b,v,g} u^{1 - d_{\\mathcal {D}}(u)},$ which is a Laurent expression.", "The statement follows.", "$\\Box $ Remark 9.4 Both Theorem REF and Corollary REF can be deduced directly from Urban Renewal Theorem, see, for example, [33]." ], [ "Conjectures", "Let $v$ be any vertex of any of the quivers in the classification of Theorem REF .", "We formulate here conjectures generalizing the results of this paper from $A_m \\otimes {\\hat{A}}_{2n-1}$ , which is a special case of the first family, to all families from our classification.", "The following conjecture generalizes Corollaries REF and REF .", "In light of Theorem REF this conjecture is a stronger version of [29].", "Conjecture 10.1 For every vertex $v$ , there exist numbers $i$ and $N$ and rational functions $J_0 = 1, J_1, \\dots , J_{N}, J_{N+1}=1$ in $\\mathbb {Q}({ \\mathbf {x}})$ such that The $J_k$ -s are the conserved quantities of the $T$ -system; For any $t$ we have $J_0 T_v(t+i(N+1)) - J_1 T_v(t+iN) + \\cdots \\pm J_{N+1} T_v(t) = 0.$ Note that among the linear recurrences satisfied by the sequences there is a minimal one.", "This is because if two recurrences are satisfied, then so is one given by the greatest common divisor of their characteristic polynomials.", "Let us from now on assume that the choices of $i,N$ and $J_k$ -s are made so that the resulting recurrence is minimal.", "The following conjecture generalizes Theorem REF and Corollary REF .", "Conjecture 10.2 The $J_k$ -s belong to the upper cluster algebra of the cluster algebra associated with $Q$ .", "In particular, they are Laurent polynomials in variables at any moment $t$ .", "The following conjecture generalizes Conjecture REF .", "Conjecture 10.3 The $J_k$ -s are positive Laurent expressions in terms of any cluster of the cluster algebra.", "In other words, they are elements of the positive cone inside the upper cluster algebra, as defined by Sherman and Zelevinsky [32].", "Our next conjecture is open even in Type ${\\hat{A}}_{2n-1}\\otimes A_m$ .", "Let $Q$ be any affine $\\boxtimes $ finite quiver and let the $J_k$ -s be as above.", "Consider an infinite Toeplitz matrix $\\mathcal {J}= \\mathcal {J}(m,n)$ where the entries are defined as follows: $\\mathcal {J}_{i,j} ={\\left\\lbrace \\begin{array}{ll}J_{j-i} & \\text{if $0 \\le j-i \\le m+1$;}\\\\0 & \\text{otherwise.}\\end{array}\\right.", "}$ Conjecture 10.4 All minors of $\\mathcal {J}$ are either identically 0 or positive Laurent polynomials in ${ \\mathbf {x}}$ .", "In other words, we conjecture that the $J_i$ -s form a totally positive sequence, or Pólya frequency sequence, see [3] for the background.", "We also state the following weaker version of Conjecture REF : Conjecture 10.5 The roots of the recurrence polynomial $J(z) = z^{N+1}-J_1 z^N + \\cdots \\pm J_N z \\mp 1$ are positive real numbers.", "Each of the types $A_m$ , $D_m$ and $E_6$ has a canonical involution on the Dynkin diagram, sending the diagram to itself.", "Denote this involution $\\eta $ .", "Assume our $T$ -system is of the tensor product type, and more specifically of the form $\\Lambda ^{\\prime } \\otimes \\hat{\\Lambda }$ , where $\\Lambda ^{\\prime }$ is a finite type Dynkin diagram of type $A_m$ , $D_{2m+1}$ or $E_6$ , and $\\hat{\\Lambda }$ is an arbitrary extended Dynkin diagram.", "Let $v^{\\prime }$ be the vertex of $\\Lambda ^{\\prime } \\otimes \\hat{\\Lambda }$ having the same $\\hat{\\Lambda }$ coordinate, but whose $\\Lambda ^{\\prime }$ coordinate is obtained from that of $v$ via involution $\\eta $ .", "The following conjecture generalizes Corollary REF .", "Conjecture 10.6 The recurrence polynomials of $v$ and $v^{\\prime }$ have the same coefficients but in the opposite order.", "In particular, if $v=v^{\\prime }$ , then the recurrence polynomial $J(z)$ is palindromic.", "Assume now we are in any other case, i.e.", "either our $T$ -system belongs to a different family of the classification, or it is a tensor product but $\\Lambda ^{\\prime }$ is not of type $A_m$ , $D_{2m+1}$ or $E_6$ .", "The following conjecture again generalizes Corollary REF .", "Conjecture 10.7 The recurrence polynomial $J(z)$ of $v$ is palindromic." ], [ "Tropical $T$ -systems: definition", "Each bipartite recurrent quiver $Q$ has the corresponding $T$ -system which we will call the geometric $T$ -system associated with $Q$ in order to distinguish it from another system which we introduce in this section.", "We refer the reader to Example REF for an illustration of most of the statements that we prove in Sections REF -.", "Definition 11.1 Let $Q$ be a bipartite recurrent quiver, and let ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ be any map.", "Then the tropical $T$ -system associated with $Q$ is a family of integers $ \\mathfrak {t}^{ \\lambda }_v(t)\\in \\mathbb {Z}$ for every $v\\in { { \\operatorname{Vert}}(Q)},t\\in \\mathbb {Z}$ with $t+{\\epsilon }_v$ even satisfying the following relations: $ \\mathfrak {t}^{ \\lambda }_v(t+1)+ \\mathfrak {t}^{ \\lambda }_v(t-1)&=&\\max \\left(\\sum _{u\\rightarrow v} \\mathfrak {t}^{ \\lambda }_u(t),\\sum _{v\\rightarrow w} \\mathfrak {t}^{ \\lambda }_w(t)\\right);\\\\ \\mathfrak {t}^{ \\lambda }_v({\\epsilon }_v)&=&{ \\lambda }(v).$ It is apparent from the definition that $ \\mathfrak {t}^{ \\lambda }_v(t)$ is the tropicalization of $T_v(t)$ .", "One can define a tropical $T$ -system with values in $\\mathbb {Q}$ or $\\mathbb {R}$ , but for our purposes it is sufficient to consider only the integer-valued version (see also Remark REF ).", "The defining recurrence relation can be translated into the language of bigraphs as follows: if $G(\\Gamma ,\\Delta )$ is a bipartite bigraph then the relation becomes $ \\mathfrak {t}^{ \\lambda }_v(t+1)+ \\mathfrak {t}^{ \\lambda }_v(t-1)=\\max \\left(\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t),\\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t)\\right).$ If $P({ \\mathbf {x}})\\in \\mathbb {Z}[{ \\mathbf {x}}^{\\pm 1}]$ is a multivariate Laurent polynomial in variables $(x_v)_{v\\in { { \\operatorname{Vert}}(Q)}}$ then define $P\\mid _{{ \\mathbf {x}}=q^{ \\lambda }}\\in \\mathbb {Z}[q^{\\pm 1}]$ to be the univariate Laurent polynomial in $q$ obtained from $P$ by substituting $x_v=q^{{ \\lambda }(v)}$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "Further, define ${ \\operatorname{deg}_{\\max }}(q,P\\mid _{{ \\mathbf {x}}=q^{ \\lambda }})$ to be the maximal degree of $q$ in $P\\mid _{{ \\mathbf {x}}=q^{ \\lambda }}$ .", "The following claim gives a connection between the geometric and tropical $T$ -systems: Proposition 11.2 (see [12]) For every $v\\in { { \\operatorname{Vert}}(Q)},t\\in \\mathbb {Z}$ with $t+{\\epsilon }_v$ even and any ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ , we have $ \\mathfrak {t}^{ \\lambda }_v(t)={ \\operatorname{deg}_{\\max }}\\left(q,T_v(t)\\mid _{{ \\mathbf {x}}=q^{ \\lambda }}\\right).$" ], [ "Linear algebraic properties of the affine Coxeter transformation", "Let ${\\hat{\\Lambda }}$ be a bipartite affine $ADE$ Dynkin diagram, and let $w$ and $b$ be the numbers of white and black vertices in ${\\hat{\\Lambda }}$ respectively.", "One can view a map $\\mathbf {u}:{\\operatorname{Vert}({\\hat{\\Lambda }})}\\rightarrow \\mathbb {Z}$ as a vector $\\begin{pmatrix}\\mathbf {u}_W\\\\\\mathbf {u}_B\\end{pmatrix}\\in \\mathbb {Z}^{w+b}$ .", "Then the adjacency matrix $A_{\\hat{\\Lambda }}$ of ${\\hat{\\Lambda }}$ has the form $A_{\\hat{\\Lambda }}=\\begin{pmatrix}0 & A\\\\A^t & 0\\end{pmatrix}$ where $A$ is a $w\\times b$ matrix and $ ^t$ denotes matrix transpose.", "Define the mutation matrices $\\omega _W:=\\begin{pmatrix}-I_w & A\\\\0 & I_b\\end{pmatrix};\\quad \\omega _B=:=\\begin{pmatrix}I_w & 0\\\\A^t & -I_b\\end{pmatrix}.$ Here $I_k$ is the identity $k\\times k$ matrix.", "Finally, the Coxeter transformation for ${\\hat{\\Lambda }}$ is defined as a product $\\mathbf {C}=\\omega _B\\omega _W$ .", "By Lemma REF , the matrix $A_{\\hat{\\Lambda }}$ has a dominant eigenvector $\\mathbf {v}=\\begin{pmatrix}\\mathbf {v}_W\\\\\\mathbf {v}_B\\end{pmatrix}$ corresponding to the eigenvalue 2.", "This means $A\\mathbf {v}_B=2\\mathbf {v}_W;\\quad A^t\\mathbf {v}_W=2\\mathbf {v}_B.$ Just as in Part , all the coordinates of $\\mathbf {v}$ are assumed to be positive integers with greatest common divisor equal to 1.", "For a vector $\\mathbf {u}=\\begin{pmatrix}\\mathbf {u}_W\\\\\\mathbf {u}_B\\end{pmatrix}$ , we define three linear functionals as follows: $ \\operatorname{SPEED}_W(\\mathbf {u}):={\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }-{\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle };\\quad \\operatorname{SPEED}_B(\\mathbf {u}):=- \\operatorname{SPEED}_W(\\mathbf {u});$ $ \\operatorname{SUM}(\\mathbf {u}):={\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle }+{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }.$ Here ${\\langle }\\cdot ,\\cdot {\\rangle }$ denotes the standard inner product in $\\mathbb {R}^w$ and in $\\mathbb {R}^b$ .", "Proposition 11.3 For any vector $\\mathbf {u}=\\begin{pmatrix}\\mathbf {u}_W\\\\\\mathbf {u}_B\\end{pmatrix}$ , the following holds: $ \\operatorname{SPEED}_B(\\omega _W(\\mathbf {u}))&=& \\operatorname{SPEED}_W(\\mathbf {u});\\\\ \\operatorname{SPEED}_W(\\omega _B(\\mathbf {u}))&=& \\operatorname{SPEED}_B(\\mathbf {u});\\\\ \\operatorname{SUM}(\\omega _W(\\mathbf {u}))&=& \\operatorname{SUM}(\\mathbf {u})+2 \\operatorname{SPEED}_W(\\mathbf {u});\\\\ \\operatorname{SUM}(\\omega _B(\\mathbf {u}))&=& \\operatorname{SUM}(\\mathbf {u})+2 \\operatorname{SPEED}_B(\\mathbf {u}).$ We will only prove the equalities for $\\omega _W$ , and the argument is a pretty straightforward calculation: $ \\operatorname{SPEED}_B(\\omega _W(\\mathbf {u}))&=&{\\langle }\\mathbf {v}_W,A\\mathbf {u}_B-\\mathbf {u}_W{\\rangle }-{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }={\\langle }A^t\\mathbf {v}_W,\\mathbf {u}_B{\\rangle }-{\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle }-{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }\\\\&=&{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }-{\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle }= \\operatorname{SPEED}_W(\\mathbf {u});\\\\ \\operatorname{SUM}(\\omega _W(\\mathbf {u}))&=&{\\langle }\\mathbf {v}_W,A\\mathbf {u}_B-\\mathbf {u}_W{\\rangle }+{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }={\\langle }A^t\\mathbf {v}_W,\\mathbf {u}_B{\\rangle }-{\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle }+{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }\\\\&=&{\\langle }A^t\\mathbf {v}_W,\\mathbf {u}_B{\\rangle }-{\\langle }\\mathbf {v}_W,\\mathbf {u}_W{\\rangle }+{\\langle }\\mathbf {v}_B,\\mathbf {u}_B{\\rangle }= \\operatorname{SUM}(\\mathbf {u})+2 \\operatorname{SPEED}_W(\\mathbf {u}).\\\\$ Proposition REF says that $ \\operatorname{SPEED}$ is preserved while $ \\operatorname{SUM}$ grows linearly as we mutate.", "It turns out that up to a shift by $\\mathbf {v}$ , the mutation action is periodic: Proposition 11.4 For any affine $ADE$ Dynkin diagram ${\\hat{\\Lambda }}$ , there exists an integer $h_a({\\hat{\\Lambda }})$ called the affine Coxeter number and an integer $\\theta ({\\hat{\\Lambda }})$ such that for any vector $\\mathbf {u}=\\begin{pmatrix}\\mathbf {u}_W\\\\\\mathbf {u}_B\\end{pmatrix}$ , we have $\\mathbf {C}^{h_a({\\hat{\\Lambda }})}\\mathbf {u}=\\mathbf {u}+\\theta ({\\hat{\\Lambda }}) \\operatorname{SPEED}_W(\\mathbf {u})\\mathbf {v}.$ Moreover, $\\theta ({\\hat{\\Lambda }})=\\frac{4h_a({\\hat{\\Lambda }})}{{\\langle }\\mathbf {v},\\mathbf {v}{\\rangle }},$ and the values of $h_a({\\hat{\\Lambda }})$ and $\\theta ({\\hat{\\Lambda }})$ are given in Table REF .", "Stekolshchik [34] gives complete information on the Jordan normal form of $\\mathbf {C}$ : all eigenvalues of $\\mathbf {C}$ are roots of unity and the greatest common divisor of their periods is $h_a({\\hat{\\Lambda }})$ .", "Moreover, all of them have multiplicity one except for one of them ($\\lambda =1$ ) which has multiplicity 2.", "In our notation, the eigenvector attached to eigenvalue 1 is precisely $\\mathbf {v}$ and the adjoint vector is $\\mathbf {v}^{\\prime }:=\\frac{1}{4}\\begin{pmatrix}\\mathbf {v}_W\\\\-\\mathbf {v}_B\\end{pmatrix}$ (see [34]).", "We have $\\mathbf {C}\\mathbf {v}^{\\prime }=\\mathbf {v}+\\mathbf {v}^{\\prime }$ , and they are orthogonal to each other and to all other eigenvectors.", "The result follows.", "Table: (see Table 4.1) Affine Coxeter numbers of affine ADEADE Dynkin diagramsExample 11.5 Let ${\\hat{\\Lambda }}={\\hat{D}}_4$ .", "Then $h_a({\\hat{\\Lambda }})=2$ and the dominant eigenvector is given by $\\mathbf {v}={\\begin{matrix}1& &1\\\\&2& \\\\1& &1\\end{matrix}}.$ Thus $\\theta ({\\hat{\\Lambda }})=\\frac{4\\cdot 2}{1^2+1^2+1^2+1^2+2^2}=1.$ Consider the following vector of initial values: $\\mathbf {u}={\\begin{matrix}1& &1\\\\&2& \\\\1& &2\\end{matrix}}.$ Let us assume that the vertex in the middle is white.", "Then $ \\operatorname{SPEED}_W(\\mathbf {u})=1+1+1+2-2\\cdot 2=1$ .", "Since $h_a({\\hat{\\Lambda }})=2$ , we need to calculate $\\mathbf {C}^2(\\mathbf {u})=\\omega _B\\omega _W\\omega _B\\omega _W\\mathbf {u}.$ The sequence of vectors that we will get is: $\\mathbf {u}={\\begin{matrix}1& &1\\\\&2& \\\\1& &2\\end{matrix}}\\xrightarrow{}{\\begin{matrix}1& &1\\\\&3& \\\\1& &2\\end{matrix}}\\xrightarrow{}{\\begin{matrix}2& &2\\\\&3& \\\\2& &1\\end{matrix}}\\xrightarrow{}{\\begin{matrix}2& &2\\\\&4& \\\\2& &1\\end{matrix}}\\xrightarrow{}{\\begin{matrix}2& &2\\\\&4& \\\\2& &3\\end{matrix}}=\\mathbf {u}+\\mathbf {v}.$ We indeed see that $\\mathbf {C}^{h_a({\\hat{\\Lambda }})}(\\mathbf {u})=\\mathbf {C}^2(\\mathbf {u})=\\mathbf {u}+\\mathbf {v}=\\mathbf {u}+\\theta ({\\hat{\\Lambda }}) \\operatorname{SPEED}(\\mathbf {u})\\mathbf {v}.$ We would like to apply these observations to the tropical $T$ -system $ \\mathfrak {t}^{ \\lambda }_v(t)$ defined above.", "Let $G=(\\Gamma ,\\Delta )$ be a bigraph, and let ${\\hat{\\Lambda }}$ be a connected component of $\\Gamma $ isomorphic to an affine $ADE$ Dynkin diagram.", "We define $ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t)$ to be the vector in $\\mathbb {R}^w$ for $t$ even and in $\\mathbb {R}^b$ for $t$ odd which sends $v\\in {\\operatorname{Vert}({\\hat{\\Lambda }})}$ to $ \\mathfrak {t}^{ \\lambda }_v(t)$ when $t+{\\epsilon }_v$ is even.", "In particular, the vectors $\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}$ and $\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+2)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}$ belong to $\\mathbb {R}^{w+b}$ .", "Moreover, they satisfy the following inequality: $\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+2)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}\\ge \\omega _W\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}.$ Here $\\ge $ means that each coordinate of the vector on the left hand side is at least the corresponding coordinate of the vector on the right hand side.", "This inequality holds trivially by the definition of the tropical $T$ -system.", "Moreover, it is an equality if and only if for every white vertex $v$ of ${\\hat{\\Lambda }}$ , we have $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(2t+1)\\ge \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(2t+1).$ Now, using the positivity of the coordinates of $\\mathbf {v}$ and Proposition REF , we get that $ \\operatorname{SPEED}_B\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+2)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}&\\ge & \\operatorname{SPEED}_W\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}; \\\\ \\operatorname{SUM}\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+2)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}&\\ge & \\operatorname{SUM}\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}+2 \\operatorname{SPEED}_W\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(2t+1)\\end{pmatrix}.$ And again, each inequality becomes an equality if and only if (REF ) holds for every white vertex $v$ of ${\\hat{\\Lambda }}$ .", "Define the following functions of $t$ : $ \\operatorname{SPEED}_{\\hat{\\Lambda }}(t)&:=&{\\left\\lbrace \\begin{array}{ll} \\operatorname{SPEED}_W\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t+1)\\end{pmatrix}, & \\text{if $t$ is even};\\\\ \\operatorname{SPEED}_B\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t+1)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t)\\end{pmatrix}, & \\text{if $t$ is odd};\\end{array}\\right.", "}\\\\ \\operatorname{SUM}_{\\hat{\\Lambda }}(t)&:=&{\\left\\lbrace \\begin{array}{ll} \\operatorname{SUM}\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t+1)\\end{pmatrix}, & \\text{if $t$ is even};\\\\ \\operatorname{SUM}\\begin{pmatrix} \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t+1)\\\\ \\mathfrak {t}^{ \\lambda }_{\\hat{\\Lambda }}(t)\\end{pmatrix}, & \\text{if $t$ is odd}.\\end{array}\\right.", "}$ We have thus shown the following: Proposition 11.6 Let $G=(\\Gamma ,\\Delta )$ be a bipartite bigraph, and let ${\\hat{\\Lambda }}$ be a connected component of $\\Gamma $ isomorphic to an affine $ADE$ Dynkin diagram.", "Then for every $t\\in \\mathbb {Z}$ we have $ \\operatorname{SPEED}_{\\hat{\\Lambda }}(t+1)\\ge \\operatorname{SPEED}_{\\hat{\\Lambda }}(t);\\quad \\operatorname{SUM}_{\\hat{\\Lambda }}(t+1)\\ge \\operatorname{SUM}_{\\hat{\\Lambda }}(t)+2 \\operatorname{SPEED}_{\\hat{\\Lambda }}(t).$ Moreover, either (REF ) holds or both inequalities are strict.", "$\\Box $" ], [ "Solitonic behavior: soliton resolution", "It turns out that for Zamolodchikov integrable quivers, the tropical $T$ -system behaves linearly for all but finitely many moments of time.", "Namely, let $Q$ be a bipartite recurrent quiver and assume $Q$ is Zamolodchikov integrable but not Zamolodchikov periodic.", "Let $G(Q)=(\\Gamma ,\\Delta )$ be the corresponding bipartite bigraph.", "Then Corollary REF together with Remark REF imply that all connected components of $\\Gamma $ are affine $ADE$ Dynkin diagrams.", "The following proposition will be later illustrated by Example REF .", "Proposition 12.1 Assume that $Q$ is Zamolodchikov integrable and all connected components of $\\Gamma $ are affine $ADE$ Dynkin diagrams as above.", "Then for every map ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ there exists an integer $t_0$ such that for every $\|t\|>t_0$ and for every $v\\in { { \\operatorname{Vert}}(Q)}$ with $t+{\\epsilon }_v$ even we have $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t+1)\\ge \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t+1).$ In other words, for any initial data ${ \\lambda }$ , the inequality (REF ) is violated only finitely many times.", "If the inequality (REF ) is violated infinitely many times, then there exists a connected component ${\\hat{\\Lambda }}$ of $\\Gamma $ such that $ \\operatorname{SPEED}_{\\hat{\\Lambda }}(t)\\rightarrow +\\infty $ as $t\\rightarrow +\\infty $ , because each time (REF ) is violated, $ \\operatorname{SPEED}_{\\hat{\\Lambda }}(t)$ increases by at least 1 (see Proposition REF ).", "In this case, again, by Proposition REF , $ \\operatorname{SUM}_{\\hat{\\Lambda }}(t)$ grows superlinearly.", "By Proposition REF , $ \\operatorname{SUM}_{\\hat{\\Lambda }}(t)$ is just a linear combination of ${ \\operatorname{deg}_{\\max }}(q,T_v(t)\\mid _{{ \\mathbf {x}}=q^{ \\lambda }})$ for $v\\in {\\hat{\\Lambda }}$ , and thus there is a vertex $v\\in {\\hat{\\Lambda }}$ for which ${ \\operatorname{deg}_{\\max }}(q,T_v(t)\\mid _{{ \\mathbf {x}}=q^{ \\lambda }})$ grows superlinearly.", "But the values of $T_v(t)$ satisfy a linear recurrence, and thus ${ \\operatorname{deg}_{\\max }}(q,T_v(t)\\mid _{{ \\mathbf {x}}=q^{ \\lambda }})$ cannot grow faster than linearly.", "Remark 12.2 This proof works exactly the same way if the values of $ \\mathfrak {t}^{ \\lambda }$ are assumed to lie in $\\mathbb {Q}$ instead of $\\mathbb {Z}$ .", "We do not know whether the result of Proposition REF holds when the values of $ \\mathfrak {t}^{ \\lambda }$ belong to $\\mathbb {R}$ .", "Now we are finally able to deduce Theorem REF : By Corollary REF , Remark REF and Proposition REF , we need to show that if all components of $\\Gamma $ and of $\\Delta $ are affine $ADE$ Dynkin diagrams then $Q$ cannot be recurrent.", "By Proposition REF , the inequality $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t+1)\\ge \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t+1)$ is violated finitely many times.", "By symmetry between $\\Gamma $ and $\\Delta $ , the reverse inequality $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t+1)\\le \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t+1)$ is also violated finitely many times.", "Therefore after finitely many steps we will have $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t+1)= \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t+1)$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "To see that this is impossible, consider the following integers $ y^{ \\lambda }_v(t)$ defined for $t+{\\epsilon }_v$ even: $ y^{ \\lambda }_v(t)=\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t+1) - \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t+1).$ It is well-known that the numbers $ y^{ \\lambda }_v(t)$ give (up to a sign) a solution to the tropical $Y$ -system associated with $Q$ , see, for example, [16].", "Since the mutations for the tropical $Y$ -system are involutions as well, they are invertible, so we get a contradiction with (REF ) because it states that for all initial data ${ \\lambda }$ , the tropical $Y$ -system $ y^{ \\lambda }_v(t)$ eventually becomes zero.", "Combining Proposition REF with Proposition REF , we get the following corollary which we call “soliton resolution”: Corollary 12.3 Let $Q$ be a Zamolodchikov integrable quiver and let ${\\hat{\\Lambda }}$ be a component of $\\Gamma $ isomorphic to an affine $ADE$ Dynkin diagram.", "Then for every map ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ , there exist integers $ \\operatorname{SPEED}^+_{\\hat{\\Lambda }}({ \\lambda })$ and $ \\operatorname{SPEED}^-_{\\hat{\\Lambda }}({ \\lambda })$ such that for all $t\\gg 0$ and all $v\\in {\\hat{\\Lambda }}$ we have $ \\mathfrak {t}^{ \\lambda }_v(t+2h_a({\\hat{\\Lambda }}))= \\mathfrak {t}^{ \\lambda }_v(t)+\\theta ({\\hat{\\Lambda }}) \\operatorname{SPEED}^+_{\\hat{\\Lambda }}({ \\lambda })\\mathbf {v}(v);$ for all $t\\ll 0$ and all $v\\in {\\hat{\\Lambda }}$ we have $ \\mathfrak {t}^{ \\lambda }_v(t-2h_a({\\hat{\\Lambda }}))= \\mathfrak {t}^{ \\lambda }_v(t)+\\theta ({\\hat{\\Lambda }}) \\operatorname{SPEED}^-_{\\hat{\\Lambda }}({ \\lambda })\\mathbf {v}(v);$ In other words, the values of $ \\mathfrak {t}^{ \\lambda }_v$ grow linearly for $\|t\|\\gg 0$ .", "For instance, the integers $ \\operatorname{SPEED}^+_{\\hat{\\Lambda }}({ \\lambda })$ and $ \\operatorname{SPEED}^-_{\\hat{\\Lambda }}({ \\lambda })$ are calculated in example REF .", "Let us explain the soliton terminology.", "Assume $Q$ is an affine $\\boxtimes $ finite $ADE$ bigraph and consider the associated tropical $T$ -system $ \\mathfrak {t}^{ \\lambda }$ .", "Its restriction to each affine slice ${\\hat{\\Lambda }}$ behaves independently of other slices when $\|t\|\\gg 0$ .", "We treat it as a particle (a 1-soliton).", "Then what happens is that when $t$ grows from $-\\infty $ , the particles move independently with constant speeds given by (REF ).", "Then for small values of $t$ they start interacting with each other and eventually they again start moving independently with constant speeds given by REF ).", "Such a phenomenon is commonly called soliton resolution, see [37].", "Corollary 12.4 Let $Q$ be a Zamolodchikov integrable quiver and let ${\\hat{\\Lambda }}$ be a component of $\\Gamma $ isomorphic to an affine $ADE$ Dynkin diagram.", "Then for every map ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ , the following are equivalent: $ \\operatorname{SPEED}^+_{\\hat{\\Lambda }}({ \\lambda })=0$ ; $ \\operatorname{SPEED}^-_{\\hat{\\Lambda }}({ \\lambda })=0$ ; $ \\mathfrak {t}^{ \\lambda }_v$ is a periodic sequence for every $v\\in { { \\operatorname{Vert}}(Q)}$ .", "It is obvious that (REF ) implies (REF ) and (REF ).", "The fact that each of them implies (REF ) follows from Corollary REF : if $ \\operatorname{SPEED}^+_{\\hat{\\Lambda }}({ \\lambda })=0$ then $ \\mathfrak {t}^{ \\lambda }_v$ is periodic for $t\\gg 0$ , but then it is periodic for all $t$ ." ], [ "Solitonic behavior: speed conservation", "In this section we show that the speeds with which affine slices move get preserved after the scattering process is over, in the sense of Corollary REF .", "This can be viewed as a tropical version of Corollary REF .", "Specifically, let ${\\hat{\\Lambda }}_1,\\dots ,{\\hat{\\Lambda }}_m$ be the $m$ affine slices of $A_m \\otimes {\\hat{A}}_{2n-1}$ , and for $r=1,2,\\dots ,m$ denote $ \\operatorname{SPEED}_r^+:= \\operatorname{SPEED}_{{\\hat{\\Lambda }}_r}^+({ \\lambda });\\quad \\operatorname{SPEED}_r^-:= \\operatorname{SPEED}_{{\\hat{\\Lambda }}_r}^-,$ where ${ \\lambda }:{ { \\operatorname{Vert}}(Q)}\\rightarrow \\mathbb {Z}$ is fixed throughout this section.", "Theorem 13.1 For any $1 \\le r \\le m$ we have $ \\operatorname{SPEED}_r^{+} = \\operatorname{SPEED}_{m+1-r}^{-}.$ Example 13.2 Let us give an example of the kind of phenomenon in Theorem REF .", "Let us say that $m=3$ and $n=2$ , so our quiver $Q$ is $A_3\\otimes {\\hat{A}}_1$ depicted in FigureREF .", "Figure: The bigraph A 3 ⊗A ^ 1 A_3\\otimes {\\hat{A}}_1We will compactly draw this quiver as ${\\begin{matrix}a&b&c\\\\d&e&f\\end{matrix}}$.", "Let ${\\hat{\\Lambda }}_1,{\\hat{\\Lambda }}_2,{\\hat{\\Lambda }}_3$ be the three red connected components, and assume we start our mutation sequence with black vertices.", "Then we have $ \\operatorname{SPEED}_{{\\hat{\\Lambda }}_1}(t)=d-a;\\quad \\operatorname{SPEED}_{{\\hat{\\Lambda }}_2}(t)=b-e;\\quad \\operatorname{SPEED}_{{\\hat{\\Lambda }}_3}(t)=f-c.$ We denote ${\\mathbf {S}}(t)=( \\operatorname{SPEED}_{{\\hat{\\Lambda }}_1}(t), \\operatorname{SPEED}_{{\\hat{\\Lambda }}_2}(t), \\operatorname{SPEED}_{{\\hat{\\Lambda }}_3}(t))$ .", "Now, for $t\\ll 0$ , $ \\operatorname{SPEED}_{{\\hat{\\Lambda }}_r}(t)=- \\operatorname{SPEED}^-_r$ and for $t\\gg 0$ , $ \\operatorname{SPEED}_{{\\hat{\\Lambda }}_r}(t)= \\operatorname{SPEED}^+_r$ for $r=1,2,3$ .", "The mutations and speeds for initial values ${\\begin{matrix}6&6&7\\\\3&10&5\\end{matrix}}$ are given in Table REF .", "It is clear from the table that $ \\operatorname{SPEED}^-_1= \\operatorname{SPEED}_3^+=3,\\quad \\operatorname{SPEED}^-_2= \\operatorname{SPEED}_2^+=4,\\quad \\operatorname{SPEED}^-_3= \\operatorname{SPEED}^+_1=2.$ This agrees with the statement of Theorem REF .", "Next, it is also apparent from the table that the entries of ${\\mathbf {S}}(t)$ weakly increase, and each of them changes if and only if for at least one vertex in the corresponding connected component, the sum of blue neighbors is strictly larger than the sum of red neighbors.", "This is precisely the statement of Proposition REF .", "Finally, observe that for every vertex $v\\in {\\hat{\\Lambda }}_r$ , we have $ \\mathfrak {t}^{ \\lambda }_v(4)= \\mathfrak {t}^{ \\lambda }_v(2)+2 \\operatorname{SPEED}^+_r,$ which is an application of Corollary REF .", "Table: The evolution of the tropical TT-system of type A 3 ⊗A ^ 1 A_3\\otimes {\\hat{A}}_1.", "The blue boldface numbers are the ones for which the sum of blue neighbors was strictly larger than the sum of red neighbors.Let $H_r^{\\oplus } = \\max _{\\mathfrak {h}(\\mathcal {D}) = r} \\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}}(u)){ \\lambda }(u),$ be the tropicalizations of Goncharov-Kenyon Hamiltonians $H_r$ .", "Here $d_{\\mathcal {D}}(u)$ is as before the degree of $u$ in the associated graph $G_{\\mathcal {D}}$ on the cylinder, and the sum is taken over all tilings $\\mathcal {D}$ of height $4r + \\mathfrak {h}(\\mathfrak {S})$ .", "Here on the boundary we always have $u=0$ .", "Lemma 13.3 The $H_r^{\\oplus }$ are conserved quantities of the tropical $T$ -system of type $A_m \\otimes {\\hat{A}}_{2n-1}$ .", "Follows from Proposition REF combined with the fact that $H_r$ 's themselves are conserved quantities of the corresponding geometric $T$ -system, see Corollary REF .", "Our strategy consists of proving the following proposition.", "Proposition 13.4 Both $ \\operatorname{SPEED}_r^{+}$ and $ \\operatorname{SPEED}_{m+1-r}^{-}$ are equal to $H_r^{\\oplus }$ for respectively $t\\gg 0$ and $t\\ll 0$ .", "Example 13.5 Let us continue Example REF .", "Recall the formula for $H_1$ calculated in Example REF : $H_1 = \\frac{ab}{de}+\\frac{a}{be}+\\frac{b}{ad}+\\frac{c}{f}+\\frac{d}{a}+\\frac{ef}{bc}+\\frac{e}{cf}.$ Thus, $H_1^\\oplus =\\max (a+b-d-e,a-b-e,b-a-d,c-f,d-a,e+f-b-c,e-c-f).$ For instance, at $t=-4$ we have $H_1^\\oplus =\\max (2,-11,-2,-3,2,-1,-9)=2.$ Or we can take $t=0$ instead and get $H_1^\\oplus =\\max (0,1,2,2,1,-3,2)=2.$ We encourage the reader to check that for other moments of time, $H_1^\\oplus $ is always equal to 2, which is a statement of Lemma REF .", "In agreement with Proposition REF , we have $H_1^\\oplus = \\operatorname{SPEED}_1^+= \\operatorname{SPEED}_3^-.$ Theorem REF follows trivially from Proposition REF and Lemma REF , since as a conserved quantity $H_r^{\\oplus }$ is the same at any point in time, including $t\\gg 0$ and $t\\ll 0$ .", "Let us prove Proposition REF .", "We are going to prove the $ \\operatorname{SPEED}_r^{+} = H_r^{\\oplus }$ part, the other part is essentially verbatim.", "Let us formulate several key lemmas.", "Consider the time $t\\gg 0$ large enough for all speeds to have stabilized.", "Adopt the convention $ \\operatorname{SPEED}_0^+ = \\operatorname{SPEED}_{m+1}^+ = 0$ .", "Lemma 13.6 The speeds form a weakly subadditive function, i.e.", "for any $1 < r < m$ we have $2 \\operatorname{SPEED}_r^+ \\ge \\operatorname{SPEED}_{r-1}^+ + \\operatorname{SPEED}_{r+1}^+.$ Informally, our strategy is to show that the maximum in the definition of $H_r^{\\oplus }$ is achieved on the term equal to $ \\operatorname{SPEED}_r^{+}$ .", "Note that we do not claim that this is the only term where the maximum is achieved, just that it is one of such terms.", "The following lemma is a major step.", "We postpone its proof, and first show how to use it to imply Proposition REF .", "Lemma 13.7 The maximum in the expression $\\max _{\\mathfrak {h}(\\mathcal {D}) = r} \\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}}(u)){ \\lambda }(u)$ is achieved at one of the tilings $\\mathcal {D}$ consisting entirely of horizontal dominos.", "Recall that we can compute $h(\\mathcal {D})$ by walking up from vertex $O$ to vertex $O^{\\prime }$ on the cylinder, collecting a contribution of $\\pm 1$ or $\\pm 3$ on each step.", "Let $\\epsilon _i = +1$ if the $i$ -th step along this path contributes a positive value and let $\\epsilon _i = -1$ if it contributes a negative value.", "It is easy to see that $\\mathfrak {h}(\\mathcal {D}) = \\frac{m+1+\\sum _{i=1}^{m+1} \\epsilon _i}{2}.$ If all dominos of $\\mathcal {D}$ are horizontal, each layer of the cylinder $\\mathfrak {C}_{m,2n}$ has exactly two ways to be tiled, one contributing $\\epsilon _i = +1$ and the other contributing $\\epsilon _i = -1$ .", "Lemma 13.8 If all dominos in $\\mathcal {D}$ are horizontal, we have $\\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}}(u)){ \\lambda }(u) = \\frac{1}{2} \\sum _{i=1}^{m+1} \\epsilon _i \\cdot \\left( \\operatorname{SPEED}_i^+ - \\operatorname{SPEED}_{i-1}^+\\right).$ Depending on which of the two ways to tile the $i$ -th layer of $\\mathfrak {C}_{m,2n}$ is used, this part of the tiling contributes into degrees $d_{\\mathcal {D}}(u)$ for exactly half of vertices on each of the affine slices $i$ and $i-1$ .", "Specifically, it either contributes to degrees of white $u$ -s on the $i$ -th affine slice and degrees of black $u$ -s for the $(i-1)$ -st affine slice, or the other way around.", "The statement of the lemma is the numerical expression of this observation.", "Now we are ready to prove Proposition REF .", "By Lemma REF we know that the maximum in the definition of $H_r^{\\oplus }$ is achieved at one of the tilings $\\mathcal {D}$ with all dominos horizontal.", "There are $2^{m+1}$ such tilings, corresponding to $2^{m+1}$ choices one can make for each $\\epsilon _i$ : either $\\epsilon _i=1$ or $\\epsilon _i=-1$ .", "Furthermore, the maximum is taken over $\\mathcal {D}$ -s with $\\mathfrak {h}(\\mathcal {D}) = r$ , which means exactly $r$ among $\\epsilon $ -s are $+1$ .", "According to Lemma REF we know that $ \\operatorname{SPEED}_1^+ = \\operatorname{SPEED}_1^+ - \\operatorname{SPEED}_{0}^+ \\ge \\operatorname{SPEED}_2^+ - \\operatorname{SPEED}_{1}^+ \\ge \\ldots $ $\\ge \\operatorname{SPEED}_{m+1}^+ - \\operatorname{SPEED}_{m}^+ = - \\operatorname{SPEED}_{m}^+.$ Thus the maximum is obviously achieved when the first $r$ among $\\epsilon $ -s are equal to $+1$ , and the rest of them are equal to $-1$ .", "The terms cancel out resulting in $\\frac{1}{2} \\sum _{i=1}^{m+1} \\epsilon _i \\cdot \\left( \\operatorname{SPEED}_i^+ - \\operatorname{SPEED}_{i-1}^+\\right) = \\operatorname{SPEED}_r^+.$ Thus, the maximal term in the expression for $H_r^{\\oplus }$ is equal to $ \\operatorname{SPEED}_r^+$ , as desired." ], [ "Proof of Lemma ", "Consider a domino tiling $\\mathcal {D}$ of the cylinder $\\mathfrak {C}_{m,2n}$ which has vertical dominos.", "Our strategy will be to construct a different tiling $\\mathcal {D}^{\\prime }$ , which has strictly less vertical dominos than $\\mathcal {D}$ , and such that $\\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}}(u)){ \\lambda }(u) \\;\\; \\le \\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}^{\\prime }}(u)){ \\lambda }(u).$ Recall that there is a distinguished tiling $\\mathfrak {S}$ which we call the sea, such that superposition of $\\mathfrak {S}$ with any other tiling does not contain contractible closed cycles, except for possibly double dominos.", "Consider the double dimer $\\mathcal {D}\\cup \\mathfrak {S}$ obtained by taking superposition of our $\\mathcal {D}$ and $\\mathfrak {S}$ .", "Several hula hoops are formed.", "In fact, if $\\mathcal {D}$ is contributing to $H_r^{\\oplus }$ , then $\\mathfrak {h}(\\mathcal {D})=r$ and exactly $r$ hula hoops are formed.", "Figure: The superpositions 𝔖∪𝒟\\mathfrak {S}\\cup \\mathcal {D} (top) and 𝔖∪𝒟 ' \\mathfrak {S}\\cup \\mathcal {D}^{\\prime } (bottom) at the local part where hula hoop HH is being straightened.Consider the lowest of the hula hoops $H$ which has vertical dominos.", "It exists because we assume $\\mathcal {D}$ has some vertical dominos, and the part of $\\mathcal {D}\\cup \\mathfrak {S}$ not covered by hula hoops consists of horizontal double dominos.", "Choose one of the highest points in $H$ and consider the horizontal part of $H$ that contains this point together with two vertical dominos on its ends, see Figure REF .", "Note that $H$ may have several such parts, we just pick one of them.", "We create $\\mathcal {D}^{\\prime }$ by straightening $H$ in this local spot, as shown at the bottom of Figure REF .", "It is clear that $\\mathcal {D}^{\\prime }$ has strictly less vertical dominos than $\\mathcal {D}$ does.", "Proposition 13.9 We have $\\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}}(u)){ \\lambda }(u) \\le \\sum _{u \\in \\mathfrak {C}_{m,2n}} (1 - d_{\\mathcal {D}^{\\prime }}(u)){ \\lambda }(u).$ .", "Denote the vertices surrounding this part of $H$ at time $t$ by $a_i$ -s, $b_i$ -s, and $c_i$ -s, $0 \\le i \\le 2k+2$ as shown in Figure REF .", "Let us put $\\mu (v):= \\mathfrak {t}^{ \\lambda }_v(t)$ for all $v\\in { { \\operatorname{Vert}}(Q)}$ .", "We assume $t\\gg 0$ is sufficiently large for the claim of Corollary REF to hold.", "Then the time evolution of $a$ -s depends only on the values of $\\mu $ at $a$ -s, etc.", "More formally, the following lemma holds, describing the values of the tropical $T$ -system at time $t+k$ for all $0\\le k\\le n-1$ : Lemma 13.10 Define ${\\epsilon }_k$ to be 0 if $k$ is even and 1 if $k$ is odd.", "Then we have $ \\mathfrak {t}^{ \\lambda }_{a_{k+1}}(t+k) &=& \\sum _{i=1}^{k+1} \\mu (a_{2i-1}) - \\sum _{i=1}^{k} \\mu (a_{2i}),\\\\ \\mathfrak {t}^{ \\lambda }_{b_k}(t+k) &=& \\sum _{i=0}^{k} \\mu (b_{2i}) - \\sum _{i=1}^{k} \\mu (b_{2i-1}),\\\\ \\mathfrak {t}^{ \\lambda }_{b_{k+2}}(t+k) &=& \\sum _{i=1}^{k+1} \\mu (b_{2i}) - \\sum _{i=1}^{k} \\mu (b_{2i+1}),\\\\ \\mathfrak {t}^{ \\lambda }_{c_{k+1}}(t+k) &=& \\sum _{i=1}^{k+1} \\mu (c_{2i-1}) - \\sum _{i=1}^{k} \\mu (c_{2i}).$ A straightforward application of recurrences $ \\mathfrak {t}^{ \\lambda }_{a_i}(t+j+1) = \\mathfrak {t}^{ \\lambda }_{a_{i-1}}(t+j) + \\mathfrak {t}^{ \\lambda }_{a_{i+1}}(t+j) - \\mathfrak {t}^{ \\lambda }_{a_i}(t+j-1), \\text{ etc.", "}$ which hold due to Proposition REF and our choice of large enough initial time $t$ .", "Now we are ready to prove Proposition REF .", "Each edge of $G(\\mathcal {D})$ subtracts from the corresponding term of $H_r^{\\oplus }$ two variables on its ends.", "Thus, we will compare those contributions for $G(\\mathcal {D})$ and $G(\\mathcal {D})$ .", "We want to show that one is bigger than the other, which translates into $\\mu (b_0) + \\mu (b_1) + \\sum _{i=1}^k (\\mu (a_{2i}) + 2 \\mu (b_{2i}) + \\mu (c_{2i})) + \\mu (b_{2k+1}) + \\mu (b_{2k+2}) \\ge \\sum _{i=1}^k (\\mu (a_{2i-1}) + 2 \\mu (b_{2i-1}) + \\mu (c_{2i-1})).$ By Lemma REF this is easily seen to be equivalent to $ \\mathfrak {t}^{ \\lambda }_{b_k}(t+k) + \\mathfrak {t}^{ \\lambda }_{b_{k+2}}(t+k) \\ge \\mathfrak {t}^{ \\lambda }_{a_{k+1}}(t+k) + \\mathfrak {t}^{ \\lambda }_{c_{k+1}}(t+k),$ which holds by Proposition REF and our choice of large enough $t$ ." ], [ "Conjectures", "We conjecture that both soliton resolution and speed conservation properties hold for all families of our classification in Theorem REF .", "For soliton resolution, we make the following conjecture, generalizing Proposition REF .", "It can also be viewed as a tropical analog of Conjecture REF .", "Conjecture 14.1 For any quiver $Q$ in our affine $\\boxtimes $ finite classification and any initial conditions either over $\\mathbb {Z}$ , or more generally over $\\mathbb {R}$ , there exists $t_0$ such that for $\|t\| > t_0$ the edges of finite component graph $\\Delta $ do not affect the dynamics, i.e.", "for any vertex $v \\in Q$ we have $\\sum _{(u,v)\\in \\Gamma } \\mathfrak {t}^{ \\lambda }_u(t)\\ge \\sum _{(v,w)\\in \\Delta } \\mathfrak {t}^{ \\lambda }_w(t).$ In other words, for large enough time in both directions the affine slices of $Q$ evolve as separate particles.", "For speed conservation, we need to consider two cases, just as we did in Conjectures REF and REF .", "Each of the types $A_m$ , $D_m$ and $E_6$ has a canonical involution on the Dynkin diagram, sending the diagram to itself.", "As before, denote this involution $\\eta $ .", "Assume our tropical $T$ -system is of the tensor product type, and more specifically of the form $\\Lambda ^{\\prime } \\otimes \\hat{\\Lambda }$ , where $\\Lambda ^{\\prime }$ is a finite type Dynkin diagram of type $A_m$ , $D_{2m+1}$ or $E_6$ , and $\\hat{\\Lambda }$ is an arbitrary extended Dynkin diagram.", "Let $\\hat{\\Lambda }_v$ and $\\hat{\\Lambda }_{\\eta (v)}$ be two affine slices of $\\Lambda ^{\\prime } \\otimes \\hat{\\Lambda }$ such that their $\\Lambda ^{\\prime }$ coordinates are related by $\\eta $ .", "Let $ \\operatorname{SPEED}_{v}^{\\pm }$ and $ \\operatorname{SPEED}_{\\eta (v)}^{\\pm }$ be the corresponding speeds for $t\\gg 0$ and $t\\ll 0$ .", "Here we assume that Conjecture REF holds and thus the speeds are well-defined.", "The following conjecture generalizes Theorem REF .", "It can also be viewed as a tropical analog of Conjecture REF .", "Conjecture 14.2 We have $ \\operatorname{SPEED}_v^{+} = \\operatorname{SPEED}_{\\eta (v)}^{-}.$ Assume now we are in any other case, i.e.", "either our tropical $T$ -system belongs to a different family of the classification, or it is a tensor product but $\\Lambda ^{\\prime }$ is not of types $A_m$ , $D_{2m+1}$ or $E_6$ .", "Let $\\hat{\\Lambda }$ be any affine slice of the quiver, and let $ \\operatorname{SPEED}_{\\hat{\\Lambda }}^{\\pm }$ be the corresponding speeds as $t\\gg 0$ and $t\\ll 0$ .", "The following conjecture again generalizes Corollary REF .", "Conjecture 14.3 We have $ \\operatorname{SPEED}_{\\hat{\\Lambda }}^{+} = \\operatorname{SPEED}_{\\hat{\\Lambda }}^{-}.$" ] ]	1606.04878
[ [ "Understanding the Factors that Impact the Popularity of GitHub\n Repositories" ], [ "Abstract Software popularity is a valuable information to modern open source developers, who constantly want to know if their systems are attracting new users, if new releases are gaining acceptance, or if they are meeting user's expectations.", "In this paper, we describe a study on the popularity of software systems hosted at GitHub, which is the world's largest collection of open source software.", "GitHub provides an explicit way for users to manifest their satisfaction with a hosted repository: the stargazers button.", "In our study, we reveal the main factors that impact the number of stars of GitHub projects, including programming language and application domain.", "We also study the impact of new features on project popularity.", "Finally, we identify four main patterns of popularity growth, which are derived after clustering the time series representing the number of stars of 2,279 popular GitHub repositories.", "We hope our results provide valuable insights to developers and maintainers, which can help them on building and evolving systems in a competitive software market." ], [ "Introduction", "GitHub is the world's largest collection of open source software, with around 9 million users and 17 million public repositories.https://github.com/search/advanced, verified on 04/04/2016 In addition to a git-based version control system, GitHub integrates many features for social coding.", "For example, developers can fork their own copy of a repository, work and improve the code locally, and then submit a pull request to integrate their changes in the main repository.", "The key characteristics and challenges of this pull-based development model is recently explored in many studies [1], [2], [3], [4].", "However, GitHub also supports other typical features from social networks.", "For example, users can star a repository to manifest their interest or satisfaction with the hosted project.", "Consequently, the number of stars of a GitHub repository can be seen as a proxy of its popularity.", "Currently, the two most popular repositories on GitHub are FreeCodeCamp/FreeCodeCamp (a coding education software, which claims to have more than 300K usershttps://www.freecodecamp.com/about, verified on 04/04/2016) and twbs/bootstrap (a library of HTML and CSS templates, which is used by almost 7M web siteshttp://trends.builtwith.com/docinfo/Twitter-Bootstrap, verified 04/04/2016).", "A deep understanding of the factors that impact the number of stars of GitHub repositories is important to software developers because they want to know whether their systems are attracting new users, whether the new releases are gaining acceptance, whether their systems are as popular as competitor systems, etc.", "Unfortunately, we have few studies about the popularity of GitHub systems.", "The exceptions are probably an attempt to differentiate popular and unpopular Python repositories using machine learning techniques [5] and a study on the effect of project's popularity on documentation quality [6].", "By contrast, popularity is extensively studied on other social platforms, like YouTube [7], [8] and Twitter [9], [10].", "These studies are mainly conducted to guide content generators on producing successful social media content.", "Similarly, knowledge on software popularity might also provide valuable insights on how to build and evolve systems in a competitive market.", "This paper presents an in-depth investigation on the popularity of GitHub repositories.", "We first collected historical data about the number of stars of 2,500 popular repositories.", "We use this dataset to answer four research questions: RQ #1: How popularity varies per programming language, application domain, and repository owner?", "The goal is to provide an initial view about the popularity of the studied systems, by comparing the number of stars according to programming language, application domain, and repository owner (user or organization).", "RQ #2: Does popularity correlate with other characteristics of a repository, like age, number of commits, number of contributors, and number of forks?", "This investigation is important to check whether there are factors that can be worked to increase a project's popularity.", "RQ #3: How early do repositories get popular?", "With this research question, we intend to check whether gains of popularity are concentrated in specific phases of a repository's lifetime, specifically in early releases.", "RQ #4: What is the impact of new features on popularity?", "This investigation can show if relevant gains in popularity happen due to new features (implemented in new releases).", "In the second part of the paper, we identify four patterns of popularity growth in GitHub, which are derived after clustering the time series that describe the growth of the number of stars of the systems in our dataset.", "These patterns can help developers to understand how their systems have grown in the past and to predict future growth trends.", "Finally, in the third part of the paper, we present a qualitative study with GitHub developers to clarify some findings and themes of our study.", "A total of 44 developers participated to the study.", "The main contribution of this paper is an investigation of factors that may impact the popularity of GitHub repositories, including the identification of the major patterns that can be used to describe popularity trends.", "Although similar studies exist for social networks, to our knowledge we are the first to focus on the popularity of systems hosted in an ultra-large repository of open source code.", "Organization: The rest of this paper is organized as follows.", "Section describes and characterizes the dataset used in this study.", "Section uses this dataset to provide answers to four questions about the popularity of GitHub repositories.", "Section documents four patterns that describe the popularity growth of GitHub systems.", "Section reports the feedback of GitHub developers about three specific themes of our study.", "Section discusses threats to validity and Section presents related work.", "Finally, Section concludes the paper and lists future work." ], [ "Dataset", "The dataset used in this paper includes the top-2,500 public repositories with more stars in GitHub.", "We limit the study to 2,500 repositories for two major reasons.", "First, to focus on the characteristics of the highly popular GitHub systems.", "Second, because we investigate the impact of application domain on popularity, which demands a manual classification of the domain of each system.", "All data was obtained using the GitHub API, which provides services to search public repositories and to retrieve specific data about them (e.g., stars, commits, contributors, and forks).", "The data was collected on March 28th, 2016.", "Besides retrieving the number of stars on this date for each system, we also relied on GitHub API to collect historical data about the number of stars.", "For this purpose, we used a service from the API that returns all star events of a given repository.", "For each star, these events store the date and the user who starred the repository.", "However, GitHub API returns at most 100 events by request (i.e., a page) and at most 400 pages.", "For this reason, it is not possible to retrieve all stars events of systems with more than 40K stars, as is the case of FreeCodeCamp, Bootstrap, AngularJS, D3, and Font-Awesome.", "Therefore, these five systems are not considered when answering the third and fourth research questions (that depend on historical data) and also on the study about common growth patterns (Section ).", "Table REF shows descriptive statistics on the number of stars of the repositories in our dataset.", "The number of stars ranges from 2,150 (for CyberAgent/android-gpuimage) to 97,948 stars (for FreeCodeCamp/FreeCodeCamp).", "The median number of stars is 3,441.", "Table: Descriptive statistics on the number of stars of the repositories in our dataset of 2,500 popular GitHub systemsAge, Commits, Contributors, and Forks: Figure REF shows the distribution of the age (in number of weeks), number of commits, number of contributors, and number of forks for the 2,500 systems in the dataset.", "For age, the first, second, and third quartiles are 101, 169, and 250 weeks, respectively.", "For number of commits, the first, second, and third quartiles are 228, 608, and 1,721, respectively.", "For number of contributors, the first, second, and third quartiles are 17, 41, and 96, respectively;We report data from contributors as retrieved by GitHub API.", "This data may be different from the one presented on the project's page at GitHub, which only counts contributors with GitHub account.", "and for number of forks, the first, second, and third quartiles are 298, 533, and 1,045, respectively.", "Therefore, the systems in our dataset are mature and have many commits and contributors.", "Figure: Age, number of commits, number of contributors, and number of Forks (outliers are omitted)Programming Language: As returned by GitHub API, the language of a repository is the one with the highest percentage of source code, considering the files in the repository.", "Figure REF shows the distribution of the systems per programming language.", "JavaScript is the most popular language (855 repositories, 34.2%), followed by Python (203 repositories, 8.1%), Java (202 repositories, 8.0%), Objective-C (188 repositories, 7.5%), and Ruby (178 repositories, 7.1%).", "Despite a concentration of systems in these languages, the dataset includes systems in 53 languages, including Groovy, R, Julia, and XSLT (all with just one repository).", "Figure: Top-10 languages by number of repositoriesOwner: We also provide results grouped by repository owner.", "In GitHub, a repository can be owned by a user (e.g., torvalds/linux) or by an organization (e.g., facebook/react).", "In our dataset, 1,263 repositories (50.5%) are owed by organizations and 1,237 repositories (49.5%) by users.", "Application Domain: In the study reported in this paper, results are also grouped by application domain.", "However, different from other source code repositories, like SourceForge, GitHub does not include information about the application domain of a project.", "For this reason, we manually classified the domain of each system in our dataset.", "Initially, the first and third authors of this paper inspected the description of the top-200 repositories to provide a first list of application domains.", "After this initial classification, the first author inspected the short description (and in many cases the GitHub page and the project's page) of the remaining 2,300 repositories.", "During this process, he also marked the repositories with dubious classification decisions.", "These particular cases were discussed by the first and second authors, to reach a consensus decision.", "The spreadsheet with the proposed classification is publicly available at https://goo.gl/73Sbvz.", "The systems are classified in the following six domains: Application software: systems that provide functionalities to end-users, like browsers and text editors (e.g., WordPress/WordPress and adobe/brackets).", "System software: systems that provide services and infrastructure to other systems, like operating systems, middleware, servers, and databases (e.g., torvalds/linux and mongodb/mongo).", "Web libraries and frameworks (e.g., twbs/bootstrap and angular/angular.js).", "Non-web libraries and frameworks (e.g., google/guava and facebook/fresco).", "Software tools: systems that support software development tasks, like IDEs, package managers, and compilers (e.g., Homebrew/homebrew and git/git).", "Documentation: repositories with documentation, tutorials, source code examples, etc.", "(e.g., iluwatar/java-design-patterns).", "Figure REF shows the number of systems in each domain.", "The top-3 domains are web libraries and frameworks (837 repositories, 33%), non-web libraries and frameworks (641 repositories, 25%), and software tools (470 repositories, 18%).", "Figure: Number of repositories by domain" ], [ "Results", "In this section, we use the described dataset to answer the four research questions listed in the paper's introduction.", "RQ #1: How popularity varies per programming language, application domain, and repository owner?", "Figure REF shows the distribution of the number of stars for the top-10 languages with more repositories.", "The top-3 languages whose repositories have the highest median number of stars are: JavaScript (3,697 stars), Go (3,549 stars), and HTML (3,513 stars).", "The three languages whose repositories have the lowest median number of stars are PHP (3,245 stars), Java (3,224 stars), and Python (3,099 stars).", "By applying the Kruskal-Wallis test to compare multiple samples, we find that the distribution of the number of stars per language is different (p-value $=$ 0.001).", "Thus, we can consider that programming language may impact on system popularity.", "Figure: Stars by programming language (considering only the top-10 languages with more repositories)Figure REF shows the distribution of the number of stars for the repositories in each application domain.", "The median number of stars varies as follow: systems software (3,807 stars), web libraries and frameworks (3,596 stars), documentation (3,547 stars), software tools (3,538 stars), applications (3,443 stars), and now-web libraries and frameworks (3,204 stars).", "By applying the Kruskal-Wallis test, we find that the distribution of the number of stars by domain is different (p-value $<$ 0.001).", "Therefore, application domain is also an important factor that may impact on system popularity.", "Figure: Popularity by application domainFinally, Figure REF shows how popularity varies depending on the repository owner (i.e., user or organization).", "The median number of stars is 3,622 stars for repositories owned by organizations and 3,298 stars for repositories owned by users.", "By applying the Mann-Whitney test, we detect that indeed these distributions are different (p-value $<$ 0.001).", "We hypothesize that repositories owned by organizations—specially major software companies and free software foundations—have more funding and resources, which somehow explains their higher popularity.", "Figure: Popularity by repository ownerSummary: The top-5 languages with more stars are JavaScript, Python, Java, Objective-C, and Ruby (Figure REF ).", "However, the top-5 languages whose systems have the highest median number of stars are JavaScript, Go, HTML, CSS, and C (Figure REF ).", "The top-3 application domains whose repositories have more stars are systems software, web libraries and frameworks, and documentation.", "Repositories owned by organizations are more popular than the ones owned by individuals.", "Figure: Correlation analysis.", "In subfigures (c) and (d), the line is the identity relationRQ #2: Does popularity correlate with repository's age, number of commits, number of contributors, and number of forks?", "Figure REF shows scatterplots correlating the number of stars with the age (in number of weeks), number of commits, number of contributors, and number of forks of a repository.", "First, the plots suggest that stars are not correlated with the repository's age (Figure REF ).", "We have old repositories with few stars and new repositories with many stars.", "For example, apple/swift has only five months and 28,105 stars, while mojombo/chronic has more than 8 years and 2,440 stars.", "Essentially, this result shows that repositories gain stars at different speeds.", "We ran Spearman's rank correlation test and the resulting correlation coefficient is close to zero ($rho$ $=$ 0.0757 and p-value $<$ 0.001).", "The scatterplot in Figure REF suggests that stars are weakly correlated with number of commits ($rho$ $=$ 0.249 with p-value $<$ 0.001).", "Similarly, as presented in Figure REF stars are weakly correlated with contributors ($rho$ $=$ 0.341 with p-value $<$ 0.001).", "In this figure, a logarithm scale is used in both axes; the line represents the identity relation: below the line are the systems with more contributors than stars.", "Interestingly, two systems indeed have more contributors than stars: raspberrypi/linux (17,766 contributors and 2,739 stars) and Linuxbrew/linuxbrew (7,304 contributors and 2,241 stars).", "This happens because they are forks of highly successful repositories (torvalds/linux and Homebrew/brew, respectively).", "The top-3 systems with more stars per contributor are shadowsocks/shadowsocks (12,287 stars/contributor), octocat/Spoon-Knife (9,944 stars/contributor), and wg/wrk (7,923 stars/contributor).", "All these systems have just one contributor.", "The three systems with less stars per contributor are android/platform_frameworks_base (2.28 stars/contributor), FFmpeg/FFmpeg (2.39 stars/contributor), and DefinitelyTyped/DefinitelyTyped (2.68 stars/ contributor).", "Finally, Figure REF shows plots correlating a system popularity and its number of forks.", "As visually suggested by the figure, there is a strong positive correlation between stars and forks ($rho$ $=$ 0.549 and p-value $<$ 0.001).", "For example, twbs/bootstrap is the second repository with the highest number of stars and the second one with more forks.", "angular/angular.js is the third repository in number of stars and the third one with more forks.", "In Figure REF , we can also see that only nine systems (0.36%) have more forks than stars.", "As examples, we have a repository that just provides a tutorial for forking a repository (octocat/SpoonKnife) and a popular puzzle game (gabrielecirulli/2048), whose success motivated many forks with variations of the original implementation.", "Since the game can be downloaded directly from the web, we hypothesize that it receives most users' feedback in the web and not on GitHub.", "Summary: There is no correlation between numbers of stars and the repository's age; however, there is a weak correlation with commits and contributors.", "Moreover, a strong correlation with forks was found.", "RQ #3: How early do repositories get popular?", "Figure REF shows the cumulative distribution of the fraction of time a repository takes to receive at least 10%, at least 50%, and at least 90% of its stars.", "Specifically, the y-axis shows the fraction of repositories that achieved 10%, 50%, and 90% of their stars in a period of time that does not exceed the fraction of time shown in the x-axis.", "Around 40% of the repositories receive 10% of their stars very early, in the first days after the initial release (label A, in Figure REF ).", "We hypothesize that many of these initial stars come from early adopters, who start commenting and using novel open source software immediately after they are out.", "After this initial burst of popularity, the growth of half of the repositories tend to stabilize.", "For example, half of the repositories take 51% of their age to receive 50% of their stars (label B); and half of the repositories take 91% of their age to receive 90% of their total number of stars (label C).", "Figure: Cumulative distribution of the fraction of time a repository takes to receive 10%, 50%, and 90% of its starsSummary: Repositories have a tendency to receive more stars right after their first public release.", "After this period, for half of the repositories the growth rate tends to stabilize.", "RQ #4: What is the impact of new features on popularity?", "In this research question, we investigate the impact of new features on the popularity of GitHub repositories.", "The goal is to check whether the implementation of new features (resulting in new releases of the projects) contribute to a boost in popularity.", "Specifically, we selected 834 repositories from our dataset (33.3%) that follow a semantic versioning convention to number releases.http://semver.org In such systems, versions are identified by three integers, in the format $x.y.z$ , with the following semantics: increments in ${x}$ denote major releases, which can be incompatible with old versions; increments in $y$ denote minor releases, which add functionality in a backwards-compatible manner; and increments in $z$ denote patches implementing bug fixes.", "In our sample, we identified 580 major releases and 4,343 minor releases.", "First, we counted the fraction of stars received by each repository in the week following all releases (major or minor) and just after major releases.", "As mentioned, the goal is to check the impact of new releases in the number of stars.", "Figure REF shows the distribution of these fractions.", "When considering all releases, the fraction of stars gained in the first week after the releases is 1.1% (first quartile), 3.2% (second quartile), and 10.2% (third quartile).", "For the major releases only, it is 0.5% (first quartile), 1.4% (second quartile), and 4.3% (third quartile).", "so-fancy/diff-so-fancy (a visualization for git diffs) is the repository with the highest fraction of stars received after releases.", "The repository has 53 days and 4,402 stars.", "Since it has a fast releasing rate (one new release per week, on average), it gained almost of its stars (89.1%) in the weeks after releases.", "Figure: Fraction of stars gained in the first week after all releases and just after the major releasesWe computed a second ratio for each repository: fraction of stars in the week following all releases or just major releases (FS) $/$ fraction of time represented by these weeks (FT).", "When $\\mathit {FS}/\\mathit {FT} > 1$ , the repository gains proportionally more stars after the releases.", "Figure REF shows boxplots with the results of $\\mathit {FS}/\\mathit {FT}$ for all repositories.", "When considering all releases, we have that $\\mathit {FS}/\\mathit {FT}$ is 0.80 (first quartile), 1.25 (second quartile), and 1.98 (third quartile).", "For major releases only, we have that $\\mathit {FS}/\\mathit {FT}$ is 0.81 (first quartile), 1.53 (second quartile), and 2.98 (third quartile).", "Figure: Fraction of stars in the week following all releases (or just the major releases) // fraction of time represented by these weeksFigure REF shows the median values of $\\mathit {FS}/\\mathit {FT}$ computed using stars gained after $n$ weeks ($1 \\le n \\le 4$ ).", "This ratio decreases, both for major and for all releases.", "Therefore, although there is some gains of stars after releases, they tend to decrease after few weeks.", "Figure: Fraction of stars by fraction of time (median values), computed using different time intervals (in weeks)Summary: There is an acceleration in the number of stars gained just after releases.", "For example, half of the repositories gain at least 53% more stars in the week following major releases, than in the other weeks (see Figure REF ).", "However, because repositories usually have much more weeks without releases than with releases, this phenomenon is not sufficient to generate a major concentration of popularity gain after releases.", "For example, 75% of the systems gain at most 4.3% of their stars in the week following major releases (see Figure REF ).", "Figure: Clusters of time series representing the growth of the number of starts of 2,279 GitHub repositories" ], [ "Popularity Growth Patterns", "In this section, we investigate common patterns of popularity growth concerning the GitHub repositories in our dataset.", "To this purpose, we use the KSC algorithm [11].", "This algorithm clusters time series with similar shapes using a metric that is invariant to scaling and shifting.", "The algorithm is used in other studies to cluster time series representing the popularity of YouTube videos [12] and Twitter [9].", "Like K-means [13], KSC requires as input the number of clusters $k$ .", "Because the time series provided as input to KSC must have the same length, we only consider data regarding the last 52 weeks (one year).", "Due to this restriction, we exclude 216 repositories (8.6%) that have less than 52 weeks.", "We use the $\\beta _{CV}$ heuristic [14] to define the best number $k$ of clusters.", "$\\beta _{CV}$ is defined as the ratio of the coefficient of variation of the intracluster distances and the coefficient of variation of the intercluster distances.", "The smallest value of $k$ after which the $\\beta _{CV}$ ratio remains roughly stable should be selected.", "This means that new added clusters affect only marginally the intra and intercluster variations [8].", "In our dataset, the values of $\\beta _{CV}$ stabilize for $k=4$ (see Figure REF ).", "Figure: β CV \\beta _{CV} for 2≤k≤152 \\le k \\le 15Table: Popularity Growth Patterns" ], [ "Proposed Growth Patterns", "Figure REF shows plots with the time series in each cluster.", "The time series representing the clusters' centroids are presented in Figure REF .", "The time series in clusters C1, C2, and C3 suggest a linear growth, but at different speeds.", "On the other hand, the series in cluster C4 suggest repositories with a sudden growth on the number of stars.", "We refer to these clusters as including systems with Slow, Moderate, Fast, and Viral Growth, respectively.", "Figure: Time series representing the centroids of each clusterSlow growth is the dominant pattern, including 65.7% of the repositories in our sample, as presented in Table REF .", "The table also shows the number of repositories in each cluster and the percentage of stars gained by the cluster's centroids in the period under analysis (52 weeks).", "The speed in which the stars are gained by repositories on cluster C1 is the lowest one (27.3% of new stars in one year).", "Moderate growth is the second pattern with more repositories (26.9% of the repositories and 94% of new stars in one year).", "5.7% of the repositories have a fast growth (469.2% of new stars in the analyzed year).", "The last cluster (Viral Growth) describes repositories with a massive growth in their number of stars in a short period of time.", "It is a less common pattern, including 1.6% of the repositories.", "Figure REF shows two examples of systems with a viral growth: nylas/N1 (an email client, with a peak of more than 7,300 stars in a single week) and Soundnode/soundnode-app (a desktop client for SoundCloud, which received almost 1,400 stars in a single week).", "Figure: Examples of viral growthFigure: Percentage of systems following the proposed growth patterns, for the most popular programming languages, application domains, and repository owners" ], [ "Growth Patterns vs Repositories Properties", "Figure REF shows the percentage of systems following the proposed growth patterns, for the top-10 programming languages in number of repositories, application domains, repository owners, and age.", "The three languages with the highest percentage of systems with slow growth are Ruby (92%), CSS (82%), and HTML (79%).", "By contrast, the languages with the highest percentage of systems with fast growth are Go (7.6%) and Java (7.6%).", "Go is a new language that is attracting a lot of interest.https://www.thoughtworks.com/radar/languages-and-frameworks/go-language, verified on 04/07/2016.", "Regarding Java, 61 out of 95 repositories with fast growth are Android applications.", "JavaScript is the language with the highest number of repositories with viral growth (10 repositories), followed by C++ (5 repositories) and Python (5 repositories).", "In relative terms, 2.7% of the Python systems have a viral growth, followed by 1.6% of the systems implemented in Ruby.", "When we group the systems by application domain, 75% of the web libraries and frameworks have a slow growth.", "Interestingly, the two domains with the highest percentage of systems following a fast growth are documentation (9.8%) and non-web libraries and frameworks (6.9%).", "Regarding the repository owners, there is no substantial difference between users and organizations.", "For slow growth, the percentage of systems is 66.4% and 64.9%, for users and organizations, respectively.", "For fast growth, the percentage is 5.8% and 5.6%, respectively.", "Finally, the last bars in Figure REF show that old repositories tend to present a slow growth.", "The percentage of such repositories ranges from 30.8% (age $<$ 2 years) to 87.2% (age $\\ge $ 4 years).", "As mentioned, we found a high percentage of web frameworks and libraries—especially the ones implemented in Ruby, CSS, and HTML—with a slow growth.", "We hypothesize two main reasons to explain this result.", "First, web libraries and frameworks are the dominant applications in our dataset of popular applications (837 repositories, 33%).", "This implies in a high competition, with many systems disputing the same users.", "For example, we found a list of JavaScript MVC-based frameworks with slow growth, including systems like knockout/knockout, spine/spine, quirkey/sammy, and sproutcore/sproutcore.", "These systems have to compete with “blockbusters”, like angular/angular.js, which is certainly a challenging task.", "The second reason is that there are many highly popular web frameworks and libraries in our dataset.", "For example, among the top-10% repositories in number of stars, 42.8% are web libraries and frameworks.", "We cannot assume that these systems will present the same growth rates of less popular ones.", "For instance, if angular/angular.js starts to grow at 469.2% per year (the growth rate observed for the centroid of the repositories with fast growth) it will have almost 1.5M stars in two years." ], [ "Feedback from Developers", "We contacted the main developers of some GitHub repositories to clarify the results presented in the previous sections.", "Specifically, we surveyed developers about three themes: (a) the impact on popularity of repositories owned by users (Section REF ); (b) the main characteristics of successful releases (Section REF ); (c) the reasons for the peaks of popularity observed in systems with viral growth (Section REF ).", "The surveys were performed by means of follow-up emails." ], [ "Impact on Popularity of Repositories Owned by Users", "In Section (RQ #1), we found that repositories owned by organizations are more popular than the ones owned by individuals.", "For example, among the top-100 most popular repositories, only 30 repositories are owned by users.", "The developers of 17 of such systems have a public mail address in their GitHub profile.", "We sent a short survey to these developers and received responses from five of them (29.8%).", "In this survey, we asked two questions.", "First, we asked the developers about possible plans to migrate their repositories to an organization account.", "All developers answered negatively this question.", "Two developers mentioned they want to explicitly appear as the repository owner, like in this answer: “I worked hard to create the project, and having it under my personal username is necessary to have proper credit for it.” To complement the first question, we asked the developers if they agree that migrating the repositories to an organization account would help to attract more users.", "Four developers (80%) answered negatively to this question and only one participant provided the following answer: “It depends on what organization it is.", "If it's a well known org I'm sure it helps, otherwise I don't think it makes a difference.” Therefore, although it seems “easier” to organizations to reach the top positions of GitHub popularity ranking, some systems owned by individual developers also reach these positions.", "These developers usually do not want to move to organizational accounts, basically to keep full control and credit for their repositories." ], [ "Characteristics of Successful Releases", "To reveal the characteristics of the most successful releases in our dataset (see RQ #4, Section ), we perform a survey with the main developers of 60 releases with the highest fraction of stars gained on the week after the release (and whose developers have a public mail address on their GitHub profile).", "We received answers from 25 developers, which corresponds to a response ratio of 41.6%.", "First, we asked the developers about the type of features implemented in these releases.", "As presented in Table REF , the releases usually include both functional and non-functional requirements (14 answers), followed by releases with mostly functional requirements (9 answers).", "We did not receive answers about releases including non-functional requirements.", "Two developers provide other types of answers (“complete rewrite” and “maintenance release”, respectively).", "Table: Features implemented in successful releasesWe also asked the developers to explain how the features implemented in these releases were selected (answers including multiple items are possible in this question).", "As presented in Table REF , the features usually come from ideas of the repository' maintainers (23 answers) and from user's suggestions (11 answers).", "Table: How the features are selected?" ], [ "Reasons for Viral Growth", "To expose the reasons for viral growth, we sent a message to the main developer of 22 systems with viral growth and who have a public mail address on their GitHub profile.", "In the message, we asked the developers to explain the peaks observed in the number of stars of their repositories.", "We received answers from 14 developers, which corresponds to a response ratio of 63%.", "As presented in Table REF , 11 developers (78.5%) linked the peaks to posts in social media sites, mostly Hacker News.https://news.ycombinator.com/ For example, we received the following answer: “I posted about this project on HackerNews.", "It quickly got a lot of attention and remained on the front page of HackerNews (a very high traffic tech site) for over 24 hours.", "It subsequently made it onto the github.com/explore as one of the top starred repositories for around a week.", "Because the repo was highlighted in these two high-profile locations for so much time, it received an incredible amount of traffic, which translated to a considerable number of stars.” Table: Sources of popularity" ], [ "Threats to Validity", "Number of stars as a proxy for popularity: In the paper, we consider that stars are proxies for a project popularity, as common in studies about the popularity of social media content [7], [8], [9], [10].", "However, a developer can star a repository for other reasons, for example, when she in fact finds problems in the system and wants to create a bookmark for later access and analysis.", "Dataset.", "GitHub has millions of repositories.", "We build our dataset by collecting the top-2,500 repositories with more stars, which represents a small fraction in comparison to the GitHub's universe.", "However, our goal is exactly to investigate the popularity of the most starred repositories.", "Furthermore, most GitHub repositories are forks and have very low activity [15], [16].", "Application domains.", "Because GitHub does not classify the hosted applications in domains, we performed this classification manually.", "Therefore, it is subjected to errors and inaccuracies.", "To mitigate this threat, the dubious classification decisions were discussed by two paper's authors.", "Growth patterns.", "The selection of the number of clusters is a key parameter in algorithms like KSC.", "To mitigate this threat, we employed a heuristic that considers the intra/intercluster distance variations [14].", "Furthermore, the analysis of growth patterns was based on the stars obtained on the last year.", "The stars before this period are not considered, since the KSC algorithm requires time series with the same length." ], [ "Related Work", "Several studies examine the relationship between popularity of mobile apps and their code properties [17], [18], [19], [20], [21], [22], [23], [24], [25].", "Yuan et al.", "investigate 28 factors along eight dimensions to understand how high-rated Android applications are different from low-rated ones [22].", "Their result shows that external factors, like number of promotional images, are the most influential factors.", "Guerrouj and Baysal explore the relationships between mobile apps' success and API quality [26].", "They found that changes and bugs in API methods are not strong predictors of apps' popularity.", "Ruiz et al.", "examine the relationship between the number of ad libraries and app's user ratings [20].", "Their results show that there is no relationship between the number of ad libraries in an app and its rating.", "Linares-Vásquez et al.", "investigate how the fault- and change-proneness of Android API elements relate to applications' lack of success [19].", "They state that making heavy use of fault- and change-prone APIs can negatively impact the success of these apps.", "Other studies examine source code repositories in order to understand what makes a project popular.", "Weber and Luo attempt to differentiate popular and unpopular Python projects on GitHub using machine learning techniques [5].", "They found that in-code features are more important than author metadata features.", "Zho et al.", "study the frequency of folders used by 140 thousands GitHub projects and the results suggest that the use of standard folders (e.g., doc, test, examples) may have an impact on project popularity [27].", "Bissyande et al.", "analyze the popularity, interoperability, and impact of various programming languages, using a dataset of 100K open source software projects [28].", "Aggarwal et al.", "study the effect of social interactions on GitHub projects' documentation [6].", "They conclude that popular projects tend to attract more documentation collaborators.", "By analyzing usage of Java APIs, Mileva states that popularity trend is a method for displaying the users preferences and for predicting their future [29].", "Finally, other studies analyze the relationship between popularity and software quality.", "Sajnani et.", "al.", "study the relationship between component popularity and component quality in Maven [30], finding that, in most cases, there is no correlation.", "Capra et.", "al.", "evaluate the effect of firms' participation on communities of open source projects and conclude that firms' involvement improves the popularity, but leads to lower software quality [31].", "To our knowledge, we are the first study to track popularity over time on social code sharing sites, like GitHub.", "However, there are similar studies in other contexts, like App Stores [23], video sharing sites [32], and social platforms [10].", "Chatzopoulou et al.", "[32] analyze popularity of Youtube videos by looking at properties and patterns metrics.", "They report that many of the popularity metrics are highly correlated.", "In our study, we also report correlations between stars and other popularity metrics (e.g., forks).", "Lehmann et al.", "[9] analyze popularity peaks of hashtags in Twitter.", "They found four usage patterns restricted to a two-week period centered on the peak time whereas the popularity patterns presented in this study are based on the last year data." ], [ "Conclusion", "In this paper, we first studied the popularity of GitHub repositories aiming to answer four research questions.", "We concluded that three most common domains on GitHub are web libraries and frameworks, non-web libraries and frameworks, and software tools.", "However, the three domains whose repositories have more stars are systems software, web libraries and frameworks, and documentation.", "Additionally, we found that repositories owned by organizations are more popular than the ones owned by individuals (RQ #1).", "We also reported the existence of a strong correlation between stars and forks, a weak correlation between stars and commits, and a weak correlation between stars and contributors (RQ #2), confirming the importance of a large base of contributors to the success of open source software [33].", "We concluded that repositories have a tendency to receive more stars right after their first public release.", "After this period, for half of the repositories the growth rate tends to stabilize (RQ #3).", "In other words, bursts of popularity do not explain the popularity growth of most repositories.", "We showed that there is an acceleration in the number of stars gained just after releases (RQ #4), which confirms the importance of developers constantly evolving and improving their systems.", "We identified four patterns of popularity growth, which were derived after clustering the time series that describe the number of stars of the systems in our dataset.", "We found that slow growth is the most common pattern (65.7%) and that very few systems present a viral behavior (1.6%).", "Slow growth is more common in case of overpopulated application domains (as web libraries and frameworks) and for old repositories.", "As future work, we plan to investigate repositories that are not popular yet and compare them with the popular ones.", "We also plan to correlate repository and language popularity to provide relative measures of popularity.", "For example, if we restrict the analysis to developers from a given language, a Scala repository can be considered more popular than a JavaScript one, although having less stars.", "Moreover, we plan to investigate models for predicting software popularity, which can be used for example to warn developers when signs of stagnation are detected in their repositories." ], [ "Acknowledgments", "This research is supported by FAPEMIG and CNPq." ] ]	1606.04984
[ [ "Electrons at the monkey saddle: a multicritical Lifshitz point" ], [ "Abstract We consider 2D interacting electrons at a monkey saddle with dispersion $\\propto p_x^3-3p_xp_y^2$.", "Such a dispersion naturally arises at the multicritical Lifshitz point when three van Hove saddles merge in an elliptical umbilic elementary catastrophe, which we show can be realized in biased bilayer graphene.", "A multicritical Lifshitz point of this kind can be identified by its signature Landau level behavior $E_m\\propto (Bm)^{3/2}$ and related oscillations in thermodynamic and transport properties, such as de Haas-van Alphen and Shubnikov-de Haas oscillations, whose period triples as the system crosses the singularity.", "We show, in the case of a single monkey saddle, that the non-interacting electron fixed point is unstable to interactions under the renormalization group flow, developing either a superconducting instability or non-Fermi liquid features.", "Biased bilayer graphene, where there are two non-nested monkey saddles at the $K$ and $K^\\prime$ points, exhibits an interplay of competing many-body instabilities, namely s-wave superconductivity, ferromagnetism, and spin- and charge-density wave." ], [ "Introduction", "Systems of two-dimensional (2D) electrons close to van Hove (vH) singularities[1], [2], [3], [4], [5], [6], [7], [8], [9] are of interest because of their displayed logarithmic enhancement of the electron density of states (DoS), which translates into a propensity to many-body instabilities[1].", "Among many exciting possibilities opened by proximity to vH singularities is that unconventional $d+id$ chiral superconductivity could occur in strongly doped graphene monolayer[10].", "Figure: Pictorial representation of Fermi surface families in abiased bilayer graphene system for three different values of theinterlayer voltage bias δ\\delta .", "Three van Hove saddles withdispersions ∝(p x 2 -p y 2 )\\propto (p_x^2-p_y^2) are shown with black dots(δ≠δ c \\delta \\ne \\delta _c) while arrows indicate their displacement uponincreasing the value of δ\\delta .", "At the critical value of the biasδ c \\delta _c they merge into a monkey saddle∝(p x 3 -3p x p y 2 )\\propto (p_x^3-3p_xp_y^2) that closes into a trifolium-shaped Fermisurface.The transition of the Fermi level through a vH singularity can be interpreted essentially as a Lifshitz transition of a neck-narrowing type[11], wherein two disconnected regions of the Fermi surface (FS) merge together.", "Alternatively, if the touching occurs at the edge of the Brillouin zone, as it happens for the square lattice, it may be interpreted as a FS turning inside out (from electron-like to hole-like).", "A multicritical Lifshitz point (MLP) arises as both a crossing of several Lifshitz transition lines, and as a singularity in the electronic dispersion $\\xi ({p})$ .", "MLPs of bosonic type have been analyzed and classified in the context of phase transitions, where terms in the free-energy-density functional with higher-order derivatives of an order parameter, say the magnetization, need to be kept at special points in the phase diagram [12], [13], [14].", "Yet, MLPs of fermionic type, with a singularity in the fermionic dispersion $\\xi ({p})$ , have been largely unexplored, only in a scenario involving Majorana fermions and spin liquids [15] where the monkey saddle was produced because of symmetries of the low energy Hamiltonian as opposed to a merging of several vH singularities.", "In this paper we study fermionic MLPs, using biased bilayer graphene (BLG) as a concrete example of a physical realization.", "In the case of BLG, three vH saddles merge into a monkey saddle at critical value of the interlayer voltage bias (see Figs.", "REF ,REF ).", "Mathematically, the monkey saddle is a genuine mathematical singularity with a degenerate quadratic form as opposed to vH saddle, which is not a true singularity in a mathematical sense, having a non-degenerate quadratic form of the $(+-)$ signature, $\\propto p_x^2-p_y^2$ .", "Physically, we identify key differences between the case of a MLP and that of the usual vH singularity.", "First, the monkey-saddle-like dispersion $\\propto p_x^3-3p_xp_y^2$ at the MLP exhibits a stronger, power-law divergence in the DoS and thus leads to even stronger many-body instabilities, with higher transition temperatures as a result.", "These stronger DoS divergences greatly simplify the renormalization group (RG) analysis of the problem, yielding a super-renormalizable theory.", "We find that the non-interacting electron fixed point is unstable to interactions, developing either a superconducting instability or non-Fermi liquid behavior.", "In the case of BLG, which has two non-nested monkey saddles at the $K$ and $K^{\\prime }$ points, interactions lead to instabilities to $s$ -wave superconducting state, ferromagnetism, spin-, and charge density wave, depending on the nature of interactions.", "Second, the monkey saddle possesses a signature Landau level (LL) structure with energy levels $E_m\\propto (Bm)^{3/2}$ .", "In addition, oscillations in different thermodynamic and transport properties, such as de Haas-van Alphen and Shubnikov de Haas oscillations, for example, are sensitive to the presence of the multicritical point.", "The monkey saddle can be identified by the scaling of the period of these oscillations with the Fermi energy as $\\Delta (1/B)\\propto E_F^{2/3}$ and with an abrupt tripling of the period as Fermi level goes from below to above the saddle, due to a change of the FS topology.", "The presentation of the results in the paper is organized as follows.", "In Sec.", "we present how the monkey saddle arises in voltage-biased BLG.", "We show how four different FS topologies can be attained by varying the bias voltage and the chemical potential, and identify the MLP in the phase diagram as the location where these four different phases meet at a point.", "There we also discuss the nature of the divergence in the density of states for the monkey saddle dispersion.", "In Sec.", "we obtain the energies of the quantized Landau orbits within a quasiclassical approximation, and present arguments for the period tripling of the magnetic oscillations as the system undergoes a FS topology change; these features may serve as clear experimental telltales of the MLP in BLG.", "In Sec.", "we present an RG analysis of the case when interactions are present in a system with an isolated monkey saddle, where we show that the system is either unstable to superconductivity or flows to a non-Fermi liquid, depending on the sign of the interactions.", "The RG analysis for the case of BLG with two monkey saddles at the $K$ and $K^{\\prime }$ points is studied in Sec.", ", where we discuss the possible instabilities of the system.", "We close the paper by summarizing the results and discussing open problem in Sec.", "." ], [ "Hamiltonian and dispersion", "Here we explicitly show how the monkey saddle arises in BLG.", "We consider AB-type stacked BLG, with the layers labeled by 1 and 2, and the two sublattices within each layer labeled by $A$ and $B$ .", "The spinor representing the electronic amplitudes is chosen in the order $({A1}, {B1}, {A2}, {B2})$ .", "We consider an extended tight-binding model that includes next-nearest neighbor hopping, where the Hamiltonian of the system linearized near the $K$ point is [16] $\\check{H}_0=\\begin{pmatrix}\\frac{1}{2}V & vp_- & 0 & v_3p_+\\\\vp_+ & \\frac{1}{2}V & \\gamma _1 & 0\\\\0 & \\gamma _1 & -\\frac{1}{2}V & vp_-\\\\v_3p_- & 0 & vp_+ & -\\frac{1}{2}V\\end{pmatrix}.$ Here $v$ is the band velocity of monolayer graphene, $\\gamma _1=0.4\\text{ eV}$ is an interlayer coupling constant and $v_3\\approx 0.1v$ describes trigonal warping that arises as a result of the next-nearest-neighbor hopping.", "$V$ is an interlayer voltage bias and $p_\\pm =p_x\\pm ip_y$ is the momentum.", "BLG has four energy bands and in this paper we are focused solely on the lowest upper band with an electron dispersion[17] $\\begin{split}\\xi ^2({p})=\\frac{V^2}{4}\\left(1-2\\frac{v^2p^2}{\\gamma _1^2}\\right)^2+v_3^2p^2+\\dots \\\\\\dots +2\\frac{v_3v^2}{\\gamma _1}p^3\\cos 3\\phi +\\frac{v^4p^4}{\\gamma _1^2}.\\end{split}$ For voltage biases $V$ of the order of the trigonal warping energy scale $\\gamma _1$ the $\\propto p^4$ contribution arising from the first term can be safely neglected.", "It is convenient to introduce dimensionless variables, redefining energies as $\\xi \\rightarrow (v_3\\gamma _1/v)\\xi $ and momenta as ${p}\\rightarrow (v_3\\gamma _1/v^2){p}$ , $\\xi ^2({p})=(\\delta /2)^2+u_3^2\\left[(1-\\delta ^2)p^2+2p^3\\cos 3\\phi +p^4\\right],$ where $u_3\\equiv v_3/v\\approx 0.1$ is a dimensionless measure of the warping strength and $\\delta \\equiv V/(v_3\\gamma _1/v)$ .", "The dispersion near the $K^\\prime $ point can be obtained from the one near the $K$ point by inversion, ${p}\\rightarrow -{p}$ .", "Unlike in the case of a monolayer graphene, where the warping merely distorts the Dirac cone with low-energy dispersion unaffected, BLG behaves in a very different way.", "In the absence of interlayer voltage bias, the trigonal warping destroys the parabolic dispersion, breaking it down into four Dirac cones.", "A non-zero interlayer voltage $V$ gaps out these Dirac cones while also gradually inverting the central electron pocket into a hole-like pocket at the critical value of the bias $V_c=(v_3/v)\\gamma _1$ ($\\delta _c=1$ in dimensionless units introduced above).", "This critical value of the bias marks a singularity in the electronic dispersion $\\xi ({p})$ .", "At the subcritical interlayer voltage bias $\\delta <1$ the electronic dispersion $\\xi ({p})$ has seven extremal points, four electronic pockets and three vH saddle points.", "While the three outer electronic pockets are robust and are present at all voltage biases, the central extremum and three vH saddle points merge at the critical voltage falling apart again into three saddles and a hole-like pocket at the supercritical bias $\\delta >1$ , see Fig.", "REF .", "In the vicinity of the singular point the electronic dispersion behavior is governed by the lowest powers of the momentum: $\\xi ({p})\\propto \\underbrace{(1-\\delta ^2)p^2}_{\\text{Pert(2,1)}}+\\underbrace{p^3\\cos 3\\phi }_{\\text{CG(2)}}.$ This momentum behavior corresponds exactly to the symmetry-restricted elliptic umbilic elementary catastrophe ($D_4^-$ within $ADE$ classification) [18].", "From the point of view of the catastrophe theory the cubic term $p^3\\cos 3\\phi \\equiv \\text{CG(2)}$ is a catastrophe germ defining the nature of the singularity in $\\xi ({p})$ function, while the quadratic term $(1-\\delta ^2)p^2\\equiv \\text{Pert(2,1)}$ is a lattice-symmetry restricted perturbation, with one parameter $\\delta $ , which regularizes the singularity.", "Qualitatively the behavior of the system can be viewed as a bifurcation of a monkey saddle $p^3\\cos 3\\phi \\equiv p_x^3-3p_xp_y^2$ into three vH(ordinary) saddles and a maximum/minimum: $\\underbrace{p_x^3-3p_xp_y^2}_{\\text{monkey saddle}} \\longleftrightarrow 3\\times \\underbrace{(p_x^2-p_y^2)}_{{\\text{vH saddle}}}+1\\times \\underbrace{p^2}_{{\\text{e/h pocket}}}.$" ], [ "Strong density of states divergence", "The monkey saddle leads to a strong IR divergence in the DoS.", "While the vH saddle has a logarithmic DoS, any generic higher order saddle $\\xi ({p},n)=ap^n\\cos n\\phi $ has a power-law divergence in the DoS.", "To obtain the DoS for a higher order saddle, it is convenient to work on generalized hyperbolic coordinates $(\\xi ,\\eta )=a(p^n\\cos n\\phi , p^n\\sin n\\phi )$ (where $n=1,2$ correspond to polar and hyperbolic coordinates, respectively).", "The dispersion of the saddle is given by the $\\xi $ variable, while $\\eta $ plays the role of the hyperbolic angle, parametrizing displacements along the FS.", "The density of states is given by $\\begin{split}\\nu (\\xi ,n)=\\oint _{FS}\\frac{(\\mathrm {d}{p})}{\\mathrm {d}\\xi }=\\frac{a^{-2/n}}{(2\\pi )^2n}\\int _{-\\infty }^{+\\infty }\\frac{\\mathrm {d}\\eta }{(\\xi ^2+\\eta ^2)^{\\frac{n-1}{n}}}\\\\=\\frac{a^{-2/n}}{4n\\pi ^{3/2}}\\frac{\\Gamma \\left(\\frac{1}{2}-\\frac{1}{n}\\right)}{\\Gamma \\left(1-\\frac{1}{n}\\right)}\\xi ^{-\\frac{n-2}{n}},\\end{split}$ where the $(\\mathrm {d}{p})\\equiv d^2p/(2\\pi )^2$ , and we set Planck's constant to unit ($\\hbar =1$ ).", "Figure: Quasiclassical LL orbits in momentum space for energiesslightly below and slightly above the monkey saddle (and criticalvoltage bias).", "The number of connected FS components changes fromthree to one as the Fermi level crosses zero." ], [ "Fermi surface topology phase diagram", "The electron FS at a given Fermi energy is defined as a cross-section of the electron dispersion $\\xi ({p},\\delta )=E_F$ .", "There are four distinct Fermi surface topology phases within the $(\\delta ,E_F)$ plane (see Fig.", "REF ).", "All of them have the same three-fold symmetry but can be discerned by their topological invariants, the number of connected components and the number of holes.", "Namely, in our case the four phases can be labeled uniquely by the first two Betti numbers of their FS $(b_{0},b_{1})$ as (1,0), (4,0), (3,0), and (1,1).", "These four phases are separated by two lines of topological phase transitions.", "One of the lines is of a weaker, band-edge transition type, while another is of a stronger vH type (the former has a jump in the DoS while the latter has a log-divergence).", "The multicritical Lifshitz point lies at the intersection of these two lines." ], [ "Magnetic oscillations at the monkey saddle", "Within a quasiclassical approximation, the LLs can be obtained by quantization of the area enclosed by quasiparticle orbit in momentum space, $\\int (d{p})=\\frac{m}{2\\pi l_B^2},$ where $l_B=\\sqrt{c/eB}$ is a magnetic length and $m$ is the LL index.", "For a system tuned exactly to the monkey saddle (or any higher order saddle), the behavior is dominated by the singularity itself, so that $\\begin{split}\\int _0^{E_m}\\nu (\\xi )d\\xi =\\frac{1}{8\\pi ^{1/2}}\\frac{\\Gamma \\left(\\frac{1}{2}-\\frac{1}{n}\\right)}{\\Gamma \\left(1-\\frac{1}{n}\\right)}\\left(\\frac{E_m}{a}\\right)^{\\frac{2}{n}}\\\\\\Rightarrow E_m=\\alpha \\left(\\frac{a}{l_B^n} \\right)m^{n/2}\\propto (Bm)^{n/2}\\end{split}$ with a numerical coefficient $\\alpha =\\left(4\\sqrt{\\pi }\\frac{\\Gamma \\left(1-\\frac{1}{n}\\right)}{\\Gamma \\left(\\frac{1}{2}-\\frac{1}{n}\\right)}\\right)^{\\frac{n}{2}}\\underset{(n=3)}{=}2.27.$ As always, LLs imply oscillations of various transport and thermodynamic properties in an applied magnetic field.", "Since such oscillations happen as LLs cross the Fermi level of the system.", "At the critical voltage bias $\\delta _c=1$ but with a small positive detuning from the energy of the saddle point, i.e., $E_F$ slightly higher than $\\delta _c/2$ , we can see from Eq.", "(REF ) that we have a periodicity in inverse magnetic field with a period $\\Delta \\left(\\frac{1}{B}\\right)=\\frac{e\\hbar }{c}\\left(\\frac{E_F}{\\alpha a}\\right)^{2/n},$ where we reinserted Planck's constant $\\hbar $ .", "Eqs.", "(REF ,REF ) are given for positive LL energies, when $E_F$ slightly higher than $\\delta _c/2$ , and the FS consists of one connected component, see Figs.", "REF ,REF .", "The situation is different for negative energies, when when $E_F$ slightly lower than $\\delta _c/2$ and the Fermi surface has three disconnected components.", "In this case the LLs are triply degenerate (on top of the valley degeneracy), and are three times as sparse, $E_{-m}=-\\alpha al_B^{-n}(3m)^{n/2},$ and oscillations period in inverse magnetic field is three times smaller as well.", "(All equations above are for spinless electrons: in a real system Zeeman splitting should be taken into account as well.)", "The tripling of the periodicity of oscillation is a telltale of the Fermi surface topology change, and can be viewed physically as follows.", "The area of the Fermi surface is not very different slightly before or slightly after it undergoes the topology change.", "At the critical point, the area that fits just one electron orbit is brought inside the Fermi surface upon insertion of a flux quantum.", "When there is a single surface, one can indeed fit a physical electron within that orbit.", "However, when the Fermi surface contains the three pockets, the additional area brought inside each pocket due to a single flux quantum insertion is only 1/3 of what is needed to fit one electron.", "If there were quasiparticles with charge 1/3, then they could fill separately the area in the three pockets; but there are no such particles in the system.", "Hence, the flux periodicity is tripled when the Fermi surfaces are disconnected, as one can only add a full electron at each pocket, requiring the addition of three flux quanta.", "This is the physical origin of the period tripling.", "Figure: Left: A Fermi surface near a van Hove saddle callsfor a two-cutoff RG scheme.", "The grey area represents occupiedelectron states.", "The hatched region of the phase space correspondsto a step dξd\\xi in electron energy.", "Normally, one cutoffdΛ 1 ∼dξd\\Lambda _1\\sim d\\xi is sufficient, but here we see that thelogarithmic DoS at the van Hove saddle together with an openhyperbolic Fermi surface lead to tails of the hatched region thatreach out to the rest of the Fermi surface away from the van Hovesaddle.", "The purpose of the second cutoff Λ 2 \\Lambda _2 is to cut thesetails and isolate van Hove saddle.Right: No second cutoff is needed at themonkey saddle." ], [ "RG flow at the monkey saddle", "Here we analyze a single monkey saddle within a one-loop RG framework.", "Assuming short-range interaction, an electron action is given by ${\\cal S}=\\int (\\mathrm {d}\\tau \\mathrm {d}{r})\\left[\\psi ^\\dagger [\\partial _\\tau -\\xi (-i{\\nabla })+\\mu ]\\psi -\\frac{g}{2}(\\psi ^\\dagger \\psi )^2\\right]$ with interaction $\\frac{g}{2}(\\psi ^\\dagger \\psi )^2=g(\\psi ^\\dagger _{\\uparrow }\\psi ^\\dagger _{\\downarrow }\\psi _{\\downarrow }\\psi _{\\uparrow }).$ We focus on the system tuned exactly to the monkey saddle, so that the dispersion is determined by the catastrophe germ $\\xi ({p})=p^3\\cos 3\\phi $ and the non-singular part of FS is irrelevant (see Fig.", "REF ).", "Tree-level RG involves rescaling of frequency and momenta as $\\omega \\rightarrow s^{-1}\\omega ,\\,{p}\\rightarrow s^{-1/3}{p},\\,\\psi \\rightarrow s^{-1/3}\\psi ,$ and results in the interaction constant scaling as $g\\rightarrow gs^{+1/3},$ entailing super-renormalizability of the theory.", "Super-renormalizability brings crucial simplifications with respect to the case of the ordinary vH saddle: while the separation of the saddle from the non-singular part of the FS requires two cut-offs in the case of vH singularities ($n=2$ ), it requires only one cut-off for higher order singularities ($n>2$ ), see Fig.", "REF .", "This difference can be traced back to the behavior of DoS obtained in Eq.", "(REF ).", "In the case of the vH saddle ($n=2$ ), the integral over the angle-like variable $\\eta $ diverges logarithmically, requiring an additional cut-off in the problem that is interpreted as a Fermi velocity cut-off in Refs. ghamari,kapustin.", "In contrast, for any higher-order saddle with $n>2$ , the DoS at a given energy is well-defined and is determined solely by the saddle and does not require a large momentum cut-off.", "This means that for $n>2$ the theory is free of UV divergences and contains only (meaningful) IR divergences that are regularized by temperature $T$ and chemical potential $\\mu $ .", "We introduce a dimensionless coupling constant in a natural way as $\\lambda (\\Upsilon )=\\nu (\\Upsilon ) g(\\Upsilon ), $ with a smooth infrared cutoff $\\Upsilon $ that we take to be either $\\mu $ or $T$ , so that the beta function for the dimensionless coupling constant is (see appendix) $\\frac{\\mathrm {d}\\lambda }{\\mathrm {d}\\ln \\nu (\\Upsilon )}=\\lambda -c\\lambda ^2$ with a non-negative coefficient $c=\\frac{\\mathrm {d}\\Pi _{pp}}{\\mathrm {d}\\nu (\\Upsilon )}-\\frac{\\mathrm {d}\\Pi _{ph}}{\\mathrm {d}\\nu (\\Upsilon )}\\ge 0,$ where $\\Pi _{pp}$ and $\\Pi _{ph}$ are particle-particle and particle-hole polarization operators.", "The scaling behavior of the system strongly resembles that of 1D interacting electrons.", "Namely, exactly at the monkey saddle at $\\mu =0$ the one-loop contribution to beta function vanishes, leaving a critical theory with tree-level scaling only $\\frac{\\mathrm {d}\\lambda }{\\mathrm {d}\\ln \\nu (T)}=\\lambda \\quad (\\mu =0,\\forall T).$ This behavior is linked to an additional symmetry[9] that arises exactly at the monkey saddle, and is a combination of time-reversal transformation $(\\varepsilon ,{p})\\rightarrow (-\\varepsilon ,-{p})$ plus a particle-hole transformation $\\psi ^\\dagger \\rightleftharpoons \\psi $ .", "This symmetry is present only for odd saddles with $\\xi (-{p})=-\\xi ({p})$ and is absent for even saddles that have a dispersion that is invariant under spatial inversion.", "Figure: Phase diagram (blue solid line) for an isolated monkey saddleand attractive coupling constant.", "Critical chemical potential isdetermined by the equation g 0 ν(μ c )=2\\left\|g_0\\right\|\\nu (\\mu _c)=2 and the plot isgiven in units of μ c \\mu _c for both temperature and chemicalpotential.", "Any odd saddle (n=3,5,⋯n=3,5,\\dots ) has qualitatively samephase diagram, but the situation is different for even saddles(n=2,4,⋯n=2,4,\\dots ).", "Even case is illustrated with red dashed line forn=4n=4.At the same time, away from the monkey saddle $\\frac{\\mathrm {d}\\lambda }{\\mathrm {d}\\ln \\nu (\\mu )}=\\lambda -\\frac{1}{2}\\lambda ^2\\quad (T\\ll \\left\|\\mu \\right\|\\ne 0),$ and the system either flows to a non-trivial fixed point $\\lambda =2$ for any positive initial coupling constant $\\lambda _0>0$ or develops a superconducting instability with $\\lambda $ diverging as (for $\\lambda _0<0$ ) $\\lambda (\\mu )=\\frac{\\nu (\\mu )g_0}{1+2g_0[\\nu (\\mu )-\\nu _0]}\\simeq \\frac{3\\mu _c}{2(\\mu _c-\\mu )}.$ Here $\\nu _0$ and $g_0$ are the DoS and coupling constant at the initial energy scale, while $\\mu _c$ marks the energy scale corresponding to the instability.", "This leads to a non-BCS type of behavior for the critical energy scale $\\mu _c,T_c\\propto g_0^{\\frac{n}{n-2}}\\underset{(n=3)}{=}g_0^3.$ In fact, the one-loop RG equations can be integrated out for any $\\mu ,T$ and the solution is equivalent to resummation of a leading diagrammatic series in the language of Feynman diagrams.", "The resulting expression for a dimensional coupling constant $g$ reads as $g^{-1}\|_{(\\mu ,T)}=\\left(\\Pi _{pp}-\\Pi _{ph}\\right)\|_{(\\mu ,T)}+g_0^{-1},$ Thus, within a one-loop approximation, the phase transition line for attractive interaction $g<0$ is determined by the equation $g_0\\left(\\Pi _{pp}-\\Pi _{ph}\\right)\|_{(\\mu ,T)}+1=0$ and the resulting phase diagram is given in Fig.REF .", "As to the quasiparticle width, it is zero within the one-loop approximation.", "A non-zero result can be obtained from a two-loop diagram that yields a quasiparticle width at the monkey saddle ($\\mu =0$ ) that signals non-Fermi-liquid behavior $\\Gamma \\sim \\lambda ^2(T)\\;T\\propto T^{1/3},$ since for $\\mu =0$ there is only a tree-level scaling and $\\lambda (T)=g\\,\\nu (T)\\propto T^{-1/3}$ for an invariant value of the dimensionful coupling constant $g$ .", "This implies that our analysis breaks down at energy scales $T^\\sim \\Gamma (T^)$ , or equivalently when dimensionless coupling constant $\\lambda (T^*)\\gtrsim 1$ becomes too large.", "The situation is the same for any odd saddle, $n=3,5,\\dots $ , but is very different for even saddles.", "For even saddles there is no cancellation of the one-loop contribution, so that $c\\ne 0$ at $\\mu =0$ and the dimensionless coupling constant flows to a fixed point $\\lambda =1/c$ yielding marginal Fermi liquid behavior with decay rate $\\Gamma \\sim T$ .", "While this implies a dimensionless coupling constant of order one, the existence of this fixed point could be justified within $1/N$ expansion techniques." ], [ "RG flow for bilayer graphene", "In BLG there are two copies of the monkey saddle at the $K$ and $K^\\prime $ points, which are related by time-reversal symmetry, with dispersions $\\xi _\\pm ({p})=\\pm \\xi ({p})$ .", "The four-fermion interaction now has three coupling constants: $\\frac{g}{2}(\\psi ^\\dagger \\psi )^2=g_1(\\psi ^\\dagger _{+ i}\\psi ^\\dagger _{- j}\\psi _{+ j}\\psi _{- i}) + g_2(\\psi ^\\dagger _{+i}\\psi ^\\dagger _{- j}\\psi _{- j}\\psi _{+ i} )\\nonumber \\\\+ {g_3(\\psi ^\\dagger _{+ i}\\psi ^\\dagger _{+ j}\\psi _{- j}\\psi _{- i})} + g_4(\\psi ^\\dagger _{\\alpha \\uparrow }\\psi ^\\dagger _{\\alpha \\downarrow }\\psi _{\\alpha \\downarrow }\\psi _{\\alpha \\uparrow }),$ where $i,j=\\uparrow \\downarrow $ indices stand for spin and $\\alpha =\\pm $ corresponds to $K/K^{\\prime }$ valley isospin, respectively.", "Our notation for coupling constants is the same as in Refs. furukawa,nandkishore.", "The Umklapp $g_3$ coupling is forbidden because the $K$ and $K^\\prime $ points are inequivalent in the sense of momentum conservation modulo reciprocal lattice vector, ${Q}=2{p}_{KK^\\prime }\\lnot \\simeq {0}$ .", "There are now four polarization operators that drive the RG flow, particle-particle and particle-hole at zero and ${Q}$ momentum transfer.", "We focus on BLG tuned exactly at the monkey saddle with both critical voltage bias $\\delta =1$ and chemical potential $\\mu =0$ .", "The relative roles of polarization operators are $&d_0\\equiv \\frac{\\mathrm {d}\\Pi _{pp}({Q})}{\\mathrm {d}\\Pi _{pp}({Q})}=1,\\, &d_2\\equiv \\frac{\\mathrm {d}\\Pi _{ph}({0})}{\\mathrm {d}\\Pi _{pp}({Q})}=1,\\\\&d_1\\equiv \\frac{\\mathrm {d}\\Pi _{ph}({Q})}{\\mathrm {d}\\Pi _{pp}({Q})}=3,\\, &d_3\\equiv \\frac{\\mathrm {d}\\Pi _{pp}({0})}{\\mathrm {d}\\Pi _{pp}({Q})}=3.$ Since $\\Pi _{pp}({Q})\\sim \\nu (T)$ , it is reasonable to define dimensionless interaction constants as $\\lambda _i=g_i\\Pi _{pp}({Q})$ and take $\\mathrm {d}[\\ln \\Pi _{pp}({Q})]$ as RG time.", "This gives RG equations $\\dot{\\lambda }_1=&\\lambda _1-6\\lambda _1^2+2\\lambda _1\\lambda _4,\\\\\\dot{\\lambda }_2=&\\lambda _2+2(\\lambda _1-\\lambda _2)\\lambda _4-3\\lambda _1^2,\\\\\\dot{\\lambda }_4=&\\lambda _4+\\lambda _1^2+2\\lambda _1\\lambda _2-2\\lambda _2^2,$ and the RG flow is in fact similar to that of the square lattice[1] with parameters $d_i$ given by Eqs.", "(REF ,) and one interaction channel turned off, $g_3\\equiv 0$ .", "The crucial difference with the case of a single monkey saddle is that the solution $\\lambda _1=\\lambda _2=0$ describing two decoupled saddles is now always unstable.", "The analysis of the RG flow is presented in the appendix and it shows that there are four possible many-body instabilities, $s$ -wave superconducting (SC), ferromagnetic (FM), charge-density-wave (CDW) and a competing spin/charge-density-wave (SDW/CDW).", "However, only three instabilities, SC, FM, and SDW/CDW are possible for initially repulsive interactions, as is shown in a Fig.", "REF .", "For Hubbard model the initial conditions correspond to all interaction constants being equal and positive, $\\lambda _i=(\\lambda )_0>0$ , and lead to FM phase.", "Figure: RG phase diagram showing a leading instability as afunction of initial coupling constants.", "The figure on the leftshows the case of positive λ 1 >0\\lambda _1>0, while the one on theright corresponds to λ 1 <0\\lambda _1<0.", "(λ 1 \\lambda _1 never changessign under the RG flow.)", "There are four possible instabilities:SC superconducting, ferromagnetic (FM), charge-density wave (CDW) and a competing spin/charge-density-wave(SDW/CDW).", "The Hubbard model initial conditionsλ 1 =λ 2 =λ 4 >0\\lambda _1=\\lambda _2=\\lambda _4>0 lead to the development of FMinstability." ], [ "Conclusions", "We studied the properties of electronic systems tuned to a monkey saddle singularity, where the dispersion is $\\propto p_x^3-3p_xp_y^2$ .", "We showed that such a situation occurs in a MLP where three vH singularities merge.", "We showed that such a singular point is accessible in BLG by controlling two parameters, the interlayer bias voltage and the chemical potential.", "We identified a number of experimentally accessible features associated with the monkey saddle dispersion when the system is subject to a magnetic field.", "The Landau level structure has a trademark behavior where $E_m\\propto (Bm)^{3/2}$ , different from the behavior of both linearly or quadratically dispersing systems.", "The oscillations of either thermodynamic or transport properties with the applied magnetic field (de Haas-van Alphen or Shubnikov-de Haas oscillations) contain a signature tripling of the oscillation period when the Fermi energy crosses the saddle point energy.", "This tripling, associated with the topological transition between a single- and three-sheeted FS, can be viewed as a smoking gun of the monkey saddle singularity.", "Generically, the singular electronic dispersion in such MLP implies a strong tendency towards development of many-body instabilities.", "We found that the stronger divergence of the DoS in monkey saddle singularities ($n=3$ ), as compared to the case of ordinary vH singularities ($n=2$ ), brings about crucial simplifications in the field theoretical analysis of the effect of interactions.", "We showed that the theory for systems with higher order singularities ($n>2$ ) is super-renormalizable.", "Thus, in contrast to the case of vH singularities where an RG analysis requires two cut-off scales to properly account for the singular and non-singular parts of the FS, the analysis of higher order saddles requires no large momentum (UV) cut-off, since there are only IR divergences, which are regularized by temperature $T$ and chemical potential $\\mu $ .", "Via an RG analysis of the super-renormalizable theory, we showed that the non-interacting electron fixed point of a system with a single monkey saddle is unstable to interactions, developing either a superconducting instability or non-Fermi liquid behavior.", "We also showed that the electronic lifetime depends crucially on the symmetry of the dispersion, with odd and even saddles displaying non-Fermi-liquid and marginal Fermi liquid behavior, respectively.", "For BLG, which has two two non-nested monkey saddles at the $K$ and $K^{\\prime }$ points, we showed that interactions (depending on their nature) lead to $s$ -wave superconductivity, ferromagnetism, charge-density wave, or spin-density wave.", "The studies of MLP in electronic systems suggest an exciting link to catastrophe and singularity theories.", "Namely, the monkey saddle could be considered as a lattice-symmetry-restricted elliptical umbilic elementary catastrophe $D_4^-$ .", "Catastrophe theory may be a useful language to classify the different possible singularities where FS topology changes.", "The relevant classification at criticality is not that of the FS topologies, but of the singularity itself.", "Controlling the chemical potential and the interlayer bias voltage in BLG is a clear example of how to engineer a catastrophe in an electronic system, the monkey saddle.", "Crystalline symmetries may reduce the possible types of catastrophes in the ADE classification that could be realized in solid state systems.", "Which other singularities could occur in electronic systems remains an open problem.", "However, our analysis of the physical consequences of such singularities should be applicable to other types of catastrophe in systems of electrons." ], [ "Acknowledgements", "This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) Grant No.", "EP/M007065/1 (G.G.)", "and by the DOE Grant DEF-06ER46316 (C.C.", ")." ], [ "RG flow", "The RG flow equation for the dimensionless interaction $\\lambda $ constant is connected to renormalization of the dimensional coupling constant $g$ as $\\frac{\\mathrm {d}\\lambda }{\\mathrm {d}\\ln \\nu }=\\frac{\\mathrm {d}(\\nu g)}{\\mathrm {d}\\ln \\nu }=\\lambda +\\nu ^2\\;\\frac{\\mathrm {d}g}{\\mathrm {d}\\nu }.$ The one-loop renormalization of $g$ is given by two diagrams shown in Fig.", "REF and yield $\\delta g= -g^2\\Pi _{pp}(\\mu ,T)+g^2\\Pi _{ph}(\\mu ,T).$ Combining Eqs.", "(REF ) and (REF ) we obtain the RG equation for $\\lambda $ , $\\frac{\\mathrm {d}\\lambda }{\\mathrm {d}\\ln \\nu (\\Upsilon )}=\\lambda -c\\lambda ^2,\\quad c=\\frac{\\mathrm {d}\\Pi _{pp}}{\\mathrm {d}\\nu (\\Upsilon )}-\\frac{\\mathrm {d}\\Pi _{ph}}{\\mathrm {d}\\nu (\\Upsilon )}\\ge 0\\;,$ presented in the main text as Eqs.", "(REF ) and (REF ).", "The polarization operators are defined as $\\Pi _{ph}({q},\\mu ,T)=&-T_{l,{p}}G(i\\varepsilon _{l},{p}+{q})G(i\\varepsilon _{l},{p}),\\\\\\Pi _{pp}({q},\\mu ,T)=&T_{l,{p}}G(i\\varepsilon _{l},{p}+{q})G(-i\\varepsilon _{l},-{p}),$ and the particle-hole polarization operator can be evaluated to be $\\Pi _{ph}=&-T\\int _{{p}}\\sum _l\\frac{1}{i\\varepsilon _l-\\xi _{{p}+{q}}+\\mu }\\frac{1}{i\\varepsilon _l-\\xi _{{p}}+\\mu }\\\\=&\\;\\frac{1}{2}\\int _{{p}}\\frac{f(\\xi _{{p}+{q}}-\\mu )-f(\\xi _{{p}}-\\mu )}{\\xi _{{p}+{q}}-\\xi _{{p}}}\\\\\\underset{{q}\\rightarrow 0}{=}&\\;\\frac{1}{2}\\int \\nu (\\xi )\\,f^\\prime (\\xi -\\mu )\\;\\mathrm {d}\\xi \\;,$ where $f(\\xi )=\\tanh \\xi /2T$ .", "Similarly, the particle-particle polarization operator is $\\Pi _{pp}=&\\frac{1}{2}\\int _{{p}}\\frac{f(\\xi _{{p}}-\\mu )+f(\\xi _{-{p}}-\\mu )}{\\xi _{{p}}+\\xi _{-{p}}-2\\mu }\\\\=&\\frac{1}{2}\\int \\nu (\\xi )\\,\\frac{f(\\xi +\\mu )-f(\\xi -\\mu )}{2\\mu }\\;\\mathrm {d}\\xi \\;.$ The difference of polarization operators that drives RG flow has the following asymptotic behavior: $\\Pi _{pp}-\\Pi _{ph}={\\left\\lbrace \\begin{array}{ll}0 & \\mu =0,\\,T\\ne 0\\\\\\dfrac{1}{2}\\nu (\\mu ) & T = 0,\\,\\mu \\ne 0\\end{array}\\right.", "},$ where the cancellation at $\\mu =0$ in fact holds for any external frequency and momentum.", "Figure: Top: one-loop contribution to renormalization of the interaction constant for an isolated monkey saddle.Bottom: two-loop contribution to the quasiparticle decay rate.The chemical potential also has a correction due to a Hartree-type diagram, $\\delta \\mu =g_{l,{p}} G(i\\varepsilon _l,{p}),$ corresponding to the shift in the monkey saddle's Fermi energy.", "(This contribution is the equivalent of the fluctuational renormalization of the critical temperature in thermodynamic phase transitions.)", "Finally, we point out that the cancellation of the one-loop contribution at $\\mu =0$ is a feature specific to odd saddles.", "For an $n$ -th order saddle with a dispersion $\\xi =p^n\\cos n\\phi $ the DoS behaves as $\\nu (\\varepsilon )\\propto \\varepsilon ^{-(n-2)/n}$ , while the polarization operators behave as $\\Pi _{pp}-\\Pi _{ph}={\\left\\lbrace \\begin{array}{ll}\\frac{1+(-1)^n}{n}C_n\\nu (\\mu ) & \\mu =0,\\,T\\ne 0\\\\\\frac{n-2}{2}\\nu (\\mu ) & T = 0,\\,\\mu \\ne 0\\end{array}\\right.", "},$ with a (positive) numerical constant $\\begin{split}C_n=&\\int _0^\\infty \\mathrm {d}x\\,x^{-(n-2)/n}(2\\cosh ^2(x/2))^{-1}\\\\=&2(2^{2/n}-1)\\Gamma \\left(2-\\frac{2}{n}\\right)\\left[-\\zeta \\left(1-\\frac{2}{n}\\right)\\right].\\end{split}$ As we mentioned in the main text, this difference leads to non-Fermi-liquid and marginal Fermi liquid behavior for odd and even saddles, respectively." ], [ "Quasiparticle decay rate", "The quasiparticle decay rate is related to the imaginary part of the electron self-energy, which can be written (using the real-time Keldysh technique) as $\\begin{split}\\Delta \\Sigma (\\varepsilon ,{p})=&-\\int _{\\omega ,{q}}\\left[B(\\omega )+f(\\varepsilon -\\omega )\\right]\\times \\\\&\\times \\Delta G(\\varepsilon -\\omega ,{p}-{q})\\Delta L(\\omega ,{q})\\\\=&-i\\int _{{q}}\\left[B(\\varepsilon -\\xi _{{p}-{q}})+f(\\xi _{{p}-{q}})\\right]\\times \\\\&\\times \\Delta L(\\varepsilon -\\xi _{{p}-{q}},{q}),\\end{split}$ where $B(x)=\\coth (x/2T)$ and $f(x)=\\tanh (x/2T)$ are bosonic and fermionic distribution functions, $L$ is an interaction propagator, and $\\Delta (\\dots ) =(\\dots )^R-(\\dots )^A$ stands for the difference between retarded and advanced components.", "The interaction propagator within the one-loop approximation is essentially $\\begin{split}\\Delta \\Sigma (\\varepsilon ,{p})=&-ig^2\\int _{{k},{q}}\\delta (\\varepsilon +\\xi _{{k}+{q}-{p}}-\\xi _{{q}}-\\xi _{{k}})\\times \\\\&\\times \\Big (f(\\xi _{{q}})[f(\\xi _{{k}})-f(\\xi _{{k}}+\\xi _{{q}}-\\varepsilon )]+\\\\&+1-f(\\xi _{{k}})f(\\xi _{{k}}+\\xi _{{q}}-\\varepsilon )\\Big ),\\end{split}$ where we made use of the relation between equilibrium distribution functions $[f(x+y)-f(x)]B(y)=1-f(x+y)f(x)$ , and redefined integration variables ${k},{q}$ .", "This equation is essentially a statement of Fermi's golden rule.", "Rescaling momenta as $({k},{q})\\rightarrow T^{1/3}({k},{q})$ we see that the quasiparticle width at the monkey saddle for zero chemical potential and zero external frequency and momenta behaves as $\\Gamma =\\frac{i}{2}\\Delta \\Sigma (0,{0})\\Big \|_{\\mu =0}\\sim [\\underbrace{\\nu (T)g}_{\\lambda (T)}]^2T\\propto T^{1/3}.$ On the other hand, for non-zero chemical potential we find regular Fermi-liquid-like behavior[19], $\\Gamma \\sim \\lambda ^2(\\mu )\\frac{\\varepsilon ^2}{\\mu }\\ln \\frac{\\mu }{\\varepsilon },\\quad T\\ll \\varepsilon \\ll \\left\|\\mu \\right\|.$" ], [ "Polarization operators", "In BLG there are two additional polarization operators, with non-zero momentum transfer ${Q}$ $\\Pi _{ph}({Q},\\mu ,T)=&-T_{l,{p}}G(i\\varepsilon _{l},{p})G(i\\varepsilon _{l},{Q}+{p}),\\\\\\Pi _{pp}({Q},\\mu ,T)=&T_{l,{p}}G(i\\varepsilon _{l},{p})G(-i\\varepsilon _{l},{Q}-{p}).$ Once calculated, they yield $\\Pi _{ph}({Q})=&\\frac{1}{2}\\int \\nu (\\xi )\\,\\frac{f(\\xi -\\mu )-f(-\\xi -\\mu )}{2\\xi }\\;\\mathrm {d}\\xi ,\\\\\\Pi _{pp}({Q})=&\\frac{1}{2}\\int \\nu (\\xi )\\,\\frac{f(\\xi +\\mu )}{\\xi +\\mu }\\;\\mathrm {d}\\xi .$ In this paper we focus on the case when the system is tuned to the monkey saddle, $\\mu =0$ , where $\\Pi _{ph}({0})&=\\Pi _{pp}({0})=C_3\\;\\nu (T),\\quad &(\\mu =0)\\\\\\Pi _{ph}({Q})&=\\Pi _{pp}({Q})=3C_3\\;\\nu (T),\\quad &$ with numerical constant $C_3=\\int _0^\\infty \\mathrm {d}x\\,x^{-1/3}\\frac{1}{2\\cosh ^2(x/2)}=1.14.$" ], [ "RG equations", "The RG flow equations for a square lattice with two hot spots were derived in Ref.", "furukawa.", "These equations are very general and in their infinitesimal form, after an elementary RG step, they give $\\begin{split}\\delta {g}_1=&2g_1(g_2-g_1)\\delta \\Pi _{ph}({Q})+2g_1g_4\\delta \\Pi _{ph}({0})\\\\&-2g_1g_2\\delta \\Pi _{pp}({Q}),\\\\\\delta {g}_2=&(g_2^2+g_3^2)\\delta \\Pi _{ph}({Q})+2(g_1-g_2)g_4\\delta \\Pi _{ph}({0})\\\\&-(g_1^2+g_2^2)\\delta \\Pi _{pp}({Q}),\\\\\\delta {g}_3=&-2g_3g_4\\delta \\Pi _{pp}({0})+2(2g_2-g_1)g_3\\delta \\Pi _{ph}({Q})\\\\\\delta {g}_4=&-(g_3^2+g_4^2)\\delta \\Pi _{pp}({0})\\\\&+(g_1^2+2g_1g_2-2g_2^2+g_4^2)\\delta \\Pi _{ph}({0}).\\end{split}$ In the case of BLG there is no Umklapp scattering between $K$ and $K^\\prime $ points, and thus we set $g_3\\equiv 0$ .", "The coupling constants $g_i$ are dimensionful, but we introduce dimensionless coupling constants as follows.", "Since $\\Pi _{pp}({0})\\propto \\nu $ (see Eq.", "REF ), it is appropriate and convenient to define the dimensionless constants as $\\lambda _i=\\Pi _{pp}({0})g_i$ , and take $\\mathrm {d}\\ln \\Pi _{pp}({0})$ for RG time $\\mathrm {d}s$ : $\\begin{split}\\dot{\\lambda }_1=&\\lambda _1+2d_1\\lambda _1(\\lambda _2-\\lambda _1)+2d_2\\lambda _1\\lambda _4-2d_3\\lambda _1\\lambda _2,\\\\\\dot{\\lambda }_2=&\\lambda _2+d_1\\lambda _2^2+2d_2(\\lambda _1-\\lambda _2)\\lambda _4-d_3(\\lambda _1^2+\\lambda _2^2),\\\\\\dot{\\lambda }_4=&\\lambda _4-d_0\\lambda _4^2+d_2(\\lambda _1^2+2\\lambda _1\\lambda _2-2\\lambda _2^2+\\lambda _4^2),\\end{split}$ where for the sake of generality we introduced an additional parameter $d_0$ .", "Parameters $d_i$ are defined in the main text by Eqs.", "REF , and their explicit numerical values follow from Eqs.", "REF ,.", "This scheme gives the RG flow presented in the main text, $\\dot{\\lambda }_1=&\\lambda _1-6\\lambda _1^2+2\\lambda _1\\lambda _4,\\\\\\dot{\\lambda }_2=&\\lambda _2+2(\\lambda _1-\\lambda _2)\\lambda _4-3\\lambda _1^2,\\\\\\dot{\\lambda }_4=&\\lambda _4+\\lambda _1^2+2\\lambda _1\\lambda _2-2\\lambda _2^2.$ At the brink of a many-body instability the coupling constants diverge as $\\lambda _i=\\frac{\\lambda _i^{(0)}}{s_c-s},$ where $s_c$ is a critical RG time corresponding to the instability.", "By seeking solutions of this form we get a system of algebraic equations $\\begin{split}\\lambda _1^{(0)}=&-6\\left(\\lambda _1^{(0)}\\right)^2+2\\lambda _1^{(0)}\\lambda _4^{(0)},\\\\\\lambda _2^{(0)}=&2\\left(\\lambda _1^{(0)}-\\lambda _2^{(0)}\\right)\\lambda _4^{(0)}-3\\left(\\lambda _1^{(0)}\\right)^2,\\\\\\lambda _4^{(0)}=&\\left(\\lambda _1^{(0)}\\right)^2+2\\lambda _1^{(0)}\\lambda _2^{(0)}-2\\left(\\lambda _2^{(0)}\\right)^2.\\end{split}$ This system has the following four stable solutions $\\lambda _1:\\lambda _2:\\lambda _4=&2:1:(3+\\sqrt{12}) &\\text{(FM)}\\\\=&0:1:(-1) &\\text{([S/C]DW)}\\\\=&(-2):(-1):(\\sqrt{12}-3)&\\text{(CDW)}\\\\=&0:(-1):(-1) &\\text{(SC)}$ that correspond to ferromagnetic (FM), competing spin- and charge-density-wave ([S/C]DW), charge-density-wave (CDW) and $s$ -wave SC instabilities respectively.", "The nature of instabilities is identified with the help of the susceptibilities calculated in Refs.", "[1], [2], [20].", "Susceptibilities to different order parameters diverge as $\\chi _j\\propto (s_c-s)^{\\alpha _j}$ , so the leading instability is the one with the most negative value of $\\alpha _j$ , given by $\\alpha _{sP_{Q}}=&2\\lambda _4^0\\\\\\alpha _{s_\\pm P_{Q}}=&2\\lambda _4^0\\\\\\alpha _{CDW}=&6(2\\lambda _1^0-\\lambda _2^0)\\\\\\alpha _{SDW}=&-6\\lambda _2^0\\\\\\alpha _{\\text{spin}}=&-2(\\lambda _1^0+\\lambda _4^0)\\\\\\alpha _{\\text{charge}}=&2(-\\lambda _1^0+2\\lambda _2^0+\\lambda _4^0)\\\\\\alpha _{sP}=&6(-\\lambda _1^0+\\lambda _2^0)$ for finite momentum $s$ -wave and $s_\\pm $ -wave superconducting, charge density wave, spin density wave, ferromagnetic (uniform spin), uniform charge ($\\kappa $ ), and $s$ -wave superconducting instabilities respectively.", "Susceptibilities can be calculated by studying renormalization of test vertices[10].", "The first group of instabilities correspond to uniform densities with a test Lagrangian density $\\delta \\mathcal {L}=\\sum _{i=\\uparrow \\downarrow }\\sum _{\\alpha =+-}n_{i\\alpha }\\psi ^\\dagger _{i\\alpha }\\psi _{i\\alpha },$ where renormalization of test vertices $n_{i\\alpha }$ within one-loop approximation is given by equation $\\frac{\\mathrm {d}}{\\mathrm {d}s}\\begin{pmatrix}n_{+\\uparrow }\\\\n_{+\\downarrow }\\\\n_{-\\uparrow }\\\\n_{-\\downarrow }\\end{pmatrix}=d_2\\begin{pmatrix}0 & -\\lambda _4 &\\lambda _1-\\lambda _2 & - \\lambda _2\\\\-\\lambda _4 & 0 & -\\lambda _2 & \\lambda _1-\\lambda _2\\\\\\lambda _1-\\lambda _2 & -\\lambda _2 & 0 & -\\lambda _4\\\\-\\lambda _2 & \\lambda _1-\\lambda _2 & -\\lambda _4 & 0\\end{pmatrix}\\begin{pmatrix}n_{+\\uparrow }\\\\n_{+\\downarrow }\\\\n_{-\\uparrow }\\\\n_{-\\downarrow }\\end{pmatrix}$ and susceptibilities are equal to $\\alpha =-2\\gamma $ , where $\\gamma $ is an eigenvalue of (REF ).", "Solving for eigensystem of (REF ) we find four instabilities with susceptibilities $\\alpha _{\\text{spin}}=&-2(\\lambda _1^0+\\lambda _4^0),\\\\\\alpha _{\\text{charge}}=&2(-\\lambda _1^0+2\\lambda _2^0+\\lambda _4^0),\\\\\\alpha _\\text{{valley}}=&2(\\lambda _1^0-2\\lambda _2^0+\\lambda _4^0),\\\\\\alpha _{\\text{spin-valley}}=&2(\\lambda _1^0-\\lambda _4^0).$ The second group of instabilities is a charge- and spin-density waves, $\\delta \\mathcal {L}=\\sum _{i=\\uparrow \\downarrow }n_{{Q}i}\\psi ^\\dagger _{-i}\\psi _{+i}+\\text{h.c.}$ $\\frac{\\mathrm {d}}{\\mathrm {d}s}\\begin{pmatrix}n_{{Q}\\uparrow }\\\\n_{{Q}\\downarrow }\\end{pmatrix}=d_1\\begin{pmatrix}\\lambda _2-\\lambda _1 & -\\lambda _1\\\\-\\lambda _1 & \\lambda _2-\\lambda _1\\end{pmatrix}\\begin{pmatrix}n_{{Q}\\uparrow }\\\\n_{{Q}\\downarrow }\\end{pmatrix},$ $\\alpha _{CDW}=&6(2\\lambda _1^0-\\lambda _2^0),\\\\\\alpha _{SDW}=&-6\\lambda _2^0.$ The third group represents superconducting $s$ - and $s_\\pm $ -wave instabilities, $\\delta \\mathcal {L}=\\Delta _{1}\\psi ^\\dagger _{+\\uparrow }\\psi ^\\dagger _{-\\downarrow }+\\Delta _{2}\\psi ^\\dagger _{-\\uparrow }\\psi ^\\dagger _{+\\downarrow }+\\text{h.c.},$ $\\frac{\\mathrm {d}}{\\mathrm {d}s}\\begin{pmatrix}\\Delta _{1}\\\\\\Delta _{2}\\end{pmatrix}=d_3\\begin{pmatrix}-\\lambda _2 & -\\lambda _1\\\\-\\lambda _1 & -\\lambda _2\\end{pmatrix}\\begin{pmatrix}\\Delta _{1}\\\\\\Delta _{2}\\end{pmatrix},$ $\\alpha _{sP}=&6(\\lambda _2^0+\\lambda _1^0),\\\\\\alpha _{s_{\\pm }P}=&6(\\lambda _2^0-\\lambda _1^0).$ Finally, the last group corresponds to finite momentum superconductivities, $\\delta \\mathcal {L}=\\Delta _{s{Q}+}\\psi ^\\dagger _{+\\uparrow }\\psi ^\\dagger _{+\\downarrow }+\\Delta _{s{Q}-}\\psi ^\\dagger _{-\\uparrow }\\psi ^\\dagger _{-\\downarrow }+\\text{h.c.},$ $\\frac{\\mathrm {d}}{\\mathrm {d}s}\\begin{pmatrix}\\Delta _{s1}\\\\\\Delta _{s2}\\end{pmatrix}=d_0\\begin{pmatrix}-\\lambda _4 & 0\\\\0 & -\\lambda _4\\end{pmatrix}\\begin{pmatrix}\\Delta _{s1}\\\\\\Delta _{s2}\\end{pmatrix},$ $\\alpha _{sP_{Q}}=&\\lambda _4,\\\\\\alpha _{s_{\\pm }P_{Q}}=&\\lambda _4.$ Going back to the analysis of RG flow (REF ), since $\\lambda _1$ cannot change sign (RHS for $\\dot{\\lambda }_1$ is equal to zero when $\\lambda _1=0$ ), it is convenient to analyze the RG flow in $y_2=\\lambda _2/\\lambda _1$ vs. $y_4=\\lambda _4/\\lambda _1$ coordinates, $\\dot{y}_2=&\\lambda _1\\left(-3+6y_2+2y_4-4y_2y_4\\right),\\\\\\dot{y}_4=&\\lambda _1\\left(1+2y_2+6y_4-2(y_2^2+y_4^2)\\right).$ We can then reparametrize the RG flow eliminating $\\lambda _1$ to get a system of equations $y_2^\\prime =&-3+6y_2+2y_4-4y_2y_4,\\\\y_4^\\prime =&1+2y_2+6y_4-2(y_2^2+y_4^2),$ that can be solved exactly in the coordinates $y_\\pm $ , $y_\\pm =(y_4-3/2)\\pm (y_2-1/2):\\quad y_\\pm ^\\prime =6-y_\\pm ^2.$ This allows us to identify all phases and phase boundaries on the $y_2y_4$ plane.", "Thus, the plot in $\\lambda _2/\\lambda _1$ vs. $\\lambda _4/\\lambda _1$ coordinates explicitly shows the fate of the system for different initial coupling constants.", "Fig.", "REF (left) shows the phase diagram of RG flow for $\\lambda _1>0$ .", "FM, SC or competing [S/C]DW instabilities are possible with phase boundaries $\\lambda _2-\\lambda _1/2=0, &\\quad (\\text{SC/SDW})\\\\\\lambda _2+\\lambda _4-(2-\\sqrt{3})\\lambda _1=0, &\\quad (\\text{FM/SC})\\\\\\lambda _2-\\lambda _4-(\\sqrt{3}-1)\\lambda _1=0, &\\quad (\\text{FM/[S/C]DW})$ and the lines cross at the point $\\lambda _1:\\lambda _2:\\lambda _4=2:1:(3-\\sqrt{12}).$ For negative values $\\lambda _1<0$ we get options of SC, CDW, and [S/C]DW and Fig.", "REF (right).", "The phase boundaries are now $\\lambda _2+\\left\|\\lambda _1\\right\|/2=0, &\\quad (\\text{SC/[S/C]DW})\\\\\\lambda _2+\\lambda _4-(2-\\sqrt{3})\\left\|\\lambda _1\\right\|=0, &\\quad (\\text{SC/CDW})\\\\\\lambda _2-\\lambda _4-(\\sqrt{3}-1)\\left\|\\lambda _1\\right\|=0, &\\quad (\\text{CDW/[S/C]DW})$ crossing at the point $\\left\|\\lambda _1\\right\|:\\lambda _2:\\lambda _4=2:1:(3-\\sqrt{12}).$" ] ]	1606.04950
[ [ "Tracing dark energy with quasars" ], [ "Abstract The nature of dark energy, driving the accelerated expansion of the Universe, is one of the most important issues in modern astrophysics.", "In order to understand this phenomenon, we need precise astrophysical probes of the universal expansion spanning wide redshift ranges.", "Quasars have recently emerged as such a probe, thanks to their high intrinsic luminosities and, most importantly, our ability to measure their luminosity distances independently of redshifts.", "Here we report our ongoing work on observational reverberation mapping using the time delay of the Mg II line, performed with the South African Large Telescope (SALT)." ], [ "Introduction", "Dark energy (DE), estimated to constitute about 70% of the universal mass-energy content today, is one of the crucial ingredients of the standard cosmological model.", "Since the first observational evidence of the accelerated expansion thanks to Supernovae Ia (SNeIa), various efforts have been undertaken to understand the nature of this phenomenon.", "On the observational side this requires designing ever more precise measurements, and various probes have been proposed and used to study DE.", "Quasars, currently detected at redshifts up to $z\\sim 7$ , are ideally suited to this task thanks to their very high intrinsic luminosity and large number density.", "However, using them as standard candles in a similar way to SNeIa, requires knowing their individual absolute luminosities [11].", "Determining the latter is possible through the measurement of the time delay between the variable nuclear continuum and emission lines [5], or by analysing the shape of the lines to measure the BLR size.", "Here we focus on the first of these methods, which is possible by measuring the delay of the H$\\beta $ line, performed first for nearby Active Galactic Nuclei (AGN) [3], [8], [9], [2], [1].", "The time delays in quasars are of the order of a few years, so the observations require sparse monitoring over an extended period of time." ], [ "Method", "The work reported here is based on a simple mechanism of the broad line region (BLR) formation, presented in [4], which explains how the size of BLR depends on the absolute monochromatic luminosity: $R_{\\rm BLR} = {\\rm const}\\; L_v^{1/2}$ .", "In our method we assume that: (i) we know the redshift of the source; (ii) the optical/UV continuum is generated in the inner part of the accretion disk surrounding the central black hole, while the broad emission lines are produced in another disk region, in clouds above the accretion disk as shown in Fig.", "REF .", "Dust leads to outflow and forces the material to rise high above the disk, the dust cannot however survive in temperatures much higher than 1000 K. Strong radiation field destroys the dust (through evaporation) and the material falls back without a driving force.", "More accurately, this mechanism is proposed for the low ionization line part of the BLR, like H$\\beta $ and MgII, which do not show a systematic shift in velocity with respect to the narrow line region (NLR).", "Using the observational method of reverberation mapping we measure the BLR size, which allows us to determine the absolute monochromatic luminosity of the quasar.", "Comparing the latter with the observed monochromatic flux from photometry, we obtain the luminosity distance and are able to locate the source on the distance–redshift diagram.", "Until present, we have collected observations of three quasars at redshift $z\\sim 1$ , and part of this work is already published [6], [7].", "Spectra of two sources, CTS C30.10 and HE 0435-4312, are shown in Fig.", "REF .", "This project requires using the MgII line, and our studies are pioneer in this respect, as such monitoring has never been done before.", "Data obtained so far with the South African Large Telescope (SALT) show that we achieve the required accuracy (below 2%) of the MgII measurement to determine its variability [7], [10], and simulations indicate that the program can give the accuracy of 0.06–0.32 mag in the distance modulus for each of the concerned quasars [5].", "Figure: SALT spectroscopy of the MgII emission line for the quasar CTS C30.10 (left) and HE 0435-4312 (right).", "The collected data cover respectively 15 months and 3 years of observations." ], [ "Discussion", "Our method is very promising for DE studies, as it can easily reach to much higher redshifts than available with supernovae (ground-based observations of SNeIa hardly reach beyond $z = 1$ ).", "On the other hand, such reverberation studies require very large time spans, as at least 5 years of systematic observations are needed to estimate the emission line delay with respect to the continuum.", "High-quality data from SALT give us the opportunity to accurately model the emission line shape.", "This provides us with a new tool for cosmological analyses which aim at understanding the mystery of dark energy.", "This work was supported by Polish grants #719/N-SALT/2010/0 and #UMO-2012/07/B/ST9/04425.", "We acknowledge the support from the Foundation for Polish Science through the Master/Mistrz program 3/2012." ] ]	1606.05130
[ [ "Statistical Mechanics of Avalanches" ], [ "Abstract Statistical mechanics of infinite avalanches is studied in the framework of nonequilibrium random-field Ising model.", "Critical behavior of the model on a random graph (dilute Bethe lattice) is analyzed in detail.", "We show that sites with a minimum coordination number 4 play a key role in the occurrence of infinite avalanches.", "Earlier results which did not seem to fit together very well are explained." ], [ "Introduction", "Zero-temperature nonequilibrium random-field Ising model has been used extensively to understand hysteresis in systems with quenched disorder [1], [2], [3], [4], [5], [6], [7], [8].", "Analysis and simulations of the model reveal that character of hysteresis loop depends upon several factors including probability distribution of the random-field, dimensionality and connectivity of the lattice.", "A common choice for the random-field distribution is a Gaussian with average zero and standard deviation $\\sigma $ .", "In some cases depending upon the dimensionality and connectivity of the lattice, there exists a critical value $\\sigma =\\sigma _c$ such that each half of the hysteresis loop has a critical point characterized by diverging susceptibility of the system.", "For $\\sigma < \\sigma _c$ , there is a discontinuity in each half of the loop arising from a massive flipping of spins triggered by an infinitesimal change in the applied field, i.e.", "an infinite avalanche.", "Infinite avalanches and a critical point go hand in hand in the present model.", "Infinite avalanches end in a critical point as $\\sigma $ is increased to its critical value.", "While the critical behavior in the vicinity of $\\sigma _c$ has been investigated extensively, the criteria for the existence of $\\sigma _c$ , its dependence on the dimensionality and connectivity of the lattice has received less attention.", "As discussed below, general conditions for the occurrence of infinite avalanches in this simple model remain unclear so far.", "This brief report addresses this issue drawing upon extant results as well as a new result presented here." ], [ "The model, extant results, and related issues", "We begin by describing the model briefly, listing known results and related issues.", "The Hamiltonian of the model is, $H=-J\\sum _{i,j}s_is_j-h\\sum _i s_i -\\sum _i h_i s_i $ Here $s_i=\\pm 1$ is an Ising spin at site $i$ , $h$ is a uniform applied field, and $h_i$ is a quenched field drawn from a Gaussian distribution of mean zero and standard deviation $\\sigma $ ; $J$ is ferromagnetic coupling between nearest neighbors on a lattice.", "The applied field $h$ is assumed to vary infinitely slowly, and the dynamics of the model is taken to be the adiabatic zero-temperature single-spin-flip Glauber dynamics.", "At each value of $h$ , spins in the system are flipped as needed till each spin $s_i$ is aligned along the net local field $\\ell _i$ at its site; $\\ell _i = J \\sum _{j \\ne i} s_j + h +h_i$ .", "We start at $h=-\\infty $ with the stable state of the system having all spins down, and the magnetization per site $m(h)=-1$ .", "Now $h$ is increased till some spin flips up, say at $h=\\tilde{h}$ .", "A spin that flips up increases the field on its neighbors by $2J$ , and some of them may flip up and so on.", "Thus a spin flipping up may initiate an avalanche of flipped up spins.", "Sites in an avalanche lie on a connected cluster whose size equals the size of the avalanche.", "Each avalanche of size $s$ increases $m(h)$ by an amount $2s/N$ where N is the size of the system.", "When an avalanche is finished, the field $h$ is increased again till the next avalanche occurs.", "This process is continued till all spins are up.", "As $h$ is ramped up from $h=-\\infty $ to $h=\\infty $ , $m(h)$ increases in tiny irregular steps separated by random quiescent intervals along the applied field.", "This is Barkhausen noise but not the main concern of the present paper.", "Our concern is with a discontinuity in $m(h)$ in the thermodynamic limit, i.e.", "a macroscopic avalanche of the order of $N$ and the criteria for its occurrence.", "What we know so far is that if a macroscopic avalanche occurs, it occurs for $\\sigma < \\sigma _c$ where $\\sigma _c$ is a critical value.", "If there is no $\\sigma _c$ , there is no avalanche.", "Also, there is only one avalanche, say at $h=\\tilde{h}$ .", "As $\\sigma \\rightarrow \\sigma _c$ , $\\tilde{h}$ decreases and so does the size of the avalanche.", "The size goes to zero at $\\sigma = \\sigma _c$ , $h=\\tilde{h}_c$ but fluctuations at this point are anomalously large.", "This is a nonequilibrium critical point with behavior similar to that of an equilibrium Ising model at its critical temperature $T_c$ .", "Indeed, $\\sigma _c$ plays a role analogous to $T_c$ .", "It is not clear why this should be so because $T_c$ takes into account thermal relaxation of all states of the system, but $\\sigma _c$ is based only on one initial state (all spins down) and its zero-temperature Glauber dynamics.", "The question we ask is whether $T_c$ and $\\sigma _c$ are determined by similar criteria.", "Our minimal model is characterized by a small set of parameters: $J, h, \\sigma $ , and two implicit parameters $d$ and $z$ denoting the dimension and coordination number of the lattice.", "$J$ sets the energy scale, $h$ and $\\sigma $ are used as tuning parameters to locate a critical point if it exists.", "This means that the existence of a critical point $(\\sigma _c, \\tilde{h}_c)$ must depend on $d$ , or $z$ , or both.", "For thermal model, the existence of $T_c$ is decided by $d$ alone which should be above the lower critical dimension $d_\\ell $ for the system; $d_\\ell =1$ for the pure Ising model, $d_\\ell =2$ for the random-field Ising model [9].", "For $d > d_\\ell $ , the temperature-driven critical behavior does not depend on $z$ .", "For example, if there is a $T_c$ on a square lattice ($d=2,z=4$ ), there is also a $T_c$ on the honeycomb lattice ($d=2,z=3$ ).", "One may expect the same for the disorder-driven critical behavior at $T=0$ on grounds that both types of critical phenomena are caused by a diverging length; diverging correlation length in one case and a diverging avalanche in the other.", "However, there is a $\\sigma _c$ on the square lattice [8] and the triangular lattice [10] but no $\\sigma _c$ on the honeycomb lattice [11].", "We would like to understand why?", "The coordination number $z=4$ has a special connection with $\\sigma _c$ .", "The square lattice ($z=4$ ) is commonly used for studying the behavior of a model in $d=2$ .", "Numerical efforts to find $\\sigma _c$ for the square lattice were inconclusive initially.", "This was thought to have a bearing on $d_\\ell $ for the random-field Ising model which was in question initially.", "Eventually theory settled $d_\\ell =2$ , and numerical work on large systems indicated the presence of $\\sigma _c$ on the square lattice.", "However, as mentioned above there appear to be other factors beside $d_\\ell $ that determine $\\sigma _c$ .", "An analytic solution of the model on a Bethe lattice of integer coordination number $z$ shows that infinite avalanches and critical phenomena occur only if $z\\ge 4$ [12].", "This is surprising because in all other cases, as far as we know, the critical behavior of an Ising model on a Bethe lattice does not depend upon $z$ if $z>2$ .", "Evidently, there is not a very simple and clear physical reason for the absence of infinite avalanches on a $z=3$ Bethe lattice.", "It has been explained by mapping the problem to a branching process in population dynamics [13].", "The unusual dependence of $\\sigma _c$ on the coordination number $z$ of a Bethe lattice is seen on periodic lattices as well.", "For example, an infinite avalanche does not occur on any periodic lattice with $z=3$ irrespective of the dimension $d$ of the space in which the lattice is embedded [11].", "One way to gain insight into the effect of $z$ on $\\sigma _c$ is to study the model on lattices whose average coordination number $z_{av}$ varies continuously between integer values.", "With this in mind infinite avalanches were studied on a dilute triangular lattice [14].", "The study suggested that infinite avalanches occur when $z_{av} \\ge 4$ but did not rule out a lower value.", "Next, the problem was studied on a Bethe lattice of a mixed coordination number such that a fraction $c_4$ of the sites had $z=4$ and the remaining fraction $1-c_4$ had $z=3$ .", "An exact solution was obtained and verified by numerical simulations.", "The result turned out to be somewhat surprising.", "Infinite avalanches can occur in the entire range $0<c_4\\le 1$ if $\\sigma < \\sigma _c$ where $\\sigma _c \\rightarrow 0$ continuously as $c_4 \\rightarrow 0$ [15].", "This suggests that the presence of a small fraction of $z=4$ sites suffices to produce an infinite avalanche.", "However, the physical reason for this is not clear." ], [ "Criticality on a dilute $z=4$ Bethe lattice", "A dilute $z=4$ Bethe lattice with only a fraction $c$ of sites occupied by spins provides another example of interest that can be solved analytically.", "On such a lattice, there are sites with coordination numbers $z=0,1,2,3,4$ .", "This problem was studied earlier in the limit $c\\rightarrow 1$ , and $c \\rightarrow 0$ to show how a tiny fraction of magnetic grains in geological rocks transforms familiar hysteresis loops into wasp-waisted loops [16].", "In the following, we revisit this problem in the regime of moderate $c$ to examine the dependence of $\\sigma _c$ on $c$ .", "A key quantity is the conditional probability $Q^(h,\\sigma )$ that a nearest neighbor of an occupied site in the deep interior of a Cayley tree (the central site) is down at $h$ before the central site is relaxed.", "The dynamics of the model is abelian i.e.", "the same final state is reached irrespective of the order in which the sites are relaxed.", "We start with all spins down on a Cayley tree and relax them in the following order: first we relax spins on the surface, then move towards the center relaxing spins on one level at a time.", "This amounts to calculating $Q^n(h,\\sigma )$ for increasing $n$ where $Q^n(h,\\sigma )$ is the probability that a site on level $n$ is down before its neighbor at level $n+1$ is relaxed.", "Let us take the surface to be at level 0.", "A spin on the surface experiences a quenched random field $h_i$ , an external field $h$ , and a field $-J$ from the unrelaxed neighbor at level 1.", "When relaxed, it may flip up or stay down depending on the value of the net field $h_i+h-J$ on it.", "The probability $Q^0(h,\\sigma )$ that it stays down is the probability that $h_i+h-J \\le 0$ .", "Taking into account the Gaussian distribution of $h_i$ , we get $Q^0(h,\\sigma )=0.5 [1 + \\operatorname{erf}{ \\lbrace (J-h)/\\sqrt{2 \\sigma ^2}\\rbrace }]$ .", "An equation for $ Q^n(h,\\sigma )$ , $n > 1 $ , is obtained similarly if we keep in mind that each spin to be relaxed at level $n$ has one unrelaxed neighbor at level $n+1$ and $z-1$ relaxed neighbors at level $n-1$ .", "We get, $Q^n(h,\\sigma )=\\sum _{m=0}^{z-1}[Q^{n-1}(h,\\sigma )]^m [1 -Q^{n-1}(h,\\sigma )]^{z-1-m} q_{z,m+1}(h,\\sigma ) $ Here $q_{z,m}(h,\\sigma )$ is the probability that a $z$ -coordinated spin with $m$ neighbors down is down at applied field $h$ .", "$q_{z,m}(h,\\sigma )=\\frac{1}{\\sqrt{2\\pi \\sigma ^2}} \\int _{-\\infty }^{(2m-z)J-h} e^{\\frac{-h_i^2}{2\\sigma ^2}} dh_i = \\frac{1}{2} \\left[1 +\\operatorname{erf}{\\left\\lbrace \\frac{(2m-z)J-h}{\\sqrt{2 \\sigma ^2}}\\right\\rbrace }\\right];\\hspace{7.11317pt} (m=0,\\ldots ,z) $ The fixed-point $Q^(h,\\sigma )$ is given by, $Q^(h,\\sigma )= \\lim _{n\\rightarrow \\infty } Q^n(h,\\sigma )$ .", "We find $Q^(h,\\sigma ) > 1/2$ if $h < J$ , $Q^(h,\\sigma ) < 1/2$ if $h > J$ .", "There is a discontinuity in $Q^(h,\\sigma )$ at $h=J$ and therefore an infinite avalanche in the system if $\\sigma < \\sigma _c$ .", "The discontinuity deceases in size with increasing $\\sigma $ and vanishes at $\\sigma =\\sigma _c$ .", "At $\\sigma _c$ , the two solutions at $h=J$ merge into $Q^(J,\\sigma ) =1/2$ .", "The equation determining $\\sigma _c$ is, $A (A+4B)=0.", "$ where $A=c^3 \\lbrace 1 + q_{4,4}(J,\\sigma )- 3q_{2,2}(J,\\sigma ) \\rbrace \\mbox{ and } B=1- \\lbrace c^3 q_{4,4}(J,\\sigma )+3 c^2(1-c) q_{3,3}(J,\\sigma )+3 c (1-c)^2 q_{2,2}(J,\\sigma )+(1-c)^3q_{1,1}(J,\\sigma )\\rbrace $ The factor $A$ is negative for all values of $\\sigma $ of interest.", "Therefore $\\sigma _c$ is effectively determined by the equation $A+4B=0$ .", "No real positive value of $\\sigma _c$ satisfies this equation if $c$ is less than a critical value $c_{min}$ .", "Numerically, $c_{min}\\approx 0.557$ .", "The exact value (argument to be presented below) is $c_{min} =2^{1/3} / (1+2^{1/3}) \\approx 0.5575$ .", "For $c > c_{min}$ , $\\sigma _c$ increases continuously with increasing $c$ starting from $\\sigma _c=0$ at $c=c_{min}$ .", "The increase is remarkably steep in a narrow region adjacent to $c_{min}$ .", "We may designate this region as the critical region.", "The width of the critical region is very small but the increase of $\\sigma _c$ in this region is substantial.", "Thus a plot of $\\sigma _c$ vs. $c$ appears almost vertical at $c=c_{min}$ .", "Theoretically, the slope of $\\sigma _c$ vs. $c$ curve is infinite at $c_{min}$ .", "It gradually decreases as one moves away from $c_{min}$ but remains very large over the entire critical region.", "Figure (1) shows a plot of $\\sigma _c$ vs. $c$ ; $\\sigma _c$ appears to rise vertically from $\\sigma _c=0$ to $\\sigma _c\\approx 0.275$ i.e.", "all values in the range $0 < \\sigma _c <0.275$ satisfy the equation at $c \\approx 0.557$ .", "Thereafter $\\sigma _c$ increases more gradually.", "At $c=1$ we recover the known result $\\sigma _c=1.781$ for the undiluted $z=4$ Bethe lattice.", "The data plotted in figure (1) was obtained using a standard numerical recipe for evaluating error functions in the expression for $\\sigma _c$ .", "The error in using this recipe is $\\epsilon \\le 10^{-7}$ .", "Within this error, $\\sigma _c$ at $c_{min}$ rises vertically as shown in figure (1).", "However, if one uses another tool ( Mathematica ) to calculate the error functions with a greater precision ($\\epsilon \\le 10^{-16}$ ), the vertical portion of the $\\sigma _c$ vs. $c$ curve is replaced by a curve that bends slightly to the right in the narrow critical region; $\\sigma _c$ increases continuously from 0.0 to 0.307275 as $c$ increases from $c_{min}\\approx 0.5575$ to $c=0.558$ [17].", "This continuous but sharp increase in a narrow region would also appear to be nearly vertical when plotted on the scale of figure (1).", "The important point is that mathematical tools with higher precision as well as theoretical analysis agree that $\\sigma _c$ is a continuous, monotonic, but very steeply increasing function of $c$ immediately above the threshold $c=c_{min}$ .", "The situation brings to mind some (not so well understood) transitions in liquid crystals where the order parameter appears to jump discontinuously as in a first-order transition but the entropy changes continuously as in a second-order transition.", "In order to confirm this sharp change at $c \\approx 0.557$ , we performed simulations for $m(h,\\sigma )$ at $\\sigma =0.4$ for $c=0.55$ and $c=0.57$ .", "Theory predicts a discontinuity in $m(h,\\sigma =0.4)$ for $c=0.57$ but no discontinuity for $c=0.55$ .", "This is what we observed in the simulations.", "Figure (2) shows the closeness between the numerical and corresponding theoretical results.", "The numerical results are obtained on a random graph rather than a Cayley tree in order to eliminate large surface effects.", "The initial state of the random graph is taken as all spins down.", "These simulations match the theoretical result on a Cayley tree if the surface spins are kept down, rather than relaxed at $h$ .", "This procedure does not alter $\\sigma _c$ , but shifts the discontinuity from $h=J$ to to $h > J$ .", "The discontinuity moves closer to $h=J$ as $\\sigma $ increases, and vanishes at $h=J$ as $\\sigma \\rightarrow \\sigma _c$ .", "If the surface of the Cayley tree is relaxed at $h$ rather than held in a fixed state, then the discontinuity occurs at $h=J$ only." ], [ "General criteria for infinite avalanches", "To recapitulate, (i) infinite avalanches occur if $z \\ge 4$ and $\\sigma < \\sigma _c(z)$ but do not occur if $z=2, 3$ (ii) on a lattice with $c:1-c$ mixture of $z=4$ and $z=3$ sites, infinite avalanches occur for all $c$ ($0 < c \\le 1$ ) if $\\sigma < \\sigma _c(c)$ , (iii) on a $z=4$ lattice with a fraction $c$ of sites occupied, infinite avalanches occur if $c > 0.557$ and $\\sigma < \\sigma _c(c)$ .", "The reason why infinite avalanches do not occur for large $\\sigma $ irrespective of other considerations is simple.", "Spins tend to flip up independently in the presence of large disorder, hence no infinite avalanche.", "Results (ii) and (iii) are puzzling at first sight; (ii) suggests that an arbitrarily small fraction of $z=4$ sites is sufficient to cause an infinite avalanche but (iii) contradicts it because nearly $5\\%$ sites have $z=4$ at $c \\approx 0.557$ .", "This requires further discussion.", "First, we look at the reason for (i).", "Absence of an infinite avalanche means $Q^(h,\\sigma )$ is continuous at $h=J$ .", "It is easy to verify that $Q^(J,\\sigma ) = 1/2$ is a fixed point irrespective of $z$ and $\\sigma $ .", "The absence or presence of an infinite avalanche depends on the stability of this fixed point.", "If $Q^(J,\\sigma ) = 1/2$ is stable, there is no discontinuity in $Q^(h,\\sigma )$ at $h=J$ .", "An unstable $Q^(J,\\sigma ) = 1/2$ splits into two stable fixed points, one larger and the other smaller than $1/2$ at $h=J$ .", "Consequently the system jumps from a small magnetization state to a large magnetization via an infinite avalanche.", "The stability of $Q^(J,\\sigma ) = 1/2$ is examined by turning down a small fraction of up sites on the surface of the Cayley tree and examining its effect on the next layer of sites.", "In other words, we increase the fraction of down sites on the surface from $1/2$ to $1/2 + \\delta Q^0$ , and calculate the fraction $1/2 + \\delta Q^1$ of down sites on the layer next to the surface.", "Focus on a set of $z-1$ sites on the surface which have a common neighbor, say $B$ at the higher level.", "Consider the case when at least one of the $z-1$ sites, say $A$ is up and $B$ is also up.", "Now if $A$ is turned down, the local field at $B$ gets reduced by $2J$ .", "The probability that $B$ will turn down as a result of it is given by $\\delta Q^1(J,\\sigma ) = B_z \\delta Q^0(J,\\sigma )$ .", "Using $q_{z,k}=q_{z,k}(h,\\sigma )$ defined earlier, we obtain $B_z = (z-1)\\frac{1}{2^{z-2}} \\sum _{m=0}^{z-2}{{z-2}{m}}(q_{z,m+2}-q_{z,m+1}) $ Above equation is understood as follows: site $A$ can be chosen in $z-1$ ways, remaining $z-2$ sites are down with probability $\\frac{1}{2}$ , $(q_{z,m+2}-q_{z,m+1})$ is the probability that site $B$ is up if $m+1$ of its neighbors are down but flips down if $m+2$ neighbors are down, $q_{z,k}$ is the probability that a $z$ -coordinated spin is down if $k$ of its neighbors are down.", "The quantities $B_2 =q_{2,2}-q_{2,1}$ , $B_3 = q_{3,3}-q_{3,1}$ , and $B_4 = q_{4,4} + q_{4,3}- q_{4,2} - q_{4,1}$ are of special interest.", "$B_2$ and $B_3$ are less than unity for $\\sigma > 0$ ; as $\\sigma \\rightarrow 0$ , $B_2\\rightarrow 1$ and also $B_3 \\rightarrow 1$ .", "Hence the fixed point $Q^(J,\\sigma ) = 1/2$ is stable and the possibility of an infinite avalanche is ruled out on a $z=2$ or a $z=3$ lattice.", "At $h=J$ , $B_4$ simplifies to $B_4=\\frac{3}{2}(q_{4,4}-q_{4,2})$ ; $B_4 \\rightarrow 3/2$ as $\\sigma \\rightarrow 0$ .", "It decreases continuously with increasing $\\sigma $ ; $B_4 \\rightarrow 1$ as $\\sigma \\rightarrow \\sigma _c \\approx 1.781$ .", "Thus $Q^(J,\\sigma ) = 1/2$ is unstable on a $z=4$ lattice if $\\sigma < \\sigma _c$ and consequently there is an infinite avalanche in this case.", "This also confirms that $\\sigma _c$ obtained from considering the stability of $Q^(J,\\sigma ) = 1/2$ is the same as obtained from requiring two roots of the fixed point equation to merge into each other.", "Figure (3) shows the initial value $Q^0(h,\\sigma = J)$ and corresponding fixed-point value $Q^(h,\\sigma = J)$ in the neighborhood of $h=J$ for $z=2, 3,$ and 4.", "For $h < J$ , the bottom line represents surface $Q^0(h,J)$ ; higher curves show fixed points $Q^(h,J)$ for $z=2, 3,$ and 4 respectively.", "The relative position of $Q^0(h,J)$ and $Q^(h,J)$ gets reversed for $h > J$ ; $Q^(h,J) > Q^0(h,J)$ if $h < J$ , but $Q^(h,J) < Q^0(h,J)$ if $h > J$ .", "$Q^0(h,J)$ is of course continuous at $h=J$ and $Q^0(J,J)=0.5$ ; $Q^(h,J)$ is continuous at $h=J$ if $z \\le 3$ but discontinuous if $z=4$ .", "Figure (4) shows the growth for $z=4$ , and decay for $z=2, 3$ of a small perturbation $\\delta Q^0(h = J,\\sigma = J)$ under successive iterations.", "Next, we turn our attention to the dilute $z=4$ lattice.", "In this case as well, $Q^(J,\\sigma ) = 1/2$ is a fixed point.", "This fixed point must be unstable in the limit $\\sigma \\rightarrow 0$ if there is to be an infinite avalanche.", "In the limit $\\sigma \\rightarrow 0$ , $B_2\\rightarrow 1$ , $B_3 \\rightarrow 1$ , and $B_4 \\rightarrow 3/2$ .", "Consider a perturbation $\\delta Q^0(J,0)$ to the fixed point $Q^(J,\\sigma ) = 1/2$ .", "As we move from the surface of the tree towards its center, a $z=4$ site increases the perturbation by a factor $3/2$ , but $z=3$ and $z=2$ sites keep it unchanged.", "The $z=1$ sites produce a new effect on the dilute lattice.", "They break the continuity of the path from the surface to the center.", "In our algorithm for relaxing sites, it is assumed that one of the neighbors of the site being relaxed, the one at a higher level, is present and unrelaxed.", "The bond with this neighbor ensures connection between adjacent levels of the tree.", "A $z=1$ site breaks this connection with probability $\\frac{3}{4} z_1$ , where $z_1$ is the fraction of sites with one nearest neighbor only.", "If $z_4$ is the fraction of sites with 4 neighbors, then at every level of relaxation of the lattice, the perturbation is boosted with the probability $\\frac{3}{2} z_4$ , and terminated with probability $\\frac{3}{4} z_1$ .", "The critical point occurs when the two opposing effects balance each other, i.e.", "$z_1 = 2 z_4$ .", "Using $z_4=c^5$ , $z_1=4c^2 (1-c)^3$ , the critical value of $c$ is given by the equation $c^3=2(1-c)^3$ , or $c=2^{1/3}/(1+2^{1/3}) \\approx 0.5575$ .", "The observed infinite avalanche on a mixed lattice with a fraction $c_4$ of $z=4$ sites and $1-c_4$ of $z=3$ sites for $c_4 > 0$ is also understood in this light.", "The path from the surface to the center is never broken on the mixed lattice, and therefore an arbitrarily small presence of $z_4$ sites creates a gap in $Q^(J,\\sigma )$ in the deep interior of the tree." ], [ "Conclusion", "To conclude, we have presented general criteria for the occurrence of infinite avalanches in the zero-temperature nonequilibrium random-field Ising model on a Bethe lattice.", "We find that infinite avalanches occur when all of the following conditions are fulfilled: (i) $\\sigma $ is sufficiently small, (ii) there is a spanning cluster of occupied sites on the lattice, and (iii) the spanning cluster has a fraction of sites, even an arbitrarily small fraction, with connectivity $z \\ge 4$ .", "We have explained the reason for these conditions.", "The presence of an infinite avalanche on a mixed coordination lattice ($z=3$ or 4) with an arbitrarily small fraction of $z=4$ sites, and its absence on a dilute $z=4$ lattice in a certain regime of dilution is now easily understood.", "Our analysis also shows that disorder in the form of dilution of magnetic ions on a lattice affects hysteresis differently from disorder in the form of on-site random-fields.", "This is important because positional disorder in the form of vacancies is quite common in materials.", "We find a peculiar geometry driven transition near $c=0.5575$ on a dilute $z=4$ Bethe lattice.", "Similar behavior may be expected for $z>4$ as well.", "Infinite avalanches vanish at this critical point continuously, but the slope of the continuous curve is nearly infinite.", "It appears as a first-order jump in the order parameter for all practical purposes.", "Bethe lattices often approximate real systems reasonably well.", "So this feature of the model may be observable in appropriate hysteresis experiments and useful in understanding other weakly first-order phase transitions as well.", "Finally, we wish to end with a caution.", "We have made a case that a lower critical coordination number rather than a lower critical dimension determines critical hysteresis.", "Our suggestion is based on exact results on Bethe lattices and simulations on some periodic lattices.", "It conflicts with a widely accepted view in statistical physics community in favor of a lower critical dimension.", "Further work may be required to settle this issue but we mention two factors that may invalidate our suggestion.", "Results on Bethe lattices are essentially mean field results and do not necessarily have a bearing on criticality on periodic lattices in finite dimensions.", "Secondly, subtle corrections to scaling may explain the extant numerical results on periodic lattices without doing away with the importance of a lower critical dimension.", "Figure: Critical value of the standard deviation of the quenched random-fieldσ c \\sigma _c on a 4-coordinated random graph with a fractional occupancy cc of itssites.", "The magnetization m(h,σ)m(h,\\sigma ) has a discontinuity if σ<σ c \\sigma < \\sigma _c.There is an almost vertical drop in σ c \\sigma _c at c=0.557c=0.557 approximately.Figure: Magnetization m(h,σ)m(h,\\sigma ) in increasing field hh on a4-coordinated dilute random graph for c=0.55c=0.55 (open circles) andc=0.57c=0.57 (filled circles) where cc is the fraction of occupied siteson the graph.", "The quenched random field on occupied sites has meanvalue equal to zero, and standard deviation σ=0.40\\sigma =0.40.", "Theoreticalpredictions are superimposed on the respective simulations (a singlerun on N=10 7 N=10^7 graph) and fit them quite well.", "The magnetization issmooth for c=0.55c=0.55 and has a jump for c=0.57c=0.57 as predicted by thetheory.Figure: The figure shows Q 0 (h,σ=J)Q^0(h,\\sigma =J) and Q (h,σ=J)Q^(h,\\sigma =J) in increasingfield hh (lower half of the hysteresis loop) on a Cayley tree of coordinationnumber z=2,3,4z=2, 3, 4.", "The broken black line shows Q 0 (h,σ=J)Q^0(h,\\sigma =J) which is commonto all zz because a surface site has only one neighbor irrespective of zz;Q 0 (h,σ=J)Q^0(h,\\sigma =J) decreases continuously with increasing hh and passes through thepoint Q 0 (h,σ=J)=0.5Q^0(h,\\sigma =J)=0.5 at h=Jh=J.", "Q (h,σ=J)Q^(h,\\sigma =J) for z=2z=2 (pink curveclosest to the black broken line) and z=3z=3 (red curve next closest to the blackbroken line) behave similarly; both decrease continuously and pass throughQ (h,σ=J)=0.5Q^(h,\\sigma =J)=0.5 at h=Jh=J.", "However, as we go from z=2z=2 to z=3z=3,Q (h,σ=J)Q^(h,\\sigma =J) becomes steeper at h=Jh=J, and generally moves farther away fromQ 0 (h,σ=J)Q^0(h,\\sigma =J).", "For z≥4z\\ge 4, Q (h,σ=J)Q^(h,\\sigma =J) acquires a discontinuity at h=Jh=Jas shown in the figure by the blue curve.Figure: The conditional probability Q t (J,J)Q^t(J,J) (see text) at tt successive levels ofthe Cayley tree starting from the surface (t=0t=0).", "If Q 0 (J,J)=0.5Q^0(J,J)=0.5, Q t (J,J)Q^t(J,J)remains equal to 0.50.5 as shown by the horizontal line.", "A small perturbation δQ 0 \\delta Q^0 added to Q 0 (J,J)Q^0(J,J) gradually decreases to zero if Q (J,J)=0.5Q^(J,J)=0.5 is stable, butincreases with tt if Q (J,J)=0.5Q^(J,J)=0.5 is unstable; perturbed Q t (J,J)Q^t(J,J) lies on thesame side of the horizontal line Q (J,J)=0.5Q^(J,J)=0.5 as the initial perturbation δQ 0 \\delta Q^0.", "Figure shows Q (J,J)=0.5Q^*(J,J)=0.5 is stable for z=2z=2 (green triangles) and z=3z=3 (redsquares), but unstable for z=4z=4 (blue circles).", "Filled symbols correspond to δQ 0 >0\\delta Q^0 > 0 while the empty symbols correspond to δQ 0 <0\\delta Q^0 <0.", "The relaxed state onthe first twelve levels of the tree is shown which suffices to make the trends clear." ] ]	1606.05066
[ [ "Boundary-induced spin density waves in linear Heisenberg\n antiferromagnetic spin chains with $\\mathbf{S \\ge 1}$" ], [ "Abstract Linear Heisenberg antiferromagnets (HAFs) are chains of spin-$S$ sites with isotropic exchange $J$ between neighbors.", "Open and periodic boundary conditions return the same ground state energy in the thermodynamic limit, but not the same spin $S_G$ when $S \\ge 1$.", "The ground state of open chains of N spins has $S_G = 0$ or $S$, respectively, for even or odd N. Density matrix renormalization group (DMRG) calculations with different algorithms for even and odd N are presented up to N = 500 for the energy and spin densities $\\rho(r,N)$ of edge states in HAFs with $S = 1$, 3/2 and 2.", "The edge states are boundary-induced spin density waves (BI-SDWs) with $\\rho(r,N)\\propto(-1)^{r-1}$ for $r=1,2,\\ldots N$.", "The SDWs are in phase when N is odd, out of phase when N is even, and have finite excitation energy $\\Gamma(N)$ that decreases exponentially with N for integer $S$ and faster than 1/N for half integer $S$.", "The spin densities and excitation energy are quantitatively modeled for integer $S$ chains longer than $5 \\xi$ spins by two parameters, the correlation length $\\xi$ and the SDW amplitude, with $\\xi = 6.048$ for $S = 1$ and 49.0 for $S = 2$.", "The BI-SDWs of $S = 3/2$ chains are not localized and are qualitatively different for even and odd N. Exchange between the ends for odd N is mediated by a delocalized effective spin in the middle that increases $\|\\Gamma(N)\|$ and weakens the size dependence.", "The nonlinear sigma model (NL$\\sigma$M) has been applied the HAFs, primarily to $S = 1$ with even N, to discuss spin densities and exchange between localized states at the ends as $\\Gamma(N) \\propto (-1)^N \\exp(-N/\\xi)$..." ], [ "Introduction", "The Hilbert space of a system of $N$ spins $S$ has dimension $(2S + 1)^N$ .", "The total spin $S_T \\le N S$ and its $z$ components are conserved for isotropic (Heisenberg) exchange interactions between spins.", "The simplest case is a chain with equal exchange $J$ between nearest neighbors.", "A great many theoretical and experimental studies have been performed on the linear Heisenberg antiferromagnet (HAF), Eq.", "REF below, with $S = 1/2$ and $J > 0$ .", "There are multiple reasons why.", "First, there are good physical realizations of spin-1/2 chains in inorganic crystals with localized spins on metal ions and in organic crystals based on one-dimensional (1D) stacks of radical ions.", "Second, the Hilbert space is smallest for $S = 1/2$ for any choice of exchange interactions, small enough to access the full spectrum and thermal physics for comparison with experiment.", "Third, long ago Bethe and Hulthen obtained the exact ground state [1], of the infinite chain with antiferromagnetic exchange between nearest neighbors, a prototypical gapless many-body system with quasi-long-range order.", "HAFs and related chains with $S \\ge 1$ came to the fore with Haldane's conjecture based on field theory that integer $S$ chains are gapped.", "[3] Shortly thereafter, White introduced the density matrix renormalization group (DMRG) method that made possible accurate numerical calculation of the ground state properties of $S \\ge 1$ chains.", "[4], The thermodynamic limit of spin chains with exchange interactions leads to quantum phase diagrams with many interesting correlated phases.", "According to the valence bond solid (VBS) analysis, [6] integer $S$ chains have localized edge states with spin $s = S/2$ .", "DMRG studies of finite chains have confirmed edge states in both integer [7], [8] $S$ and half integer [9], [10] $S$ chains.", "Machens et al., [11] have recently discussed short $S \\ge 1$ HAFs with comparable energies for bulk excitations and edge states.", "They summarize previous studies such as the relation of $S \\ge 1$ HAFs to the nonlinear $\\sigma $ model (NL$\\sigma $ M), its application to edge states, the VBS model and its valence bond diagrams.", "Qin et al., [9] applied DMRG to HAFs up to 100 spins to discuss the energies of edge states and to distinguish between chains of integer and half integer $S$ .", "DMRG is quantitative for $S = 1$ HAFs of $N \\le 100$ spins with correlation length $\\xi \\sim 6$ and large Haldane gap.", "Longer chains are necessary for the $S = 2$ HAF with $\\xi \\sim 50$ or for the gapless $S = 3/2$ HAF.", "In this paper we consider edge states of HAFs with $S = 1$ , 3/2 and 2 in systems of up to 500 spins.", "We use conventional DMRG for chains with an even number of spins and another algorithm for chains with an odd number of spins.", "We compute and model the spin densities of edge states as well as their excitation energies.", "The Hamiltonian of the spin-$S$ HAF chain with open boundary conditions (OBC) is $H_S(N) = J \\sum _{r=1}^{N-1} \\vec{S}_r \\cdot \\vec{S}_{r+1}.$ The spin at site $r$ is $S_r$ , the total spin $S_T$ and its $z$ component $S^z$ are conserved, and $J = 1$ is a convenient unit of energy.", "The terminal spins $r = 1$ and $N$ are coupled to only one spin in Eq.", "REF .", "Periodic boundary conditions (PBC) also has $J$ between sites 1 and $N$ .", "Every spin is then coupled to two neighbors, the system has translational symmetry, and the smallest $S_T$ is expected in the ground state (GS) for AF exchange.", "Indeed, the GS of PBC chains is a singlet, spin $S_G = 0$ , except for odd $N$ and half integer $S$ , when $S_G = 1/2$ .", "The sectors of integer and half integer $S$ are disjoint, and even $N$ is conventionally taken for the thermodynamic limit.", "As noted by Faddeev and Takhtajan, the thermodynamic limit of the $S = 1/2$ HAF with odd $N$ is not well understood.", "[12] HAFs with OBC are fundamentally different because there is no energy penalty for parallel spins at sites 1 and $N$ .", "The GS of Eq.", "REF remains a singlet for even $N$ , but it becomes a multiplet with $S_G = S$ and Zeeman degeneracy $(2S_G + 1)$ for odd $N$ .", "The lowest-energy triplet is necessarily an excited state when $N$ is even.", "For integer $S$ , the singlet is an excited state when $N$ is odd, while for half integer $S > 1/2$ , the doublet is an excited state when $N$ is odd.", "Except in the $S = 1/2$ case, $S_G$ depends on the boundary conditions for arbitrarily large systems.", "It follows that HAFs with OBC support edge states with $S_G \\ge 1$ whose energies become degenerate in the thermodynamic limit with those of PBC systems with $S_G = 0$ or 1/2.", "We define the energy gaps of edge states in chains of $N$ spins as $\\Gamma _S(N) = E_0(S,N) - E_0(0,N),$ where $E_0(S,N)$ is the lowest energy in the sector with total spin $S$ .", "Even $N$ leads to $\\Gamma _S(N) > 0$ .", "Odd $N$ leads to $\\Gamma _S(N) < 0$ for integer $S$ and to $\\Gamma _S(N) < 0$ relative to $E_0(1/2,N)$ for half integer $S$ .", "Since DMRG algorithms conserve $S^z$ rather than $S$ , the most accurate results are the GS in sectors with increasing $S^z$ and $\\Gamma _S(N) > 0$ .", "Otherwise, the singlet or doublet is an excited state in the $S^z = 0$ sector for integer $S$ or in the $S^z = 1/2$ sector for half integer $S$ .", "The size dependence of $\\Gamma _S(N)$ is faster than $1/N$ , which distinguishes gap states from bulk excitations that may also have zero gap in the thermodynamic limit.", "We shall characterize edge states using spin densities and call them boundary-induced spin density waves (BI-SDWs).", "BI-SDW is more descriptive than edge state and is more accurate than localized state, since BI-SDWs are not localized in half integer $S$ chains.", "By convention, we choose the Zeeman level $S^z = S$ when $S \\ge 1$ and define the spin density at site $r$ as $\\rho (r,N) = \\langle S^z_r \\rangle , \\qquad \\qquad r=1, 2 \\ldots N.$ The expectation value is with respect to the state of interest.", "Singlet states have $\\rho (r,N) = 0$ at all sites.", "SDWs with $S \\ge 1$ have equal spin density at $r$ and $N+1-r$ by symmetry in chains, $\\rho (N) = \\rho (1) > 0$ by construction and $\\rho (r,N)\\propto (-1)^{r-1}$ .", "It is advantageous to focus on spin densities rather than energy gaps.", "Spin densities are exclusively associated with $S > 0$ states while the $\\Gamma _S(N)$ in Eq.", "REF are small differences between extensive energies.", "The NL$\\sigma $ M is a good approximation for $S \\ge 1$ HAFs, and theoretical discussions have focused as much on field theory as on spin chains.", "[13], , [15], [16] The model for integer $S$ chains relates edge states to an effective Hamiltonian between spins $s^{\\prime } = S/2$ at the ends, [11] $H_{eff}(N) = (-1)^N J_e\\exp (-N/\\xi ) \\vec{s^{\\prime }}_1\\cdot \\vec{s^{\\prime }}_N.$ The correlation length $\\xi $ and exchange $J_e$ are fit to DMRG results for $H_S$ .", "An interesting point is that $\\xi $ refers to the bulk, the singlet GS in the thermodynamic limit, as has been confirmed within numerical accuracy in $S = 1$ chains.", "[7] The $S = 2$ chain has two gap states that afford more stringent tests of Eq.", "REF .", "For example, the ratio of the two gaps is necessarily 3:1 for $s^{\\prime }_1 = s^{\\prime }_N = 1$ .", "Edge states in HAFs with half integer $S \\ge 3/2$ have been discussed [17], [9], [11] using $H_{eff}$ with effective spins $s^{\\prime } = (S - 1/2)/2$ and effective exchange $J^{\\prime }(N)$ that decreases faster than $1/N$ but not exponentially.", "Our principal goal is the quantitative description of edge states in HAFs that are sufficiently long to neglect bulk excitations in $S = 3/2$ or 2 chains.", "The paper is organized as follows.", "Section summarizes conventional DMRG algorithm for even $N$ and a different algorithm for odd $N$ that is related to $Y$ junctions.", "Section presents BI-SDWs spin densities and gaps for $S = 1$ and $S = 2$ chains with finite Haldane gaps and finite correlation lengths $\\xi $ .", "DMRG returns $\\xi = 6.048$ for $S = 1$ chains, in agreement with 6.03(1) reported previously, [7] and $\\xi = 49.0$ for $S = 2$ chains.", "DMRG spin densities are fit quantitatively by BI-SDWs that are in phase for odd $N$ , out of phase for even $N$ .", "The coupling $H_{eff}(N)$ between ends is quantitative for $S = 1$ chains and is semi quantitative for $S = 2$ chains, in qualitative agreement with the VBS picture of localized spins.", "Section presents the BI-SDWs and gaps of the $S = 3/2$ chain.", "The BI-SDWs are not localized in this case.", "The singlet-triplet gap $\\Gamma _1(N)$ for even $N$ decreases faster than $1/N$ , as anticipated by Ng.", "[17] The gap $\\Gamma _{3/2}(N)$ for odd $N$ requires a modified $H_{eff}(N)$ with a delocalized spin in central part in addition to spins at the ends.", "The delocalized spin rationalizes $\|\\Gamma _{3/2}(N)\| > \\Gamma _1(N)$ and a weaker size dependence.", "The Discussion summarizes the limited nature of connections to the NL$\\sigma $ M or to VBS." ], [ "DMRG algorithms", "By now, DMRG is a mature numerical method for 1D systems.", "[18], [19] It gives excellent GS properties and has been widely applied to spin chains.", "Conventional DMRG starts with a superblock that consists of four sites: one site in the left block, one in the right block and two new sites, the central sites.", "The left and right blocks increase by one site as two new sites are added at every step.", "The procedure generates a chain with OBC and an even number of sites $N$ .", "The vast majority of DMRG calculations been performed on chains with even $N$ .", "White has discussed [20] an algorithm with one rather than two central sites that speeds up the computational time by a factor of two to four.", "The method was tested on an $S = 1$ HAF of 100 spins.", "We use conventional DMRG for spin chains with even $N$ and adapt an algorithm for odd $N$ that was developed for $Y$ junctions.", "[21] $Y$ junctions of $N = 3n + 1$ spins have three arms of $n$ spins plus a central site for which we recently presented an efficient DMRG algorithm.", "Fig.", "2 of Ref.", "mk2016 shows the growth of the infinite DMRG algorithm.", "A chain of $N = 2n + 1$ spins can be viewed as two arms of $n$ spins plus a central site.", "The algorithm takes the system as an arm plus the central site and the environment as the other arm.", "Since the system of $n + 1$ spins at step $n$ becomes the environment at the next step, the chain grows by two spins at each step.", "The procedure described for $Y$ junctions [21] holds with one fewer arm for chains of $N = 2n + 1$ spins.", "The accuracy of the algorithm for odd $N$ is comparable to conventional DMRG, as has already been shown for $Y$ functions.", "[21] In either algorithm, new sites are coupled to the most recently added sites and the superblock Hamiltonian contains only new and once renormalized operators.", "Table REF has representative DMRG results for the ground states of $S = 2$ and 3/2 chains with $N \\ge 100$ spins.", "The index $m$ is the number of states kept per block.", "The truncation error is $P(m) = 1 - \\sum _j^m \\omega _j$ where the sum is over the eigenvalues $\\omega _j$ of the density matrix.", "Several sweeps of finite DMRG calculations are required for $S = 3/2$ or 2, with $N$ calculations per sweep, and finite DMRG is necessary for accurate spin densities.", "Increasing $m$ rapidly increases the required computer resources for long chains and involves trade offs.", "We have checked our results against previous studies in Table REF as well as against $S = 1$ chains and find comparably small or smaller $P(m)$ that amount to evolutionary improvements for even $N$ .", "The algorithm for odd $N$ returns equally small $P(m)$ .", "Table: Representative previous and present DMRGcalculations for HAF chains with spin S=2S = 2 or 3/2 andN≥100N \\ge 100 sites.", "The truncation error is P(m)=a×10 -b P(m) = a \\times 10^{-b}when mm states are kept per block.In the following we have set $m$ according to Table REF and performed 5-10 sweeps of finite DMRG for $S = 2$ and 3/2 chains.", "We estimate that GS energies per site are accurate to $10^{-8}$ for $S = 1$ chains, to $10^{-6}$ for $S = 3/2$ and to $10^{-5}$ for $S = 2$ .", "The energy gaps $\\Gamma _S(N)$ between the GS in sectors with different total spin are accurate to $10^{-5}$ for $S = 1$ and to $10^{-4}$ for $S = 3/2$ or 2.", "The spin densities are estimated to be accurate to better than $10^{-4}$ based, for example, on DMRG calculations with different algorithms for $N$ and $N - 1$ .", "Accurate $\\rho (r,N)$ are readily obtained in large systems whose $\\Gamma _S(N)$ are not accessible." ], [ "Integer spin, $\\mathbf {S = 1}$ and 2", "We start with the extensively studied $S = 1$ HAF with OBC and even $N$ .", "The large Haldane gap [7], [23] $\\Delta (1) = 0.4105$ reduces the computational effort.", "The singlet-triplet gap $\\Gamma _1(N)$ in Eq.", "REF decreases rapidly with system size.", "The GS alternates between $S_G = 0$ and 1 for even and odd $N$ , respectively.", "We evaluate $\\Gamma _1(N)$ for even $N$ as the difference of the total energy in the sectors $S^z = 0$ and 1.", "In addition, we also obtain $\\Gamma _1(N) < 0$ for odd $N$ using the first excited state in the $S^z = 0$ sector.", "The excited state is accurate to $10^{-6}$ for $m > 300$ .", "As shown in Fig.", "REF , upper panel, with different symbols for even and odd $N$ , $\|\\Gamma _1(N)\|$ decreases as $J_e \\exp (-N/\\xi )$ with $\\xi = 6.048$ .", "The effective exchange between the ends is $J_e = 0.7137$ in Eq.", "REF with spins $s^{\\prime } = 1/2$ .", "The effective Hamiltonian is quantitative for $S = 1$ chains.", "The gap at $N = 80$ is $1.5 \\times 10^{-6}$ , which still exceeds the estimated numerical accuracy.", "The inset shows the relevant VBS valence bond diagram.", "[6] Each line is a singlet pair, $(\\alpha \\beta - \\beta \\alpha )/\\sqrt{2}$ , between $S =1/2$ spins, two per $S = 1$ site, and the circles are unpaired spins at the ends.", "Comparable DMRG accuracy for $S = 1$ chains with even $N$ has been discussed previously.", "S$ø$ rensen and Affleck found $\\xi = 6.07$ for $\\Gamma _1(N)$ and 6.028(3) for spin densities.", "[24] White and Huse obtained [7] the GS energy per site very accurately and reported $\\xi = 6.03(1)$ for the spin densities of a 60-site chain with an auxiliary spin-1/2 at one end (site $N + 1$ in Eq.", "REF ).", "Schollwöck et al., [8] discussed the same procedure for $S = 2$ chains with even $N$ and an auxiliary spin-1 at one end.", "Auxiliary spins at both ends with adjustable exchange to sites 1 and $N$ can be used to study bulk excitations.", "[7] In this paper, we shall not resort to auxiliary spins.", "We always consider BI-SDWs at both ends of chains.", "Figure: Upper panel: Singlet-triplet gap\|Γ 1 (N)\|\|\\Gamma _1(N)\| of S=1S = 1 chains with NN spins, even or odd,in Eq.", ";inset: VBS valence bond diagram for N=8N = 8.", "Lower panel:Spin densities ρ(r,65)\\rho (r,65) of the S=1S = 1 chain with 65 spins.The spin densities $\\rho (r,65)$ in Fig.", "REF , lower panel, are for the GS of the 65-spin chain.", "We take $S^z = 1$ and obtain positive $\\rho (r)$ at odd numbered sites and negative $\\rho (r)$ at even numbered sites, respectively.", "All chains with $S > 0$ have $\\rho (r,N) \\propto (-1)^{r-1}$ , which is why call them as BI-SDWs.", "Table REF lists the spin densities of the first 10 sites in chains of 66/65 spins and 48/47 spins.", "The 66/65 spin densities clearly refer to the same triplet and speak to the numerical accuracy since different algorithms are used.", "The spin density at site 1 is slightly greater than 1/2, and so is the total spin density to odd-numbered sites.", "The total spin density to an even-numbered site approaches 1/2 from below and exceeds 0.45 at $r = 10$ .", "The apparent exponential decrease of $\|\\rho (r,N)\|$ does not hold for the first few sites since, for example, $\|\\rho (2)\| < \\rho (3)$ .", "The triplets are identical near the ends but of course differ at the middle of the chain, where $\\rho (33,65) = 4.82 \\times 10^{-3}$ becomes $\\rho (33,66) = \\rho (34,66) = 3.7 \\times 10^{-4}$ .", "The 48/47 data illustrate the weak size dependence of spin densities at the ends.", "Well-defined edge states must become size independent.", "The first 10 sites of $N = 65$ or 66 chains are close to the thermodynamic limit of BI-SDWs.", "Table: DMRG results for spin densities at thefirst ten sites of S=1S = 1 chains of NN spins.To minimize the even-odd variations of spin densities and to divide out an overall scale factor, we consider the function $f(r,N) = \\frac{\\rho (r-1)-\\rho (r+1)}{\\rho (r-1)+\\rho (r+1)}\\approx -\\frac{\\partial }{\\partial r} \\ln \\rho (r,N).$ $f(r,N)$ is odd with respect to the chain's midpoint while $\\rho (r,N)$ is even.", "Figure REF shows $\\ln \|f(r,N)\|$ for $S = 1$ chains up to the middle, $r \\le (N + 1)/2$ .", "The DMRG points near the edge become size independent.", "Except for the first few ($\\sim 10$ ) sites, $f(r,N)$ is constant up to about $N/2 - 2\\xi $ .", "The difference between even and odd $N$ is clearly seen in the middle region, and $f(r,N)$ for even, odd pairs are a convenient way to present spin densities directly without making any assumptions about the appropriate model or interpretation.", "It follows that the thermodynamic limit is $\\ln \|f(r)\| = -1.801$ .", "The lines are fits as discussed below using the correlation length $\\xi = 6.048$ from the gap $\\Gamma _1(N)$ , in accord with the NL$\\sigma $ M's expectation of equal $\\xi $ for gaps and spin densities.", "Figure: Symbols are DMRG spin densities inf(r,N)f(r,N), Eq.", ", to the middle of S=1S = 1 chains of NNspins.", "The lines are Eq.", "with correlation lengthξ=6.048\\xi = 6.048.", "The horizontal dashed line is the thermodynamic limit.Figure: Symbols are DMRG results for \|ρ(r,N)\|\|\\rho (r,N)\|to the middle of S=1S = 1 chains.", "Lines are Eq.", "withξ=6.048\\xi = 6.048 and A=0.566A = 0.566.", "Even and odd NN deviate fromAexp(-r/ξ)A \\exp (-r/\\xi ) near the middle of chains.The magnitudes of the spin densities are shown in Fig.", "REF as a function of $r/\\xi $ up to the middle of the chains.", "They decrease as $A \\exp (-r/\\xi )$ and deviate upward in the middle for odd $N$ , downwards for even $N$ .", "The amplitude $A$ is independent of system size when $N/\\xi > 5$ .", "To model the spin densities of integer $S$ chains, we introduce SDWs at the left and right ends, $\\rho (r,N) &=& A (-1)^{r-1} \\left[ \\exp (-r/\\xi ) \\right.", "\\nonumber \\\\& & \\left.", "- (-1)^N \\exp \\left(-(N+1-r)/\\xi \\right) \\right].$ The SDWs are in phase for odd $N$ when all odd-numbered sites have $\\rho > 0$ ; they are out of phase for even $N$ with equal $\\rho $ at sites $N/2$ and $N/2 + 1$ .", "Except for Ref.", "sorensen94, the spin densities have been assumed to decrease exponentially, thereby ignoring contributions from the other end.", "While that is the case in the thermodynamic limit, $N > 10\\xi $ is minimally required to neglect contributions from the other BI-SDW in the middle.", "Since the system size in DMRG calculations rarely exceeds $10 \\xi $ , it is advantageous to consider both ends.", "We have $\\rho (r,N) = 2 A (-1)^{r-1} \\exp \\left(-(N+1)/2\\xi \\right){\\left\\lbrace \\begin{array}{ll}\\cosh \\left((N+1-2r)/2\\xi \\right), & \\quad (odd N) \\\\\\sinh \\left((N+1-2r)/2\\xi \\right), & \\quad (even N) \\\\\\end{array}\\right.", "}$ The postulated BI-SDWs lead to $f(r,N) = \\tanh (1/\\xi ){\\left\\lbrace \\begin{array}{ll}\\tanh \\left( (N+1-2r)/2\\xi \\right), \\quad (odd N) \\\\\\coth \\left( (N+1-2r)/2\\xi \\right), \\quad (odd N) \\\\\\end{array}\\right.", "}$ The relative phase of the SDWs matters within $\\pm 2\\xi $ of the middle.", "The range of $r$ is the same for $N$ and $N - 1$ when $N$ is even.", "The lines in Fig.", "REF are $\\ln \|f(r,N)\|$ for $\\xi = 6.048$ and continuous $r$ in Eq.", "REF .", "The thermodynamic limit is $f(r) = \\tanh (1/\\xi )$ , the dashed line in Fig.", "REF , and it reduces to $1/\\xi $ for a continuous chain.", "The correlation length $\\xi $ is accurately obtained using both even and odd chains.", "The $N = 119/120$ spin densities indicate a gap of $\\Gamma _1(120) = 1.72 \\times 10^{-9}$ that is far below the accuracy of the energy difference.", "DMRG spin densities are also limited, however, to less than 149/150; there the $\\rho (r,N)$ show considerable scatter where $f(r,N)$ has even-odd variations.", "The SDW amplitude $A = 0.656$ in Fig.", "REF accounts quantitatively for spin densities aside from the first few.", "The parameters $\\xi $ and $A$ suffice for all fits in Figs.", "REF and REF .", "The $S = 2$ chain has a smaller Haldane gap [23] of $\\Delta (2) = 0.088$ .", "Numerical analysis is more difficult since (i) there are more degrees of freedom per site; (ii) $N > 5\\xi $ requires longer chains; and (iii) gaps $\\Gamma _S(N) < \\Delta (2)$ also require longer chains to distinguish between edge and bulk excitations.", "Results are fewer and less accurate.", "The nature of BI-SDWs in $S = 2$ or 3/2 chains was the motivation for DMRG calculations on even and odd chains of hundreds of spins.", "According to the NL$\\sigma $ M, edge states for $S = 2$ are associated with spin $s^{\\prime } = S/2 = 1$ in $H_{eff}$ , Eq.", "REF .", "Even chains have a singlet GS and gaps to two edge states, $\\Gamma _1(N)$ to the triplet ($S = 1$ ) and $\\Gamma _2(N)$ to the quintet ($S = 2$ ).", "The correlation length $\\xi $ is the same for both and $\\Gamma _2(N) = 3 \\Gamma _1(N)$ .", "The VBS valence bond diagram corresponds to two $S = 1$ diagrams in Fig.", "REF (a): There are four lines per interior $S = 2$ site and two lines, two unpaired spin at the ends.", "The BI-SDW analysis of $S = 1$ chains is equally applicable to integer $S$ chains.", "Increasing $S$ leads to longer $\\xi $ and to gaps $\\Gamma _S(N)$ whose relative magnitudes are fixed in advance by Eq.", "REF .", "A chain with $J = 1$ and 200 spins $S = 2$ or 400 spins $S = 3/2$ has a GS energy of roughly $-10^3$ .", "The corresponding $\\Gamma (N)$ in Table REF are less than $10^{-3}$ and their estimated accuracy is $\\pm 1\\times 10^{-4}$ .", "Our $S = 2$ and 3/2 gaps are consequently limited to $N \\sim 200$ and 450, respectively.", "They are differences between total energies.", "Spin densities, by contrast, are exclusively related to the GS in a sector with $S > 0$ .", "The representative gaps in Table REF cover more than a decade.", "We studied the $m$ dependence of gaps in $S = 2$ and 3/2 chains, summarized in Table REF , in order to find the largest accessible systems.", "The gap $\\Delta (2)$ of the infinite $S = 2$ chain is slightly larger than $\\Gamma _2(32)$ .", "The competition between edge and bulk excitations in short HAFs with $S \\ge 1$ is discussed elsewhere.", "[9], [10], [11], [25] Table: Edge-state energy gaps Γ(N)\\Gamma (N),Eq.", ", of HAFs with NN spins SS and J=1J = 1 inEq.", ".Figure REF shows $\\Gamma _1(N)$ and $\\Gamma _2(N)$ for $S = 2$ and even $N$ .", "The gaps are exponential in $N/\\xi $ , as expected for integer $S$ .", "The correlation length, $\\xi \\sim 49$ , is the same within our numerical accuracy.", "The ratio is $\\Gamma _2/\\Gamma _1 = 3.45$ based on the fitted lines and it varies between 3.27 and 3.56 for individual points.", "Although $\\Gamma _2/\\Gamma _1 = 3.45$ is approximate, the ratio is larger than the NL$\\sigma $ M value of 3 based on Eq.", "REF .", "We return to gaps after presenting results for spin densities.", "Figure: Edge-state gaps Γ 1 (N)\\Gamma _1(N) andΓ 2 (N)\\Gamma _2(N) of S=2S = 2 chains of NN spins in Eq.", ".$S = 2$ chains with odd $N$ have a quintet GS and excitations to the triplet and singlet.", "We again use $f(r,N)$ and the BI-SDWs analysis.", "Figure REF shows $\\ln \|f(r,N)\|$ in the $S^z = 2$ sector up to the middle of the chains.", "For the sake of clarity, not all points are shown.", "Even-odd effects now extend to about the first 25 sites and become size independent in long chains.", "The thermodynamic limit is $f(r) = \\tanh (1/\\xi )$ with $\\xi = 49.0$ .", "The magnitudes of spin-densities in Fig.", "REF are fit as a function of $r/\\xi $ with the same $\\xi $ and $A = 0.90$ in Eq.", "REF .", "Two parameters are nearly quantitative aside from sites $r < 25$ .", "The triplet is an excited state for either even or odd $N$ .", "It is the lowest state in the $S^z = 1$ sector for even $N$ and the first excited state in that sector for odd $N$ .", "DMRG calculations for even $N$ converge slowly for reasons we do not understand in detail.", "The triplet spin densities return the same $f(r,N)$ as the quintets in Fig.", "REF .", "Figure: Open symbols are DMRG spin densitiesin the S z =2S^z = 2 sector for f(r,N)f(r,N), Eq.", ", to themiddle of S=2S = 2 chains of NN spins.", "Solid lines are Eq.", "with correlation length ξ=49.0\\xi = 49.0.", "The horizontal dashedline in the thermodynamic limit.Figure: Open symbols are DMRG results for\|ρ(r,N)\|\|\\rho (r,N)\| in the S z =2S^z = 2 sector to the middle of S=2S = 2chains.", "Lines are Eq.", "with ξ=49.0\\xi = 49.0and A=0.90A = 0.90.", "Even and odd NN deviate from Aexp(-r/ξ)A \\exp (-r/\\xi )in the middle.The correlation length $\\xi = 49.0$ based on spin densities is more accurate than $\\xi $ from energy gaps.", "The $\\xi = 49.0$ fits account for $\\rho (r,N)$ of even and odd chains that extend to 500 spins, whereas numerical accuracy limits $\\Gamma _S(N)$ to $N \\sim 200$ .", "Schollwöck et al., [8] argued that the thermodynamic limit requires $N > 5\\xi $ and obtained (Fig.", "6 of [schollwock96]) $\\xi = 49(1)$ for $N = 270$ with an auxiliary spin-1 at the other end using the local correlation length $\\xi (r) = 2/[ln(\\rho (r-1)/\\rho (r+1))]$ .", "Qin et al., [9] estimated that $\\xi \\sim 33$ for S = 2 chains up to $N = 100$ and remarked that the accuracy was much worse than for $S = 1$ chains.", "Indeed, spin densities for $N = 127/128$ return $\\xi \\sim 36$ .", "As seen in Figs.", "REF and REF for $S = 1$ and 2, respectively, the thermodynamic limit requires $N > 5\\xi $ even when the contribution of the BI-SDW at the other end is included.", "The present results for $S = 2$ chains offer more stringent comparisons of the NL$\\sigma $ M. The model is semi quantitative: The ratio $\\Gamma _2/\\Gamma _1 = 3.45$ is greater than 3.", "We note that BI-SDWs with exponentially decreasing $\\rho (r,N)$ would assumed on general grounds and follows directly from $f(r, N)$ , but not the same $\\xi $ for gaps and spin densities." ], [ "Half integer spin, $\\mathbf {S = 3/2}$", "HAF chains with half integer $S \\ge 3/2$ are gapless and their edge states are fundamentally different.", "Even chains have a singlet GS and BI-SDWs with integer $S$ ; odd chains have $S_G = S$ and BI-SDWs with half integer $S > 1/2$ .", "The even $S = 3/2$ chain has a singlet-triplet gap $\\Gamma _1(N)$ that decreases faster than $1/N$ and has been studied by Qin et al., [9] and in greater detail by Fáth et al.", "[10] The NL$\\sigma $ M gap goes as [10] $N \\Gamma _1(N) = \\frac{a}{\\ln B N} + O\\left( \\frac{\\ln \\ln N}{(\\ln N)^2} \\right)$ Fath et al.", "[10] used DMRG to compute $\\Gamma _1(N)$ for $S = 3/2$ chains from $N = 12$ to 192 in steps of 12 spins.", "The first term of Eq.", "REF leads to parameters $a(N)$ and $B(N)$ whose size dependence was obtained from successive gaps $\\Gamma _1(N+12)$ and $\\Gamma _1(N)$ .", "Extrapolation in $1/N$ gave the thermodynamic values of $a = 1.58$ and $B = 0.11$ with $\\pm 15\\%$ uncertainties.", "In the present study, we are characterizing BI-SDWs in spin chains and take the first term with constant $a$ , $B$ as a two-parameter approximation.", "Figure REF shows the calculated gaps of $S = 3/2$ HAFs as $N\|\\Gamma (N)\|$ .", "The gaps decrease faster than $1/N$ as expected for edge states.", "The NL$\\sigma $ M size dependence for even $N$ is Eq.", "REF with $J_e(N) = \\Gamma _1 = a/(N \\ln B N)$ .", "The dashed line has $a = 1.58$ and $B = 0.11$ as inferred by Fáth et al.", "[10] The solid line for even $N$ is a power law with two parameters, $\\Gamma _1 = J_e(N) = 4.79 N^{-1.42}$ .", "Either fit is adequate over this range of system sizes, and neither accounts the decrease at $N > 400$ .", "The shortest chains in which edge and bulk excitations are decoupled are probably in the range $N = 30$ to 60, and the desired $\\Gamma _1(N)$ fits are for long chains.", "The gaps $\|\\Gamma _{3/2}(N)\|$ for odd $N$ are several times larger and their size dependence is weaker.", "They can be approximation by a different logarithm or power law.", "The gaps $\\Gamma _{3/2}(N)$ and $\\Gamma _1(N)$ of the $S = 3/2$ chain are in marked contrast to equal $\|\\Gamma _1(N)\|$ in Fig.", "REF for $S = 1$ chains with even and odd $N$ .", "Figure: Open symbols are DMRG results forthe gaps Γ 1 (N)\\Gamma _1(N) and \|Γ 3/2 (N)\|\|\\Gamma _{3/2}(N)\| for S=3/2S = 3/2chains with NN spins in Eq. .", "The solid and dashedlines are power law and logarithmic fits, respectively, with twoparameters.", "For even NN, the NLσ\\sigma M parameters in Eq.", "are a=1.58a = 1.58, B=0.11B = 0.11.The BI-SDWs of even chains are triplets.", "The ratios $f(r,N)$ in Eq.", "REF are quite different for the $S = 3/2$ chain, either even or odd, and are not shown.", "The upper panel of Fig.", "REF shows the magnitude of spin densities up to the middle of chains.", "The SDWs converge at small $r$ but are not localized in the $S = 3/2$ chain.", "The spin densities add up to $S^z = 1$ for even $N$ .", "They decrease slowly and the sum over $\|\\rho (r,N)\|$ diverges in the thermodynamic limit.", "The lines are fits that are discussed below.", "The lower panel of Fig.", "REF shows the cumulative spin density to site $R$ that we define as $T(R,N) = \\sum _{r=1}^R \\rho (r,N) + \\rho (R+1,N)/2.$ The total spin density is $S^z = 1 = T(N - 1,N) + \\rho (1,N)/2$ .", "$T(R,N)$ increases rapidly to 0.5 around $R_{1/2} \\sim 15$ , reaches a broad maximum that depends on system size and decreases as required by symmetry to $0.5$ in the middle of the chain.", "The VBS valence bond diagram in the inset has unpaired spins at each end that correspond [17] to $s^{\\prime } = (S - 1/2)/2 = 1/2$ .", "Each $S = 3/2$ site forms three singlet-paired spins to a neighbor.", "The middle and either the top or bottom line corresponds to the VBS diagram of the $S = 1$ chain with a localized spin at the ends.", "The remaining line with paired spins is a singlet valence bond diagram of the $S = 1/2$ chain.", "The slow variation of $T(R,N)$ in the middle and no net spin between $R_{1/2}$ and $N - R_{1/2}$ is consistent with singlet-paired spins.", "Figure: Upper panel: Open symbols are DMRGspin densities \|ρ(r,N)\|\|\\rho (r,N)\| to the middle of S=3/2S = 3/2 HAFs witheven NN in Eq. .", "The lines are fits based onEq.", "with B=2B = 2 and the indicated scale factorsC N C_N.", "Lower panel: Cumulative spin densities, Eq.", ",up to site RR; inset: VBS valence bond diagram for even NN.The GS of odd chains is a quartet, $S = 3/2$ .", "Figure REF , upper panel, shows $\|\\rho (r,N)\|$ to the middle of chains.", "The large amplitude of in-phase BI-SDWs in the middle decreases slowly with system size.", "The cumulative spin density $T(R,N)$ in the lower panel is again given by Eq.", "REF except that the $r = (N + 1)/2$ spin density is shared equally between the two halves.", "The total is $S^z = 3/2$ for the entire chain, or 0.75 for the half chain.", "The rapid initial increase to $T(R_{1/2},N) = 0.5$ by $R_{1/2} \\sim 15$ suggests a spin-1/2, as does the gradual increase to $0.75$ in the middle.", "The VBS valence bond diagram in the lower panel has three unpaired spins, two at one end, one at the other end; the diagram with reversed unpaired spins at the ends contributes equally by symmetry.", "The middle and either top or bottom line is again the $S = 1$ VBS diagram.", "The remaining line is an $S = 1/2$ valence bond diagram with an unpaired spin at either end.", "Although the diagram correctly has three unpaired spins, the DMRG spin densities clearly show one spin in the central region rather than at the ends.", "The $S = 1/2$ HAF with odd $N$ does not support edge states.", "The spin density is delocalized over the entire chain.", "[26] Even more simply, a half-filled tight binding or Hückel band of $N = 2n + 1$ sites has spin density $1/(n + 1)$ at odd numbered sites and $\\rho = 0$ at even numbered sites; in that case, $T(R,N)$ goes as $R/(n + 1)$ and immediately rationalizes the linear increase in Fig.", "REF , lower panel.", "We attribute the larger gap $\|\\Gamma _{3/2}(N)\| > \\Gamma _1(N)$ in Fig.", "REF and its weaker dependence of system size to enhanced coupling between the ends by the delocalized spin in the middle.", "Figure: Upper panel: Open symbols areDMRG spin densities \|ρ(r,N)\|\|\\rho (r,N)\| to the middle of S=3/2S = 3/2HAFs with odd NN in Eq. .", "The lines are fitsbased on Eq.", "with B=2B = 2 and the indicatedscale factors C N C_N.", "Lower panel: Cumulative spin densities,Eq.", ", up to site RR; inset: one of twoequivalent VBS valence bond diagrams for odd NN).The BI-SDW amplitude at the middle in the upper panel of Fig.", "REF decreases slightly faster than $N^{-1/2}$ .", "The size dependence of the amplitude suggests modeling the spin densities as $\\rho (r,N) &=& (-1)^{r-1} C_N \\left( \\left(\\frac{\\ln B r}{r}\\right)^{1/2}\\right.\\nonumber \\\\& & \\left.", "-(-1)^N \\left(\\frac{\\ln B(N+1-r)}{N+1-r}\\right)^{1/2}\\right).$ The amplitude $C_N$ depends on system size because the SDWs are not localized.", "We took $\|\\rho (r,N)\|$ with $B = 2$ and the indicated $C_N$ to generate the lines in the upper panels of Figs.", "REF and REF .", "The spin densities are adequately fit in the central region in either case.", "Deviations are seen for $r < 10$ when $N$ is even and for $r < 15$ when $N$ is odd.", "To some extent, Eq.", "REF can be understood in terms of the NL$\\sigma $ M. In the thermodynamic limit, the GS spin correlation functions $C(r)$ depend only on the separation $r$ between spins.", "The NL$\\sigma $ M result is [24] $C(r) \\equiv \\langle S_0^z S_r^z \\rangle \\propto (-1)^r r^{-1/2}\\exp (-r/\\xi ),$ for integer spin HAFs and $r \\gg \\xi $ .", "Several authors [27], [7], [24] have remarked that DMRG results for $r^{1/2}\|C(r)\|$ are noticeably closer to exponential in $S = 1$ chains of 60 or 100 spins.", "Since converged $C(r)$ are limited to about $r < N/4$ , such agreement is promising but not forced.", "White and Huse discuss [7] the point explicitly and show (Fig.", "4 of [white-huse-prb93]) that the ratio $\|C(r)\|$ to the NL$\\sigma $ M correlation function becomes constant at $r \\sim 2 \\xi \\sim 12$ .", "The first few sites where $C(r)$ can be computed most accurately are inevitably excluded from direct comparison since the NL$\\sigma $ M describes a continuous rather than a discrete system.", "The $r^{-1/2}$ factor in $C(r)$ does not appear in the spin densities of integer $S$ chains, [24] whose exponential decrease with $r/\\xi $ is shown in Figs.", "REF and REF .", "The spin correlations of the $S = 1/2$ HAF go as $\|C(r)\| \\propto (\\ln r/ r_0)^{1/2}/r$ according to field theory [16] and Monte Carlo calculations [28] up to $N = 4096$ return $r_0 = 0.08$ .", "But exact results for $C(r)$ in finite PBC systems [28] still show significant deviations at $N = 32$ .", "Hallberg et al.", "[29] applied the NL$\\sigma $ M and DMRG to the $S = 3/2$ chain and confirmed that it belongs to the same universality class as the $S = 1/2$ chain.", "They report $\|C(r)\| \\propto (\\ln B r)^{1/2}/r$ and estimate $B = 0.60$ from $r = 4$ to 25 in a 60-spin chain.", "We find $B = 0.45$ in similar calculations for $N$ = 200.", "Fáth et al.", "[10] extrapolate to $B = 0.11$ for $\\Gamma _1(N)$ in the thermodynamic limit.", "The differences are negligible in the context of spin densities.", "Then $r^{1/2} \|C(r)\|$ gives Eq.", "REF when contributions from both ends are taken into account.", "DMRG results for $\|\\rho (r,N)\|$ deviate from Eq.", "REF near the ends of $S = 3/2$ chains and from Eq.", "REF in integer $S$ chains.", "The choice of $B$ changes the fits at small $r$ .", "Since small $r$ is not modeled quantitatively in either case and does not concern us here, we took $B = 2$ in Eq.", "REF for the spin densities of $S = 3/2$ chains.", "Three effective spins are needed for the $S = 3/2$ spin densities when $N$ is odd, a spin $s^{\\prime }$ in the middle in addition to spins at the ends.", "The generalization of Eq.", "REF to half integer $S$ and odd $N$ is $H_{eff}(N) = -J_1(N)\\left(\\vec{s^{\\prime }} \\cdot \\left(\\vec{s^{\\prime }}_1 + \\vec{s^{\\prime }}_N\\right) \\right) - J_2(N) \\vec{s^{\\prime }}_1 \\cdot \\vec{s^{\\prime }}_N.$ The eight microstates of $H_{eff}$ correspond to the GS quartet and two doublets.", "Both the total effective spin $S^{\\prime }$ and $S^{\\prime }_{1N} = s^{\\prime }_1 + s^{\\prime }_N$ are conserved, with $S^{\\prime } = 3/2$ , $S^{\\prime }_{1N} = 1$ in the GS.", "The doublets have $S^{\\prime } = 1/2$ and $S^{\\prime }_{1N} = 0$ or 1.", "The spectrum is $E_{eff}(S^{\\prime },S^{\\prime }_{1N}) &=& -\\frac{J_1}{2} S^{\\prime }(S^{\\prime }+1)+ \\frac{J_1 - J_2}{2} S^{\\prime }_{1N}(S^{\\prime }_{1N} + 1) \\nonumber \\\\& & \\quad + \\frac{3(J_1+2J_2)}{8}.$ The gap $\\Gamma _{3/2} = -J_1/2 - J_2$ is to the doublet with singlet-paired spins at the ends; the gap to parallel spins is $-3J_1/2$ .", "The effective exchanges in Eq.", "REF can be fit to DMRG results for the doublets with the lowest and second lowest energy in the $S^z = 1/2$ sector.", "We find $J_1(99) = 0.04214$ , $J_2(99) = -0.00346$ and $J_1(199) = 0.01836$ , $J_2(199) = -0.00176$ .", "Large $\|\\Gamma _{3/2}(N)\|$ in Fig.", "REF for odd $N$ is due to $J_1(N)$ and coupling through the delocalized effective spin $s^{\\prime }$ .", "The small effective exchange $J_2(N)$ is antiferromagnetic.", "To conclude the discussion of $S = 3/2$ chains, we recall that the GS for PBC and odd $N$ has $S_G = 1/2$ .", "Since $J_{1N} = J$ is between sites in the same sublattice, the system is not bipartite, and the GS has a domain wall or topological soliton.", "The OBC system is bipartite.", "The doublet $S = S^z = 1/2$ with the lowest energy has positive spin densities at odd-numbered sites and negative spin densities at even-numbered sites, respectively, with singlet paired $s^{\\prime }_1$ and $s^{\\prime }_N$ in Eq.", "REF .", "Figure REF shows $\|\\rho (r,N)\|$ for $S = S^z = 1/2$ to the middle of $S = 3/2$ chains in the upper panel and the cumulative spin density $T(R,N)$ in the lower panel.", "The magnitude of the spin density at the middle decreases roughly as $N^{-0.42}$ .", "There are no boundary-induced edge states.", "The spin is delocalized as expected on general grounds and becomes the effective spin $s^{\\prime }$ in Eq.", "REF .", "By contrast, the spin densities are entirely associated with BI-SDWs in OBC systems with even $N$ or integer $S$ since singlet states have $\\rho (r,N) = 0$ at all sites.", "Figure: Upper panel: DMRG spin densities\|ρ(r,N)\|\|\\rho (r,N)\| to the middle of S=3/2S = 3/2 HAFs with odd NNin Eq.", "for the lowest energy doublet state,S=S z =1/2S = S^z = 1/2.", "Lower panel: Cumulativespin densities, Eq.", ", up to site RR." ], [ "Discussion", "We have applied different DMRG algorithms to spin-$S$ HAFs, Eq.", "REF , with even and odd number sites in order to obtain accurate edge states in chains of several hundred spins.", "The principal results are the energy gaps $\\Gamma _S(N)$ , Eq.", "REF , and the spin densities $\\rho (r,N)$ , Eq.", "REF , that are modeled as boundary-induced spin density waves (BI-SDWs) at both ends.", "For the $S = 1$ HAF, we reproduce and refine previous studies on even chains of 60 or 100 spins that exceed the correlation length $\\xi = 6.048$ by an order of magnitude.", "We confirm that the gap goes as $(-1)^N J_e \\exp (-N/\\xi )$ in chains with odd $N$ .", "Two parameters, $\\xi $ and the SDW amplitude, account quantitatively for $\\Gamma _1(N)$ and $\\rho (r,N)$ for chains from $N = 35$ to at least 120.", "The BI-SDWs are in phase for odd $N$ , out of phase for even $N$ .", "The smaller Haldane gap of the $S = 2$ HAF or the gapless $S = 3/2$ HAF requires substantially longer chains, here up to 500 spins, whose edge states have previously been studied in shorter chains $N < 200$ .", "The spin densities of $S = 2$ HAFs beteen $N = 199$ and 502 are modeled by BI-SDWs with correlation length $\\xi = 49.0$ and amplitude $A = 0.90$ .", "There are now two gaps, $\\Gamma _1(N)$ and $\\Gamma _2(N)$ , that decrease exponentially as $r / \\xi $ up to the $N \\sim 220$ limit of our numerical accuracy.", "The gap ratio is $\\Gamma _2(N)/\\Gamma _1(N) = 3.45$ .", "The gap $\\Gamma _1(N)$ of the $S = 3/2$ HAF with even $N$ decreases faster than $1/N$ , roughly as $N^{-1.42}$ or as $1/\\ln (0.11 N)$ .", "The gap $\\Gamma _{3/2}(N)$ for odd $N$ has larger amplitude and weaker size dependence.", "The BI-SDWs of the $S = 3/2$ chain have maximum spin density at the ends but are not localized.", "The $S = 3/2$ spin densities in chains of more than 100 spins have not been previously reported to the best of our knowledge.", "The $S = 3/2$ ground state for odd $N$ can be modeled as a spin-1/2 at each end and a spin-$1/2$ in between.", "DMRG calculations can be performed on longer chains of $N \\sim 1000$ and/or larger $S$ .", "But the condition $N > 5 \\xi $ for integer $S$ is increasingly difficult to satisfy for small Haldane gaps $\\Delta (S)$ whose rapid decrease has been reported [23] to $S = 5$ .", "Moreover, the gaps will require extraordinary accuracy since, as shown in Table REF , $\\Gamma (N) < 10^{-3}$ is reached at $N = 200$ for $S = 2$ or at $N = 400$ for $S = 3/2$ .", "Spin densities are more promising probes of long chains in terms of the $S^z > 0$ sectors of $N$ and $N - 1$ spins.", "But the required system size for half integer $S$ is poorly known and may not have been reached in the present work.", "The nonlinear sigma model (NL$\\sigma $ M) and valence bond solid (VBS) have been applied to spin chains, primarily to the $S = 1$ HAF in the thermodynamic limit.", "Machens et al.", "[11] summarize and critically evaluate both the NL$\\sigma $ M and VBS in connection with short chains of less than 20 spins.", "In partial disagreement with earlier works, they find that the effective coupling between edge states in Eq.", "REF in short chains is influenced by the comparably small finite-size gaps of bulk excitations.", "We have characterized long chains whose prior modeling has mainly been for $S = 1$ .", "Accurate DMRG results for $S = 2$ or $S = 3/2$ HAFs are a prerequisite for comparisons, mainly via $H_{eff}$ in Eq.", "REF , with either the NL$\\sigma $ M or VBS.", "Good agreement in $S = 1$ chains carries over to some extent to $S = 2$ chains and less so to $S = 3/2$ chains.", "Spin densities to $N = 500$ yield $\\xi = 49.0$ for the correlation length of the $S = 2$ chain.", "The gaps in shorter chains return the same $\\xi $ , but the ratio $\\Gamma _2(N)/\\Gamma _1(N)$ is 3.45 instead of 3.", "The deviation is real.", "The 3/2 chain does not follow the [25] $(-1)^NJ_e(N)$ pattern of $H_{eff}$ in Eq.", "REF .", "The BI-DWIs are not localized.", "Two effective spins $s^{\\prime } = 1/2$ at the ends account for $\\Gamma _1(N)$ when $N$ is even.", "A third $s^{\\prime } = 1/2$ in the middle leads to $\\Gamma _{3/2}(N)$ and $H_{eff}$ in Eq.", "REF for odd $N$ .", "In other ways, however, comparisons are simply not possible.", "Since field theory starts with a continuous system rather than a discrete chain, the ends can be distinguished from the bulk but not sites at a finite distance from the ends.", "Similarly, VBS deals with special Hamiltonians, [6], [11], [8], [30] that contain, in addition to Eq.", "REF , terms that go as $B_p({\\mathbf {S}}_r \\cdot {\\mathbf {S}}_{r+1})^p$ with $2 \\le p \\le 2S$ and coefficients $B_p$ .", "Exact GS are obtained in the thermodynamic limit of these models.", "The relevant valence bond diagrams have paired spins, as shown, except at the first and last sites.", "Either the NL$\\sigma $ M or VBS correctly places localized states or unpaired spins for integer $S$ , but neither describes the BI-SDWs found in DMRG calculations spin-$S$ HAFs.", "The BI-SDWs are not localized in half integer $S$ chains and have different effective coupling between ends.", "Direct solution of Eq.", "REF for $S \\ge 1$ chains inevitably leads to edge states whose features are blurred or lost in the NL$\\sigma $ M or VBS.", "Comparisons may well be limited to effective spins and exchange at the ends.", "The occurrence of edge states in HAFs with $S \\ge 1$ follows directly from Eq.", "REF , as shown in the Introduction.", "PBC systems with $J_{1N} = J$ have $S_G = 0$ except for half integer $S$ and odd $N$ , when $S_G = 1/2$ .", "OBC systems with $J_{1N} = 0$ have $S_G = 0$ for even $N$ and $S_G = S$ for odd $N$ .", "The energy per site in the thermodynamic limit cannot depend on boundary conditions for short-range interactions.", "Different $S_G$ under OBC and PBC implies edge states, or BI-SDWs, in HAF with $S \\ge 1$ and gaps $\\Gamma _S(N)$ relative to $S_G = 0$ for even $N$ or integer S or to $S_G = 1/2$ for odd $N$ and half integer S. The size dependence and interpretation of gaps or spin densities are standard for integer $S$ .", "The spin densities and gaps of the $S = 3/2$ chain lead to different BI-SDWs for even and odd $N$ .", "The NL$\\sigma $ M or VBS provides useful guidance for quantitative modeling of BI-SDWs obtained by DMRG for HAFs with $S \\ge 1$ .", "We thank S. Ramasesha and D. Huse for discussions and the NSF for partial support of this work through the Princeton MRSEC (DMR-0819860).", "MK thanks DST for a Ramanujan Fellowship SR/S2/RJN-69/2012 and DST for funding computation facility through SNB/MK/14-15/137." ] ]	1606.05054
[ [ "Coexistence of Weyl Physics and Planar Defects in Semimetals TaP and\n TaAs" ], [ "Abstract We report a structural study of the Weyl semimetals TaAs and TaP, utilizing diffraction and imaging techniques, where we show that they contain a high density of defects, leading to non-stoichiometric single crystals of both semimetals.", "Despite the observed defects and non-stoichiometry on samples grown using techniques already reported in the literature, de Haas-van Alphen measurements on TaP reveal quantum oscillations and a high carrier mobility, an indication that the crystals are of quality comparable to those reported elsewhere.", "Electronic structure calculations on TaAs reveal that the position of the Weyl points relative to the Fermi level shift with the introduction of vacancies and stacking faults.", "In the case of vacancies the Fermi surface becomes considerably altered, while the effect of stacking faults on the electronic structure is to allow the Weyl pockets to remain close to the Fermi surface.", "The observation of quantum oscillations in a non-stoichiometric crystal and the persistence of Weyl fermion pockets near the Fermi surface in a crystal with stacking faults point to the robustness of these quantum phenomena in these materials." ], [ "Introduction", "Weyl fermions, massless fermions predicted by Hermann Weyl in 1929[1] as solutions to the Dirac equation, have not yet been observed as fundamental particles in high energy physics.", "In 2011, however, it was predicted that Weyl fermions can be realized in condensed matter physics as electronic quasi-particles in the family of pyrochlore iridates[2] and the ferromagnetic spinel compound HgCr$_2$ Se$_4$ .", "[3] Following recent theoretical predictions of Weyl fermions in the simple semimetal TaAs and its isostructural compounds TaP, NbAs, and NbP,[4], [5] Weyl fermions were discovered experimentally shortly thereafter in TaAs[6], [7] and was quickly confirmed by additional studies,[8], [9] along with the discovery of Weyl fermions in the isostructural NbAs[10] and TaP.", "[11] Several studies on these semimetals have emerged: detailed investigations of the Fermi surface topology,[12], [13], [14] the observation of large magnetoresistance and high carrier mobility,[15], [16], [17], [18], [19], [20], [21] the report of a quantum phase transition in TaP,[22] a Raman study of the lattice dynamics identifying all optical phonon modes in TaAs,[23] a magnetization study of TaAs,[24] and pressure studies of NbAs.", "[25], [26] Until now, all four transition metal pnictide semimetals – TaP, TaAs, NbP, and NbAs – have been studied with an assumed nominal 1:1 stoichiometric ratio between the transition metal and the pnictide.", "However, it is well known that the thermodynamic and transport properties of a material depend on the actual stoichiometry: e.g., the magnetoresistance and the Fermi surface topology might be modified by disorder.", "In fact, a recent study observed quantum interference patterns arising from quasi-particle scattering near point defects on the surface of a single crystalline NbP,[27] followed by a theoretical investigation of surface state quasi-particle interference patterns in TaAs and NbP.", "[28] It is therefore pertinent that a structural study be carried out on these materials to determine what, if any, defects are present.", "The four semimetals have been extensively studied prior to the recent surge of interest.", "In fact, a non-stoichiometric composition for one of these compounds was already reported in 1954:[29] niobium phosphide was found as NbP$_{0.95}$ , and an assumption was made that TaP would have a similar composition: TaP$_{0.95}$ .", "The reported symmetry, the centrosymmetric group $I4_{1}/amd$ , was later corrected to the non-centrosymmetric space group $I4_{1}md$ (# 109) for all isostructural semimetals.", "[30], [31] The phase relations were further explored in a number of reports.", "[32], [33], [34], [35], [36], [37] Stacking disorder was reported by Willerström in all four semimetals,[38] where the disorder originates from a formation of a metastable WC-type (hexagonal, $P\\bar{6}m2$ ) during the early stages of the synthesis reaction that would partially transform into the stable NbAs-type structure, resulting in a structure containing variable fractions of NbAs- and WC-type moieties.", "Depending on the temperature at which the powder samples were removed from the furnace, the nominal composition changed.", "Xu et al.", "[39] performed an extensive study on the crystal structure, electrical transport, and magnetic properties of single crystalline NbP.", "However, it was reported as stoichiometric with no defects of any kind.", "Saparov et al.", "[40] reported on extensive structure, thermodynamic and transport properties on a series of transition metal arsenides, including TaAs and NbAs, but the samples were not single crystalline.", "Here, we report a structural study on the tantalum pnictide semimetals TaAs and TaP utilizing single crystal x-ray diffraction (XRD), energy dispersive spectroscopy (EDS), and transmission electron microscopy/scanning transmission electron microscopy (TEM/STEM).", "Figure: (Color online) Crystal structure of TMPn (TM=Ta, Nb, and Pn=As, P), which crystallizes in the non-centrosymmetric space group I4 1 mdI4_{1}md, depicting (a) the unit cell, and the local environment of (b) the transition metal, and (c) the pnictide.XRD is essentially a measure of electron density and provides detailed information on the stoichiometry of a single crystal.", "However, it is not as sensitive to defects as other methods, as it assumes that all intensity is located in Bragg peaks.", "Therefore, point defects show up as reduced electron density, and stacking faults can result in apparent twinning or in the formation of anti-sites or anti-domains.", "Both of these will affect the measured electron density.", "EDS cannot give any information on possible defects, but it gives a value of the composition of a single crystal and therefore the overall stoichiometry, although it often has a large margin of error.", "In order to be able to detect defects, TEM and STEM can give a detailed picture of the atomic arrangement in a crystal, and, therefore, this technique can clearly identify any possible point and planar defects.", "Utilizing these three experimental methods, we show that these semimetals – synthesized using techniques recently reported in the literature – are, in fact, non-stoichiometric, and display substantial defect densities.", "The defects manifest themselves as site vacancies, anti-site disorder and anti-domains due to stacking faults.", "In TaP, we observe a phosphorous deficiency in both XRD and EDS accompanied by stacking faults, anti-site disorder and vacancies.", "In contrast, for TaAs, we observe a tantalum deficiency in XRD but a range of deficiencies in EDS, accompanied by a high density of stacking faults only.", "In addition, we present electronic structure calculations for TaAs where we show the effect of vacancies and stacking faults on the band structure and Fermi surface.", "Furthermore, we present de Haas-van Alphen measurements (magnetic torque) on TaP to demonstrate that the specimens studied here are of similar quality to those discussed elsewhere." ], [ "Results & Discussion", "The (Nb,Ta)(P,As) materials crystallize in the space group $I4_{1}md$ with the structure built up by a three dimensional network of trigonal prisms of TMPn$_6$ and PnTM$_6$ (TM=Nb, Ta and Pn=P, As), as can be seen in Fig.", "REF .", "The results of the structural refinements from the single crystal x-ray diffraction of the two tantalum semimetals are summarized in Table REF .", "Both semimetals refine with a Flack parameter[41] of approximately 0.5, indicating that these materials are racemic compounds.", "What follows are results for each semimetal.", "Table: Single crystal x-ray diffraction refinement parameters of the TaPn semimetals, collected at ambient temperature.", "The semimetals crystallize in I4 1 mdI4_{1}md with Z=4Z=4.", "The atomic parameters and anisotropic displacement parameters (in ×10 4 \\times 10^{4} Å 2 ^2) are at the bottom.", "The atomic xx- and yy-coordinates are the same for all atoms.", "The Ta zz-coordinate at 4a (0,0,z0,0,z) is fixed at z=0z=0, whereas the Pn zz-coordinate is refined.Figure: (Color online) TEM/STEM images of TaP (the tilt in the images is due to drift).", "(a) TEM bright field image view of a TaP crystal with arrows highlighting the stacking faults.", "The faults are stacked along the cc-axis.", "(b) Atomic resolution HAADF-STEM ZZ-contrast image viewed along [010][010], with a region of stacking faults.", "As indicated by arrows, the bright spheres are Ta atoms while the smaller, less intense spheres are P atoms.", "Atoms along a strip in the cc-direction have been highlighted with different colors to display the atomic models of the known I4 1 mdI4_{1}md structure and the stacking fault region.", "For clarity, the model atoms of Ta and P are of the same size.", "(c) An HAADF-STEM image of a different region, highlighting anti-sites, a site vacancy, and a stacking fault.", "A selection of atoms has been colored to display the atomic model across the anti-sites.", "(d) Diffraction pattern of the [010][010] direction showing streaks arising due to the stacking faults.TaP.", "The single crystal refinement of the TaP crystal structure displayed larger than expected anisotropic displacement parameters (ADPs) for phosphorus when compared to tantalum ADPs, indicating P-site deficiency.", "Refining the site occupancy factor (SOF) of the P-site indeed yielded a significant drop in occupancy to 0.83, and a commensurate reduction of the anisotropic displacement parameters to match those of Ta.", "This scenario was repeated for a different crystal and gave identical results within errors.", "Elemental analysis via EDS on single crystals reveals a stoichiometric range of TaP$_{0.82-0.84}$ , in excellent agreement with the XRD results.", "Figure: The de Haas-van Alphen signal (via magnetic torque measurements) of TaP.", "The signal has been normalized by the field and the background has been subtracted to reveal only the oscillatory signal.This non-stoichiometry initially hints at a large number of point defects (site vacancies) on the anionic P-sites, and could also be a sign of a potential superstructure, Ta$_6$ P$_5$ , in TaP, with 6 times the volume of the substructure.", "However, TEM/STEM images of a crushed crystal of TaP (Fig.", "REF ) reveal a somewhat different picture: defects in the form of planar defects (stacking faults), anti-sites and vacancies.", "The planar defects are prevalent with a high density of stacking faults while point defects (anti-sites and vacancies) have much less density.", "In Fig.", "REF (a), the arrows indicate stacking faults along the $c$ -axis.", "Fig.", "REF (b) is a STEM high angle annular dark field (HAADF) $Z$ -contrast image which shows the atomic arrangement of Ta ($Z=73$ , bright spheres) and P ($Z=15$ , small low-intensity spheres).", "The image clearly displays the expected stacking of the known $I4_{1}md$ structure, disrupted by \"shifts\" (indicated by yellow lines) of one half lattice width in $a$ (or $b$ ), creating a region with a different stacking arrangement.", "Fig.", "REF (c) is an HAADF-STEM image of a different region clearly displaying anti-site regions.", "These are areas where the P sites are occupied by Ta and Ta sites occupied by P. The normally low-intensity P sites have a much higher intensity than expected in these regions, thus indicating a substantial number of Ta atoms in these locations.", "In addition, the Ta sites in these regions have a lower intensity as expected, a sign that there are P atoms in those columns.", "Furthermore, the anti-sites in Fig.", "REF (c) also show point defects in the form of Ta vacancies, an \"empty\" column of Ta atoms.", "It is possible that the column may contain some P atoms.", "Fig.", "REF (d) shows an electron diffraction pattern of the single crystal in the $(h0l)$ plane with streaks along the $c^$ direction for $h=2n+1$ .", "Due to the stacking faults, the streaks appear along selected directions in reciprocal space and are pronounced in the $(h0l)$ plane with $h=2n+1$ positions (or, equivalently, in the $(0kl)$ plane with $k=2n+1$ ) only.", "The reason for this can be understood by examining the structure (Fig.", "REF ) and its layering of Ta atoms along the $c$ -axis: as the structure is incrementally built up along $c$ , Ta atoms are located in $(0,0,0)$ , $(0,\\frac{1}{2},\\frac{1}{4})$ , $(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})$ , $(\\frac{1}{2},0,\\frac{3}{4})$ , and $(0,0,1)$ , and so on for the next unit cell.", "One immediately notices that the vectors between these positions includes alternating shifts of $\\frac{1}{2}a$ and $\\frac{1}{2}b$ as the layers progress.", "When a stacking fault occurs, this alternating shift pattern is disrupted, resulting in the next Ta-atom to be stacked directly above the previous one.", "Additional layers of the same alignment may add on to create a slab consisting of a different stacking arrangement than the original stacking.", "To return to the original stacking, an additional fault would be needed in order to bring the next Ta-atom into the alternating shifts of $\\frac{1}{2}a$ and $\\frac{1}{2}b$ stacking pattern.", "Therefore, since the stacking faults involve shifts by $\\frac{1}{2}a$ (or equivalently, $\\frac{1}{2}b$ ), only $(h0l)$ with $h$ odd (or $k$ odd in $(0kl)$ ) show streaks.", "The clear evidence of defects and non-stoichiometry in TaP raises the question of crystal quality.", "However, magnetic torque measurements on a single crystal of TaP clearly display an oscillatory de Haas-van Alphen signal (Fig.", "REF ).", "Oscillations are observed as low as 1 T, thus indicating a minimum drift mobility of $10^4$ cm$^2$ /Vs at 1.4 K, comparable to other reports of ultrahigh carrier mobility in TaP.", "[20] A Dingle plot[42] suggests that the drift mobility even exceeds $10^5$ cm$^2$ /Vs, despite the large number of defects.", "In fact, high frequency oscillations are also observed, most likely arising due a shift of the Fermi level produced by the defects and/or non-stoichiometry of the samples.", "Further, the oscillations differ from sample to sample, a feature that can be attribute to different stoichiometries for different crystals.", "[42] The observed quantum oscillations are a sign that, electronically, these crystals are of comparable quality to those reported elsewhere, and it is likely that all experiments performed so far on these materials used non-stoichiometric samples with defects.", "TaAs.", "For TaAs, initially the same assumption was made as in TaP, viz.", "a pnictide deficiency.", "However, the single crystal refinement clearly showed that the As site is not deficient.", "In fact, refining the As site occupancy factor (SOF), it increased above 1.", "Fixing the arsenic SOF at 1.0 while refining the tantalum SOF, the occupancy of the Ta site, as expected, dropped to 0.92, yielding a stoichiometry of Ta$_{0.92}$ As.", "In contrast, EDS analysis on different single crystals showed a wide range of elemental composition: from an As-deficiency (TaAs$_{0.91}$ ) to a Ta-deficiency (Ta$_{0.7}$ As).", "This wide range observed with EDS and the discrepancy between the XRD and EDS measurements are the result of a high density of stacking faults, producing anti-domains.", "Figure: (Color online) TEM/STEM images of TaAs (the tilt in the images is due to drift).", "(a) TEM bright field image view of a TaAs crystal.", "The faults in this case are also stacked along the cc-axis.", "(b) Atomic resolution HAADF-STEM ZZ-contrast image viewed along [010][010].", "A strip has been highlighted to indicate unit cells of the known I4 1 mdI4_{1}md structure and two regions of stacking faults.", "(c) Diffraction pattern of the [010][010] direction with streaks arising due to the stacking faults.", "(d) Illustration of two unit cells separated by a region of stacking faults.TEM/STEM images of a crushed crystal of TaAs (Fig.", "REF ) clearly show the high density of stacking faults.", "As in the case of TaP, the stacking faults occur along the $c$ -axis (Fig.", "REF (a)).", "Fig.", "REF (b) shows the HAADF-STEM image where the Ta atoms are the bigger and brighter spheres.", "The atomic number of As ($Z=33$ ) is sufficient to show the As atoms as less bright and smaller spheres.", "An area along the $c$ -axis has been highlighted to show where the structure is broken up by regions of stacking faults (between the yellow lines).", "Fig.", "REF (c) shows the electron diffraction pattern in the $(h0l)$ plane of a TaAs single crystal, similarly to the TaP crystal, with streaks along the $c^$ direction for $h=2n+1$ .", "Fig.", "REF (d) is an illustration of the atomic arrangement showing two unit cells separated by stacking faults.", "Electronic structure calculations on stoichiometric TaAs, TaAs containing vacancies, and TaAs containing stacking faults were performed in order to elucidate the effects these structural features have on the Fermi surface, the Weyl points and the density of states (DOS).", "The calculations confirm that Ta-deficient TaAs behaves as an electron-doped system and As-deficient TaAs as a hole-doped system when compared to stoichiometric TaAs (the band structures are displayed in the Appendix in Fig.", "REF ).", "Notice that the electronic structure is significantly altered when introducing vacancies: the position of the Weyl points relative to the Fermi level has shifted dramatically in the presence of vacancies when compared to the stoichiometric case.", "The DOS results are displayed in Fig.", "REF .", "In the case of As vacancies, the DOS from $-2$ eV to 0 eV is still of similar shape to the DOS of stoichiometric TaAs near the $E_F$ , but with non-zero states at $E_F$ , as expected.", "In contrast, Ta vacancies distort the DOS at $E_F$ by about 0.5 states/eV/atom.", "Fig.", "REF shows the Fermi surfaces of three cases: stoichiometric TaAs, TaAs with 12.5 % Ta vacancies, and TaAs with 12.5 % As vacancies, calculated using a $2\\times 2\\times 1$ supercell (primitive unit cell), i.e., a supercell twice as long along the $a$ - and the $b$ -axis.", "A number of small electron and hole pockets of similar size to those found in the regular unit cell of stoichiometric TaAs exist (Fig.", "REF (a)), pockets associated with the Weyl points.", "[4], [5] However, when introducing Ta and As vacancies in the form of point defects, the small electron and hole pockets appear to form away from the Weyl points (Fig.", "REF (b) and (c)).", "Notice that in addition to the small pockets, much larger Fermi surface sheets emerge upon introducing vacancies (Fig.", "REF (d) and (e)).", "The presence of small electron/hole pockets can explain the similar behavior in Shubnikov-de Haas (SdH) oscillations where the larger frequencies may be harder to detect.", "[16] Figure: (Color online) Fermi surfaces of TaAs, TaAs with a 12.5 % Ta vacancy concentration, and TaAs with a 12.5 % As vacancy concentration, calculated using a 2×2×12\\times 2\\times 1 supercell (primitive unit cell).", "The electron and hole pockets are shown in blue and red color, respectively.", "In general the colors used in this figure are the same as used in Fig.", "for the corresponding bands.", "Figure (a) shows the small electron and hole pockets in the case of TaAs without any vacancy, (b) the small electron and hole pockets that form when TaAs contains 12.5 % Ta vacancies, and (c) the small electron and hole pockets that form when TaAs contains 12.5 % As vacancies.", "The corresponding Fermi surface sheets obtained in the vacancy cases are shown in (d) and (e).However, as noticed from the TEM results (Fig.", "REF ), the majority of defects are stacking faults, i.e., planar defects.", "Fig.", "REF compares the Fermi surface sheets of stoichiometric, ideal, TaAs with that of a stoichiometric TaAs containing stacking faults (as depicted in Fig.", "REF (d)).", "The calculations were performed on a $1\\times 1\\times 3$ supercell, using a simple tetragonal unit cell (conventional unit cell) for the stoichiometric TaAs, and the stacking fault is accommodated by taking three times the size of this unit cell along the $c$ -axis.", "If the conventional body-centered tetragonal unit cell (primitive unit cell) would be used, the smallest unit cell which includes a stacking fault would be larger.", "Figs.", "REF (a) and (b) depicts the electron and hole pockets of the stoichiometric, ideal, supercell, which include the Weyl points.", "Figs.", "REF (c) and (d) show the corresponding Fermi sheets when stacking faults are introduced.", "However, when shifting the Fermi level by a rather small amount of energy (60 meV upwards for the electron pockets and 60 meV downwards for the hole pockets) the pockets containing the Weyl points emerge again.", "Thus, while the calculation corresponds to a periodic array of stacking faults with a quite large stacking fault density of $1/3$ , the Fermi surface structure is more or less preserved.", "Therefore, a different explanation for the robust nature of the SdH behavior over several samples may be that the main defects present in the crystals are stacking faults as opposed to As or Ta vacancies.", "Band structure calculations comparing stoichiometric, ideal, TaAs with TaAs containing stacking faults are displayed in Fig.", "REF .", "The effect of stacking faults near the Weyl point is to slightly move the Fermi level, but the main features of the band structure near the Weyl point are preserved.", "Figure: (Color online) Figures (a) and (b) show the electron (blue) and hole (red) pockets which include the Weyl points in the case of a stoichiometric TaAs supercell which is three times the unit cell along the cc-axis (1×1×31\\times 1\\times 3 supercell).", "Figures (c) and (d) show the corresponding Fermi sheets in the case of a TaAs 1×1×31\\times 1\\times 3 supercell containing stacking faults.", "Figures (e) and (f) show the same electron and hole pockets as in (c) and (d) but with the Fermi level shifted by 60 meV upwards and downwards for the electron and hole pocket, respectively.Vacancy formation energy calculations were performed to shed some light on the defects in TaAs.", "Using a $2\\times 2\\times 1$ supercell (conventional body-centered unit cell), calculations for the compositions Ta$_{16}$ As$_{15}$ (As vacancy) and Ta$_{15}$ As$_{16}$ (Ta vacancy) were performed.", "The formation energy of an As vacancy was 4.09 eV per Ta$_{16}$ As$_{15}$ unit, while only 2.90 eV per Ta$_{15}$ As$_{16}$ unit for a Ta vacancy.", "This shows that the formation of Ta vacancies is more energetically favorable than the formation of As vacancies, in agreement with the results from the single crystal x-ray diffraction.", "In addition, a calculation of stacking fault energy yields $\\Delta E=(E_{SF}-E_{0})/A=0.791$ eV/Å$^{2}=126.7$ mJ/m$^{2}$ .", "Here, $E_{SF}$ denotes the energy of a stacking fault structure, $E_{0}$ the energy of a perfect structure, and $A$ the surface area.", "The moderate value of the stacking fault energy suggest the likelihood of formation of stacking faults in these compounds." ], [ "Conclusion", "We have shown that the two tantalum pnictide semimetals TaAs and TaP grow with stoichiometry deviations, and display a high number of defect densities in the form of stacking faults, anti-site disorder, and vacancies.", "The differences between the two samples is striking: while TaP displays the full range of defects and grows with a large anionic pnictide deficiency, TaP$_{0.83(3)}$ , TaAs shows only stacking faults, although at a higher density than TaP, accompanied by cationic transition metal deficiency, Ta$_{0.92(2)}$ As.", "As the two semimetals were grown using the same techniques reported in the literature, this indicates that, most likely, all experiments performed so far on these materials have used non-stoichiometric samples containing defects.", "It is, however, clear that our crystals are electronically of comparable quality to those studied previously, despite the high number of defects and large stoichiometry variations, since our magnetic torque measurements on TaP reveal oscillatory de Haas-van Alphen signals and an ultrahigh carrier mobility, two observations normally associated with high quality crystals.", "Our electronic structure calculations of TaAs show that while the Fermi surface is considerably altered when introducing vacancies, there still exist electron and hole pockets of similar size to those found in stoichiometric TaAs.", "These small electron and hole pockets, however, appear to form away from the Weyl points and the position of the Weyl points relative to the Fermi level is shifted.", "Introducing a periodic array of stacking faults show that the Fermi surface structure for the electron and hole pockets near the Weyl points is similar to the stoichiometric, ideal, TaAs, and the main effect of the stacking faults is to change the location of the Fermi level: by slightly raising or lowering the energy, Weyl fermion pockets appear again, and of the same size as seen in quantum oscillations.", "In other words, while vacancies significantly alter the location of the Weyl pockets, stacking faults seem to preserve the main features (such as location and dispersion) of the electronic structure near the Weyl points.", "With quantum oscillations experimentally observed in a single crystal containing a high number of stacking faults and defects, and Weyl fermion pockets appearing in the Fermi surface sheets of TaAs with stacking faults, it is clear that our results illustrate the robustness of these quantum phenomena in this family of semimetals.", "Future studies of these materials need therefore to carefully analyze the stoichiometry, since it is clear that structural defects are present and may shift the actual stoichiometry.", "Interestingly, these defects do not seem to couple to the electronic transport properties, as reflected in the observed high mobilities, but do affect the Weyl point positions and should have consequences for the Weyl-related physics.", "It remains to be determined what these effects will be." ], [ "Acknowledgments", "T.B.", "and T.S.", "are supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under Award #DE-SC0008832.", "K.-W.C. and R.E.B.", "acknowledge support from the National High Magnetic Field Laboratory UCGP program.", "L.B.", "is supported by DOE-BES through Award #DE-SC0002613.", "Work at the University of Missouri (J.S.", "and D.J.S) is supported by the U.S. Department of Energy, Basic Energy Sciences through the S3TEC Energy Frontier Research Center, Award No.", "DE-SC0001299/DE-FG02-09ER46577.", "This work was performed at the National High Magnetic Field Laboratory, which is supported by the NSF cooperative agreement DMR-1157490 and the State of Florida." ], [ "Vacancies", "Figure REF displays the results of band structure calculations of TaAs in the case of introduced vacancies.", "Fig.", "REF (a) shows the band structure of TaAs obtained in the body-centered tetragonal phase which is consistent with published results.", "Fig.", "REF (b) reproduces the band structure of the stoichiometric TaAs using a supercell twice as long along the $a$ - and the $b$ -axis (four times the volume of the unit cell), i.e., a $2\\times 2\\times 1$ supercell (primitive unit cell).", "This is in order to identify the pockets near the Weyl points and to compare the band structure of the stoichiometric case with the cases of 12.5 % vacancy concentration which require a similar supercell.", "Fig.", "REF (c) shows the results obtained for the band structure of TaAs with 12.5 % Ta vacancy concentration, while Fig.", "REF (d) shows the band structure of TaAs with 12.5 % As vacancy concentration.", "The band structures are consistent with the fact that TaAs with Ta vacancies behaves as an electron-doped system whereas TaAs with As vacancies behaves as a hole-doped system.", "Figure: (a) The band structure of TaAs in the body centered tetragonal phase consistent with published results.", "(b) The band structure of TaAs supercell containing four times the volume of the unit cell, twice along the aa- and the bb-axis.", "(c) The band structure of TaAs with 12.5 % Ta vacancy concentration.", "(d) The band structure of TaAs with 12.5 % As vacancy concentration.Figure REF shows the total and partial density of states (DOS) of stoichiometric TaAs, along with slightly Ta-vacant TaAs, Ta$_{15}$ As$_{16}$ , and As-vacant TaAs, Ta$_{16}$ As$_{15}$ , calculated using a $2\\times 2\\times 1$ supercell (conventional unit cell).", "The As vacancy case has a DOS that is still of similar shape to the DOS of stoichiometric TaAs near the $E_F$ , from $-2$ eV to 0 eV but with non-zero states at $E_F$ , as expected.", "In contrast, the Ta vacancy case seems to have distorted the DOS near $E_F$ , with about 0.5 states/eV/atom at $E_F$ .", "Figure: Total and partial density of states of stoichiometric TaAs (bottom panel), TaAs with As vacancies (middle panel), and TaAs with Ta vacancies (top panel)." ], [ "Stacking Faults", "Figure REF compares the band structure of a stoichiometric, ideal, TaAs to that of TaAs containing a stacking fault.", "In these calculations, a simple tetragonal unit cell for the stoichiometric TaAs was used and the stacking fault are accommodated by taking three times the size of this unit cell along the $c$ -axis, a $1\\times 1\\times 3$ supercell (conventional unit cell).", "The main effect of the stacking fault is the significant changes in the bands in directions $\\Gamma \\rightarrow X^{\\prime }$ and $X^{\\prime }\\rightarrow Z^{\\prime }$ which do not contain the Weyl point (i.e., $\\Sigma ^{\\prime }$ ).", "The effect of the stacking fault near the Weyl point, however, is mainly to slightly move the Fermi level, and the main features of the band structure near the Weyl point are preserved.", "Figure: (a) The band structure along certain high symmetry points of a stoichiometric TaAs supercell three times the unit cell along the cc-axis.", "(b) The band structure in the case of a TaAs supercell containing one stacking fault." ], [ "Methods", "Sample Preparation.", "Single crystals of TaAs were grown by chemical vapor transport, as previously described.", "[18] Polycrystalline precursor specimens were first prepared by sealing elemental Ta and As mixtures under vacuum in quartz ampoules and heating the mixtures at a rate of 100$$ C/hr to 700$$ C, followed by a dwell at this temperature for 3 days.", "The polycrystalline TaAs boules were subsequently sealed under vacuum in quartz ampoules with 3 mg/cm$^3$ of iodine to serve as the transporting agent.", "The ampoules had diameters 1.4 cm and lengths 10 cm and were placed in a horizontal tube furnace such that a temperature gradient would be established during firing.", "The ampoules were slowly heated at a rate of 18$$ C/hr, and put into a temperature gradient with $\\Delta T=850$ C$-950$ C. The ampoules were maintained under this condition for 3 weeks and were finally rapidly cooled to room temperature.", "This process produced a large number of single crystal specimens with typical dimensions of 0.5 mm on a side.", "TaP single crystals were synthesized through a chemical vapor transport technique using iodine as the transport agent.", "99.98% pure Ta powder and 99% pure P lumps were introduced into quartz tubes together with 99.999% pure iodine serving as the transporting agent.", "The quartz tubes were evacuated, sealed, brought to 500$$ C, held at this temperature for 1 day, then brought to 650$$ C, held for 12 hours, and then finally raised to 975$$ C and held there for 5 days.", "Subsequently, they were cooled to 800$$ C and held there for 1 day, followed by an air quench.", "Single Crystal X-ray Diffraction.", "Crystals of the semimetals were structurally characterized by single crystal x-ray diffraction using an Oxford-Diffraction Xcalibur2 CCD system with graphite-monochromated Mo$K\\alpha $ radiation.", "Data was collected to a resolution of 0.4 Å, equivalent to $2\\theta = 125$ .", "Reflections were recorded, indexed and corrected for absorption using the Agilent CrysAlisPro software.", "[43] Subsequent structure refinements were carried out using CRYSTALS,[44] using atomic positions from the literature.", "[45] The data quality for all samples allowed for an unconstrained full matrix refinement against $F^2$ , with anisotropic displacement parameters for all atoms.", "Crystallographic information files (CIFs) have been deposited with ICSD (CSD Nos.", "430436 and 430437 for TaP and TaAs, respectively).", "[46] EDS.", "EDS was performed with a field-emission scanning electron microscopy (Zeiss 1540 XB), on 6 to 12 spots each on the several single crystals studied.", "The EDS stoichiometries quoted here result from average values.", "TEM.", "The TEM samples were prepared by crushing single crystals that were previously checked by XRD, in Ethyl Alcohol 200 Proof in a pestle and mortar.", "The suspension was then dropped onto a carbon/formvar TEM grid (Ted Pella, Inc.) using a 1.5 ml pipette.", "TEM/STEM images were collected using the probe aberration corrected JEOL JEM-ARM200cF with a cold field emission gun at 80 kV to avoid beam damage.", "The STEM high angle annular dark field (STEM-HAADF) images were taken with the JEOL HAADF detector using the following experimental conditions: probe size 7c, CL aperture 30 $\\mu $ m, scan speed 32 $\\mu $ s/pixel, and camera length 8 cm, which corresponds to a probe convergence semi-angle of 11 mrad and collection angles of $76-174.6$ mrad.", "Qualitatively, the intensity of atomic columns in STEM-HAADF images is proportional to the atomic number $Z^n$ , where $n$ is close to 2, i.e., they are $Z$ -number-sensitive images ($Z$ -contrast).", "The STEM resolution of the microscope is 0.78 Å.", "Electronic structure calculations.", "Electronic structure calculations were performed by using the Vienna ab-initio simulation package[47], [48], [49], [50] (VASP) within the generalized gradient approximation (GGA).", "We have included the contribution of spin-orbit coupling in our calculations.", "The Perdew-Burke-Ernzerhof (PBE) exchange correlation functional [51] and the projected augmented wave (PAW) methodology[52] were used to describe the core electrons.", "The 5s, 5p and 5d electrons for Ta and the 3d, 4s and 4p electrons for As were treated as valence electrons in all our calculations.", "The energy cut off for the plane-wave basis was chosen to be 600 meV.", "A total of 208 bands and a $k-$ point mesh of $8\\times 8\\times 8$ were used for the self-consistent ground state calculations.", "A total of 100 $k-$ points were chosen between each pair of special $k-$ points in the Brillouin-zone for the band-structure calculations.", "The Fermi surfaces were generated using a $k-$ point mesh of $10\\times 10\\times 16$ .", "The Fermi surfaces were generated using the eigenvalues obtained from VASP and were visualized using the XCrysden software.", "[53] In the case of the formation energy calculations, the calculations were carried out using the linearized augmented plane-wave (LAPW) method as implemented in the WIEN2K code, with the fully relaxed structures generated from VASP.", "The LAPW sphere radii were 2.35 bohr for both Ta and As.", "The cut-off parameter for the basis was $R_{min}K_{max}=9$ .", "We used well converged $k$ -point sampling for the total energy calculations which are sensitive to $k$ -points, especially in the formation energy calculations.", "For the vacancy formation energy calculation, a $2\\times 2\\times 1$ supercell was generated, which includes 31 Ta and As atoms and one Ta or As vacancy, yielding a composition of Ta$_{15}$ As$_{16}$ and Ta$_{16}$ As$_{15}$ , respectively.", "Then the vacancy formation energy was calculated using $\\Delta H_{f}=E_{\\alpha }-E_{\\textrm {host}}+\\sum _{\\alpha }n_{\\alpha }\\mu _{\\alpha }$ , where $E_{\\alpha }$ and $E_{\\textrm {host}}$ are the energies with and without vacancy $\\alpha $ .", "Here, $n_{\\alpha }$ and $\\mu _{\\alpha }$ are the number of vacancies and the chemical potential of vacancy $\\alpha $ in the elemental phase, respectively.", "The formation energies shown in the results are calculated within WIEN2K with spin-orbit coupling." ] ]	1606.05178
[ [ "On the optimality of grid cells" ], [ "Abstract Grid cells, discovered more than a decade ago [5], are neurons in the brain of mammals that fire when the animal is located near certain specific points in its familiar terrain.", "Intriguingly, these points form, for a single cell, a two-dimensional triangular grid, not unlike our Figure 3.", "Grid cells are widely believed to be involved in path integration, that is, the maintenance of a location state through the summation of small displacements.", "We provide theoretical evidence for this assertion by showing that cells with grid-like tuning curves are indeed well adapted for the path integration task.", "In particular we prove that, in one dimension under Gaussian noise, the sensitivity of measuring small displacements is maximized by a population of neurons whose tuning curves are near-sinusoids -- that is to say, with peaks forming a one-dimensional grid.", "We also show that effective computation of the displacement is possible through a second population of cells whose sinusoid tuning curves are in phase difference from the first.", "In two dimensions, under additional assumptions it can be shown that measurement sensitivity is optimized by the product of two sinusoids, again yielding a grid-like pattern.", "We discuss the connection of our results to the triangular grid pattern observed in animals." ], [ "Abstract", "Grid cells, discovered more than a decade ago [5], are neurons in the brain of mammals that fire when the animal is located near certain specific points in its familiar terrain.", "Intriguingly, these points form, for a single cell, a two-dimensional triangular grid, not unlike our Figure 3.", "Grid cells are widely believed to be involved in path integration, that is, the maintenance of a location state through the summation of small displacements.", "We provide theoretical evidence for this assertion by showing that cells with grid-like tuning curves are indeed well adapted for the path integration task.", "In particular we prove that, in one dimension under Gaussian noise, the sensitivity of measuring small displacements is maximized by a population of neurons whose tuning curves are near-sinusoids — that is to say, with peaks forming a one-dimensional grid.", "We also show that effective computation of the displacement is possible through a second population of cells whose sinusoid tuning curves are in phase difference from the first.", "In two dimensions, under additional assumptions it can be shown that measurement sensitivity is optimized by the product of two sinusoids, again yielding a grid-like pattern.", "We discuss the connection of our results to the triangular grid pattern observed in animals." ], [ "Introduction", "Grid cells [5] are neurons in the dorsocodal medial entorhinal cortex of mammals that fire when the animal is near specific locations in its familiar environment; intriguingly, these locations form, for a single cell, a two-dimensional regular triangular grid [5].", "Ever since their discovery, grid cells have been hypothesized to be involved in space representation [5], [4], and in particular in neural algorithms “that integrate information about place, distance, and direction” [5], a task usually referred to as path integration [5], [8].", "But why are neurons with grid-like tuning curves well adapted for the task of path integration?", "This is the question we address in this paper.", "Path integration presumably entails the measurement of small displacements.", "Therefore, for path integration to be effective, measurement of small displacements has to be as accurate as possible.", "What is the tuning curve, for neurons measuring small displacements, that has the highest possible sensitivity, that is, the smallest possible variance?", "We show that, in one dimension, optimal measurement sensitivity is achieved through a one-dimensional grid.", "In particular, we consider a population of neurons measuring small displacements on the circle.", "Working on the circle instead of the line segment, or the infinite line, simplifies the analysis by avoiding edge effects.", "We assume that the tuning curves of these neurons are cyclical shifts of one another, and that the noise of the measurement is Gaussian.", "We seek the tuning curve maximizing the accuracy of the measurement.", "A useful surrogate of accuracy is the Fisher information [6], which upper-bounds the accuracy of any estimator.", "We establish that the tuning curve maximizing Fisher information is a sinusoidal-like wave (see Figure 1) — that is to say, a tuning curve whose peaks form a grid.", "The frequency of the wave is that of the eigenvector corresponding to the smallest positive eigenvalue of the noise correlation matrix.", "We say “sinusoidal-like” because, mathematically, the optimal solutions form a family of near-sinusoidal functions parametrized by any function $\\psi :[0,1]\\mapsto [0,1]$ , with the common sinusoid corresponding to the identity function (notice the differences between the sinusoid-like waves in Figure 2).", "Figure: The optimum tuning curveBut how can the displacement be read out from the change in spiking rates in such a population?", "We notice that the actual displacement can be computed with the help of a second cell population which is identical to the first, albeit with tuning curves phase-shifted by $90^{\\rm o}$ .", "This last observation about the computation of displacement is relevant when exploring how our analysis can be extended to two dimensions.", "Again, in order to avoid edge effects we are working on the torus (the unit square with its opposite sides identified).", "Under quite hefty assumptions (of independence of both correlation and tuning curves, as well as of independent optimization of sensitivity in each direction) it can be shown that the optimum tuning curve in two dimensions is indeed a grid.", "We speculate that the triangular grid may be the result of optimization of measurement sensitivity under an additional constraint, and that constraint may be computational: In two dimensions, the problem of inferring the displacement from tuning curve change requires not one additional phase-shifted population of cells, but three such populations.", "The interface of the four phase-shifted populations is best achieved through a triangular grid (Figure 3)." ], [ "Related work:", "Over the past decade, there has been much theoretical investigation of grid cells, their origin, and their role.", "It has been noted that a grid-like pattern can result from the interference of two, or possibly three, sinusoidal waves [2], while periodic tuning curves on the circle [11] and the torus [8] can be generated by neural networks (see Figure 3A and B in [4]); however, these models were not proposed in the context of optimizing the accuracy of measurement.", "It has also been shown that continuous attractor models can generate triangular grid-like responses [1], while experimental data are consistent with a 2-dimensional response of the population [15].", "Grid cells were interpreted in [12] and, in a different way, in [3], as very efficient novel neural codes for encoding position and velocity, and in [7] it is shown that in this arena grid cells are more apt than place cells; these works are methodologically close to ours in that they also employ Fisher information for their analysis and comparisons — without, however, seeking the tuning curve design that maximizes it.", "Cells with one-dimensional grid-like firing patterns have also very recently come up in the analysis of the responses of animal grid cells to one-dimensional environments [14], characterized as projections (“slices”) of a two-dimensional lattice to the one-dimensional circumstances of the experiment.", "Finally, recently it was claimed in [10] that the tuning curve in one dimension with maximum Fisher information is a sinusoid curve, while the product of two such curves is optimum in two dimensions, results very similar to ours; unfortunately (as pointed out in Sections 2 and 3) the mathematical development in that manuscript contains significant gaps." ], [ "The optimal tuning curve is periodic", "Consider a population of $N$ neurons measuring a small angular displacement at a point on the circle.", "We assume that the tuning curves of the $N$ neurons are identical, albeit shifted by multiples of the angle $2\\pi \\over N$ .", "We seek the tuning curve that maximizes the sensitivity of measuring small displacements.", "The neurons respond to an angular stimulus $\\theta \\in [0,2\\pi ]$ , and the tuning curve of the $i$ th neuron is denoted by $f_i(\\theta )$ ; we assume that the tuning curves are identical but shifted by multiples of ${2\\pi \\over N}$ , that is, $f_{i+1\\bmod N}(\\theta ) = f_i(\\theta + {2\\pi \\over n})$ .", "The average population activity caused by a stimulus is thus the vector ${\\hbox{\\bf }}{f}(\\theta )=(f_1(\\theta ), f_2(\\theta ),...,f_N(\\theta ))$ .", "The derivative $d{f}_i\\over d\\theta $ is denoted $\\dot{f}_i$ , and we denote by ${\\hbox{\\bf }}{\\dot{f}}(\\theta )$ the vector $(\\dot{f}_1(\\theta ), \\dot{f}_2(\\theta ),...,\\dot{f}_N(\\theta ))$ .", "A stimulus $\\theta $ results in the response $r_i(\\theta )=f_i(\\theta )+\\eta _i$ , for $i=1,\\ldots ,N$ , where $\\eta _i$ is Gaussian noise.", "The values of noise at different neurons, $\\eta _i$ and $\\eta _j$ , are correlated, and this correlation is assumed to be independent of $\\theta $ , and denoted $C_{i,j}$ , a quantity that depends on the distance on the ring of the neurons $i$ and $j$ .", "It follows that the noise correlation matrix $C$ is both circulant and symmetric.", "Importantly, we also assume that the total signal power — the sum of the squares of the slopes of the tuning curves — is bounded from above by a constant, which we take to be one.", "Thus $\\dot{{\\hbox{\\bf }}{f}} (\\theta )^T \\dot{{\\hbox{\\bf }}{f}} (\\theta )\\le 1$ for all $\\theta $ .", "If the stimulus changes from $\\theta $ to $\\theta + \\Delta \\theta $ , this results in population activity ${\\hbox{\\bf }}{r}(\\theta +\\Delta \\theta )$ .", "The goal of the decoding system is to estimate $\\Delta \\theta $ from the change in the population response — that is, from ${\\hbox{\\bf }}{r}(\\theta ) - {\\hbox{\\bf }}{r}(\\theta + \\Delta \\theta )$ .", "To do this as effectively as possible, the overall variance of the measurement must be as small as possible.", "Instead of this “overall variance”, it is convenient in this context to work with the Fisher information of the population, a function of the tuning curves and the correlation matrix which is known, by the Cramer-Rao theorem [6], to bound from above the accuracy of any any unbiased estimator.", "We seek the tuning curves $f_i(\\theta )$ with the largest Fisher information under correlation matrix $C$ .", "Under Gaussian noise, and the assumption that $C$ is nonsingular, it is well known [6], [13] that the Fisher information can be written as follows: ${\\hbox{\\bf }}{I}(\\theta ) = {{\\hbox{\\bf }}{\\dot{f}}} (\\theta )^{T} {\\hbox{\\bf }}{C}^{-1} {{\\hbox{\\bf }}{\\dot{f}}} (\\theta )\\qquad \\mathrm {{(1)}}$ Thus, we seek the vector ${\\hbox{\\bf }}{\\dot{f}}$ satisfying ${\\hbox{\\bf }}{\\dot{f}}^T{\\hbox{\\bf }}{\\dot{f}}\\le 1$ that maximizes the right-hand side of (1).", "Furthermore, ${\\hbox{\\bf }}{C^{-1}}$ is also symmetric, and its eigenvalues are the inverses of the eigenvalues of ${\\hbox{\\bf }}{C}$ , while its eigenvectors are the same as those of ${\\hbox{\\bf }}{C}$ .", "Recall now the Courant-Fischer theorem [9] (stated below for the case of real symmetric matrices and the largest eigenvalue only): Theorem 1 (Courant-Fischer, 1953) If $A$ is symmetric, then the vector $x$ in the unit ball $x^Tx\\le 1$ that maximizes $x^TAx$ is the eigenvector corresponding to the largest eigenvalue of $A$ .", "Comparing with equation (1), we conclude that the optimum tuning curve vector ${\\hbox{\\bf }}{f}$ has derivative ${\\hbox{\\bf }}{\\dot{f}}$ equal to the eigenvector corresponding to the smallest positive eigenvalue of ${\\hbox{\\bf }}{C}$ (the inverse of the largest eigenvalue of $C^{-1}$ ).", "What is this eigenvector?", "Since ${\\hbox{\\bf }}{C}$ is circulant and symmetric, it is well known [9] that each eigenvalue $\\lambda _k$ , for $k=0,\\ldots , {N\\over 2}-1$ , has multiplicity two, and the two corresponding eigenvectors are the two sinusoidal waves $v_k$ and $w_k$ : $v_k=[\\cos (0),\\cos (k\\delta ),\\cos (2k\\delta ),\\ldots , \\cos ((N-1)k\\delta )] \\hbox{\\rm \\ \\ and}$ $w_{k}=[\\sin (0),\\sin (k\\delta ),\\sin (2k\\delta ),\\ldots , \\sin ((N-1)k\\delta )],$ where $\\delta ={2\\pi \\over N}$ and $k=0,1,\\ldots ,{N\\over 2}-1$ .", "We conclude that the optimum tuning curve vector has derivative of the form ${\\hbox{\\bf }}{\\dot{f}}(\\theta )=\\alpha _k(\\theta )v_k + \\beta _{k}(\\theta )w_{k},$ where $\\alpha _k^2(\\theta ) + \\beta ^2_{k}=1$ and the smallest positive eigenvalue of $C$ is $\\lambda _k$ .", "We now apply the change of variables The otherwise similar argument in [10] does not contain this step, and as a result it is incomplete and the full spectrum of optimal solutions (see Figure 2) is missed.", "$\\alpha _k(\\theta ) =\\sin (\\phi (\\theta )), \\beta _{k}(\\theta ) =\\cos (\\phi (\\theta ))$ to obtain $\\dot{{\\hbox{\\bf }}{f}}_i(\\theta )=\\sin (\\phi (\\theta ))\\cos (ik\\delta )+\\cos (\\phi (\\theta ))\\sin (ik\\delta ) =\\ \\sin (\\phi (\\theta )+ik\\delta ).\\qquad \\mathrm {{(2)}}$ Recall that we are assuming that the tuning curves of the $N$ neurons are identical, albeit shifted by $\\delta ={2\\pi \\over N}$ ; that is, for all $i$ and $\\theta $ , $f_{i+1}(\\theta ) = f_i(\\theta +\\delta )$ , and thus $\\dot{f}_{i+1}(\\theta ) = \\dot{f}_i(\\theta +\\delta )$ .", "Substituting into (2) we conclude that $\\sin (\\phi (\\theta )+(i+1)k\\delta ) = \\sin (\\phi (\\theta +\\delta )+ik\\delta )$ for all $\\theta $ , or equivalently $\\phi (\\theta +\\delta ) = \\phi (\\theta ) +k\\delta + 2n\\pi , \\qquad \\mathrm {{(3)}}$ for some integer $n$ .", "The simplest solution of (3) is $\\phi (\\theta )=(k+nN)\\theta +c$ for all $\\theta $ and for some constant $c$ (which we take zero without loss of generality) and integer $n$ .", "It follows that ${\\dot{f}}_i(\\theta )= \\sin (K\\theta +ik\\delta ),$ and thus ${f}_i(\\theta )= {1\\over K}\\cos (K\\theta +ik\\delta )+\\left\|{1\\over K}\\right\|,\\qquad \\mathrm {{(4)}}$ where $K=k+nN$ for some integer $n$ (positive, zero, or negative), and we took the constant of the integration to be $\|{1\\over K}\|$ so the tuning curve takes only positive values.", "However, (3) has many more solutions.", "Let $\\psi (\\theta )$ be any function mapping $[0,\\delta ]$ to the reals, and, for any $\\theta \\in [0,2\\pi ]$ , define $\\theta ^{\\prime }\\in [0,\\delta ]$ and $\\theta ^{\\prime \\prime }\\in \\lbrace 0,\\delta , 2\\delta ,\\ldots 2\\pi -\\delta \\rbrace $ by the equation $\\theta = \\theta ^{\\prime }+\\theta ^{\\prime \\prime }$ .", "Then the function $\\dot{f}(\\theta ) = \\sin (\\psi (\\theta ^{\\prime })+ \\theta ^{\\prime \\prime })$ is also the derivative of an optimal tuning curve.", "Indicatively, in Figure 2 we show the tuning curves resulting from six simple functions $\\psi $ , including $\\psi (x)=x$ (the true sinusoid).", "Figure: Six optimum tuning curves" ], [ "Two Dimensions", "In view of the one-dimensional result, one may suspect that grid-like structures may also be optimal in two dimensions.", "Intuitively, it is tempting to try and reduce the two-dimensional case to the one-dimensional case just solved, and show that the optimum two-dimensional tuning curve must be the product of two sinusoids, and therefore a grid.", "Unfortunately, this matter turns out to be quite a bit more complicated.", "Assume a population of neurons measuring displacements on the torus (the product of two circles, or, equivalently, a square with opposite edges identified in the parallel way), whose tuning curves are shifts of one another along some lattice on the torus defined by the unit vectors along two different directions $x$ and $y$ (not necessarily orthogonal).", "We make some additional assumptions: The noise correlations in the $x$ and $y$ directions have the same form and are independent of each other.", "That is, the correlation tensor decomposes into the product of two identical correlation matrices.", "We further assume that the tuning curve of the neurons can also be decomposed as the product of identical one-dimensional tuning curves in the $x$ and $y$ dimensions.", "It then follows that the Fisher information in the $x$ dimension has the form ${I_x}(x) = \\left[\\dot{f_x}(x)^T C^{-1}\\dot{f_x}(x)\\right]\\cdot \\left[(f_y(y)^T C^{-1} f_y(y))\\right],\\qquad \\mathrm {{(5)}}$ and similarly for the $y$ direction.", "Identifying the optimum tuning curve, even under these assumptions, is still ill-defined, because of the possibly unbounded second factor in (5): $f_y$ must be obtained by integrating $\\dot{f_y}$ , a step that introduces an unbounded integration constant.", "We could of course impose an upper bound on $f_y$ — a justified assumption since neurons cannot fire at arbitrarily high rates, — but then the optimization problem becomes an intractable one, involving integral inequality constraintsThis difficulty is ignored in [10].. We can obtain a meaningful solution only under one additional assumption: That the Fisher information in each of the $x$ and $y$ directions is maximized independently of the other direction.", "That is, the overall sensitivity is not maximized, and instead the sensitivities along the two directions $x$ and $y$ are maximized separately, yielding an overall suboptimal solution.", "This is not implausible, if one considers the independent evolution of two separate modules, each measuring displacement in one of the two directions.", "Under these assumptions, the result for the one-dimensional case does generalize immediately, and the tuning curve maximizing the Fisher information at all stimuli turns out to be the outer product of near-sinusoidal waves in the $x$ and $y$ directions.", "If the two directions form an angle of $120^{{\\rm o}}$ , the familiar triangular grid results.", "The idea that the triangular firing field structure can result from the interference of two oscillations has been suggested before [2]; however, the advantage of this structure was unclear.", "But why should the two directions $x$ and $y$ be at an angle of $120^{{\\rm o}}$ to form the familiar triangular grid observed in [5]?", "One possible answer comes from algorithmic considerations, discussed next." ], [ "Computing the displacement", "Consider a population of cells around the circle as in the previous section with the sinusoid tuning curve $f(\\theta )$ in (4) above, measuring (under noise) the change in firing rate $\\Delta f = f(\\theta )-f(\\theta +\\Delta \\theta )$ .", "What is the mechanism whereby the displacement $\\Delta \\theta $ is inferred from the measurement of $\\Delta f$ ?", "This seems problematic, since the value of the stimulus $\\theta $ appears to be needed, and keeping track of $\\theta $ is the purpose of path integration...", "But upon closer consideration, we note that the decoding mechanism does not quite need the stimulus $\\theta $ , but just its cosine.", "Recall that, as $\\Delta \\theta $ goes to zero, $\\Delta f = \\dot{f} \\Delta \\theta $ , where $f = \\sin (K\\theta + \\hbox{\\rm const})$ .", "Hence, $\\Delta \\theta = \\Delta f{1\\over K\\cos (K\\theta )}.\\qquad \\mathrm {{(6)}}$ In the one-dimensional case, this additional information can be obtained with a very simple architecture: Suppose that there is a second population of neurons, with identical tuning curve $g(\\theta )$ with the primary population, albeit shifted by an angle $\\alpha \\ne 0, \\pi $ ; say $\\alpha = 90^{{\\rm o}}$ .", "Then this new population yields a similar equation, with $\\sin $ replacing $\\cos $ because of the phase shift: $\\Delta \\theta = \\Delta g{1\\over K\\sin (K\\theta )}.\\qquad \\mathrm {{(7)}}$ Thus we have two trigonometric equations for the two unknowns $\\Delta \\theta $ and $\\cos (K\\theta )$ , which can be easily solved: By dividing (6) by (7) we note that $\\tan (\\theta )={\\Delta g\\over \\Delta f}$ , and hence $\\Delta \\theta = \\Delta f{1+({\\Delta g\\over \\Delta f})^2\\over K}.$ That is, the displacement measurement can be computed from the two populations.", "We conclude that, in one dimension, a second identical population of neurons shifted by $90^{{\\rm o}}$ suffices for an effective readout.", "Now in two dimensions, the equivalent of (6) is $\\Delta \\theta _x K \\cos (K\\theta _x) + \\Delta \\theta _y K \\cos (K\\theta _y)= \\Delta f.$ Notice that now there are four unknowns ($\\Delta \\theta _x$ , $\\cos (K\\theta _x)$ , $\\Delta \\theta _y$ , and $\\cos (K\\theta _y)$ ).", "We conclude that three additional populations of neurons seem to be required, with different two-dimensional shifts from the original one.", "The most natural way to implement such a scheme is through shifts in three directions, forming equal angles of $120^{{\\rm o}}$ with each other (see Figure 3).", "Hence the familiar triangular grid may be the most natural way to implement this mechanism.", "Further analytical articulation of this point is the subject of future work.", "Figure: Four interlaced populations of grid cells" ], [ "Discussion", "What is the origin and utility of the grid cells' distinctive firing field, and what does it have to do with path integration?", "We have shown that, in one dimension, the tuning curve that optimizes the accuracy of displacement measurements is a near-sinusoid wave, whose peaks naturally form a one-dimensional grid.", "In two dimensions, we needed several further assumptions in order to show that the optimum tuning curve is the product of two sinusoidal waves in two non-parallel directions.", "If these two directions form an angle of $120^{\\hbox{{\\small o}}}$ , the familiar triangular grid results.", "We have also presented ideas about a possible mechanism for computing the change in position from the change in the response.", "For two dimensions, our proposed mechanism predicts the existence of four populations of grid cells with regularly displaced firing fields, and suggests that the triangular architecture may be the optimum solution of the joint problem of maximizing both sensitivity and accuracy of decoding.", "One may further speculate about grid cells for three dimensions (relevant for animals such as bats and sea mammals).", "Here, our analysis predicts a total of six populations, and these must be interlaced in a way analogous to that in Figure 3.", "The arrangement in Figure 3 works because the simple triangular lattice of one population, say population A, if copied three times and appropriately shifted, has the property that each point has at least one neighbor from each of the other three populations.", "This suggests the following technical question in 3-dimensional geometry: Is there a lattice in three dimensions with the property that, if it is duplicated six times, each point of each copy of the lattice is adjacent to at least one point from each of the other five copies?", "It turns out that the lattice generated by the vectors $(0,3,0), (0,0,2), (1,1,1)$ has this property: Notice that the corresponding matrix has determinant 6, suggesting that six copies of the lattice can be arranged in space, and it is easy to check that each point of each copy is at distance one from a point from each of the other copies.", "Therefore, intriguing compromises do exist between the design of a neural architecture for path integration in three dimensions and the realities of three-dimensional geometry." ], [ "Acknowledgment:", "Many thanks to Reza Moazzezi for fruitful collaboration during the early stages of this work, to Umesh Vazirani for illuminating discussions, and to Christos-Alexandros Psomas for help with the figures." ] ]	1606.04876
[ [ "Actor of a crossed module of Leibniz algebras" ], [ "Abstract We extend to the category of crossed modules of Leibniz algebras the notion of biderivation via the action of a Leibniz algebra.", "This results into a pair of Leibniz algebras which allow us to construct an object which is the actor under certain circumstances.", "Additionally, we give a description of an action in the category of crossed modules of Leibniz algebras in terms of equations.", "Finally, we check that, under the aforementioned conditions, the kernel of the canonical map from a crossed module to its actor coincides with the center and we introduce the notions of crossed module of inner and outer biderivations." ], [ "Introduction", "In the category of groups it is possible to describe an action via an object called the actor, which is given by the group of automorphisms.", "Its analogue in the category of Lie algebras is the Lie algebra of derivations.", "Groups and Lie algebras are examples of categories of interest, introduced by Orzech in [14].", "For these categories (see [12] for more examples), Casas, Datuashvili and Ladra [5] gave a procedure to construct an object that, under certain circumstances, plays the role of actor.", "For the particular case of Leibniz algebras (resp.", "associative algebras) that object is the Leibniz algebra of biderivations (resp.", "the algebra of bimultipliers).", "In [13], Norrie extended the definition of actor to the 2-dimensional case by giving a description of the corresponding object in the category of crossed modules of groups.", "The analogue construction for the category of crossed modules of Lie algebras is given in [8].", "Regarding the category of crossed modules of Leibniz algebras, it is not a category of interest, but it is equivalent to the category of $cat^1$ -Leibniz algebras (see for example [7]), which is itself a modified category of interest in the sense of [3].", "Therefore it makes sense to study representability of actions in such category under the context of modified categories of interest, as it is done in [3] for crossed modules of associative algebras.", "Bearing in mind the ease of the generalization of the actor in the category of groups and Lie algebras to crossed modules, together with the role of the Leibniz algebra of biderivations, it makes sense to assume that the analogous object in the category of crossed modules of Leibniz algebras will be the actor only under certain hypotheses.", "In [6] the authors gave an equivalent description of an action of a crossed module of groups in terms of equations.", "A similar description is done for an action of a crossed module of Lie algebras (see [4]).", "In order to extend the notion of actor to crossed modules of Leibniz algebras, we generalize the concept of biderivation to the 2-dimensional case, describe an action in that category in terms of equations and give sufficient conditions for the described object to be the actor.", "The article is organized as follows: In Section we recall some basic definitions on actions and crossed modules of Leibniz algebras.", "In Section we construct an object that extends the Leibniz algebra of biderivations to the category of crossed modules of Leibniz algebras (Theorem REF ) and give a description of an action in such category in terms of equations.", "In Section we find sufficient conditions for the previous object to be the actor of a given crossed module of Leibniz algebras (Theorem REF ).", "Finally, in Section we prove that the kernel of the canonical homomorphism from a crossed module of Leibniz algebras to its actor coincides with the center of the given crossed module.", "Additionally, we introduce the notions of crossed module of inner and outer biderivations and show that, given a short exact sequence in the category of crossed modules of Leibniz algebras, it can be extended to a commutative diagram including the actor and the inner and outer biderivations." ], [ "Preliminaries", "In this section we recall some needed basic definitions.", "Throughout the paper we fix a commutative ring with unit $\\operatorname{\\mathbf {k}}$ .", "All algebras are considerer over $\\operatorname{\\mathbf {k}}$ .", "Definition 2.1 ([10]) A Leibniz algebra $\\mathfrak {p}$ is a $\\operatorname{\\mathbf {k}}$ -module together with a bilinear operation $[ \\ , \\ ]\\colon \\mathfrak {p}\\times \\mathfrak {p}\\rightarrow \\mathfrak {p}$ , called the Leibniz bracket, which satisfies the Leibniz identity: $[[p_1,p_2],p_3]=[p_1,[p_2,p_3]]+[[p_1,p_3],p_2],$ for all $p_1,p_2,p_3\\in \\mathfrak {p}$ .", "A homomorphism of Leibniz algebras is a $\\operatorname{\\mathbf {k}}$ -linear map that preserves the bracket.", "We denote by $\\operatorname{Ann}(\\mathfrak {p})$ (resp.", "$[\\mathfrak {p},\\mathfrak {p}]$ ) the annihilator (resp.", "commutator) of $\\mathfrak {p}$ , that is the subspace of $\\mathfrak {p}$ generated by $\\lbrace p_1 \\in \\mathfrak {p} \\ \| \\ [p_1,p_2] = [p_2,p_1] = 0, \\ \\text{for all } p_2 \\in \\mathfrak {p}\\rbrace $ $(\\text{resp.", "}\\lbrace [p_1,p_2] \\ \| \\ \\text{for all } p_1, p_2 \\in \\mathfrak {p}\\rbrace )$ It is obvious that both $\\operatorname{Ann}(\\mathfrak {p})$ and $[\\mathfrak {p},\\mathfrak {p}]$ are ideals of $\\mathfrak {p}$ .", "Definition 2.2 ([11]) Let $\\mathfrak {p}$ and $\\mathfrak {m}$ be two Leibniz algebras.", "An action of $\\mathfrak {p}$ on $\\mathfrak {m}$ consists of a pair of bilinear maps, $\\mathfrak {p}\\times \\mathfrak {m}\\rightarrow \\mathfrak {m}$ , $(p,m)\\mapsto \\left[p,m\\right]$ and $\\mathfrak {m}\\times \\mathfrak {p}\\rightarrow \\mathfrak {m}$ , $(m,p)\\mapsto \\left[m,p\\right]$ , such that $[p,[m,m^{\\prime }]] & = [[p,m],m^{\\prime }]-[[p,m^{\\prime }],m], \\\\ [m,[p,m^{\\prime }]] & = [[m,p],m^{\\prime }]-[[m,m^{\\prime }],p], \\\\[m,[m^{\\prime },p]] & = [[m,m^{\\prime }],p]-[[m,p],m^{\\prime }], \\\\ [m,[p,p^{\\prime }]] & = [[m,p],p^{\\prime }]-[[m,p^{\\prime }],p], \\\\[p,[m,p^{\\prime }]] & = [[p,m],p^{\\prime }]-[[p,p^{\\prime }],m], \\\\[p,[p^{\\prime },m]] & = [[p,p^{\\prime }],m]-[[p,m],p^{\\prime }],$ for all $m, m^{\\prime } \\in \\mathfrak {m}$ and $p, p^{\\prime }\\in \\mathfrak {p}$ .", "Given an action of a Leibniz algebra $\\mathfrak {p}$ on $\\mathfrak {m}$ , we can consider the semidirect product Leibniz algebra $\\mathfrak {m}\\rtimes \\mathfrak {p}$ , which consists of the $\\operatorname{\\mathbf {k}}$ -module $\\mathfrak {m}\\oplus \\mathfrak {p}$ together with the Leibniz bracket given by $[(m,p),(m^{\\prime },p^{\\prime })]=([m,m^{\\prime }]+[p,m^{\\prime }]+[m,p^{\\prime }], [p,p^{\\prime }]),$ for all $(m,p), (m^{\\prime },p^{\\prime })\\in \\mathfrak {m}\\oplus \\mathfrak {p}$ .", "Definition 2.3 ([11]) A crossed module of Leibniz algebras (or Leibniz crossed module, for short) $(\\mathfrak {m}, \\mathfrak {p}, \\eta )$ is a homomorphism of Leibniz algebras $\\eta \\colon \\mathfrak {m}\\rightarrow \\mathfrak {p}$ together with an action of $\\mathfrak {p}$ on $\\mathfrak {m}$ such that $& \\eta ([p,m])=[p,\\eta (m)] \\quad \\text{and} \\quad \\eta ([m,p])=[\\eta (m),p], \\\\& [\\eta (m),m^{\\prime }]=[m,m^{\\prime }]=[m,\\eta (m^{\\prime })],$ for all $m,m^{\\prime } \\in \\mathfrak {m}$ , $p \\in \\mathfrak {p}$ .", "A homomorphism of Leibniz crossed modules $(\\varphi , \\psi )$ from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $(\\mathfrak {n},\\mathfrak {q}, \\mu )$ is a pair of Leibniz homomorphisms, $\\varphi \\colon \\mathfrak {m} \\rightarrow \\mathfrak {n}$ and $\\psi \\colon \\mathfrak {p} \\rightarrow \\mathfrak {q}$ , such that they commute with $\\eta $ and $\\mu $ and they respect the actions, that is $\\varphi ([p,m]) = [\\psi (p),\\varphi (m)]$ and $\\varphi ([m,p]) = [\\varphi (m),\\psi (p)]$ for all $m\\in \\mathfrak {m}$ , $p\\in \\mathfrak {p}$ .", "Identity (REF ) will be called equivariance and () Peiffer identity.", "We will denote by $\\operatorname{\\textbf {\\textsf {XLb}}}$ the category of Leibniz crossed modules and homomorphisms of Leibniz crossed modules.", "Since our aim is to construct a 2-dimensional generalization of the actor in the category of Leibniz algebras, let us first recall the following definitions.", "Definition 2.4 ([10]) Let $\\mathfrak {m}$ be a Leibniz algebra.", "A biderivation of $\\mathfrak {m}$ is a pair $(d,D)$ of $\\operatorname{\\mathbf {k}}$ -linear maps $d,D \\colon \\mathfrak {m} \\rightarrow \\mathfrak {m}$ such that $d([m,m^{\\prime }]) & = [d(m),m^{\\prime }] + [m,d(m^{\\prime })],\\\\D([m,m^{\\prime }]) & = [D(m),m^{\\prime }] - [D(m^{\\prime }),m],\\\\[m,d(m^{\\prime })] & = [m, D(m^{\\prime })],$ for all $m,m^{\\prime } \\in \\mathfrak {m}$ .", "We will denote by $\\operatorname{Bider}(\\mathfrak {m})$ the set of all biderivations of $\\mathfrak {m}$ .", "It is a Leibniz algebra with the obvious $\\operatorname{\\mathbf {k}}$ -module structure and the Leibniz bracket given by $[(d_1,D_1), (d_2,D_2)] = (d_1 d_2 - d_2 d_1, D_1 d_2 - d_2 D_1).$ It is not difficult to check that, given an element $m\\in \\mathfrak {m}$ , the pair $(\\operatorname{ad}(m), \\operatorname{Ad}(m))$ , with $\\operatorname{ad}(m)(m^{\\prime })=-[m^{\\prime },m]$ and $\\operatorname{Ad}(m)(m^{\\prime })=[m,m^{\\prime }]$ for all $m^{\\prime } \\in \\mathfrak {m}$ , is a biderivation.", "The pair $(\\operatorname{ad}(m), \\operatorname{Ad}(m))$ is called inner biderivation of $m$ ." ], [ "The main construction", "In this section we extend to crossed modules the Leibniz algebra of biderivations.", "First we need to translate the notion of a biderivation of a Leibniz algebra into a biderivation between two Leibniz algebras via the action.", "Definition 3.1 Given an action of Leibniz algebras of $\\mathfrak {q}$ on $\\mathfrak {n}$ , the set of biderivations from $\\mathfrak {q}$ to $\\mathfrak {n}$ , denoted by $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ , consists of all the pairs $(d,D)$ of $\\operatorname{\\mathbf {k}}$ -linear maps, $d,D \\colon \\mathfrak {q} \\rightarrow \\mathfrak {n}$ , such that $d([q,q^{\\prime }]) & = [d(q),q^{\\prime }] + [q,d(q^{\\prime })], \\\\D([q,q^{\\prime }]) & = [D(q),q^{\\prime }] - [D(q^{\\prime }),q], \\\\[q,d(q^{\\prime })] & = [q,D(q^{\\prime })], $ for all $q,q^{\\prime } \\in \\mathfrak {q}$ .", "Given $n \\in \\mathfrak {n}$ , the pair of $\\operatorname{\\mathbf {k}}$ -linear maps $(\\operatorname{ad}(n),\\operatorname{Ad}(n))$ , where $\\operatorname{ad}(n) (q) = - [q,n]$ and $\\operatorname{Ad}(n) (q) = [n,q]$ for all $q \\in \\mathfrak {q}$ , is clearly a biderivation from $\\mathfrak {q}$ to $\\mathfrak {n}$ .", "Observe that $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {q})$ , with the action of $\\mathfrak {q}$ on itself defined by its Leibniz bracket, is exactly $\\operatorname{Bider}(\\mathfrak {q})$ .", "Let us assume for the rest of the article that $(\\mathfrak {n},\\mathfrak {q},\\mu )$ is a Leibniz crossed module.", "One can easily check the following result.", "Lemma 3.2 Let $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "Then $(d \\mu , D \\mu ) \\in \\operatorname{Bider}(\\mathfrak {n})$ and $(\\mu d, \\mu D) \\in \\operatorname{Bider}(\\mathfrak {q})$ .", "We also have the following result.", "Lemma 3.3 Let $(d_1,D_1)$ , $(d_2,D_2) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "Then $[D_1 \\mu d_2 (q), q^{\\prime }] & = [D_1 \\mu D_2(q), q^{\\prime }], \\\\[q, D_1 \\mu d_2 (q^{\\prime })] & = [q, D_1 \\mu D_2 (q^{\\prime })],$ for all $q,q^{\\prime } \\in \\mathfrak {q}$ .", "Let $q, q^{\\prime } \\in \\mathfrak {q}$ and $(d_1,D_1), (d_2,D_2) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "According to the identity () for $(d_2,D_2)$ , $[q^{\\prime },d_2(q)]=[q^{\\prime },D_2(q)]$ , so $D_1 \\mu ([q^{\\prime },d_2(q)])=D_1 \\mu ([q^{\\prime },D_2(q)])$ .", "Due to () and the equivariance of $(\\mathfrak {q}, \\mathfrak {n}, \\mu )$ , one can easily derive that $[D_1(q^{\\prime }),\\mu d_2(q)] - [D_1 \\mu d_2(q),q^{\\prime }] = [D_1(q^{\\prime }),\\mu D_2(q)] - [D_1 \\mu D_2(q),q^{\\prime }].$ By the Peiffer identity and () for $(d_2,D_2)$ , $[D_1(q^{\\prime }),\\mu d_2(q)] = [D_1(q^{\\prime }),\\mu D_2(q)]$ .", "Therefore $[D_1 \\mu d_2(q),q^{\\prime }] = [D_1 \\mu D_2(q),q^{\\prime }]$ .", "The other identity can be proved similarly by using (REF ) and ().", "$\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ has an obvious $\\operatorname{\\mathbf {k}}$ -module structure.", "Regarding its Leibniz structure, it is described in the next proposition.", "Proposition 3.4 $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ is a Leibniz algebra with the bracket given by $[(d_1,D_1),(d_2,D_2)] = (d_1 \\mu d_2 - d_2 \\mu d_1, D_1 \\mu d_2 - d_2 \\mu D_1)$ for all $(d_1,D_1), (d_2,D_2) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "It follows directly from Lemma REF .", "Now we state the following definition.", "Definition 3.5 The set of biderivations of the Leibniz crossed module $(\\mathfrak {n},\\mathfrak {q},\\mu )$ , denoted by $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , consists of all quadruples $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2))$ such that $& (\\sigma _1,\\theta _1) \\in \\operatorname{Bider}(\\mathfrak {n}) \\quad \\text{and} \\quad (\\sigma _2,\\theta _2) \\in \\operatorname{Bider}(\\mathfrak {q}),\\\\& \\mu \\sigma _1 = \\sigma _2 \\mu \\quad \\text{and} \\quad \\mu \\theta _1 = \\theta _2 \\mu ,\\\\& \\sigma _1([q,n]) = [\\sigma _2(q),n] + [q,\\sigma _1(n)],\\\\& \\sigma _1([n,q]) = [\\sigma _1(n),q] + [n,\\sigma _2(q)],\\\\& \\theta _1([q,n]) = [\\theta _2(q),n] - [\\theta _1(n),q],\\\\& \\theta _1([n,q]) = [\\theta _1(n),q] - [\\theta _2(q),n],\\\\& [q,\\sigma _1(n)] = [q,\\theta _1(n)],\\\\& [n,\\sigma _2(q)] = [n,\\theta _2(q)],$ for all $n \\in \\mathfrak {n}$ , $q \\in \\mathfrak {q}$ .", "Given $q \\in \\mathfrak {q}$ , it can be readily checked that $((\\sigma ^{q}_1,\\theta ^{q}_1),(\\sigma ^{q}_2,\\theta ^{q}_2))$ , where $\\sigma ^{q}_1 (n) & = - [n,q], \\qquad \\theta ^{q}_1 (n) = [q,n],\\\\\\sigma ^{q}_2 (q^{\\prime }) & = - [q^{\\prime },q], \\qquad \\theta ^{q}_2 (q^{\\prime }) = [q,q^{\\prime }],$ is a biderivation of the crossed module $(\\mathfrak {n}, \\mathfrak {q}, \\mu )$ .", "The following lemma is necessary in order to prove that $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ is indeed a Leibniz algebra.", "Lemma 3.6 Let $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)), ((\\sigma ^{\\prime }_1,\\theta ^{\\prime }_1),(\\sigma ^{\\prime }_2,\\theta ^{\\prime }_2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ and $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "Then $\\begin{split}[D \\sigma _2 (q), q^{\\prime }] & = [D \\theta _2(q), q^{\\prime }], \\\\[q, D \\sigma _2 (q^{\\prime })] & = [q, D \\theta _2 (q^{\\prime })], \\\\[\\theta _1 d (q), q^{\\prime }] & = [\\theta _1 D (q), q^{\\prime }], \\\\[q, \\theta _1 d (q^{\\prime })] & = [q, \\theta _1 D (q^{\\prime })],\\end{split}\\qquad \\begin{split}[D \\sigma _2 (q), n] & = [D \\theta _2(q), n], \\\\[n, D \\sigma _2 (q)] & = [n, D \\theta _2 (q)], \\\\[\\theta _1 d (q), n] & = [\\theta _1 D (q), n], \\\\[n, \\theta _1 d (q)] & = [n, \\theta _1 D (q)],\\end{split}\\qquad \\begin{split}[\\theta _1 \\sigma ^{\\prime }_1 (n), q] & = [\\theta _1 \\theta ^{\\prime }_1(n), q], \\\\[q, \\theta _1 \\sigma ^{\\prime }_1 (n)] & = [q, \\theta _1 \\theta ^{\\prime }_1(n)], \\\\[\\theta _2 \\sigma ^{\\prime }_2 (q), n] & = [\\theta _2 \\theta ^{\\prime }_2(q), n], \\\\[n, \\theta _2 \\sigma ^{\\prime }_2 (q)] & = [n, \\theta _2 \\theta ^{\\prime }_2(q)],\\end{split}$ for all $n \\in \\mathfrak {n}$ , $q,q^{\\prime } \\in \\mathfrak {q}$ .", "Let us show how to prove the first identity; the rest of them can be checked similarly.", "Let $q, q^{\\prime } \\in \\mathfrak {q}$ , $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ and $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "Since $(\\sigma _2, \\theta _2)$ is a biderivation of $\\mathfrak {q}$ , we have that $[q^{\\prime },\\sigma _2(q)]=[q^{\\prime },\\theta _2(q)]$ .", "Therefore $D([q^{\\prime },\\sigma _2(q)])=D([q^{\\prime },\\theta _2(q)])$ .", "Directly from (), we get that $[D(q^{\\prime }),\\sigma _2(q)] - [D \\sigma _2(q),q^{\\prime }] = [D(q^{\\prime }),\\theta _2(q)] - [D \\theta _2(q),q^{\\prime }].$ Thus, due to (), $[D(q^{\\prime }),\\sigma _2(q)] = [D(q^{\\prime }),\\theta _2(q)]$ .", "Hence, $[D \\sigma _2(q),q^{\\prime }] = [D \\theta _2(q),q^{\\prime }]$ .", "The $\\operatorname{\\mathbf {k}}$ -module structure of $\\operatorname{Bider}(\\mathfrak {n}, \\mathfrak {q}, \\mu )$ is evident, while its Leibniz structure is described as follows.", "Proposition 3.7 $\\operatorname{Bider}(\\mathfrak {n}, \\mathfrak {q}, \\mu )$ is a Leibniz algebra with the bracket given by $[((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)), ((\\sigma ^{\\prime }_1,\\theta ^{\\prime }_1),(\\sigma ^{\\prime }_2,\\theta ^{\\prime }_2))] = ([(\\sigma _1,\\theta _1),(\\sigma ^{\\prime }_1,\\theta ^{\\prime }_1)],[(\\sigma _2,\\theta _2),(\\sigma ^{\\prime }_2,\\theta ^{\\prime }_2)]) \\\\= ((\\sigma _1 \\sigma ^{\\prime }_1 - \\sigma ^{\\prime }_1 \\sigma _1, \\theta _1 \\sigma ^{\\prime }_1 - \\sigma ^{\\prime }_1 \\theta _1),(\\sigma _2 \\sigma ^{\\prime }_2 - \\sigma ^{\\prime }_2 \\sigma _2, \\theta _2 \\sigma ^{\\prime }_2 - \\sigma ^{\\prime }_2 \\theta _2)),$ for all $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)), ((\\sigma ^{\\prime }_1,\\theta ^{\\prime }_1),(\\sigma ^{\\prime }_2,\\theta ^{\\prime }_2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "It follows directly from Lemma REF .", "Proposition 3.8 The $\\operatorname{\\mathbf {k}}$ -linear map $\\Delta \\colon \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}) \\rightarrow \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , given by $(d,D) \\mapsto ((d \\mu , D \\mu ),(\\mu d, \\mu D))$ is a homomorphism of Leibniz algebras.", "$\\Delta $ is well defined due to Lemma REF , while checking that it is a homomorphism of Leibniz algebras is a matter of straightforward calculations.", "Since we aspire to make $\\Delta $ into a Leibniz crossed module, we need to define an action of $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ on $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "Theorem 3.9 There is an action of $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ on $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ given by: $[((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)),(d,D)] & = (\\sigma _1 d - d \\sigma _2, \\theta _1 d - d \\theta _2), \\\\[(d,D),((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2))] & = (d \\sigma _2 - \\sigma _1 d, D \\sigma _2 - \\sigma _1 D),$ for all $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "Moreover, the Leibniz homomorphism $\\Delta $ (see Proposition REF ) together with the above action is a Leibniz crossed module.", "Let $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ and $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "Checking that both $(\\sigma _1 d - d \\sigma _2, \\theta _1 d - d \\theta _2)$ and $(d \\sigma _2 - \\sigma _1 d, D \\sigma _2 - \\sigma _1 D)$ satisfy conditions (REF ) and () requires the combined use of the properties satisfied by the elements in $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ and $(d,D)$ , but calculations are fairly straightforward.", "As an example, we show how to prove that $(\\sigma _1 d - d \\sigma _2, \\theta _1 d - d \\theta _2)$ verifies (REF ).", "Let $q, q^{\\prime } \\in \\mathfrak {q}$ .", "Then $(\\sigma _1 d - d \\sigma _2) ([q,q^{\\prime }]) = & \\sigma _1 ([d(q),q^{\\prime }] + [q,d(q^{\\prime })]) - d([\\sigma _2(q), q^{\\prime }] + [q,\\sigma _2(q^{\\prime })]) \\\\= & [\\sigma _1 d (q), q^{\\prime }] + [d(q), \\sigma _2(q^{\\prime })] + [\\sigma _2(q),d(q^{\\prime })] + [q,\\sigma _1 d(q^{\\prime })] \\\\& - [d \\sigma _2(q), q^{\\prime }] - [\\sigma _2(q), d(q^{\\prime })] - [d(q),\\sigma _2(q^{\\prime })] - [q,d \\sigma _2(q^{\\prime })] \\\\= & [(\\sigma _1 d - d \\sigma _2) (q), q^{\\prime }] + [q, (\\sigma _1 d - d \\sigma _2) (q^{\\prime })].$ As for condition (), in the case of $(\\sigma _1 d - d \\sigma _2, \\theta _1 d - d \\theta _2)$ , it follows from (), the identity () for $(d,D)$ and the second identity in the first column from Lemma REF .", "Namely, $[q, (\\sigma _1 d - d \\sigma _2) (q^{\\prime })] & = [q, \\sigma _1 d(q^{\\prime })] - [q, d \\sigma _2(q^{\\prime })] = [q, \\theta _1 d(q^{\\prime })] - [q, D \\sigma _2(q^{\\prime })] \\\\& = [q, \\theta _1 d(q^{\\prime })] - [q, D \\theta _2(q^{\\prime })] = [q, \\theta _1 d(q^{\\prime })] - [q, d \\theta _2(q^{\\prime })],$ for all $q,q^{\\prime } \\in \\mathfrak {q}$ .", "A similar procedure allows to prove that $(d \\sigma _2 - \\sigma _1 d, D \\sigma _2 - \\sigma _1 D)$ satisfies condition () as well.", "Routine calculations show that (REF ) and () together with the definition of the brackets in $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ and $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ provide an action of Leibniz algebras.", "It only remains to prove that $\\Delta $ satisfies the equivariance and the Peiffer identity.", "It is immediate to check that $\\Delta ([((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)),(d,D)]) = ((\\sigma _1 d \\mu - d \\sigma _2 \\mu , \\theta _1 d \\mu - d \\theta _2 \\mu ),\\\\ (\\mu \\sigma _1 d - \\mu d \\sigma _2, \\mu \\theta _1 d - \\mu d \\theta _2)),$ while $[((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)),\\Delta (d,D)] = ((\\sigma _1 d \\mu - d \\mu \\sigma _1, \\theta _1 d \\mu - d \\mu \\theta _1),\\\\ (\\sigma _2 \\mu d - \\mu d \\sigma _2, \\theta _2 \\mu d - \\mu d \\theta _2)).$ Condition () guarantees that $(\\ref {equiv_Xlb_actor_1}) = (\\ref {equiv_Xlb_actor_2})$ .", "The other identity can be checked similarly.", "The Peiffer identity follows immediately from (REF ) and () along the definition of $\\Delta $ and the bracket in $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ ." ], [ "The actor", "In [14], Orzech introduced the notion of category of interest, which is nothing but a category of groups with operations verifying two extra conditions.", "$\\operatorname{\\textbf {\\textsf {Lb}}}$ is a category of interest, although $\\operatorname{\\textbf {\\textsf {XLb}}}$ is not.", "Nevertheless, it is equivalent to the category of $cat^1$ -Leibniz algebras (see for example [7]), which is itself a modified category of interest in the sense of [3].", "So it makes sense to study representability of actions in $\\operatorname{\\textbf {\\textsf {XLb}}}$ under the context of modified categories of interest, as it is done in [3] for crossed modules of associative algebras.", "However, since $\\operatorname{\\textbf {\\textsf {XLb}}}$ is an example of semi-abelian categories, and an action is the same as a split extension in any semi-abelian category [2], we choose a different, more combinatorial approach to the problem, by constructing the semidirect product (split extension) of Leibniz crossed modules.", "We use the term actor (as in [3], [5]) for an object which represents actions in a semi-abelian category, the general definition of which is known from [2] under the name split extension classifier.", "We need to remark that, given a Leibniz algebra $\\mathfrak {m}$ , $\\operatorname{Bider}(\\mathfrak {m})$ is the actor of $\\mathfrak {m}$ under certain conditions.", "In particular, the following result is proved in [5].", "Proposition 4.1 ([5]) Let $\\mathfrak {m}$ be a Leibniz algebra such that $\\operatorname{Ann}(\\mathfrak {m})=0$ or $[\\mathfrak {m},\\mathfrak {m}]=\\mathfrak {m}$ .", "Then $\\operatorname{Bider}(\\mathfrak {m})$ is the actor of $\\mathfrak {m}$ .", "Bearing in mind the ease of the generalization of the actor in the category of groups and Lie algebras to crossed modules, together with the role of $\\operatorname{Bider}(\\mathfrak {m})$ in regard to any Leibniz algebra $\\mathfrak {m}$ , it makes sense to consider $(\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}),\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu ),\\Delta )$ as a candidate for actor in $\\operatorname{\\textbf {\\textsf {XLb}}}$ , at least under certain conditions (see Proposition REF ).", "However, it would be reckless to define an action of a Leibniz crossed module $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ as a homomorphism from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to the Leibniz crossed module $(\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}),\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu ),\\Delta )$ , since we cannot ensure that the mentioned homomorphism induces a set of actions of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ from which we can construct the semidirect product.", "In [6] the authors give an equivalent description of an action of a crossed module of groups in terms of equations.", "A similar description can be done for an action of a crossed module of Lie algebras (see [4]).", "This determines our approach to the problem.", "We consider a homomorphism from a Leibniz crossed module $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $(\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}),\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu ),\\Delta )$ , which will be denoted by $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ from now on, and unravel all the properties satisfied by the mentioned homomorphism, transforming them into a set of equations.", "Then we check that the existence of that set of equations is equivalent to the existence of a homomorphism of Leibniz crossed modules from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ only under certain conditions.", "Finally we prove that those equations indeed describe a comprehensive set of actions by constructing the associated semidirect product, which is an object in $\\operatorname{\\textbf {\\textsf {XLb}}}$ .", "Lemma 4.2 Let $\\mathfrak {q}$ be a Leibniz algebra and $(\\sigma ,\\theta ), (\\sigma ^{\\prime },\\theta ^{\\prime }) \\in \\operatorname{Bider}(\\mathfrak {q})$ .", "If $\\operatorname{Ann}(\\mathfrak {q}) = 0$ or $[\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}$ , then $\\theta \\sigma ^{\\prime } (q) = \\theta \\theta ^{\\prime } (q),$ for all $q \\in \\mathfrak {q}$ .", "Let $(\\mathfrak {n},\\mathfrak {q},\\mu )$ be a Leibniz crossed module, $((\\sigma _1,\\theta _1),(\\sigma _2,\\theta _2)) \\in \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ and $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ .", "If $\\operatorname{Ann}(\\mathfrak {n}) = 0$ or $[\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}$ , then $D \\sigma _2 (q) & = D \\theta _2 (q), \\\\\\theta _1 d (q) & = \\theta _1 D (q), $ for all $q \\in \\mathfrak {q}$ .", "Calculations in order to prove (i) are straightforward.", "Regarding (ii), $D \\sigma _2 (q) - D \\theta _2 (q)$ and $\\theta _1 d (q) - \\theta _1 D (q)$ are elements in $\\operatorname{Ann}(\\mathfrak {n})$ , immediately from the identities in the second column from Lemma REF .", "Therefore, if $\\operatorname{Ann}(\\mathfrak {n}) = 0$ , it is clear that (REF ) and () hold.", "Let us now assume that $[\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}$ .", "Given $q,q^{\\prime } \\in \\mathfrak {q}$ , directly from the fact that $(\\sigma _2,\\theta _2) \\in \\operatorname{Bider}(\\mathfrak {q})$ and $(d,D) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ , we get that $D \\theta _2([q,q^{\\prime }]) & = [D \\theta _2(q),q^{\\prime }] - [D(q^{\\prime }),\\theta _2(q)] - [D \\theta _2 (q^{\\prime }),q] + [D (q),\\theta _2 (q^{\\prime })], \\\\D \\sigma _2([q,q^{\\prime }]) & = [D \\sigma _2(q),q^{\\prime }] - [D(q^{\\prime }),\\sigma _2(q)] + [D(q),\\sigma _2(q^{\\prime })] - [D \\sigma _2(q^{\\prime }),q].$ Due to () and the first identity in the first column from Lemma REF , $D \\theta _2([q,q^{\\prime }]) = D \\sigma _2([q,q^{\\prime }])$ .", "By hypothesis, every element in $\\mathfrak {q}$ can be expressed as a linear combination of elements of the form $[q,q^{\\prime }]$ .", "This fact together with the linearity of $D$ , $\\sigma _2$ and $\\theta _2$ , guarantees that $D \\theta _2(q) = D \\sigma _2(q)$ for all $q \\in \\mathfrak {q}$ .", "The identity () can be checked similarly by making use of (), (), () and the third identity in the first column from Lemma REF .", "Theorem 4.3 Let $(\\mathfrak {m},\\mathfrak {p},\\eta )$ and $(\\mathfrak {n},\\mathfrak {q},\\mu )$ in $\\operatorname{\\textbf {\\textsf {XLb}}}$ .", "There exists a homomorphism of crossed modules from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $(\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}), \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu ), \\Delta )$ , if the following conditions hold: (i) There are actions of the Leibniz algebra $\\mathfrak {p}$ (and so $\\mathfrak {m}$ ) on the Leibniz algebras $\\mathfrak {n}$ and $\\mathfrak {q}$ .", "The homomorphism $\\mu $ is $\\mathfrak {p}$ -equivariant, that is $\\mu ([p,n]) & =[p,\\mu (n)], \\\\\\mu ([n,p]) & =[\\mu (n),p], $ and the actions of $\\mathfrak {p}$ and $\\mathfrak {q}$ on $\\mathfrak {n}$ are compatible, that is $[n,[p,q]] & = [[n,p],q]-[[n,q],p], \\\\[p,[n,q]] & = [[p,n],q]-[[p,q],n], \\\\[p,[q,n]] & = [[p,q],n]-[[p,n],q], \\\\[n,[q,p]] & = [[n,q],p]-[[n,p],q], \\\\[q,[n,p]] & = [[q,n],p]-[[q,p],n], \\\\[q,[p,n]] & = [[q,p],n]-[[q,n],p], $ for all $n\\in \\mathfrak {n}$ , $p\\in \\mathfrak {p}$ and $q\\in \\mathfrak {q}$ .", "(i) There are two $\\operatorname{\\mathbf {k}}$ -bilinear maps $\\xi _1 \\colon \\mathfrak {m}\\times \\mathfrak {q}\\rightarrow \\mathfrak {n}$ and $\\xi _2 \\colon \\mathfrak {q}\\times \\mathfrak {m}\\rightarrow \\mathfrak {n}$ such that $\\mu \\xi _2(q,m) & =[q,m], \\\\\\mu \\xi _1(m,q) & =[m,q], \\\\\\xi _2 (\\mu (n),m) & = [n,m], \\\\\\xi _1 (m,\\mu (n)) & = [m,n], \\\\\\xi _2 (q,[p,m]) & = \\xi _2 ([q,p],m) - [\\xi _2(q,m),p], \\\\\\xi _1 ([p,m],q) & = \\xi _2 ([p,q],m) - [p,\\xi _2(q,m)], \\\\\\xi _2 (q,[m,p]) & = [\\xi _2(q,m),p] - \\xi _2 ([q,p],m), \\\\\\xi _1 ([m,p],q) & = [\\xi _1(m,q),p] - \\xi _1 (m,[q,p]), \\\\\\xi _2 (q,[m,m^{\\prime }]) & = [\\xi _2(q,m),m^{\\prime }] - [\\xi _2(q,m^{\\prime }),m], \\\\\\xi _1 ([m,m^{\\prime }],q) & = [\\xi _1(m,q),m^{\\prime }] - [m,\\xi _2(q,m^{\\prime })], \\\\\\xi _2 ([q,q^{\\prime }],m) & = [\\xi _2(q,m),q^{\\prime }] + [q,\\xi _2(q^{\\prime },m)], \\\\\\xi _1 (m,[q,q^{\\prime }]) & = [\\xi _1(m,q),q^{\\prime }] - [\\xi _1(m,q^{\\prime }),q], \\\\[q,\\xi _1(m,q^{\\prime })] & = - [q,\\xi _2(q^{\\prime },m)], \\\\\\xi _1 (m,[p,q]) &= - \\xi _1(m,[q,p]), \\\\[p,\\xi _1(m,q)] & = - [p,\\xi _2(q,m)], $ for all $m,m^{\\prime }\\in \\mathfrak {m}$ , $n\\in \\mathfrak {n}$ , $p\\in \\mathfrak {p}$ , $q,q^{\\prime }\\in \\mathfrak {q}$ .", "Additionally, the converse statement is also true if one of the following conditions holds: $& \\operatorname{Ann}(\\mathfrak {n})=0=\\operatorname{Ann}(\\mathfrak {q}), \\\\& \\operatorname{Ann}(\\mathfrak {n})=0 \\quad \\text{and} \\quad [\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}, \\\\& [\\mathfrak {n},\\mathfrak {n}] = \\mathfrak {n} \\quad \\text{and} \\quad [\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}.", "$ Let us suppose that (i) and (ii) hold.", "It is possible to define a homomorphism of crossed modules $(\\varphi ,\\psi )$ from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ as follows.", "Given $m \\in \\mathfrak {m}$ , $\\varphi (m)=(d_{m},D_{m})$ , with $d_{m} (q) = - \\xi _{2}(q,m), \\qquad D_{m} (q) = \\xi _{1}(m,q),$ for all $q \\in \\mathfrak {q}$ .", "On the other hand, for any $p \\in \\mathfrak {p}$ , $\\psi (p)=((\\sigma ^{p}_1,\\theta ^{p}_1), (\\sigma ^{p}_2,\\theta ^{p}_2))$ , with $\\sigma ^{p}_1(n) & = - [n,p], \\qquad \\theta ^{p}_1(n) = [p,n], \\\\\\sigma ^{p}_2(q) & = - [q,p], \\qquad \\theta ^{p}_2(q) = [p,q],$ for all $n \\in \\mathfrak {n}$ , $q \\in \\mathfrak {q}$ .", "It follows directly from ()–() that $(d_{m},D_{m}) \\in \\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ for all $m \\in \\mathfrak {m}$ .", "Besides, $\\varphi $ is clearly $\\operatorname{\\mathbf {k}}$ -linear and given $m,m^{\\prime } \\in \\mathfrak {m}$ , $[\\varphi (m),\\varphi (m^{\\prime })] = [(d_{m},D_{m}),(d_{m^{\\prime }},D_{m^{\\prime }})] = [d_{m} \\mu d_{m^{\\prime }} - d_{m^{\\prime }} \\mu d_{m}, D_{m} \\mu d_{m^{\\prime }} - d_{m^{\\prime }} \\mu D_{m}].$ For any $q \\in \\mathfrak {q}$ , $d_{m} \\mu d_{m^{\\prime }} (q) - d_{m^{\\prime }} \\mu d_{m} (q) & = -\\xi _{2} (\\mu d_{m^{\\prime }} (q), m) + \\xi _2 (\\mu d_{m} (q), m^{\\prime }) \\\\& = - [d_{m^{\\prime }} (q),m] + [d_{m}(q),m^{\\prime }] \\\\ & = [\\xi _{2} (q,m^{\\prime }),m] - [\\xi _{2}(q,m), m^{\\prime }] \\\\& = -\\xi _{2} (q,[m,m^{\\prime }]) = d_{[m,m^{\\prime }]}(q),$ due to () and ().", "Analogously, it can be easily checked the identity $(D_{m} \\mu d_{m^{\\prime }} - d_{m^{\\prime }} \\mu D_{m}) (q) = D_{[m,m^{\\prime }]} (q)$ by making use of (), () and ().", "Hence, $\\varphi $ is a homomorphism of Leibniz algebras.", "As for $\\psi $ , it is necessary to prove that $((\\sigma ^{p}_1,\\theta ^{p}_1), (\\sigma ^{p}_2,\\theta ^{p}_2))$ satisfies all the axioms from Definition REF for any $p \\in \\mathfrak {p}$ .", "The fact that $(\\sigma ^{p}_1,\\theta ^{p}_1)$ (respectively $(\\sigma ^{p}_2,\\theta ^{p}_2)$ ) is a biderivation of $\\mathfrak {n}$ (respectively $\\mathfrak {q}$ ) follows directly from the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ .", "The identities $\\mu \\theta ^{p}_1 = \\theta ^{p}_2 \\mu $ and $\\mu \\sigma ^{p}_1 = \\sigma ^{p}_2 \\mu $ are immediate consequences of (REF ) and () respectively.", "Observe that the combinations of the identities (REF ) and () and the identities () and () yield the equalities $-[n,[q,p]] = [n,[p,q]] \\qquad \\text{and} \\qquad -[q,[n,p]] = [q,[p,n]].$ These together with ()–() allow us to prove that $((\\sigma ^{p}_1,\\theta ^{p}_1), (\\sigma ^{p}_2,\\theta ^{p}_2))$ does satisfy conditions ()–() from Definition REF .", "Therefore, $\\psi $ is well defined, while it is obviously $\\operatorname{\\mathbf {k}}$ -linear.", "Moreover, due to (REF ) we know that $[\\psi (p),\\psi (p^{\\prime })] = ((\\sigma ^{p}_1 \\sigma ^{p^{\\prime }}_1 - \\sigma ^{p^{\\prime }}_1 \\sigma ^{p}_1, \\theta ^{p}_1 \\sigma ^{p^{\\prime }}_1 - \\sigma ^{p^{\\prime }}_1 \\theta ^{p}_1),(\\sigma ^{p}_2 \\sigma ^{p^{\\prime }}_2 - \\sigma ^{p^{\\prime }}_2 \\sigma ^{p}_2, \\theta ^{p}_2 \\sigma ^{p^{\\prime }}_2 - \\sigma ^{p^{\\prime }}_2 \\theta ^{p}_2)),$ and by definition $\\psi ([p,p^{\\prime }]) = ((\\sigma ^{[p,p^{\\prime }]}_1,\\theta ^{[p,p^{\\prime }]}_1), (\\sigma ^{[p,p^{\\prime }]}_2,\\theta ^{[p,p^{\\prime }]}_2)).$ One can easily check that the corresponding components are equal by making use of the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ .", "Hence, $\\psi $ is a homomorphism of Leibniz algebras.", "Recall that $\\Delta \\varphi (m) & = ((d_{m} \\mu , D_{m} \\mu ),(\\mu d_{m}, \\mu D_{m})), \\\\\\psi \\eta (m) & = ((\\sigma ^{\\eta (m)}_1,\\theta ^{\\eta (m)}_1),(\\sigma ^{\\eta (m)}_2,\\theta ^{\\eta (m)}_2)),$ for any $m \\in \\mathfrak {m}$ , but $d_m \\mu (n) & = -\\xi _2 (\\mu (n),m) = -[n,m] = -[n,\\eta (m)] = \\sigma ^{\\eta (m)}_{1} (n),\\\\D_m \\mu (n) & = \\xi _1 (m,\\mu (n)) = [m,n] = [\\eta (m),n] = \\theta ^{\\eta (m)}_{1} (n),\\\\\\mu d_m (q) & = -\\mu \\xi _2(q,m) = - [q,m] = - [q,\\eta (m)] = \\sigma ^{\\eta (m)}_{2} (q), \\\\\\mu D_m (q) & = \\mu \\xi _1(m,q) = [m,q] = [\\eta (m),q] = \\theta ^{\\eta (m)}_{2} (q),$ for all $n \\in \\mathfrak {n}$ , $q \\in \\mathfrak {q}$ , due to (REF ), (), (), ().", "Therefore, $\\Delta \\varphi = \\psi \\eta $ .", "It only remains to check the behaviour of $(\\varphi ,\\psi )$ regarding the action of $\\mathfrak {p}$ on $\\mathfrak {m}$ .", "Let $m \\in \\mathfrak {m}$ and $p \\in \\mathfrak {p}$ .", "Due to (REF ) and (), $[\\psi (p),\\varphi (m)] & = (\\sigma ^{p}_1 d_{m} - d_{m} \\sigma ^{p}_2, \\theta ^{p}_1 d_{m} - d_{m} \\theta ^{p}_2), \\\\[\\varphi (m),\\psi (p)] & = (d_{m} \\sigma ^{p}_2 - \\sigma ^{p}_1 d_{m}, D_{m} \\sigma ^{p}_2 - \\sigma ^{p}_1 D_{m}).$ On the other hand, by definition, we know that $\\varphi ([p,m]) & = (d_{[p,m]},D_{[p,m]}), \\\\\\varphi ([m,p]) & = (d_{[m,p]},D_{[m,p]}).$ Directly from (), (), () and () one can easily confirm that the required identities between components hold.", "Hence, we can finally ensure that $(\\varphi ,\\psi )$ is a homomorphism of Leibniz crossed modules.", "Now let us show that it is necessary that at least one of the conditions (REF )–() holds in order to prove the converse statement.", "Let us suppose that there is a homomorphism of crossed modules ${\\mathfrak {m} [d]_{\\varphi }[r]^{\\eta } & \\mathfrak {p} [d]^{\\psi } \\\\\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n}) [r]_{\\Delta } & \\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )}$ Given $m \\in \\mathfrak {m}$ and $p \\in \\mathfrak {p}$ , let us denote $\\varphi (m)$ by $(d_{m},D_{m})$ and $\\psi (p)$ by $((\\sigma ^{p}_1,\\theta ^{p}_1),(\\sigma ^{p}_2,\\theta ^{p}_2))$ , which satisfy conditions (REF )–() from Definition REF and conditions (REF )–() from Definition REF respectively.", "Also, due to the definition of $\\Delta $ (see Proposition REF ), the commutativity of (REF ) can be expressed by the identity $((d_{m} \\mu ,D_{m} \\mu ),(\\mu d_{m},\\mu D_{m})) = ((\\sigma ^{\\eta (m)}_1,\\theta ^{\\eta (m)}_1),(\\sigma ^{\\eta (m)}_2,\\theta ^{\\eta (m)}_2)),$ for all $m \\in \\mathfrak {m}$ .", "It is possible to define four bilinear maps, all of them denoted by $[-,-]$ , from $\\mathfrak {p} \\times \\mathfrak {n}$ to $\\mathfrak {n}$ , $\\mathfrak {n} \\times \\mathfrak {p}$ to $\\mathfrak {n}$ , $\\mathfrak {p} \\times \\mathfrak {q}$ to $\\mathfrak {q}$ and $\\mathfrak {q} \\times \\mathfrak {p}$ to $\\mathfrak {q}$ , given by $[p,n] & = \\theta ^{p}_1(n), \\qquad [n,p] = -\\sigma ^{p}_1(n), \\\\[p,q] & = \\theta ^{p}_2(q), \\qquad [q,p] = -\\sigma ^{p}_2(q),$ for all $n\\in \\mathfrak {n}$ , $p \\in \\mathfrak {p}$ , $q \\in \\mathfrak {q}$ .", "These maps define actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ .", "The first three identities for the action on $\\mathfrak {n}$ (respectively $\\mathfrak {q}$ ) follow easily from the fact that $(\\sigma ^{p}_1,\\theta ^{p}_1)$ (respectively $(\\sigma ^{p}_2,\\theta ^{p}_2)$ ) is a biderivation of $\\mathfrak {n}$ (respectively $\\mathfrak {q}$ ).", "Since $\\psi $ is a Leibniz homomorphism, we get that $((\\sigma ^{[p,p^{\\prime }]}_1,\\theta ^{[p,p^{\\prime }]}_1),(\\sigma ^{[p,p^{\\prime }]}_2,\\theta ^{[p,p^{\\prime }]}_2)) = ((\\sigma ^{p}_1 \\sigma ^{p^{\\prime }}_1 - \\sigma ^{p^{\\prime }}_1 \\sigma ^{p}_1, \\theta ^{p}_1 \\sigma ^{p^{\\prime }}_1 - \\sigma ^{p^{\\prime }}_1 \\theta ^{p}_1),\\\\(\\sigma ^{p}_2 \\sigma ^{p^{\\prime }}_2 - \\sigma ^{p^{\\prime }}_2 \\sigma ^{p}_2, \\theta ^{p}_2 \\sigma ^{p^{\\prime }}_2 - \\sigma ^{p^{\\prime }}_2 \\theta ^{p}_2)).$ The identities between the first and the second (respectively the third and the fourth) components in those quadruples allow us to confirm the fourth and fifth identities for the action of $\\mathfrak {p}$ on $\\mathfrak {n}$ (respectively $\\mathfrak {q}$ ).", "As for the last condition for both actions, it is fairly straightforward to check that $[[p,p^{\\prime }],n] - [[p,n],p^{\\prime }] & = \\theta ^{p}_1 \\sigma ^{p^{\\prime }}_1 (n),\\\\[[p,p^{\\prime }],q] - [[p,q],p^{\\prime }] & = \\theta ^{p}_2 \\sigma ^{p^{\\prime }}_2 (q),$ while $[p,[p^{\\prime },n]] & = \\theta ^{p}_1 \\theta ^{p^{\\prime }}_1 (n),\\\\[p,[p^{\\prime },q]] & = \\theta ^{p}_2 \\theta ^{p^{\\prime }}_2 (q),$ for all $n \\in \\mathfrak {n}$ , $p,p^{\\prime } \\in \\mathfrak {p}$ , $q \\in \\mathfrak {q}$ .", "However, if at least one of the conditions (REF )–() holds, due to Lemma REF (i), $\\theta ^{p}_1 \\sigma ^{p^{\\prime }}_1 (n)=\\theta ^{p}_1 \\theta ^{p^{\\prime }}_1 (n)$ and $\\theta ^{p}_2 \\sigma ^{p^{\\prime }}_2 (q)=\\theta ^{p}_2 \\theta ^{p^{\\prime }}_2 (q)$ .", "Therefore, we can ensure that there are Leibniz actions of $\\mathfrak {p}$ on both $\\mathfrak {n}$ and $\\mathfrak {q}$ , which induce actions of $\\mathfrak {m}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ via $\\eta $ .", "The reader might have noticed that a fourth possible condition on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ could have been considered in order to guarantee the existence of the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ from the existence of the homomorphism of Leibniz crossed modules $(\\varphi ,\\psi )$ .", "In fact, if $[\\mathfrak {n},\\mathfrak {n}] = \\mathfrak {n}$ and $\\operatorname{Ann}(\\mathfrak {q}) = 0$ , the problem with the last condition for the actions could have been solved in the same way.", "Nevertheless, this fourth condition does not guarantee that (ii) holds, as we will prove immediately below.", "Regarding (REF ) and (), they follow directly from () (observe that, by hypothesis, $((\\sigma ^{p}_1,\\theta ^{p}_1),(\\sigma ^{p}_2,\\theta ^{p}_2))$ is a biderivation of $(\\mathfrak {n},\\mathfrak {q},\\mu )$ for any $p \\in \\mathfrak {p}$ ).", "Similarly, (REF )–() follow almost immediately from ()–().", "Hence, (i) holds.", "Concerning (ii), we can define $\\xi _1(m,q)=D_{m}(q)$ and $\\xi _2(q,m)=-d_{m}(q)$ for any $m \\in \\mathfrak {m}$ , $q \\in \\mathfrak {q}$ .", "In this way, $\\xi _1$ and $\\xi _2$ are clearly bilinear.", "(REF ), (), () and () follow immediately from the identity (REF ) and the fact that the actions of $\\mathfrak {m}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ are induced by the actions of $\\mathfrak {p}$ via $\\eta $ .", "Identities (), () and () are a direct consequence of (REF )–() (recall that, by hypothesis, $(d_{m},D_{m})$ is a biderivation from $\\mathfrak {q}$ to $\\mathfrak {n}$ for any $m \\in \\mathfrak {m}$ ).", "Since $\\varphi $ is a Leibniz homomorphism, we have that $(d_{[m,m^{\\prime }]}, D_{[m,m^{\\prime }]}) = (d_{m} \\mu d_{m^{\\prime }} - d_{m^{\\prime }} \\mu d_{m}, D_{m} \\mu d_{m^{\\prime }} - d_{m^{\\prime }} \\mu D_{m}).$ This identity, together with () and (), allows to easily prove that () and () hold.", "Note that, since $(\\varphi ,\\psi )$ is a homomorphism of Leibniz crossed modules, $\\varphi ([p,m]) = [\\psi (p),\\varphi (m)]$ and $\\varphi ([m,p]) = [\\varphi (m),\\psi (p)]$ for all $m \\in \\mathfrak {m}$ , $p \\in \\mathfrak {p}$ .", "Due to the definition of the action of $\\operatorname{Bider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ on $\\operatorname{Bider}(\\mathfrak {q},\\mathfrak {n})$ (see Theorem REF ), we can write $(d_{[p,m]}, D_{[p,m]}) & = (\\sigma ^{p}_1 d_{m} - d_{m} \\sigma ^{p}_2, \\theta ^{p}_1 d_{m} - d_{m} \\theta ^{p}_2), \\\\(d_{[m,p]}, D_{[m,p]}) & = (d_{m} \\sigma ^{p}_2 - \\sigma ^{p}_1 d_{m}, D_{m} \\sigma ^{p}_2 - \\sigma ^{p}_1 D_{m}).$ Identities (), (), () and () follow immediately from the previous identities.", "Regarding () and (), directly from the definition of $\\xi _1$ , $\\xi _2$ and the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ , we have that $\\xi _1(m,[p,q]) & = D_{m} \\theta ^{p}_2 (q), \\qquad [p,\\xi _1(m,q)] = \\theta ^{p}_1 D_{m} (q), \\\\- \\xi _1(m,[q,p]) & = D_{m} \\sigma ^{p}_2 (q), \\qquad - [p,\\xi _2(q,m)] = \\theta ^{p}_1 d_{m} (q),$ for all $m \\in \\mathfrak {m}$ , $p\\in \\mathfrak {p}$ , $q \\in \\mathfrak {q}$ .", "Nevertheless, if at least one of the conditions (REF )–() holds, due to Lemma REF (ii), $D_{m} \\theta ^{p}_2 (q) = D_{m} \\sigma ^{p}_2 (q)$ and $\\theta ^{p}_1 D_{m} (q) = \\theta ^{p}_1 d_{m} (q)$ .", "Hence, (ii) holds.", "Remark 4.4 A closer look at the proof of the previous theorem shows that neither conditions () and (), nor the identities $[p,[p^{\\prime },n]] = [[p,p^{\\prime }],n]-[[p,n],p^{\\prime }]$ and $[p,[p^{\\prime },q]] = [[p,p^{\\prime }],q]-[[p,q],p^{\\prime }]$ (which correspond to the sixth axiom satisfied by the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ respectively) are necessary in order to prove the existence of a homomorphism of crossed modules $(\\varphi ,\\psi )$ from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , under the hypothesis that (i) and (ii) hold.", "Actually, if we remove those conditions from (i) and (ii), the converse statement would be true for any Leibniz crossed module $(\\mathfrak {n},\\mathfrak {q},\\mu )$ , even if it does not satisfy any of the conditions (REF )–().", "The problem is that () and (), together with the sixth identity satisfied by the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ are essential in order to prove that (i) and (ii) as in Theorem REF describe a set of actions of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ , as we will show immediately below.", "This agrees with the idea of $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ not being “good enough” to be the actor of $(\\mathfrak {n},\\mathfrak {q},\\mu )$ in general, just as $\\operatorname{Bider}(\\mathfrak {m})$ is not always the actor of a Leibniz algebra $\\mathfrak {m}$ .", "Example 4.5 Let $(\\mathfrak {m},\\mathfrak {p},\\eta ) \\in \\operatorname{\\textbf {\\textsf {XLb}}}$ , there is a homomorphism $(\\varphi ,\\psi ) \\colon (\\mathfrak {m},\\mathfrak {p},\\eta ) \\rightarrow \\overline{\\operatorname{Act}}(\\mathfrak {m},\\mathfrak {p},\\eta )$ , with $\\varphi (m) = (d_{m},D_{m})$ and $\\psi (p) = ((\\sigma ^{p}_1,\\theta ^{p}_1), (\\sigma ^{p}_2,\\theta ^{p}_2))$ , where $d_{m} (p) = - [p,m], \\qquad D_{m} (p) = [m,p],$ and $\\sigma ^{p}_1(m) & = - [m,p], \\qquad \\theta ^{p}_1(m) = [p,m], \\\\\\sigma ^{p}_2(p^{\\prime }) & = - [p^{\\prime },p], \\qquad \\theta ^{p}_2(p^{\\prime }) = [p,p^{\\prime }],$ for all $m \\in \\mathfrak {m}$ , $p,p^{\\prime } \\in \\mathfrak {p}$ .", "Calculations in order to prove that $(\\varphi ,\\psi )$ is indeed a homomorphism of Leibniz crossed modules are fairly straightforward.", "Of course, this homomorphism does not necessarily define a set of actions from which it is possible to construct the semidirect product.", "Theorem REF , along with the result immediately bellow, shows that if $(\\mathfrak {m},\\mathfrak {p},\\eta )$ satisfies at least one of the conditions (REF )–(), then the previous homomorphism does define an appropriate set of actions of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on itself.", "Let $(\\mathfrak {m},\\mathfrak {p},\\eta )$ and $(\\mathfrak {n},\\mathfrak {q},\\mu )$ be Leibniz crossed modules such that (i) and (ii) from Theorem REF hold.", "Therefore, there are Leibniz actions of $\\mathfrak {m}$ on $\\mathfrak {n}$ and of $\\mathfrak {p}$ on $\\mathfrak {q}$ , so it makes sense to consider the semidirect products of Leibniz algebras $\\mathfrak {n} \\rtimes \\mathfrak {m}$ and $\\mathfrak {q} \\rtimes \\mathfrak {p}$ .", "Furthermore, we have the following result.", "Theorem 4.6 There is an action of the Leibniz algebra $\\mathfrak {q}\\rtimes \\mathfrak {p}$ on the Leibniz algebra $\\mathfrak {n}\\rtimes \\mathfrak {m}$ , given by $[(q,p),(n,m)] = ([q,n] + [p,n] + \\xi _2(q,m),[p,m]), \\\\[(n,m),(q,p)] = ([n,q] + [n,p] + \\xi _1(m,q),[m,p]), $ for all $(q,p) \\in \\mathfrak {q} \\rtimes \\mathfrak {p}$ , $(n,m) \\in \\mathfrak {n} \\rtimes \\mathfrak {m}$ , with $\\xi _1$ and $\\xi _2$ as in Theorem REF .", "Moreover, the Leibniz homomorphism $(\\mu ,\\eta ) \\colon \\mathfrak {n}\\rtimes \\mathfrak {m}\\rightarrow \\mathfrak {q}\\rtimes \\mathfrak {p}$ , given by $(\\mu ,\\eta )(n,m)=(\\mu (n),\\eta (m)),$ for all $(n,m)\\in \\mathfrak {n} \\rtimes \\mathfrak {m}$ , together with the previous action, is a Leibniz crossed module.", "Identities (REF ) and () follow easily from the conditions satisfied by $(\\mathfrak {m},\\mathfrak {p},\\eta )$ and $(\\mathfrak {n},\\mathfrak {q},\\mu )$ (see Theorem REF ).", "Nevertheless, as an example, we show how to prove the third one.", "Calculations for the rest of the identities are similar.", "Let $(n,m),(n^{\\prime },m^{\\prime }) \\in \\mathfrak {n} \\rtimes \\mathfrak {m}$ and $(q,p) \\in \\mathfrak {q} \\rtimes \\mathfrak {p}$ .", "By routine calculations we get that $[(n,m),[(n^{\\prime },m^{\\prime }),(q,p)]] = (\\underbrace{[n,[n^{\\prime },q]]}_{(1)} \\underbrace{+ [n,[n^{\\prime },p]]}_{(2)} \\underbrace{+ [n,\\xi _1(m^{\\prime },q)]}_{(3)} \\underbrace{+ [m,[n^{\\prime },q]]}_{(4)} \\\\ \\underbrace{+ [m,[n^{\\prime },p]]}_{(5)} \\underbrace{+ [m,\\xi _1(m^{\\prime },q)]}_{(6)} \\underbrace{+ [n,[m^{\\prime },p]]}_{(7)}, \\underbrace{[m,[m^{\\prime },p]]}_{(8)}),\\\\[[(n,m),(n^{\\prime },m^{\\prime })],(q,p)] = (\\underbrace{[[n,n^{\\prime }],q]}_{(1^{\\prime })} \\underbrace{+ [[n,n^{\\prime }],p]}_{(2^{\\prime })} \\underbrace{+ [[n,m^{\\prime }],q]}_{(3^{\\prime })} \\underbrace{+ [[m,n^{\\prime }],q]}_{(4^{\\prime })} \\\\ \\underbrace{+ [[m,n^{\\prime }],p]}_{(5^{\\prime })} \\underbrace{+ \\xi _1([m,m^{\\prime }],q)}_{(6^{\\prime })} \\underbrace{+ [[n,m^{\\prime }],p]}_{(7^{\\prime })}, \\underbrace{[[m,m^{\\prime }],p]}_{(8^{\\prime })}),\\\\[[(n,m),(q,p)],(n^{\\prime },m^{\\prime })] = (\\underbrace{[[n,q],n^{\\prime }]}_{(1^{\\prime \\prime })} \\underbrace{+ [[n,p],n^{\\prime }]}_{(2^{\\prime \\prime })} \\underbrace{+ [[n,q],m^{\\prime }]}_{(3^{\\prime \\prime })} \\underbrace{+ [\\xi _1(m,q),n^{\\prime }]}_{(4^{\\prime \\prime })} \\\\ \\underbrace{+ [[m,p],n^{\\prime }]}_{(5^{\\prime \\prime })} \\underbrace{+ [\\xi _1(m,q),m^{\\prime }]}_{(6^{\\prime \\prime })} \\underbrace{+ [[n,p],m^{\\prime }]}_{(7^{\\prime \\prime })}, \\underbrace{[[m,p],m^{\\prime }]}_{(8^{\\prime \\prime })}).$ Let us show that $(i) = (i^{\\prime }) - (i^{\\prime \\prime })$ for $i=1,\\dots ,8$ .", "It is immediate for $i=1,2,8$ due to the action of $\\mathfrak {q}$ on $\\mathfrak {n}$ and the actions of $\\mathfrak {p}$ on $\\mathfrak {n}$ and $\\mathfrak {m}$ .", "For $i=5$ , the identity follows from the fact that the action of $\\mathfrak {m}$ on $\\mathfrak {n}$ is defined via $\\eta $ together with the equivariance of $\\eta $ .", "Namely, $[m,[n^{\\prime },p]] & = [\\eta (m),[n^{\\prime },p]] = [[\\eta (m),n^{\\prime }],p] - [[\\eta (m),p],n^{\\prime }] \\\\& = [[m,n^{\\prime }],p] - [\\eta ([m,p]),n^{\\prime }] = [[m,n^{\\prime }],p] - [[m,p],n^{\\prime }].$ The procedure is similar for $i=7$ .", "For $i=3$ , it is necessary to make use of the Peiffer identity of $\\mu $ , (), the definition of the action of $\\mathfrak {m}$ on $\\mathfrak {n}$ and $\\mathfrak {q}$ via $\\eta $ and (REF ): $[n,\\xi _1(m^{\\prime },q)] & = [n,\\mu \\xi _1(m^{\\prime },q)] = [n,[m^{\\prime },q]] = [n,[\\eta (m^{\\prime }),q]] \\\\& = [[n,\\eta (m^{\\prime })],q] - [[n,q],\\eta (m^{\\prime })] = [[n,m^{\\prime }],q] - [[n,q],m^{\\prime }].$ The conditions required in order to prove the identity for $i=4$ are the same used for $i=3$ except (REF ), which is replaced by ().", "Finally, for $i=6$ , due to () and the definition of the action of $\\mathfrak {m}$ on $\\mathfrak {n}$ via $\\eta $ , we know that $\\xi _1([m,m^{\\prime }],q) = [\\xi _1(m,q),m^{\\prime }] - [m,\\xi _2(q,m^{\\prime })] = [\\xi _1(m,q),m^{\\prime }] - [\\eta (m),\\xi _2(q,m^{\\prime })],$ but applying (), we get $\\xi _1([m,m^{\\prime }],q) = [\\xi _1(m,q),m^{\\prime }] + [\\eta (m),\\xi _1(m^{\\prime },q)] = [\\xi _1(m,q),m^{\\prime }] + [m,\\xi _1(m^{\\prime },q)],$ so $(6) = (6^{\\prime }) - (6^{\\prime \\prime })$ and the third identity holds.", "Note that () and () are necessary in order to check the fourth and fifth identities respectively.", "Checking that $(\\mu ,\\eta )$ is indeed a Leibniz homomorphism follows directly from the definition of the action of $\\mathfrak {m}$ on $\\mathfrak {n}$ via $\\eta $ together with the conditions (REF ) and ().", "Regarding the equivariance of $(\\mu ,\\eta )$ , given $(n,m)\\in \\mathfrak {n} \\rtimes \\mathfrak {m}$ and $(q,p) \\in \\mathfrak {q} \\rtimes \\mathfrak {p}$ , $(\\mu ,\\eta ) ([(q,p),(n,m)]) & = (\\mu ,\\eta ) ([q,n] + [p,n] + \\xi _2(q,m), [p,m])\\\\& = (\\mu ([q,n]) + \\mu ([p,n]) + \\mu \\xi _2(q,m), \\eta ([p,m]))\\\\& = ([q,\\mu (n)] + [p, \\mu (n)] + [q,m],[p,\\eta (m)])\\\\& = ([q,\\mu (n)] + [p, \\mu (n)] + [q,\\eta (m)],[p,\\eta (m)])\\\\& = [(q,p),(\\mu (n),\\eta (m))],$ due to the equivariance of $\\mu $ and $\\eta $ , (REF ), (REF ) and the definition of the action of $\\mathfrak {m}$ on $\\mathfrak {q}$ via $\\eta $ .", "Similarly, but using () and () instead of (REF ) and (REF ), it can be proved that $(\\mu ,\\eta ) ([(n,m),(q,p)]) = [(\\mu (n),\\eta (m)),(q,p)]$ .", "The Peiffer identity of $(\\mu ,\\eta )$ follows easily from the homonymous property of $\\mu $ and $\\eta $ , the definition of the action of $\\mathfrak {m}$ on $\\mathfrak {n}$ via $\\eta $ and the conditions () and ().", "Definition 4.7 The Leibniz crossed module $(\\mathfrak {n}\\rtimes \\mathfrak {m}, \\mathfrak {q}\\rtimes \\mathfrak {p},(\\mu , \\eta ))$ is called the semidirect product of the Leibniz crossed modules $(\\mathfrak {n},\\mathfrak {q},\\mu )$ and $(\\mathfrak {m},\\mathfrak {p},\\eta )$ .", "Note that the semidirect product determines an obvious split extension of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ by $(\\mathfrak {n},\\mathfrak {q},\\mu )$ ${(0,0,0) [r] & (\\mathfrak {n},\\mathfrak {q},\\mu ) [r] & (\\mathfrak {n}\\rtimes \\mathfrak {m}, \\mathfrak {q}\\rtimes \\mathfrak {p},(\\mu , \\eta )) @<0.7ex>[r] & (\\mathfrak {m},\\mathfrak {p},\\eta ) [r] @<0.7ex>[l] & (0,0,0)}$ Conversely, any split extension of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ by $(\\mathfrak {n},\\mathfrak {q},\\mu )$ is isomorphic to their semidirect product, where the action of $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ is induced by the splitting homomorphism.", "Remark 4.8 If $(\\mathfrak {m},\\mathfrak {p},\\eta )$ and $(\\mathfrak {n},\\mathfrak {q},\\mu )$ are Leibniz crossed modules and at least one of the following conditions holds, $\\operatorname{Ann}(\\mathfrak {n})=0=\\operatorname{Ann}(\\mathfrak {q})$ , $\\operatorname{Ann}(\\mathfrak {n})=0$ and $[\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}$ , $[\\mathfrak {n},\\mathfrak {n}] = \\mathfrak {n}$ and $[\\mathfrak {q},\\mathfrak {q}] = \\mathfrak {q}$ , an action of the crossed module $(\\mathfrak {m},\\mathfrak {p},\\eta )$ on $(\\mathfrak {n},\\mathfrak {q},\\mu )$ can be also defined as a homomorphism of Leibniz crossed modules from $(\\mathfrak {m},\\mathfrak {p},\\eta )$ to $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "In other words, under one of those conditions, $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\mu )$ is the actor of $(\\mathfrak {n},\\mathfrak {q},\\mu )$ and it can be denoted simply by $\\operatorname{Act}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "Example 4.9 (i) Let $\\mathfrak {n}$ be an ideal of a Leibniz algebra $\\mathfrak {q}$ and consider the crossed module $(\\mathfrak {n},\\mathfrak {q},\\iota )$ , where $\\iota $ is the inclusion.", "It is easy to check that $\\overline{\\operatorname{Act}}(\\mathfrak {n},\\mathfrak {q},\\iota ) = (X,Y,\\iota )$ , where $X$ is a Leibniz algebra isomorphic to $\\lbrace (d,D) \\in \\operatorname{Bider}(\\mathfrak {q}) \\, \| \\, d(q), D(q) \\in \\mathfrak {n} \\text{ for all } q \\in \\mathfrak {q} \\rbrace $ and $Y$ is a Leibniz algebra isomorphic to $\\lbrace (d,D) \\in \\operatorname{Bider}(\\mathfrak {q}) \\, \| \\, (d_{\|\\mathfrak {n}},D_{\|\\mathfrak {n}}) \\in \\operatorname{Bider}(\\mathfrak {n}) \\rbrace $ .", "(ii) Given a Leibniz algebra $\\mathfrak {q}$ , it can be regarded as a Leibniz crossed module in two obvious ways, $(0,\\mathfrak {q},0)$ and $(\\mathfrak {q},\\mathfrak {q},\\operatorname{id}_{\\mathfrak {q}})$ .", "As a particular case of the previous example, one can easily check that $\\overline{\\operatorname{Act}}(0,\\mathfrak {q},0)\\cong (0,\\operatorname{Bider}(\\mathfrak {q}),0)$ and $\\overline{\\operatorname{Act}}(\\mathfrak {q},\\mathfrak {q},\\operatorname{id}_{\\mathfrak {q}})\\cong (\\operatorname{Bider}(\\mathfrak {q}),\\operatorname{Bider}(\\mathfrak {q}),\\operatorname{id})$ .", "(iii) Every Lie crossed module $(\\mathbb {n},\\mathbb {q},\\mu )$ can be regarded as a Leibniz crossed module (see for instance [7]).", "Note that in this situation, both the multiplication and the action are antisymmetric.", "The actor of $(\\mathbb {n},\\mathbb {q},\\mu )$ is $(\\operatorname{Der}(\\mathbb {q},\\mathbb {n}), \\operatorname{Der}(\\mathbb {n},\\mathbb {q},\\mu ), \\Delta )$ , where $\\operatorname{Der}(\\mathbb {q},\\mathbb {n})$ is the Lie algebra of all derivations from $\\mathbb {q}$ to $\\mathbb {n}$ and $\\operatorname{Der}(\\mathbb {n},\\mathbb {q},\\mu )$ is the Lie algebra of derivations of the crossed module $(\\mathbb {n},\\mathbb {q},\\mu )$ (see [8] for the details).", "Given $(d,D) \\in \\operatorname{Bider}(\\mathbb {q},\\mathbb {n})$ , both $d$ and $D$ are elements in $\\operatorname{Der}(\\mathbb {q},\\mathbb {n})$ .", "Additionally, if we assume that at least one of the conditions from the previous lemma holds, then either $\\operatorname{Ann}(\\mathbb {n})=0$ or $[\\mathbb {q},\\mathbb {q}]=\\mathbb {q}$ .", "In this situation, one can easily derive from () that $\\operatorname{Bider}(\\mathbb {q},\\mathbb {n})=\\lbrace (d,d) \\ \| \\ d \\in \\operatorname{Der}(\\mathbb {q},\\mathbb {n})\\rbrace $ .", "Besides, the bracket in $\\operatorname{Bider}(\\mathbb {q},\\mathbb {n})$ becomes antisymmetric and, as a Lie algebra, it is isomorphic to $\\operatorname{Der}(\\mathbb {q},\\mathbb {n})$ .", "Similarly, $\\operatorname{Bider}(\\mathbb {n},\\mathbb {q},\\mu )$ is a Lie algebra isomorphic to $\\operatorname{Der}(\\mathbb {n},\\mathbb {q},\\mu )$ and $\\overline{\\operatorname{Act}}(\\mathbb {n},\\mathbb {q},\\mu )$ is a Lie crossed module isomorphic to $\\operatorname{Act}(\\mathbb {n},\\mathbb {q},\\mu )$ ." ], [ "Center of a Leibniz crossed module", "Let us assume in this section that $(\\mathfrak {n},\\mathfrak {q},\\mu )$ is a Leibniz crossed module that satisfies at least one of the conditions (REF )–().", "Denote by $\\operatorname{Z}(\\mathfrak {q})$ the center of the Leibniz algebra $\\mathfrak {q}$ , which in this case coincides with its annihilator (note that the center and the annihilator are not the same object in general).", "Consider the canonical homomorphism $(\\varphi ,\\psi )$ from $(\\mathfrak {n},\\mathfrak {q},\\mu )$ to $\\operatorname{Act}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , as in Example REF .", "It is easy to check that $\\operatorname{Ker}(\\varphi ) & = \\mathfrak {n}^{\\mathfrak {q}} \\quad \\text{and} \\quad \\operatorname{Ker}(\\psi ) = \\operatorname{st}_{\\mathfrak {q}}({\\mathfrak {n}}) \\cap \\operatorname{Z}(\\mathfrak {q}),$ where $\\mathfrak {n}^{\\mathfrak {q}} = \\lbrace n \\in \\mathfrak {n} \\, \| \\, [q,n] = [n,q] = 0, \\ \\text{for all} \\ q \\in \\mathfrak {q}\\rbrace $ and $\\operatorname{st}_{\\mathfrak {q}}({\\mathfrak {n}}) = \\lbrace q \\in \\mathfrak {q} \\, \| \\, [q,n] = [n,q] = 0, \\ \\text{for all} \\ n \\in \\mathfrak {n}\\rbrace $ .", "Therefore, the kernel of $(\\varphi ,\\psi )$ is the Leibniz crossed module $(\\mathfrak {n}^{\\mathfrak {q}}, \\operatorname{st}_{\\mathfrak {q}}({\\mathfrak {n}}) \\cap \\operatorname{Z}(\\mathfrak {q}),\\mu )$ .", "Thus, the kernel of $(\\varphi ,\\psi )$ coincides with the center of the crossed module $(\\mathfrak {n},\\mathfrak {q},\\mu )$ , as defined in the preliminary version of [1] for crossed modules in modified categories of interest.", "This definition of center agrees with the categorical notion of center by Huq [9] and confirms that our construction of the actor for a Leibniz crossed module is consistent.", "Example 5.1 Consider the crossed module $(\\mathfrak {n},\\mathfrak {q},\\iota )$ , where $\\mathfrak {n}$ is an ideal of $\\mathfrak {q}$ and $\\iota $ is the inclusion.", "Then, its center is given by the Leibniz crossed module $(\\mathfrak {n} \\cap \\operatorname{Z}(\\mathfrak {q}), \\operatorname{Z}(\\mathfrak {q}), \\iota )$ .", "In particular, the center of $(0,\\mathfrak {q},0)$ is $(0, \\operatorname{Z}(\\mathfrak {q}), 0)$ and the center of $(\\mathfrak {q},\\mathfrak {q},\\operatorname{id}_{\\mathfrak {q}})$ is $(\\operatorname{Z}(\\mathfrak {q}), \\operatorname{Z}(\\mathfrak {q}), \\operatorname{id})$ .", "By analogy to the definitions given for crossed modules of Lie algebras (see [8]), we can define the crossed module of inner biderivations of $(\\mathfrak {n},\\mathfrak {q},\\mu )$ , denoted by $\\operatorname{InnBider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , as $\\operatorname{Im}(\\varphi ,\\psi )$ , which is obviously an ideal.", "The crossed module of outer biderivations, denoted by $\\operatorname{OutBider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ , is the quotient of $\\operatorname{Act}(\\mathfrak {n},\\mathfrak {q},\\mu )$ by $\\operatorname{InnBider}(\\mathfrak {n},\\mathfrak {q},\\mu )$ .", "Let ${(0,0,0) [r] & (\\mathfrak {n},\\mathfrak {q},\\mu ) [r] & (\\mathfrak {n}^{\\prime },\\mathfrak {q}^{\\prime },\\mu ^{\\prime }) [r] & (\\mathfrak {n}^{\\prime \\prime },\\mathfrak {q}^{\\prime \\prime },\\mu ^{\\prime \\prime }) [r] & (0,0,0)}$ be a short exact sequence of crossed modules of Leibniz algebras.", "Then, there exists a homomorphism of Leibniz crossed modules $(\\alpha , \\beta ) \\colon (\\mathfrak {n}^{\\prime },\\mathfrak {q}^{\\prime },\\mu ^{\\prime }) \\rightarrow \\operatorname{Act}(\\mathfrak {n},\\mathfrak {q},\\mu )$ so that the following diagram is commutative: $@C=1cm{(0,0,0) [r] & (\\mathfrak {n},\\mathfrak {q},\\mu ) [r] [d] & (\\mathfrak {n}^{\\prime },\\mathfrak {q}^{\\prime },\\mu ^{\\prime }) [r] [d]^{(\\alpha ,\\beta )} & (\\mathfrak {n}^{\\prime \\prime },\\mathfrak {q}^{\\prime \\prime },\\mu ^{\\prime \\prime }) [r] [d] & (0,0,0) \\\\(0,0,0) [r] & \\operatorname{InnBider}(\\mathfrak {n},\\mathfrak {q},\\mu ) [r] & \\operatorname{Act}(\\mathfrak {n},\\mathfrak {q},\\mu ) [r] & \\operatorname{OutBider}(\\mathfrak {n},\\mathfrak {q},\\mu ) [r] & (0,0,0)}$ where $(\\alpha , \\beta )$ is defined as $\\alpha (n^{\\prime }) = (d_{n^{\\prime }},D_{n^{\\prime }})$ and $\\beta (q^{\\prime }) = ((\\sigma ^{q^{\\prime }}_1,\\theta ^{q^{\\prime }}_1), (\\sigma ^{q^{\\prime }}_2,\\theta ^{q^{\\prime }}_2))$ , with $d_{n^{\\prime }} (q) = - [q,n^{\\prime }], \\qquad D_{n^{\\prime }} (q) = [n^{\\prime },q],$ and $\\sigma ^{q^{\\prime }}_1(n) & = - [n,q^{\\prime }], \\qquad \\theta ^{q^{\\prime }}_1(n) = [q^{\\prime },n], \\\\\\sigma ^{q^{\\prime }}_2(q) & = - [q,q^{\\prime }], \\qquad \\theta ^{q^{\\prime }}_2(q) = [q^{\\prime },q],$ for all $n^{\\prime } \\in \\mathfrak {n}^{\\prime }$ , $q^{\\prime } \\in \\mathfrak {q}^{\\prime }$ , $n \\in \\mathfrak {n}$ , $q \\in \\mathfrak {q}$ ." ] ]	1606.04871
[ [ "Convex lattice polygonal lines with a constrained number of vertices" ], [ "Abstract A detailed combinatorial analysis of planar convex lattice polygonal lines is presented.", "This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained." ], [ "Introduction", "In 1979, Arnold considered the question of the number of equivalence classes of convex lattice polygons having a given integer as area (we say that two polygons having their vertices on $\\mathbb {Z}^2$ are equivalent if one is the image of the other by an automorphism of $\\mathbb {Z}^2$ ).", "Later, Vershik changed the constraint in this problem and raised the question of the number, and typical shape, of convex lattice polygons included in a large box $[-n,n]^2$ .", "The stepping stone in this problem lies in the understanding of the number and shape of polygonal lines having integer vertices, starting from the origin and forming a sequence of increasing slopes.", "In 1994, three different solutions to this problem were found by Bárány , Vershik and Sina .", "Namely, they proved that, when $n$ goes to infinity: The number of convex polygonal lines with vertices in $(\\mathbb {Z}\\cap [0,n])^2$ joining $(0,0)$ to $(n,n)$ is equal to $\\displaystyle \\exp (3(\\zeta (3)/\\zeta (2))^{1/3}\\,n^{2/3}+o(n^{2/3}))$ .", "The number of vertices constituting a typical line is equivalent to $(\\zeta (3)^2\\zeta (2))^{-1/3}\\,n^{2/3}$ .", "There is a limit shape for a typical convex polygonal line, which is an arc of a parabola.", "It turns out that these problems are related to an earlier family of works we shall discuss now.", "In 1926, Jarník found an asymptotic equivalent of the maximal number of integral points that can be interpolated by a convex curve of Euclidean length $n$ .", "He obtained also an explicit number-theoretic constant times $n^{2/3}$ .", "This article was at the origin of many works of Diophantine analysis, and we refer the reader to the papers of Schmidt and Bombieri and Pila for more recent results, discussions and open questions on this subject.", "One may slightly change Jarník's framework, and consider the set of integral points which are interpolated by the graph on $[0,n]$ of an increasing and strictly convex function satisfying $f(0)=0$ and $f(n)=n$ .", "In 1995, Acketa and Žunić proved the following box analog of Jarník's result: the largest number of vertices for an increasing convex polygonal line on $\\mathbb {Z}_+^2$ joining $(0,0)$ to $(n,m)$ is asymptotically equivalent to $3\\pi ^{-2/3}({nm})^{1/3}$ .", "They derived the asymptotic value of the maximal number of vertices for a lattice polygon included in a square.", "The nature of the results shows that these problems are related to both affine differential geometry and geometry of numbers.", "Indeed, the parabola found as limit shape coincides with the convex curve inside the square having the largest affine perimeter.", "Furthermore, the appearance of the values of the Riemann zeta function underlines the arithmetic aspects of the problem.", "One could show indeed, by using Valtr's formula , that if the lattice $\\mathbb {Z}^2$ was replaced by a Poisson Point Process having intensity one (which can be thought as the most isotropic “lattice” one can imagine), the constants $ (\\zeta ^2(3)\\zeta (2))^{-1/3} \\approx 0.749$ and $3(\\zeta (3)/\\zeta (2))^{1/3} \\approx 2.702$ would be merely raised respectively to 1 and 3 asymptotically almost surely.", "The link with number theory was made even more clear by the authors who proved in that Riemann's Hypothesis is actually equivalent to the fact that the remainder term $o(n^{2/3})$ in point (a) is $o(n^{1/6+\\varepsilon })$ for all $\\varepsilon >0$ .", "As we said above, various strategies have been considered for Vershik's problem.", "Bárány and Vershik use generating functions and an affine perimeter maximization problem.", "Later, Vershik and Zeitouni made result (c) more precise and general by proving a large deviation principle whose rate function involves this affine perimeter.", "Sina 's approach was very different.", "His proof is based on a statistical mechanical description of the problem.", "It was recently made fully rigorous and extended by Bogachev and Zarbaliev ." ], [ "Main results", "Our aim in this paper is to improve the three results (a),(b),(c) described above.", "In particular, we shall address the following natural extension of (c) which appears as an open question in Vershik's 1994 article: “Theorem 3.1 shows how the number of vertices of a typical polygonal line grows.", "However, one can consider some other fixed growth, say, $\\sqrt{n}$ , and look for the limit shapes for uniform distributions connected with this growth [...]\" One of our results is that, not only there still exists a limit shape when the number of vertices is constrained, but also the parabolic limit shape is actually universal for all growth rates.", "The following theorem is a consequence of Theorem REF of section and Theorem REF of section which concern respectively limit shape results for lines with many and few vertices.", "Theorem The Hausdorff distance between a random convex polygonal line on $(\\frac{1}{n}\\mathbb {Z}\\cap [0,1])^2$ joining $(0,0)$ to $(1,1)$ with at most $k$ vertices, and the arc of parabola $\\left\\lbrace (x,y) \\in [0,1]^2 \\mid \\sqrt{\\vphantom{1} y}\\,+ \\sqrt{\\vphantom{y}1-x} = 1\\right\\rbrace ,$ converges in probability to 0 when both $n$ and $k$ tend to infinity.", "The proof of this theorem requires a detailed combinatorial analysis of convex polygonal lines with a constrained number of vertices.", "This is the purpose of Theorem REF and Theorem REF which together complete results (a) and (b) in the following way: Theorem Let $p(n;k)$ denote the number of convex polygonal lines in $\\mathbb {Z}_+^2$ joining $(0,0)$ to $(n,n)$ and having $k$ vertices.", "There exist two functions ${\\bf c}$ and ${\\bf e}$ (which are explicitly computed in Theorem REF ) such that, for all $\\ell \\in (0,+\\infty )$ , if $k$ is asymptotically equivalent to ${\\bf c}(\\ell )\\,n^{2/3}$ , then $\\log p(n;k) \\sim {\\bf e}(\\ell )\\, n^{2/3}.$ If $k$ is asymptotically negligible compared to $n^{2/3}$ , then $p(n;k) = \\left(\\frac{n^2}{k^3}\\right)^{k+o(k)}.$ If $k$ is asymptotically negligible compared to $n^{1/2}(\\log n)^{-1/4}$ , then $p(n;k) \\sim \\frac{1}{k!", "}\\binom{n-1}{k-1}^2.$ Let us mention that the question of the number of vertices is reminiscent of other ones considered, for instance, by Erdős and Lehner , Arratia and Tavaré , or Vershik and Yakubovich who were studying combinatorial objects (integer partitions, permutations, polynomials over finite field, Young tableaux, etc.)", "having a specified number of summands (according to the setting, we call summands, cycles, irreducible divisors, etc.", ")." ], [ "Organization of the paper", "In section , we present the detailed combinatorial analysis in the case of many vertices $k \\gg \\log \|n\|$ .", "Following Sina 's approach, the method, borrowed from classical ideas of statistical physics, relies on the introduction of a grand canonical ensemble which endows the considered combinatorial object with a parametrized probability measure.", "Then, the strategy consists in calibrating the parameters of the probability in order to fit with the constraints one has to deal with.", "Namely, in our question, it turns out that one can add one parameter in Sina 's probability distribution that makes it possible to take into account, not only the location of the extreme point of the polygonal line but also the number of vertices it contains.", "In this model, we are able to establish a contour-integral representation of the logarithmic partition function in terms of Riemann's and Barnes' zeta functions.", "The residue analysis of this representation leads to precise estimates of this function as well as of its derivatives, which correspond to the moments of the random variables of interest such as the position of the terminal point and the number of vertices of the line.", "Using a local limit theorem, we finally obtain the asymptotic behavior of the number of lines having ${\\bf c}(\\ell )\\, (n_1n_2)^{1/3}$ vertices in terms of the polylogarithm functions $\\operatorname{Li}_1,\\operatorname{Li}_2,\\operatorname{Li}_3$ .", "We also obtain an asymptotic formula for the number of lines having a number $k$ of vertices satisfying $\\log \|n\| \\ll k \\ll \|n\|^{2/3}$ .", "In section , we derive results about the limit shape of lines having a fixed number of vertices $k \\gg \\log \|n\|$ , answering the question of Vershik in a wide range.", "In section , we extend the results about combinatorics and limit shape beyond $\\log \|n\|$ .", "The approach here is radically different and more elementary.", "It allows us to recover the results of sections and , up to $k \\ll \|n\|^{1/3}$ .", "It relies on the comparison with a continuous setting which has been studied by Bárány and Bárány, Rote, Steiger, Zhang .", "In section , we go back to Jarník's original problem.", "In addition to Jarník's result that we recover, we give the asymptotic number of lines, typical number of vertices, and limit shape, which is an arc of a circle, in this different framework.", "In section , we mix both types of conditions.", "The statistical physical method still applies and we obtain, for the convex lines joining $(0,0)$ to $(n,n)$ and having a given total length, a continuous family of convex limit shapes that interpolates the diagonal of the square and the two sides of the square, going through the above arc of parabola and arc of circle." ], [ "A one-to-one correspondence", "We start this paper by reminding the correspondence between finite convex polygonal lines issuing from 0 whose vertices define increasing sequences in both coordinates and finite distributions of multiplicities on the set of pairs of coprime positive integers.", "This correspondence is a discrete analogue of the Gauss-Minkowski transformation in convex geometry.", "More precisely, let $\\Pi $ denote the set of finite planar convex polygonal lines $\\Gamma $ issuing from 0 such that the vertices of $\\Gamma $ are points of the lattice $\\mathbb {Z}^2$ and the angle between each side of $\\Gamma $ and the horizontal axis is in the interval $[0,\\pi /2]$ .", "Now consider the set $\\mathbb {X}$ of all vectors $x = (x_1,x_2)$ whose coordinates are coprime positive integers including the pairs $(0,1)$ and $(1,0)$ .", "Jarník observed that the space $\\Pi $ admits a simple alternative description in terms of distributions of multiplicities on $\\mathbb {X}$ .", "Lemma The space $\\Pi $ is in one-to-one correspondence with the space $\\Omega $ of nonnegative integer-valued functions $x \\mapsto \\omega (x)$ on $\\mathbb {X}$ with finite support (that is $\\omega (x)\\ne 0$ for only finitely many $x \\in \\mathbb {X}$ ).", "The inverse map $\\Omega \\rightarrow \\Pi $ corresponds to the following simple construction: for a given multiplicity distribution $\\omega \\in \\Omega $ and for all $\\theta \\in [0,\\infty ]$ , let us define $X_i^\\theta (\\omega ) := \\sum _{\\begin{array}{c}(x_1,x_2) \\in \\mathbb {X}\\\\ x_2 \\le \\theta x_1\\end{array}} \\omega (x)\\cdot x_i, \\qquad i \\in \\lbrace 1,2\\rbrace .$ When $\\theta $ ranges over $[0,\\infty ]$ , the function $\\theta \\mapsto X^\\theta (\\omega ) = (X^\\theta _1(\\omega ),X^\\theta _2(\\omega ))$ takes a finite number of values which are points of the lattice quadrant $\\mathbb {Z}_+^2$ .", "These points are in convex position since we are adding vectors in increasing slope order.", "The convex polygonal curve $\\Gamma \\in \\Pi $ associated to $\\omega $ is simply the linear interpolation of these points starting from $ (0,0) $ ." ], [ "A detailed combinatorial analysis", "For every $n = (n_1,n_2) \\in \\mathbb {Z}_+^2$ and $k\\in \\mathbb {Z}_+$ , define $\\Pi (n; k)$ the subset of $\\Pi $ consisting of polygonal lines $\\Gamma \\in \\Pi $ with endpoint $n$ and having $k$ vertices, and denote by $p(n;k) := \\left\|\\Pi (n;k)\\right\|$ its cardinality.", "Before we can state our first theorem, let us recall that the polylogarithm $\\operatorname{Li}_s(z)$ is defined for all complex number $s$ with $\\Re (s) > 0$ and $\|z\| < 1$ by $\\operatorname{Li}_s(z) = \\sum _{k=1}^\\infty \\frac{z^k}{k^s} = \\frac{1}{\\Gamma (s)}\\int _0^\\infty \\frac{z t^{s-1}}{e^t - z} dt.$ The integral in the last term is a holomorphic function of $z$ in $\\mathbb {C}\\setminus [1,+\\infty )$ .", "We will work with this analytic continuation of $\\operatorname{Li}_s$ in the sequel.", "We now define ${\\bf c}(\\ell )$ and ${\\bf e}(\\ell )$ for all $\\ell \\in (0,+\\infty )$ by ${\\bf c}(\\ell ) =\\frac{\\ell }{1-\\ell }\\times \\frac{\\rm {Li}_2(1-\\ell )}{\\zeta (2)^{1/3}(\\zeta (3)-\\rm {Li}_3(1-\\ell ))^{2/3}},\\quad {\\bf e}(\\ell ) =3\\left(\\frac{\\zeta (3)-\\rm {Li}_3(1-\\ell )}{\\zeta (2)}\\right)^{1/3}- \\log (\\ell ){\\bf c}(\\ell ).$ The following statement indicates the asymptotic exponential behavior of $p(n;k)$ in the case of many vertices, that is to say, when $k$ is not too small with respect to $\|n\|$ .", "Theorem 1 Suppose that $\|n\|$ and $k$ tend to $+\\infty $ such that $n_1 \\asymp n_2$ and $\\log \|n\|$ is asymptotically negligible compared to $k$ .", "If there exists $\\ell \\in (0,+\\infty )$ such that $k \\sim {\\bf c}(\\ell )(n_1n_2)^{1/3}$ , then $\\log p(n;k) \\sim {\\bf e}(\\ell ) (n_1n_2)^{1/3}.$ If $k$ is asymptotically negligible compared to $(n_1n_2)^{1/3}$ , then $p(n;k) = \\left(\\frac{n_1n_2}{k^3}\\right)^{k+o(k)}.$ Figure: Distribution of the number of vertices of a random convex polygonal line.", "The point of maximal 𝐞{\\bf e}-coordinate corresponds to typical lines.", "The point of maximal 𝐜{\\bf c}-coordinate corresponds to lines with a maximal number of vertices.", "Note that the curve is not symmetric.Remark The function $\\ell \\mapsto {\\bf e}(\\ell )$ is maximal for $\\ell =1$ and the corresponding coefficients are ${\\bf c}(1)=\\dfrac{1}{(\\zeta (2)\\zeta (3)^2)^{1/3} },&{\\bf e}(1)=3\\left(\\dfrac{\\zeta (3)}{\\zeta (2)}\\right)^{1/3},$ which already recovers results (a) and (b).", "Remark As a byproduct of Theorem REF , one can deduce the asymptotic behavior of the maximal number $M(n)$ of integral points that an increasing convex function satisfying $f(0)=0$ and $f(n)=n$ can interpolate.", "This question and its counterpart, concerning the maximal convex lattice polygons inscribed in a convex set was solved by Acketa and Žunić who proved that $M_n \\sim 3\\pi ^{-2/3}\\,n^{2/3}$ .", "Starting from Theorem REF , the proof goes as follows.", "We first notice that ${\\bf e}(\\lambda )$ tends to 0 when $\\lambda $ goes to infinity.", "In the same time, $ {\\bf c}(\\lambda ) \\sim -\\zeta (2)^{-1/3}\\rm {Li}_2(1-\\lambda )(-\\rm {Li}_3(1-\\lambda ))^{-2/3}$ which tends to ${3}\\pi ^{-2/3}$ .", "Since ${\\bf e}(\\lambda )$ remain strictly positive, we get $\\liminf n^{-2/3}M(n) \\ge 3\\pi ^{-2/3}$ .", "Now, let $\\varepsilon >0$ and suppose $ \\limsup n^{-2/3} M(n) \\ge 3\\pi ^{-2/3}(1+2\\varepsilon )$ .", "Then, for arbitrary large $n$ , there is a polygonal line $\\Gamma \\in \\Pi (n,n)$ having at least $3\\pi ^{-2/3}\\,n^{2/3}(1+\\varepsilon )$ vertices.", "By choosing $k = 3\\pi ^{-2/3}\\,n^{2/3}$ vertices among the vertices of this line, we get already a subset of $\\Pi (n; k)$ whose cardinality is larger than $e^{cn^{2/3}}$ with $c > 0$ .", "This enters in contradiction with the fact that $\\lim _{\\lambda \\rightarrow \\infty } {\\bf e}(\\lambda ) = 0$ ." ], [ "Modification of Sina 's model and proof of Theorem ", "Recall from section the set $\\mathbb {X}= \\lbrace (x_1,x_2) \\in \\mathbb {Z}_+^2 \\mid \\gcd (x_1,x_2) = 1\\rbrace $ of primitive vectors and the set $\\Omega $ of functions $\\omega \\colon \\mathbb {X}\\rightarrow \\mathbb {Z}_+$ with finite support.", "The restriction of Jarník's correspondence to the subspace $\\Pi (n;k)$ induces a bijection with the subset $\\Omega (n;k)$ of $\\Omega $ consisting of multiplicity distributions $\\omega \\in \\Omega $ such that the “observables” $X_1(\\omega ) := \\sum _{x \\in \\mathbb {X}} \\omega (x)\\cdot x_1,\\quad X_2(\\omega ) := \\sum _{x \\in \\mathbb {X}} \\omega (x)\\cdot x_2,\\quad K(\\omega ) := \\sum _{x \\in \\mathbb {X}} \\mathbf {1}_{\\lbrace \\omega (x) > 0\\rbrace }$ are respectively equal to $n_1,n_2$ and $k$ .", "Notice that $X_1 = X_1^\\infty $ and $X_2 = X_2^\\infty $ with the previous notations.", "The random variables $X_1$ and $X_2$ correspond to the coordinates of the endpoint of the polygonal chain while $K$ counts its number of vertices.", "For all $\\lambda > 0$ and for every couple of parameters $\\beta = (\\beta _1,\\beta _2) \\in (0,+\\infty )^2$ , we endow $\\Omega $ with the probability measure defined for $\\omega \\in \\Omega $ by $\\mathbb {P}_{\\beta ,\\lambda }(\\omega ) & := \\frac{1}{Z(\\beta ,\\lambda )}\\exp \\left[- \\sum _{x \\in \\mathbb {X}} \\omega (x)\\, \\beta \\cdot x\\right] \\lambda ^{K(\\omega )}\\\\& = \\frac{1}{Z(\\beta ,\\lambda )} e^{-\\beta _1 X_1(\\omega )} e^{-\\beta _2 X_2(\\omega )} \\lambda ^{K(\\omega )},$ where the partition function $Z(\\beta ,\\lambda )$ is chosen as the normalization constant $Z(\\beta ,\\lambda )= \\sum _{n \\in \\mathbb {Z}_+^2} \\sum _{k\\ge 1} p(n ; k)\\, e^{-\\beta \\cdot n} \\lambda ^k.$ Note that $Z(\\beta ,\\lambda )$ is finite for all values of the parameters $(\\beta ,\\lambda ) \\in (0,+\\infty )^3$ .", "Indeed, if we denote by $p(n) = \\sum _{k\\ge 1} p(n;k)$ the total number of convex polygonal lines of $\\Pi $ with end point $n = (n_1,n_2)$ and $M_n$ the maximal number of edges of such a line, the following bound holds: $Z(\\beta ,\\lambda )\\le \\sum _{n \\in \\mathbb {Z}_+^2} p(n)\\, \\max (1,\\lambda )^{M_n} \\, e^{-\\beta \\cdot n}.$ We use now the results of baranylimit1995,vershiklimit1994,sinaiprobabilistic1994 according to which $\\log p(n) = O(\|n\|^{2/3})$ and of where Acketa and Žunić have proven that $M_n = O(\|n\|^{2/3})$ .", "We will use in the sequel the additional remark that $Z(\\beta ,\\lambda )$ is an analytic function of $\\lambda $ for all $\\beta > 0$ .", "The partition function $Z$ is of crucial interest since its partial logarithmic derivatives are equal to expectations of macroscopic characteristics of the polygonal line.", "Namely, the expected coordinates of the endpoint of the line are given by: $\\mathbb {E}_{\\beta ,\\lambda }[X_i]= \\sum _{n \\in \\mathbb {Z}_+^2} \\sum _{k\\ge 1} n_i\\frac{p(n ; k)\\, e^{-\\beta \\cdot n} \\lambda ^k}{Z(\\beta ,\\lambda )}=-\\frac{\\partial }{\\partial \\beta _i} \\log Z(\\beta ,\\lambda ), \\qquad i \\in \\lbrace 1,2\\rbrace .$ Similarly, for $i,j \\in \\lbrace 1,2\\rbrace $ , $\\mathbb {E}_{\\beta ,\\lambda }[K]=\\lambda \\frac{\\partial }{\\partial \\lambda } \\log Z(\\beta ,\\lambda ), \\qquad \\operatorname{Cov}_{\\beta ,\\lambda }(X_i,X_j)=\\frac{\\partial ^2}{\\partial \\beta _i\\partial \\beta _j} \\log Z(\\beta ,\\lambda ).$ Taking $\\lambda =1$ , the probability $\\mathbb {P}_{\\beta ,\\lambda }$ is nothing but the two-parameter probability distribution introduced by Sina .", "Under the measure $\\mathbb {P}_{\\beta ,\\lambda }$ , the variables $(\\omega (x))_{x\\in \\mathbb {X}}$ are still independent, as in Sina 's framework, but follow a geometric distribution only for $\\lambda =1$ .", "In the general case, the measure $\\mathbb {P}_{\\beta ,\\lambda }$ is absolutely continuous with respect to Sina 's measure with density proportional to $\\lambda ^{K(\\cdot )}$ and the distribution of $\\omega (x)$ is a biased geometric distribution.", "Loosely speaking, $\\mathbb {P}_{\\beta ,\\lambda }$ corresponds to the introduction of a penalty of the probability by a factor $\\lambda $ each time a vertex appears.", "Strictly speaking, it is only a penalty when $\\lambda < 1$ and a reward when $\\lambda > 1$ .", "Since $\\mathbb {P}_{\\beta ,\\lambda }(\\omega )$ depends only on the values of $X_1(\\omega )$ , $X_2(\\omega )$ , and $K(\\omega )$ , we deduce that the conditional distribution it induces on $\\Omega (n_1,n_2; k)$ is uniform.", "For instance, we have the following formula for all $(\\beta ,\\lambda ) \\in (0,+\\infty )^2\\times (0,+\\infty )$ which will be instrumental in the proof: $p(n_1,n_2;k)=Z(\\beta ,\\lambda )\\, e^{\\beta _1 n_1} e^{\\beta _2 n_2} \\lambda ^{-k}\\,\\mathbb {P}_{\\beta ,\\lambda }[X_1 = n_1, X_2 = n_2, K = k].$ In order to get a logarithmic equivalent of $ p(n_1,n_2;k)$ , our strategy is to choose the three parameters so that $\\mathbb {E}_{\\beta ,\\lambda }\\left[X_1\\right] = n_1,\\quad \\mathbb {E}_{\\beta ,\\lambda }\\left[X_2\\right] = n_2,\\quad \\mathbb {E}_{\\beta ,\\lambda }\\left[K\\right]=k.$ This will indeed lead to an asymptotic equivalent of $\\mathbb {P}_{\\beta ,\\lambda }[X_1 = n_1, X_2 = n_2, K = k]$ due to a local limit result.", "This equivalent having polynomial decay, it will not interfere with the estimation of $\\log p(n_1,n_2;k)$ .", "The analysis of the partition function in the next subsection leads to a calibration of the parameters $\\beta _1, \\beta _2, \\lambda $ satisfying the above conditions.", "Lemma REF shows that, for this calibration, $k\\sim {\\bf c}(\\lambda ) (n_1n_2)^{1/3}$ and $\\log Z(\\beta ,\\lambda )\\sim \\beta _1 n_1\\sim \\beta _2 n_2 \\sim \\left(\\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)}\\right)^{1/3}(n_1n_2)^{1/3}.$ Furthermore, Theorem REF implies that $\\log \\mathbb {P}_{\\beta ,\\lambda }[X_1 = n_1, X_2 = n_2, K = k]=O(\\log n)$ .", "So finally, Theorem REF follows readily by plugging these estimates into (REF ).", "Note that the case $k = o(\|n\|^{2/3})$ corresponds to $\\lambda $ going to 0, and the above asymptotics become, as stated in Theorem REF , $\\lambda \\sim \\frac{k^3}{n_1n_2}, \\qquad \\text{and}\\qquad \\log Z \\sim \\beta _1n_1 \\sim \\beta _2 n_2 \\sim k.$ As a consequence, the term $\\lambda ^{-k}$ dominates the asymptotic in (REF ), which concludes the proof." ], [ "Estimates of the logarithmic partition function and its derivatives", "We need in the following, the analogue to the Barnes bivariate zeta function defined for $\\beta =(\\beta _1,\\beta _2) \\in (0,+\\infty )^2$ by $\\zeta _2^(s ; \\beta ) := \\sum _{x \\in \\mathbb {X}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s},$ this series being convergent for $\\Re (s) > 2$ .", "The following preliminary lemma gives useful properties of this function.", "This will be done by expressing this function in terms of the Barnes zeta function $\\zeta _2(s,w ; \\beta )$ which is defined by analytic continuation of the series $\\zeta _2(s,w ; \\beta ) = \\sum _{n \\in \\mathbb {Z}_+^2} (w + \\beta _1 n_1 + \\beta _2 n_2)^{-s}, \\qquad \\Re (s) > 2, \\Re (w) > 0.$ It is well known that $\\zeta _2(s,w ; \\beta )$ has a meromorphic continuation to the complex $s$ -plane with simple poles at $s = 1$ and 2, and that the residue at $s = 2$ is simply $(\\beta _1\\beta _2)^{-1}$ .", "In the next lemma, we derive the relation between $\\zeta _2$ and $\\zeta _2^$ , and we also establish an explicit meromorphic continuation of $\\zeta _2$ to the half-plane $\\Re (s) > 1$ in order to obtain later polynomial bounds for $\|\\zeta _2^(s)\|$ as $\|\\Im (s)\| \\rightarrow +\\infty $ .", "Before the statement, let us recall that the fractional part $\\lbrace x\\rbrace \\in [0,1)$ of a real number $x \\in \\mathbb {R}$ is defined as $\\lbrace x\\rbrace = x - \\lfloor x \\rfloor $ .", "Lemma 1 The functions $\\zeta _2(s,w ; \\beta )$ and $\\zeta _2^(s ; \\beta )$ have a meromorphic continuation to the complex plane.", "The meromorphic continuation of $\\zeta _2(s,w ; \\beta )$ to the half-plane $\\Re (s) > 1$ is given by $\\zeta _2(s,w ;\\beta ) &= \\frac{1}{\\beta _1\\beta _2}\\frac{w^{-s+2}}{(s-1)(s-2)} + \\frac{(\\beta _1 + \\beta _2)w^{-s+1}}{2\\beta _1\\beta _2(s-1)} + \\frac{w^{-s}}{4}\\\\& - \\frac{\\beta _2}{\\beta _1} \\int _0^{+\\infty } \\frac{\\lbrace y\\rbrace -\\frac{1}{2}}{(w+\\beta _2 y)^s}\\,dy - \\frac{\\beta _1}{\\beta _2} \\int _0^{+\\infty } \\frac{\\lbrace x\\rbrace -\\frac{1}{2}}{(w+\\beta _1 x)^s}\\,dx\\\\& - s\\frac{\\beta _2}{2} \\int _0^{+\\infty } \\frac{\\lbrace y\\rbrace -\\frac{1}{2}}{(w+\\beta _2 y)^{s+1}}dy- s\\frac{\\beta _1}{2} \\int _0^{+\\infty } \\frac{\\lbrace x\\rbrace -\\frac{1}{2}}{(w+\\beta _1 x)^{s+1}}dx\\\\& + s(s+1)\\beta _1\\beta _2 \\int _0^{+\\infty }\\int _0^{+\\infty } \\frac{(\\lbrace x\\rbrace - \\frac{1}{2})(\\lbrace y\\rbrace -\\frac{1}{2})}{(w+\\beta _1 x +\\beta _2 y)^{s+2}}\\,dxdy.$ The meromorphic continuation of $\\zeta _2^(s;\\beta )$ is given for all $s \\in \\mathbb {C}$ by $\\zeta _2^(s;\\beta ) = \\frac{1}{\\beta _1^s} + \\frac{1}{\\beta _2^s} + \\frac{\\zeta _2(s,\\beta _1+\\beta _2 ; \\beta )}{\\zeta (s)}.$ We apply the Euler-Maclaurin formula to the partial summation defined by $F(x) = \\sum _{n_2\\ge 0} (w +\\beta _1 x + \\beta _2 n_2)^{-s}$ , leading to $\\sum _{n_1 \\ge 1} F(n_1) = \\int _0^{\\infty } F(x)\\,dx - \\frac{F(0)}{2} + \\int _0^\\infty (\\lbrace x\\rbrace -\\frac{1}{2})F^{\\prime }(x)\\,dx.$ We use again the Euler-Maclaurin formula for each of the summations in $n_2$ to obtain (i).", "In order to prove (ii), we express $\\zeta _2^(s;\\beta )$ in terms of $\\zeta _2(s,\\beta _1+\\beta _2;\\beta )$ for all $s$ with real part $\\Re (s) > 2$ .", "The result will follow from the analytic continuation principle.", "By definition of $\\zeta _2^(s;\\beta )$ , $\\zeta (s)\\left[\\zeta _2^(s;\\beta ) - \\frac{1}{\\beta _1^s} - \\frac{1}{\\beta _2^s}\\right] = \\left[\\sum _{d\\ge 1}\\frac{1}{d^s}\\right]\\left[\\sum _{\\begin{array}{c}x_1,x_2 \\ge 1\\\\ \\gcd (x_1,x_2)=1\\end{array}} \\frac{1}{(\\beta _1 x_1 + \\beta _2 x_2)^s}\\right]=\\sum _{x_1,x_2 \\ge 1} \\frac{1}{(\\beta _1 x_1 + \\beta _2 x_2)^s}.$ Now we make the connection between these zeta functions and the logarithmic partition function of our modified Sina's model.", "Lemma 2 Let $c > 2$ .", "For all parameters $(\\beta ,\\lambda ) \\in (0,+\\infty )^2 \\times (0,+\\infty )$ , $\\log Z(\\beta ,\\lambda )= \\frac{1}{2i\\pi } \\int _{c - i\\infty }^{c + i\\infty } (\\zeta (s+1) - \\operatorname{Li}_{s+1}(1-\\lambda ))\\zeta _2^(s;\\beta )\\Gamma (s)ds.$ Given the product form of the distribution $\\mathbb {P}_{\\beta ,\\lambda }$ , we see that the random variables $\\omega (x)$ for $x \\in \\mathbb {X}$ are mutually independent.", "Moreover, the marginal distribution of $\\omega (x)$ is a biased geometric distribution.", "It is absolutely continuous with respect to the geometric distribution of parameter $e^{-\\beta \\cdot x}$ with density proportional to $k \\mapsto \\lambda ^{1_{k > 0}}$ .", "In other words, for all $k \\in \\mathbb {Z}_+$ , $\\mathbb {P}_{\\beta ,\\lambda }[\\omega (x) = k] = Z_x(\\beta , \\lambda )^{-1} e^{-k \\beta \\cdot x} \\lambda ^{1_{k > 0}}$ where the normalization constant $Z_x(\\beta , \\lambda ) = 1 + \\lambda \\dfrac{e^{-\\beta \\cdot x}}{1-e^{-\\beta \\cdot x}}$ is easily computed.", "We can now deduce the following product formula for the partition function: $Z(\\beta ,\\lambda )= \\prod _{x\\in \\mathbb {X}} Z_x(\\beta , \\lambda ) = \\prod _{x\\in \\mathbb {X}} \\left(1 + \\lambda \\frac{e^{-\\beta \\cdot x}}{1-e^{-\\beta \\cdot x}}\\right).$ For now, we assume that $\\lambda \\in (0,1)$ .", "Taking the logarithm of the product above $\\log Z(\\beta ,\\lambda )& = \\sum _{x \\in {\\mathbb {X}}} \\log \\left(1 + \\lambda \\frac{e^{-\\beta \\cdot x}}{1-e^{-\\beta \\cdot x}}\\right) \\\\&= \\sum _{x\\in {\\mathbb {X}}} \\log (1-(1-\\lambda )e^{-\\beta \\cdot x}) - \\sum _{x\\in {\\mathbb {X}}} \\log (1-e^{-\\beta \\cdot x}) \\\\&= \\sum _{x\\in {\\mathbb {X}}} \\sum _{r \\ge 1} \\frac{1 - (1-\\lambda )^r}{r} e^{-r\\beta \\cdot x}.$ Now we use the fact that the Euler gamma function $\\Gamma (s)$ and the exponential function are related through Mellin's inversion formula $e^{-z} = \\frac{1}{2i\\pi } \\int _{c - i\\infty }^{c + i\\infty } \\Gamma (s) z^{-s} ds,$ for all $c > 0$ and $z \\in \\mathbb {C}$ with positive real part.", "Choosing $c > 2$ so that the series and the integral all converge and applying the Fubini theorem, this yields $\\log Z(\\beta ,\\lambda )& = \\frac{1}{2i\\pi } \\sum _{x\\in {\\mathbb {X}}} \\sum _{r\\ge 1} \\int _{c - i\\infty }^{c + i\\infty }\\frac{1-(1-\\lambda )^r}{r} r^{-s} (\\beta \\cdot x)^{-s} \\Gamma (s) ds \\\\& = \\frac{1}{2i\\pi } \\int _{c - i\\infty }^{c + i\\infty } (\\zeta (s+1) - \\operatorname{Li}_{s+1}(1-\\lambda ))\\zeta _2^(s;\\beta )\\Gamma (s)\\,ds.$ The lemma is proven for all $\\lambda \\in (0,1)$ .", "The extension to $\\lambda > 0$ will now result from analytic continuation.", "We already noticed that the left hand term is analytic in $\\lambda $ for all fixed $\\beta $ .", "Proving the analyticity of the right hand term requires only to justify the absolute convergence of the integral on the vertical line.", "From Lemma REF , we know that $\\zeta _2^(c + i\\tau ; \\beta )$ is polynomially bounded as $\|\\tau \|$ tends to infinity.", "Taking $s = c - 1 + i\\tau $ , successive integrations by parts of the formula $(\\zeta (s+1) - \\operatorname{Li}_{s+1} (1-\\lambda ))\\Gamma (s+1) = \\lambda \\int _0^\\infty \\frac{e^x x^s}{(e^x - 1)(e^x - 1 + \\lambda )} \\,dx$ show for all integer $N > 0$ , there exists a constant $C_N > 0$ such that, uniformly in $\\tau $ , $\\bigl \| (\\zeta (s+1) - \\operatorname{Li}_{s+1} (1-\\lambda ))\\Gamma (s+1) \\bigr \| \\le \\frac{ C_N\\lambda }{(1+\|\\tau \|)^N}.$ Finally, the next Lemma makes use of the contour integral representation of $\\log Z(\\beta ,\\lambda )$ to derive at the same time an asymptotic formula for each one of its derivatives.", "Lemma 3 Let $(p,q_1,q_2) \\in \\mathbb {Z}_+^3$ .", "For all $\\varepsilon > 0$ , there exists $C > 0$ such that $\\left\|\\left[\\lambda \\frac{\\partial }{\\partial \\lambda }\\right]^p\\left[\\frac{\\partial }{\\partial \\beta _1}\\right]^{q_1}\\left[\\frac{\\partial }{\\partial \\beta _2}\\right]^{q_2} \\left( \\log Z(\\beta ,\\lambda )- \\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)\\beta _1\\beta _2}\\right)\\right\| \\le \\frac{C\\,\\lambda }{\|\\beta \|^\\kappa }$ with $\\kappa = q_1 + q_2 + 1 + \\varepsilon $ , uniformly in the region $\\lbrace (\\beta ,\\lambda ) \\mid \\varepsilon < \\frac{\\beta _1}{\\beta _2} < \\frac{1}{\\varepsilon } \\text{ and } 0 < \\lambda < \\frac{1}{\\varepsilon }\\rbrace $ .", "Lemma REF provides an integral representation of the logarithmic partition function $\\log Z(\\beta ,\\lambda )$ .", "We will use the residue theorem to shift the contour of integration from the vertical line $\\Re (s) = 3$ to the line $\\Re (s) = 1 + \\varepsilon $ .", "Lemma REF shows that the function $M(s) := (\\zeta (s+1)-\\operatorname{Li}_{s+1}(1-\\lambda )))\\zeta _2^(s;\\beta )\\Gamma (s)$ is meromorphic in the strip $1 < \\Re (s) < 3$ with a single pole at $s=2$ , where the residue is given by $\\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)} \\cdot \\frac{1}{\\beta _1\\beta _2}$ From the inequality (REF ), Lemma REF and the fact that $\|\\zeta (s)\|$ has no zero with $\\Re (s) > 1$ , we see that $M(s)$ vanishes uniformly in $1 + \\varepsilon \\le \\Re (s) \\le 3$ when $\|\\Im (s)\|$ tends to $+ \\infty $ .", "By the residue theorem, $\\log Z(\\beta ,\\lambda )= \\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda ))}{\\zeta (2)\\beta _1\\beta _2}+ \\frac{1}{2i\\pi } \\int _{1 + \\varepsilon - i\\infty }^{1 + \\varepsilon + i\\infty } M(s)\\,ds.$ From the Leibniz rule applied in the formula of Lemma REF (i), we obtain directly the meromorphic continuation of $\\frac{\\partial ^{q_1}}{\\partial \\beta _1^{q_1}}\\frac{\\partial ^{q_2}}{\\partial \\beta _2^{q_2}}\\zeta _2(s,\\beta _1+\\beta _2;\\beta )$ in the half-plane $\\Re (s) > 1$ .", "We also obtain the existence of a constant $C > 0$ such that $\\left\|\\left[\\frac{\\partial }{\\partial \\beta _1}\\right]^{q_1}\\left[\\frac{\\partial }{\\partial \\beta _2}\\right]^{q_2}\\zeta _2(1 + \\varepsilon + i\\tau ,\\beta _1+\\beta _2;\\beta )\\right\| \\le \\frac{C\\, \|\\tau \|^{2 + q_1 + q_2}}{\|\\beta \|^\\kappa }$ with $\\kappa = q_1 + q_2 + 1 + \\varepsilon $ .", "A reasoning similar to the one we have used in order to derive (REF ) shows that for all integers $p$ and $N > 0$ , there exists a constant $C_{p,N}$ such that, uniformly in $\\tau $ , $\\left\| \\left[\\lambda \\frac{\\partial }{\\partial \\lambda }\\right]^p (\\zeta (s+1) - \\operatorname{Li}_{s+1} (1-\\lambda ))\\Gamma (s+1) \\right\| \\le \\frac{ C_{p,N}\\,\\lambda }{(1+\|\\tau \|)^N}.$ In order to differentiate both sides of equation (REF ) and permute the partial derivatives and the integral sign, we have to mention the fact that the Riemann zeta function is bounded from below on the line $\\Re (s) = 1 + \\varepsilon $ and that the derivatives of $\\operatorname{Li}_s(1-\\lambda )$ with respect to $\\lambda $ are all bounded.", "This also gives the announced bound on the error term." ], [ "Calibration of the shape parameters", "When governed by the Gibbs measure $\\mathbb {P}_{\\beta ,\\lambda }$ , the expected value of the random vector with components $X_1(\\omega ) = \\sum _{x\\in \\mathbb {X}} \\omega (x) x_1, \\quad X_2(\\omega ) = \\sum _{x\\in \\mathbb {X}} \\omega (x) x_2, \\quad K(\\omega ) = \\sum _{x\\in \\mathbb {X}} \\mathbf {1}_{\\lbrace \\omega (x) > 0\\rbrace },$ is simply given by the logarithmic derivatives of the partition function $Z(\\beta ,\\lambda )$ .", "Remember that we planned to choose $ \\lambda $ and $ \\beta _1, \\beta _2 $ as functions of $n = (n_1,n_2)$ an $k$ in order for the probability $\\mathbb {P}[X_1 = n_1, X_2 = n_2, K = k]$ to be maximal, which is equivalent to $\\mathbb {E}(X_1)=n_1$ , $\\mathbb {E}(X_2)=N_2$ and $\\mathbb {E}(K)=k$ .", "We address this question in the next lemma.", "Lemma 4 Assume that $n_1,n_2,k$ tend to infinity with $n_1 \\asymp n_2$ and $\|k\| = O(\|n\|^{2/3})$ .", "There exists a unique choice of $(\\beta _1,\\beta _2,\\lambda )$ as functions of $(n,k)$ such that $\\mathbb {E}_{\\beta ,\\lambda }[X_1] = n_1, \\quad \\mathbb {E}_{\\beta ,\\lambda }[X_2] = n_2, \\quad \\mathbb {E}_{\\beta ,\\lambda }[K] = k.$ Moreover, they satisfy $n_1 \\sim \\frac{\\zeta (3) - \\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)({\\beta _1})^2{\\beta _2}}, \\quad n_2 \\sim \\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2){\\beta _1}({\\beta _2})^2}, \\quad k \\sim -\\frac{\\lambda \\partial _\\lambda \\operatorname{Li}_3(1-\\lambda )}{\\zeta (2){\\beta _1}{\\beta _2}}.$ If $k = o(\|n\|^{2/3})$ , then $\\lambda $ goes to 0 and the above relations yield $\\beta _1 \\sim \\frac{k}{n_1}, \\quad \\beta _2 \\sim \\frac{k}{n_2}, \\quad \\lambda \\sim \\frac{k^3}{n_1n_2}.$ With the change of variable $\\lambda = e^{-\\gamma }$ , the existence and uniqueness of $(\\beta ,\\lambda )$ are equivalent to the fact that the function $f \\colon (\\beta _1,\\beta _2,\\gamma ) \\mapsto \\beta _1 n_1 + \\beta _2 n_2 + \\gamma k + \\log Z(\\beta , e^{-\\gamma })$ has a unique critical point in the open domain $D = (0,+\\infty )^2 \\times \\mathbb {R}$ .", "First observe that $f$ is smooth and strictly convex since its Hessian matrix is actually the covariance matrix of the random vector $(X_1,X_2,K)$ .", "In addition, from the very definition (REF ) of $Z(\\beta ,\\lambda )$ , we can see that $f$ converges to $+\\infty $ in the neighborhood of any point of the boundary of $D$ as well as when $\|\\beta _1\| + \|\\beta _2\| + \|\\gamma \|$ tends to $+\\infty $ .", "The function being continuous in $D$ , this implies the existence of a minimum, which by convexity is the unique critical point $(\\beta ^,\\gamma ^)$ of $f$ .", "From now on, we will be concerned and check along the proof that we stay in the regime $\\beta _1,\\beta _2\\rightarrow 0$ , $\\beta _1 \\asymp \\beta _2$ , and $\\gamma $ bounded from below.", "From Lemma REF , we can approximate $f$ by the simpler function $g \\colon (\\beta _1,\\beta _2,\\gamma ) \\mapsto \\beta _1 n_1 + \\beta _2 n_2 + \\gamma k + \\dfrac{\\zeta (3)-\\operatorname{Li}_3(1-e^{-\\gamma })}{\\beta _1\\beta _2}$ with $\|f(\\beta ,\\gamma ) - g(\\beta ,\\gamma )\| \\le \\dfrac{Ce^{-\\gamma }}{\|\\beta \|^{3/2}}$ for some constant $C > 0$ .", "The unique critical point $(\\tilde{\\beta },\\tilde{\\gamma })$ of $g$ satisfies $n_1 = \\frac{\\zeta (3) - \\operatorname{Li}_3(1-e^{-\\tilde{\\gamma }})}{\\zeta (2)(\\tilde{\\beta _1})^2\\tilde{\\beta _2}},\\quad n_2 = \\frac{\\zeta (3)-\\operatorname{Li}_3(1-e^{-\\tilde{\\gamma }})}{\\zeta (2)\\tilde{\\beta _1}(\\tilde{\\beta _2})^2}, \\quad k = -\\frac{e^{-\\tilde{\\gamma }}\\partial _\\lambda \\operatorname{Li}_3(1-e^{-\\tilde{\\gamma }})}{\\zeta (2)\\tilde{\\beta _1}\\tilde{\\beta _2}}.$ The goal now is to prove that $(\\beta ^,\\gamma ^)$ is close to $(\\tilde{\\beta },\\tilde{\\gamma })$ .", "To this aim, we find a convex neighborhood $C$ of $(\\tilde{\\beta },\\tilde{\\gamma })$ such that $g\|_{\\partial C} \\ge g(\\tilde{\\beta },\\tilde{\\gamma }) + \\frac{Ce^{-\\tilde{\\gamma }}}{\\tilde{\\beta _1}\\tilde{\\beta _2}}$ .", "In the neighborhood of $(\\tilde{\\beta },\\tilde{\\gamma })$ the expression of the Hessian matrix of $g$ yields $g(\\tilde{\\beta }_1 + t_1, \\tilde{\\beta }_2 + t_2, \\tilde{\\gamma } + u) \\ge g(\\tilde{\\beta }_1 , \\tilde{\\beta }_2 , \\tilde{\\gamma } ) + \\frac{\\tilde{C}e^{-\\tilde{\\gamma }} }{(\\tilde{\\beta }_1\\tilde{\\beta }_2)^2}(\\Vert t\\Vert ^2 + \\tilde{\\beta }_1\\tilde{\\beta }_2 \|u\|^2)$ .", "Therefore we need only take $C = [\\tilde{\\beta }_1 - C_1\\tilde{\\beta }_1^{5/4}, \\tilde{\\beta }_1 + C_1\\tilde{\\beta }_1^{5/4}] \\times [\\tilde{\\beta }_2 - C_2\\tilde{\\beta }_2^{5/4}, \\tilde{\\beta }_2 + C_2\\tilde{\\beta }_2^{5/4}] \\times [\\tilde{\\gamma } - C_3 \|\\beta \|^{1/4}, \\tilde{\\gamma } + C_3 \|\\beta \|^{1/4}].$ Therefore, $f\|_{\\partial C} > f(\\tilde{\\beta },\\tilde{\\gamma })$ .", "By convexity of $f$ and $C$ this implies $(\\beta ^,\\gamma ^) \\in C$ .", "Hence $\\beta _1^ \\sim \\tilde{\\beta }_1, \\quad \\beta _2^* \\sim \\tilde{\\beta }_2, \\quad e^{-\\gamma ^} \\sim e^{-\\tilde{\\gamma }},$ concluding the proof." ], [ "A local limit theorem", "In this section, we show that the random vector $(X_1,X_2,K)$ satisfies a local limit theorem when the parameters are calibrated as above.", "Let $\\Gamma _{\\beta ,\\lambda }$ be the covariance matrix under the measure $\\mathbb {P}_{\\beta ,\\lambda }$ of the random vector $(X_1,X_2, K)$ .", "Theorem 2 (Local limit theorem) Let us assume that $n_1,n_2,k$ tend to infinity such that $n_1 \\asymp n_2 \\asymp \|n\|$ , $\\log \|n\| = o(k)$ , and $k = O(\|n\|^{2/3})$ .", "For the choice of parameters made in Lemma REF , $\\mathbb {P}_{\\beta ,\\lambda }[X = n, K = k] \\sim \\frac{1}{(2\\pi )^{3/2}}\\frac{1}{\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}}.$ Moreover, $\\det \\Gamma _{\\beta ,\\lambda }\\asymp \\frac{\|n\|^4}{k}$ If $k = o(\|n\|^{2/3})$ , $\\mathbb {P}_{\\beta ,\\lambda }[X = n, K = k] \\sim \\frac{1}{(2\\pi )^{3/2}}\\frac{\\sqrt{k}}{n_1n_2}$ This result is actually an application of a more general lemma proven by the first author in .", "In order to state the lemma, we introduce some notations.", "Let $\\sigma _{\\beta ,\\lambda }^2$ be the smallest eigenvalue of $\\Gamma _{\\beta ,\\lambda }$ .", "Introducing $X_{1,x} = \\omega (x) \\cdot x_1$ , $X_{2,x} = \\omega (x) \\cdot x_2$ and $K_x = 1_{\\lbrace \\omega (x) > 0\\rbrace }$ as well as $\\overline{X_{1,x}} , \\overline{X_{2,x}},\\overline{K_x}$ their centered counterparts, let $L_{\\beta ,\\lambda }$ be the Lyapunov ratio $L_{\\beta ,\\lambda }:= \\sup _{(t_1,t_2,u)\\in \\mathbb {R}^3} \\sum _{x \\in \\mathbb {X}} \\frac{\\mathbb {E}_{\\beta ,\\lambda }\\left\|t_1 \\overline{X_{1,x}} + t_2 \\overline{X_{2,x}} + u\\overline{K_x}\\right\|^3}{\\Gamma _{\\beta ,\\lambda }(t_1,t_2,u)^{3/2}}.$ where $\\Gamma _{\\beta ,\\lambda }(\\cdot )$ stands for the quadratic form canonically associated to $\\Gamma _{\\beta ,\\lambda }$ .", "Let $ \\phi _{\\beta ,\\lambda }(t,u) = \\mathbb {E}_{\\beta ,\\lambda }(e^{i(t_1 X_1 + t_2 X_2 + uK})$ for all $(t_1,t_2,u) \\in \\mathbb {R}^3$ .", "Finally, we consider the ellipsoid $\\mathcal {E}_{\\beta ,\\lambda }$ defined by $\\mathcal {E}_{\\beta ,\\lambda } := \\left\\lbrace (t_1,t_2,u) \\in \\mathbb {R}^3 \\mid \\Gamma _{\\beta ,\\lambda }(t_1,t_2,u) \\le (4L_{\\beta ,\\lambda })^{-2}\\right\\rbrace .$ The following lemma is a reformulation of Proposition 7.1 in .", "It gives three conditions on the product distributions $\\mathbb {P}_{\\beta ,\\lambda }$ that entail a local limit theorem with given speed of convergence.", "Lemma 5 With the notations introduced above, suppose that there exists a family of number $(a_{\\beta ,\\lambda })$ such that $\\frac{1}{\\sigma _{\\beta ,\\lambda }\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}} = {O}(a_{\\beta ,\\lambda }), \\\\\\frac{L_{\\beta ,\\lambda }}{\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}} = {O}(a_{\\beta ,\\lambda }), \\\\\\sup _{(t,u) \\in [-\\pi ,\\pi ]^3\\setminus \\mathcal {E}_{\\beta ,\\lambda }} \\left\|\\phi _{\\beta ,\\lambda }(t,u)\\right\| = {O}(a_{\\beta ,\\lambda }).$ Then, a local limit theorem holds uniformly for $\\mathbb {P}_{\\beta ,\\lambda }$ with rate $a_{\\beta ,\\lambda }$ : $\\sup _{(n,k)\\in \\mathbb {Z}^3}\\; \\left\|\\mathbb {P}_{\\beta ,\\lambda }[X = n, K = k] - \\frac{\\exp \\left[-\\frac{1}{2}\\Gamma _{\\beta ,\\lambda }^{-1}\\bigl ((n,k) - \\mathbb {E}_{\\beta ,\\lambda }(X,K)\\bigr )\\right]}{(2\\pi )^{3/2}\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}}\\right\| = {O}(a_{\\beta ,\\lambda }).$ When governed by the Gibbs measure $\\mathbb {P}_{\\beta ,\\lambda }$ , the covariance matrix $\\Gamma _{\\beta ,\\lambda }$ of the random vector $(X_1,X_2,K)$ is simply given by the Hessian matrix of the log partition function $\\log Z(\\beta ,\\lambda )$ .", "Let $u(\\lambda ) := (\\zeta (3) - \\operatorname{Li}_3(1-\\lambda ))/\\zeta (2)$ for $\\lambda > 0$ .", "Applications of Lemma REF for all $(p,q_1,q_2) \\in \\mathbb {Z}_+^3$ such that $p+q_1+q_2 = 2$ imply that this covariance matrix is asymptotically equivalent to $\\begin{bmatrix}\\beta _1\\beta _2 & 0 & 0\\\\0 & \\beta _1^3 \\beta _2 & 0\\\\0 & 0 & \\beta _1\\beta _2^3\\end{bmatrix}^{-\\frac{1}{2}}\\begin{bmatrix}\\lambda ^2 u^{\\prime \\prime }(\\lambda ) + \\lambda u^{\\prime }(\\lambda ) &\\lambda u^{\\prime }(\\lambda ) &\\lambda u^{\\prime }(\\lambda ) \\\\\\end{bmatrix}\\lambda u^{\\prime }(\\lambda ) &2u(\\lambda ) &u(\\lambda ) \\\\$ u'() u() 2u() 12 0 0 0 13 2 0 0 0 123 -12.", "$A straightforward calculation shows that this matrix is positive definite for all $ > 0$.$ Lemma 6 The random vector $(X_1,X_2, K)$ has a covariance matrix $\\Gamma _{\\beta ,\\lambda }$ satisfying $\\Gamma _{\\beta ,\\lambda }(t,u) \\asymp \\frac{(n_1)^{5/3}}{(\\lambda n_2)^{1/3}} \|t_1\|^2 + \\frac{(n_2)^{5/3}}{(\\lambda n_1)^{1/3}} \|t_2\|^2 + (\\lambda n_1n_2)^{1/3} \|u\|^2, \\qquad \|n\| \\rightarrow +\\infty .$ All the coefficients of the previous matrix $u(\\lambda ), \\lambda u^{\\prime }(\\lambda ), \\lambda ^2 u^{\\prime \\prime }(\\lambda )$ are of order $\\lambda $ in the neighborhood of 0, and the determinant is equivalent to $\\lambda ^3$ .", "Therefore, the eigenvalues are also of order $\\lambda $ .", "The result follows from the fact that the values of $\\beta _1$ and $\\beta _2$ are given by (REF ) and that $\\zeta (3)-\\operatorname{Li}_3(1-\\lambda ) \\asymp \\zeta (2) \\lambda $ .", "Lemma 7 The Lyapunov coefficient satisfies $L_{\\beta ,\\lambda }= O(\\lambda ^{-1/6} \\vert n \\vert ^{-1/3})$ .", "Using Lemma REF , there exists a constant $C > 0$ such that $L_{\\beta ,\\lambda }\\le C \\sum _{x \\in \\mathbb {X}} \\left[\\frac{\\mathbb {E}_{\\beta ,\\lambda }\\vert \\overline{X_{1,x}} \\vert ^3 }{\\lambda ^{-1/2}}\\frac{n_2^{1/2}}{n_1^{5/2}} + \\frac{\\mathbb {E}_{\\beta ,\\lambda }\\vert \\overline{X_{2,x}} \\vert ^3 }{\\lambda ^{-1/2}}\\frac{n_1^{1/2}}{n_2^{5/2}} + \\frac{\\mathbb {E}_{\\beta ,\\lambda }\\vert \\overline{K_x} \\vert ^3 }{\\lambda ^{1/2}(n_1n_2)^{1/2}}\\right].$ Therefore, we need only prove that $\\sum _{x \\in \\mathbb {X}} \\mathbb {E}_{\\beta ,\\lambda }\\left\|\\overline{K_x} \\right\|^3 = O(\\vert n\\vert ^{2/3}), \\qquad \\sum _{x \\in \\mathbb {X}} \\mathbb {E}_{\\beta ,\\lambda }\\left\|\\overline{X_{i,x}} \\right\|^3 = O(\\vert n\\vert ^{5/3}).$ Notice that for a Bernoulli random variable $B(p)$ of parameter $p$ , one has $\\mathbb {E}[\\vert B(p) - p\\vert ^3] \\le 4 (\\mathbb {E}[B(p)^3] + p^3) \\le 8 p$ .", "This implies $\\sum _{x \\in \\mathbb {X}} \\mathbb {E}_{\\beta ,\\lambda }\\left\|\\overline{K_x} \\right\|^3 \\le \\sum _{x \\in \\mathbb {X}} \\frac{8 \\lambda e^{-\\beta \\cdot x}}{1 - (1-\\lambda ) e^{-\\beta \\cdot x}} \\le \\sum _{x \\in \\mathbb {X}} \\frac{8 \\lambda e^{-\\beta \\cdot x}}{1 - e^{-\\beta \\cdot x}} = O(\\frac{\\lambda }{\\beta _1\\beta _2}).$ Similarly, we obtain $ \\sum _{x \\in \\mathbb {X}} \\mathbb {E}_{\\beta ,\\lambda }\\left\|\\overline{X_{1,x}} \\right\|^3 = O(\\frac{\\lambda }{\\beta _1^4\\beta _2}),\\quad \\sum _{x \\in \\mathbb {X}} \\mathbb {E}_{\\beta ,\\lambda }\\left\|\\overline{X_{2,x}} \\right\|^3 = O(\\frac{\\lambda }{\\beta _1\\beta _2^4}).$ Lemma 8 Condition (REF ) of Lemma REF is satisfied.", "More precisely, $\\limsup _{\|n\| \\rightarrow +\\infty } \\quad \\sup _{(t,u) \\in [-\\pi ,\\pi ]^3\\setminus \\mathcal {E}_{\\beta ,\\lambda }}\\quad \\frac{1}{\\lambda ^{1/3} \|n\|^{2/3}} \\log \|\\phi _n(t,u)\| < 0.$ From Lemmas REF and REF , there exists a constant $c > 0$ depending on $\\lambda $ such that for all $n=(n_1,n_2)$ with $\|n\|$ large enough, $[-\\pi ,\\pi ]^3 \\setminus \\mathcal {E}_{\\lambda ,n} \\subset \\lbrace (t,u) \\in \\mathbb {R}^3 \\mid c < \|u\| \\le \\pi \\text{ or } c \\lambda ^{1/3} \|n\|^{-1/3} < \|t\|\\rbrace .$ The strategy of the proof is to deal separately with the cases $\|u\| > c$ and $\|t\| > c \\lambda ^{1/3} \|n\|^{-1/3}$ , which requires to find first adequate bounds for $\|\\phi _n(t,u)\|$ in both cases.", "For all $(t_1,t_2, u) \\in \\mathbb {R}^3$ and $x\\in \\mathbb {X}$ , let us write $t = (t_1,t_2)$ and $\\rho ^x = e^{-\\beta \\cdot x}$ .", "The “partial” characteristic function $\\phi _n^x(t,u) = \\mathbb {E}[e^{i(t\\cdot X_x + uK_x )}]$ is given by $\\phi _n^x(t,u) = \\left(1 + \\lambda e^{iu}\\dfrac{e^{it\\cdot x}\\rho ^x}{1- e^{it\\cdot x}\\rho ^x}\\right)\\left(1 + \\lambda \\dfrac{\\rho ^x}{1- \\rho ^x}\\right)^{-1},$ hence a straightforward calculation yields $\\left\|\\phi _n^x(t,u)\\right\|^2 & = 1 - \\frac{\\frac{4\\lambda \\rho ^x}{(1-(1-\\lambda )\\rho ^x)^2}\\left[\\frac{\\rho ^x(2+(\\lambda -2)\\rho ^x)}{(1-\\rho ^x)^2}\|\\sin (\\frac{t\\cdot x}{2})\|^2 + \|\\sin (\\frac{t\\cdot x + u}{2})\|^2 - \\rho ^x \|\\sin (\\frac{u}{2})\|^2 \\right]}{1+\\frac{4\\rho ^x}{(1-\\rho ^x)^2}\|\\sin (\\frac{t\\cdot x}{2})\|^2}\\\\& \\le \\exp \\left\\lbrace - \\frac{\\frac{4\\lambda \\rho ^x}{(1-(1-\\lambda )\\rho ^x)^2}\\bigl (2\\rho ^x\|\\sin (\\frac{t\\cdot x}{2})\|^2 + \|\\sin (\\frac{t\\cdot x + u}{2})\|^2 - \\rho ^x \|\\sin (\\frac{u}{2})\|^2 \\bigr )}{1+\\frac{4\\rho ^x}{(1-\\rho ^x)^2}\|\\sin (\\frac{t\\cdot x}{2})\|^2}\\right\\rbrace $ Using the law of sines in a triangle with angles $\\frac{t\\cdot x}{2}$ , $\\frac{u}{2}$ and $\\frac{2\\pi - t\\cdot x+u}{2}$ , we see that the numerator inside the bracket is proportional (with positive constant) to $2\\rho ^x \\Vert a\\Vert ^2 + \\Vert {b}\\Vert ^2 - \\rho ^x \\Vert {a} + {b}\\Vert ^2$ where $a$ and $b$ are two-dimensional vectors.", "Since the real quadratic form $(a_i,b_i) \\mapsto 2\\rho \\, a_i^2 + b_i^2 - \\frac{2\\rho }{1+2\\rho } \\,(a_i + b_i)^2$ is positive for all $\\rho \\in (0,1)$ and for $i \\in \\lbrace 1,2\\rbrace $ , we deduce that $\|\\phi _n^x(t,u)\| \\le \\exp \\left\\lbrace - \\frac{\\frac{2\\lambda \\rho ^x}{(1-(1-\\lambda )\\rho ^x)^2}}{1+\\frac{4\\rho ^x}{(1-\\rho ^x)^2}}\\left(\\frac{2\\rho ^x}{1+2\\rho ^x}-\\rho ^x\\right)\\left\|\\sin (\\tfrac{u}{2})\\right\|^2 \\right\\rbrace $ for all $x$ such that $\\rho _x \\le \\frac{1}{2}$ .", "In the same way, the positivity of the quadratic form $(a_i,b_i) \\mapsto \\frac{\\rho }{1-\\rho }\\, a_i^2 + b_i^2 - \\rho \\, (a_i+b_i)^2$ yields $\|\\phi _n^x(t,u)\| \\le \\exp \\left\\lbrace - \\frac{\\frac{2\\lambda \\rho ^x}{(1-(1-\\lambda )\\rho ^x)^2}}{1+\\frac{4\\rho ^x}{(1-\\rho ^x)^2}}\\left(2\\rho ^x-\\frac{\\rho ^x}{1-\\rho ^x}\\right)\\left\|\\sin (\\tfrac{t\\cdot x}{2})\\right\|^2 \\right\\rbrace $ for all $x$ such that $\\rho _x \\le \\frac{1}{2}$ .", "Let us begin with the region $\\lbrace (t,u) \\in \\mathbb {R}^3 \\mid c < \|u\| \\le \\pi \\rbrace $ .", "In this case $\|\\sin (\\tfrac{u}{2})\|$ is uniformly bounded from below by $\|\\sin (\\tfrac{c}{2})\|$ .", "Hence using (REF ) for the $x \\in \\mathbb {X}$ such that $\\frac{1}{4} < \\rho ^x \\le \\frac{1}{3}$ and the bound $\|\\phi _n^x(t,u)\| \\le 1$ for all other $x$ , we obtain $\\log \|\\phi _n(t,u)\| \\le - \\frac{1}{160}\\frac{\\lambda \|\\sin (\\tfrac{c}{2})\|^2}{(1+\\frac{1}{3}\|\\lambda -1\|)^2}\\left\|\\left\\lbrace x \\in \\mathbb {X}\\mid \\frac{1}{4} < \\rho ^x \\le \\frac{1}{3}\\right\\rbrace \\right\|.$ To conclude, let us recall that the number of integral points with coprime coordinates such that $\\frac{1}{4} < e^{-\\beta \\cdot x} \\le \\frac{1}{3}$ is asymptotically equal to $\\frac{1}{\\zeta (2)}\\frac{\\log (4/3)}{2\\beta _1\\beta _2} \\asymp \\lambda ^{-2/3} \|n\|^{2/3}$ .", "We now turn to the region $\\lbrace (t,u) \\in [-\\pi ,\\pi ]^3 \\mid c\\lambda ^{1/3} \|n\|^{-1/3} < \|t\|\\rbrace $ .", "Without loss of generality, we can assume $\|t_1\| > c^{\\prime } \\lambda ^{1/3} \|n\|^{-1/3}$ for some universal constant $c^{\\prime } \\in (0; c)$ .", "Using the inequality (REF ) for the elements $x \\in \\mathbb {X}$ such that $\\frac{1}{4} < \\rho ^x \\le \\frac{1}{3}$ and the bound $\|\\phi _n^x(t,u)\| \\le 1$ for all other $x$ , we obtain for all $\\varepsilon \\in (0,1)$ , $\\log \|\\phi _n(t,u)\| \\le - \\frac{\\varepsilon ^2}{64} \\frac{\\lambda }{(1 + \\frac{1}{3}\|\\lambda -1\|)^2} \\left\|\\left\\lbrace x \\in \\mathbb {X}\\mid \\frac{1}{4} < \\rho ^x \\le \\frac{1}{3} \\text{ and } \|\\sin (\\tfrac{t \\cdot x}{2})\| \\ge \\varepsilon \\right\\rbrace \\right\|.$ Since the number of $x \\in \\mathbb {X}$ such that $\\frac{1}{4} < e^{-\\beta \\cdot x} \\le \\frac{1}{3}$ is asymptotically equal to $\\frac{\\log (4/3)}{2\\zeta (2)\\beta _1\\beta _2}$ , it is enough to prove that we can find $\\varepsilon $ such that the set of vectors $x \\in \\mathbb {Z}_+^2$ with $\|\\sin (\\tfrac{t\\cdot x}{2})\| < \\varepsilon $ has density strictly smaller than $\\frac{1}{\\zeta (2)}$ in $\\lbrace x \\in \\mathbb {Z}_+^2 \\mid \\frac{1}{4} < \\rho ^x \\le \\frac{1}{3}\\rbrace $ .", "We split up this region according to horizontal lines, that is to say with $\\frac{t_2x_2}{2}$ constant.", "The set $\\lbrace x_1 \\in \\mathbb {R}\\mid \|\\sin (\\frac{t_2x_2}{2}+\\tfrac{t_1x_1}{2})\| < \\varepsilon \\rbrace $ is a periodic union of strips of period $\\tau _1 = \\frac{2\\pi }{t_1} \\ge 2$ and width bounded by $4\\varepsilon \\tau _1$ .", "Hence the number of $x_1 \\in \\mathbb {Z}_+$ satisfying this condition and lying in any bounded finite interval $I$ is at most $\\left(\\frac{\|I\|}{\\tau _1} + 2\\right)(4\\varepsilon \\tau _1+1)$ .", "Summing up the contributions of the horizontal lines, this shows the existence of some positive constant $C > 0$ independent of $\\varepsilon $ such that for all $\\varepsilon \\in (0,1)$ , the number of $x \\in \\mathbb {Z}_+^2$ satisfying both $\\frac{1}{4} < e^{-\\beta \\cdot x} \\le \\frac{1}{3}$ and $\|\\sin (\\tfrac{t\\cdot x}{2})\| < \\varepsilon $ is bounded by $(\\tfrac{1}{2} + C\\varepsilon ) \\frac{\\log (4/3)}{2\\beta _1\\beta _2} + C \|n\|^{1/3}\\log \|n\|.$ To achieve our goal, we can therefore choose $\\varepsilon = \\frac{1}{2C}(\\frac{1}{\\zeta (2)}-\\frac{1}{2}) > 0$ .", "We simply check that the hypotheses of Lemma REF are satisfied.", "From Lemma REF , we have $\\sigma _{\\beta ,\\lambda }^2 \\asymp k$ and $\\det (\\Gamma _{\\beta ,\\lambda }) \\asymp k^{-1}\|n\|^4$ , hence $\\frac{1}{\\sigma _{\\beta ,\\lambda }\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}} \\asymp \\frac{1}{\|n\|^2}.$ Using in addition Lemma REF , we have also $\\frac{L_{\\beta ,\\lambda }}{\\sqrt{\\det \\Gamma _{\\beta ,\\lambda }}} = O\\left(\\frac{1}{\|n\|^2}\\right).$ Finally, Lemma REF shows the existence of some constant $c > 0$ such that for all $(n,k)$ large enough, $\\sup _{(t,u) \\in [-\\pi ,\\pi ]^3\\setminus \\mathcal {E}_{\\beta ,\\lambda }}\\; \|\\phi _n(t,u)\| \\le e^{-c k}$ Since we have made the assumption $\\log \|n\| = o(k)$ , the quantity $e^{-ck}$ is also bounded from above by $\|n\|^{-2}$ .", "Therefore, all hypotheses of Lemma REF are satisfied.", "As a consequence, $\\mathbb {P}_{\\beta ,\\lambda }$ satisfies a local limit theorem with speed rate $a_{\\beta ,\\lambda } \\asymp \|n\|^{-2}$ ." ], [ "Limit shape", "We start by proving the existence of a limit shape in the modified Sina model, which is the aim of the next two lemmas.", "The natural normalization for the convex polygonal line is to divide each coordinate by the corresponding expectations for the final point.", "The first lemma shows that the arc of parabola is the limiting curve of the expectation of the random convex polygonal line $m_i^\\theta (\\beta ,\\lambda ) = \\mathbb {E}_{\\beta ,\\lambda }[X_i^\\theta ]$ for $i\\in \\lbrace 1,2\\rbrace , \\theta \\in [0,\\infty ]$ under the $\\mathbb {P}_{\\beta ,\\lambda }$ distribution.", "Lemma 9 Suppose that $\\beta _1$ and $\\beta _2$ tend to 0 such that $\\beta _1 \\asymp \\beta _2$ and $\\lambda $ is bounded from above.", "Then $\\lim _{\|\\beta \| \\rightarrow 0} \\sup _{\\theta \\in [0,\\infty ]} \\left\|\\left[\\frac{m_1^\\theta (\\beta ,\\lambda )}{m_1^\\infty (\\beta ,\\lambda )},\\frac{m_2^\\theta (\\beta ,\\lambda )}{m_2^\\infty (\\beta ,\\lambda )}\\right] -\\left[\\frac{\\theta (\\theta + 2\\frac{\\beta _1}{\\beta _2})}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2},\\frac{\\theta ^2}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2}\\right] \\right\| = 0.$ Since we are dealing with continuous increasing functions, the uniform convergence convergence will follow from the simple convergence.", "We mimic the proof of Lemma REF , except that the domain of summation $\\mathbb {X}$ is replaced by the subset of vectors $x$ such that $x_2 \\le \\theta x_1$ .", "The expectations are given by the first derivatives of the partial logarithmic partition function $\\log Z^\\theta (\\beta , \\lambda ) = \\frac{1}{2i\\pi }\\int _{c-i\\infty }^{c+i\\infty } (\\zeta (s+1) - \\operatorname{Li}_{s+1}(1-\\lambda )) {\\zeta _2^{\\theta ,}}(s)\\Gamma (s)\\,ds$ where $\\zeta _2^{\\theta ,}$ is the restricted zeta function defined by analytic continuation of the series $\\zeta _2^{\\theta ,}(s) & = \\sum _{\\begin{array}{c}x \\in \\mathbb {X}\\\\ x_2 \\le \\theta x_1\\end{array}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s} \\\\& = \\frac{1}{\\beta _1^s} + \\frac{1_{\\lbrace \\theta = \\infty \\rbrace } }{\\beta _2^s} + \\frac{1}{\\zeta (s)}\\sum _{\\begin{array}{c}x_1,x_2 \\ge 1\\\\ x_2 \\le \\theta x_1\\end{array}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}.$ The continuation of the underlying restricted Barnes zeta function is obtained using the Euler-Maclaurin formula several times: $\\sum _{x_2 = 1}^{\\lfloor \\theta x_1 \\rfloor } (\\beta _1 x_1 + \\beta _2 x_2)^{-s} & =\\int _1^{\\lfloor \\theta x_1 \\rfloor } (\\beta _1 x_1 + \\beta _2 x_2)^{-s}\\,dx_2 + \\frac{(\\beta _1 x_1 + \\beta _2)^{-s}}{2} + \\frac{(\\beta _1 x_1 + \\beta _2 \\lfloor \\theta x_1 \\rfloor )^{-s}}{2} \\\\& \\qquad -s \\beta _2 \\int _1^{\\lfloor \\theta x_1 \\rfloor }(\\lbrace x_2\\rbrace - \\frac{1}{2}) (\\beta _1 x_1 + \\beta _2 x_2)^{-(s+1)}\\,dx_2\\\\& = \\int _1^{\\theta x_1} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}\\,dx_2 + \\frac{(\\beta _1 x_1 + \\beta _2)^{-s}}{2} + \\frac{(\\beta _1 x_1 + \\beta _2 \\lfloor \\theta x_1 \\rfloor )^{-s}}{2} \\\\& \\qquad -s \\beta _2 \\int _1^{\\lfloor \\theta x_1 \\rfloor }(\\lbrace x_2\\rbrace - \\frac{1}{2}) (\\beta _1 x_1 + \\beta _2 x_2)^{-(s+1)}\\,dx_2 \\\\& \\qquad - \\int _{\\lfloor \\theta x_1 \\rfloor }^{\\theta x_1} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}\\,dx_2\\\\& = \\frac{(\\beta _1 x_1 + \\beta _2)^{-s+1}}{\\beta _2(s-1)}- \\frac{(\\beta _1 x_1 + \\beta _2 \\theta x_1)^{-s+1}}{\\beta _2(s-1)} + R(s,x_1,\\beta _1,\\beta _2,\\theta )$ where $R(s,x_1,\\beta _1,\\beta _2,\\theta ) & = \\frac{(\\beta _1 x_1 + \\beta _2)^{-s}}{2} + \\frac{(\\beta _1 x_1 + \\beta _2 \\lfloor \\theta x_1 \\rfloor )^{-s}}{2} \\\\& \\qquad -s \\beta _2 \\int _1^{\\lfloor \\theta x_1 \\rfloor }(\\lbrace x_2\\rbrace - \\frac{1}{2}) (\\beta _1 x_1 + \\beta _2 x_2)^{-(s+1)}\\,dx_2 \\\\& \\qquad - \\int _{\\lfloor \\theta x_1 \\rfloor }^{\\theta x_1} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}\\,dx_2\\\\$ is such that $\\sum _{x_1 \\ge 1} R(s,x_1,\\beta _1,\\beta _2,\\theta )$ converges absolutely for all $s$ with $\\Re (s) > 1$ .", "Therefore the latter series defines a holomorphic function in the half-plane $\\Re (s) > 1$ .", "Finally, $\\sum _{\\begin{array}{c}x_1,x_2 \\ge 1\\\\ x_2 \\le \\theta x_1\\end{array}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}& = \\frac{(\\beta _1+\\beta _2)^{-s + 2}}{\\beta _1\\beta _2(s-1)(s-2)} - \\frac{(\\beta _1+\\theta \\beta _2)^{-s+2}}{(\\beta _1+\\theta \\beta _2)\\beta _2(s-1)(s-2)}\\\\& \\qquad + \\widetilde{R}(s,\\beta _1,\\beta _2,\\theta )$ where $\\widetilde{R}$ is holomorphic in $s$ for $\\Re (s) > 1$ .", "Hence, the residue at $s = 2$ is $\\frac{\\theta }{\\beta _1(\\beta _1 + \\theta \\beta _2)}$ .", "Taking the derivatives with respect to $\\beta _1$ and $\\beta _2$ , we obtain, $- \\frac{\\partial }{\\partial \\beta _1}\\sum _{\\begin{array}{c}x_1,x_2 \\ge 1\\\\ x_2 \\le \\theta x_1\\end{array}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}& = \\frac{1}{\\beta _1^2\\beta _2}\\frac{\\theta (\\theta + 2 \\frac{\\beta _1}{\\beta _2})}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2} \\frac{1}{s-2} + R_1(s,\\beta _1,\\beta _2,\\theta )$ and similarly $- \\frac{\\partial }{\\partial \\beta _2}\\sum _{\\begin{array}{c}x_1,x_2 \\ge 1\\\\ x_2 \\le \\theta x_1\\end{array}} (\\beta _1 x_1 + \\beta _2 x_2)^{-s}& = \\frac{1}{\\beta _1 \\beta _2^2} \\frac{\\theta ^2}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2} \\frac{1}{s-2} + R_2(s,\\beta _1,\\beta _2,\\theta )$ where both remainder terms $R_1$ and $R_2$ are holomorphic in $s$ in the half-plane $\\sigma := \\Re (s) > 1$ and are bounded, up to positive constants, by $\\frac{\|s\|^2}{\\sigma - 1} \\min (\\beta _1,\\beta _2)^{-\\sigma -1}.$ This decrease makes it possible to apply the residue theorem in order to shift to the left the vertical line of integration from $\\sigma = 3$ to $\\sigma = \\frac{3}{2}$ .", "When $\\beta _1$ and $\\beta _2$ tend to 0 and $\\frac{\\beta _1}{\\beta _2}$ tends to $\\ell $ , we thus find $\\mathbb {E}_{\\beta ,\\lambda }[X_1^\\theta ] &= \\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)}\\left[ \\frac{1}{\\beta _1^2\\beta _2} \\frac{\\theta (\\theta + 2\\frac{\\beta _1}{\\beta _2})}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2} + O\\left(\\frac{1}{\|\\beta \|^{5/2}}\\right)\\right],\\\\\\mathbb {E}_{\\beta ,\\lambda }[X_2^\\theta ] &= \\frac{\\zeta (3)-\\operatorname{Li}_3(1-\\lambda )}{\\zeta (2)} \\left[\\frac{1}{\\beta _1^2\\beta _2} \\frac{\\theta ^2}{(\\theta + \\frac{\\beta _1}{\\beta _2})^2} + O\\left(\\frac{1}{\|\\beta \|^{5/2}}\\right)\\right].$ We obtain the announced result by normalizing these quantities by their limits when $\\theta $ goes to infinity.", "Lemma 10 (Uniform exponential concentration) Suppose that $\\beta _1$ and $\\beta _2$ tend to 0 such that $\\beta _1 \\asymp \\beta _2$ and $\\lambda $ is bounded from above.", "For all $\\eta \\in (0,1)$ , we have $\\mathbb {P}_{\\beta ,\\lambda }\\left[\\sup _{1\\le i \\le 2}\\sup _{\\theta \\in [0,\\infty ]} \\frac{\|X^\\theta _i - m_i^\\theta (\\beta ,\\lambda )\|}{m_i^\\infty (\\beta ,\\lambda )} > \\eta \\right] \\le \\exp \\left\\lbrace -\\frac{c(\\lambda )\\eta ^2}{8\\beta _1\\beta _2}\\left(1 + o(1)\\right)\\right\\rbrace .$ Fix $i \\in \\lbrace 1,2\\rbrace $ and let $M_\\theta = X_i^\\theta - m_i^\\theta (\\beta ,\\lambda )$ for all $\\theta \\ge 0$ .", "The stochastic process $(M_\\theta )_{\\theta \\ge 0}$ is a $\\mathbb {P}_{\\beta ,\\lambda }$ -martingale, therefore $(e^{t M_\\theta })_{\\theta \\ge 0}$ is a positive $\\mathbb {P}_{\\beta ,\\lambda }$ -submartingale for any choice of $t \\ge 0$ such that $\\mathbb {E}_{\\beta ,\\lambda }[e^{tX_i}]$ is finite.", "This condition is satisfied when $t < \\beta _1$ .", "Doob's martingale inequality implies for all $\\eta > 0$ , $\\mathbb {P}_{\\beta ,\\lambda }\\left[\\sup _{\\theta \\in [0,\\infty ]} M_\\theta > \\eta \\,m_i^\\infty (\\beta ,\\lambda )\\right]& = \\mathbb {P}_{\\beta ,\\lambda }\\left[\\sup _{\\theta \\in [0,\\infty ]} e^{tM_\\theta }> e^{t\\eta m_i^\\infty (\\beta ,\\lambda )}\\right]\\\\& \\le e^{-t\\eta m_i^\\infty (\\beta ,\\lambda )} \\,\\mathbb {E}_{\\beta ,\\lambda }\\left[e^{tM_\\infty }\\right] = e^{-t(\\eta +1)m_i^\\infty (\\beta ,\\lambda )} \\,\\mathbb {E}_{\\beta ,\\lambda }[e^{tX_i}]$ For $i = 1$ , Lemma REF shows that the logarithm of the right-hand side satisfies $-t(1+\\eta ) m_1^\\infty (\\beta ,\\lambda ) + \\log \\frac{Z(\\beta _1 - t, \\beta _2 ; \\lambda )}{Z (\\beta _1,\\beta _2;\\lambda )} = \\frac{c(\\lambda )}{\\beta _1\\beta _2}\\left[-\\frac{t(1+\\eta )}{\\beta _1} - 1 + \\frac{\\beta _1}{\\beta _1-t} + o(1)\\right]$ asymptotically when $t$ and $\\beta _1$ are of the same order.", "The same holds for $i=2$ .", "This is roughly optimized for the choice $t = \\beta _i\\left(1-(1+\\eta )^{-1/2}\\right)$ , which gives $\\mathbb {P}_{\\beta ,\\lambda } \\left[\\sup _{\\theta \\in [0,\\infty ]} M_\\theta > \\eta \\, m_i^\\infty (\\beta ,\\lambda )\\right] \\le \\exp \\left\\lbrace -\\frac{2c(\\lambda )}{\\beta _1\\beta _2}\\left(1+\\frac{\\eta }{2}-\\sqrt{1+\\eta } + o(1)\\right)\\right\\rbrace .$ When considering the martingale defined by $N_\\theta = m_i^\\theta (\\beta ,\\lambda ) - X_i^\\theta $ , one obtains with the same method $\\mathbb {P}_{\\beta ,\\lambda }\\left[\\sup _{\\theta \\in [0,\\infty ]} N_\\theta > \\eta \\,m_i^\\infty (\\beta ,\\lambda )\\right] \\le \\exp \\left\\lbrace -\\frac{2c(\\lambda )}{\\beta _1\\beta _2}\\left(1 - \\frac{\\eta }{2} - \\sqrt{1-\\eta } + o(1)\\right)\\right\\rbrace .$ Since the previous inequalities hold for both $i\\in \\lbrace 1,2\\rbrace $ , a simple union bound now yields $\\mathbb {P}_{\\beta ,\\lambda }\\left[\\sup _{1\\le i \\le 2}\\sup _{\\theta \\in [0,\\infty ]} \\frac{\|X^\\theta _i - m_i^\\theta (\\beta ,\\lambda )\|}{m_i^\\infty (\\beta ,\\lambda )} > \\eta \\right] \\le 4 \\exp \\left\\lbrace -\\frac{c(\\lambda )\\eta ^2}{8\\beta _1\\beta _2}\\left(1+ o(1)\\right)\\right\\rbrace .$ We introduce the following parametrization of the arc of parabola $\\sqrt{y} + \\sqrt{1-x} = 1$ : $x_1(\\theta ) = \\frac{\\theta (\\theta +2)}{(\\theta + 1)^2}, \\quad x_2(\\theta ) = \\frac{\\theta ^2}{(\\theta + 1)^2}, \\qquad \\theta \\in [0,\\infty ].$ Theorem 3 (Limit shape for numerous vertices) Assume that $n_1 \\asymp n_2 \\rightarrow +\\infty $ , and $k = O(\|n\|^{2/3})$ , and $\\log \|n\| = o(k)$ .", "There exists $c > 0$ such that for all $\\eta \\in (0,1)$ , $\\mathbb {P}_{n,k}\\left[\\sup _{1\\le i \\le 2}\\sup _{\\theta \\in [0,\\infty ]} \\frac{\|X^\\theta _i - x_i(\\frac{\\beta _2}{\\beta _1}\\theta )\|}{n_i} > \\eta \\right] \\le \\exp \\left\\lbrace -c\\eta ^2 k\\left(1+ o(1)\\right)\\right\\rbrace .$ In particular, the Hausdorff distance between a random convex polygonal line on $\\frac{1}{n}\\mathbb {Z}_+^2$ joining $(0,0)$ to $(1,1)$ with at most $k$ vertices and the arc of parabola $\\sqrt{y}+ \\sqrt{1-x} = 1$ converges in probability to 0.", "Using the triangle inequality and Lemma REF , we need only prove the analogue of Lemma REF for the uniform probability $\\mathbb {P}_{n,k}$ .", "Remind that the measure $\\mathbb {P}_{\\beta ,\\lambda }$ conditional on the event $\\lbrace X=n,K=k\\rbrace $ is nothing but the uniform probability $\\mathbb {P}_{n,k}$ .", "Hence for all event $E$ , $\\mathbb {P}_{n,k}(E) \\le \\frac{\\mathbb {P}_{\\beta ,\\lambda }(E)}{\\mathbb {P}_{\\beta ,\\lambda }(X=n,K=k)}.$ Applying this with the deviation event above for the parameters $(\\beta ,\\lambda )$ defined in section REF and using the Local Limit Theorem REF as well as the concentration bound provided by Lemma REF , the right-hand side reads, up to constants, $\\frac{\|n\|^2}{\\sqrt{k}} \\,\\exp \\left\\lbrace -c\\eta ^2 k (1+o(1))\\right\\rbrace .$ Since $\\log \|n\| = o(k)$ , the result follows." ], [ "Combinatorial analysis", "The previous machinery does not apply in the case of very few vertices but it can be completed by an an elementary approach that we present now which will actually work up to a number of vertices negligible compared to $n^{1/3}$ .", "It is based on the following heuristics: when $n$ tends to $+\\infty $ and the number of edges $k$ is very small compared to $n$ , one can expect that choosing an element of $\\Pi (n;k)$ at random is somewhat similar to choosing $k-1$ vertices from $[0,1]^2$ in convex position at random.", "Bárány and Bárány, Rote, Steiger, Zhang proved by two different methods the existence of a parabolic limit shape in this continuous setting.", "These works are based on Valtr's observation that each convex polygonal line with $k$ edges is associated, by permutation of the edges, to exactly $k!$ increasing North-East polygonal lines with pairwise different slopes.", "Our first theorem is the convex polygonal line analogue to a result of Erdös and Lehner on integer partitions .", "Theorem 4 The number of convex polygonal lines joining $(0,0)$ to $(n,n)$ with $k$ edges satisfies $p(n;k) = \\frac{1}{k!", "}\\binom{n-1}{k-1}^2\\left(1+o(1)\\right),$ this formula being valid uniformly in $k$ for $k = o(n^{1/2}/(\\log n)^{1/4})$ .", "Let us start by proving an upper bound.", "This is done by considering the inequality $\\left\| \\Pi (n;k) \\right\| \\le \\frac{1}{k!", "}\\binom{n-1}{k-1}^2 + \\frac{2}{(k-1)!", "}\\binom{n-1}{k-2}\\binom{n-1}{k-1} + \\frac{1}{(k-2)!}", "\\binom{n-1}{k-2}^2$ where the first term bounds the number of convex polygonal lines which are associated to strictly North-East lines, the second term bounds the number of lines having either a first horizontal vector or a last vertical one, and the third term bounds the numbers of convex polygonal lines having both a horizontal and a vertical vector.", "We now turn to a lower bound.", "Let $\\lbrace U_1, U_2, \\dots , U_{k-1}\\rbrace $ and $\\lbrace V_1, V_2, \\dots , V_{k-1}\\rbrace $ be two independent and uniformly distributed random subsets of $\\lbrace 1,\\dots ,n-1\\rbrace $ of size $k-1$ whose elements are indexed in increasing order $U_1 < U_2 < \\cdots < U_{k-1}$ and $V_1 < V_2 < \\cdots < V_{k-1}$ .", "Let $M_0 = (0,0)$ , $M_k = (n,n)$ and $M_i = (U_i,V_i)$ for $1 \\le i \\le k-1$ .", "Obviously, the polygonal line $(M_0,M_1,\\dots ,M_n)$ has uniform distribution among all increasing polygonal line from $(0,0)$ to $(n,n)$ .", "We claim that the distribution of $(\\overrightarrow{M_{0}M_1},\\overrightarrow{M_1M_2},\\dots ,\\overrightarrow{M_{k-1}M_k})$ conditioned on the event that no two of these vectors are parallel is uniform among the lines of $\\Pi (n,k)$ such that no side is parallel to the $x$ -axis or the $y$ -axis.", "Moreover, since the vectors are exchangeable, the probability that we can find $i < j$ such that $\\overrightarrow{M_{i-1}M_i}$ and $\\overrightarrow{M_{j-1}M_j}$ are parallel is bounded from above by $\\binom{k}{2}$ times the probability that $Y = \\overrightarrow{M_0M_1}$ and $Z = \\overrightarrow{M_1M_2}$ are parallel.", "Using the simple estimate $\\binom{n-1}{k-1} \\ge \\frac{n^{k-1}}{(k-1)!", "}(1-o(1))$ which is asymptotically true since $k = o(\\sqrt{n})$ , we find that for all $(y,z) \\in (\\mathbb {N}^2)^2$ , the probability that $Y=y$ and $Z=y$ is $\\mathbb {P}(Y = y, Z=z) &= \\frac{\\binom{n-y_1-z_1}{k-3}\\binom{n-y_2-z_2}{k-3}}{\\binom{n-1}{k-1}^2}\\\\& \\le \\frac{4k^2}{n^2}\\left(1 - \\frac{y_1+z_1}{n}\\right)_+^{k-3}\\left(1-\\frac{y_2+z_2}{n}\\right)_+^{k-3}\\\\& \\le \\frac{4k^2}{n^2} \\exp \\left\\lbrace -\\frac{k-3}{n}\\left(y_1+y_2+z_1+z_2\\right)\\right\\rbrace .$ We can therefore dominate the probability that $Y$ and $Z$ are parallel by the probability that geometrically distributed random vectors are parallel, which is exactly estimated in the following lemma applied with $\\beta = \\frac{k}{n}$ .", "In conclusion, the probability that at least two vectors are parallel is bounded by $\\frac{k^4}{n^2}\\log (n)$ up to a constant.", "Lemma 11 Let $Y_1,Y_2,Z_1,Z_2$ be independent and identically distributed geometric random variables of parameter $1-e^{\\beta }$ with $\\beta > 0$ .", "When $\\beta $ goes to 0, the probability that the vectors $Y=(Y_1,Y_2)$ and $Z = (Z_1,Z_2)$ are parallel is asymptotically equal to $\\frac{\\beta ^2}{\\zeta (2)}\\log \\frac{1}{\\beta }.$ The probability that $Y$ and $Z$ are parallel is $\\sum _{x\\in \\mathbb {X}}\\sum _{i,j \\ge 1} \\mathbb {P}(Y = i\\,x, Z = j \\,x) = (1-e^{-\\beta })^4 \\sum _{x\\in \\mathbb {X}}\\sum _{i,j \\ge 1} e^{-\\beta (i+j)(x_1 + x_2)}.$ The Mellin transform of the double summation in the right-hand side with respect to $\\beta > 0$ is well-defined for all $s \\in \\mathbb {C}$ with $\\Re (s) > 2$ and it is equal to $\\sum _{x\\in \\mathbb {X}}\\sum _{i,j \\ge 1} \\frac{\\Gamma (s)}{(x_1+x_2)^s(i+j)^s} = \\frac{\\Gamma (s)}{\\zeta (s)}(\\zeta (s-1)-\\zeta (s))^2.$ Expanding this Mellin transform in Laurent series at the pole $s = 2$ of order 2 and using the residue theorem to express the Mellin inverse, one finds $\\sum _{x\\in \\mathbb {X}}\\sum _{i,j \\ge 1} e^{-\\beta (i+j)(x_1+x_2)} = \\frac{1}{\\zeta (2)}\\frac{\\log \\frac{1}{\\beta }}{\\beta ^2} - \\frac{C}{\\beta ^2} + O\\left(\\frac{1}{\\beta }\\right),\\qquad \\text{as }\\beta \\rightarrow 0.$ where $C = \\frac{2\\zeta (2) - \\zeta ^{\\prime }(2) - 1 - \\gamma }{\\zeta (2)} \\approx 0.471207$ ." ], [ "Limit shape", "Theorem 5 (Limit shape for few vertices) The Hausdorff distance between a random convex polygonal line in $(\\frac{1}{n}\\mathbb {Z}\\cap [0,1])^2$ joining $(0,0)$ to $(1,1)$ having at most $k$ vertices and the arc of parabola $\\sqrt{\\vphantom{x}y}+ \\sqrt{\\vphantom{y}1-x} = 1$ converges in probability to 0 when both $n$ and $k$ tend to $+\\infty $ with $k = o(n^{1/3})$ .", "Bárány and Bárány, Rote, Steiger, Zhang proved by two different methods the existence of a limit shape in the following continuous setting: if one picks at random $k-1$ points uniformly from the square $[0,1]^2$ , then conditional on the event that these points are in convex position, the Hausdorff distance between the convex polygonal line thus defined and the parabolic arc goes to 0 in probability as $k$ goes to $+\\infty $ .", "Our strategy is to show that this result can be extended to the discrete setting $([0,1] \\cap \\frac{1}{n}\\mathbb {Z})^2$ if $k$ is small enough compared to $n$ by using a natural embedding of the discrete model into the continuous model.", "For this purpose, we first observe that the distribution of the above continuous model can be described as follows: pick uniformly at random $k-1$ points from both the $x$ -axis and the $y$ -axis, rank them in increasing order and let $0 = U_0 < U_1 <U_2 < \\dots < U_{k-1} < U_k = 1$ and $0 = V_0 < V_1 < V_2 < \\dots < V_{k-1} < V_k = 1$ denote this ranking.", "The points $(U_i,V_i)$ define an increasing North-East polygonal line joining $(0,0)$ to $(1,1)$ .", "Reordering the segment lines of this line by increasing slope order, exchangeability arguments show that we obtain a convex line with $k$ edges that follows the desired distribution.", "This is analogous to the discrete construction of strictly North-East convex lines from $(0,0)$ to $(n,n)$ that occurs in the proof of Theorem REF .", "Now, we define the lattice-valued random variables $\\tilde{U}_0 \\le \\tilde{U}_1 \\le \\tilde{U}_2 \\le \\dots \\le \\tilde{U}_{k-1} \\le \\tilde{U}_{k}$ and $\\tilde{V}_0 \\le \\tilde{V}_1 \\le \\tilde{V}_2 \\le \\dots \\le \\tilde{V}_{k-1} \\le \\tilde{V}_k$ by discrete approximation: ${\\left\\lbrace \\begin{array}{ll}\\tilde{U}_i \\in \\frac{1}{n}\\mathbb {Z}, \\quad U_i \\le \\tilde{U}_i < U_i + \\frac{1}{n}\\\\\\tilde{V}_i \\in \\frac{1}{n}\\mathbb {Z}, \\quad V_i - \\frac{1}{n} < \\tilde{V}_i \\le V_i,\\end{array}\\right.", "}\\qquad \\text{for }1 \\le i \\le k-1.$ Remark that we still have $(\\tilde{U}_0,\\tilde{V}_0) = (0,0)$ and $(\\tilde{U}_k,\\tilde{V}_k) = (1,1)$ .", "Let $X_i = (U_i - U_{i-1} ,V_i - V_{i-1})$ and let $\\tilde{X}_i = (\\tilde{U}_i - \\tilde{U}_{i-1} ,\\tilde{V}_i - \\tilde{V}_{i-1})$ be the discrete approximation of $X_i$ for $1 \\le i \\le k$ .", "Conditional on the event that the slopes of $(X_1,\\dots ,X_k)$ and $(\\tilde{X}_1,\\dots , \\tilde{X}_k)$ are pairwise distinct and ranked in the same order, the Hausdorff distance between the associated convex polygonal lines is bounded by $\\frac{k}{n}$ , which goes asymptotically to 0.", "Since a direct application of shows that the distance between the convex line associated to $X$ and the parabolic arc converges to 0 in probability as $k$ tends to $+\\infty $ , we deduce that the Hausdorff distance between the convex line associated to $\\tilde{X}$ and the parabolic arc also converges in probability to 0 on this event.", "As in the proof of Theorem REF , the joint density of $(X_i,X_j)$ is dominated by the density of a couple of independent vectors whose coordinates are independent exponential variables with parameter $k$ .", "These vectors being of order of magnitude $\\frac{1}{k}$ , the order of the slopes of $(X_i,X_j)$ and $(\\tilde{X}_i,\\tilde{X}_j)$ may be reversed only if the angle between $X_i$ and $X_j$ is smaller than $\\frac{ck}{n}$ for some $c > 0$ , which happens with probability of order $\\frac{k}{n}$ .", "Consequently, the probability that there exists $i < j$ for which the slopes of $(X_i,X_j)$ and $(\\tilde{X}_i,\\tilde{X}_j)$ are ranked in opposite is bounded, up to a constant, by $\\binom{k}{2} \\frac{k}{n}$ .", "Therefore, the Hausdorff distance between the convex line associated to $\\tilde{X}$ and the parabolic arc also converges to 0 in probability if $k = o(n^{1/3})$ .", "The final step is to compare the distribution of the increasing reordering of $(\\tilde{X}_1,\\dots ,\\tilde{X}_k)$ with the uniform distribution on $\\Pi (n;k)$ .", "As a consequence of Theorem REF , the probability that a uniformly random element of $\\Pi (n;k)$ is strictly North-East tends to 1.", "The key point, which follows from Valtr's observation, is that the uniform distribution on strictly North-East convex lines with $k$ edges coincides with the distribution of the line obtained by reordering the vectors $(\\tilde{X}_1,\\dots ,\\tilde{X}_k)$ , conditional on the event that these vectors are pairwise linearly independent and strictly North-East.", "Since we showed in the previous paragraph that all the angles between two vectors of $(\\tilde{X}_1,\\dots ,\\tilde{X}_k)$ are at least $\\frac{ck}{n}$ with probability $1 - O(\\frac{k^3}{n})$ , the linear independence condition occurs with probability tending to 1.", "On the other hand, $(\\tilde{X}_1,\\dots ,\\tilde{X}_k)$ are strictly North-East with probability $1 - O(\\frac{k^2}{n})$ .", "Therefore, the event we conditioned on has a probability tending to 1, which proves that the total variation distance between the two distributions tends to 0." ], [ "Back to Jarník's problem", "In , Jarník gives an asymptotic formula of the maximum possible number of vertices of a convex lattice polygonal line having a total Euclidean length smaller than $n$ , and whose segments make an angle with the $x$ -axis between 0 and $\\frac{\\pi }{4}$ .", "What he finds is $\\frac{3}{2}\\,\\frac{n^{2/3}}{(2\\pi )^{1/3}}$ .", "If, in order to be closer to our setting, we ask the segments to make an angle with the $x$ -axis between 0 and $\\frac{\\pi }{2}$ , Jarník's formula is changed into $\\frac{3}{2}\\frac{n^{2/3}}{\\pi ^{1/3}}$ (which is twice the above result for $\\frac{n}{2}$ ).", "In this section, we want to present a detailed combinatorial analysis of this set of lines, which leads to Jarník's result as well as to the asymptotic of the typical number of vertices of such lines.", "It is the analog of Bárány, Sina and Vershik's result when the constraint concerns the total length.", "Let us first describe Jarník's argument, which is a good application of the correspondence described in section .", "It says the following: the function $\\omega $ realizing the maximum can be taken among the functions taking their values in $\\lbrace 0,1\\rbrace $ .", "Indeed, by changing the non-zero values of a function $\\nu $ into 1, one can obtain a polygonal line with the same number of vertices, but with a shorter length.", "Now, if the number of vertices $k$ is given, the convex line having minimal length, will be defined by the function $\\omega $ which associates 1 to the $k$ points of $\\mathbb {X}$ which are the closest to the origin.", "Since the set $X$ has an asymptotic density $\\frac{6}{\\pi ^2}$ , when $N$ is big, this set of points is asymptotically equivalent to the intersection of $X$ with the disc of center $O$ having radius $R$ satisfying $\\frac{6}{\\pi ^2}\\cdot \\frac{\\pi R^2}{4}=N$ i.e.", "$R=(\\frac{2\\pi }{3} N)^{1/2}$ .", "The total length of the line is equivalent to $L= \\int _0^R r\\times \\frac{6}{\\pi ^2}\\frac{\\pi }{2}r dr=\\frac{R^3}{\\pi }=\\frac{(\\frac{2\\pi }{3} N)^{3/2}}{\\pi }$ .", "This yields precisely $N=\\frac{3}{2} \\frac{L^{2/3}}{\\pi ^{1/3}}\\simeq 1.02\\, L^{2/3}$ .", "In order to get finer results, we introduce the probability distribution on the space $\\Omega $ proportional to $\\exp \\left(-\\beta \\sum _{x \\in \\mathbb {X}} \\omega (x) \\sqrt{\|x_1\|^2 + \|x_2\|^2}\\right) \\lambda ^{\\sum _{x\\in \\mathbb {X}} 1_{\\lbrace \\omega (x) > 0\\rbrace }}$ which depends on two parameters $\\beta ,\\lambda $ .", "In this set-up, the partition function turns out to be $Z = \\prod _{x\\in \\mathbb {X}} \\frac{1-(1-\\lambda )e^{-\\beta \\sqrt{\|x_1\|^2 + \|x_2\|^2}}}{1-e^{-\\beta \\sqrt{\|x_1\|^2+\|x_2\|^2}}}.$ The Mellin transform representation for $\\log Z$ now involves $\\frac{\\Gamma (s)(\\operatorname{Li}_{s+1}(1-\\lambda )-\\zeta (s+1))}{\\zeta (s)}\\sum _{x_1,x_2 \\ge 1} {(\|x_1\|^2+\|x_2\|^2)}^{-s/2}, \\qquad \\Re (s) > 2.$ The factors $\\zeta (s)^{-1}$ and $\\operatorname{Li}_{s+1}(1-\\lambda )-\\zeta (s+1)$ , which correspond respectively to the coprimality condition on the lattice and to the penalty of vertices, are still present.", "The main difference relies in the replacement of the Barnes zeta function by the Epstein zeta function which comes from the penalty by length in the model.", "With the help of the residue analysis of this Mellin transform and a local limit theorem, we obtain: Theorem 6 Let $p_J(n;k)$ denote the number of convex polygonal lines on $\\mathbb {Z}_+^2$ issuing from $(0,0)$ with $k$ vertices and length between $n$ and $n+1$ .", "As $n$ tends to $+\\infty $ , $\\text{if}\\qquad \\frac{k}{n^{2/3}} \\longrightarrow \\frac{\\pi ^{1/3}}{2}{\\bf c}(\\lambda ), \\qquad \\text{then}\\qquad \\frac{1}{n^{2/3}} \\log p_J(n;k) \\longrightarrow \\frac{\\pi ^{1/3}}{2}{\\bf e}(\\lambda ),$ where ${\\bf e}$ and ${\\bf c}$ are the functions introduced in Theorem REF .", "Moreover, the Hausdorff distance between a random element of this set normalized by $\\frac{1}{n}$ , and the arc of circle $\\lbrace (x,y) \\in [0,1]^2 \\mid x^2 + (y-1)^2 = 1\\rbrace $ converges to 0 in probability.", "From this result, we deduce that the typical number of vertices of such a line which is achieved for $\\lambda =1$ is asymptotically equal to $\\left(\\frac{3}{4\\pi \\zeta (3)^2}\\right)^{1/3} n^{2/3}.$ Similarly, the total number of convex lattice polygonal lines having length between $n$ and $n+1$ is asymptotically equal to $\\exp \\left(\\frac{3^{4/3}\\zeta (3)^{1/3}}{(4\\pi )^{1/3}}\\, n^{2/3} (1+o(1))\\right).$ In addition, we can derive Jarník's result in the lines of Remark ." ], [ "Mixing constraints and finding new limit shapes", "In this section we introduce a family of convex lattice polygonal line models which achieves a continuous interpolation of limit shapes between the diagonal of the square and the South-East corner sides of the square, passing through the arc of circle and the arc of parabola.", "Let $\\Vert \\cdot \\Vert _1$ and $\\Vert \\cdot \\Vert _2$ denote respectively the Taxicab norm and the Euclidean norm on $\\mathbb {R}^2$ .", "Recall that for all $x \\in \\mathbb {R}^2$ , $\\Vert x\\Vert _1 = \|x_1\| + \|x_2\| \\ge \\Vert x\\Vert _2 = \\sqrt{\|x_1\|^2+\|x_2\|^2} \\ge \\frac{1}{\\sqrt{2}}\\Vert x\\Vert _1.$ The Gibbs distribution we consider on the space $\\Omega $ involves both these norms in order to take into account both the extreme point of the line and its length: $\\frac{1}{Z}\\exp \\left(-\\beta \\sum _{x\\in \\mathbb {X}} \\omega (x) (\\Vert x\\Vert _1 + \\lambda \\sqrt{2} \\Vert x\\Vert _2)\\right), \\quad Z = \\prod _{x \\in \\mathbb {X}} \\left(1 - e^{-\\beta (\\Vert x\\Vert _1 + \\lambda \\sqrt{2}\\Vert x\\Vert _2)}\\right).$ This infinite product is convergent if $\\beta >0$ and $\\lambda > -\\frac{1}{\\sqrt{2}}$ or if $\\beta < 0$ and $\\lambda < -1$ .", "In both cases, the Mellin transform representation of $\\log Z$ involves $\\frac{\\Gamma (s)\\zeta (s+1)}{\\zeta (s)} \\sum _{x_1,x_2 \\ge 1} (\\Vert x\\Vert _1 + \\lambda \\sqrt{2} \\Vert x\\Vert _2)^{-s},\\qquad \\Re (s) > 2.$ As usual, the leading term of the expansion of $\\log Z$ when $\\beta \\rightarrow 0$ is obtained by computing the residue of this function at $s = 2$ .", "It turns out to be $\\frac{\\zeta (3)}{2\\zeta (2)} \\int _{-\\pi /4}^{\\pi /4} \\frac{d\\theta }{(\\lambda + \\cos (\\theta ))^2}.$ An application of the residue theorem shows that the expected length of the curve is asymptotically equivalent to $\\frac{1}{\\beta ^3}\\frac{\\zeta (3)}{\\sqrt{2}\\zeta (2)} \\int _{-\\pi /4}^{\\pi /4} \\frac{d\\theta }{(\\lambda + \\cos (\\theta ))^3}$ and that the coordinates of the ending point have asymptotic expected value $\\frac{1}{\\beta ^3}\\frac{\\zeta (3)}{2\\zeta (2)} \\int _{-\\pi /4}^{\\pi /4} \\frac{\\cos (\\theta ) d\\theta }{(\\lambda + \\cos (\\theta ))^3}.$ As in previous sections, a local limit theorem gives a correspondence between this Gibbs measure and the uniform distribution on a specific set of convex lines, namely the convex polygonal line with endpoint $(n,n)$ and total length belonging to $[L\\cdot n, L\\cdot n + 1]$ for some $L \\in ]\\sqrt{2}, 2[$ which is a function of $\\lambda $ , $L(\\lambda ) = \\sqrt{2} \\dfrac{\\int _0^{\\frac{\\pi }{4}}{\\frac{1}{(\\lambda +\\cos u)^3}}du}{\\int _0^{\\frac{\\pi }{4}}\\frac{\\cos u}{(\\lambda +\\cos u)^3}du}.$ By computations analogous to section , one can show that the uniform distribution on lines with length between $L(\\lambda )\\cdot n$ and $L(\\lambda ) \\cdot n+1$ concentrates around the curve described by the parametrization $x_\\lambda (\\phi )=\\sqrt{2}\\dfrac{\\int _{0}^\\phi {\\frac{\\cos u}{(\\lambda +\\cos (u-{\\frac{\\pi }{4}}))^3}}du}{\\int _{-\\pi /4}^{\\pi /4}{\\frac{\\cos u}{(\\lambda +\\cos u)^3}du}},\\quad y_\\lambda (\\phi )=\\sqrt{2}\\dfrac{\\int _{0}^\\phi {\\frac{\\sin u}{(\\lambda +\\cos (u-{\\frac{\\pi }{4}}))^3}}du}{\\int _{-\\pi /4}^{\\pi /4}{\\frac{\\cos u}{(\\lambda +\\cos u)^3}}du} \\quad (0\\le \\phi \\le {\\frac{\\pi }{2}}).$ The table provided in Figure REF summarizes the limit shapes that we obtain for some limit values of $\\lambda $ .", "See also Figure REF for a plot showing the interpolation of those limit shapes.", "Figure: Critical and special values in the spectrum of limit shapes for the model of convex lattice lines with mixed constraints.Figure: Limit shapes of different Euclidean lengths.Successively: black!502\\sqrt{2} (diagonal); red!801.481.48, blue!80π 2\\frac{\\pi }{2} (circle), 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[ [ "Critical slowing down and hyperuniformity on approach to jamming" ], [ "Abstract Hyperuniformity characterizes a state of matter that is poised at a critical point at which density or volume-fraction fluctuations are anomalously suppressed at infinite wavelengths.", "Recently, much attention has been given to the link between strict jamming and hyperuniformity in frictionless hard-particle packings.", "Doing so requires one to study very large packings, which can be difficult to jam properly.", "We modify the rigorous linear programming method of Donev et al.", "[J. Comp.", "Phys.", "197, 139 (2004)] in order to test for jamming in putatively jammed packings of hard-disks in two dimensions.", "We find that various standard packing protocols struggle to reliably create packings that are jammed for even modest system sizes; importantly, these packings appear to be jammed by conventional tests.", "We present evidence that suggests that deviations from hyperuniformity in putative maximally random jammed (MRJ) packings can in part be explained by a shortcoming in generating exactly-jammed configurations due to a type of \"critical slowing down\" as the necessary rearrangements become difficult to realize by numerical protocols.", "Additionally, various protocols are able to produce packings exhibiting hyperuniformity to different extents, but this is because certain protocols are better able to approach exactly-jammed configurations.", "Nonetheless, while one should not generally expect exact hyperuniformity for disordered packings with rattlers, we find that when jamming is ensured, our packings are very nearly hyperuniform, and deviations from hyperuniformity correlate with an inability to ensure jamming, suggesting that strict jamming and hyperuniformity are indeed linked.", "This raises the possibility that the ideal MRJ packings have no rattlers.", "Our work provides the impetus for the development of packing algorithms that produce large disordered strictly jammed packings that are rattler-free." ], [ "Introduction", "Dense packings of hard (nonoverlapping) spheres in $d$ -dimensional Euclidean space $\\mathbb {R}^d$ have been a source of fascination to scientists across the physical and mathematical sciences.", "Particle packings have served as simple, yet powerful models for a wide variety of condensed matter systems including liquids, glasses, colloids, particulate composites, granular materials, and biological systems, to name a few [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].", "Of particular interest are mechanically stable or jammed packings [4], [7], [10], [15], [27], [18], [19].", "In order to make the notion of jamming rigorous, Torquato et al.", "introduced the following rigorous hierarchical jamming categories for frictionless spheres in $\\mathbb {R}^d$ [4], [5]: a locally jammed packing is one in which no particle may be displaced while all others are fixed in place.", "A collectively jammed packing is one in which no subset of particles may be displaced while fixing the shape of the system boundary.", "A strictly jammed packing is a packing in which no subset of particles may be displaced while allowing volume-preserving deformations of the system boundary [5].", "Thus, all strictly jammed packings are collectively jammed, and all collectively jammed packings are locally jammed (each particle is locally trapped by at least $d+1$ contacting spheres not all in the same hemisphere).", "Collectively jammed packings are stable against uniform compression, i.e., their bulk modulus is positive; and strictly jammed packings are additionally stable against shear, implying they also have a positive shear modulus.", "In the limit of exact strict jamming, the bulk and shear moduli of hard-particle packings both diverge to infinity [5].", "Torquato and Stillinger have conjectured that any strictly jammed saturated infinite packing of identical spheres is hyperuniform [6], [28].", "A saturated packing of hard spheres is one in which there is no space available to add another sphere.", "Any hyperuniform point pattern is poised at a “critical point” because it is characterized by an anomalously large suppression of large-scale density fluctuations such that the direct correlation function is long-ranged [6], which is manifested by a local number variance $\\sigma ^2(R)$ that grows more slowly than $R^d$ for a spherical observation window of radius $R$ or, equivalently, a structure factor $S(k)$ that tends to zero as the wavenumber $\|k\|$ tends to zero.", "More generally, for two-phase media, hyperuniformity is manifested by a local volume-fraction variance that decays more rapidly than $R^{-d}$ or, equivalently, by a spectral density ${\\tilde{\\chi }}(k)$ [29], [30] that tends to zero in the limit $\|k\| \\rightarrow 0$ [31].", "To date, there is no known counterexample to this conjecture, notwithstanding a recent study that calls into question the link between jamming and hyperuniformity [32].", "What is the rationale for such a conjecture?", "First, we note that a packing can be hyperuniform without being strictly jammed.", "For example, a honeycomb lattice packing of identical circular disks in two dimensions is only locally jammed but is hyperuniform with positive bulk modulus and zero shear modulus.", "This example stresses the importance of the strict jamming constraint.", "Indeed, appropriate deformations and compressions of a packing that is only locally or collectively jammed (and hence hypostatic with respect to strict jamming) can lead to a denser strictly jammed packing that is isostatic or hyperstatic [33], filling space more uniformly.", "We also know that there are infinite periodic packings, such as the triangular lattice of identical circular disks in $\\mathbb {R}^2$ and face-centered-cubic lattice in $\\mathbb {R}^3$ that are rigorously known to be strictly jammed [5] under periodic or hard-wall boundary conditions, and hyperuniform.", "In such situations, randomly removing a finite fraction of particles such that there are no “di-vacancies” in the two-dimensional example and no “tri-vacancies” in the three-dimensional example [34] while maintaining strict jamming results in non-hyperuniform packings, i.e., $S(0) \\ne 0$ .", "This example illustrates vividly that hyperuniformity is degraded by “defects”—an issue that we will discuss in more detail in the Conclusions.", "Therefore, the conjecture includes the saturation condition.", "Moreover, we know that collisions in equilibrium hard-sphere configurations on the approach to jammed ordered states are not strictly hyperuniform due to vibrational fluctuations and only become exactly hyperuniform when the ideal jammed state without any defects is attained.", "We expect this to be the case on the approach to disordered jammed states.", "Thus, based on these considerations, one expects that statistically homogeneous disordered strictly jammed saturated packings of identical spheres are hyperuniform.", "Importantly, the conjecture eliminates packings that may have a rigid backbone but possess “rattlers” (particles that not locally jammed but are free to move about a confining cage) because a strictly jammed packing cannot contain rattlers [35], [27].", "Typical packing protocols that have generated disordered jammed packings tend to contain a small concentration of rattlers; because of these particles, one cannot say that the whole (saturated) packing is “jammed”.", "Therefore, the conjecture cannot apply to these packings—a subtle point that has not been fully appreciated.", "Nonetheless, it is an open question what effect the rattlers have on hyperuniformity in the context of strict jamming and whether there exists a maximally random jammed (MRJ) state with no rattlers in the infinite-volume limit that is exactly hyperuniform.", "Donev et al.", "[36] set out to see to what extent relatively large MRJ-like sphere packings (with system sizes of up to $N=10^6$ particles) in $\\mathbb {R}^3$ were hyperuniform, even though there was a small concentration of rattlers (about 2.5%), precluding them from the conjecture as noted above.", "Nonetheless, they found that a packing of $10^6$ particles that included the rattlers was nearly hyperuniform with $\\lim _{k \\rightarrow 0} S(k) = 6.1 \\times 10^{-4}$ and first peak value $S_{max} = 4.1$ [37].", "(When the rattlers were removed, the structure factor at the origin had a substantially larger value, showing that the backbone alone is far from hyperuniform.)", "This numerical finding supporting the link between effective hyperuniformity of an isostatic disordered packing to its mechanical rigidity spurred a number of subsequent numerical and experimental investigations that reached similar conclusions [38], [39], [40], [41], [42], [43], [44].", "In all cases, effective or near hyperuniformity is conferred because the majority of the particles are contained in the strictly-jammed backbone and there are few rattlers.", "Indeed, it has been systematically shown that as a hard-sphere system, substantially away from a jammed state, is driven toward strict jamming through densification, $S(0)$ monotonically decreases until effective hyperuniformity is achieved at the putative MRJ state.", "Specifically, $S(0)$ was found to approach zero approximately linearly as a function of density from 93% to 99% of jamming density, where extrapolating the linear trend in $S(0)$ to jamming density yielded $S(0) = -1 \\times 10^{-4}$ [43].", "This study clearly establishes a correlation between distance to jamming and hyperuniformity, and additionally introduces a “nonequilibrium index” describing the interplay between hyperuniformity and a dynamic measure of distance to jamming [45].", "In $\\mathbb {R}^2$ , disordered, MRJ-like packings of equal-sized disks are very hard to observe, and it has only recently been shown that highly-disordered, isostatic jammed states exist at all [46].", "Therefore, it is common to introduce a size dispersity in order to induce geometrical frustration and increase the degree of disorder in the resulting packings.", "However, examining the point configurations derived from the centers of such polydisperse packings could lead one to incorrectly conclude that the packings were not hyperuniform.", "Zachary et al.", "demonstrated [39] that the proper means of investigating hyperuniformity in this case is through a packing's spectral density $\\tilde{\\chi }(k)$ ; that is, making an extrapolation towards the origin to estimate $\\lim _{k \\rightarrow 0} \\tilde{\\chi }(k) = 0$ .", "They found that MRJ-like binary systems of disks with size ratio $\\alpha = 1.4$ and small disk mole fraction $x=0.75$ exhibited near hyperuniform behavior with $\\lim _{k \\rightarrow 0} \\tilde{\\chi }(k) = 1.0 \\times 10^{-5}$ .", "Thus, even though polydispersity is not part of the original conjecture [6], effective hyperuniformity can be observed in polydisperse packings as well, provided that the size distribution is suitably constrained.", "It is even possible that the conjecture can be extended to polydisperse strictly jammed saturated packings; however, necessary and sufficient conditions for this criterion are highly nontrivial.", "Nonetheless, jamming is again a crucial necessary property to attain near-hyperuniformity in $\\mathbb {R}^2$ as it was in $\\mathbb {R}^3$ .", "A fascinating open question remains as to whether putative MRJ packings can be made to be even more hyperuniform than established to date or exactly hyperuniform with numerical protocols as the system size is made large enough.", "This is an extremely delicate question to answer because one must be able to ensure that true jamming is achieved to within a controlled tolerance as the system size increases without bound.", "The latter condition is required to ascertain the infinite-wavelength hyperuniformity property and yet any packing algorithm necessarily must treat a finite system and hence the smallest accessible positive wavenumber at which $S(k)$ or ${\\tilde{\\chi }}(k)$ can be measured is of the order of $2\\pi /N^{1/d}$ , where $N$ is the number of particles.", "The situation is further complicated by noise at the smallest wavenumbers, numerical and protocol-dependent errors, and the reliance on extrapolations of such uncertain data to the zero-wavenumber limit.", "To make matters even more complex, we will present evidence that current packing algorithms stop short of hyperuniformity—and jamming—because requisite collective rearrangements of the particles become practically impossible as criticality is approached, i.e., a type of “critical slowing down” [47], [48].", "In this regard, it is noteworthy that general nearly hyperuniform point configurations can be made to be exactly hyperuniform by very tiny collective displacements via the collective-coordinate approach [49], which by construction enables the structure factor to be constrained to take exact targeted values at a range of wavevectors, as shown recently in Ref.", "[50].", "Figure REF vividly illustrates this point using an initial nearly hyperuniform configuration in which $S(0)=1 \\times 10^{-4}$ (which is comparable to the value obtained in MRJ-like states) and then collectively displacing the particles by tiny amounts until the structure factor $S(k)$ vanishes linearly with $k$ in the limit $k \\rightarrow 0$ , as in the case of disordered jammed packings.", "While these particles are not jammed, this example serves to emphasize that it only requires very tiny displacements to make a nearly hyperuniform system exactly hyperuniform.", "Thus, a critical slowing down implies that it becomes increasingly difficult numerically to drive the value of $S(0)$ down to its lowest possible value if a true jammed critical state could be attained.", "Figure: (Color online.)", "A disordered nonhyperuniform configuration with S(0)=1×10 -4 S(0)=1 \\times 10^{-4} (left panel)and a disordered hyperuniform configuration in which the structure factor S(k)S(k) vanishes linearlywith kk in the limit k→0k \\rightarrow 0 (right panel).", "The configuration on the right is obtained by very small collective displacements of the particles on the leftvia the collective-coordinate methods described in Ref.", ".", "Visually, these configurations look very similar to one another, vividlyrevealing the “hidden order” that can characterize disordered hyperuniform systems .Indeed, each particle in the left panel on average moves a root-mean-square distance that is about 0.2% of thethe mean-nearest-neighbor distance as measured by the configuration proximity metric to produce the configurationin the right panel.In this paper, we will investigate the role of jamming and present evidence that suggests that deviations from hyperuniformity in MRJ-like packings can in part be explained by a shortcoming of the numerical protocols to generate exactly-jammed configurations as a result of the type of critical slowing down mentioned above.", "In order to attempt to observe jammed states, we will utilize a variety of standard hard-sphere packing protocols including the Lubachevsky-Stillinger (LS) event-driven molecular dynamics algorithm [52], [53] and the Torquato-Jiao (TJ) sequential linear programming algorithm [54] to obtain putatively collectively jammed and strictly jammed MRJ packings [55], respectively.", "We will focus on frictionless binary disk packings in two dimensions because it allows us to study the behavior at smaller wavenumbers than three dimensions, assuming $N$ is held constant.", "In addition, it is computationally easier to ensure proper jamming at a given system size, further increasing our ability to query the long-wavelength-behavior of the MRJ state.", "Studying such systems will enable us to make contact with the recent investigation of Wu et al.", "[56] who examined binary packings of soft-disks above the jamming transition.", "They found that at finite positive pressures the spectral density exhibited a local minimum at a finite wavenumber and proceeded to grow for smaller wavenumbers, calling into question whether hyperuniformity is observed when approaching the jamming transition from above.", "However, they also recognized that the presence of hyperuniformity at jamming may be sensitive to the specific protocol used to construct the jammed configurations.", "The rest of the paper is organized as follows: in Sec.", ", we present a variety of methods to test whether a packing, ordered or not, is truly jammed.", "These tests include a modification of the rigorous linear programming method of Donev et al.", "[57] and so-called “pressure-leak” tests.", "We then apply them to our packings and find that standard protocols fail to produce jammed packings at surprisingly low system sizes.", "In Sec.", ", we investigate the subtleties related to system size and numerical protocol that affect one's ability to observe hyperuniform configurations in putatively jammed disordered packings using computer simulations.", "Discussion and concluding remarks are given in Sec.", "." ], [ "Methods to test for jamming", "A variety of methods have been used to test for jamming in hard-particle packings in the past; we will begin by reviewing some common methods, then introduce the linear programming method that we use in this work to rigorously test packings for jamming.", "We will show that even relatively small 2D packings with a high reduced pressure $P = pV/(Nk_BT)$ may not be jammed, even though some methods may imply otherwise." ], [ "Pressure-Leak Method", "An effective heuristic means of testing for collective jamming is a so-called “pressure leak” test [52], [53], [4], [57], in which the spheres are subjected to standard molecular dynamics for some relatively large time [58].", "If the system pressure begins to drop substantially, then one may conclude that the packing was not collectively jammed; the pressure leak indicates that the particles have discovered an unjamming motion.", "While this test is effective for packings that are not well-jammed or have a large interparticle gap, it struggles with packings that are at a high reduced pressure and packings that are nearly jammed, but for which an unjamming motion requires the cooperative motion of many spheres.", "In configuration space, this scenario is analogous to the $Nd$ -dimensional configuration point being locally confined to a high-dimensional “bottleneck” from which escape may only occur in very specific directions.", "In such cases, the algorithm may require to process prohibitively many collisions per particle.", "Animations demonstrating the pressure test on an ordered and a disordered configuration are included in the Supplemental Material [59]." ], [ "Linear Programming-Based Approach", "Donev et al.", "introduced a method that uses randomized sequential linear programming to rigorously test for collective or strict jamming in packings of frictionless spheres [57], [33].", "Speaking physically, the algorithm applies random body forces to the spheres in the packing and seeks to maximize the work due to those forces by displacing them in the direction of their applied forces while obeying the constraint that no spheres overlap.", "Spheres that displace as a result of the optimization can be identified as rattlers; if every sphere is a rattler, then the packing is unjammed.", "This is implemented using sequential linear programming techniques.", "Let $R = (r_1 , \\dots , r_N)^T$ describe the position of the $N$ spheres at the beginning of an iteration, and let $\\Delta R = (\\Delta r_1 , \\dots , \\Delta r_N)^T$ be a vector of design variables describing how the spheres displace.", "The vector $B \\in \\mathbb {R}^{Nd}$ contains the body forces which will attempt to displace the spheres; the scalar objective function $Z=B^T \\Delta R$ to be maximized is, physically speaking, the work performed on the packing due to $B$ .", "For two spheres $i$ and $j$ not to overlap, we require $\\left\|\\left\| (r_i + \\Delta r_i) - (r_j + \\Delta r_j) \\right\|\\right\| \\ge D_{ij}$ , where $D_{ij} = (D_i+D_j)/2$ is the additive diameter between spheres $i$ and $j$ with diameters $D_i$ and $D_j$ .", "Linearizing this gives the following linear program (LP): $&&{\\rm maximize}~ Z=B^T \\Delta R\\nonumber \\\\&&{\\rm subject~to}\\nonumber \\\\&&\\Delta r_i - \\Delta r_j \\le r_{ij}-D_{ij} ~ \\forall ~i,j \\ne i$ where $r_{ij} = \\left\|\\left\| r_i - r_j \\right\|\\right\|$ .", "In the case where one wishes to check for strict jamming, one adds $d(d+1)/2$ strain variables to deform the fundamental cell and a constraint that the fundamental cell volume $V$ does not increase.", "For a packing with periodic boundary conditions (as we consider throughout the current work), these variables enter the constraints through pairs of spheres interacting through periodic boundary conditions; in the case where one is considering a packing with hard walls, these variables would show up in constraint terms involving the boundary.", "When dealing with non-ideal packings (i.e.", "packings thought to be very close to exact jamming but with $\\phi _c-\\phi >0$ where $\\phi _c$ is the jamming density), one must provide for the fact that even backbone spheres will be able to move by some small amount.", "Therefore, one is forced to relax the criterion that any sphere that moves must be a rattler.", "This was done by introducing a tolerance, i.e., any sphere that moves more than $\\Delta _{tol}$ is a rattler.", "In addition, one must now ask not whether the packing is exactly jammed or not, but whether or not it is confined to a jamming basin.", "Given a jammed configuration $R_J$ at packing fraction $\\phi _c$ , a jamming basin $\\mathcal {J}(R_J,\\phi _c)$ is defined as the set of points in configuration space for which the only accessible local packing fraction maximum under continuous displacements corresponds to $R_J$ (modulo rattlers).", "The density $\\phi ^<\\phi _c$ is defined as the highest density at which $R$ may be continuously displaced to arrive at at least one other local maximum; the quantity $\\phi _c-\\phi ^$ is the “depth” of the jamming basin.", "It can be difficult in practice to pick a value for $\\Delta _{tol}$ , or to answer the “jamming basin” question definitively.", "This is because the available configuration space to a packing is, in general, very complicated, and the impression one obtains of it through this algorithm is dependent on the particular choice of $B$ .", "Traditionally, $B$ is generated randomly, and the LP is solved iteratively several times in order to begin exploring in the direction of $B$ ; thus, one might obtain a sense of the distance over which spheres in the packing might displace despite being in a very closely-packed configuration [33].", "In addition, the variations in the geometry of various jamming basins (even those corresponding to an ensemble of similar packings) are considerable.", "Between these two factors, it is very difficult in practice to answer the binary question of whether or not a packing is truly within a jamming basin using the standard LP jamming test.", "We overcome this difficulty by choosing $B$ in a special manner designed to elucidate a particular local rearrangement that we call a “pop”.", "We pick a backbone sphere $i$ and a $d$ -combination of backbone spheres contacting it $\\mathcal {C}=\\lbrace j_1,\\dots ,j_d\\rbrace $ .", "We then determine the plane containing the spheres in $\\mathcal {C}$ and $b$ , the unit vector orthogonal to this plane facing away from sphere $i$ .", "The load applied to the packing is $B ={\\left\\lbrace \\begin{array}{ll}b & {\\rm for~sphere~}i\\\\-b/d & {\\rm for~spheres~in~} \\mathcal {C}\\\\0 & {\\rm otherwise.}\\end{array}\\right.", "}\\nonumber $ This is illustrated in Fig.", "REF .", "Physically, we are asking for sphere $i$ to “pop” through the plane described by $\\mathcal {C}$ , thus leaving the current jamming basin and entering a new one with a distinct contact network.", "It is important that the body forces sum to zero so that trivial uniform translations of the packing are not favored; such movements can obscure whether progress is made in realizing a “pop”.", "Animations demonstrating the pop test on an ordered and a disordered configuration are included in the Supplemental Material [59].", "Figure: (Color online.)", "Illustration of the body forces (arrows) chosen for a single iteration of the “pop test”.", "The green dashed line connects the two blue (dark gray when viewed in black and white) spheres that make up 𝒞\\mathcal {C}.", "The forces are chosen such that the red (light gray) sphere “pops” through the two blue spheres; the sum of the forces on the packing is zero so that uniform translations of the entire packing are not favored.", "If the center of the red sphere crosses the green dashed line, then the packing is not confined to a jamming basin.It is important to note that while $B$ is nonzero for only a small number of spheres, the linear program is free to displace all of the spheres in the packing as it carries out its optimization, meaning that global rearrangements are being considered.", "In other words, while the salient characteristic of a “pop” might be a local rearrangement, very complicated collective movements of many particles may take place in effecting it.", "We solve the linear program in Eq.", "(REF ) iteratively until either sphere $i$ passes “pops” through the plane or the packing stops rearranging.", "The latter may be inferred by monitoring the objective value associated with an optimization iteration.", "In the former case, we have found an unjamming motion, and the packing is unjammed; this may be confirmed by compressing the packing from this new configuration and comparing the ensuing contact network to the original network.", "If a pop is not found, we continue by picking another combination of contacts or another sphere until all possibilities are exhausted, at which point we conclude that the packing must be jammed.", "By focusing on particle displacements instead of density increases, this test “inverts” the jamming problem and is able to efficiently determine if a given packing is within a jamming basin, even when improving the packing density is computationally difficult or the “pressure test” fails to find an unjamming motion even after considerable computational time." ], [ "Comparison of Jamming Tests", "In order to establish a benchmark for the pressure test's performance, we investigated the case of the square lattice and found that when the interparticle gap is on the order of $10^{-12}$ disk diameters, evidence of a developing pressure leak might only become visible after up to $10^6$ collisions per sphere; see Appendix for details.", "In order to illustrate the difference between the “pressure test” and “pop test”, we consider putatively collectively jammed disordered bidisperse disk packings made using a standard compression schedule with the Lubachevsky-Stillinger (LS) algorithm within a square box with periodic boundary conditions.", "The LS algorithm is capable of creating packings that are collectively jammed, but only if one is careful to design a compression schedule that is slow enough to allow the packing to escape unstable mechanical equilibria by discovering the requisite collective particle rearrangements.", "On the other hand, compression rates that are too slow allow the system to equilibrate towards less disordered states, which is at odds with our desire to probe the MRJ state [60].", "To accomplish this, we start from initial conditions produced by random sequential addition at a density of $\\phi _0 = 0.40$ .", "We use an initial dimensionless expansion rate of $\\gamma = dD_{max}/dt \\sqrt{m/(k_BT)} = 10^{-3}$ where $D_{max}$ is the diameter of the largest disk and $m$ is the mass of a disk and proceed until the reduced pressure exceeds $P=10^6$ ; at that point, the expansion rate is reduced to $\\gamma = 10^{-6}$ and the procedure continues until the reduced pressure reaches at least $P=10^8$ .", "We considered system sizes of $N=10^2~{\\rm to}~10^4$ , and we produce between $10^2$ and $10^3$ packings for each system size as is needed for reliable statistics.", "In order to encourage the protocol to generate disordered configurations, we consider binary packings with a size ratio $a=1.4$ and number ratio $x=0.5$ .", "An example is shown in Fig.", "REF .", "At $N=10000$ , the mean density of our packings is $\\overline{\\phi }= 0.8474$ with a standard deviation of $\\sigma (\\phi ) = 1.8 \\times 10^{-4}$ [61].", "To provide a basis for comparison, we also used the Torquato-Jiao (TJ) sequential linear programming method to generate collectively jammed MRJ packings with the same size ratio, number ratio, and system size.", "Details of the algorithm, which solves the “adaptive shrinking-cell” optimization problem, can be found in Ref.", "[54].", "For our current work, we use an influence sphere of radius $\\gamma _{ij}=D_{max}/10$ , translation limit $\\left\| \\Delta x \\right\| \\le D_{max}/20$ , and global strain limit $\\left\| \\epsilon _{ij} \\right\| \\le D_{max}/20$ .", "Initial conditions are created using random sequential addition at a density of $\\phi = 0.40$ inside a square fundamental cell of unit volume with periodic boundary conditions.", "The packings are compressed while holding the box shape fixed until the volume changes by less than $10^{-12}$ over two successive iterations.", "In order to control for distance to jamming, the density of the terminal configurations are decreased so that their short-time reduced pressure (measured over 1000 collisions per disk) is $P=10^9$ .", "Figure: (Color online.)", "A collectively jammed packing produced by the LS algorithm under the compression schedule specified in Sec.", ".We subject our packings to both a pressure test lasting for $10^6$ collisions per disk as well as the “pop” test outlined above.", "Importantly, nearly all of the LS and TJ packings that we produced pass the pressure test.", "Figure REF shows the probability that a packing generated using this compression schedule passes the pop test.", "A curve is also included for the TJ algorithm's pass rate for the pressure test.", "Our results are consistent with the intuition that jammed packings are more difficult to produce as $N$ increases.", "However, our results illustrate vividly that standard methods to assess jamming can give misleading results even for modest system sizes, i.e., as the system size becomes on the order of one thousand disks.", "Figure: (Color online.)", "The probability that a packing of a given system size NN, created using the LS and TJ algorithms, will be jammed, as determined by the “pop” and “pressure” tests.", "Error bars correspond to a 95% confidence interval as calculated using the Clopper-Pearson method .Our results show that one must be careful in assuming that a given protocol is producing packings that are [nearly] truly jammed.", "Observing a high reduced pressure that persists for an extended period of time is not a reliable means of demonstrating that a packing is jammed.", "The test we demonstrate here is an efficient means of determining collective motions that unjam packings, revealing that even packings of modest size which were previously thought to be jammed may not actually be so.", "As the system size increases, this becomes an increasingly subtle, yet crucial point to which traditional methods like the pressure test are not sensitive.", "Given this, one must be careful when relying on numerical results when drawing conclusions about the nature of the MRJ state.", "Our results also suggest that previous studies of large, disordered, putatively jammed packings were not carried out on truly-jammed configurations." ], [ "Considerations That Prevent Numerical Packings From Being Exactly Hyperuniform", "In the following subsections, we will examine the effect of system size as well as the packing protocol used when measuring hyperuniformity in nearly-jammed, finite packings of disks.", "To do this, we quantify density fluctuations in real and reciprocal space using the local volume fraction variance $\\sigma _\\tau ^2(R)$ and isotropic spectral density $\\tilde{\\chi }(k)$ , respectively.", "In packings of equal-sized spheres, one may investigate the presence of hyperuniformity by considering the sphere centers and computing either their local number density variance $\\sigma ^2(R)$ or structure factor $S(k)$ .", "However, these approaches fail to take into account the effect of polydispersity and have been shown [38], [39], [40] to incorrectly suggest that MRJ packings of polydisperse or anisotropic particles are not hyperuniform, whereas $\\sigma _\\tau ^2(R)$ and $\\tilde{\\chi }(k)$ , which properly account for these particle characteristics, show otherwise.", "We begin by reviewing the procedure for computing the spectral density $\\tilde{\\chi }(k)$ [29] and local volume fraction variance $\\sigma _\\tau ^2(R)$ [63], [31] for a packing of polydisperse spheres in $\\mathbb {R}^d$ .", "For complete derivations, see Ref.", "[39].", "For a system of hard-spheres with periodic boundary conditions, the spectral density may be defined via discrete Fourier transform as $\\tilde{\\chi }(k) = \\frac{\\left\| \\sum _{j=1}^N \\exp (-i k \\cdot r_j) \\tilde{m}(k;D_j/2) \\right\|^2}{V} ~ (k \\ne 0),$ where $\\tilde{m} (k;R) =\\left( \\frac{2 \\pi }{k R} \\right)^{d/2} R^d J_{d/2}(kR)$ is the Fourier transform of the indicator function for a $d$ -dimensional sphere of radius $R$ [6]; $J_\\nu (x)$ is the Bessel function of the first kind of order $\\nu $ .", "The vectors $k$ at which this may be evaluated are integer combinations of the reciprocal basis vectors, defined as the columns of the matrix $\\mathbf {\\Lambda }_R = [(2\\pi ) \\mathbf {\\Lambda }^{-1}]^T$ , where the columns of $\\mathbf {\\Lambda }\\in \\mathbb {R}^{d \\times d}$ span the fundamental cell of our simulation box.", "In order to investigate the nature of density fluctuations in real space, one may consider the variance of the local volume fraction, defined as [63] $\\sigma _\\tau ^2(R) = \\frac{1}{v_1(R)} \\int _{\\mathbb {R}^d} \\chi (r) \\alpha (r;R) d r,$ where $v_1(R)$ is the volume of a $d$ -dimensional sphere of radius $R$ , $\\chi (r)$ is the autocovariance function, and $\\alpha (r;R)$ is the scaled intersection volume, which is as the intersection volume of two spheres of radius $R$ separated by a distance $r$ divided by $v_1(R)$ .", "In practice, $\\sigma _\\tau ^2$ may be computed by randomly placing a sufficiently large number of spherical windows of radius $R$ within the packing." ], [ "Hyperuniformity and System Size", "In order to study the relation between jamming and hyperuniformity in disordered packings that are putatively jammed, we use the LS algorithm with the compression schedule described above to produce packings of binary disks with system sizes up to $N=2 \\times 10^4$ .", "In particular, we will establish that not only does jamming become less common as $N$ increases (as shown above), but that hyperuniformity is concomitantly lost to a “saturation” in $\\tilde{\\chi }(k)$ at small wavenumbers.", "The spectral densities of our packings are plotted in Fig.", "REF ; the curves drawn are ensemble averages with 1000 packings per curve; data is binned according to wavenumber $k = \\left\|\\left\| k \\right\|\\right\|$ with a bin width of $\\Delta k = 0.01$ .", "Raw data for the spectral density of the packings in these ensembles is available in the Supplemental Material [59].", "There are significant variations within any ensemble from packing to packing; this variation and its effect on determining hyperuniformity will be discussed for a set of 1000 packings of $N = 500$ binary disks produced using the LS protocol in this section.", "To quantify this, we fit the unbinned spectral density of each packing individually with a polynomial of order $n$ (i.e.", "$f(\\left\| k \\right\|;a_0,\\dots ,a_n) = \\sum _{j=0}^n a_j \\left\| k \\right\|^j$ ) for all wavenumbers within $0 < \\left\| k \\right\| \\langle D \\rangle / 2 \\pi \\le k_{max}$ , where $\\langle D \\rangle = 1/N \\sum _{i=1}^N D_i$ is the number-averaged diameter.", "For $n=1$ and 2, we use $k_{max} = 0.11$ to investigate the behavior below the kink seen in Fig.", "REF .", "For $n=3$ , we use $k_{max}=0.40$ for more complete data.", "We pick this particular value of $k_{max}$ for $n=3$ because the standard deviation of the value of the fit's intercept is minimized for this range.", "To illustrate the goodness of fit typical of our fits, the inset of Fig.", "REF shows binned data for the $N=20000$ ensemble along with its fitted cubic polynomial.", "For our ensemble of $N=500$ fitted with a cubic polynomial, the intercept at the origin $a_0$ has a mean of $6.5 \\times 10^{-5}$ and a standard deviation of $1.8 \\times 10^{-3}$ , meaning that these packings can be considered effectively hyperuniform in the sense that if one were to consider a single packing from this ensemble and ask whether is is hyperuniform, then the data imply that the random variations from packing to packing are large enough that one could conclude that the answer is “yes” within this noise.", "Interestingly, the volume fraction fluctuations in direct space strongly corroborate this conclusion, suggesting that it is of considerable utility when diagnosing hyperuniformity in small systems; we will look at this presently.", "The mean extrapolated values of $\\tilde{\\chi }(0)$ for the other ensembles of packings shown in Fig.", "REF are presented in Table REF for various $n$ and $k_{max}$ .", "Figure: (Color online.)", "Spectral density for LS packings with various system sizes.", "Curves shown are binned ensemble averages with 1000 packings per curve and bin width Δk=0.01\\Delta k=0.01.", "While ensembles of smaller system sizes seem to be hyperuniform, a saturation in χ ˜(k)\\tilde{\\chi }(k) is observed as the system size increases and jamming is no longer ensured.", "Error bars are shown for a 95% confidence interval.", "The inset shows the cubic fit reported in Table (orange dashed line) on top of the binned data for N=20000N=20000.Table: Extrapolated spectral densities at k=0k=0 for ensembles of N p N_p packings created with the LS algorithm with system size NN, fitted using a polynomial of order nn for data at wavenumbers 0<k〈D〉/2π≤k max 0 < k \\langle D \\rangle / 2 \\pi \\le k_{max}.", "Reported values are the mean a 0 ¯\\overline{a_0} and standard deviation σ(a 0 )\\sigma (a_0) for ensembles of 1000 packings for N≤2000N \\le 2000 and 100 packings for N=20000N=20000.As the system size grows, a deviation from hyperuniformity becomes increasingly apparent in the spectral density curves, i.e., a “saturation” appears at low wavenumbers; the curvature of the trend inflects at about $k\\langle D \\rangle / (2 \\pi ) = 0.05$ .", "Interestingly, this wavenumber roughly corresponds to the largest wavelength that is typically observable in a jammed system of 500 disks.", "At the same time, this system size lies approximately at the point where a given packing will be jammed with a likelihood of about 50%.", "This prompts us to suggest that the physical origin of the “saturating” behavior may be fundamentally linked to the ability of the packing protocol to resolve collective rearrangements on a corresponding length scale that may be necessary to reach a truly-jammed state.", "A point in configuration space may become locally trapped within a “bottleneck” from which escape may only happen in very few directions.", "The result in practice is a “critical slowing down”[47] in which exact jamming takes increasingly long to resolve, as we elaborate in Sec.", ".", "We have also found that this “saturating” behavior can be misrepresented when care is not taken in extrapolating the effective value of $\\tilde{\\chi }(0)$ from the $\\tilde{\\chi }(k)$ for which data is available.", "Specifically, the effect of binning data to obtain an ensemble average can artificially increase or decrease the perceived value of $\\tilde{\\chi }(0)$ , where we have observed mostly increases in $\\tilde{\\chi }(0)$ when bin size is substantially larger than the smallest wavenumber.", "Moreover, the spectral density of individual packings tend to vary by a significant amount from that of the ensemble average.", "This begs the question: how close to zero must the extrapolated spectral density at $k=0$ be in order to be considered “effectively hyperuniform”?", "We address these questions in detail in Appendix .", "Our estimates for a single packing in the $N = 2000$ ensemble described in Appendix indicate that $a_0 = \\tilde{\\chi }(0)$ will be randomly distributed about zero with a standard deviation of about $4 \\times 10^{-4}$ .", "Averaging over an entire 100 packing set yields a better estimate of the mean value $\\overline{a_0}$ of the average of the set, and tighter zero-hypothesis confidence intervals.", "Our methods show that with about 50% probability, a mean value of $\\overline{a_0}$ such that $-4 \\times 10^{-5} \\le \\overline{a_0} \\le 4 \\times 10^{-5}$ indicates effective hyperuniformity, given the noise inherent in the calculation.", "Referencing Table REF values for the $N=2000$ packings and applying Student's t-distribution to the standard deviation reported, we see that with 68% probability the mean is within $6.1 \\times 10^{-5}$ of $4.6 \\times 10^{-5}$ .", "This is on the higher side of the range $-4 \\times 10^{-5}$ to $4 \\times 10^{-5}$ , indicating that the packings together could be hyperuniform within the error, but suggesting that individually many of them are not.", "This result corresponds well with the prediction that jamming and hyperuniformity are linked, since the majority of the $N=2000$ packings are not jammed according to the pop test, yet they are close to jamming since they all pass the pressure test.", "We also compute the local packing fraction variance $\\sigma _\\tau ^2$ for these ensembles with various system sizes, as shown in Fig.", "REF .", "For hyperuniform packings, $\\sigma _\\tau ^2(R)$ will scale towards zero more quickly than $R^{-2}$ as $R$ tends towards infinity.", "Equivalently, $R^2 \\sigma _\\tau ^2(R)$ will be a decreasing quantity with growing $R$ .", "This is clearly the case with our data, implying that these ensembles are hyperuniform by this metric.", "We see a sudden decrease for sufficiently large $R$ for each $N$ , but our studies on larger samples imply that this is a finite-size effect similar to that observed in Ref.", "[44].", "However, note that the hyperuniformity trends are apparent even for the smaller system sizes.", "These results indicate that there is a maximum length scale $R_{max}$ (smaller than the half-width of the simulation box) that can be considered when diagnosing hyperuniformity.", "In addition, while the spectral density calculation may require hundreds or even thousands of packings to converge to well-resolved curves, the direct-space curves converge much more quickly.", "It has been noted before that if one must ascertain hyperuniformity from either a small system or an ensemble with a limited population, the direct-space computation is particularly effective at diagnosing hyperuniformity [44].", "Figure: (Color online.)", "Window packing fraction variance scaled by the window volume (R/〈D〉) 2 (R/\\langle D \\rangle )^2 for LS packings for various NN.", "Curves shown are binned ensemble averages.", "If the ordinate scales towards zero, then the systems are hyperuniform.", "The sudden decrease at the highest values of RR for each curve are due to finite-size effects, but hyperuniformity is apparent even for the smaller system sizes.", "The uncertainty in the curves is very small, on the order of the line width." ], [ "Hyperuniformity and Protocol Dependence", "We now consider the effect that one's choice of packing protocol has on the degree of hyperuniformity.", "In particular, we will present evidence that different protocols, which come near to exact jamming to different degrees, create packings with correspondingly different degrees of hyperuniformity.", "Importantly, while all of the protocols considered have the capacity to generate jammed packings given arbitrarily high numerical precision and computing time, practical considerations like computational cost and the rate of convergence force one to terminate any algorithm before exact jamming is attained.", "It is this distance to jamming that we would like to focus on.", "To do this, we begin with our LS-generated binary disk packings as well as packings made using a standard soft-sphere protocol following the procedure in [56].", "For the latter algorithm, particles interact through the standard harmonic pair potential [7] $\\psi _{ij}(r_{ij}) ={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2} \\left(1 - r_{ij}/D_{ij} \\right)^2 & {\\rm for}~r_{ij} < D_{ij} \\\\0 & {\\rm for}~r_{ij} \\ge D_{ij}\\end{array}\\right.", "}$ where $D_{ij} = (D_i + D_j)/2$ and $D_i$ is the diameter of particle $i$ .", "An enthalpy-like function $H=\\Gamma _N \\log (V)+\\sum _{i,j} \\psi _{ij}$ is minimized for $\\Gamma _N/N=1.373 \\times 10^{-4}$ using a conjugate gradient method, terminating when the gradient of the objective function falls below $10^{-13}$ or its value remains constant (as expressed in double precision) over 10 optimization steps.", "Initial configurations are Poisson point processes at reduced density $\\sum _i v_i/V = 0.84$ , where $v_i$ is the volume of particle $i$ .", "We input the resulting packings from these two protocols as initial conditions [64] for the TJ algorithm, and proceed to generate putatively strictly-jammed packings.", "An animation demonstrating this combination protocol using the soft-sphere algorithm followed by the TJ algorithm is provided in the Supplemental Material [59].", "Since TJ seeks out local packing fraction maxima, the threshold state being approached in the soft-sphere protocol is the same state that is being approached using TJ.", "While it is beyond the scope of this work to investigate the similarities and differences in the approach to the jamming transition from these two protocols, the limiting configuration is essentially unchanged when the packings are given to TJ.", "Figure REF shows the ensemble-averaged spectral density of ensembles of 100 packings of system size $N=10^4$ for both protocols before and after being input to TJ; the corresponding direct-space measurement of the window volume fraction variance $\\sigma _\\tau ^2(R)$ is shown in Fig.", "REF .", "Raw data for the spectral density of these ensembles is available in the Supplemental Material [59].", "The LS-generated configurations do not change by a significant amount upon being fed to the TJ algorithm, suggesting that the difference between collective and strict jamming is minor at sufficiently large system sizes [33], [35].", "Figure: (Color online.)", "The spectral density of disordered binary disk packings created by a variety of protocols.", "Curves shown are binned ensemble averages with bin width Δk=0.01\\Delta k=0.01.", "Feeding the packings produced by the soft-sphere protocol (identified as “SS” in the legend) into the TJ algorithm increases the hyperuniformity of the packings significantly as they are brought closer to exact jamming.", "The putatively collectively-jammed packings generated by LS are almost unchanged upon being given to TJ for strict jamming, implying that the difference between collective and strict jamming is small at large system sizes.", "Error bars are shown for a 95% confidence interval.Figure: (Color online.)", "Window number variance scaled by the window volume (R/〈D〉) 2 (R/\\langle D \\rangle )^2 for binary disk packings with system size N=10 4 N=10^4 created by a variety of protocols.", "Feeding the packings produced by the soft-sphere protocol (identified as “SS” in the legend) into the TJ algorithm increases the range over which hyperuniform scaling behavior is observed as the packings are brought closer to exact jamming.", "The LS packings exhibit hyperuniform scaling over a significant range of length scales (i.e.", "the ordinate scales towards zero as RR increases).", "The uncertainty in the curves is on the order of the line width.Of particular importance is that the sharp increase in $\\tilde{\\chi }$ near the origin that is observed in soft-sphere packings at a positive pressure [56] vanishes upon subsequent packing using the TJ algorithm.", "This is seen in the direct-space calculations as well, since the non-hyperuniform $R^{-2}$ scaling in $\\sigma _\\tau ^2(R)$ seen with the soft-sphere protocol at length scales beyond $R=7 \\langle D \\rangle $ becomes hyperuniform upon subsequent packing with TJ.", "This ability to obtain length scales over which hyperuniformity is observed is one of the main strengths of the direct-space calculation [44].", "TJ is allowing us to get several orders of magnitude closer to exact jamming and more accurately discern its spectral density: for TJ, $\\phi _J-\\phi \\approx 10^{-12}$ , whereas in the soft-sphere case, $\\phi -\\phi _J \\approx 10^{-3}$ .", "This reinforces the idea that bringing a packing closer to exact jamming will improve its hyperuniformity.", "Additionally, this result again emphasizes the subtleties in characterizing the long-wavelength nature of the MRJ state and underscores the need to be mindful of the nature of the numerical insensitivities in packing protocols—particularly as they approach highly-disordered states." ], [ "Conclusions and Discussion", "We have introduced a method that uses sequential linear programming to test for jamming in packings of frictionless hard-spheres.", "We applied this algorithm to disordered packings of bidisperse disks and found that standard protocols struggle to create packings that are truly confined to a jamming basin when the system size grows to be on the order of one thousand disks or more.", "Importantly, heuristic tests like the pressure test fail to find these unjamming motions.", "We then examined the spectral density of the packings we generated for a variety of system sizes and found an inflection at a wavenumber corresponding to the largest accessible length scale in a typical packing with system size $N=500$ —the same system size for which the probability of producing a truly-jammed configuration is approximately 50%.", "Given this, we conclude that our inability to observe exactly-hyperuniform configurations at larger system sizes is directly linked to the difficulty of producing exactly-jammed configurations.", "We also found that, by bringing soft-sphere packings closer to their jamming transition point by using the TJ algorithm, the degree of hyperuniformity was also increased by several orders of magnitude as measured by the volume fraction fluctuations in direct space.", "This ability to obtain length scales over which hyperuniformity is observed is one of the main strengths of the direct-space calculation [44].", "This finding also suggests that the break from hyperuniformity at large length scales, previously thought to be inherent to packings above the jamming transition, is in reality due to an excessive distance from the jamming transition, and that subsequent resolution makes this feature disappear.", "Because of our careful consideration of packings' distance to jamming and the corresponding evolution of their structure, we conclude that one cannot rely solely on current technology to say that there is no connection between jamming in disordered packings and hyperuniformity since jamming cannot be ensured.", "In both of the cases studied above, we point out an evident “critical slowing down” in that the collective particle movements required to reach an exactly-jammed state take longer to resolve as the system size grows.", "A point in configurational space may become locally trapped within a “bottleneck” from which escape may only happen in very few directions, corresponding to the collective rearrangements that the packing must undergo.", "For an event-driven MD protocol such as LS, it may take many collisions per particle before this escape is discovered.", "As the system size increases, the dimensionality of configuration space increases as well and this “escape” becomes an increasingly rare event.", "The result in practice is a “critical slowing down” in which exact jamming takes increasingly long to resolve.", "This dynamic critical behavior is well-known in other physical systems, the most well-known of which is perhaps the kinetic Ising model [47].", "The fact that it is also observable in packing contributes additional evidence to support the idea that the MRJ state lies at a special type of critical point, namely, one in which the direct correlation function $c(r)$ , rather than the total correlation function $h(r)$ , is long-ranged due to the fact that the appropriate spectral function is zero at $k=0$ [6].", "This is to be contrasted with a thermal critical point in which density fluctuations diverge because $h(r)$ is long-ranged.", "It is important to notice that the protocols that we are aware of tend to produce configurations possessing a positive rattler fraction.", "As these particles do not contribute to the rigidity of the backbone, they might be regarded as “defects” within the disordered configuration.", "Their location is also not uniquely specified, in contrast to the positions of the backbone particles.", "Therefore, we expect that any packing containing rattlers cannot necessarily be exactly hyperuniform due to the freedom the rattlers possess.", "However, one must also be aware that the backbone configurations that give rise to rattler cages are also interesting in that the cages surrounding the rattlers tend to have significantly different local structures from that of the rest of the packing [65].", "Therefore, it is not enough to expect that “optimizing” the rattlers' positions will necessarily yield a hyperuniform configuration.", "In general, one should not expect that a packing possessing a jammed backbone will be hyperuniform unless it is saturated and does not possess any rattlers.", "It has been observed previously [65] that the TJ algorithm produces packings of equal-sized spheres in three dimensions that are simultaneously more disordered and have significantly fewer rattlers than other known protocols (e.g., [53]), leaving open the possibility that packings that are more disordered may have even fewer rattlers still.", "This raises the possibility that the ideal MRJ (most disordered, strictly jammed) packings of identical spheres have no rattlers.", "This possibility is currently being further investigated [66].", "Indeed, if this is true, then one might reasonably expect that this would also hold for two-dimensional systems of identical particles [46] and certain polydisperse packings in 2D and 3D (given qualifications on the distribution of particle sizes).", "Devising algorithms that produce large rattler-free disordered jammed packings is an outstanding, challenging task.", "According to the Torquato-Stillinger conjecture, any MRJ-like strict jammed packing without any rattlers would be hyperuniform in the infinite-volume limit [6].", "It has been found that the average rattler fraction $N_R/N$ observed in a disordered packing is dependent upon the protocol being used; in three dimensions, putative MRJ packings produced with the LS algorithm tend to have a rattler fraction of approximately $N_R/N \\approx 0.025$ [36].", "On the other hand, the TJ algorithm produces packings with $N_R/N \\approx 0.015$ [65].", "For the binary systems considered in this work with $(\\alpha ,x) = (1.4,0.5)$ , the LS algorithm produces an average rattler fraction of $N_R/N=0.063 \\pm 0.001$ , whereas the TJ algorithm produces a mean of $N_R/N=0.048 \\pm 0.001$ , mirroring the story in three dimensions.", "Given that the packings produced by TJ are significantly more disordered as measured by standard order metrics [65], we ask whether the true MRJ state has no rattlers.", "The existence of such a jammed state remains an open question, and addressing it would presumably require a novel packing protocol.", "For $d=3$ , similar difficulties to those observed in this work exist in producing truly-jammed packings.", "For example, previous investigations have suggested that it is difficult to produce disordered packings of $N=10^4$ spheres using the LS algorithm that can pass even a pressure-leak test [35].", "Simulations using TJ take increasingly long amounts of time in producing jammed packings at comparable system sizes.", "Thus, we point out that there is an issue that is practical in nature associated with producing jammed packings of monodisperse spheres at large system sizes.", "An apparent “saturation” was observed in the structure factor of disordered soft-sphere packings in which the trend $S(k) \\propto k$ breaks down and becomes constant and positive for smaller wavenumbers, implying that jamming and hyperuniformity may not be connected [32].", "The wavenumber associated with this “turnover” corresponds to wavelengths on the order of approximately 20 spheres, which mirrors the “critical” system size above which it seems to be difficult to generate truly-jammed packings.", "This parallels the observations we have made in our current study.", "We suggest, therefore, that this observed departure from exact hyperuniformity may be due to an inability to resolve the particle displacements necessary to approach the jamming threshold–particularly given the system sizes that were considered ($N=5 \\times 10^5$ ).", "Our results demonstrate the particular difficulty of producing systems of either hard or soft particles that fall precisely at the jamming transition.", "Moreover, ensuring that a packing is truly in a jamming basin is a highly nontrivial task.", "Is there hope for producing improved algorithms to yield higher quality MRJ-like states?", "Of critical importance is that the algorithm be efficient at determining and resolving the collective movements that allow the packing to escape configurations that are not true local density maxima.", "Evidently, some of these motions may involve all of the particles in the system, and may achieve very small immediate changes in density.", "Thus, an intelligent way of identifying and carrying out these displacements is warranted.", "We have suggested that reaching an exactly-jammed state requires particles to “slide\" by each other in very specific ways when the packing is barely inside a jamming basin.", "Nonetheless, we point out that near-hyperuniformity may be readily observed and quantified in a number of meaningful ways.", "One is to measure the ratio between the value of the structure factor at the origin (e.g., as obtained by scattering experiments) and at the first peak [37].", "A second technique is to measure the volume fraction fluctuations in real space as a function of observation window size and look for the appropriate scaling relation as above [44], [67].", "This latter technique has indicated the existence of real-world systems that are hyperuniform over length scales spanning as much as four orders of magnitude [67].", "Thus, it is of interest to investigate the physical consequences of near-hyperuniformity as observed in such systems.", "The authors thank F. H. Stillinger for many insightful discussions.", "This work was supported in part by the National Science Foundation under Grant No.", "DMS-1211087." ], [ "Jamming tests on the square lattice", "In order to assess the reliability of the pop test for discovering unjamming motions, we consider the case of a square lattice in 2D, using both (a) a monodisperse disk packing and (b) a binary disk packing with size ratio $(\\alpha ,x)=(1.4,0.5)$ ; examples of the starting configurations are shown in Fig.", "REF .", "Figure: (Color online.)", "Square lattice packings of (a) monodisperse and (b) binary disks.We considered system sizes of $N=10^2$ and $10^4$ and size ratios $\\alpha =1$ and $1.4$ (the former referring to the monodisperse limit).", "We perform pressure pressure tests for $\\phi _c^-\\phi = 10^{-8}$ and $10^{-11}$ , where $\\phi _c^(\\alpha ) = [ \\pi (1+\\alpha ^2)] / [2(1+\\alpha )^2]$ is the close-packing density of the lattice [68].", "The reduced pressure is plotted as a function of the number of events per sphere in Fig.", "REF .", "It is important to note that the results of the pressure test will vary with different starting velocities, so we include several representative runs for each case.", "As expected, the pressure drops quickly after some amount of time, indicating that these packings are not jammed and that an unjamming motion has been discovered.", "Note that more events per sphere are required to observe a pressure “leak” when $\\phi _c^*-\\phi $ is smaller.", "In addition, the pressure leak takes slightly longer to show up in the bidisperse case, presumably due to the lessened degree of symmetry in the packing and correspondingly smaller degeneracy of its unjamming motions.", "Figure: (Color online.)", "Pressure tests for a square lattice of (a) monodisperse and (b) binary spheres with (α,x)=(1.4,0.5)(\\alpha ,x) = (1.4,0.5) with N=10 2 N=10^2 disks at putative jamming gaps of φ J -φ=10 -8 \\phi _J-\\phi =10^{-8} (lower curves in each subfigure) and 10 -11 10^{-11} (upper curves).", "As the jamming gap approaches zero, longer simulations are required to observe unjamming.", "For systems with P≈10 8 P \\approx 10^8, a simulation of one million collisions per particle is usually long enough to observe an unjamming motion.The pop test was also able to demonstrate the existence of “popping” motions for all of the aforementioned square lattices.", "We also note that the pop test did not perform differently for different jamming gaps, suggesting that it is robust against differences in interparticle distance." ], [ "Considerations Regarding Ensemble-Averaging Spectral Density Data", "Great care must be taken when numerically extrapolating $\\tilde{\\chi }(k)$ values to $\\tilde{\\chi }(0)$ .", "Additionally, claims about hyperuniformity for small sets of packings with extrapolated $\\tilde{\\chi }(0)$ values near zero must be accompanied by reasonable estimates of confidence intervals in order to be valid.", "In this Appendix, we show one way in which this might be accomplished.", "We mentioned above that the effect of binning data to obtain an ensemble average can give misleading results about the perceived value of $\\tilde{\\chi }(k)$ , particularly when $\\tilde{\\chi }$ is small.", "This can be the case when: (i) binning strategies are used to average data from a group of packings produced using the same protocol (but different initial conditions), then (ii) a curve is fit to the binned $\\tilde{\\chi }(k)$ values, and finally (iii) the curve is extrapolated to k = 0 to derive $\\tilde{\\chi }(0)$ .", "The choice of bin width has a significant effect on the appearance of the ensemble averaged-curve.", "For example, considering a set of 100 packings of $N = 2000$ binary disks produced using the LS protocol, binning the $\\tilde{\\chi }(k)$ values with bin width equal to $\\Delta k=0.01$ and then fitting a 3rd order polynomial for $0 \\le k\\langle D \\rangle / 2\\pi \\le 0.4$ yields an extrapolated $\\tilde{\\chi }(0)$ value of $-4.6 \\times 10^{-5}$ .", "However, fitting individual 3rd order polynomials to each packing's $\\tilde{\\chi }(k)$ and then averaging the 100 values of $a_0$ yields $\\overline{a_0} = 4.6 \\times 10^{-5}$ .", "This difference between a single extrapolated $\\tilde{\\chi }(0)$ value and the average $\\overline{a_0}$ of individual fits is small but significant.", "It begs the question, how close to zero must $a_0$ or $\\overline{a_0}$ be to be “effectively hyperuniform”?", "To this end, we begin by considering the probability distribution associated with $\\tilde{\\chi }(k)$ as a function of the wavenumber $k$ .", "We take an ensemble of 100 packings of $N=2000$ bidisperse disks and consider the set of spectral densities observed at each wavenumber within the range $0 \\le k / 2 \\pi \\le 0.5$ .", "From the data at these wavenumbers (normalized by their respective means), probability density functions are obtained; these are shown in Fig.", "REF as sets of filled circles.", "The solid black line is found by aggregating the data together to obtain an average over all wavenumbers.", "The data are well-fitted by an exponential distribution, indicating that for wavenumbers $k$ that are near to one another [69] for all of the packings studied, the expected value of any single $\\tilde{\\chi }(k)$ is equal to the standard deviation of the distribution.", "Figure: (Color online.)", "Probability density functions of spectral density values observed in nearly-jammed packings of N=2000N=2000 bidisperse disks.", "One dataset is shown (as a series of filled circles) for each wavenumber, and the thick black line shows the average over all wavenumbers.", "Normalizing the distributions with respect to their means makes them collapse to a single master curve, showing that χ ˜(k)\\tilde{\\chi }(k) across an ensemble of packings is approximately exponentially-distributed at any given wavenumber.To determine an estimate for the standard deviation of any extrapolated value $a_0 = \\tilde{\\chi }(0)$ for a single packing, we considered the following toy problem: suppose that for a hypothetical $\\tilde{\\chi }(k)$ curve, the spectral density corresponding to each wavenumber $0 \\le k/ 2 \\pi \\le 0.4$ for a hypothetical packing of side length $45 \\langle D \\rangle $ is chosen from an exponential distribution with a mean given by $\\overline{\\tilde{\\chi }}(k) = \\beta k$ , where $\\beta $ is chosen empirically to reflect the data from MRJ bidisperse packings.", "The standard deviation of the constant term in a linear fit to this hypothetical $\\tilde{\\chi }(k)$ curve over the range $0 \\le k/ 2 \\pi \\le 0.4$ is $\\sigma (a_0) = 4.2 \\times 10^{-4}$ .", "This provides an estimate for the standard deviation of an $a_0$ value extrapolated from the spectral density of any single packing in the aforementioned set of 100, assuming that the packings in the set exhibit roughly linear behavior in $\\tilde{\\chi }(k)$ for small $k$ with a near-zero extrapolated $\\tilde{\\chi }(0)$ .", "However, the $\\overline{a_0}$ value of an ensemble should still converge toward zero as the population of the ensemble grows towards infinity, provided that all of the packings were in fact effectively hyperuniform.", "To establish to what extent this convergence would occur, we consider a second toy problem in which we generate 20 sets of 100 such hypothetical $\\tilde{\\chi }(k)$ curves with each $\\tilde{\\chi }(k)$ for each curve derived from an exponential distribution as just described.", "We calculate linear fits and determine $a_0$ for each curve.", "We average these values to obtain $\\overline{a_0}$ for each set, then compute the mean and standard deviation of these twenty measurements.", "We find values of $1 \\times 10^{-6}$ and $5 \\times 10^{-5}$ , respectively.", "This suggests that a good estimate for a $\\overline{a_0}$ of a set of 100 packings exhibiting linear and hyperuniform spectral density would be within $\\pm 5 \\times 10^{-5}$ of zero about 68% of the time, within $\\pm 1.0 \\times 10^{-4}$ about 95% of the time, and within $\\pm 1.5 \\times 10^{-4}$ about 99% of the time.", "For our 100 packings with $N=2000$ , $\\overline{a_0} = 4.6 \\times 10^{-5} \\pm 6.1 \\times 10^{-4}$ , falling at about the 64th percentile of the distribution we obtained in the above toy problem (in which the ensemble was exactly hyperuniform by construction), indicating that with about 64% probability, not all the packings in the set are hyperuniform.", "This conclusion suggests that the numerical precision to which we can determine $\\tilde{\\chi }(0)$ for this set of 100 packings is not sufficient to rule out effective hyperuniformity with reasonable certainty.", "A larger sample set might yield more certainty.", "However, this conclusion supports the notion that jamming is associated with hyperuniformity: only one of the packings in the set passed the pop test and is therefore jammed.", "In summary, using the distribution of $\\tilde{\\chi }(k)$ values near a given wavenumber $k$ , we are able to provide an estimate, for a system of $N=2000$ binary disks produced as described, of a range of extrapolated $\\tilde{\\chi }(0) = a_0$ values that might be considered hyperuniform.", "That range is $0.0 \\pm 4.2 \\times 10^{-4}$ , within one standard deviation.", "Using this previously described method, estimates for confidence intervals over which a set of such packings might be considered effectively hyperuniform can be derived.", "For 100 packings of $N=2000$ such disks, an $\\overline{a_0}$ value of $0.0 \\pm 1.5 \\times 10^{-4}$ could be considered hyperuniform.", "Estimates for larger and smaller sets could be performed as well.", "These findings, based on the binning study and the study of the distribution of $\\tilde{\\chi }(k)$ values, lead us to suggest that great care must be taken when numerically extrapolating $\\tilde{\\chi }(k)$ values to $\\tilde{\\chi }(0)$ .", "Additionally, claims about hyperuniformity for small sets of packings with extrapolated $\\tilde{\\chi }(0)$ values near zero must be accompanied by reasonable estimates of confidence intervals, perhaps derived from the method described in this Appendix, in order to be valid." ] ]	1606.05227

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